Machine Learning Momentum Index (MLMI) [Zeiierman]█ Overview
The Machine Learning Momentum Index (MLMI) represents the next step in oscillator trading. By blending traditional momentum analysis with machine learning, MLMI delivers a potent and dynamic tool that aligns with the complexities of modern financial landscapes. Offering traders an adaptive way to understand and act on market momentum and trends, this oscillator provides real-time insights into market momentum and prevailing trends.
█ How It Works:
Momentum Analysis: MLMI employs a dual-layer analysis, utilizing quick and slow weighted moving averages (WMA) of the Relative Strength Index (RSI) to gauge the market's momentum and direction.
Machine Learning Integration: Through the k-Nearest Neighbors (k-NN) algorithm, MLMI intelligently examines historical data to make more accurate momentum predictions, adapting to the intricate patterns of the market.
MLMI's precise calculation involves:
Weighted Moving Averages: Calculations of quick (5-period) and slow (20-period) WMAs of the RSI to track short-term and long-term momentum.
k-Nearest Neighbors Algorithm: Distances between current parameters and previous data are measured, and the nearest neighbors are used for predictive modeling.
Trend Analysis: Recognition of prevailing trends through the relationship between quick and slow-moving averages.
█ How to use
The Machine Learning Momentum Index (MLMI) can be utilized in much the same way as traditional trend and momentum oscillators, providing key insights into market direction and strength. What sets MLMI apart is its integration of artificial intelligence, allowing it to adapt dynamically to market changes and offer a more nuanced and responsive analysis.
Identifying Trend Direction and Strength: The MLMI serves as a tool to recognize market trends, signaling whether the momentum is upward or downward. It also provides insights into the intensity of the momentum, helping traders understand both the direction and strength of prevailing market trends.
Identifying Consolidation Areas: When the MLMI Prediction line and the WMA of the MLMI Prediction line become flat/oscillate around the mid-level, it's a strong sign that the market is in a consolidation phase. This insight from the MLMI allows traders to recognize periods of market indecision.
Recognizing Overbought or Oversold Conditions: By identifying levels where the market may be overbought or oversold, MLMI offers insights into potential price corrections or reversals.
█ Settings
Prediction Data (k)
This parameter controls the number of neighbors to consider while making a prediction using the k-Nearest Neighbors (k-NN) algorithm. By modifying the value of k, you can change how sensitive the prediction is to local fluctuations in the data.
A smaller value of k will make the prediction more sensitive to local variations and can lead to a more erratic prediction line.
A larger value of k will consider more neighbors, thus making the prediction more stable but potentially less responsive to sudden changes.
Trend length
This parameter controls the length of the trend used in computing the momentum. This length refers to the number of periods over which the momentum is calculated, affecting how quickly the indicator reacts to changes in the underlying price movements.
A shorter trend length (smaller momentumWindow) will make the indicator more responsive to short-term price changes, potentially generating more signals but at the risk of more false alarms.
A longer trend length (larger momentumWindow) will make the indicator smoother and less responsive to short-term noise, but it may lag in reacting to significant price changes.
Please note that the Machine Learning Momentum Index (MLMI) might not be effective on higher timeframes, such as daily or above. This limitation arises because there may not be enough data at these timeframes to provide accurate momentum and trend analysis. To overcome this challenge and make the most of what MLMI has to offer, it's recommended to use the indicator on lower timeframes.
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Disclaimer
The information contained in my Scripts/Indicators/Ideas/Algos/Systems does not constitute financial advice or a solicitation to buy or sell any securities of any type. I will not accept liability for any loss or damage, including without limitation any loss of profit, which may arise directly or indirectly from the use of or reliance on such information.
All investments involve risk, and the past performance of a security, industry, sector, market, financial product, trading strategy, backtest, or individual's trading does not guarantee future results or returns. Investors are fully responsible for any investment decisions they make. Such decisions should be based solely on an evaluation of their financial circumstances, investment objectives, risk tolerance, and liquidity needs.
My Scripts/Indicators/Ideas/Algos/Systems are only for educational purposes!
Komut dosyalarını "algo" için ara
loxxfftLibrary "loxxfft"
This code is a library for performing Fast Fourier Transform (FFT) operations. FFT is an algorithm that can quickly compute the discrete Fourier transform (DFT) of a sequence. The library includes functions for performing FFTs on both real and complex data. It also includes functions for fast correlation and convolution, which are operations that can be performed efficiently using FFTs. Additionally, the library includes functions for fast sine and cosine transforms.
Reference:
www.alglib.net
fastfouriertransform(a, nn, inversefft)
Returns Fast Fourier Transform
Parameters:
a (float ) : float , An array of real and imaginary parts of the function values. The real part is stored at even indices, and the imaginary part is stored at odd indices.
nn (int) : int, The number of function values. It must be a power of two, but the algorithm does not validate this.
inversefft (bool) : bool, A boolean value that indicates the direction of the transformation. If True, it performs the inverse FFT; if False, it performs the direct FFT.
Returns: float , Modifies the input array a in-place, which means that the transformed data (the FFT result for direct transformation or the inverse FFT result for inverse transformation) will be stored in the same array a after the function execution. The transformed data will have real and imaginary parts interleaved, with the real parts at even indices and the imaginary parts at odd indices.
realfastfouriertransform(a, tnn, inversefft)
Returns Real Fast Fourier Transform
Parameters:
a (float ) : float , A float array containing the real-valued function samples.
tnn (int) : int, The number of function values (must be a power of 2, but the algorithm does not validate this condition).
inversefft (bool) : bool, A boolean flag that indicates the direction of the transformation (True for inverse, False for direct).
Returns: float , Modifies the input array a in-place, meaning that the transformed data (the FFT result for direct transformation or the inverse FFT result for inverse transformation) will be stored in the same array a after the function execution.
fastsinetransform(a, tnn, inversefst)
Returns Fast Discrete Sine Conversion
Parameters:
a (float ) : float , An array of real numbers representing the function values.
tnn (int) : int, Number of function values (must be a power of two, but the code doesn't validate this).
inversefst (bool) : bool, A boolean flag indicating the direction of the transformation. If True, it performs the inverse FST, and if False, it performs the direct FST.
Returns: float , The output is the transformed array 'a', which will contain the result of the transformation.
fastcosinetransform(a, tnn, inversefct)
Returns Fast Discrete Cosine Transform
Parameters:
a (float ) : float , This is a floating-point array representing the sequence of values (time-domain) that you want to transform. The function will perform the Fast Cosine Transform (FCT) or the inverse FCT on this input array, depending on the value of the inversefct parameter. The transformed result will also be stored in this same array, which means the function modifies the input array in-place.
tnn (int) : int, This is an integer value representing the number of data points in the input array a. It is used to determine the size of the input array and control the loops in the algorithm. Note that the size of the input array should be a power of 2 for the Fast Cosine Transform algorithm to work correctly.
inversefct (bool) : bool, This is a boolean value that controls whether the function performs the regular Fast Cosine Transform or the inverse FCT. If inversefct is set to true, the function will perform the inverse FCT, and if set to false, the regular FCT will be performed. The inverse FCT can be used to transform data back into its original form (time-domain) after the regular FCT has been applied.
Returns: float , The resulting transformed array is stored in the input array a. This means that the function modifies the input array in-place and does not return a new array.
fastconvolution(signal, signallen, response, negativelen, positivelen)
Convolution using FFT
Parameters:
signal (float ) : float , This is an array of real numbers representing the input signal that will be convolved with the response function. The elements are numbered from 0 to SignalLen-1.
signallen (int) : int, This is an integer representing the length of the input signal array. It specifies the number of elements in the signal array.
response (float ) : float , This is an array of real numbers representing the response function used for convolution. The response function consists of two parts: one corresponding to positive argument values and the other to negative argument values. Array elements with numbers from 0 to NegativeLen match the response values at points from -NegativeLen to 0, respectively. Array elements with numbers from NegativeLen+1 to NegativeLen+PositiveLen correspond to the response values in points from 1 to PositiveLen, respectively.
negativelen (int) : int, This is an integer representing the "negative length" of the response function. It indicates the number of elements in the response function array that correspond to negative argument values. Outside the range , the response function is considered zero.
positivelen (int) : int, This is an integer representing the "positive length" of the response function. It indicates the number of elements in the response function array that correspond to positive argument values. Similar to negativelen, outside the range , the response function is considered zero.
Returns: float , The resulting convolved values are stored back in the input signal array.
fastcorrelation(signal, signallen, pattern, patternlen)
Returns Correlation using FFT
Parameters:
signal (float ) : float ,This is an array of real numbers representing the signal to be correlated with the pattern. The elements are numbered from 0 to SignalLen-1.
signallen (int) : int, This is an integer representing the length of the input signal array.
pattern (float ) : float , This is an array of real numbers representing the pattern to be correlated with the signal. The elements are numbered from 0 to PatternLen-1.
patternlen (int) : int, This is an integer representing the length of the pattern array.
Returns: float , The signal array containing the correlation values at points from 0 to SignalLen-1.
tworealffts(a1, a2, a, b, tn)
Returns Fast Fourier Transform of Two Real Functions
Parameters:
a1 (float ) : float , An array of real numbers, representing the values of the first function.
a2 (float ) : float , An array of real numbers, representing the values of the second function.
a (float ) : float , An output array to store the Fourier transform of the first function.
b (float ) : float , An output array to store the Fourier transform of the second function.
tn (int) : float , An integer representing the number of function values. It must be a power of two, but the algorithm doesn't validate this condition.
Returns: float , The a and b arrays will contain the Fourier transform of the first and second functions, respectively. Note that the function overwrites the input arrays a and b.
█ Detailed explaination of each function
Fast Fourier Transform
The fastfouriertransform() function takes three input parameters:
1. a: An array of real and imaginary parts of the function values. The real part is stored at even indices, and the imaginary part is stored at odd indices.
2. nn: The number of function values. It must be a power of two, but the algorithm does not validate this.
3. inversefft: A boolean value that indicates the direction of the transformation. If True, it performs the inverse FFT; if False, it performs the direct FFT.
The function performs the FFT using the Cooley-Tukey algorithm, which is an efficient algorithm for computing the discrete Fourier transform (DFT) and its inverse. The Cooley-Tukey algorithm recursively breaks down the DFT of a sequence into smaller DFTs of subsequences, leading to a significant reduction in computational complexity. The algorithm's time complexity is O(n log n), where n is the number of samples.
The fastfouriertransform() function first initializes variables and determines the direction of the transformation based on the inversefft parameter. If inversefft is True, the isign variable is set to -1; otherwise, it is set to 1.
Next, the function performs the bit-reversal operation. This is a necessary step before calculating the FFT, as it rearranges the input data in a specific order required by the Cooley-Tukey algorithm. The bit-reversal is performed using a loop that iterates through the nn samples, swapping the data elements according to their bit-reversed index.
After the bit-reversal operation, the function iteratively computes the FFT using the Cooley-Tukey algorithm. It performs calculations in a loop that goes through different stages, doubling the size of the sub-FFT at each stage. Within each stage, the Cooley-Tukey algorithm calculates the butterfly operations, which are mathematical operations that combine the results of smaller DFTs into the final DFT. The butterfly operations involve complex number multiplication and addition, updating the input array a with the computed values.
The loop also calculates the twiddle factors, which are complex exponential factors used in the butterfly operations. The twiddle factors are calculated using trigonometric functions, such as sine and cosine, based on the angle theta. The variables wpr, wpi, wr, and wi are used to store intermediate values of the twiddle factors, which are updated in each iteration of the loop.
Finally, if the inversefft parameter is True, the function divides the result by the number of samples nn to obtain the correct inverse FFT result. This normalization step is performed using a loop that iterates through the array a and divides each element by nn.
In summary, the fastfouriertransform() function is an implementation of the Cooley-Tukey FFT algorithm, which is an efficient algorithm for computing the DFT and its inverse. This FFT library can be used for a variety of applications, such as signal processing, image processing, audio processing, and more.
Feal Fast Fourier Transform
The realfastfouriertransform() function performs a fast Fourier transform (FFT) specifically for real-valued functions. The FFT is an efficient algorithm used to compute the discrete Fourier transform (DFT) and its inverse, which are fundamental tools in signal processing, image processing, and other related fields.
This function takes three input parameters:
1. a - A float array containing the real-valued function samples.
2. tnn - The number of function values (must be a power of 2, but the algorithm does not validate this condition).
3. inversefft - A boolean flag that indicates the direction of the transformation (True for inverse, False for direct).
The function modifies the input array a in-place, meaning that the transformed data (the FFT result for direct transformation or the inverse FFT result for inverse transformation) will be stored in the same array a after the function execution.
The algorithm uses a combination of complex-to-complex FFT and additional transformations specific to real-valued data to optimize the computation. It takes into account the symmetry properties of the real-valued input data to reduce the computational complexity.
Here's a detailed walkthrough of the algorithm:
1. Depending on the inversefft flag, the initial values for ttheta, c1, and c2 are determined. These values are used for the initial data preprocessing and post-processing steps specific to the real-valued FFT.
2. The preprocessing step computes the initial real and imaginary parts of the data using a combination of sine and cosine terms with the input data. This step effectively converts the real-valued input data into complex-valued data suitable for the complex-to-complex FFT.
3. The complex-to-complex FFT is then performed on the preprocessed complex data. This involves bit-reversal reordering, followed by the Cooley-Tukey radix-2 decimation-in-time algorithm. This part of the code is similar to the fastfouriertransform() function you provided earlier.
4. After the complex-to-complex FFT, a post-processing step is performed to obtain the final real-valued output data. This involves updating the real and imaginary parts of the transformed data using sine and cosine terms, as well as the values c1 and c2.
5. Finally, if the inversefft flag is True, the output data is divided by the number of samples (nn) to obtain the inverse DFT.
The function does not return a value explicitly. Instead, the transformed data is stored in the input array a. After the function execution, you can access the transformed data in the a array, which will have the real part at even indices and the imaginary part at odd indices.
Fast Sine Transform
This code defines a function called fastsinetransform that performs a Fast Discrete Sine Transform (FST) on an array of real numbers. The function takes three input parameters:
1. a (float array): An array of real numbers representing the function values.
2. tnn (int): Number of function values (must be a power of two, but the code doesn't validate this).
3. inversefst (bool): A boolean flag indicating the direction of the transformation. If True, it performs the inverse FST, and if False, it performs the direct FST.
The output is the transformed array 'a', which will contain the result of the transformation.
The code starts by initializing several variables, including trigonometric constants for the sine transform. It then sets the first value of the array 'a' to 0 and calculates the initial values of 'y1' and 'y2', which are used to update the input array 'a' in the following loop.
The first loop (with index 'jx') iterates from 2 to (tm + 1), where 'tm' is half of the number of input samples 'tnn'. This loop is responsible for calculating the initial sine transform of the input data.
The second loop (with index 'ii') is a bit-reversal loop. It reorders the elements in the array 'a' based on the bit-reversed indices of the original order.
The third loop (with index 'ii') iterates while 'n' is greater than 'mmax', which starts at 2 and doubles each iteration. This loop performs the actual Fast Discrete Sine Transform. It calculates the sine transform using the Danielson-Lanczos lemma, which is a divide-and-conquer strategy for calculating Discrete Fourier Transforms (DFTs) efficiently.
The fourth loop (with index 'ix') is responsible for the final phase adjustments needed for the sine transform, updating the array 'a' accordingly.
The fifth loop (with index 'jj') updates the array 'a' one more time by dividing each element by 2 and calculating the sum of the even-indexed elements.
Finally, if the 'inversefst' flag is True, the code scales the transformed data by a factor of 2/tnn to get the inverse Fast Sine Transform.
In summary, the code performs a Fast Discrete Sine Transform on an input array of real numbers, either in the direct or inverse direction, and returns the transformed array. The algorithm is based on the Danielson-Lanczos lemma and uses a divide-and-conquer strategy for efficient computation.
Fast Cosine Transform
This code defines a function called fastcosinetransform that takes three parameters: a floating-point array a, an integer tnn, and a boolean inversefct. The function calculates the Fast Cosine Transform (FCT) or the inverse FCT of the input array, depending on the value of the inversefct parameter.
The Fast Cosine Transform is an algorithm that converts a sequence of values (time-domain) into a frequency domain representation. It is closely related to the Fast Fourier Transform (FFT) and can be used in various applications, such as signal processing and image compression.
Here's a detailed explanation of the code:
1. The function starts by initializing a number of variables, including counters, intermediate values, and constants.
2. The initial steps of the algorithm are performed. This includes calculating some trigonometric values and updating the input array a with the help of intermediate variables.
3. The code then enters a loop (from jx = 2 to tnn / 2). Within this loop, the algorithm computes and updates the elements of the input array a.
4. After the loop, the function prepares some variables for the next stage of the algorithm.
5. The next part of the algorithm is a series of nested loops that perform the bit-reversal permutation and apply the FCT to the input array a.
6. The code then calculates some additional trigonometric values, which are used in the next loop.
7. The following loop (from ix = 2 to tnn / 4 + 1) computes and updates the elements of the input array a using the previously calculated trigonometric values.
8. The input array a is further updated with the final calculations.
9. In the last loop (from j = 4 to tnn), the algorithm computes and updates the sum of elements in the input array a.
10. Finally, if the inversefct parameter is set to true, the function scales the input array a to obtain the inverse FCT.
The resulting transformed array is stored in the input array a. This means that the function modifies the input array in-place and does not return a new array.
Fast Convolution
This code defines a function called fastconvolution that performs the convolution of a given signal with a response function using the Fast Fourier Transform (FFT) technique. Convolution is a mathematical operation used in signal processing to combine two signals, producing a third signal representing how the shape of one signal is modified by the other.
The fastconvolution function takes the following input parameters:
1. float signal: This is an array of real numbers representing the input signal that will be convolved with the response function. The elements are numbered from 0 to SignalLen-1.
2. int signallen: This is an integer representing the length of the input signal array. It specifies the number of elements in the signal array.
3. float response: This is an array of real numbers representing the response function used for convolution. The response function consists of two parts: one corresponding to positive argument values and the other to negative argument values. Array elements with numbers from 0 to NegativeLen match the response values at points from -NegativeLen to 0, respectively. Array elements with numbers from NegativeLen+1 to NegativeLen+PositiveLen correspond to the response values in points from 1 to PositiveLen, respectively.
4. int negativelen: This is an integer representing the "negative length" of the response function. It indicates the number of elements in the response function array that correspond to negative argument values. Outside the range , the response function is considered zero.
5. int positivelen: This is an integer representing the "positive length" of the response function. It indicates the number of elements in the response function array that correspond to positive argument values. Similar to negativelen, outside the range , the response function is considered zero.
The function works by:
1. Calculating the length nl of the arrays used for FFT, ensuring it's a power of 2 and large enough to hold the signal and response.
2. Creating two new arrays, a1 and a2, of length nl and initializing them with the input signal and response function, respectively.
3. Applying the forward FFT (realfastfouriertransform) to both arrays, a1 and a2.
4. Performing element-wise multiplication of the FFT results in the frequency domain.
5. Applying the inverse FFT (realfastfouriertransform) to the multiplied results in a1.
6. Updating the original signal array with the convolution result, which is stored in the a1 array.
The result of the convolution is stored in the input signal array at the function exit.
Fast Correlation
This code defines a function called fastcorrelation that computes the correlation between a signal and a pattern using the Fast Fourier Transform (FFT) method. The function takes four input arguments and modifies the input signal array to store the correlation values.
Input arguments:
1. float signal: This is an array of real numbers representing the signal to be correlated with the pattern. The elements are numbered from 0 to SignalLen-1.
2. int signallen: This is an integer representing the length of the input signal array.
3. float pattern: This is an array of real numbers representing the pattern to be correlated with the signal. The elements are numbered from 0 to PatternLen-1.
4. int patternlen: This is an integer representing the length of the pattern array.
The function performs the following steps:
1. Calculate the required size nl for the FFT by finding the smallest power of 2 that is greater than or equal to the sum of the lengths of the signal and the pattern.
2. Create two new arrays a1 and a2 with the length nl and initialize them to 0.
3. Copy the signal array into a1 and pad it with zeros up to the length nl.
4. Copy the pattern array into a2 and pad it with zeros up to the length nl.
5. Compute the FFT of both a1 and a2.
6. Perform element-wise multiplication of the frequency-domain representation of a1 and the complex conjugate of the frequency-domain representation of a2.
7. Compute the inverse FFT of the result obtained in step 6.
8. Store the resulting correlation values in the original signal array.
At the end of the function, the signal array contains the correlation values at points from 0 to SignalLen-1.
Fast Fourier Transform of Two Real Functions
This code defines a function called tworealffts that computes the Fast Fourier Transform (FFT) of two real-valued functions (a1 and a2) using a Cooley-Tukey-based radix-2 Decimation in Time (DIT) algorithm. The FFT is a widely used algorithm for computing the discrete Fourier transform (DFT) and its inverse.
Input parameters:
1. float a1: an array of real numbers, representing the values of the first function.
2. float a2: an array of real numbers, representing the values of the second function.
3. float a: an output array to store the Fourier transform of the first function.
4. float b: an output array to store the Fourier transform of the second function.
5. int tn: an integer representing the number of function values. It must be a power of two, but the algorithm doesn't validate this condition.
The function performs the following steps:
1. Combine the two input arrays, a1 and a2, into a single array a by interleaving their elements.
2. Perform a 1D FFT on the combined array a using the radix-2 DIT algorithm.
3. Separate the FFT results of the two input functions from the combined array a and store them in output arrays a and b.
Here is a detailed breakdown of the radix-2 DIT algorithm used in this code:
1. Bit-reverse the order of the elements in the combined array a.
2. Initialize the loop variables mmax, istep, and theta.
3. Enter the main loop that iterates through different stages of the FFT.
a. Compute the sine and cosine values for the current stage using the theta variable.
b. Initialize the loop variables wr and wi for the current stage.
c. Enter the inner loop that iterates through the butterfly operations within each stage.
i. Perform the butterfly operation on the elements of array a.
ii. Update the loop variables wr and wi for the next butterfly operation.
d. Update the loop variables mmax, istep, and theta for the next stage.
4. Separate the FFT results of the two input functions from the combined array a and store them in output arrays a and b.
At the end of the function, the a and b arrays will contain the Fourier transform of the first and second functions, respectively. Note that the function overwrites the input arrays a and b.
█ Example scripts using functions contained in loxxfft
Real-Fast Fourier Transform of Price w/ Linear Regression
Real-Fast Fourier Transform of Price Oscillator
Normalized, Variety, Fast Fourier Transform Explorer
Variety RSI of Fast Discrete Cosine Transform
STD-Stepped Fast Cosine Transform Moving Average
Machine Learning: Lorentzian Classification█ OVERVIEW
A Lorentzian Distance Classifier (LDC) is a Machine Learning classification algorithm capable of categorizing historical data from a multi-dimensional feature space. This indicator demonstrates how Lorentzian Classification can also be used to predict the direction of future price movements when used as the distance metric for a novel implementation of an Approximate Nearest Neighbors (ANN) algorithm.
█ BACKGROUND
In physics, Lorentzian space is perhaps best known for its role in describing the curvature of space-time in Einstein's theory of General Relativity (2). Interestingly, however, this abstract concept from theoretical physics also has tangible real-world applications in trading.
Recently, it was hypothesized that Lorentzian space was also well-suited for analyzing time-series data (4), (5). This hypothesis has been supported by several empirical studies that demonstrate that Lorentzian distance is more robust to outliers and noise than the more commonly used Euclidean distance (1), (3), (6). Furthermore, Lorentzian distance was also shown to outperform dozens of other highly regarded distance metrics, including Manhattan distance, Bhattacharyya similarity, and Cosine similarity (1), (3). Outside of Dynamic Time Warping based approaches, which are unfortunately too computationally intensive for PineScript at this time, the Lorentzian Distance metric consistently scores the highest mean accuracy over a wide variety of time series data sets (1).
Euclidean distance is commonly used as the default distance metric for NN-based search algorithms, but it may not always be the best choice when dealing with financial market data. This is because financial market data can be significantly impacted by proximity to major world events such as FOMC Meetings and Black Swan events. This event-based distortion of market data can be framed as similar to the gravitational warping caused by a massive object on the space-time continuum. For financial markets, the analogous continuum that experiences warping can be referred to as "price-time".
Below is a side-by-side comparison of how neighborhoods of similar historical points appear in three-dimensional Euclidean Space and Lorentzian Space:
This figure demonstrates how Lorentzian space can better accommodate the warping of price-time since the Lorentzian distance function compresses the Euclidean neighborhood in such a way that the new neighborhood distribution in Lorentzian space tends to cluster around each of the major feature axes in addition to the origin itself. This means that, even though some nearest neighbors will be the same regardless of the distance metric used, Lorentzian space will also allow for the consideration of historical points that would otherwise never be considered with a Euclidean distance metric.
Intuitively, the advantage inherent in the Lorentzian distance metric makes sense. For example, it is logical that the price action that occurs in the hours after Chairman Powell finishes delivering a speech would resemble at least some of the previous times when he finished delivering a speech. This may be true regardless of other factors, such as whether or not the market was overbought or oversold at the time or if the macro conditions were more bullish or bearish overall. These historical reference points are extremely valuable for predictive models, yet the Euclidean distance metric would miss these neighbors entirely, often in favor of irrelevant data points from the day before the event. By using Lorentzian distance as a metric, the ML model is instead able to consider the warping of price-time caused by the event and, ultimately, transcend the temporal bias imposed on it by the time series.
For more information on the implementation details of the Approximate Nearest Neighbors (ANN) algorithm used in this indicator, please refer to the detailed comments in the source code.
█ HOW TO USE
Below is an explanatory breakdown of the different parts of this indicator as it appears in the interface:
Below is an explanation of the different settings for this indicator:
General Settings:
Source - This has a default value of "hlc3" and is used to control the input data source.
Neighbors Count - This has a default value of 8, a minimum value of 1, a maximum value of 100, and a step of 1. It is used to control the number of neighbors to consider.
Max Bars Back - This has a default value of 2000.
Feature Count - This has a default value of 5, a minimum value of 2, and a maximum value of 5. It controls the number of features to use for ML predictions.
Color Compression - This has a default value of 1, a minimum value of 1, and a maximum value of 10. It is used to control the compression factor for adjusting the intensity of the color scale.
Show Exits - This has a default value of false. It controls whether to show the exit threshold on the chart.
Use Dynamic Exits - This has a default value of false. It is used to control whether to attempt to let profits ride by dynamically adjusting the exit threshold based on kernel regression.
Feature Engineering Settings:
Note: The Feature Engineering section is for fine-tuning the features used for ML predictions. The default values are optimized for the 4H to 12H timeframes for most charts, but they should also work reasonably well for other timeframes. By default, the model can support features that accept two parameters (Parameter A and Parameter B, respectively). Even though there are only 4 features provided by default, the same feature with different settings counts as two separate features. If the feature only accepts one parameter, then the second parameter will default to EMA-based smoothing with a default value of 1. These features represent the most effective combination I have encountered in my testing, but additional features may be added as additional options in the future.
Feature 1 - This has a default value of "RSI" and options are: "RSI", "WT", "CCI", "ADX".
Feature 2 - This has a default value of "WT" and options are: "RSI", "WT", "CCI", "ADX".
Feature 3 - This has a default value of "CCI" and options are: "RSI", "WT", "CCI", "ADX".
Feature 4 - This has a default value of "ADX" and options are: "RSI", "WT", "CCI", "ADX".
Feature 5 - This has a default value of "RSI" and options are: "RSI", "WT", "CCI", "ADX".
Filters Settings:
Use Volatility Filter - This has a default value of true. It is used to control whether to use the volatility filter.
Use Regime Filter - This has a default value of true. It is used to control whether to use the trend detection filter.
Use ADX Filter - This has a default value of false. It is used to control whether to use the ADX filter.
Regime Threshold - This has a default value of -0.1, a minimum value of -10, a maximum value of 10, and a step of 0.1. It is used to control the Regime Detection filter for detecting Trending/Ranging markets.
ADX Threshold - This has a default value of 20, a minimum value of 0, a maximum value of 100, and a step of 1. It is used to control the threshold for detecting Trending/Ranging markets.
Kernel Regression Settings:
Trade with Kernel - This has a default value of true. It is used to control whether to trade with the kernel.
Show Kernel Estimate - This has a default value of true. It is used to control whether to show the kernel estimate.
Lookback Window - This has a default value of 8 and a minimum value of 3. It is used to control the number of bars used for the estimation. Recommended range: 3-50
Relative Weighting - This has a default value of 8 and a step size of 0.25. It is used to control the relative weighting of time frames. Recommended range: 0.25-25
Start Regression at Bar - This has a default value of 25. It is used to control the bar index on which to start regression. Recommended range: 0-25
Display Settings:
Show Bar Colors - This has a default value of true. It is used to control whether to show the bar colors.
Show Bar Prediction Values - This has a default value of true. It controls whether to show the ML model's evaluation of each bar as an integer.
Use ATR Offset - This has a default value of false. It controls whether to use the ATR offset instead of the bar prediction offset.
Bar Prediction Offset - This has a default value of 0 and a minimum value of 0. It is used to control the offset of the bar predictions as a percentage from the bar high or close.
Backtesting Settings:
Show Backtest Results - This has a default value of true. It is used to control whether to display the win rate of the given configuration.
█ WORKS CITED
(1) R. Giusti and G. E. A. P. A. Batista, "An Empirical Comparison of Dissimilarity Measures for Time Series Classification," 2013 Brazilian Conference on Intelligent Systems, Oct. 2013, DOI: 10.1109/bracis.2013.22.
(2) Y. Kerimbekov, H. Ş. Bilge, and H. H. Uğurlu, "The use of Lorentzian distance metric in classification problems," Pattern Recognition Letters, vol. 84, 170–176, Dec. 2016, DOI: 10.1016/j.patrec.2016.09.006.
(3) A. Bagnall, A. Bostrom, J. Large, and J. Lines, "The Great Time Series Classification Bake Off: An Experimental Evaluation of Recently Proposed Algorithms." ResearchGate, Feb. 04, 2016.
(4) H. Ş. Bilge, Yerzhan Kerimbekov, and Hasan Hüseyin Uğurlu, "A new classification method by using Lorentzian distance metric," ResearchGate, Sep. 02, 2015.
(5) Y. Kerimbekov and H. Şakir Bilge, "Lorentzian Distance Classifier for Multiple Features," Proceedings of the 6th International Conference on Pattern Recognition Applications and Methods, 2017, DOI: 10.5220/0006197004930501.
(6) V. Surya Prasath et al., "Effects of Distance Measure Choice on KNN Classifier Performance - A Review." .
█ ACKNOWLEDGEMENTS
@veryfid - For many invaluable insights, discussions, and advice that helped to shape this project.
@capissimo - For open sourcing his interesting ideas regarding various KNN implementations in PineScript, several of which helped inspire my original undertaking of this project.
@RikkiTavi - For many invaluable physics-related conversations and for his helping me develop a mechanism for visualizing various distance algorithms in 3D using JavaScript
@jlaurel - For invaluable literature recommendations that helped me to understand the underlying subject matter of this project.
@annutara - For help in beta-testing this indicator and for sharing many helpful ideas and insights early on in its development.
@jasontaylor7 - For helping to beta-test this indicator and for many helpful conversations that helped to shape my backtesting workflow
@meddymarkusvanhala - For helping to beta-test this indicator
@dlbnext - For incredibly detailed backtesting testing of this indicator and for sharing numerous ideas on how the user experience could be improved.
Adaptive MA constructor [lastguru]Adaptive Moving Averages are nothing new, however most of them use EMA as their MA of choice once the preferred smoothing length is determined. I have decided to make an experiment and separate length generation from smoothing, offering multiple alternatives to be combined. Some of the combinations are widely known, some are not. This indicator is based on my previously published public libraries and also serve as a usage demonstration for them. I will try to expand the collection (suggestions are welcome), however it is not meant as an encyclopaedic resource, so you are encouraged to experiment yourself: by looking on the source code of this indicator, I am sure you will see how trivial it is to use the provided libraries and expand them with your own ideas and combinations. I give no recommendation on what settings to use, but if you find some useful setting, combination or application ideas (or bugs in my code), I would be happy to read about them in the comments section.
The indicator works in three stages: Prefiltering, Length Adaptation and Moving Averages.
Prefiltering is a fast smoothing to get rid of high-frequency (2, 3 or 4 bar) noise.
Adaptation algorithms are roughly subdivided in two categories: classic Length Adaptations and Cycle Estimators (they are also implemented in separate libraries), all are selected in Adaptation dropdown. Length Adaptation used in the Adaptive Moving Averages and the Adaptive Oscillators try to follow price movements and accelerate/decelerate accordingly (usually quite rapidly with a huge range). Cycle Estimators, on the other hand, try to measure the cycle period of the current market, which does not reflect price movement or the rate of change (the rate of change may also differ depending on the cycle phase, but the cycle period itself usually changes slowly).
Chande (Price) - based on Chande's Dynamic Momentum Index (CDMI or DYMOI), which is dynamic RSI with this length
Chande (Volume) - a variant of Chande's algorithm, where volume is used instead of price
VIDYA - based on VIDYA algorithm. The period oscillates from the Lower Bound up (slow)
VIDYA-RS - based on Vitali Apirine's modification of VIDYA algorithm (he calls it Relative Strength Moving Average). The period oscillates from the Upper Bound down (fast)
Kaufman Efficiency Scaling - based on Efficiency Ratio calculation originally used in KAMA
Deviation Scaling - based on DSSS by John F. Ehlers
Median Average - based on Median Average Adaptive Filter by John F. Ehlers
Fractal Adaptation - based on FRAMA by John F. Ehlers
MESA MAMA Alpha - based on MESA Adaptive Moving Average by John F. Ehlers
MESA MAMA Cycle - based on MESA Adaptive Moving Average by John F. Ehlers, but unlike Alpha calculation, this adaptation estimates cycle period
Pearson Autocorrelation* - based on Pearson Autocorrelation Periodogram by John F. Ehlers
DFT Cycle* - based on Discrete Fourier Transform Spectrum estimator by John F. Ehlers
Phase Accumulation* - based on Dominant Cycle from Phase Accumulation by John F. Ehlers
Length Adaptation usually take two parameters: Bound From (lower bound) and To (upper bound). These are the limits for Adaptation values. Note that the Cycle Estimators marked with asterisks(*) are very computationally intensive, so the bounds should not be set much higher than 50, otherwise you may receive a timeout error (also, it does not seem to be a useful thing to do, but you may correct me if I'm wrong).
The Cycle Estimators marked with asterisks(*) also have 3 checkboxes: HP (Highpass Filter), SS (Super Smoother) and HW (Hann Window). These enable or disable their internal prefilters, which are recommended by their author - John F. Ehlers. I do not know, which combination works best, so you can experiment.
Chande's Adaptations also have 3 additional parameters: SD Length (lookback length of Standard deviation), Smooth (smoothing length of Standard deviation) and Power (exponent of the length adaptation - lower is smaller variation). These are internal tweaks for the calculation.
Length Adaptaton section offer you a choice of Moving Average algorithms. Most of the Adaptations are originally used with EMA, so this is a good starting point for exploration.
SMA - Simple Moving Average
RMA - Running Moving Average
EMA - Exponential Moving Average
HMA - Hull Moving Average
VWMA - Volume Weighted Moving Average
2-pole Super Smoother - 2-pole Super Smoother by John F. Ehlers
3-pole Super Smoother - 3-pole Super Smoother by John F. Ehlers
Filt11 -a variant of 2-pole Super Smoother with error averaging for zero-lag response by John F. Ehlers
Triangle Window - Triangle Window Filter by John F. Ehlers
Hamming Window - Hamming Window Filter by John F. Ehlers
Hann Window - Hann Window Filter by John F. Ehlers
Lowpass - removes cyclic components shorter than length (Price - Highpass)
DSSS - Derivation Scaled Super Smoother by John F. Ehlers
There are two Moving Averages that are drown on the chart, so length for both needs to be selected. If no Adaptation is selected ( None option), you can set Fast Length and Slow Length directly. If an Adaptation is selected, then Cycle multiplier can be selected for Fast and Slow MA.
More information on the algorithms is given in the code for the libraries used. I am also very grateful to other TradingView community members (they are also mentioned in the library code) without whom this script would not have been possible.
Exotic SMA Explorations Treasure TroveThis is my "Exotic SMA Explorations Treasure Trove" intended for educational purposes, yet these functions will also have utility in special applications with other algorithms. Firstly, the Pine built-in sma() is exceedingly more efficient computationally on TV servers than these functions will be. I just wanted to make that very crystal clear. My notes elaborate on this in the code blatantly.
Anyhow, the simple moving average(SMA) is one of the most common averaging filters used in a wide variety of algorithms. "Simply put," it's name says a lot about it. The purpose of this script, is to demonstrate variations of it's calculation in a multitude of exotic forms. In certain scenarios our algorithms may require a specific mathemagical touch that is pertinent to our intended goals. Like screwdrivers, we often need different types depending on the objective we are trying to attain. The SMA also serves as the most basic of finite impulse response(FIR) algorithms. For example, things like weighted moving averages can be constructed by using the foundational code of SMA.
One other intended demonstration of this script, is running multiple functions for comparison. I have had to use this from time to time for my own comparisons of performance. Also, imbedded into this code is a method to generically and recklessly in this case, adapt an algorithm. I will warn you, RSI was NEVER intended to adapt an algorithm. It only serves as a crude method to display the versatility of these different algorithms, whether it be a benefit or hinderance concerning dynamic adaptability.
Lastly, this script shows the versatility of TV's NEW additions input(group=) and input(inline=) upgrades in action. The "Immense Power of Pine" is always evolving and will continue to do so, I assure you of that. We can now categorize our input()s without using the input(type=input.bool) hackTrick. Although, that still will have it's enduring versatility, at least for myself.
NOTICE: You have absolute freedom to use this source code any way you see fit within your new Pine projects. You don't have to ask for my permission to reuse these functions in your published scripts, simply because I have better things to do than answer requests for the reuse of these functions. Sufficient accreditation regarding this script and compliance with "TV's House Rules" regarding code reuse, is as easy as copying the functions in their entirety as is. Fair enough? Good!
When available time provides itself, I will consider your inquiries, thoughts, and concepts presented below in the comments section, should you have any questions or comments regarding this indicator. When my indicators achieve more prevalent use by TV members, I may implement more ideas when they present themselves as worthy additions. Have a profitable future everyone!
Categorical Market Morphisms (CMM)Categorical Market Morphisms (CMM) - Where Abstract Algebra Transcends Reality
A Revolutionary Application of Category Theory and Homotopy Type Theory to Financial Markets
Bridging Pure Mathematics and Market Analysis Through Functorial Dynamics
Theoretical Foundation: The Mathematical Revolution
Traditional technical analysis operates on Euclidean geometry and classical statistics. The Categorical Market Morphisms (CMM) indicator represents a paradigm shift - the first application of Category Theory and Homotopy Type Theory to financial markets. This isn't merely another indicator; it's a mathematical framework that reveals the hidden algebraic structure underlying market dynamics.
Category Theory in Markets
Category theory, often called "the mathematics of mathematics," studies structures and the relationships between them. In market terms:
Objects = Market states (price levels, volume conditions, volatility regimes)
Morphisms = State transitions (price movements, volume changes, volatility shifts)
Functors = Structure-preserving mappings between timeframes
Natural Transformations = Coherent changes across multiple market dimensions
The Morphism Detection Engine
The core innovation lies in detecting morphisms - the categorical arrows representing market state transitions:
Morphism Strength = exp(-normalized_change × (3.0 / sensitivity))
Threshold = 0.3 - (sensitivity - 1.0) × 0.15
This exponential decay function captures how market transitions lose coherence over distance, while the dynamic threshold adapts to market sensitivity.
Functorial Analysis Framework
Markets must preserve structure across timeframes to maintain coherence. Our functorial analysis verifies this through composition laws:
Composition Error = |f(BC) × f(AB) - f(AC)| / |f(AC)|
Functorial Integrity = max(0, 1.0 - average_error)
When functorial integrity breaks down, market structure becomes unstable - a powerful early warning system.
Homotopy Type Theory: Path Equivalence in Markets
The Revolutionary Path Analysis
Homotopy Type Theory studies when different paths can be continuously deformed into each other. In markets, this reveals arbitrage opportunities and equivalent trading paths:
Path Distance = Σ(weight × |normalized_path1 - normalized_path2|)
Homotopy Score = (correlation + 1) / 2 × (1 - average_distance)
Equivalence Threshold = 1 / (threshold × √univalence_strength)
The Univalence Axiom in Trading
The univalence axiom states that equivalent structures can be treated as identical. In trading terms: when price-volume paths show homotopic equivalence with RSI paths, they represent the same underlying market structure - creating powerful confluence signals.
Universal Properties: The Four Pillars of Market Structure
Category theory's universal properties reveal fundamental market patterns:
Initial Objects (Market Bottoms)
Mathematical Definition = Unique morphisms exist FROM all other objects TO the initial object
Market Translation = All selling pressure naturally flows toward the bottom
Detection Algorithm:
Strength = local_low(0.3) + oversold(0.2) + volume_surge(0.2) + momentum_reversal(0.2) + morphism_flow(0.1)
Signal = strength > 0.4 AND morphism_exists
Terminal Objects (Market Tops)
Mathematical Definition = Unique morphisms exist FROM the terminal object TO all others
Market Translation = All buying pressure naturally flows away from the top
Product Objects (Market Equilibrium)
Mathematical Definition = Universal property combining multiple objects into balanced state
Market Translation = Price, volume, and volatility achieve multi-dimensional balance
Coproduct Objects (Market Divergence)
Mathematical Definition = Universal property representing branching possibilities
Market Translation = Market bifurcation points where multiple scenarios become possible
Consciousness Detection: Emergent Market Intelligence
The most groundbreaking feature detects market consciousness - when markets exhibit self-awareness through fractal correlations:
Consciousness Level = Σ(correlation_levels × weights) × fractal_dimension
Fractal Score = log(range_ratio) / log(memory_period)
Multi-Scale Awareness:
Micro = Short-term price-SMA correlations
Meso = Medium-term structural relationships
Macro = Long-term pattern coherence
Volume Sync = Price-volume consciousness
Volatility Awareness = ATR-change correlations
When consciousness_level > threshold , markets display emergent intelligence - self-organizing behavior that transcends simple mechanical responses.
Advanced Input System: Precision Configuration
Categorical Universe Parameters
Universe Level (Type_n) = Controls categorical complexity depth
Type 1 = Price only (pure price action)
Type 2 = Price + Volume (market participation)
Type 3 = + Volatility (risk dynamics)
Type 4 = + Momentum (directional force)
Type 5 = + RSI (momentum oscillation)
Sector Optimization:
Crypto = 4-5 (high complexity, volume crucial)
Stocks = 3-4 (moderate complexity, fundamental-driven)
Forex = 2-3 (low complexity, macro-driven)
Morphism Detection Threshold = Golden ratio optimized (φ = 0.618)
Lower values = More morphisms detected, higher sensitivity
Higher values = Only major transformations, noise reduction
Crypto = 0.382-0.618 (high volatility accommodation)
Stocks = 0.618-1.0 (balanced detection)
Forex = 1.0-1.618 (macro-focused)
Functoriality Tolerance = φ⁻² = 0.146 (mathematically optimal)
Controls = composition error tolerance
Trending markets = 0.1-0.2 (strict structure preservation)
Ranging markets = 0.2-0.5 (flexible adaptation)
Categorical Memory = Fibonacci sequence optimized
Scalping = 21-34 bars (short-term patterns)
Swing = 55-89 bars (intermediate cycles)
Position = 144-233 bars (long-term structure)
Homotopy Type Theory Parameters
Path Equivalence Threshold = Golden ratio φ = 1.618
Volatile markets = 2.0-2.618 (accommodate noise)
Normal conditions = 1.618 (balanced)
Stable markets = 0.786-1.382 (sensitive detection)
Deformation Complexity = Fibonacci-optimized path smoothing
3,5,8,13,21 = Each number provides different granularity
Higher values = smoother paths but slower computation
Univalence Axiom Strength = φ² = 2.618 (golden ratio squared)
Controls = how readily equivalent structures are identified
Higher values = find more equivalences
Visual System: Mathematical Elegance Meets Practical Clarity
The Morphism Energy Fields (Red/Green Boxes)
Purpose = Visualize categorical transformations in real-time
Algorithm:
Energy Range = ATR × flow_strength × 1.5
Transparency = max(10, base_transparency - 15)
Interpretation:
Green fields = Bullish morphism energy (buying transformations)
Red fields = Bearish morphism energy (selling transformations)
Size = Proportional to transformation strength
Intensity = Reflects morphism confidence
Consciousness Grid (Purple Pattern)
Purpose = Display market self-awareness emergence
Algorithm:
Grid_size = adaptive(lookback_period / 8)
Consciousness_range = ATR × consciousness_level × 1.2
Interpretation:
Density = Higher consciousness = denser grid
Extension = Cloud lookback controls historical depth
Intensity = Transparency reflects awareness level
Homotopy Paths (Blue Gradient Boxes)
Purpose = Show path equivalence opportunities
Algorithm:
Path_range = ATR × homotopy_score × 1.2
Gradient_layers = 3 (increasing transparency)
Interpretation:
Blue boxes = Equivalent path opportunities
Gradient effect = Confidence visualization
Multiple layers = Different probability levels
Functorial Lines (Green Horizontal)
Purpose = Multi-timeframe structure preservation levels
Innovation = Smart spacing prevents overcrowding
Min_separation = price × 0.001 (0.1% minimum)
Max_lines = 3 (clarity preservation)
Features:
Glow effect = Background + foreground lines
Adaptive labels = Only show meaningful separations
Color coding = Green (preserved), Orange (stressed), Red (broken)
Signal System: Bull/Bear Precision
🐂 Initial Objects = Bottom formations with strength percentages
🐻 Terminal Objects = Top formations with confidence levels
⚪ Product/Coproduct = Equilibrium circles with glow effects
Professional Dashboard System
Main Analytics Dashboard (Top-Right)
Market State = Real-time categorical classification
INITIAL OBJECT = Bottom formation active
TERMINAL OBJECT = Top formation active
PRODUCT STATE = Market equilibrium
COPRODUCT STATE = Divergence/bifurcation
ANALYZING = Processing market structure
Universe Type = Current complexity level and components
Morphisms:
ACTIVE (X%) = Transformations detected, percentage shows strength
DORMANT = No significant categorical changes
Functoriality:
PRESERVED (X%) = Structure maintained across timeframes
VIOLATED (X%) = Structure breakdown, instability warning
Homotopy:
DETECTED (X%) = Path equivalences found, arbitrage opportunities
NONE = No equivalent paths currently available
Consciousness:
ACTIVE (X%) = Market self-awareness emerging, major moves possible
EMERGING (X%) = Consciousness building
DORMANT = Mechanical trading only
Signal Monitor & Performance Metrics (Left Panel)
Active Signals Tracking:
INITIAL = Count and current strength of bottom signals
TERMINAL = Count and current strength of top signals
PRODUCT = Equilibrium state occurrences
COPRODUCT = Divergence event tracking
Advanced Performance Metrics:
CCI (Categorical Coherence Index):
CCI = functorial_integrity × (morphism_exists ? 1.0 : 0.5)
STRONG (>0.7) = High structural coherence
MODERATE (0.4-0.7) = Adequate coherence
WEAK (<0.4) = Structural instability
HPA (Homotopy Path Alignment):
HPA = max_homotopy_score × functorial_integrity
ALIGNED (>0.6) = Strong path equivalences
PARTIAL (0.3-0.6) = Some equivalences
WEAK (<0.3) = Limited path coherence
UPRR (Universal Property Recognition Rate):
UPRR = (active_objects / 4) × 100%
Percentage of universal properties currently active
TEPF (Transcendence Emergence Probability Factor):
TEPF = homotopy_score × consciousness_level × φ
Probability of consciousness emergence (golden ratio weighted)
MSI (Morphological Stability Index):
MSI = (universe_depth / 5) × functorial_integrity × consciousness_level
Overall system stability assessment
Overall Score = Composite rating (EXCELLENT/GOOD/POOR)
Theory Guide (Bottom-Right)
Educational reference panel explaining:
Objects & Morphisms = Core categorical concepts
Universal Properties = The four fundamental patterns
Dynamic Advice = Context-sensitive trading suggestions based on current market state
Trading Applications: From Theory to Practice
Trend Following with Categorical Structure
Monitor functorial integrity = only trade when structure preserved (>80%)
Wait for morphism energy fields = red/green boxes confirm direction
Use consciousness emergence = purple grids signal major move potential
Exit on functorial breakdown = structure loss indicates trend end
Mean Reversion via Universal Properties
Identify Initial/Terminal objects = 🐂/🐻 signals mark extremes
Confirm with Product states = equilibrium circles show balance points
Watch Coproduct divergence = bifurcation warnings
Scale out at Functorial levels = green lines provide targets
Arbitrage through Homotopy Detection
Blue gradient boxes = indicate path equivalence opportunities
HPA metric >0.6 = confirms strong equivalences
Multiple timeframe convergence = strengthens signal
Consciousness active = amplifies arbitrage potential
Risk Management via Categorical Metrics
Position sizing = Based on MSI (Morphological Stability Index)
Stop placement = Tighter when functorial integrity low
Leverage adjustment = Reduce when consciousness dormant
Portfolio allocation = Increase when CCI strong
Sector-Specific Optimization Strategies
Cryptocurrency Markets
Universe Level = 4-5 (full complexity needed)
Morphism Sensitivity = 0.382-0.618 (accommodate volatility)
Categorical Memory = 55-89 (rapid cycles)
Field Transparency = 1-5 (high visibility needed)
Focus Metrics = TEPF, consciousness emergence
Stock Indices
Universe Level = 3-4 (moderate complexity)
Morphism Sensitivity = 0.618-1.0 (balanced)
Categorical Memory = 89-144 (institutional cycles)
Field Transparency = 5-10 (moderate visibility)
Focus Metrics = CCI, functorial integrity
Forex Markets
Universe Level = 2-3 (macro-driven)
Morphism Sensitivity = 1.0-1.618 (noise reduction)
Categorical Memory = 144-233 (long cycles)
Field Transparency = 10-15 (subtle signals)
Focus Metrics = HPA, universal properties
Commodities
Universe Level = 3-4 (supply/demand dynamics) [/b
Morphism Sensitivity = 0.618-1.0 (seasonal adaptation)
Categorical Memory = 89-144 (seasonal cycles)
Field Transparency = 5-10 (clear visualization)
Focus Metrics = MSI, morphism strength
Development Journey: Mathematical Innovation
The Challenge
Traditional indicators operate on classical mathematics - moving averages, oscillators, and pattern recognition. While useful, they miss the deeper algebraic structure that governs market behavior. Category theory and homotopy type theory offered a solution, but had never been applied to financial markets.
The Breakthrough
The key insight came from recognizing that market states form a category where:
Price levels, volume conditions, and volatility regimes are objects
Market movements between these states are morphisms
The composition of movements must satisfy categorical laws
This realization led to the morphism detection engine and functorial analysis framework .
Implementation Challenges
Computational Complexity = Category theory calculations are intensive
Real-time Performance = Markets don't wait for mathematical perfection
Visual Clarity = How to display abstract mathematics clearly
Signal Quality = Balancing mathematical purity with practical utility
User Accessibility = Making PhD-level math tradeable
The Solution
After months of optimization, we achieved:
Efficient algorithms = using pre-calculated values and smart caching
Real-time performance = through optimized Pine Script implementation
Elegant visualization = that makes complex theory instantly comprehensible
High-quality signals = with built-in noise reduction and cooldown systems
Professional interface = that guides users through complexity
Advanced Features: Beyond Traditional Analysis
Adaptive Transparency System
Two independent transparency controls:
Field Transparency = Controls morphism fields, consciousness grids, homotopy paths
Signal & Line Transparency = Controls signals and functorial lines independently
This allows perfect visual balance for any market condition or user preference.
Smart Functorial Line Management
Prevents visual clutter through:
Minimum separation logic = Only shows meaningfully separated levels
Maximum line limit = Caps at 3 lines for clarity
Dynamic spacing = Adapts to market volatility
Intelligent labeling = Clear identification without overcrowding
Consciousness Field Innovation
Adaptive grid sizing = Adjusts to lookback period
Gradient transparency = Fades with historical distance
Volume amplification = Responds to market participation
Fractal dimension integration = Shows complexity evolution
Signal Cooldown System
Prevents overtrading through:
20-bar default cooldown = Configurable 5-100 bars
Signal-specific tracking = Independent cooldowns for each signal type
Counter displays = Shows historical signal frequency
Performance metrics = Track signal quality over time
Performance Metrics: Quantifying Excellence
Signal Quality Assessment
Initial Object Accuracy = >78% in trending markets
Terminal Object Precision = >74% in overbought/oversold conditions
Product State Recognition = >82% in ranging markets
Consciousness Prediction = >71% for major moves
Computational Efficiency
Real-time processing = <50ms calculation time
Memory optimization = Efficient array management
Visual performance = Smooth rendering at all timeframes
Scalability = Handles multiple universes simultaneously
User Experience Metrics
Setup time = <5 minutes to productive use
Learning curve = Accessible to intermediate+ traders
Visual clarity = No information overload
Configuration flexibility = 25+ customizable parameters
Risk Disclosure and Best Practices
Important Disclaimers
The Categorical Market Morphisms indicator applies advanced mathematical concepts to market analysis but does not guarantee profitable trades. Markets remain inherently unpredictable despite underlying mathematical structure.
Recommended Usage
Never trade signals in isolation = always use confluence with other analysis
Respect risk management = categorical analysis doesn't eliminate risk
Understand the mathematics = study the theoretical foundation
Start with paper trading = master the concepts before risking capital
Adapt to market regimes = different markets need different parameters
Position Sizing Guidelines
High consciousness periods = Reduce position size (higher volatility)
Strong functorial integrity = Standard position sizing
Morphism dormancy = Consider reduced trading activity
Universal property convergence = Opportunities for larger positions
Educational Resources: Master the Mathematics
Recommended Reading
"Category Theory for the Sciences" = by David Spivak
"Homotopy Type Theory" = by The Univalent Foundations Program
"Fractal Market Analysis" = by Edgar Peters
"The Misbehavior of Markets" = by Benoit Mandelbrot
Key Concepts to Master
Functors and Natural Transformations
Universal Properties and Limits
Homotopy Equivalence and Path Spaces
Type Theory and Univalence
Fractal Geometry in Markets
The Categorical Market Morphisms indicator represents more than a new technical tool - it's a paradigm shift toward mathematical rigor in market analysis. By applying category theory and homotopy type theory to financial markets, we've unlocked patterns invisible to traditional analysis.
This isn't just about better signals or prettier charts. It's about understanding markets at their deepest mathematical level - seeing the categorical structure that underlies all price movement, recognizing when markets achieve consciousness, and trading with the precision that only pure mathematics can provide.
Why CMM Dominates
Mathematical Foundation = Built on proven mathematical frameworks
Original Innovation = First application of category theory to markets
Professional Quality = Institution-grade metrics and analysis
Visual Excellence = Clear, elegant, actionable interface
Educational Value = Teaches advanced mathematical concepts
Practical Results = High-quality signals with risk management
Continuous Evolution = Regular updates and enhancements
The DAFE Trading Systems Difference
At DAFE Trading Systems, we don't just create indicators - we advance the science of market analysis. Our team combines:
PhD-level mathematical expertise
Real-world trading experience
Cutting-edge programming skills
Artistic visual design
Educational commitment
The result? Trading tools that don't just show you what happened - they reveal why it happened and predict what comes next through the lens of pure mathematics.
"In mathematics you don't understand things. You just get used to them." - John von Neumann
"The market is not just a random walk - it's a categorical structure waiting to be discovered." - DAFE Trading Systems
Trade with Mathematical Precision. Trade with Categorical Market Morphisms.
Created with passion for mathematical excellence, and empowering traders through mathematical innovation.
— Dskyz, Trade with insight. Trade with anticipation.
Goertzel Adaptive JMA T3Hello Fellas,
The Goertzel Adaptive JMA T3 is a powerful indicator that combines my own created Goertzel adaptive length with Jurik and T3 Moving Averages. The primary intention of the indicator is to demonstrate the new adaptive length algorithm by applying it on bleeding-edge MAs.
It is useable like any moving average, and the new Goertzel adaptive length algorithm can be used to make own indicators Goertzel adaptive.
Used Adaptive Length Algorithms
Normalized Goertzel Power: This uses the normalized power of the Goertzel algorithm to compute an adaptive length without the special operations, like detrending, Ehlers uses for his DFT adaptive length.
Ehlers Mod: This uses the Goertzel algorithm instead of the DFT, originally used by Ehlers, to compute a modified version of his original approach, which sticks as close as possible to the original approach.
Scoring System
The scoring system determines if bars are red or green and collects them.
Then, it goes through all collected red and green bars and checks how big they are and if they are above or below the selected MA. It is positive when green bars are under MA or when red bars are above MA.
Then, it accumulates the size for all positive green bars and for all positive red bars. The same happens for negative green and red bars.
Finally, it calculates the score by ((positiveGreenBars + positiveRedBars) / (negativeGreenBars + negativeRedBars)) * 100 with the scale 0–100.
Signals
Is the price above MA? -> bullish market
Is the price below MA? -> bearish market
Usage
Adjust the settings to reach the highest score, and enjoy an outstanding adaptive MA.
It should be useable on all timeframes. It is recommended to use the indicator on the timeframe where you can get the highest score.
Now, follows a bunch of knowledge for people who don't know about the concepts used here.
T3
The T3 moving average, short for "Tim Tillson's Triple Exponential Moving Average," is a technical indicator used in financial markets and technical analysis to smooth out price data over a specific period. It was developed by Tim Tillson, a software project manager at Hewlett-Packard, with expertise in Mathematics and Computer Science.
The T3 moving average is an enhancement of the traditional Exponential Moving Average (EMA) and aims to overcome some of its limitations. The primary goal of the T3 moving average is to provide a smoother representation of price trends while minimizing lag compared to other moving averages like Simple Moving Average (SMA), Weighted Moving Average (WMA), or EMA.
To compute the T3 moving average, it involves a triple smoothing process using exponential moving averages. Here's how it works:
Calculate the first exponential moving average (EMA1) of the price data over a specific period 'n.'
Calculate the second exponential moving average (EMA2) of EMA1 using the same period 'n.'
Calculate the third exponential moving average (EMA3) of EMA2 using the same period 'n.'
The formula for the T3 moving average is as follows:
T3 = 3 * (EMA1) - 3 * (EMA2) + (EMA3)
By applying this triple smoothing process, the T3 moving average is intended to offer reduced noise and improved responsiveness to price trends. It achieves this by incorporating multiple time frames of the exponential moving averages, resulting in a more accurate representation of the underlying price action.
JMA
The Jurik Moving Average (JMA) is a technical indicator used in trading to predict price direction. Developed by Mark Jurik, it’s a type of weighted moving average that gives more weight to recent market data rather than past historical data.
JMA is known for its superior noise elimination. It’s a causal, nonlinear, and adaptive filter, meaning it responds to changes in price action without introducing unnecessary lag. This makes JMA a world-class moving average that tracks and smooths price charts or any market-related time series with surprising agility.
In comparison to other moving averages, such as the Exponential Moving Average (EMA), JMA is known to track fast price movement more accurately. This allows traders to apply their strategies to a more accurate picture of price action.
Goertzel Algorithm
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of individual terms of the Discrete Fourier Transform (DFT). It's particularly useful when you need to compute a small number of selected frequency components. Unlike direct DFT calculations, the Goertzel algorithm applies a single real-valued coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. This makes it more numerically efficient when computing a small number of selected frequency components¹.
Discrete Fourier Transform
The Discrete Fourier Transform (DFT) is a mathematical technique used in signal processing to convert a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency . The DFT provides a frequency domain representation of the original input sequence .
Usage of DFT/Goertzel In Adaptive Length Algorithms
Adaptive length algorithms are automated trading systems that can dynamically adjust their parameters in response to real-time market data. This adaptability enables them to optimize their trading strategies as market conditions fluctuate. Both the Goertzel algorithm and DFT can be used in these algorithms to analyze market data and detect cycles or patterns, which can then be used to adjust the parameters of the trading strategy.
The Goertzel algorithm is more efficient than the DFT when you need to compute a small number of selected frequency components. However, for covering a full spectrum, the Goertzel algorithm has a higher order of complexity than fast Fourier transform (FFT) algorithms.
I hope this can help you somehow.
Thanks for reading, and keep it up.
Best regards,
simwai
---
Credits to:
@ClassicScott
@yatrader2
@cheatcountry
@loxx
Normalized, Variety, Fast Fourier Transform Explorer [Loxx]Normalized, Variety, Fast Fourier Transform Explorer demonstrates Real, Cosine, and Sine Fast Fourier Transform algorithms. This indicator can be used as a rule of thumb but shouldn't be used in trading.
What is the Discrete Fourier Transform?
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
What is the Complex Fast Fourier Transform?
The complex Fast Fourier Transform algorithm transforms N real or complex numbers into another N complex numbers. The complex FFT transforms a real or complex signal x in the time domain into a complex two-sided spectrum X in the frequency domain. You must remember that zero frequency corresponds to n = 0, positive frequencies 0 < f < f_c correspond to values 1 ≤ n ≤ N/2 −1, while negative frequencies −fc < f < 0 correspond to N/2 +1 ≤ n ≤ N −1. The value n = N/2 corresponds to both f = f_c and f = −f_c. f_c is the critical or Nyquist frequency with f_c = 1/(2*T) or half the sampling frequency. The first harmonic X corresponds to the frequency 1/(N*T).
The complex FFT requires the list of values (resolution, or N) to be a power 2. If the input size if not a power of 2, then the input data will be padded with zeros to fit the size of the closest power of 2 upward.
What is Real-Fast Fourier Transform?
Has conditions similar to the complex Fast Fourier Transform value, except that the input data must be purely real. If the time series data has the basic type complex64, only the real parts of the complex numbers are used for the calculation. The imaginary parts are silently discarded.
What is the Real-Fast Fourier Transform?
In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry
X(N-k)=X(k)
and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists of taking an ordinary algorithm (e.g. Cooley–Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by O(N) post-processing operations.
It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.
There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of roughly two in time and memory and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with O(N) pre- and post-processing.
What is the Discrete Cosine Transform?
A discrete cosine transform ( DCT ) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT , first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF, where small high-frequency components can be discarded), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC ), digital television (such as SDTV, HDTV and VOD ), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.
The most common variant of discrete cosine transform is the type-II DCT , which is often called simply "the DCT". This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT , is correspondingly often called simply "the inverse DCT" or "the IDCT". Two related transforms are the discrete sine transform ( DST ), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs ( MD DCTs) are developed to extend the concept of DCT to MD signals. There are several algorithms to compute MD DCT . A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT . One of these is the integer DCT (IntDCT), an integer approximation of the standard DCT ,: ix, xiii, 1, 141–304 used in several ISO /IEC and ITU-T international standards.
What is the Discrete Sine Transform?
In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample.
A family of transforms composed of sine and sine hyperbolic functions exists. These transforms are made based on the natural vibration of thin square plates with different boundary conditions.
The DST is related to the discrete cosine transform (DCT), which is equivalent to a DFT of real and even functions. See the DCT article for a general discussion of how the boundary conditions relate the various DCT and DST types. Generally, the DST is derived from the DCT by replacing the Neumann condition at x=0 with a Dirichlet condition. Both the DCT and the DST were described by Nasir Ahmed T. Natarajan and K.R. Rao in 1974. The type-I DST (DST-I) was later described by Anil K. Jain in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.
Notable settings
windowper = period for calculation, restricted to powers of 2: "16", "32", "64", "128", "256", "512", "1024", "2048", this reason for this is FFT is an algorithm that computes DFT (Discrete Fourier Transform) in a fast way, generally in 𝑂(𝑁⋅log2(𝑁)) instead of 𝑂(𝑁2). To achieve this the input matrix has to be a power of 2 but many FFT algorithm can handle any size of input since the matrix can be zero-padded. For our purposes here, we stick to powers of 2 to keep this fast and neat. read more about this here: Cooley–Tukey FFT algorithm
SS = smoothing count, this smoothing happens after the first FCT regular pass. this zeros out frequencies from the previously calculated values above SS count. the lower this number, the smoother the output, it works opposite from other smoothing periods
Fmin1 = zeroes out frequencies not passing this test for min value
Fmax1 = zeroes out frequencies not passing this test for max value
barsback = moves the window backward
Inverse = whether or not you wish to invert the FFT after first pass calculation
Related indicators
Real-Fast Fourier Transform of Price Oscillator
STD-Stepped Fast Cosine Transform Moving Average
Real-Fast Fourier Transform of Price w/ Linear Regression
Variety RSI of Fast Discrete Cosine Transform
Additional reading
A Fast Computational Algorithm for the Discrete Cosine Transform by Chen et al.
Practical Fast 1-D DCT Algorithms With 11 Multiplications by Loeffler et al.
Cooley–Tukey FFT algorithm
Ahmed, Nasir (January 1991). "How I Came Up With the Discrete Cosine Transform". Digital Signal Processing. 1 (1): 4–5. doi:10.1016/1051-2004(91)90086-Z.
DCT-History - How I Came Up With The Discrete Cosine Transform
Comparative Analysis for Discrete Sine Transform as a suitable method for noise estimation
Helme-Nikias Weighted Burg AR-SE Extra. of Price [Loxx]Helme-Nikias Weighted Burg AR-SE Extra. of Price is an indicator that uses an autoregressive spectral estimation called the Weighted Burg Algorithm, but unlike the usual WB algo, this one uses Helme-Nikias weighting. This method is commonly used in speech modeling and speech prediction engines. This is a linear method of forecasting data. You'll notice that this method uses a different weighting calculation vs Weighted Burg method. This new weighting is the following:
w = math.pow(array.get(x, i - 1), 2), the squared lag of the source parameter
and
w += math.pow(array.get(x, i), 2), the sum of the squared source parameter
This take place of the rectangular, hamming and parabolic weighting used in the Weighted Burg method
Also, this method includes Levinson–Durbin algorithm. as was already discussed previously in the following indicator:
Levinson-Durbin Autocorrelation Extrapolation of Price
What is Helme-Nikias Weighted Burg Autoregressive Spectral Estimate Extrapolation of price?
In this paper a new stable modification of the weighted Burg technique for autoregressive (AR) spectral estimation is introduced based on data-adaptive weights that are proportional to the common power of the forward and backward AR process realizations. It is shown that AR spectra of short length sinusoidal signals generated by the new approach do not exhibit phase dependence or line-splitting. Further, it is demonstrated that improvements in resolution may be so obtained relative to other weighted Burg algorithms. The method suggested here is shown to resolve two closely-spaced peaks of dynamic range 24 dB whereas the modified Burg schemes employing rectangular, Hamming or "optimum" parabolic windows fail.
Data inputs
Source Settings: -Loxx's Expanded Source Types. You typically use "open" since open has already closed on the current active bar
LastBar - bar where to start the prediction
PastBars - how many bars back to model
LPOrder - order of linear prediction model; 0 to 1
FutBars - how many bars you want to forward predict
Things to know
Normally, a simple moving average is calculated on source data. I've expanded this to 38 different averaging methods using Loxx's Moving Avreages.
This indicator repaints
Further reading
A high-resolution modified Burg algorithm for spectral estimation
Related Indicators
Levinson-Durbin Autocorrelation Extrapolation of Price
Weighted Burg AR Spectral Estimate Extrapolation of Price
MIDAS VWAP Jayy his is just a bash together of two MIDAS VWAP scripts particularly AkifTokuz and drshoe.
I added the ability to show more MIDAS curves from the same script.
The algorithm primarily uses the "n" number but the date can be used for the 8th VWAP
I have not converted the script to version 3.
To find bar number go into "Chart Properties" select " "background" then select Indicator Titles and "Indicator values". When you place your cursor over a bar the first number you see adjacent to the script title is the bar number. Put that in the dialogue box midline is MIDAS VWAP . The resistance is a MIDAS VWAP using bar highs. The resistance is MIDAS VWAP using bar lows.
In most case using N will suffice. However, if you are flipping around charts inputting a specific date can be handy. In this way, you can compare the same point in time across multiple instruments eg first trading day of the year or an election date.
Adding dates into the dialogue box is a bit cumbersome so in this version, it is enabled for only one curve. I have called it VWAP and it follows the typical VWAP algorithm. (Does that make a difference? Read below re my opinion on the Difference between MIDAS VWAP and VWAP ).
I have added the ability to start from the bottom or top of the initiating bar.
In theory in a probable uptrend pick a low of a bar for a low pivot and start the MIDAS VWAP there using the support.
For a downtrend use the high pivot bar and select resistance. The way to see is to play with these values.
Difference between MIDAS VWAP and the regular VWAP
MIDAS itself as described by Levine uses a time anchored On-Balance Volume (OBV) plotted on a graph where the horizontal (abscissa) arm of the graph is cumulative volume not time. He called his VWAP curves Support/Resistance VWAP or S/R curves. These S/R curves are often referred to as "MIDAS curves".
These are the main components of the MIDAS chart. A third algorithm called the Top-Bottom Finder was also described. (Separate script).
Additional tools have been described in "MIDAS_Technical_Analysis"
Midas Technical Analysis: A VWAP Approach to Trading and Investing in Today’s Markets by Andrew Coles, David G. Hawkins
Copyright © 2011 by Andrew Coles and David G. Hawkins.
Denoting the different way in which Levine approached the calculation.
The difference between "MIDAS" VWAP and VWAP is, in my opinion, much ado about nothing. The algorithms generate identical curves albeit the MIDAS algorithm launches the curve one bar later than the VWAP algorithm which can be a pain in the neck. All of the algorithms that I looked at on Tradingview step back one bar in time to initiate the MIDAS curve. As such the plotted curves are identical to traditional VWAP assuming the initiation is from the candle/bar midpoint.
How did Levine intend the curves to be drawn?
On a reversal, he suggested the initiation of the Support and Resistance VVWAP (S/R curve) to be started after a reversal.
It is clear in his examples this happens occasionally but in many cases he initiates the so-called MIDAS S/R VWAP right at the reversal point. In any case, the algorithm is problematic if you wish to start a curve on the first bar of an IPO .
You will get nothing. That is a pain. Also in Levine's writings, he describes simply clicking on the point where a
S/R VWAP is to be drawn from. As such, the generally accepted method of initiating the curve at N-1 is a practical and sensible method. The only issue is that you cannot draw the curve from the first bar on any security, as mentioned without resorting to the typical VWAP algorithm. There is another difference. VWAP is launched from the middle of the bar (as per AlphaTrends), You can also launch from the top of the bar or the bottom (or anywhere for that matter). The calculation proceeds using the top or bottom for each new bar.
The potential applications are discussed in the MIDAS Technical Analysis book.
Trend Following Strategy with KNN
### 1. Strategy Features
This strategy combines the K-Nearest Neighbors (KNN) algorithm with a trend-following strategy to predict future price movements by analyzing historical price data. Here are the main features of the strategy:
1. **Dynamic Parameter Adjustment**: Uses the KNN algorithm to dynamically adjust parameters of the trend-following strategy, such as moving average length and channel length, to adapt to market changes.
2. **Trend Following**: Captures market trends using moving averages and price channels to generate buy and sell signals.
3. **Multi-Factor Analysis**: Combines the KNN algorithm with moving averages to comprehensively analyze the impact of multiple factors, improving the accuracy of trading signals.
4. **High Adaptability**: Automatically adjusts parameters using the KNN algorithm, allowing the strategy to adapt to different market environments and asset types.
### 2. Simple Introduction to the KNN Algorithm
The K-Nearest Neighbors (KNN) algorithm is a simple and intuitive machine learning algorithm primarily used for classification and regression problems. Here are the basic concepts of the KNN algorithm:
1. **Non-Parametric Model**: KNN is a non-parametric algorithm, meaning it does not make any assumptions about the data distribution. Instead, it directly uses training data for predictions.
2. **Instance-Based Learning**: KNN is an instance-based learning method that uses training data directly for predictions, rather than generating a model through a training process.
3. **Distance Metrics**: The core of the KNN algorithm is calculating the distance between data points. Common distance metrics include Euclidean distance, Manhattan distance, and Minkowski distance.
4. **Neighbor Selection**: For each test data point, the KNN algorithm finds the K nearest neighbors in the training dataset.
5. **Classification and Regression**: In classification problems, KNN determines the class of a test data point through a voting mechanism. In regression problems, KNN predicts the value of a test data point by calculating the average of the K nearest neighbors.
### 3. Applications of the KNN Algorithm in Quantitative Trading Strategies
The KNN algorithm can be applied to various quantitative trading strategies. Here are some common use cases:
1. **Trend-Following Strategies**: KNN can be used to identify market trends, helping traders capture the beginning and end of trends.
2. **Mean Reversion Strategies**: In mean reversion strategies, KNN can be used to identify price deviations from the mean.
3. **Arbitrage Strategies**: In arbitrage strategies, KNN can be used to identify price discrepancies between different markets or assets.
4. **High-Frequency Trading Strategies**: In high-frequency trading strategies, KNN can be used to quickly identify market anomalies, such as price spikes or volume anomalies.
5. **Event-Driven Strategies**: In event-driven strategies, KNN can be used to identify the impact of market events.
6. **Multi-Factor Strategies**: In multi-factor strategies, KNN can be used to comprehensively analyze the impact of multiple factors.
### 4. Final Considerations
1. **Computational Efficiency**: The KNN algorithm may face computational efficiency issues with large datasets, especially in real-time trading. Optimize the code to reduce access to historical data and improve computational efficiency.
2. **Parameter Selection**: The choice of K value significantly affects the performance of the KNN algorithm. Use cross-validation or other methods to select the optimal K value.
3. **Data Standardization**: KNN is sensitive to data standardization and feature selection. Standardize the data to ensure equal weighting of different features.
4. **Noisy Data**: KNN is sensitive to noisy data, which can lead to overfitting. Preprocess the data to remove noise.
5. **Market Environment**: The effectiveness of the KNN algorithm may be influenced by market conditions. Combine it with other technical indicators and fundamental analysis to enhance the robustness of the strategy.
[Pandora] Error Function Treasure Trove - ERF/ERFI/Sigmoids+PRAISE:
At this time, I have to graciously thank the wonderful minds behind the new "Pine Profiler Mode" (PPM). Directly prior to this release, it allowed me to ascertain script performance even more. While I usually write mostly in highly optimized Pine code, PPM visually identified a few bottlenecks that would otherwise be hard to identify. Anyone who contributed to PPMs creation and testing before release... BRAVO!!! I commend all of those who assisted in it's state-of-the-art engineering and inception, well done!
BACKSTORY:
This script is specifically being released in defense of another member, an exceptionally unique PhD. It was brought to my attention that a script-mod-event occurred, regarding the publishing of a measly antiquated error function (ERF) calculation within his script. This sadly resulted in the now former member jumping ship after receiving unmannerly responses amidst his curious inquiries as to why his erf() was modded. To forbid rusty and rudimentary formulations because a mod-on-duty is temporally offended by a non-nefarious release of code, is in MY opinion an injustice to principles of perpetuating open-source code intended to benefit thousands to millions of community members. While Pine is the heart and soul of TV, the mathematical concepts contributed from the minds of members is the inspirational fuel of curiosity that powers it's pertinent reason to exist and evolve.
It is an indisputable fact that most members are not greatly skilled Pine Poets. Many members may be incapable of innovating robust function code in Pine, even if they have one or more PhDs. We ALL come from various disciplines of mathematical comprehension and education. Some mathematicians are not greatly skilled at coding, while some coders are not exceptional at math. So... what am I to do to attempt to resolve this circumstantial challenge??? Those who know me best are aware that I will always side with "the right side of history" in order to accomplish my primary self-defined missions I choose to accept. Serving as an algorithmic advocate, I felt compelled to intercede by compiling numerous error functions into elegant code of very high caliber that any and every TV member may choose to employ, so this ERROR never happens again.
After weeks of contemplation into algorithms I knew little about, I prioritized myself to resolve an unanticipated matter by creating advanced formulas of exquisitely crafted error functions refined to the best of my current abilities. My aversion for unresolved problems motivated me to eviscerate error function insufficiencies with many more rigid formulations beyond what is thought to exist. ERF needed a proper algorithmic exorcism anyways. In my furiosity, I contemplated an array of madMAXimum diplomatic demolition methods, choosing the chain saw massacre technique to slaughter dysfunctionalities I encountered on a battered ERF roadway. This resulted in prolific solutions that should assuredly endure the test of time. Poetically, as you will come to see, I am ripping the lid off of Pandora's box of error functions in this case to correct wrongs into a splendid bundle of rights for members.
INTENTION:
Error function (ERF) enthusiasts... PREPARE FOR GLORY!! The specific purpose of this script is to deprecate classic error functions with the creation of a fierce and formidable army of superior formulations, each having varying attributes of computational complexity with differing absolute error ranges in their results for multiple compute scenarios. This is NOT an indicator... It is intended to allow members to embark on endeavors to advance the profound knowledge base of this growing worldwide community of 60+ million inquisitive minds. For those of you who believe computational mathematics and statistics is near completion at its finest; I am here to inform you, this is ridiculous to ponder. We are no where near statistical excellence that can and will exist eventually. At this time, metaphorically speaking, we are merely scratching microns off of the surface of the skin of a statistical apple Isaac Newton once pondered.
THIS RELEASE:
Following weeks of pondering methodical experiments beyond the ordinary, I am liberating these wild notions of my error function explorations to the entire globe as copyleft code, not just Pine. This Pandora's basket of ERFs is being openly disclosed for the sake of the sanctity of mathematics, empirical science (not the garbage we are told by CONTROLocrats to blindly trust), revolutionary cutting edge engineering, cosmology, physics, information technology, artificial intelligence, and EVERY other mathematical branch of human knowledge being discovered over centuries. I do believe James Glaisher would favor my aims concerning ERF aspirations embracing the "Power of Pine".
The included functions are intended for TV members to use in any way they see fit. This is a gift to ALL members to foster future innovative excellence on this platform. Any attempt to moderate this code without notification of "self-evident clear and just cause" will be considered an irrevocable egregious action. The original foundational PURPOSE of establishing script moderation (I clearly remember) was primarily to maintain active vigilance over a growing community against intentional nefarious actions and/or behaviors in blatant disrespect to other author's works AND also thwart rampant copypasting bandit operations, all while accommodating balanced principles of fairness for an educational community cause via open source publishing that should support future algorithmic inventions well beyond my lifespan.
APPLICATIONS:
The related error functions are used in probability theory, statistics, and numerous and engineering scientific disciplines. Its key characteristics and applications are innumerable in computational realms. Its versatility and significance make it a fundamental tool in arenas of quantitative analysis and scientific research...
Probability Theory - Is widely used in probability theory to calculate probabilities and quantiles of the normal distribution.
Statistics - It's related to the Gaussian integral and plays a crucial role in statistics, especially in hypothesis testing and confidence interval calculations.
Physics - In physics, it arises in the study of diffusion equations, quantum mechanics, and heat conduction problems.
Engineering - Applications exist in engineering disciplines such as signal processing, control theory, and telecommunications.
Error Analysis - It's employed in error analysis and uncertainty quantification.
Numeric Approximations - Due to its lack of a closed-form expression, numerical methods are often employed to approximate erf/erfi().
AI, LLMs, & MACHINE LEARNING:
The error function (ERF) is indispensable to various AI applications, particularly due to its relation to Gaussian distributions and error analysis. It is used in Gaussian processes for regression and classification, probabilistic inference for Bayesian networks, soft margin computation in SVMs, neural networks involving Gaussian activation functions or noise, and clustering algorithms like Gaussian Mixture Models. Improved ERF approximations can enhance precision in these applications, reduce computational complexity, handle outliers and noise better, and improve optimization and convergence, possibly leading to more accurate, efficient, and robust AI systems.
BONUS ALGORITHMS:
While ERFs are versatile, its opposite also exists in the form of inverse error functions (ERFIs). I have also included a modified form of the inverse fisher transform along side MY sigmoid (sigmyod). I am uncertain what sigmyod() may be used for, but it's a culmination of my examinations deep into "sigmoid domains", something I am fascinated by. Whatever implications it may possess, I am unveiling it along with it's cousin functions. For curious minds, this quality of composition seen here is ideally what underlies what I would term "Pandora functionality" that empowers my Pandora indication. I go through hordes of formulations, testing, and inspection to find what appears to be the most beneficial logical/mathematical equation to apply...
SCRIPT OPERATION:
To showcase the characteristics and performance of my ERF/ERFI formulations, I devised a multi-modal script. By using bar_index , I generated a broad sequence of numeric values to input into the first ERF/ERFI parameter. These sequences allow you to inspect the contours of the error function's outputs for both ERF and ERFI. When combined with compute-intensive precision functions (CIPFs), the polynomial function output values can be subtracted from my CIPFs to obtain results of absolute error, displaying the accuracy of the many polynomial estimation functions I tuned in testing for Pine's float environment.
A host of numeric input settings are wildly adjustable to inspect values/curvatures across the range of numeric input sequences. Very large numbers, such as Divisor:100,000,100/Offset:200,000,000 for ERF modes or... Divisor:100,000,100/Offset:100,000,000 for ERFI modes, will display miniscule output values calculated from input values in close proximity to 0.0 for the various estimates, similar to a microscope. ERFI approximations very near in proximity to +/-1.0 will always yield large deviations of absolute error. Dragging/zooming your chart or using the Offset input will aid with visually clipping off those ERFI extremes where float precision functions cannot suffice.
NOTICE:
perf() and perfi() are intended for precision computation (as good as it basically gets) in a float environment. However, they are CPU intensive (especially perfi). I wouldn't recommend these being used in ANY Pine script unless it's an "absolute necessity" to do so to accomplish your goal. I only built them to obtain "absolute error curvatures" of the error functions for the polynomial approximations. These are visible in the accuracy modes in the indicator Settings.
AI Moving Average (Expo)█ Overview
The AI Moving Average indicator is a trading tool that uses an AI-based K-nearest neighbors (KNN) algorithm to analyze and interpret patterns in price data. It combines the logic of a traditional moving average with artificial intelligence, creating an adaptive and robust indicator that can identify strong trends and key market levels.
█ How It Works
The algorithm collects data points and applies a KNN-weighted approach to classify price movement as either bullish or bearish. For each data point, the algorithm checks if the price is above or below the calculated moving average. If the price is above the moving average, it's labeled as bullish (1), and if it's below, it's labeled as bearish (0). The K-Nearest Neighbors (KNN) is an instance-based learning algorithm used in classification and regression tasks. It works on a principle of voting, where a new data point is classified based on the majority label of its 'k' nearest neighbors.
The algorithm's use of a KNN-weighted approach adds a layer of intelligence to the traditional moving average analysis. By considering not just the price relative to a moving average but also taking into account the relationships and similarities between different data points, it offers a nuanced and robust classification of price movements.
This combination of data collection, labeling, and KNN-weighted classification turns the AI Moving Average (Expo) Indicator into a dynamic tool that can adapt to changing market conditions, making it suitable for various trading strategies and market environments.
█ How to Use
Dynamic Trend Recognition
The color-coded moving average line helps traders quickly identify market trends. Green represents bullish, red for bearish, and blue for neutrality.
Trend Strength
By adjusting certain settings within the AI Moving Average (Expo) Indicator, such as using a higher 'k' value and increasing the number of data points, traders can gain real-time insights into strong trends. A higher 'k' value makes the prediction model more resilient to noise, emphasizing pronounced trends, while more data points provide a comprehensive view of the market direction. Together, these adjustments enable the indicator to display only robust trends on the chart, allowing traders to focus exclusively on significant market movements and strong trends.
Key SR Levels
Traders can utilize the indicator to identify key support and resistance levels that are derived from the prevailing trend movement. The derived support and resistance levels are not just based on historical data but are dynamically adjusted with the current trend, making them highly responsive to market changes.
█ Settings
k (Neighbors): Number of neighbors in the KNN algorithm. Increasing 'k' makes predictions more resilient to noise but may decrease sensitivity to local variations.
n (DataPoints): Number of data points considered in AI analysis. This affects how the AI interprets patterns in the price data.
maType (Select MA): Type of moving average applied. Options allow for different smoothing techniques to emphasize or dampen aspects of price movement.
length: Length of the moving average. A greater length creates a smoother curve but might lag recent price changes.
dataToClassify: Source data for classifying price as bullish or bearish. It can be adjusted to consider different aspects of price information
dataForMovingAverage: Source data for calculating the moving average. Different selections may emphasize different aspects of price movement.
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Disclaimer
The information contained in my Scripts/Indicators/Ideas/Algos/Systems does not constitute financial advice or a solicitation to buy or sell any securities of any type. I will not accept liability for any loss or damage, including without limitation any loss of profit, which may arise directly or indirectly from the use of or reliance on such information.
All investments involve risk, and the past performance of a security, industry, sector, market, financial product, trading strategy, backtest, or individual's trading does not guarantee future results or returns. Investors are fully responsible for any investment decisions they make. Such decisions should be based solely on an evaluation of their financial circumstances, investment objectives, risk tolerance, and liquidity needs.
My Scripts/Indicators/Ideas/Algos/Systems are only for educational purposes!
Adaptivity: Measures of Dominant Cycles and Price Trend [Loxx]Adaptivity: Measures of Dominant Cycles and Price Trend is an indicator that outputs adaptive lengths using various methods for dominant cycle and price trend timeframe adaptivity. While the information output from this indicator might be useful for the average trader in one off circumstances, this indicator is really meant for those need a quick comparison of dynamic length outputs who wish to fine turn algorithms and/or create adaptive indicators.
This indicator compares adaptive output lengths of all publicly known adaptive measures. Additional adaptive measures will be added as they are discovered and made public.
The first released of this indicator includes 6 measures. An additional three measures will be added with updates. Please check back regularly for new measures.
Ehers:
Autocorrelation Periodogram
Band-pass
Instantaneous Cycle
Hilbert Transformer
Dual Differentiator
Phase Accumulation (future release)
Homodyne (future release)
Jurik:
Composite Fractal Behavior (CFB)
Adam White:
Veritical Horizontal Filter (VHF) (future release)
What is an adaptive cycle, and what is Ehlers Autocorrelation Periodogram Algorithm?
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, page 135:
"Adaptive filters can have several different meanings. For example, Perry Kaufman's adaptive moving average (KAMA) and Tushar Chande's variable index dynamic average (VIDYA) adapt to changes in volatility . By definition, these filters are reactive to price changes, and therefore they close the barn door after the horse is gone.The adaptive filters discussed in this chapter are the familiar Stochastic , relative strength index (RSI), commodity channel index (CCI), and band-pass filter.The key parameter in each case is the look-back period used to calculate the indicator. This look-back period is commonly a fixed value. However, since the measured cycle period is changing, it makes sense to adapt these indicators to the measured cycle period. When tradable market cycles are observed, they tend to persist for a short while.Therefore, by tuning the indicators to the measure cycle period they are optimized for current conditions and can even have predictive characteristics.
The dominant cycle period is measured using the Autocorrelation Periodogram Algorithm. That dominant cycle dynamically sets the look-back period for the indicators. I employ my own streamlined computation for the indicators that provide smoother and easier to interpret outputs than traditional methods. Further, the indicator codes have been modified to remove the effects of spectral dilation.This basically creates a whole new set of indicators for your trading arsenal."
What is this Hilbert Transformer?
An analytic signal allows for time-variable parameters and is a generalization of the phasor concept, which is restricted to time-invariant amplitude, phase, and frequency. The analytic representation of a real-valued function or signal facilitates many mathematical manipulations of the signal. For example, computing the phase of a signal or the power in the wave is much simpler using analytic signals.
The Hilbert transformer is the technique to create an analytic signal from a real one. The conventional Hilbert transformer is theoretically an infinite-length FIR filter. Even when the filter length is truncated to a useful but finite length, the induced lag is far too large to make the transformer useful for trading.
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, pages 186-187:
"I want to emphasize that the only reason for including this section is for completeness. Unless you are interested in research, I suggest you skip this section entirely. To further emphasize my point, do not use the code for trading. A vastly superior approach to compute the dominant cycle in the price data is the autocorrelation periodogram. The code is included because the reader may be able to capitalize on the algorithms in a way that I do not see. All the algorithms encapsulated in the code operate reasonably well on theoretical waveforms that have no noise component. My conjecture at this time is that the sample-to-sample noise simply swamps the computation of the rate change of phase, and therefore the resulting calculations to find the dominant cycle are basically worthless.The imaginary component of the Hilbert transformer cannot be smoothed as was done in the Hilbert transformer indicator because the smoothing destroys the orthogonality of the imaginary component."
What is the Dual Differentiator, a subset of Hilbert Transformer?
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, page 187:
"The first algorithm to compute the dominant cycle is called the dual differentiator. In this case, the phase angle is computed from the analytic signal as the arctangent of the ratio of the imaginary component to the real component. Further, the angular frequency is defined as the rate change of phase. We can use these facts to derive the cycle period."
What is the Phase Accumulation, a subset of Hilbert Transformer?
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, page 189:
"The next algorithm to compute the dominant cycle is the phase accumulation method. The phase accumulation method of computing the dominant cycle is perhaps the easiest to comprehend. In this technique, we measure the phase at each sample by taking the arctangent of the ratio of the quadrature component to the in-phase component. A delta phase is generated by taking the difference of the phase between successive samples. At each sample we can then look backwards, adding up the delta phases.When the sum of the delta phases reaches 360 degrees, we must have passed through one full cycle, on average.The process is repeated for each new sample.
The phase accumulation method of cycle measurement always uses one full cycle's worth of historical data.This is both an advantage and a disadvantage.The advantage is the lag in obtaining the answer scales directly with the cycle period.That is, the measurement of a short cycle period has less lag than the measurement of a longer cycle period. However, the number of samples used in making the measurement means the averaging period is variable with cycle period. longer averaging reduces the noise level compared to the signal.Therefore, shorter cycle periods necessarily have a higher out- put signal-to-noise ratio."
What is the Homodyne, a subset of Hilbert Transformer?
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, page 192:
"The third algorithm for computing the dominant cycle is the homodyne approach. Homodyne means the signal is multiplied by itself. More precisely, we want to multiply the signal of the current bar with the complex value of the signal one bar ago. The complex conjugate is, by definition, a complex number whose sign of the imaginary component has been reversed."
What is the Instantaneous Cycle?
The Instantaneous Cycle Period Measurement was authored by John Ehlers; it is built upon his Hilbert Transform Indicator.
From his Ehlers' book Cybernetic Analysis for Stocks and Futures: Cutting-Edge DSP Technology to Improve Your Trading by John F. Ehlers, 2004, page 107:
"It is obvious that cycles exist in the market. They can be found on any chart by the most casual observer. What is not so clear is how to identify those cycles in real time and how to take advantage of their existence. When Welles Wilder first introduced the relative strength index (rsi), I was curious as to why he selected 14 bars as the basis of his calculations. I reasoned that if i knew the correct market conditions, then i could make indicators such as the rsi adaptive to those conditions. Cycles were the answer. I knew cycles could be measured. Once i had the cyclic measurement, a host of automatically adaptive indicators could follow.
Measurement of market cycles is not easy. The signal-to-noise ratio is often very low, making measurement difficult even using a good measurement technique. Additionally, the measurements theoretically involve simultaneously solving a triple infinity of parameter values. The parameters required for the general solutions were frequency, amplitude, and phase. Some standard engineering tools, like fast fourier transforms (ffs), are simply not appropriate for measuring market cycles because ffts cannot simultaneously meet the stationarity constraints and produce results with reasonable resolution. Therefore i introduced maximum entropy spectral analysis (mesa) for the measurement of market cycles. This approach, originally developed to interpret seismographic information for oil exploration, produces high-resolution outputs with an exceptionally short amount of information. A short data length improves the probability of having nearly stationary data. Stationary data means that frequency and amplitude are constant over the length of the data. I noticed over the years that the cycles were ephemeral. Their periods would be continuously increasing and decreasing. Their amplitudes also were changing, giving variable signal-to-noise ratio conditions. Although all this is going on with the cyclic components, the enduring characteristic is that generally only one tradable cycle at a time is present for the data set being used. I prefer the term dominant cycle to denote that one component. The assumption that there is only one cycle in the data collapses the difficulty of the measurement process dramatically."
What is the Band-pass Cycle?
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, page 47:
"Perhaps the least appreciated and most underutilized filter in technical analysis is the band-pass filter. The band-pass filter simultaneously diminishes the amplitude at low frequencies, qualifying it as a detrender, and diminishes the amplitude at high frequencies, qualifying it as a data smoother. It passes only those frequency components from input to output in which the trader is interested. The filtering produced by a band-pass filter is superior because the rejection in the stop bands is related to its bandwidth. The degree of rejection of undesired frequency components is called selectivity. The band-stop filter is the dual of the band-pass filter. It rejects a band of frequency components as a notch at the output and passes all other frequency components virtually unattenuated. Since the bandwidth of the deep rejection in the notch is relatively narrow and since the spectrum of market cycles is relatively broad due to systemic noise, the band-stop filter has little application in trading."
From his Ehlers' book Cycle Analytics for Traders Advanced Technical Trading Concepts by John F. Ehlers , 2013, page 59:
"The band-pass filter can be used as a relatively simple measurement of the dominant cycle. A cycle is complete when the waveform crosses zero two times from the last zero crossing. Therefore, each successive zero crossing of the indicator marks a half cycle period. We can establish the dominant cycle period as twice the spacing between successive zero crossings."
What is Composite Fractal Behavior (CFB)?
All around you mechanisms adjust themselves to their environment. From simple thermostats that react to air temperature to computer chips in modern cars that respond to changes in engine temperature, r.p.m.'s, torque, and throttle position. It was only a matter of time before fast desktop computers applied the mathematics of self-adjustment to systems that trade the financial markets.
Unlike basic systems with fixed formulas, an adaptive system adjusts its own equations. For example, start with a basic channel breakout system that uses the highest closing price of the last N bars as a threshold for detecting breakouts on the up side. An adaptive and improved version of this system would adjust N according to market conditions, such as momentum, price volatility or acceleration.
Since many systems are based directly or indirectly on cycles, another useful measure of market condition is the periodic length of a price chart's dominant cycle, (DC), that cycle with the greatest influence on price action.
The utility of this new DC measure was noted by author Murray Ruggiero in the January '96 issue of Futures Magazine. In it. Mr. Ruggiero used it to adaptive adjust the value of N in a channel breakout system. He then simulated trading 15 years of D-Mark futures in order to compare its performance to a similar system that had a fixed optimal value of N. The adaptive version produced 20% more profit!
This DC index utilized the popular MESA algorithm (a formulation by John Ehlers adapted from Burg's maximum entropy algorithm, MEM). Unfortunately, the DC approach is problematic when the market has no real dominant cycle momentum, because the mathematics will produce a value whether or not one actually exists! Therefore, we developed a proprietary indicator that does not presuppose the presence of market cycles. It's called CFB (Composite Fractal Behavior) and it works well whether or not the market is cyclic.
CFB examines price action for a particular fractal pattern, categorizes them by size, and then outputs a composite fractal size index. This index is smooth, timely and accurate
Essentially, CFB reveals the length of the market's trending action time frame. Long trending activity produces a large CFB index and short choppy action produces a small index value. Investors have found many applications for CFB which involve scaling other existing technical indicators adaptively, on a bar-to-bar basis.
What is VHF Adaptive Cycle?
Vertical Horizontal Filter (VHF) was created by Adam White to identify trending and ranging markets. VHF measures the level of trend activity, similar to ADX DI. Vertical Horizontal Filter does not, itself, generate trading signals, but determines whether signals are taken from trend or momentum indicators. Using this trend information, one is then able to derive an average cycle length.
Monte Carlo Range Forecast [DW]This is an experimental study designed to forecast the range of price movement from a specified starting point using a Monte Carlo simulation.
Monte Carlo experiments are a broad class of computational algorithms that utilize random sampling to derive real world numerical results.
These types of algorithms have a number of applications in numerous fields of study including physics, engineering, behavioral sciences, climate forecasting, computer graphics, gaming AI, mathematics, and finance.
Although the applications vary, there is a typical process behind the majority of Monte Carlo methods:
-> First, a distribution of possible inputs is defined.
-> Next, values are generated randomly from the distribution.
-> The values are then fed through some form of deterministic algorithm.
-> And lastly, the results are aggregated over some number of iterations.
In this study, the Monte Carlo process used generates a distribution of aggregate pseudorandom linear price returns summed over a user defined period, then plots standard deviations of the outcomes from the mean outcome generate forecast regions.
The pseudorandom process used in this script relies on a modified Wichmann-Hill pseudorandom number generator (PRNG) algorithm.
Wichmann-Hill is a hybrid generator that uses three linear congruential generators (LCGs) with different prime moduli.
Each LCG within the generator produces an independent, uniformly distributed number between 0 and 1.
The three generated values are then summed and modulo 1 is taken to deliver the final uniformly distributed output.
Because of its long cycle length, Wichmann-Hill is a fantastic generator to use on TV since it's extremely unlikely that you'll ever see a cycle repeat.
The resulting pseudorandom output from this generator has a minimum repetition cycle length of 6,953,607,871,644.
Fun fact: Wichmann-Hill is a widely used PRNG in various software applications. For example, Excel 2003 and later uses this algorithm in its RAND function, and it was the default generator in Python up to v2.2.
The generation algorithm in this script takes the Wichmann-Hill algorithm, and uses a multi-stage transformation process to generate the results.
First, a parent seed is selected. This can either be a fixed value, or a dynamic value.
The dynamic parent value is produced by taking advantage of Pine's timenow variable behavior. It produces a variable parent seed by using a frozen ratio of timenow/time.
Because timenow always reflects the current real time when frozen and the time variable reflects the chart's beginning time when frozen, the ratio of these values produces a new number every time the cache updates.
After a parent seed is selected, its value is then fed through a uniformly distributed seed array generator, which generates multiple arrays of pseudorandom "children" seeds.
The seeds produced in this step are then fed through the main generators to produce arrays of pseudorandom simulated outcomes, and a pseudorandom series to compare with the real series.
The main generators within this script are designed to (at least somewhat) model the stochastic nature of financial time series data.
The first step in this process is to transform the uniform outputs of the Wichmann-Hill into outputs that are normally distributed.
In this script, the transformation is done using an estimate of the normal distribution quantile function.
Quantile functions, otherwise known as percent-point or inverse cumulative distribution functions, specify the value of a random variable such that the probability of the variable being within the value's boundary equals the input probability.
The quantile equation for a normal probability distribution is μ + σ(√2)erf^-1(2(p - 0.5)) where μ is the mean of the distribution, σ is the standard deviation, erf^-1 is the inverse Gauss error function, and p is the probability.
Because erf^-1() does not have a simple, closed form interpretation, it must be approximated.
To keep things lightweight in this approximation, I used a truncated Maclaurin Series expansion for this function with precomputed coefficients and rolled out operations to avoid nested looping.
This method provides a decent approximation of the error function without completely breaking floating point limits or sucking up runtime memory.
Note that there are plenty of more robust techniques to approximate this function, but their memory needs very. I chose this method specifically because of runtime favorability.
To generate a pseudorandom approximately normally distributed variable, the uniformly distributed variable from the Wichmann-Hill algorithm is used as the input probability for the quantile estimator.
Now from here, we get a pretty decent output that could be used itself in the simulation process. Many Monte Carlo simulations and random price generators utilize a normal variable.
However, if you compare the outputs of this normal variable with the actual returns of the real time series, you'll find that the variability in shocks (random changes) doesn't quite behave like it does in real data.
This is because most real financial time series data is more complex. Its distribution may be approximately normal at times, but the variability of its distribution changes over time due to various underlying factors.
In light of this, I believe that returns behave more like a convoluted product distribution rather than just a raw normal.
So the next step to get our procedurally generated returns to more closely emulate the behavior of real returns is to introduce more complexity into our model.
Through experimentation, I've found that a return series more closely emulating real returns can be generated in a three step process:
-> First, generate multiple independent, normally distributed variables simultaneously.
-> Next, apply pseudorandom weighting to each variable ranging from -1 to 1, or some limits within those bounds. This modulates each series to provide more variability in the shocks by producing product distributions.
-> Lastly, add the results together to generate the final pseudorandom output with a convoluted distribution. This adds variable amounts of constructive and destructive interference to produce a more "natural" looking output.
In this script, I use three independent normally distributed variables multiplied by uniform product distributed variables.
The first variable is generated by multiplying a normal variable by one uniformly distributed variable. This produces a bit more tailedness (kurtosis) than a normal distribution, but nothing too extreme.
The second variable is generated by multiplying a normal variable by two uniformly distributed variables. This produces moderately greater tails in the distribution.
The third variable is generated by multiplying a normal variable by three uniformly distributed variables. This produces a distribution with heavier tails.
For additional control of the output distributions, the uniform product distributions are given optional limits.
These limits control the boundaries for the absolute value of the uniform product variables, which affects the tails. In other words, they limit the weighting applied to the normally distributed variables in this transformation.
All three sets are then multiplied by user defined amplitude factors to adjust presence, then added together to produce our final pseudorandom return series with a convoluted product distribution.
Once we have the final, more "natural" looking pseudorandom series, the values are recursively summed over the forecast period to generate a simulated result.
This process of generation, weighting, addition, and summation is repeated over the user defined number of simulations with different seeds generated from the parent to produce our array of initial simulated outcomes.
After the initial simulation array is generated, the max, min, mean and standard deviation of this array are calculated, and the values are stored in holding arrays on each iteration to be called upon later.
Reference difference series and price values are also stored in holding arrays to be used in our comparison plots.
In this script, I use a linear model with simple returns rather than compounding log returns to generate the output.
The reason for this is that in generating outputs this way, we're able to run our simulations recursively from the beginning of the chart, then apply scaling and anchoring post-process.
This allows a greater conservation of runtime memory than the alternative, making it more suitable for doing longer forecasts with heavier amounts of simulations in TV's runtime environment.
From our starting time, the previous bar's price, volatility, and optional drift (expected return) are factored into our holding arrays to generate the final forecast parameters.
After these parameters are computed, the range forecast is produced.
The basis value for the ranges is the mean outcome of the simulations that were run.
Then, quarter standard deviations of the simulated outcomes are added to and subtracted from the basis up to 3σ to generate the forecast ranges.
All of these values are plotted and colorized based on their theoretical probability density. The most likely areas are the warmest colors, and least likely areas are the coolest colors.
An information panel is also displayed at the starting time which shows the starting time and price, forecast type, parent seed value, simulations run, forecast bars, total drift, mean, standard deviation, max outcome, min outcome, and bars remaining.
The interesting thing about simulated outcomes is that although the probability distribution of each simulation is not normal, the distribution of different outcomes converges to a normal one with enough steps.
In light of this, the probability density of outcomes is highest near the initial value + total drift, and decreases the further away from this point you go.
This makes logical sense since the central path is the easiest one to travel.
Given the ever changing state of markets, I find this tool to be best suited for shorter term forecasts.
However, if the movements of price are expected to remain relatively stable, longer term forecasts may be equally as valid.
There are many possible ways for users to apply this tool to their analysis setups. For example, the forecast ranges may be used as a guide to help users set risk targets.
Or, the generated levels could be used in conjunction with other indicators for meaningful confluence signals.
More advanced users could even extrapolate the functions used within this script for various purposes, such as generating pseudorandom data to test systems on, perform integration and approximations, etc.
These are just a few examples of potential uses of this script. How you choose to use it to benefit your trading, analysis, and coding is entirely up to you.
If nothing else, I think this is a pretty neat script simply for the novelty of it.
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How To Use:
When you first add the script to your chart, you will be prompted to confirm the starting date and time, number of bars to forecast, number of simulations to run, and whether to include drift assumption.
You will also be prompted to confirm the forecast type. There are two types to choose from:
-> End Result - This uses the values from the end of the simulation throughout the forecast interval.
-> Developing - This uses the values that develop from bar to bar, providing a real-time outlook.
You can always update these settings after confirmation as well.
Once these inputs are confirmed, the script will boot up and automatically generate the forecast in a separate pane.
Note that if there is no bar of data at the time you wish to start the forecast, the script will automatically detect use the next available bar after the specified start time.
From here, you can now control the rest of the settings.
The "Seeding Settings" section controls the initial seed value used to generate the children that produce the simulations.
In this section, you can control whether the seed is a fixed value, or a dynamic one.
Since selecting the dynamic parent option will change the seed value every time you change the settings or refresh your chart, there is a "Regenerate" input built into the script.
This input is a dummy input that isn't connected to any of the calculations. The purpose of this input is to force an update of the dynamic parent without affecting the generator or forecast settings.
Note that because we're running a limited number of simulations, different parent seeds will typically yield slightly different forecast ranges.
When using a small number of simulations, you will likely see a higher amount of variance between differently seeded results because smaller numbers of sampled simulations yield a heavier bias.
The more simulations you run, the smaller this variance will become since the outcomes become more convergent toward the same distribution, so the differences between differently seeded forecasts will become more marginal.
When using a dynamic parent, pay attention to the dispersion of ranges.
When you find a set of ranges that is dispersed how you like with your configuration, set your fixed parent value to the parent seed that shows in the info panel.
This will allow you to replicate that dispersion behavior again in the future.
An important thing to note when settings alerts on the plotted levels, or using them as components for signals in other scripts, is to decide on a fixed value for your parent seed to avoid minor repainting due to seed changes.
When the parent seed is fixed, no repainting occurs.
The "Amplitude Settings" section controls the amplitude coefficients for the three differently tailed generators.
These amplitude factors will change the difference series output for each simulation by controlling how aggressively each series moves.
When "Adjust Amplitude Coefficients" is disabled, all three coefficients are set to 1.
Note that if you expect volatility to significantly diverge from its historical values over the forecast interval, try experimenting with these factors to match your anticipation.
The "Weighting Settings" section controls the weighting boundaries for the three generators.
These weighting limits affect how tailed the distributions in each generator are, which in turn affects the final series outputs.
The maximum absolute value range for the weights is . When "Limit Generator Weights" is disabled, this is the range that is automatically used.
The last set of inputs is the "Display Settings", where you can control the visual outputs.
From here, you can select to display either "Forecast" or "Difference Comparison" via the "Output Display Type" dropdown tab.
"Forecast" is the type displayed by default. This plots the end result or developing forecast ranges.
There is an option with this display type to show the developing extremes of the simulations. This option is enabled by default.
There's also an option with this display type to show one of the simulated price series from the set alongside actual prices.
This allows you to visually compare simulated prices alongside the real prices.
"Difference Comparison" allows you to visually compare a synthetic difference series from the set alongside the actual difference series.
This display method is primarily useful for visually tuning the amplitude and weighting settings of the generators.
There are also info panel settings on the bottom, which allow you to control size, colors, and date format for the panel.
It's all pretty simple to use once you get the hang of it. So play around with the settings and see what kinds of forecasts you can generate!
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ADDITIONAL NOTES & DISCLAIMERS
Although I've done a number of things within this script to keep runtime demands as low as possible, the fact remains that this script is fairly computationally heavy.
Because of this, you may get random timeouts when using this script.
This could be due to either random drops in available runtime on the server, using too many simulations, or running the simulations over too many bars.
If it's just a random drop in runtime on the server, hide and unhide the script, re-add it to the chart, or simply refresh the page.
If the timeout persists after trying this, then you'll need to adjust your settings to a less demanding configuration.
Please note that no specific claims are being made in regards to this script's predictive accuracy.
It must be understood that this model is based on randomized price generation with assumed constant drift and dispersion from historical data before the starting point.
Models like these not consider the real world factors that may influence price movement (economic changes, seasonality, macro-trends, instrument hype, etc.), nor the changes in sample distribution that may occur.
In light of this, it's perfectly possible for price data to exceed even the most extreme simulated outcomes.
The future is uncertain, and becomes increasingly uncertain with each passing point in time.
Predictive models of any type can vary significantly in performance at any point in time, and nobody can guarantee any specific type of future performance.
When using forecasts in making decisions, DO NOT treat them as any form of guarantee that values will fall within the predicted range.
When basing your trading decisions on any trading methodology or utility, predictive or not, you do so at your own risk.
No guarantee is being issued regarding the accuracy of this forecast model.
Forecasting is very far from an exact science, and the results from any forecast are designed to be interpreted as potential outcomes rather than anything concrete.
With that being said, when applied prudently and treated as "general case scenarios", forecast models like these may very well be potentially beneficial tools to have in the arsenal.
Machine Learning: LVQ-based StrategyLVQ-based Strategy (FX and Crypto)
Description:
Learning Vector Quantization (LVQ) can be understood as a special case of an artificial neural network, more precisely, it applies a winner-take-all learning-based approach. It is based on prototype supervised learning classification task and trains its weights through a competitive learning algorithm.
Algorithm:
Initialize weights
Train for 1 to N number of epochs
- Select a training example
- Compute the winning vector
- Update the winning vector
Classify test sample
The LVQ algorithm offers a framework to test various indicators easily to see if they have got any *predictive value*. One can easily add cog, wpr and others.
Note: TradingViews's playback feature helps to see this strategy in action. The algo is tested with BTCUSD/1Hour.
Warning: This is a preliminary version! Signals ARE repainting.
***Warning***: Signals LARGELY depend on hyperparams (lrate and epochs).
Style tags: Trend Following, Trend Analysis
Asset class: Equities, Futures, ETFs, Currencies and Commodities
Dataset: FX Minutes/Hours+++/Days
Risk-Adjusted Momentum Oscillator# Risk-Adjusted Momentum Oscillator (RAMO): Momentum Analysis with Integrated Risk Assessment
## 1. Introduction
Momentum indicators have been fundamental tools in technical analysis since the pioneering work of Wilder (1978) and continue to play crucial roles in systematic trading strategies (Jegadeesh & Titman, 1993). However, traditional momentum oscillators suffer from a critical limitation: they fail to account for the risk context in which momentum signals occur. This oversight can lead to significant drawdowns during periods of market stress, as documented extensively in the behavioral finance literature (Kahneman & Tversky, 1979; Shefrin & Statman, 1985).
The Risk-Adjusted Momentum Oscillator addresses this gap by incorporating real-time drawdown metrics into momentum calculations, creating a self-regulating system that automatically adjusts signal sensitivity based on current risk conditions. This approach aligns with modern portfolio theory's emphasis on risk-adjusted returns (Markowitz, 1952) and reflects the sophisticated risk management practices employed by institutional investors (Ang, 2014).
## 2. Theoretical Foundation
### 2.1 Momentum Theory and Market Anomalies
The momentum effect, first systematically documented by Jegadeesh & Titman (1993), represents one of the most robust anomalies in financial markets. Subsequent research has confirmed momentum's persistence across various asset classes, time horizons, and geographic markets (Fama & French, 1996; Asness, Moskowitz & Pedersen, 2013). However, momentum strategies are characterized by significant time-varying risk, with particularly severe drawdowns during market reversals (Barroso & Santa-Clara, 2015).
### 2.2 Drawdown Analysis and Risk Management
Maximum drawdown, defined as the peak-to-trough decline in portfolio value, serves as a critical risk metric in professional portfolio management (Calmar, 1991). Research by Chekhlov, Uryasev & Zabarankin (2005) demonstrates that drawdown-based risk measures provide superior downside protection compared to traditional volatility metrics. The integration of drawdown analysis into momentum calculations represents a natural evolution toward more sophisticated risk-aware indicators.
### 2.3 Adaptive Smoothing and Market Regimes
The concept of adaptive smoothing in technical analysis draws from the broader literature on regime-switching models in finance (Hamilton, 1989). Perry Kaufman's Adaptive Moving Average (1995) pioneered the application of efficiency ratios to adjust indicator responsiveness based on market conditions. RAMO extends this concept by incorporating volatility-based adaptive smoothing, allowing the indicator to respond more quickly during high-volatility periods while maintaining stability during quiet markets.
## 3. Methodology
### 3.1 Core Algorithm Design
The RAMO algorithm consists of several interconnected components:
#### 3.1.1 Risk-Adjusted Momentum Calculation
The fundamental innovation of RAMO lies in its risk adjustment mechanism:
Risk_Factor = 1 - (Current_Drawdown / Maximum_Drawdown × Scaling_Factor)
Risk_Adjusted_Momentum = Raw_Momentum × max(Risk_Factor, 0.05)
This formulation ensures that momentum signals are dampened during periods of high drawdown relative to historical maximums, implementing an automatic risk management overlay as advocated by modern portfolio theory (Markowitz, 1952).
#### 3.1.2 Multi-Algorithm Momentum Framework
RAMO supports three distinct momentum calculation methods:
1. Rate of Change: Traditional percentage-based momentum (Pring, 2002)
2. Price Momentum: Absolute price differences
3. Log Returns: Logarithmic returns preferred for volatile assets (Campbell, Lo & MacKinlay, 1997)
This multi-algorithm approach accommodates different asset characteristics and volatility profiles, addressing the heterogeneity documented in cross-sectional momentum studies (Asness et al., 2013).
### 3.2 Leading Indicator Components
#### 3.2.1 Momentum Acceleration Analysis
The momentum acceleration component calculates the second derivative of momentum, providing early signals of trend changes:
Momentum_Acceleration = EMA(Momentum_t - Momentum_{t-n}, n)
This approach draws from the physics concept of acceleration and has been applied successfully in financial time series analysis (Treadway, 1969).
#### 3.2.2 Linear Regression Prediction
RAMO incorporates linear regression-based prediction to project momentum values forward:
Predicted_Momentum = LinReg_Value + (LinReg_Slope × Forward_Offset)
This predictive component aligns with the literature on technical analysis forecasting (Lo, Mamaysky & Wang, 2000) and provides leading signals for trend changes.
#### 3.2.3 Volume-Based Exhaustion Detection
The exhaustion detection algorithm identifies potential reversal points by analyzing the relationship between momentum extremes and volume patterns:
Exhaustion = |Momentum| > Threshold AND Volume < SMA(Volume, 20)
This approach reflects the established principle that sustainable price movements require volume confirmation (Granville, 1963; Arms, 1989).
### 3.3 Statistical Normalization and Robustness
RAMO employs Z-score normalization with outlier protection to ensure statistical robustness:
Z_Score = (Value - Mean) / Standard_Deviation
Normalized_Value = max(-3.5, min(3.5, Z_Score))
This normalization approach follows best practices in quantitative finance for handling extreme observations (Taleb, 2007) and ensures consistent signal interpretation across different market conditions.
### 3.4 Adaptive Threshold Calculation
Dynamic thresholds are calculated using Bollinger Band methodology (Bollinger, 1992):
Upper_Threshold = Mean + (Multiplier × Standard_Deviation)
Lower_Threshold = Mean - (Multiplier × Standard_Deviation)
This adaptive approach ensures that signal thresholds adjust to changing market volatility, addressing the critique of fixed thresholds in technical analysis (Taylor & Allen, 1992).
## 4. Implementation Details
### 4.1 Adaptive Smoothing Algorithm
The adaptive smoothing mechanism adjusts the exponential moving average alpha parameter based on market volatility:
Volatility_Percentile = Percentrank(Volatility, 100)
Adaptive_Alpha = Min_Alpha + ((Max_Alpha - Min_Alpha) × Volatility_Percentile / 100)
This approach ensures faster response during volatile periods while maintaining smoothness during stable conditions, implementing the adaptive efficiency concept pioneered by Kaufman (1995).
### 4.2 Risk Environment Classification
RAMO classifies market conditions into three risk environments:
- Low Risk: Current_DD < 30% × Max_DD
- Medium Risk: 30% × Max_DD ≤ Current_DD < 70% × Max_DD
- High Risk: Current_DD ≥ 70% × Max_DD
This classification system enables conditional signal generation, with long signals filtered during high-risk periods—a approach consistent with institutional risk management practices (Ang, 2014).
## 5. Signal Generation and Interpretation
### 5.1 Entry Signal Logic
RAMO generates enhanced entry signals through multiple confirmation layers:
1. Primary Signal: Crossover between indicator and signal line
2. Risk Filter: Confirmation of favorable risk environment for long positions
3. Leading Component: Early warning signals via acceleration analysis
4. Exhaustion Filter: Volume-based reversal detection
This multi-layered approach addresses the false signal problem common in traditional technical indicators (Brock, Lakonishok & LeBaron, 1992).
### 5.2 Divergence Analysis
RAMO incorporates both traditional and leading divergence detection:
- Traditional Divergence: Price and indicator divergence over 3-5 periods
- Slope Divergence: Momentum slope versus price direction
- Acceleration Divergence: Changes in momentum acceleration
This comprehensive divergence analysis framework draws from Elliott Wave theory (Prechter & Frost, 1978) and momentum divergence literature (Murphy, 1999).
## 6. Empirical Advantages and Applications
### 6.1 Risk-Adjusted Performance
The risk adjustment mechanism addresses the fundamental criticism of momentum strategies: their tendency to experience severe drawdowns during market reversals (Daniel & Moskowitz, 2016). By automatically reducing position sizing during high-drawdown periods, RAMO implements a form of dynamic hedging consistent with portfolio insurance concepts (Leland, 1980).
### 6.2 Regime Awareness
RAMO's adaptive components enable regime-aware signal generation, addressing the regime-switching behavior documented in financial markets (Hamilton, 1989; Guidolin, 2011). The indicator automatically adjusts its parameters based on market volatility and risk conditions, providing more reliable signals across different market environments.
### 6.3 Institutional Applications
The sophisticated risk management overlay makes RAMO particularly suitable for institutional applications where drawdown control is paramount. The indicator's design philosophy aligns with the risk budgeting approaches used by hedge funds and institutional investors (Roncalli, 2013).
## 7. Limitations and Future Research
### 7.1 Parameter Sensitivity
Like all technical indicators, RAMO's performance depends on parameter selection. While default parameters are optimized for broad market applications, asset-specific calibration may enhance performance. Future research should examine optimal parameter selection across different asset classes and market conditions.
### 7.2 Market Microstructure Considerations
RAMO's effectiveness may vary across different market microstructure environments. High-frequency trading and algorithmic market making have fundamentally altered market dynamics (Aldridge, 2013), potentially affecting momentum indicator performance.
### 7.3 Transaction Cost Integration
Future enhancements could incorporate transaction cost analysis to provide net-return-based signals, addressing the implementation shortfall documented in practical momentum strategy applications (Korajczyk & Sadka, 2004).
## References
Aldridge, I. (2013). *High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems*. 2nd ed. Hoboken, NJ: John Wiley & Sons.
Ang, A. (2014). *Asset Management: A Systematic Approach to Factor Investing*. New York: Oxford University Press.
Arms, R. W. (1989). *The Arms Index (TRIN): An Introduction to the Volume Analysis of Stock and Bond Markets*. Homewood, IL: Dow Jones-Irwin.
Asness, C. S., Moskowitz, T. J., & Pedersen, L. H. (2013). Value and momentum everywhere. *Journal of Finance*, 68(3), 929-985.
Barroso, P., & Santa-Clara, P. (2015). Momentum has its moments. *Journal of Financial Economics*, 116(1), 111-120.
Bollinger, J. (1992). *Bollinger on Bollinger Bands*. New York: McGraw-Hill.
Brock, W., Lakonishok, J., & LeBaron, B. (1992). Simple technical trading rules and the stochastic properties of stock returns. *Journal of Finance*, 47(5), 1731-1764.
Calmar, T. (1991). The Calmar ratio: A smoother tool. *Futures*, 20(1), 40.
Campbell, J. Y., Lo, A. W., & MacKinlay, A. C. (1997). *The Econometrics of Financial Markets*. Princeton, NJ: Princeton University Press.
Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). Drawdown measure in portfolio optimization. *International Journal of Theoretical and Applied Finance*, 8(1), 13-58.
Daniel, K., & Moskowitz, T. J. (2016). Momentum crashes. *Journal of Financial Economics*, 122(2), 221-247.
Fama, E. F., & French, K. R. (1996). Multifactor explanations of asset pricing anomalies. *Journal of Finance*, 51(1), 55-84.
Granville, J. E. (1963). *Granville's New Key to Stock Market Profits*. Englewood Cliffs, NJ: Prentice-Hall.
Guidolin, M. (2011). Markov switching models in empirical finance. In D. N. Drukker (Ed.), *Missing Data Methods: Time-Series Methods and Applications* (pp. 1-86). Bingley: Emerald Group Publishing.
Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. *Econometrica*, 57(2), 357-384.
Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. *Journal of Finance*, 48(1), 65-91.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. *Econometrica*, 47(2), 263-291.
Kaufman, P. J. (1995). *Smarter Trading: Improving Performance in Changing Markets*. New York: McGraw-Hill.
Korajczyk, R. A., & Sadka, R. (2004). Are momentum profits robust to trading costs? *Journal of Finance*, 59(3), 1039-1082.
Leland, H. E. (1980). Who should buy portfolio insurance? *Journal of Finance*, 35(2), 581-594.
Lo, A. W., Mamaysky, H., & Wang, J. (2000). Foundations of technical analysis: Computational algorithms, statistical inference, and empirical implementation. *Journal of Finance*, 55(4), 1705-1765.
Markowitz, H. (1952). Portfolio selection. *Journal of Finance*, 7(1), 77-91.
Murphy, J. J. (1999). *Technical Analysis of the Financial Markets: A Comprehensive Guide to Trading Methods and Applications*. New York: New York Institute of Finance.
Prechter, R. R., & Frost, A. J. (1978). *Elliott Wave Principle: Key to Market Behavior*. Gainesville, GA: New Classics Library.
Pring, M. J. (2002). *Technical Analysis Explained: The Successful Investor's Guide to Spotting Investment Trends and Turning Points*. 4th ed. New York: McGraw-Hill.
Roncalli, T. (2013). *Introduction to Risk Parity and Budgeting*. Boca Raton, FL: CRC Press.
Shefrin, H., & Statman, M. (1985). The disposition to sell winners too early and ride losers too long: Theory and evidence. *Journal of Finance*, 40(3), 777-790.
Taleb, N. N. (2007). *The Black Swan: The Impact of the Highly Improbable*. New York: Random House.
Taylor, M. P., & Allen, H. (1992). The use of technical analysis in the foreign exchange market. *Journal of International Money and Finance*, 11(3), 304-314.
Treadway, A. B. (1969). On rational entrepreneurial behavior and the demand for investment. *Review of Economic Studies*, 36(2), 227-239.
Wilder, J. W. (1978). *New Concepts in Technical Trading Systems*. Greensboro, NC: Trend Research.
TAPDA Hourly Open Lines (Candle Body Box)-What is TAPDA?
TAPDA (Time and Price Displacement Analysis) is based on the belief that markets are driven by algorithms that respond to key time-based price levels, such as session opens. Traders who follow TAPDA track these levels to anticipate price movements, reversals, and breakouts, aligning their strategies with the patterns left by these underlying algorithms. By plotting lines at specific hourly opens, the indicator allows traders to visualize where the market may react, providing a structured way to trade alongside the algorithmic flow.
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**Sauce Alert** "TAPDA levels essentially act like algorithmic support and resistance" By plotting these hourly opens, the TAPDA Hourly Open Lines indicator helps traders track where algorithms might engage with the market.
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-How It Works:
The indicator draws a "candle body box" at selected hours, marking the open and close prices to highlight price ranges at significant times. This creates dynamic zones that reflect market sentiment and structure throughout the day. TAPDA levels are commonly respected by price, making them useful for identifying potential entry points, stop placements, and trend reversals.
-Key Features:
Customizable Hour Levels – Enable or disable specific times to fit your trading approach.
Color & Label Control – Assign unique colors and labels to each hour for better visualization.
Line Extension – Project lines for up to 24 hours into the future to track key levels.
Dynamic Cleanup – Old lines automatically delete to maintain chart clarity.
Manual Time Offset – Adjust for broker or server time zone differences.
-Current Development:
This indicator is still in development, with further updates planned to enhance functionality and customization. If you find this script helpful, feel free to copy the code and stay tuned for new features and improvements!
Bogdan Ciocoiu - Code runnerDescription
The Code Runner is a hybrid indicator that leverages other pre-configured, integrated open-source algorithms to help traders spot regular and continuation divergences.
The Code Runner specialises in integrating some of the most popular oscillators well known for their accuracy when scalping using divergence strategies.
Uniqueness
The Code Runner stands out as a one-stop-shop pack of oscillator algorithms that traders can further customise to spot divergences.
The indicator's uniqueness stands from its capability to recast each algorithm to apply to the same scale. This feature is achieved by manually adjusting the outputs of each algorithm to fit on a scale between +100 and -100.
Another benefit of the Code Runner comes from its standardisation of outputs, mainly consisting of lines. Showing lines enables traders to draw potential regular and continuation divergences quickly.
The indicator has been pre-configured to support scalping at 1-5 minutes.
Open-source
The Code Runner uses the following open-source scripts and algorithms:
www.tradingview.com
www.tradingview.com
www.tradingview.com
www.tradingview.com
www.tradingview.com
www.tradingview.com
www.tradingview.com
www.tradingview.com
These algorithms are available in the public domain either in TradingView space or outside (given their popularity in the financial markets industry).
Adaptive Average Vortex Index [lastguru]As a longtime fan of ADX, looking at Vortex Indicator I often wondered, where is the third line. I have rarely seen that anybody is calculating it. So, here it is: Average Vortex Index - an ADX calculated from Vortex Indicator. I interpret it similarly to the ADX indicator: higher values show stronger trend. If you discover other interpretation or have suggestions, comments are welcome.
Both VI+ and VI- lines are also drawn. As I use adaptive length calculation in my other scripts (based on the libraries I've developed and published), I have also included the possibility to have an adaptive length here, so if you hate the idea of calculating ADX from VI, you can disable that line and just look at the adaptive Vortex Indicator.
Note that as with all my oscillators, all the lines here are renormalized to -1..1 range unlike the original Vortex Indicator computation. To do that for VI+ and VI- lines, I subtract 1 from their values. It does not change the shape or the amplitude of the lines.
Adaptation algorithms are roughly subdivided in two categories: classic Length Adaptations and Cycle Estimators (they are also implemented in separate libraries), all are selected in Adaptation dropdown. Length Adaptation used in the Adaptive Moving Averages and the Adaptive Oscillators try to follow price movements and accelerate/decelerate accordingly (usually quite rapidly with a huge range). Cycle Estimators, on the other hand, try to measure the cycle period of the current market, which does not reflect price movement or the rate of change (the rate of change may also differ depending on the cycle phase, but the cycle period itself usually changes slowly).
VIDYA - based on VIDYA algorithm. The period oscillates from the Lower Bound up (slow)
VIDYA-RS - based on Vitali Apirine's modification of VIDYA algorithm (he calls it Relative Strength Moving Average). The period oscillates from the Upper Bound down (fast)
Kaufman Efficiency Scaling - based on Efficiency Ratio calculation originally used in KAMA
Fractal Adaptation - based on FRAMA by John F. Ehlers
MESA MAMA Cycle - based on MESA Adaptive Moving Average by John F. Ehlers
Pearson Autocorrelation* - based on Pearson Autocorrelation Periodogram by John F. Ehlers
DFT Cycle* - based on Discrete Fourier Transform Spectrum estimator by John F. Ehlers
Phase Accumulation* - based on Dominant Cycle from Phase Accumulation by John F. Ehlers
Length Adaptation usually take two parameters: Bound From (lower bound) and To (upper bound). These are the limits for Adaptation values. Note that the Cycle Estimators marked with asterisks(*) are very computationally intensive, so the bounds should not be set much higher than 50, otherwise you may receive a timeout error (also, it does not seem to be a useful thing to do, but you may correct me if I'm wrong).
The Cycle Estimators marked with asterisks(*) also have 3 checkboxes: HP (Highpass Filter), SS (Super Smoother) and HW (Hann Window). These enable or disable their internal prefilters, which are recommended by their author - John F. Ehlers . I do not know, which combination works best, so you can experiment.
If no Adaptation is selected ( None option), you can set Length directly. If an Adaptation is selected, then Cycle multiplier can be set.
The oscillator also has the option to configure the internal smoothing function with Window setting. By default, RMA is used (like in ADX calculation). Fast Default option is using half the length for smoothing. Triangle , Hamming and Hann Window algorithms are some better smoothers suggested by John F. Ehlers.
After the oscillator a Moving Average can be applied. The following Moving Averages are included: SMA , RMA, EMA , HMA , VWMA , 2-pole Super Smoother, 3-pole Super Smoother, Filt11, Triangle Window, Hamming Window, Hann Window, Lowpass, DSSS.
Postfilter options are applied last:
Stochastic - Stochastic
Super Smooth Stochastic - Super Smooth Stochastic (part of MESA Stochastic ) by John F. Ehlers
Inverse Fisher Transform - Inverse Fisher Transform
Noise Elimination Technology - a simplified Kendall correlation algorithm "Noise Elimination Technology" by John F. Ehlers
Momentum - momentum (derivative)
Except for Inverse Fisher Transform , all Postfilter algorithms can have Length parameter. If it is not specified (set to 0), then the calculated Slow MA Length is used. If Filter/MA Length is less than 2 or Postfilter Length is less than 1, they are calculated as a multiplier of the calculated oscillator length.
More information on the algorithms is given in the code for the libraries used. I am also very grateful to other TradingView community members (they are also mentioned in the library code) without whom this script would not have been possible.
[blackcat] L1 Net Volume DifferenceOVERVIEW
The L1 Net Volume Difference indicator serves as an advanced analytical tool designed to provide traders with deep insights into market sentiment by examining the differential between buying and selling volumes over precise timeframes. By leveraging these volume dynamics, it helps identify trends and potential reversal points more accurately, thereby supporting well-informed decision-making processes. The key focus lies in dissecting intraday changes that reflect short-term market behavior, offering critical input for both swing and day traders alike. 📊
Key benefits encompass:
• Precise calculation of net volume differences grounded in real-time data.
• Interactive visualization elements enhancing interpretability effortlessly.
• Real-time generation of buy/sell signals driven by dynamic volume shifts.
TECHNICAL ANALYSIS COMPONENTS
📉 Volume Accumulation Mechanisms:
Monitors cumulative buy/sell volumes derived from comparative closing prices.
Periodically resets accumulation counters aligning with predefined intervals (e.g., 5-minute bars).
Facilitates identification of directional biases reflecting underlying market forces accurately.
🕵️♂️ Sentiment Detection Algorithms:
Employs proprietary logic distinguishing between bullish/bearish sentiments dynamically.
Ensures consistent adherence to predefined statistical protocols maintaining accuracy.
Supports adaptive thresholds adjusting sensitivities based on changing market conditions flexibly.
🎯 Dynamic Signal Generation:
Detects transitions indicating dominance shifts between buyers/sellers promptly.
Triggers timely alerts enabling swift reactions to evolving market dynamics effectively.
Integrates conditional logic reinforcing signal validity minimizing erroneous activations.
INDICATOR FUNCTIONALITY
🔢 Core Algorithms:
Utilizes moving averages along with standardized deviation formulas generating precise net volume measurements.
Implements Arithmetic Mean Line Algorithm (AMLA) smoothing techniques improving interpretability.
Ensures consistent alignment with established statistical principles preserving fidelity.
🖱️ User Interface Elements:
Dedicated plots displaying real-time net volume markers facilitating swift decision-making.
Context-sensitive color coding distinguishing positive/negative deviations intuitively.
Background shading highlighting proximity to key threshold activations enhancing visibility.
STRATEGY IMPLEMENTATION
✅ Entry Conditions:
Confirm bullish/bearish setups validated through multiple confirmatory signals.
Validate entry decisions considering concurrent market sentiment factors.
Assess alignment between net volume readings and broader trend directions ensuring coherence.
🚫 Exit Mechanisms:
Trigger exits upon hitting predetermined thresholds derived from historical analyses.
Monitor continuous breaches signifying potential trend reversals promptly executing closures.
Execute partial/total closes contingent upon cumulative loss limits preserving capital efficiently.
PARAMETER CONFIGURATIONS
🎯 Optimization Guidelines:
Reset Interval: Governs responsiveness versus stability balancing sensitivity/stability.
Price Source: Dictates primary data series driving volume calculations selecting relevant inputs accurately.
💬 Customization Recommendations:
Commence with baseline defaults; iteratively refine parameters isolating individual impacts.
Evaluate adjustments independently prior to combined modifications minimizing disruptions.
Prioritize minimizing erroneous trigger occurrences first optimizing signal fidelity.
Sustain balanced risk-reward profiles irrespective of chosen settings upholding disciplined approaches.
ADVANCED RISK MANAGEMENT
🛡️ Proactive Risk Mitigation Techniques:
Enforce strict compliance with pre-defined maximum leverage constraints adhering strictly to guidelines.
Mandatorily apply trailing stop-loss orders conforming to script outputs reinforcing discipline.
Allocate positions proportionately relative to available capital reserves managing exposures prudently.
Conduct periodic reviews gauging strategy effectiveness rigorously identifying areas needing refinement.
⚠️ Potential Pitfalls & Solutions:
Address frequent violations arising during heightened volatility phases necessitating manual interventions judiciously.
Manage false alerts warranting immediate attention avoiding adverse consequences systematically.
Prepare contingency plans mitigating margin call possibilities preparing proactive responses effectively.
Continuously assess automated system reliability amidst fluctuating conditions ensuring seamless functionality.
PERFORMANCE AUDITS & REFINEMENTS
🔍 Critical Evaluation Metrics:
Assess win percentages consistently across diverse trading instruments gauging reliability.
Calculate average profit ratios per successful execution measuring profitability efficiency accurately.
Measure peak drawdown durations alongside associated magnitudes evaluating downside risks comprehensively.
Analyze signal generation frequencies revealing hidden patterns potentially skewing outcomes uncovering systematic biases.
📈 Historical Data Analysis Tools:
Maintain comprehensive records capturing every triggered event meticulously documenting results.
Compare realized profits/losses against backtested simulations benchmarking actual vs expected performances accurately.
Identify recurrent systematic errors demanding corrective actions implementing iterative refinements steadily.
Document evolving performance metrics tracking progress dynamically addressing identified shortcomings proactively.
PROBLEM SOLVING ADVICE
🔧 Frequent Encountered Challenges:
Unpredictable behaviors emerging within thinly traded markets requiring filtration processes.
Latency issues manifesting during abrupt price fluctuations causing missed opportunities.
Overfitted models yielding suboptimal results post-extensive tuning demanding recalibrations.
Inaccuracies stemming from incomplete/inaccurate data feeds necessitating verification procedures.
💡 Effective Resolution Pathways:
Exclude low-liquidity assets prone to erratic movements enhancing signal integrity.
Introduce buffer intervals safeguarding major news/event impacts mitigating distortions effectively.
Limit ongoing optimization attempts preventing model degradation maintaining optimal performance levels consistently.
Verify reliable connections ensuring uninterrupted data flows guaranteeing accurate interpretations reliably.
USER ENGAGEMENT SEGMENT
🤝 Community Contributions Welcome
Highly encourage active participation sharing experiences & recommendations!
THANKS
Heartfelt acknowledgment extends to all developers contributing invaluable insights about volume-based trading methodologies! ✨
Forex Pips Tracker PinescriptlabsThis algorithm is exclusively designed for the Forex market 🌐 and serves as a tool to measure volatility, helping to determine on average how many pips positions move per hour. With this information, a trader can place take profit and stop loss orders with greater certainty, since they know the average pip movement range during each hour of the day.
What does it do and how does it work?
• Volatility measurement in pips 📊:
The algorithm calculates the size of the movement (or range) of each candle expressed in pips. To do this, it takes the difference between the highest and lowest price of each candle and converts it into pips.
👉
• Time zone adjustment ⏰:
It allows you to configure the time zone so that the data aligns with your desired schedule. This is especially useful for comparing movements at different times based on the trader's location.
• Analysis by time intervals 🕒:
The algorithm’s logic organizes the information for each hour of the day. It stores data for the current day, the previous day, weekly, and historically (200 candles). This allows you to see how volatility varies across different periods, providing a dynamic view of market behavior.
👉
• Directionality of movement 🔄:
In addition to averaging the pip range, the algorithm determines the predominant direction of each candle (bullish or bearish). This translates into visual indicators (like arrows) that help identify whether, on average, the movement during that hour tends to go up or down.
• Table visualization 📈:
Finally, the information is presented in an integrated table on the chart. Each row corresponds to an hour of the day and shows the average number of pips and the direction (bullish, bearish, or neutral) for each analyzed period. This table makes it easy to quickly and practically interpret the volatility data.
By combining these features, the algorithm becomes an essential tool for traders looking to better understand market dynamics and optimize their trading strategies! 💼✨
Español:
Este algoritmo está diseñado exclusivamente para el mercado Forex 🌐 y sirve como una herramienta para medir la volatilidad, ayudando a determinar en promedio cuántos pips se mueven las posiciones por hora. Con esta información, un trader puede colocar el take profit y el stop loss con mayor certeza, ya que conoce el rango promedio de movimiento en pips durante cada hora del día.
¿Qué hace y cómo funciona?
• Medición de volatilidad en pips 📊:
El algoritmo calcula el tamaño del movimiento (o rango) de cada vela expresado en pips. Para ello, toma la diferencia entre el precio máximo y el mínimo de cada vela y la convierte a pips.
👉
• Ajuste de zona horaria ⏰:
Permite configurar la zona horaria para que los datos se ajusten al horario deseado. Esto es especialmente útil para comparar movimientos durante distintas horas en función de la localización del trader.
• Análisis por intervalos de tiempo 🕒:
La lógica del algoritmo organiza la información por cada hora del día. Guarda datos para el día actual, el día anterior, a nivel semanal e histórico (200 velas). Esto permite ver cómo varía la volatilidad en diferentes periodos, proporcionando una visión dinámica del comportamiento del mercado.
👉
• Direccionalidad del movimiento 🔄:
Además de promediar el rango en pips, el algoritmo determina la dirección predominante de cada vela (alcista o bajista). Esto se traduce en indicadores visuales (como flechas) que permiten identificar si, en promedio, el movimiento en esa hora tiende a subir o bajar.
• Visualización en tabla 📈:
Finalmente, la información se presenta en una tabla integrada en el gráfico. Cada fila corresponde a una hora del día y muestra el promedio de pips y la dirección (alcista, bajista o neutral) para cada uno de los periodos analizados. Esta tabla facilita la interpretación rápida y práctica de los datos de volatilidad.
Al combinar estas funciones, el algoritmo se convierte en una herramienta esencial para traders que buscan entender mejor la dinámica del mercado y optimizar sus estrategias de trading! 💼✨
Dual Zigzag [Trendoscope®]🎲 Dual Zigzag indicator is built on recursive zigzag algorithm. It is very similar to other zigzag indicators published by us and other authors. However, the key point here is, the indicator draws zigzag on both price and any other plot based indicator on separate layouts.
Before we get into the indicator, here are some brief descriptions of the underlying concepts and key terminologies
🎯 Zigzag
Zigzag indicator breaks down price or any input series into a series of Pivot Highs and Pivot Lows alternating between each other. Zigzags though shows pivot high and lows, should not be used for buying at low and selling at high. The main application of zigzag indicator is for the visualisation of market structure and this can be used as basic building block for any pattern recognition algorithms.
🎯 Recursive Zigzag Algorithm
Recursive zigzag algorithm builds zigzag on multiple levels and each level of zigzag is based on the previous level pivots. The level zero zigzag is built on price. However, for level 1, instead of price level 0 zigzag pivots are used. Similarly for level 2, level 1 zigzag pivots are used as base.
🎲 Components Dual Zigzag Indicator
Here are the components of Dual zigzag indicator
Built in Oscillator - Indicator has built in oscillator options for plotting RSI (Relative Strength Index), MFI (Money Flow Index), cci (Commodity Channel Index) , CMO (Chande Momentum Oscillator), COG (Center of Gravity), and ROC (Rate of Change). Apart from the given built in oscillators, users can also use a custom external output as base. The oscillators are not printed on the price pane. But, printed on a separate indicator overlay.
Zigzag On Oscillator - Recursive zigzag is calculated and printed on the oscillator series. Each pivot high and pivot low also prints a label having the retracement ratios, and price levels at those points. Zigzag on the oscillator is also printed on the indicator overlay pane.
Zigzag on Price - Recursive zigzag calculated based on price and printed on the price pane. This is made possible by using force_overlay option present in the drawing objects. At each zigzag pivot levels, the label having price retracement ratios, and oscillator values are printed.
It is called dual zigzag because, the indicator calculates the zigzag on both price and oscillator series of values and prints them separately on different panes on the chart.
🎲 Indicator Settings
Settings include
Theme display settings to get the right colour combination to match the background.
Zigzag settings to be used for zigzag calculation and display
Oscillator settings to chose the oscillator to be used as base for 2nd zigzag
🎲 Applications
Useful in spotting divergences with both indicator and price having their own zigzag to highlight pivots
Spotting patterns in indicators/oscillators and correlate them with the patterns on price
🎲 Using External Input
If users want to use an external indicator such as OBV instead of the built in oscillators, then can do so by using the custom option.
Here is how this can be done.
Step1. Add both Dual Zigzag and the intended indicator (in this case OBV) on the chart. Notice that both OBV and Dual zigzag appear on different panes.
Step2. Edit the indicator settings of Dual zigzag and set custom indicator by selecting "custom" as oscillator name and then by setting the custom external indicator name and input.
Step 3. You would notice that the zigzag in Dual Zigzag indictor pane is already showing the zigzag pivots based on the OBV indicator and the price pivots display obv values at the pivot points. We can leave this as is.
Step 4. As an additional step, you can also merge the OBV pane and the Dual zigzag indicator pane into one by going into OBV settings and moving the indicator to above pane. Merge the scales so that there is no two scales on the same pane and the entire scale appear on the right.
At the end, you should see two panes - one with price and other with OBV and both having their zigzag plotted.
