Theil–Sen Estimator

The Theil-Sen estimator is a nonparametric statistics method for robustly fitting a regression line to sample points (1,2).

As stated in the Wikipedia article (3), the method is "the most popular nonparametric technique for estimating a linear trend" in the applied sciences due to its robustness to outliers and limited assumptions regarding measurement errors.

Relation with other Methods
The Theil-Sen estimator can be significantly more accurate than simple linear regression (least squares) for skewed and heteroskedastic data.

Method Description
The script computes all the slopes between pairs of points and takes the median as the estimate of the regression slope, m. Subsequently, the intercept, b, is determined from the sample points as the median of y(i) − mx(i) values. The regression line in the slope–intercept form, y = mx + b, is then plotted along with the calculated prediction interval (estimated by means of the root-mean-square error).

I have added two options for how to handle pairs of points:

• Method == "All" to use the slopes of all pairs of points;
  
• Method == "Random" to use the slopes of randomly generated pairs of points.
  

The random choice of the pairs of points is based on the Wichmann–Hill is a pseudorandom number generator.

The reason for introducing the "Random" method is that the calculation of the median involves sorting the array of slopes (the size of N*(N-1)/2, where N is the number of sample points). This is a computationally demanding procedure, which runs into the limit on the cycle computation time (200 ms) set in TradingView. Therefore, the "All" method works only with Length < 50.

Also note that the number of lookback points is limited by by the maximum array size allowed in TradingView.

Literature
1. Sen, P. K. (1968) "Estimates of the regression coefficient based on Kendall's tau." JASA, 1379-1389.
2. Theil, H. (1950) "A rank-invariant method of linear and polynomial regression analysis." Reprinted in 1992 in Henri Theil’s contributions to economics and econometrics, Springer, 345-381.
3. https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator

Sürüm Notları:

Fixed an issue causing instability when using the "Random" method on high-resolution real-time charts (thanks @labtab for spotting it)

Sürüm Notları:

added the following features:

• the plot changes color (to orange by default) when the price is outside the prediction interval channel
  
• alert conditions for the channel break are now available
  
• the with of the prediction interval can be tuned by an input parameter (Prediction Interval Multiplier); its value should be interpreted as the quantile of the normal distribution with 2 (default) roughly corresponding to the 95% prediction interval
  

Sürüm Notları:

The built-in procedure is now used to evaluate the median.