Repeated Median Regression Channel

1 mar 2021

This script uses the Repeated Median (RM) estimator to construct a linear regression channel and thus offers an alternative to the available codes based on ordinary least squares.

The RM estimator is a robust linear regression algorithm. It was proposed by Siegel in 1982 (1) and has since found many applications in science and engineering for linear trend estimation and data filtering.

The key difference between RM and ordinary least squares methods is that the slope of the RM line is significantly less affected by data points that deviate strongly from the established trend. In statistics, these points are usually called outliers, while in the context of price data, they are associated with gaps, reversals, breaks from the trading range. Thus, robustness to outlier means that the nascent deviation from a predetermined trend will be more clearly seen in the RM regression compared to the least-squares estimate. For the same reason, the RM model is expected to better depict gaps and trend changes (2).

Input Description

• Length: Determines the length of the regression line.
  
• Channel Multiplier: Determines the channel width in units of root-mean-square deviation.
  
• Show Channel: If switched off, only the (central) regression line is displayed.
  
• Show Historical Broken Channel: If switched on, the channels that were broken in the past are displayed. Note that a certain historical broken channel is shown only when at least Length / 2 bars have passed since the last historical broken channel.
  
• Print Slope: Displays the value of the current RM slope on the graph.
  

Method
Calculation of the RM regression line is done as follows (1,3):

• For each sample point (t(i), y(i)) with i = 1..Length, the algorithm calculates the median of all the slopes of the lines connecting this point to the other Length-1 points.
  
• The regression slope is defined as the median of the set of these median slopes.
  
• The regression intercept is defined as the median of the set {y(i) – m*t(i)}.
  

Computational Time
The present implementation utilizes a brute-force algorithm for computing the RM-slope that takes O(Length^2) time. Therefore, the calculation of the historical broken channels might take a relatively long time (depending on the Length parameter). However, when the Show Historical Broken Channel option is off, only the real-time RM channel is calculated, and this is done quite fast.


References
1. A. F. Siegel (1982), Robust regression using repeated medians, Biometrika, 69, 242–244.
2. P. L. Davies, R. Fried, and U. Gather (2004), Robust signal extraction for on-line monitoring data, Journal of Statistical Planning and Inference 122, 65-78.
3. en.wikipedia.org/wiki/Repeated_median_regression

19 oct 2021

Notas de prensa

The code is converted to PineScript v5. The core algorithms are transferred to the public library.
There are no changes in the functionality of the script.

To use the RM regression algorithm in conjunction with interactive inputs, you can also try this tool: