STD-Filtered, Gaussian-Kernel-Weighted Moving Average BT [Loxx]

STD-Filtered, Gaussian-Kernel-Weighted Moving Average BT is the backtest for the following indicator

Included:

• This backtest uses a special implementation of ATR and ATR smoothing called "True Range Double" which is a range calculation that accounts for volatility skew.
  
• You can set the backtest to 1-2 take profits with stop-loss
  
• Signals can't exit on the same candle as the entry, this is coded in a way for 1-candle delay post entry
  
• This should be coupled with the INDICATOR version linked above for the alerts and signals. Strategies won't paint the signal "L" or "S" until the entry actually happens, but indicators allow this, which is repainting on current candle, but this is an FYI if you want to get serious with Pinescript algorithmic botting
  
• You can restrict the backtest by dates
  
• It is advised that you understand what Heikin-Ashi candles do to strategies, the default settings for this backtest is NON Heikin-Ashi candles but you have the ability to change that in the source selection
  
• This is a mathematically heavy, heavy-lifting strategy. Make sure you do your own research so you understand what is happening here.
  

STD-Filtered, Gaussian-Kernel-Weighted Moving Average is a moving average that weights price by using a Gaussian kernel function to calculate data points. This indicator also allows for filtering both source input price and output signal using a standard deviation filter.

Purpose
This purpose of this indicator is to take the concept of Kernel estimation and apply it in a way where instead of predicting past values, the weighted function predicts the current bar value at each bar to create a moving average that is suitable for trading. Normally this method is used to create an array of past estimators to model past data but this method is not useful for trading as the past values will repaint. This moving average does NOT repaint, however you much allow signals to close on the current bar before taking the signal. You can compare this to Nadaraya-Watson Estimator wherein they use Nadaraya-Watson estimator method with normalized kernel weighted function to model price.

What are Kernel Functions?
A kernel function is used as a weighing function to develop non-parametric regression model is discussed. In the beginning of the article, a brief discussion about properties of kernel functions and steps to build kernels around data points are presented.

Kernel Function
In non-parametric statistics, a kernel is a weighting function which satisfies the following properties.

A kernel function must be symmetrical. Mathematically this property can be expressed as K (-u) = K (+u). The symmetric property of kernel function enables its maximum value (max(K(u)) to lie in the middle of the curve.
The area under the curve of the function must be equal to one. Mathematically, this property is expressed as: integral −∞ + ∞ ∫ K(u)d(u) = 1
Value of kernel function can not be negative i.e. K(u) ≥ 0 for all −∞ < u < ∞.

Kernel Estimation
In this article, Gaussian kernel function is used to calculate kernels for the data points. The equation for Gaussian kernel is:

K(u) = (1 / sqrt(2pi)) * e^(-0.5 *(j / bw )^2)

Where xi is the observed data point. j is the value where kernel function is computed and bw is called the bandwidth. Bandwidth in kernel regression is called the smoothing parameter because it controls variance and bias in the output.

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Corrected input error.