Linear Regression & RSI Multi-Function Screener with Table-LabelHi fellow traders..
Happy to share a Linear Regression & RSI Multi-Function Custom Screener with Table-Labels...
The Screener scans for Linear Regression 2-SD Breakouts and RSI OB/OS levels for the coded tickers and gives Summary alerts
Uses Tables (dynamica resizing) for the scanner output instead of standard labels!
This Screener cum indicator collection has two distinct objectives..
1. Attempt re-entry into trending trades.
2. Attempt Counter trend trades using linear regression , RSI and Zigzag.
Briefly about the Screener functions..
a. It uses TABLES as Labels a FIRST for any Screener on TV.
b. Tables dynamically resize based on criteria..
c. Alerts for breakouts of the UPPER and the LOWER regression channels.(2 SD)
d. In addition to LinReg it also Screens RSI for OB/OS levels so a multifunction Screener.
e. Of course has the standard summary Alerts and programmable format for Custom functions.
f. Uses only the inbuilt Auto Fib and Lin Reg code for the screener.(No proprietary stuff)
g. The auto Zigzag code is derived(Auto fib).
Question what are all these doing in a single screener ??
ZigZag is very useful in determining Trend Up or Down from one Pivot to another.
So Once you have a firm view of the Current Trend for your chosen timeframe and ticker…
We can consider few possible trading scenarios..
a. Re-entry in an Up Trend - Combination of OS Rsi And a Lower Channel breach followed by a re-entry back into the regression channel CAN be used as an effective re-entry.
b. Similarily one can join a Down Trend on OB Rsi and Upper Channel line breach followed by re-entry into the regression channel.
If ZigZag signals a range-bound market, bound within channel lines then the Upper breakout can be used to Sell and vice-versa!
In short many possibilities for using these functions together with Scanner and Alerts.
This facilitates timely PROFITABLE Trending and Counter trend opportunities across multiple tickers.
You must give a thorough READ to the various available tutorials on ZigZag / Regression and Fib retracements before attempting counter trend trades using these tools!!
A small TIP – Markets are sideways or consolidating 70% of the time!!
Acknowledgements: - Thanks a lot DGTRD for the Auto ZigZag code and also for the eagerness to help wherever possible..Respect!!
Disclaimer: The Alerts and Screener are just few tools among many and not any kind of Buy/Sell recommendations. Unless you have sufficient trading experience please consult a Financial advisor before investing real money.
*The alerts are set for crossovers however for viewing tickers trading above or below the channel use code in line 343 and 344 after setting up the Alerts!
** RSI alerts are disabled by default to avoid clutter, but if needed one can activate code lines 441,442,444 and 445
Wish you all, Happy Profitable Trading!
Regression
Log Scale Linear RegressionThis indicator is basically the standard linear regression but adjusted to be suitable for log scale.
You can use 2 different standard deviation values, choose the data source and lookback length.
The colors are chosen directly on the main menu.
Enjoy!
Multiple Regression Polynomial ForecastEXPERIMENTAL:
Forecasting using a polynomial regression over the estimates of multiple linear regression forecasts.
note: on low data the estimates are skewd away of initial value, i added the i_min_estimate option in to try curve this issue with limited success "o_o.
Ethereum Logarithmic Growth Curves & ZonesThis script was modified to fit ethereum logarithmic pricing action.
Robust Channel [tbiktag]Introducing the Robust Channel indicator.
This indicator is based on a remarkable property of robust statistics , namely, the resistance to the presence of data points that deviate significantly from the established trend (generally speaking, outliers ). Being outlier-resistant, the Robust Channel indicator “remembers” a pre-existing trend and thus exhibits a very peculiar "lag" in case of a sharp price change. This allows high-confidence identification of such price actions as a trend reversal, range break, pullback, etc.
In the case of trending and range-bound market conditions, the price remains within the channel most of the time, fluctuating around the central line.
Technical details
The central line is calculated using the repeated median slope algorithm. For each data point in a lookback window of a user-specified Length , this method calculates the median slope of the lines that connect that point to all other points inside the window. The overall median of these median slopes is then calculated and used as an estimate of the trend slope. The algorithm is very efficient as it uses an on-the-fly procedure to update the array containing the slopes (new data pushed - old data removed).
The outer line is then calculated as the central line plus the Length -period standard deviation of the price data multiplied by a user-defined Channel Width Factor . The inner line is defined analogously below the central line.
Usage
As a stand-alone indicator, the Robust Channel can be applied similarly to the Bollinger Bands and the Keltner Channel:
A close above the outer line can be interpreted as a bullish signal and a close below the inner line as a bearish signal.
Likewise, a return to the channel from below after a break may serve as a bullish signal, while a return from above may indicate bearish sentiment.
Robust Channel can be also used to confirm chart patterns such as double tops and double bottoms.
If you like this indicator, feel free to leave your feedback in the comments below!
Matrix Library (Linear Algebra, incl Multiple Linear Regression)What's this all about?
Ever since 1D arrays were added to Pine Script, many wonderful new opportunities have opened up. There has been a few implementations of matrices and matrix math (most notably by TradingView-user tbiktag in his recent Moving Regression script: ). However, so far, no comprehensive libraries for matrix math and linear algebra has been developed. This script aims to change that.
I'm not math expert, but I like learning new things, so I took it upon myself to relearn linear algebra these past few months, and create a matrix math library for Pine Script. The goal with the library was to make a comprehensive collection of functions that can be used to perform as many of the standard operations on matrices as possible, and to implement functions to solve systems of linear equations. The library implements matrices using arrays, and many standard functions to manipulate these matrices have been added as well.
The main purpose of the library is to give users the ability to solve systems of linear equations (useful for Multiple Linear Regression with K number of independent variables for example), but it can also be used to simulate 2D arrays for any purpose.
So how do I use this thing?
Personally, what I do with my private Pine Script libraries is I keep them stored as text-files in a Libraries folder, and I copy and paste them into my code when I need them. This library is quite large, so I have made sure to use brackets in comments to easily hide any part of the code. This helps with big libraries like this one.
The parts of this script that you need to copy are labeled "MathLib", "ArrayLib", and "MatrixLib". The matrix library is dependent on the functions from these other two libraries, but they are stripped down to only include the functions used by the MatrixLib library.
When you have the code in your script (pasted somewhere below the "study()" call), you can create a matrix by calling one of the constructor functions. All functions in this library start with "matrix_", and all constructors start with either "create" or "copy". I suggest you read through the code though. The functions have very descriptive names, and a short description of what each function does is included in a header comment directly above it. The functions generally come in the following order:
Constructors: These are used to create matrices (empy with no rows or columns, set shape filled with 0s, from a time series or an array, and so on).
Getters and setters: These are used to get data from a matrix (like the value of an element or a full row or column).
Matrix manipulations: These functions manipulate the matrix in some way (for example, functions to append columns or rows to a matrix).
Matrix operations: These are the matrix operations. They include things like basic math operations for two indices, to transposing a matrix.
Decompositions and solvers: Next up are functions to solve systems of linear equations. These include LU and QR decomposition and solvers, and functions for calculating the pseudo-inverse or inverse of a matrix.
Multiple Linear Regression: Lastly, we find an implementation of a multiple linear regression, including all the standard statistics one can expect to find in most statistical software packages.
Are there any working examples of how to use the library?
Yes, at the very end of the script, there is an example that plots the predictions from a multiple linear regression with two independent (explanatory) X variables, regressing the chart data (the Y variable) on these X variables. You can look at this code to see a real-world example of how to use the code in this library.
Are there any limitations?
There are no hard limiations, but the matrices uses arrays, so the number of elements can never exceed the number of elements supported by Pine Script (minus 2, since two elements are used internally by the library to store row and column count). Some of the operations do use a lot of resources though, and as a result, some things can not be done without timing out. This can vary from time to time as well, as this is primarily dependent on the available resources from the Pine Script servers. For instance, the multiple linear regression cannot be used with a lookback window above 10 or 12 most of the time, if the statistics are reported. If no statistics are reported (and therefore not calculated), the lookback window can usually be extended to around 60-80 bars before the servers time out the execution.
Hopefully the dev-team at TradingView sees this script and find ways to implement this functionality diretly into Pine Script, as that would speed up many of the operations and make things like MLR (multiple linear regression) possible on a bigger lookback window.
Some parting words
This library has taken a few months to write, and I have taken all the steps I can think of to test it for bugs. Some may have slipped through anyway, so please let me know if you find any, and I'll try my best to fix them when I have time to do so. This library is intended to help the community. Therefore, I am releasing the library as open source, in the hopes that people may improving on it, or using it in their own work. If you do make something cool with this, or if you find ways to improve the code, please let me know in the comments.
Linear Regression CandlesThere are many linear regression indicators out there, most of them draw lines or channels, but this one actually draws a chart.
Moving Regression Band Breakout strategyFollowing the introduction of the Moving Regression Prediction Bands indicator (see link below), I'd like to propose how to utilize it in a simple band breakout strategy :
Go long after the candle closes above the upper band . The lower band (alternatively, the lower band minus the 14-period ATR or the central line ) will serve as a support line .
Exit as soon as the candle closes below the support line .
To manage the risk of false breakouts, a fixed stop loss is set to the value of the support line at the time of opening a position. When the support line moves above the position opening price, shift the stop loss to breakeven.
The same logic but in reverse applies to short positions.
As an option, it is possible to allow long entries only when the slope of the Moving Regression curve is positive (and short entries when the slope is negative).
Model parameters:
Length and Polynomial Order define the lag and smoothness of the model.
Multiplier specifies the width of the channel.
As the default model parameter values, I set those that I found to provide optimal risk / reward ratio on the daily timeframe (for both trending and range-bound market). However, the settings are very flexible and can be well-adjusted to particular market conditions. Feel free to play around and leave feedback in the comments!
Here's the original Moving Regression Prediction Bands script:
Moving Regression Prediction BandsIntroducing the Moving Regression Prediction Bands indicator.
Here I aimed to combine the principles of traditional band indicators (such as Bollinger Bands), regression channel and outlier detection methods. Its upper and lower bands define an interval in which the current price was expected to fall with a prescribed probability, as predicted by the previous-step result of the local polynomial regression (for the original Moving Regression script, see link below).
Algorithm
1. At every time step, the script performs local polynomial regression of the sample data within the lookback window specified by the Length input parameter.
2. The fitted polynomial is used to construct the Moving Regression time series as well as to extrapolate data, that is, to predict the next data point ( MRPrediction ).
3. The accuracy of local interpolation is estimated by means of the root-mean-square error ( RMSE ), that is, the deviation between the fitted polynomial and the observed values.
4. The MRPrediction and RMSE values calculated for the previous bar are then used to build the upper and lower bands , which I define as follows:
Upper Band = MRPrediction_prev + Multiplier *( RMSE_prev )
Lower Band = MRPrediction_prev - Multiplier *( RMSE_prev )
Here the Multiplier is a user-defined parameter that should be interpreted as a quantile in the standard normal distribution (the default value of 2.0 roughly corresponds to the 95% prediction interval).
To visualize the central line , the script offers the following options:
Previous-Period MR Prediction: MRPrediction_prev time series from the above equation.
MR: Conventional Moving Regression time series.
Ribbon: “Previous-Period MR Prediction” and “MR” curves plotted together and colored according to their relative value (green if MR > Previous MR Prediction; red otherwise).
Usage
My original idea was to use the band breakouts as potential trading signals. For example, the price crossing above the upper band is a bullish signal , being a potential sign that price is gaining momentum and is out of a previously predicted trend. The exit signal could be the crossing under the lower band or under the central line.
However, be aware that it is an experimental indicator, so you might fin some better strategies.
Feel free to play around!
Combo Backtest 123 Reversal & Line Regression Intercept This is combo strategies for get a cumulative signal.
First strategy
This System was created from the Book "How I Tripled My Money In The
Futures Market" by Ulf Jensen, Page 183. This is reverse type of strategies.
The strategy buys at market, if close price is higher than the previous close
during 2 days and the meaning of 9-days Stochastic Slow Oscillator is lower than 50.
The strategy sells at market, if close price is lower than the previous close price
during 2 days and the meaning of 9-days Stochastic Fast Oscillator is higher than 50.
Second strategy
Linear Regression Intercept is one of the indicators calculated by using the
Linear Regression technique. Linear regression indicates the value of the Y
(generally the price) when the value of X (the time series) is 0. Linear
Regression Intercept is used along with the Linear Regression Slope to create
the Linear Regression Line. The Linear Regression Intercept along with the Slope
creates the Regression line.
WARNING:
- For purpose educate only
- This script to change bars colors.
[JRL] Pivot Regression OscillatorIntroducing the Pivot Regression Oscillator. This oscillator uses a similar formula to the Stochastic Oscillator. However, instead of comparing the closing price to the lowest price of a period, it compares the distance between current price and the current pivot point. By basing our oscillator on pivot levels, we incorporate a much more relevant and consequential price point around which to base our comparisons.
The indicator can give reliable overbought and oversold signals, and it plots two exponential moving averages as output, which provides crossover signals that can be used to help time trades.
The Pivot Regression Oscillator can be effective for timing re-entries into a trend and seems to be able to avoid some of the false signals of other indicators.
Let me know if you find this useful. Cheers!
Machine Learning: Logistic RegressionMulti-timeframe Strategy based on Logistic Regression algorithm
Description:
This strategy uses a classic machine learning algorithm that came from statistics - Logistic Regression (LR).
The first and most important thing about logistic regression is that it is not a 'Regression' but a 'Classification' algorithm. The name itself is somewhat misleading. Regression gives a continuous numeric output but most of the time we need the output in classes (i.e. categorical, discrete). For example, we want to classify emails into “spam” or 'not spam', classify treatment into “success” or 'failure', classify statement into “right” or 'wrong', classify election data into 'fraudulent vote' or 'non-fraudulent vote', classify market move into 'long' or 'short' and so on. These are the examples of logistic regression having a binary output (also called dichotomous).
You can also think of logistic regression as a special case of linear regression when the outcome variable is categorical, where we are using log of odds as dependent variable. In simple words, it predicts the probability of occurrence of an event by fitting data to a logit function.
Basically, the theory behind Logistic Regression is very similar to the one from Linear Regression, where we seek to draw a best-fitting line over data points, but in Logistic Regression, we don’t directly fit a straight line to our data like in linear regression. Instead, we fit a S shaped curve, called Sigmoid, to our observations, that best SEPARATES data points. Technically speaking, the main goal of building the model is to find the parameters (weights) using gradient descent.
In this script the LR algorithm is retrained on each new bar trying to classify it into one of the two categories. This is done via the logistic_regression function by updating the weights w in the loop that continues for iterations number of times. In the end the weights are passed through the sigmoid function, yielding a prediction.
Mind that some assets require to modify the script's input parameters. For instance, when used with BTCUSD and USDJPY, the 'Normalization Lookback' parameter should be set down to 4 (2,...,5..), and optionally the 'Use Price Data for Signal Generation?' parameter should be checked. The defaults were tested with EURUSD.
Note: TradingViews's playback feature helps to see this strategy in action.
Warning: Signals ARE repainting.
Style tags: Trend Following, Trend Analysis
Asset class: Equities, Futures, ETFs, Currencies and Commodities
Dataset: FX Minutes/Hours/Days
Delta-RSI OscillatorIntroducing the Delta-RSI Oscillator.
This oscillator is a time derivative of the RSI, plotted as a histogram and serving as a momentum indicator. The derivative is calculated explicitly by means of local polynomial regression. It is designed to provide minimum false and premature buy/sell signals compared to many traditional momentum indicators such as Momentum, RSI, Rate of Change.
Application:
Potential trading signals provided by the Delta-RSI Oscillator include:
- zero crossing (negative-to-positive as a bullish sign and positive-to-negative sign as a bearish signal),
- change of direction (consider going long if the oscillator starts to advance, and short otherwise).
In addition, the strength of a particular trend can be estimated by looking at the Delta-RSI value (positive D-RSI in case of the uptrend, and negative in case of the downtrend).
Choosing the model Parameters:
-RSI Length: The timeframe of the RSI that is being differentiated.
- Frame Length: The length of the lookback frame used for local regression.
- Polynomial Order: The order of the local polynomial function.
Longer frames and lower order of polynomials will result in a " smoother " D-RSI, but at the expense of greater lag. Increasing the polynomial order while maintaining the frame length will reduce lag while producing more variance. The values set as default (Length=18, Order=2) were found to provide optimum the variance/lag tradeoff. However, other options (e.g., Length=35, Order=3) can also work well.
Relationship with other methods:
When developing this indicator, I was inspired by Connie Brown’s Derivative Oscillator. The latter pursues the same goal but evaluates the RSI derivative by means of triple smoothing. This paves the way for more clear interpretation and easier tuning of model parameters.
Moving RegressionMoving Regression is a generalization of moving average and polynomial regression.
The procedure approximates a specified number of prior data points with a polynomial function of a user-defined degree. Then, polynomial interpolation of the last data point is used to construct a Moving Regression time series.
Application:
Moving Regression allows one to smooth noise on the analyzed chart, assess momentum, confirm trends, and establish areas of support and resistance.
In addition, it can be used as a simple stand-alone forecasting method to identify trend direction and trend reversal points. When the local polynomial is predicted to move up in the next time step, the color of the Moving Regression curve will be green. Otherwise, the color of the curve is red. This function is (de)activated using the Predict Trend Direction flag.
Selecting the model parameters:
The effects of the moving window Length and the Local Polynomial Degree are confounded. This allows for finding the optimal trade-off between noise (variance) and lag (bias). Higher Length and lower Polynomial Degree (such as 1, i.e. linear), will result in "smoother" time series but at the cost of greater lag. Increasing the Polynomial Degree to, for example, 2 (squared) while maintaining the Length will diminish the lag and thus compromise the noise-lag tradeoff.
Relation to other methods:
When the degree of the local polynomial is set to 0 (i.e., fitting data to a constant level), the Moving Regression time series exactly matches the Simple Moving Average of the same length.
Linear Regression ChannelHello Traders,
There are several nice Linear Regression Channel scripts in the Public Library. and I tried to make one with some extra features too. This one can check if the Price breaks the channel and it shows where is was broken. Also it checks the momentum of the channel and shows it's increasing/decreasing/equal in a label, shape of the label also changes. The line colors change according to direction.
using the options, you can;
- Set the Source (Close, HL2 etc)
- Set the Channel length
- Set Deviation
- Change Up/Down Line colors
- Show/hide broken channels
- Change line width
meaning of arrows:
⇑ : Uptrend and moment incresing
⇗ : Uptrend and moment decreasing
⇓ : Downtrend and moment incresing
⇘ : Downtrend and moment decreasing
⇒ : No trend
An example for how color of lines, arrow direction and shape of label change.
Enjoy!
Diff to MeanSimple tool to find when to buy in bull market, and when to sell when bull ended.
26 EMA on weekly seems to be a decent combination for 2017 bull market.
Use it and find your combination, and hope it guide you through current bull market.
Logarithmic Regression (Weekly)This script is a combination of different logarithmic regression fits on weekly BTC data. It is meant to be used only on the weekly timeframe and on the BLX chart for bitcoin. The "fair value" line is still subjective, as it is only a regression and does not take into account other metrics.
GAURs Polynomial Regression ChannelsThanks to The Sweet Lord , here is the Gaur's Polynomial Regression Channel.
Its a Polynomial Regression Channel but applied a little differently. Wont go into technical details much. Overview of options is as follows-
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Channel Options
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1. Degree of Polynomial: 1/2/3
Default = 3
Defines the degree of polynomials - 1,2,3. Note here, degree 1 will not be a straight line since its applied differently.
Try different degrees for different fits and market conditions.
2. Channel Length:
Default 30 (candles)
You can go beyond 100 or 200 candle lengths but smaller is the usual preference of Poly-Reg-channel traders. It all depends on market conditions and your style of trading. Do your research. I am usually comfortable with a range of 20-50 (in crypto markets).
3. Basis of Channel height/boundries: ATR/Manual
Default: ATR
ATR provides a dynamically adjusted entry/exit bounds of the channels. As ATR changes, the channel bounds also changes its height. It can also be fixed manually. Manual heights wont change automatically.
4. Basis of Y-Value: open/close/ sma / ema / wma /hilow
Default: close
Y- value is the y value of the (x,y) coordinates used while calculating the regression coefficients. Dont worry about it, its nothing serious.
5. Apply channel smoothning using sma?: Yes/No
Default: Yes
Without smoothning, the channel does not "look" good.
6. Shaded Area Height Percentage:
Its the extra margin for the channel. Its in percentage of the total height (defined 3 above) of channels. The shaded area provides an extra allowance for your entries or exits beyond the ATR or manual heights.
7. Plot RSI?: Yes/No
Default: Yes
Plots RSI (orange line in between the channel - its different from the dotted center line) considering the downbound of channels as 0 (oversold) and upbound of channels as 100 (overbought)
8. Plot 200 sma?: Yes/No
Default: Yes
It plots a 200 period fast (green) and 225 period slow (red) sma . I usually use two MAs. Its visually very easy to understand.
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Sample Strategy
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You can develop your own strategy with the channels. But following is just one of the ways you can trade.
Best Application: Ranging markets. But can be happily used in volatile conditions, with a little experience.
1. SMA: -- (this condition is optional really)
If green (200) is above red (225) go only long. If red is above green go only short. Defines long term trend of the market.
2. Channel slope: -- (this stuff needs practice/experience)
Depending on the channel slope, like if its tending to go up or down, you can choose to take only short or long trades. It defines short term momentum of the market.
3. ATR based heights:
Since its ATR based, the channel height are our natural entry and exit points.
Long:
When price touches lower shaded area, consider possible long entry. Exit on price entering the upper shaded area.
Short:
Enter on upper bound shaded area, exit on lower.
4. RSI:
For additional conformations. Again note, the RSI considers the lower bound of channel as 0 and upper as 100. But since, the channel moves up and down, the RSI will also move not only as RSI but also with the channel. Meaning, say if the RSI is valued at 50, then it will be near the center of the channel but since the center changes as time and price changes, the RSI valued at 50 at different times will not be at the same horizontal level respect to the graph, although it will be at the same level (center) respect to the channel.
5. PRC Channel Percentage label:
This label is at the lower side a bit ahead of the current candle. Provides you info on what is the channel percentage. This is especially helpful in crypto markets to gauge your possible percentage profit where profits can be much higher than forex or other instruments. It can also helps you select a suitable market/instrument if the channels are based on ATR.
6. Extra indicators:
I usually use stochastic along with this setup for extra conformations.
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Donate
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Use freely and donate generously if you find value. Your help will really help.
I had earlier provided BTC addresses for donations but it seems to violate TV House rules.
Hope they make TV coins redeemable in future.
- Pranav Joshi
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Extra Info
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// © cpranavjoshi
// special thanks to the "Trading View" people for providing this great platform for free
// ------------------------
// MATH
// ------------------------
// special thanks to an article on the web that provided layman friendly explanation of the maths
// unfortunately i wont be able to provide the link to that article owing to TV restrictions, though i sincerely would have liked to credit the author.
// Google search this phrase, and you should be able to get it in one of the first results - "polynomialregression Mathematics of Polynomial Regression"
// my regression math calculation is a further resolution upon the generalized matrix formula given in the that article.
// the generalized matrix looks scary but in fact its much simpler than one may assume
// the summation sign things are just float numbers that can be easily found out
// so we get a matrix with number of equations equal to the number of unknowns.
// e.g. if its a 3rd degree poly, it has 4 unknowns (c0,c1,c2,c3) with 4 equations as in the generalized matrix
// it can be resolved by simple algebra
// Note: the results have been verified with excel using same input data points.
// pine was difficult for me so i coded it in python first to verify
// ------------------------
// WHY
// ------------------------
// this script was coded because Pranav badly needed Polynomial channels (had used them in mt4 earlier)
// and at the time of this coding, i could not find any readily available script in the trading view public library ( tnx public)
// the complex math was probably the hurdle
// i m not good in maths, but by the Will of the Lord, i could resolve the issue with simple algebra and logic
// ------------------------
// PINE
// ------------------------
// i am just an average (even poor probably) programmer and pine script is not my language
// this is a humble attempt to write my first pine with whatever i could do quickly
// experts - feel free to develop if needed. have used some workarounds in drawings/plottings. rectify them if possible
//
//
// - Pranav Joshi
Example: Polynomial Regression for Spread AnalysisExample of applying polynomial regression channel to spreads or hedges between 2 assets.
Polynomial Regression Bands + Channel [DW]This is an experimental study designed to calculate polynomial regression for any order polynomial that TV is able to support.
This study aims to educate users on polynomial curve fitting, and the derivation process of Least Squares Moving Averages (LSMAs).
I also designed this study with the intent of showcasing some of the capabilities and potential applications of TV's fantastic new array functions.
Polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modeled as a polynomial of nth degree (order).
For clarification, linear regression can also be described as a first order polynomial regression. The process of deriving linear, quadratic, cubic, and higher order polynomial relationships is all the same.
In addition, although deriving a polynomial regression equation results in a nonlinear output, the process of solving for polynomials by least squares is actually a special case of multiple linear regression.
So, just like in multiple linear regression, polynomial regression can be solved in essentially the same way through a system of linear equations.
In this study, you are first given the option to smooth the input data using the 2 pole Super Smoother Filter from John Ehlers.
I chose this specific filter because I find it provides superior smoothing with low lag and fairly clean cutoff. You can, of course, implement your own filter functions to see how they compare if you feel like experimenting.
Filtering noise prior to regression calculation can be useful for providing a more stable estimation since least squares regression can be rather sensitive to noise.
This is especially true on lower sampling lengths and higher degree polynomials since the regression output becomes more "overfit" to the sample data.
Next, data arrays are populated for the x-axis and y-axis values. These are the main datasets utilized in the rest of the calculations.
To keep the calculations more numerically stable for higher periods and orders, the x array is filled with integers 1 through the sampling period rather than using current bar numbers.
This process can be thought of as shifting the origin of the x-axis as new data emerges.
This keeps the axis values significantly lower than the 10k+ bar values, thus maintaining more numerical stability at higher orders and sample lengths.
The data arrays are then used to create a pseudo 2D matrix of x power sums, and a vector of x power*y sums.
These matrices are a representation the system of equations that need to be solved in order to find the regression coefficients.
Below, you'll see some examples of the pattern of equations used to solve for our coefficients represented in augmented matrix form.
For example, the augmented matrix for the system equations required to solve a second order (quadratic) polynomial regression by least squares is formed like this:
(∑x^0 ∑x^1 ∑x^2 | ∑(x^0)y)
(∑x^1 ∑x^2 ∑x^3 | ∑(x^1)y)
(∑x^2 ∑x^3 ∑x^4 | ∑(x^2)y)
The augmented matrix for the third order (cubic) system is formed like this:
(∑x^0 ∑x^1 ∑x^2 ∑x^3 | ∑(x^0)y)
(∑x^1 ∑x^2 ∑x^3 ∑x^4 | ∑(x^1)y)
(∑x^2 ∑x^3 ∑x^4 ∑x^5 | ∑(x^2)y)
(∑x^3 ∑x^4 ∑x^5 ∑x^6 | ∑(x^3)y)
This pattern continues for any n ordered polynomial regression, in which the coefficient matrix is a n + 1 wide square matrix with the last term being ∑x^2n, and the last term of the result vector being ∑(x^n)y.
Thanks to this pattern, it's rather convenient to solve the for our regression coefficients of any nth degree polynomial by a number of different methods.
In this script, I utilize a process known as LU Decomposition to solve for the regression coefficients.
Lower-upper (LU) Decomposition is a neat form of matrix manipulation that expresses a 2D matrix as the product of lower and upper triangular matrices.
This decomposition method is incredibly handy for solving systems of equations, calculating determinants, and inverting matrices.
For a linear system Ax=b, where A is our coefficient matrix, x is our vector of unknowns, and b is our vector of results, LU Decomposition turns our system into LUx=b.
We can then factor this into two separate matrix equations and solve the system using these two simple steps:
1. Solve Ly=b for y, where y is a new vector of unknowns that satisfies the equation, using forward substitution.
2. Solve Ux=y for x using backward substitution. This gives us the values of our original unknowns - in this case, the coefficients for our regression equation.
After solving for the regression coefficients, the values are then plugged into our regression equation:
Y = a0 + a1*x + a1*x^2 + ... + an*x^n, where a() is the ()th coefficient in ascending order and n is the polynomial degree.
From here, an array of curve values for the period based on the current equation is populated, and standard deviation is added to and subtracted from the equation to calculate the channel high and low levels.
The calculated curve values can also be shifted to the left or right using the "Regression Offset" input
Changing the offset parameter will move the curve left for negative values, and right for positive values.
This offset parameter shifts the curve points within our window while using the same equation, allowing you to use offset datapoints on the regression curve to calculate the LSMA and bands.
The curve and channel's appearance is optionally approximated using Pine's v4 line tools to draw segments.
Since there is a limitation on how many lines can be displayed per script, each curve consists of 10 segments with lengths determined by a user defined step size. In total, there are 30 lines displayed at once when active.
By default, the step size is 10, meaning each segment is 10 bars long. This is because the default sampling period is 100, so this step size will show the approximate curve for the entire period.
When adjusting your sampling period, be sure to adjust your step size accordingly when curve drawing is active if you want to see the full approximate curve for the period.
Note that when you have a larger step size, you will see more seemingly "sharp" turning points on the polynomial curve, especially on higher degree polynomials.
The polynomial functions that are calculated are continuous and differentiable across all points. The perceived sharpness is simply due to our limitation on available lines to draw them.
The approximate channel drawings also come equipped with style inputs, so you can control the type, color, and width of the regression, channel high, and channel low curves.
I also included an input to determine if the curves are updated continuously, or only upon the closing of a bar for reduced runtime demands. More about why this is important in the notes below.
For additional reference, I also included the option to display the current regression equation.
This allows you to easily track the polynomial function you're using, and to confirm that the polynomial is properly supported within Pine.
There are some cases that aren't supported properly due to Pine's limitations. More about this in the notes on the bottom.
In addition, I included a line of text beneath the equation to indicate how many bars left or right the calculated curve data is currently shifted.
The display label comes equipped with style editing inputs, so you can control the size, background color, and text color of the equation display.
The Polynomial LSMA, high band, and low band in this script are generated by tracking the current endpoints of the regression, channel high, and channel low curves respectively.
The output of these bands is similar in nature to Bollinger Bands, but with an obviously different derivation process.
By displaying the LSMA and bands in tandem with the polynomial channel, it's easy to visualize how LSMAs are derived, and how the process that goes into them is drastically different from a typical moving average.
The main difference between LSMA and other MAs is that LSMA is showing the value of the regression curve on the current bar, which is the result of a modelled relationship between x and the expected value of y.
With other MA / filter types, they are typically just averaging or frequency filtering the samples. This is an important distinction in interpretation. However, both can be applied similarly when trading.
An important distinction with the LSMA in this script is that since we can model higher degree polynomial relationships, the LSMA here is not limited to only linear as it is in TV's built in LSMA.
Bar colors are also included in this script. The color scheme is based on disparity between source and the LSMA.
This script is a great study for educating yourself on the process that goes into polynomial regression, as well as one of the many processes computers utilize to solve systems of equations.
Also, the Polynomial LSMA and bands are great components to try implementing into your own analysis setup.
I hope you all enjoy it!
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NOTES:
- Even though the algorithm used in this script can be implemented to find any order polynomial relationship, TV has a limit on the significant figures for its floating point outputs.
This means that as you increase your sampling period and / or polynomial order, some higher order coefficients will be output as 0 due to floating point round-off.
There is currently no viable workaround for this issue since there isn't a way to calculate more significant figures than the limit.
However, in my humble opinion, fitting a polynomial higher than cubic to most time series data is "overkill" due to bias-variance tradeoff.
Although, this tradeoff is also dependent on the sampling period. Keep that in mind. A good rule of thumb is to aim for a nice "middle ground" between bias and variance.
If TV ever chooses to expand its significant figure limits, then it will be possible to accurately calculate even higher order polynomials and periods if you feel the desire to do so.
To test if your polynomial is properly supported within Pine's constraints, check the equation label.
If you see a coefficient value of 0 in front of any of the x values, reduce your period and / or polynomial order.
- Although this algorithm has less computational complexity than most other linear system solving methods, this script itself can still be rather demanding on runtime resources - especially when drawing the curves.
In the event you find your current configuration is throwing back an error saying that the calculation takes too long, there are a few things you can try:
-> Refresh your chart or hide and unhide the indicator.
The runtime environment on TV is very dynamic and the allocation of available memory varies with collective server usage.
By refreshing, you can often get it to process since you're basically just waiting for your allotment to increase. This method works well in a lot of cases.
-> Change the curve update frequency to "Close Only".
If you've tried refreshing multiple times and still have the error, your configuration may simply be too demanding of resources.
v4 drawing objects, most notably lines, can be highly taxing on the servers. That's why Pine has a limit on how many can be displayed in the first place.
By limiting the curve updates to only bar closes, this will significantly reduce the runtime needs of the lines since they will only be calculated once per bar.
Note that doing this will only limit the visual output of the curve segments. It has no impact on regression calculation, equation display, or LSMA and band displays.
-> Uncheck the display boxes for the drawing objects.
If you still have troubles after trying the above options, then simply stop displaying the curve - unless it's important to you.
As I mentioned, v4 drawing objects can be rather resource intensive. So a simple fix that often works when other things fail is to just stop them from being displayed.
-> Reduce sampling period, polynomial order, or curve drawing step size.
If you're having runtime errors and don't want to sacrifice the curve drawings, then you'll need to reduce the calculation complexity.
If you're using a large sampling period, or high order polynomial, the operational complexity becomes significantly higher than lower periods and orders.
When you have larger step sizes, more historical referencing is used for x-axis locations, which does have an impact as well.
By reducing these parameters, the runtime issue will often be solved.
Another important detail to note with this is that you may have configurations that work just fine in real time, but struggle to load properly in replay mode.
This is because the replay framework also requires its own allotment of runtime, so that must be taken into consideration as well.
- Please note that the line and label objects are reprinted as new data emerges. That's simply the nature of drawing objects vs standard plots.
I do not recommend or endorse basing your trading decisions based on the drawn curve. That component is merely to serve as a visual reference of the current polynomial relationship.
No repainting occurs with the Polynomial LSMA and bands though. Once the bar is closed, that bar's calculated values are set.
So when using the LSMA and bands for trading purposes, you can rest easy knowing that history won't change on you when you come back to view them.
- For those who intend on utilizing or modifying the functions and calculations in this script for their own scripts, I included debug dialogues in the script for all of the arrays to make the process easier.
To use the debugs, see the "Debugs" section at the bottom. All dialogues are commented out by default.
The debugs are displayed using label objects. By default, I have them all located to the right of current price.
If you wish to display multiple debugs at once, it will be up to you to decide on display locations at your leisure.
When using the debugs, I recommend commenting out the other drawing objects (or even all plots) in the script to prevent runtime issues and overlapping displays.
My BTC log curveLogarithmic regression of the USD price of Bitcoin , calculated according to the equation:
y=A*exp(beta*x^lambda + c) + m*x + b
where x is the number of days since the genesis block. All parameters are editable in the script options.
Function Polynomial RegressionDescription:
A function that returns a polynomial regression and deviation information for a data set.
Inputs:
_X: Array containing x data points.
_Y: Array containing y data points.
Outputs:
_predictions: Array with adjusted _Y values.
_max_dev: Max deviation from the mean.
_min_dev: Min deviation from the mean.
_stdev/_sizeX: Average deviation from the mean.
Resources:
en.wikipedia.org
rosettacode.org
Function - Linear RegressionDescription:
A Function that returns a linear regression channel using (X,Y) vector points.
Inputs:
_X: Array containing x data points.¹
_Y: Array containing y data points.¹
Note:
¹: _X and _Y size must match.
Outputs:
_predictions: Array with adjusted _Y values at _X.
_max_dev: Max deviation from the mean.
_min_dev: Min deviation from the mean.
_stdev/_sizeX: Average deviation from the mean.
Resources:
www.statisticshowto.com
en.wikipedia.org
