Quadratic RegressionFit a quadratic polynomial (parabola) to the last length data points by minimizing the sum of squares between the data and the fitted results. The script can extrapolate the results in the future and can also display the R-squared of the model. Note that this script is subject to some limitations (more in the "Notes" section).
Settings
Length : Number of data points to use as input.
Offset : Determine the number of past fitted values to be displayed, if 0 only the extrapolated values are displayed, if 55 only the past fitted values are displayed.
Src : Input data of the indicator
Show R2 : Determine if the value of the R-squared must be displayed, by default true.
Usage
When the underlying trend in the price is not linear, we might use more advanced models to estimate it, this is where using a higher-degree regression model might be required, as such a quadratic model (second-degree) is appropriate when the underlying trend is parabolic.
Here we can see that the quadratic regression (in blue) offer a better fit than a linear one.
Another advantage of the quadratic regression is that a linear one will always have the same direction, that's not the case with the quadratic regression and as such, it is possible to forecast reversals.
Above a linear regression (in red) and two quadratic regression (in blue) with both length = 54. Note that for the sake of clarity, the above image uses a quadratic regression to show all the past fitted values and another one to show all the forecasted values.
The R-Squared is also extremely useful when it comes to measuring the accuracy of the model, with values closer to 1 indicating that the model is appropriate, and thus suggesting that the underlying trend in the price is parabolic. The R-squared can also measure the strength of the trend.
Notes
The script uses the function line.new , as such only a maximum of 54 observations are displayed, getting more observations can be done by using an additional quadratic regression like we did in the previous section. Another thing is that line.new use xloc.bar_time , as such it is possible to observe some errors with the displayed results of the indicator, such as:
This will happen when applying the indicator to symbols with session breaks, I apologize for this inconvenience and I'll try to find solutions. Note however that the indicator will work perfectly on cryptos.
Summary
That's an indicator I really wanted to make, even if it is important to note that such models are rarely useful in stock markets, however it is more than possible to create a quadratic regression (with severe limitations) with pinescript.
Today I turn 21, while I should be celebrating I still wanted to share something with the community, it's also some kind of present to myself that tells me that I am a bit better at using pinescript than last year, and I am glad I could progress (instead of regress, regression , got it?). Thx a lot for reading!
Parabola
Quadratic Least Squares Moving Average - Smoothing + Forecast Introduction
Technical analysis make often uses of classical statistical procedures, one of them being regression analysis, and since fitting polynomial functions that minimize the sum of squares can be achieved with the use of the mean, variance, covariance...etc, technical analyst only needed to replace the mean in all those calculations with a moving average, we then end up with a low lag filter called least squares moving average (lsma) .
The least squares moving average could be classified as a rolling linear regression, altho this sound really bad it is useful to understand the relationship of both methods, both have the same form, that is ax + b , where a and b are coefficients of the model. However in a simple linear regression a and b are constant, while the lsma use variables instead.
In a simple lsma we model the relationship of the closing price (dependent variable) with a linear sequence (independent variable), therefore x = 1,2,3,4..etc. However we can use polynomial of higher degrees to model such relationship, this is required if we want more reactivity. Therefore we can use a quadratic form, that is ax^2 + bx + c , where a,b and c are variables.
This is the quadratic least squares moving average (qlsma), a not so official term, but we'll stick with it because it still represent the aim of the filter quite well. In this indicator i make the calculations of the qlsma less troublesome, therefore one might understand how it would work, note that in general the coefficients of a polynomial regression model are found using matrix calculus.
The Indicator
A qlsma, unlike the classic lsma, will fit better to the price and will be more reactive, this is the advantage of using an higher degrees for its calculation, we can model more complex relationship.
lsma in green, qlsma in red, with both length = 200
However the over/under shoots are greater, i'll explain why in the next sections, but this is one of the drawbacks of using higher degrees.
The indicator allow to forecast future values, the ahead period of the forecast is determined by the forecast setting. The value for this setting should be lower than length, else the forecasts can easily over/under shoot which heavily damage the forecast. In order to get a view on how well the forecast is performing you can check the option "Show past predicted values".
Of course understanding the logic behind the forecast is important, in short regressions models best fit a certain curve to the data, this curve can be a line (linear regression), a parabola (quadratic regression) and so on, the type of curve is determined by the degree of the polynomial used, here 2, which is a parabola. Lets use a linear regression model as example :
ax + b where x is a linear sequence 1,2,3...and a/b are constants. Our goal is to find the values for a and b that minimize the sum of squares of the line with the dependent variable y, here the closing price, so our hypothesis is that :
closing price = ax + b + ε
where ε is white noise, a component that the model couldn't forecast. The forecast of the closing price 14 step ahead would be equal to :
closing price 14 step aheads = a(x+14) + b
Since x is a linear sequence we only need to sum it with the forecasting horizon period, the same is done here with :
a*(n+forecast)^2 + b*(n + forecast) + c
Note that the forecast proposed in the indicator is more for teaching purpose that anything else, this indicator can't possibly forecast future values, even on a meh rate.
Low lag filters have been used to provide noise free crosses with slow moving average, a bad practice in my opinion due to the ability low lag filters have to overshoot/undershoot, more interesting use cases might be to use the qlsma as input for other indicators.
On The Code
Some of you might know that i posted a "quadratic regression" indicator long ago, the original calculations was coming from a forum, but because the calculation was ugly as hell as well as extra inefficient (dogfood level) i had to do something about it, the name was also terribly misleading.
We can see in the code that we make heavy use of the variance and covariance, both estimated with :
VAR(x) = SMA(x^2) - SMA(x)^2
COV(x,y) = SMA(xy) - SMA(x)SMA(y)
Those elements are then combined, we can easily recognize the intercept element c , who don't change much from the classical lsma.
As Digital Filter
The frequency response of the qlsma is similar to the one of the lsma, those filters amplify certain frequencies in the passband, and have ripples in the stop band. There is something interesting about those filters, first using higher degrees allow to greater boost of the frequencies in the passband, which result in greater over/under shoots. Another funny thing is that the peak/valley of the ripples is equal the peak or valley in the ripples of another lsma of different degree.
The transient response of those filters, that is impulse response, step response...etc is related to the degree of the polynomial used, therefore lets denote a lsma of degree p : lsma(p) , the impulse response of lsma(p) is a polynomial of degree p, and the step response is simple a polynomial of order p+1.
This is why it was more interesting to estimate the qlsma using convolution, however we can no longer forecast future values.
Conclusion
I proposed a more usable quadratic least squares moving average, with more options, as well as a cleaner and more efficient code. The process of shrinking the original code is made easier when you know about the estimations of both variance and covariance.
I hope the proposed indicator/calculation is useful.
Thx for reading !
