XCAP iPolyFit

Originally Posted by Paul A. Griffin

XCAP iPolyFit:

Fits a time series to Legendre (orthogonal) polynomials and thereby provides the full least squares fit to a polynomial
over a time interval.

Author: Paul A. Griffin
January 2, 2007
Extrema Capital, 2007

This indicator provides as output the least square best fit to a set of polynomials of maximum power (degree) using
discrete Legendre polynomials. This is accomplished by decomposing the time series into discrete Legendre polynomials
over the interval of orthogonality given by 2 * Width + 1 and discarding the polynomial fits degrees above some maximum degree.
The remaining smoothed series,

SmoothedSeries[t] = c[0] * 1 + c[1] * (t-Width) + c[2] * Power(t-Width,2) + ... c[MaximumDegree]*Power(t-Width,MaximumDegree)

has coefficients given by the array c that are determined such that this polynomial has the minimum least square error over
the interval. The indicator provides two outputs:

ShowLeadingEdge = TRUE

As you move forward along the time series, the fit is generated. What is displayed is the leading, rightmost bar, "[0]", for
each fit. This is also obtainable via the Savitzky Golay method, see the reference 3 below.

ShowLastFit = TRUE

This displays the entire last fit for just the last bar on the chart, so you will only see an output on the last
2*Width + 1 bars of your chart.

Some Comments:

I like this method of determining the least square fit better than the Savitzky Golay method I have posted because is
provides the entire fit, not just the leading edge, at any bar. This makes for a smoothing filter that can be thought of a
generalization of simple moving average method. In fact, it can be thought of as a progression of noise reduction, starting with no
noise reduction (MaximumDegree = 2*Width) to reduction of signal down to one degree of freedom (a moving average, MaximumDegree=0).

So, if your goal is to reduce existing stochastic noise in the data, then this smoothed output is of interest because it can
in principle provide smoothing with minimal lag and best reduction of noise.

That is, instead of smoothing a function like this:

Average(Function(series), Length1)

with lag = Length/2, one should contemplate a new method:

Function(SmoothedSeries(L ength2,Degree))

as the smoothed series should have a higher signal to stochastic noise ratio than the original. Any improvement is based on the
assumption that stochastic noise exists in the chart. Since most standard models of asset pricing use a stochastic variable
model, from everything from options pricing to modern portfolio theory, it seems reasonable to apply this polynomial noise
reduction method directly to financial time series.

I will post some predictive indicators using this method soon, in this discussion topic. This post is to get the
foundations out of the way and to provide a basic starting point for further work.

For "Rocket Scientists" and as a side note: there is a beautiful history of the application of Legendre polynomials to science.
For example in the solution to the Hydrogen atom, they serve to define part of the angular dependence of the electron probability field
around the proton.


Some Plots:

For SPY daily chart with a width of 126 bars (126*2+1 = 253, about 1 trading year), I have created jpegs of the fits of order
0, 1, 2, 4, 8, 12, and 52. Order zero is the average over one year. Order 1 is linear regression. 2 though 8 are just interesting
low order fits. I also captured order 12 and 52 because of the number of months and weeks per year respectively.

References:

[1] http://en.wikipedia.org/wiki/Legendre_polynomials

[2] Peter Seffen, "On Digital Smoothing Filters: A Brief Review of Closed Form Solutions and Two New Filter Approaches",
Circuits Systems Signal Process, Vol. 5, No 2, 1986

[3] https://www.tradestation.com/Discuss...Topic_ID=58534

Note:
This indicator was written in EasyLanguage.
Please refer to your users manual for importation instructions.

Your comments and rating of this indicator is appreciated.

[Tams]

14.7

[aaa]

It is pretty

Is it an other indicator ?

it looks 2 have connection with aaa & XCAP iPolyFit ind

How to do that ?

[Tams]

related indicators:

XCAP iPolyCycle
http://www.traderslaboratory.com/for...ycle-7072.html

XCAP IPolyFitPredict
http://www.traderslaboratory.com/for...dict-7056.html