Hi folks,
I need some serious stat help. I am working on taking a general
distribution of data and trying to uniformly distribute it. This is
necessary for the learning phase of a machine learning algorithm.
I found a text speaking of a Chebychev polynomial fit (see text exerpt
below). My question is, 'Does anyone know what this Chebychev
polynomial' methodology is?' I would like to learn about it in order
to apply it to my application.
Any help would be appreciated.
Thanks
Anthony
===============
exerpt
===============
Distribution Density Function (DDF) Conversion
This conversion function builds a distribution histogram from the raw
input data stored in the training file and constructs a custom
conversion function using a Chebychev polynomial fit. This is the most
general purpose conversion function and produces a reasonable level of
symmetry (a.k.a. uniformly distributed) for most input distribution
formats. This function is also the default used for stimulus
conversion.
theta = DDF(c*(s - m)/z)
where: theta = complex phase angle
c = conversion coefficient
s = raw input data
m = mean of input distribution
z = standard deviation of input distribution
DDF = Chebychev Polynomial
The sigmoid coefficient c modifies the size of the bins within the
distribution histogram and thus effects the span and degree of
resolution for the resultant polynomial fit. The size of the bins are
modified in linear proportion to the conversion coefficient. The
default value for the conversion
coefficient is 1.0, however it may be modified to any value within the
range of 0.1 to 10. Below is illustrated the effect of varying the
conversion coefficient on an example of Gaussian distributed data.
(Some illlustrations in the text are here which show the distribution
histogram of the raw data, the 'sigmoidal' transformation funtion, I
guess the Chebychev polynomial, and finally, the resultant
'uniformally distributed' data.
Similar to the other illustrations the horizontal axis of the DDF
conversion function represents the raw input values, plotted as
standard deviations from the mean. The vertical axis of the DDF
conversion plot shows the phase angle output. As you can see from the
above plots the modification of the conversion coefficient has a
subtler effect upon the conversion function and it's efficiency. In
all of the above cases the conversion is adequate, however on closer
inspection one finds that the results from coefficient 1.0 are
slightly better (symmetry rating of 9.6 vs. approximately 9.2 for
coefficient values of 2.0 and 0.5). In most cases a conversion
coefficient of 1.0 will produce more than adequate results.
.
.
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