PFIT

Purpose:

Performs Least Squares Polynomial fitting with error statistics.

Syntax:

PFIT(series, order, mode, form)

(coef, R2, Se, res) = PFIT(series, order, mode, form)

series

-

An input series

order

-

An integer, the polynomial order.

mode

-

Optional. An integer, the error statistics flag:

0 :

no statistics

1 :

R2 and Se, Standard Error of Estimate (default)

form

-

Optional. An integer, form of the polynomial coefficients:

0 :

ascending powers, lowest degree to highest (default)

1 :

descending powers, highest degree to lowest

Returns:

A series or table.

 

(coef, R2, Se, res) = pfit(series, order, mode, form) returns the polynomial coefficients, residual squared, standard error and residual in separate variables.

Example:

W1: gsin(100, 0.01, 0.8)

W2: pfit(W1, 2)

W3: polygraph(col(W2,1),xvals(W1));overplot(W1,lred)

 

image\pfitpic.gif

 

W2 contains the table:

 

Coeff      R2        Se

 0.349702  0.896020  0.232544 

 2.744303 

-4.769116

 

W3 contains the fitted result with an overplot of the original data.

Example:

W4: pfit(W1, 4)

 

W4 contains the table:

 

Coeff       R2        Se

 -0.044900  0.999201  0.020604 

  6.293248  

 -6.989869  

-12.180623 

 12.057509  

 

The increase in R2 and the corresponding decrease in Se indicates the 4th order fit performs better in the least squares sense than the previous 2nd order fit.

Remarks:

pfit(series, N) performs a least squares fit of a series to

 

image\polyf04.gif

 

where y is the input series and N is the order of the fit.

 

PFIT returns the coefficients, a[k], of the above power series.

 

If form is 1 then:

 

image\polyf03.gif

 

R2, sometimes called the Coefficient of Determination, is an indication of how the fit accounts for the variability of the data. R2 can be thought of as:

 

image\pfit01.gif

 

An R2 of 1 indicates the model accounts for ALL the variability of the data. An R2 of 0 indicates no data variability is accounted for by the model.

 

The Standard Error of Estimate, Se, can be thought of as a normalized standard deviation of the residuals, or prediction errors. Given:

 

image\pfit04.gif

 

The residual or error between sample point i and fitted point i is:

 

image\pfit02.gif

 

The Standard Error of Estimate is defined as:

 

image\pfit03.gif

 

where L is the series length and N is the degree of the fitting polynomial. As the model fits the data better, Se approaches 0.

 

See LINFIT to fit linear combination of arbitrary basis functions to a series using the method of least squares with QR decomposition.

See Also:

LINFIT

LINREG

LINREG2

POLYFIT

POLYROOT

STDEV