# 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)

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

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:

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:

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:

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

The Standard Error of Estimate is defined as:

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.