# General Linear Fitting (G Dataflow)

Version:

Finds the k-dimension linear curve values and the set of k-dimension linear fit coefficients, which describe the k-dimension linear curve that best represents the input data set using the Least Square, Least Absolute Residual, or Bisquare method.

## covariance selector

Whether the node computes the covariance matrix.

Name Description
do not compute Covariance Does not compute the covariance matrix.
compute Covariance Computes the covariance matrix.

Default: do not compute Covariance

## y

Observed data set. The number of elements in y must equal the number of rows in H.

## H

Matrix that represents the formula you use to fit the data set.

Example of Obtaining the Matrix H

Assume you have a set of observations (x, y). For example, you can obtain a set of observations through data acquisition. Also, assume you think the relationship between x and y is of the following form:

$y={a}_{0}{f}_{0}\left(x\right)+{a}_{1}{f}_{1}\left(x\right)+{a}_{2}{f}_{2}\left(x\right)+{a}_{3}{f}_{3}\left(x\right)+{a}_{4}{f}_{4}\left(x\right)$

where

• ${f}_{0}\left(x\right)=1.0$
• ${f}_{1}\left(x\right)=\mathrm{sin}\left({x}^{2}\right)$
• ${f}_{2}\left(x\right)=3\mathrm{cos}\left(x\right)$
• ${f}_{3}\left(x\right)=\frac{1}{x+1}$
• ${f}_{4}\left(x\right)={x}^{4}$

The following equation describes matrix H.

${H}_{ij}={f}_{j}\left({x}_{i}\right)$

where

• Hij is element in the ith column and jth row of H
• fj(xi) is the function value of the ith element in x

In this example, the number of columns in H equals the number of elements in x. the number of rows in H equals 5.

## weight

Array of weights for the observations.

weight must be the same size as y. If you do not wire an input to weight, this node sets all elements of weight to 1. If an element in weight is less than 0, this node uses the absolute value of the element.

## tolerance

Value that determines when to stop the iterative adjustment of coefficients when you use the Least Absolute Residual or Bisquare methods.

If tolerance is less than or equal to 0, this node sets tolerance to 0.0001.

How tolerance Affects the Outputs with Different Fitting Methods

For the Least Absolute Residual method, if the relative difference of the weighted mean error of the polynomial fit in two successive iterations is less than tolerance, this node returns resulting polynomial coefficients. For the Bisquare method, if any relative difference between polynomial coefficients in two successive iterations is less than tolerance, this node returns the resulting polynomial coefficients.

Default: 0.0001

## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

## algorithm

Algorithm this node uses to compute the best fit.

Use the SVD for Rank Deficient H algorithm only if H is rank deficient or does not have a full rank and if all other algorithms are unsuccessful.

Name Value Description
SVD 0 Uses the SVD algorithm.
Givens 1 Uses the Givens algorithm.
Givens2 2 Uses the Givens2 algorithm.
Householder 3 Uses the Householder algorithm.
LU Decomposition 4 Uses the LU Decomposition algorithm.
Cholesky 5 Uses the Cholesky algorithm.
SVD for Rank Deficient H 6 Uses the SVD for Rank Deficient H algorithm.

Default: SVD

## method

Fitting method.

Name Value Description
Least Square 0 Uses the least square method.
Least Absolute Residual 1 Uses the least absolute residual method.
Bisquare 2 Uses the bisquare method.

Default: Least Square

Fitted data.

## coefficients

Set of coefficients that minimize chi squared.

## covariance

Matrix of covariance C with k-by-k elements. cjk is the covariance between ai and ak. cjj is the variance of aj. This node uses the following equation to compute the covariance matrix C:

$C={\left({H}_{0}^{T}{H}_{0}\right)}^{-1}$

## weight out

Actual weight of general linear fitting if the fitting method is Bisquare.

If the fitting method is Least Square or Least Absolute Residual, this output returns the value you enter for weight.

## error out

Error information. The node produces this output according to standard error behavior.

## residue

Weighted mean error of the fitted model.

If method is Least Absolute Residual, residue is the weighted mean absolute error. Otherwise, residue is the weighted mean square error.

## Algorithm for Obtaining the General Linear Fit

The following linear data model demonstrates how to configure the input parameters and how to use this node to obtain the fitted values and the set of least square coefficients that best represents the relationship of observations (x, y):

$y=f\left(a,x\right)=\underset{i=0}{\overset{n-1}{\sum }}{a}_{i}{f}_{i}\left(x\right)={a}_{0}{f}_{0}\left(x\right)+{a}_{1}{f}_{1}\left(x\right)+...+{a}_{n-1}{f}_{n-1}\left(x\right)$

where

• a = {a0, a1, a2, ..., an - 1}
• n is the total number of functions
• fi(x) are modal functions

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported