Least Squares Polynomial Fit

Compute polynomial coefficients that best fit input data in least-squares sense

• 

Libraries:
DSP System Toolbox / Math Functions / Polynomial Functions

Description

The Least Squares Polynomial Fit block computes the coefficients of the nth order polynomial that best fits the input data in the least-squares sense, where n is the value you specify in the Polynomial order parameter. The block computes a distinct set of n+1 coefficients for each column of the M-by-N input u.

Examples

Compute Polynomial Coefficients Using Least Squares Polynomial Fit Block

Open the ex_leastsquarespolyfit_ref model. The Polynomial Evaluation block uses this second-order polynomial to generate four values of dependent variable y from four values of independent variable u received at the In port.

The polynomial coefficients are supplied in the vector [-2 0 3] at the Coeffs port. Note that the coefficient of the first-order term is zero.

The Control points parameter of the Least Squares Polynomial Fit block is configured with the same four values of independent variable u that are used as input to the Polynomial Evaluation block, [1 2 3 4]. The Least Squares Polynomial Fit block uses these values together with the input values of dependent variable y to reconstruct the original polynomial coefficients.

Ports

Input

Input — Input data

vector | matrix

Specify the input data as a vector or a matrix of size M-by-N. The block treats a length-M unoriented vector input as an M-by-1 matrix.

Data Types: single | double

Output

Output — Polynomial coefficients

column vector | matrix

Coefficients of the polynomial that best fits the input data in the least-squares sense, returned as a column vector or a matrix of size (n+1)-by-N, where n is the value you specify in the Polynomial order parameter.

Each column of the (n+1)-by-N output matrix c represents a set of n+1 coefficients describing the best-fit polynomial for the corresponding column of the input. The coefficients in each column are arranged in order of descending exponents, c₁, c₂, ..., c_n+1.

For more information on how this block computes the polynomial coefficients, see Algorithms.

Data Types: single | double

Parameters

Control points (X) — Control points

(0:31)/31 (default) | vector

Specify the values of the independent variable to which the data in each input column correspond. For an M-by-N input, this parameter must be a length-M vector.

Tunable: Yes

Polynomial order (N) — Polynomial order

3 (default) | positive integer

Specify the order n of the polynomial that the block uses to construct the best fit. The number of coefficients is n+1.

Block Characteristics

Data Types	double | single
Direct Feedthrough	no
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no

Algorithms

For a given input column, the block computes the set of coefficients, c₁, c₂, ..., c_n+1, that minimizes the quantity $${\displaystyle \sum _{i=1}^M{(u_i-{\widehat{u}}_i)}^2}$$, where u_i is the ith element in the input column and is given by

$${\widehat{u}}_i=f\left(x_i\right)=c_1{x}_i^n+c_2{x}_i^{n-1}+\mathrm{...}+c_{n+1}.$$

The values of the independent variable, x₁, x₂, ..., x_M are specified as a length-M vector by the Control points parameter. The same M control points are used for all N polynomial fits, and can be equally or unequally spaced.

The equivalent MATLAB^® code is shown here.

c = polyfit(x,u,n)

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

Generated code relies on the memcpy or memset function (string.h) under certain conditions.

Version History

Introduced before R2006a