Polynomial
Regression – Least Square Fittings
This brief article will demonstrate how to work out
polynomial
regressions in Matlab (also known as polynomial least
squares
fittings). The idea is to find the polynomial function that properly
fits a
given set of data points.

For
this purpose, we’re going to use two useful builtin functions: polyfit
(for
fitting polynomial to data) and polyval (to evaluate polynomials).
This
is
the simplest way to use these functions:
p = polyfit(x,
y, n) finds the coefficients of a polynomial p(x) of degree n that fits
the
data y best in a leastsquares sense. 
p is a row vector of length n
+ 1
containing the polynomial coefficients in descending powers, p(1)*x^n +
p(2)*x^(n
 1) + ... + p(n)*x + p(n + 1).
y = polyval(p,
x) returns the value of a polynomial p evaluated at x. p is a vector of
length
n + 1 whose elements are the coefficients of the polynomial in
descending
powers, y = p(1)*x^n + p(2)*x^(n  1) + ... + p(n)*x + p(n + 1). If x
is a
matrix or vector, the polynomial is evaluated at all points in x.
Let’s
say that we’re given these points
x = [1 2 3 4 5.5 7 10]
y = [3 7 9 15 22 21 21]
and
want to explore fits of 2nd., 4th. and 5th. order. We could use the
following
code:
clear all, clc,
close all, format compact
%
define coordinates
x = [1 2
3 4 5.5 7 10];
y = [3 7
9 15 22 21 21];
%
plot given data in red
plot(x,
y, 'ro', 'linewidth', 2)
hold on
%
find polynomial fits of different orders
p2 =
polyfit(x, y, 2)
p4 =
polyfit(x, y, 4)
p5 =
polyfit(x, y, 5)
%
see interpolated values of fits
xc = 1 :
.1 : 10;
%
plot 2nd order polynomial
y2 =
polyval(p2, xc);
plot(xc,
y2, 'g.')
%
plot 4th order polynomial
y4 =
polyval(p4, xc);
plot(xc,
y4, 'linewidth', 2)
%
plot 5th order polynomial
y5 =
polyval(p5, xc);
plot(xc,
y5, 'k.', 'linewidth', 2)
grid
legend('original
data',
'2nd.
order fit', '4th.
order', '5th.
order')
The
resulting fits are displayed in this figure
The coefficients found
by Matlab are:
p2 = 0.3863
6.3983
4.1596
p4 = 0.0334
0.7377
5.0158
8.4137
7.5779
p5 = 0.0213
0.5022
4.1330 14.5708
25.3233 11.3510
which,
in other words, represent the following polynomials of 2nd.,
4th. and 5th. order, respectively.
y_{2}
= 0.3863x^{2} + 6.3983x  4.1596
y_{4}
= 0.0334x^{4}
 0.7377x^{3}
+ 5.0158x^{2}  8.4137x + 7.5779
y_{5}
= 0.0213x^{5}
 0.5022x^{4}
+ 4.1330x^{3}  14.5708x^{2} +
25.3233x  11.3510
You can see this video on
how to do it even easier with the integrated Curve Fitting Tool
within the Figure window.
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