Matlab implements least squares method

Two important functions

p=polyfit(x,y,n): The least square method calculates the coefficients of the fitted polynomial. x and y are the fitted data vectors and require the same dimensions, and n is the degree of fitting polynomial. Return the p vector to save the polynomial coefficients, arranged from the highest order to the lowest order. y=polyval(p,x): Calculate the function value of the polynomial. Returns the value of the polynomial at x, where p is the polynomial coefficient, and the elements are sorted in descending power of the polynomial.

Fit the following data

x=[12,9,23,56,43.5,33,41.3,76,63,26];
y=[-4,-3.85,-5.4,-5,-3.5,-1.75,-1.4,-0.5,0.4,-0.85];

Draw discrete points

plot(x,y,'o')

Fit the above curve with a linear function

clear
clc
x=[12,9,23,56,43.5,33,41.3,76,63,26];
y=[-4,-3.85,-5.4,-5,-3.5,-1.75,-1.4,-0.5,0.4,-0.85];
coefficient=polyfit(x,y,1);  %Use a function to fit the curve. If you want to use several functions to fit the curve, set n to that number
y1=polyval(coefficient,x);
plot(x,y,'o',x,y1,'-')

result:

coefficient=[0.04603,-4.34700]

The one-time function obtained is:

y=0.04603x-4.34700

A quadratic function fits the curve, and the results are as follows:

coefficient=[2.29779e-04,0.02709,-4.05811]

Other orders by analogy

The number m of non-repetitive elements in the vector x (with elements as independent variables) and the fitting order k need to satisfy m>=k+1. Simple analysis: k-order fitting needs to determine k+1 unknown parameters (such as The first-order fitting y = ax + b needs to determine the two parameters a and b), so at least k+1 equations are required, so at least k+1 different known number pairs (x, y) are required, because the function x can only correspond to one y, so at least k+1 different x is required.

For example, it is normal to fit the above curve with a 9th-order polynomial.

Fit the above curve with a tenth-order polynomial, and get incorrect results:

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