Overview

Least squares

The least squares method is a method of finding a relational expression by minimizing the sum of the squares of the errors in the processing of measured values with errors.

About the program

The program created this time is a process that lists the data x`` y prepared in advance, finds the matrix S`` T from the data, and derives the coefficient ʻa. Also, the degree of the polynomial you want to find is set as m. There may already be a library that can easily find the least squares method, but this time we will not use such a library, only the array manipulation library numpy`, and change the amount of data. We have created a flexible process.

program

`LeastSquares.py`


# -*- coding: utf-8 -*-
#!/usr/bin/env python3

import numpy as np

X_data = [10, 20, 50, 70, 100]
Y_data = [9.3, 9.8, 10.9, 11.9, 13.1]
m = 1

def ST_list(power):
    S, T = 0, 0
    for i, j in zip(X_data,Y_data):
        S += i**power
        T += i**power * j
    return S, T

def A_matrix(n_size, len):
    A = []
    for k in range(n_size):
        A.append(S_list[k+len-m*2-1:k+len-m*2-1+n_size])
    return A

S_list = []
T_list = []
for n in range(len(X_data)):
    S , T = ST_list(n)
    S_list.insert(0, S)
    T_list.insert(0, T)

A = np.array(A_matrix(m+1, len(S_list)))
b = np.array(T_list[-1*m-1::]).reshape(m+1, 1)
Ainv = np.linalg.inv(A)
x = np.dot(Ainv, b)

print("S ="); print(A); print()
print("T ="); print(b); print()
print("a ="); print(x)

Execution result

Find the coefficient a of the first-order polynomial

m = 1

S =
[[17900   250]
 [  250     5]]

T =
[[2977.]
 [  55.]]

a =
[[0.04203704]
 [8.89814815]]

Find the coefficient a of the quadratic polynomial

m = 2

S =
[[130430000   1477000     17900]
 [  1477000     17900       250]
 [    17900       250         5]]

T =
[[2.2141e+05]
 [2.9770e+03]
 [5.5000e+01]]

a =
[[1.22729504e-05]
 [4.07142857e-02]
 [8.92034855e+00]]

Summary

In this data, the coefficients ʻafor the first-order polynomial and the second-order polynomial were obtained. If you increase the amount of data, you can also perform cubic and quaternary orders, but since they are affected by floating point numbers like quadratic polynomials, you need to useround or format` to make the values easier to see. That's right.

At the end

That's all for this article. Sorry for the hard-to-see program. I feel like there is a better way, especially when it comes to array manipulation. If you have any advice, please leave it in the comments section. Thank you for staying with us until the end.

Find the coefficients of the least squares polynomial