The least squares method is a widely used method for approximating the solutions of systems of linear equations and for finding the best-fitting curve through a set of data points. The main advantages of the least squares method are:

  1. Simplicity: The least squares method is relatively simple to implement and understand, and it can be easily solved using matrix algebra.
  2. Robustness: The least squares method is relatively robust to errors in the data, and it can still produce good results even if the data contains outliers or noise.
  3. Versatility: The least squares method can be used to solve a wide range of problems, including linear and non-linear regression, curve fitting, and signal processing.
  4. Generalization: The least squares method can be used to find the best-fitting solution for any continuous and smooth function, this enables the generalization of the results obtained from the sample data.

However, there are also some limitations of the least squares method:

  1. Linearity: The least squares method assumes that the relationship between the independent and dependent variables is linear. If the relationship is non-linear, the least squares method may not produce accurate results.
  2. Normality: The least squares method assumes that the errors in the data are normally distributed. If the errors are not normally distributed, the least squares method may not produce accurate results.
  3. Outliers: The least squares method is sensitive to outliers in the data, which can have a large impact on the results.
  4. Overfitting: The least squares method can be used to fit a model to a set of data points, but it can also be used to fit a model that is too complex for the data set, which can lead to overfitting.
  5. No uncertainty: The least squares method does not provide an estimate of the uncertainty of the solution, so it can be difficult to know how confident you can be in the results obtained.

In summary, the least squares method is a powerful tool that can be used to find the best-fitting solutions for many types of problems, but it does have some limitations that should be taken into consideration when using it.

Below code runs on an STM32 microcontroller and uses the C programming language. The script first defines the data points x and y and the variables A, AT, ATA, ATb, m, and c that will be used for the least squares solution.

In the main() function, we first initialize the microcontroller, and then we initialize the matrices A, AT and ATA using nested loops. Then we calculate the least squares solution by multiplying the transpose of the matrix A with A and y. Finally, we calculate the slope m and y-intercept c from the solution and print the equation of the line in the form “y = mx + c” using the printf() function.

#include "stm32f4xx.h"

// Define the data points
float x[5] = {1, 2, 3, 4, 5};
float y[5] = {2, 4, 5, 4, 5};

// Variables for the least squares solution
float A[5][2], AT[2][5], ATA[2][2];
float ATb[2], m, c;

int main(void)
{
    // Initialize the microcontroller
    // ...

    // Initialize the matrices
    for (int i = 0; i < 5; i++)
    {
        A[i][0] = x[i];
        A[i][1] = 1;
    }
    for (int i = 0; i < 2; i++)
    {
        for (int j = 0; j < 5; j++)
        {
            AT[i][j] = A[j][i];
        }
    }
    for (int i = 0; i < 2; i++)
    {
        for (int j = 0; j < 2; j++)
        {
            ATA[i][j] = 0;
            for (int k = 0; k < 5; k++)
            {
                ATA[i][j] += AT[i][k] * A[k][j];
            }
        }
    }

    // Solve for the least squares solution
    for (int i = 0; i < 2; i++)
    {
        ATb[i] = 0;
        for (int j = 0; j < 5; j++)
        {
            ATb[i] += AT[i][j] * y[j];
        }
    }
    m = ATb[0] / ATA[0][0];
    c = ATb[1] / ATA[1][1];

    // Print the line equation
    printf("Line equation: y = %fx + %f\n", m, c);

    // ...
    while (1)
    {
        // Do other things
    }
}