Methods and prerequisites for implementing linear regression using normal equations

Normal equations are a simple and intuitive method for linear regression. The best-fit straight line is calculated directly through mathematical formulas without using iterative algorithms. This method is particularly suitable for small data sets.

First, let’s review the basic principles of linear regression. Linear regression is a method used to predict the relationship between a dependent variable Y and one or more independent variables X. There is only one independent variable X in simple linear regression, while two or more independent variables are included in multiple linear regression.

In linear regression, we use the least squares method to fit a straight line to minimize the sum of distances from data points to the straight line. The equation of the straight line is:

Y=β0 β1X1 β2X2 … βnXn

The goal of the equation is to find the best intercept and regression coefficient to to best fit the data.

Now, let’s see how to use the normal equation to calculate the optimal β0 to βn. The basic idea of ​​normal equations is that we can get the optimal regression coefficients by solving a system of linear equations.

The form of this system of linear equations is as follows:

(XT X)β=XT Y

Among them, X is the matrix of independent variables, Y is the vector of dependent variables, XT is the transpose of X, and β is the vector of regression coefficients. In this system of equations, we need to solve for β.

Next, we need to convert this system of equations into a form that can be solved. We can accomplish this step by multiplying both sides of the system of equations by the inverse matrix of (XT). In this way, the system of equations becomes normal. The core idea of ​​the equation is to obtain the optimal regression coefficient by solving a system of linear equations. The form of this system of equations is (XT X)β=XT Y, where X is the matrix of independent variables, Y is the vector of dependent variables, XT is the transpose of We can solve for β by multiplying both sides of the system of equations by the inverse matrix of (XT). This method is very simple and easy to understand, and works well for small data sets. However, it should be noted that the computational complexity of the normal equation is O(n^3), so this method may not be suitable when dealing with large data sets.

The advantage of the normal equation is that it can directly calculate the optimal regression coefficient without using an iterative algorithm. In addition, the solution of this method is unique, so there is no problem of multiple local optimal solutions.

However, normal equations also have some disadvantages. First, it requires computing the inverse matrix of (XT X), which may cause numerical stability problems. If the matrix (XT Furthermore, the normal equations with computational complexity of O(n^3) can become very slow when dealing with large data sets, so iterative algorithms may be more suitable for this case.

When using normal equations for linear regression, the following conditions also need to be met:

1. Linear relationship

Normal equations are only applicable to data with linear relationships, that is, the relationship between the dependent variable and the independent variable must be linear. If the data does not satisfy a linear relationship, then the normal equation cannot obtain a good fitting model.

2. No multicollinearity

Multicollinearity refers to the situation where there is a high degree of correlation between independent variables. If multicollinearity is present, the normal equation may not result in an accurate fitting model. In practical applications, multicollinearity can be checked by calculating the correlation coefficients between independent variables.

3. Data independence

The formal equation requires that the data be independent, that is, there is no correlation between the data of each sample. If the data are not independent, then the normal equation may yield a biased model fit.

4. Homogeneity of variances

Homogeneity of variances means that the variance of the dependent variable should remain equal under different values ​​of the independent variables. If the variances are not homogeneous, then the normal equation may result in an inaccurately fitted model. In practical applications, homogeneity of variances can be checked by plotting the residuals.

5. The error obeys the normal distribution

The normal equation requires that the error obeys the normal distribution, that is, the residual error should be random and consistent with Properties of the normal distribution. If the errors are not normally distributed, then the normal equation may result in an inaccurately fitted model.

It should be noted that the above conditions are not independent of each other, and they may affect each other. In practical applications, we need to comprehensively consider these conditions and select an appropriate regression model based on the characteristics of the data. If the data does not meet the conditions of the normal equation, you can consider using other regression methods, such as ridge regression, lasso regression, etc.

In summary, the normal equation is a simple and easy-to-understand linear regression method suitable for small data sets. But when dealing with large data sets, you need to pay attention to the issue of computational complexity and consider using other methods.

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