Hypothesis analysis and principle analysis of linear regression model

Linear regression is a commonly used statistical learning method used to establish a linear relationship between independent variables and dependent variables. This model is based on the least squares method and finds the optimal solution by minimizing the sum of squared errors between the dependent variable and the independent variable. This method is suitable for situations where there is a linear relationship in the data set and can be used to predict and analyze the relationship between the dependent variable and the independent variable.

The mathematical expression of the linear regression model is as follows:

y=beta_0 beta_1x_1 beta_2x_2 … beta_px_p epsilon

Among them, y represents the dependent variable, beta_0 represents the intercept, beta_1,beta_2,…,beta_p represents the coefficient of the independent variable, x_1,x_2,…,x_p represents the independent variable, and epsilon represents the error term.

The goal of the linear regression model is to solve for the optimal coefficients beta_0, beta_1, ..., beta_p by minimizing the sum of squares of the residuals so that the predicted value of the model is consistent with the actual value minimize the error between them. The least squares method is a commonly used method for estimating these coefficients. It determines the value of the coefficient by finding the minimum sum of squared errors.

In linear regression models, we usually use some performance indicators to evaluate the fit of the model, such as mean square error and coefficient of determination. MSE represents the average error between the predicted value and the actual value, and R-squared represents the proportion of variance explained by the model to the total variance.

The advantage of linear regression model is that it is simple and easy to understand, and can be used to explain the relationship between dependent variables and independent variables, but it also has some limitations, such as outliers and nonlinearity The data fit is poor.

In practical applications, when performing linear regression analysis, we will make some assumptions based on the characteristics of the actual problem and data set. These assumptions are usually based on the following aspects:

1. Linear relationship assumption: We assume that there is a linear relationship between the target variable and the independent variable, that is, a straight line can be used to describe the relationship between the two.

2. Independence assumption: We assume that each sample point is independent of each other, that is, the observation values ​​between each sample do not affect each other.

3. Normal distribution assumption: We assume that the error term obeys the normal distribution, that is, the distribution of the residuals conforms to the normal distribution.

4. Homoskedasticity assumption: We assume that the variances of the error terms are the same, that is, the variances of the residuals are stable.

5. Multicollinearity assumption: We assume that there is no high correlation between independent variables, that is, there is no multicollinearity between independent variables.

When performing linear regression analysis, we need to test these assumptions to determine whether they are true. If the assumptions are not met, corresponding data processing or other regression analysis methods need to be selected.

The above is the detailed content of Hypothesis analysis and principle analysis of linear regression model. For more information, please follow other related articles on the PHP Chinese website!