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Multiple linear regression algorithm
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Multiple linear regression algorithm
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Multiple linear regression is a statistical learning method used to explore the linear relationship between a dependent variable (the target variable) and one or more independent variables (eigenvariable). In multiple linear regression, we try to describe the relationship between the independent variable and the dependent variable through a linear model.
Multiple linear regression is a statistical learning method used to explore the linear relationship between the dependent variable (target variable) and one or more independent variables (characteristic variables). In multiple linear regression, we attempt to describe the relationship between the independent and dependent variables through a linear model.
1. Model Display:
多元线性回归模型表示为：[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon ]
1. Model representation:
The multiple linear regression model is represented as: [y=\ beta_1+\ beta_1 x_1+\ beta_2 x_2+...+\ beta_p x_2+\ epsilon]
Where: ( y ) is the dependent variable (target variable) ;( x_1, x_2, ..., x_p ) is the independent variable (eigenvariable;( \beta_0, \beta_1, ..., \beta_p ) are the parameters of the model, corresponding to the coefficients of the intercept term and the independent variable, respectively, and ( \epsilon ) is the error term, which represents the part of the model that cannot be explained.
Among them: (y) is the dependent variable (target variable); (x_1, x_2,..., xp) are independent variables (characteristic variables); (\ beta 0, \ beta 1,..., \ beta p) are the parameters of the model, corresponding to the intercept term and the coefficients of the independent variable, respectively; (\ epsilon) is the error term, representing the parts that the model cannot explain
2. Parameter Estimation:
The parameters of multiple linear regression models can be estimated by minimizing the Residual Sum of Squares (RSS), i.e., fitting the data by least squares. The goal of least squares is to minimize the sum of squares of the residuals between the observed and model predictions.
2. Parameter estimation:
The parameters of a multiple linear regression model can be estimated by minimizing the Residual Sum of Squares (RSS), i.e. fitting the data using the least squares method. The goal of the least squares method is to minimize the sum of squared residuals between observed values and model predictions.
3. Model Evaluation:
After fitting a multiple linear regression model, we need to evaluate the performance of the model. Commonly used evaluation indicators include: Goodness-of-fit (R^2): indicates how well the model interprets the variation in observations, with values ranging from 0 to 1, with closer to 1 indicating a better fit of the model. Mean Squared Error (MSE): Represents the mean of the squared difference between the observed value and the predicted value of the model, which is used to measure the prediction accuracy of the model.
3. Model evaluation:
After fitting the multiple linear regression model, we need to evaluate the performance of the model. The commonly used evaluation indicators include: goodness of fit (R ^ 2): represents the degree to which the model explains the variation in observed values, with values ranging from 0 to 1. The closer to 1, the better the model fits. Mean Squared Error (MSE): Refers to the mean squared difference between observed values and model predictions, used to measure the accuracy of the model's predictions.
4. Feature Selection and Multicollinearity:
When applying multiple linear regression models, it is often necessary to consider feature selection and multicollinearity. Feature selection refers to the selection of independent variables that have a significant impact on the target variable, which can be performed by statistical tests (such as t-tests) or feature importance assessments. Multicollinearity refers to the high correlation between independent variables, which can lead to inaccurate parameter estimation, which can be diagnosed by indicators such as Variance Inflation Factor (VIF).
4. Feature selection and multicollinearity:
When applying multiple linear regression models, it is usually necessary to consider feature selection and multicollinearity issues. Feature selection refers to the selection of independent variables that have a significant impact on the target variable, which can be conducted through statistical tests (such as t-tests) or feature importance assessments. Multicollinearity refers to the high correlation between independent variables, which can lead to inaccurate parameter estimation. It can be diagnosed through indicators such as Variance Inflation Factor (VIF).
5. Applications:
Multiple linear regression is widely used in various fields, including economics, sociology, biology, engineering, and more. It can be used for forecasting, controlling, and optimizing problems in a variety of scenarios, such as stock price prediction, sales volume forecasting, and house price forecasting.
5. Application areas:
Multiple linear regression is widely used in various fields, including economics, sociology, biology, engineering, etc. It can be used for various scenarios such as predicting, controlling, and optimizing problems, such as stock price prediction, sales volume prediction, housing price prediction, etc.
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References: Bilibili, Baidu Encyclopedia, Zhihu, CNKI
Translation: Baidu Translation