Detailed explanation of linear regression model in Python

Detailed explanation of linear regression model in Python

Linear regression is a classic statistical model and machine learning algorithm. It is widely used in the fields of prediction and modeling, such as stock market prediction, weather prediction, housing price prediction, etc. As an efficient programming language, Python provides a rich machine learning library, including linear regression models. This article will introduce the linear regression model in Python in detail, including model principles, application scenarios and code implementation.

Linear Regression Principle

The linear regression model is based on the linear relationship between variables. In a univariate linear regression model, we consider a linear relationship between an independent variable and a dependent variable. For example, when we want to predict the selling price of a certain house, we can use the area of ​​the house as the independent variable and the selling price as the dependent variable to build a univariate linear regression model. Assuming that the area of ​​the house is x and the selling price is y, the univariate linear regression model is expressed as:

y = β0 β1x

where, β0 and β1 are the coefficients to be solved, and y is The dependent variable, x is the independent variable.

The multivariable linear regression model needs to consider the linear relationship between multiple independent variables and the dependent variable. Suppose we want to predict the selling price of a house. At this time, we need to consider the impact of multiple independent variables such as the area of ​​the house, the location of the house, and the age of the building on the selling price. At this time, the multivariable linear regression model is expressed as:

y = β0 β1x1 β2x2 β3x3 ... βnxn

where β0 and β1~βn are the coefficients to be solved, and y is the dependent variable , x1~xn are multiple independent variables.

Solution of the linear regression model

The solution of the linear regression model is the process of solving the coefficients β0 and β1~βn. In multivariable linear regression models, the least squares method is usually used to solve for the coefficients.

The least squares method is a statistical method whose basic idea is to minimize the sum of squares of the distances from all data points to the regression line. Therefore, we need to minimize the following loss function:

J(β0, β1,...,βn) = Σ(yi - f(xi))^2

where, yi represents the actual value, and f(xi) represents the predicted value. The loss function J represents the sum of squared errors between all actual values ​​and predicted values.

The solution process of the least squares method is to find the partial derivatives of the loss function for the coefficients β0 and β1~βn respectively, and make the partial derivative equal to 0 to solve for the value of the coefficient. Specifically, the process of minimizing the loss function can be implemented using normal equations or stochastic gradient descent.

Normal equations solve the coefficients by solving equations with a derivative of 0. Specifically, we can use the following formula to solve for the coefficients:

β = (X.TX)^{-1}X.Ty

where X is the independent variable matrix and y is the factor A vector of variables, T represents the transpose of the matrix. Due to the high computational complexity of the inversion, other methods are usually used to solve the coefficients in practical applications.

The stochastic gradient descent method is an iterative solution method that minimizes the loss function by iteratively updating the coefficients. Specifically, we need to select a random sample for calculation in each iteration and then update the coefficients. As the number of iterations increases, the loss function gradually decreases and finally converges to a stable value.

Application scenarios

Linear regression models are widely used in practical applications, mainly in the fields of prediction and modeling. The following are some common application scenarios:

1. House price prediction: Predict the market selling price of a house by considering the linear relationship of multiple independent variables, such as area, location, construction age, etc.

2. Stock market prediction: Predict the rise and fall of stocks by considering the linear relationship of multiple independent variables, such as economic indicators, policy changes, market sentiment, etc.

3. Weather prediction: Predict the weather conditions in the future by considering the linear relationship of multiple independent variables, such as temperature, humidity, rainfall, etc.

Python code implementation

The following is an example of using Python to implement a linear regression model. We use the LinearRegression model from the Scikit-learn library to build a multivariable linear regression model.

First, we need to install the Scikit-learn library:

pip install -U scikit-learn

Then, we can use the following code to build a multivariable linear regression model:

#导入库
import numpy as np
from sklearn.linear_model import LinearRegression

#生成数据
np.random.seed(0)
X = np.random.rand(100, 3) #自变量，100个样本，3个特征
y = 0.5 + np.dot(X, [1.5, -2.0, 1.0]) + np.random.normal(size=100) #因变量，加入随机误差

#训练模型
model = LinearRegression().fit(X, y)

#输出模型系数
print(model.intercept_) #截距
print(model.coef_) #斜率

In the above code, we used three randomly generated independent variables and one dependent variable, then used the LinearRegression model to train the data and output the coefficients of the model. Running the above code can get the following results:

0.49843856268038534
[ 1.48234604 -1.97351656 0.99594992]

Among them, the intercept is 0.4984, and the slopes are 1.482, -1.974, and 0.996 respectively, indicating three Linear relationships between independent variables and between dependent variables.

Conclusion

The linear regression model is a classic machine learning algorithm and has a wide range of application scenarios in practical applications. As an efficient programming language, Python provides a sufficient machine learning library, making it very easy for us to use linear regression models to achieve prediction and modeling tasks. If you are interested in the application of linear regression models, it is recommended to have an in-depth understanding of the theory and code implementation to better apply it to solving practical problems.

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