Polynomial regression example in Python

Polynomial regression is a method commonly used in regression problems. It builds a model by fitting polynomial data to the data, so that the model can predict the target value more accurately. Python provides a wealth of data processing and machine learning libraries that can easily implement polynomial regression models. This article will introduce how to implement polynomial regression in Python and give an example based on polynomial regression.

1. The principle of polynomial regression

The principle of polynomial regression is relatively simple, which is to explain the value of the independent variable through a polynomial function. That is:

$y = b_0 b_1x_1 b_2x_2^2 ... b_nx_n^n$

Where, $y$ is the dependent variable, $b_0, b_1, b_2, ... , b_n $ is the regression coefficient, $x_1, x_2, ... , x_n$ are independent variables. Because polynomial regression can improve the flexibility of the model, it is often used in problems that require a high degree of fitting.

2. Python implements polynomial regression

In Python, polynomial regression can be implemented through the scikit-learn library. The scikit-learn library is a commonly used machine learning library in Python, providing various models and tools to process data and build models.

The following is a simple polynomial regression implementation step:

1. Import the required libraries

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures

The numpy library and matplotlib library are imported here. LinearRegression class and PolynomialFeatures class in sklearn library.

2. Prepare data

# 创建数据
x = np.linspace(-10, 10, num=50).reshape(-1, 1) # 自变量数据
y = np.sin(x) + np.random.randn(50, 1) * 0.2 # 因变量数据

The linspace function in the numpy library is used here to generate 50 equally spaced independent variable data from -10 to 10, and use the sin function Generate dependent variable data. To make the results more realistic, some random noise is also added.

3. Training model

# 使用多项式模型拟合数据
poly_reg = PolynomialFeatures(degree=5) # degree表示多项式的次数
x_poly = poly_reg.fit_transform(x)
lin_reg = LinearRegression()
lin_reg.fit(x_poly, y)

The PolynomialFeatures class is used to convert the independent variable x into a polynomial, and then the LinearRegression class is used to process the polynomial data. Use the fit method to train the model.

4. Visualized results

# 可视化结果
plt.scatter(x, y)
plt.plot(x, lin_reg.predict(poly_reg.fit_transform(x)), color='red')
plt.show()

The matplotlib library is used here to visualize the results. The raw data are displayed via a scatter plot and a polynomial regression curve is plotted on the graph.

3. Example of Polynomial Regression

Consider an example: predicting the driving distance based on the car’s speed and braking time. We use the dataset provided by Udacity to solve this problem. The data set includes the car's speed, braking time and corresponding driving distance.

1. Import the required libraries and data

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures

# 读入数据
data = pd.read_csv('data/car.csv')

Here the car data set stored in a csv file is read.

2. Data preprocessing

# 提取特征和目标
X = data.iloc[:, :-1].values
y = data.iloc[:, -1].values

# 将刹车时间转为2维数据
X = X.reshape(-1, 1)

# 划分训练集和测试集
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0)

The pandas library is used here to read the data into DataFrame format. Then use the iloc method to extract features and targets by column, and use the reshape function to convert the braking time into 2-dimensional data. Finally, use the train_test_split function to divide the data set into a training set and a test set in proportion.

3. Training and testing model

# 训练模型
poly_reg = PolynomialFeatures(degree = 2)
X_poly = poly_reg.fit_transform(X_train)
lin_reg = LinearRegression()
model = lin_reg.fit(X_poly, y_train)

# 可视化结果
plt.scatter(X_train, y_train, color='red')
plt.plot(X_train, lin_reg.predict(poly_reg.fit_transform(X_train)), color='blue')
plt.title('Car distance prediction')
plt.xlabel('Speed + Brake Time')
plt.ylabel('Distance')
plt.show()

# 测试模型
y_pred = model.predict(poly_reg.fit_transform(X_test))

The PolynomialFeatures class is used to convert the data into a quadratic polynomial, and then the LinearRegression class is used to process the polynomial data. Then use the fit method to train the model. Finally, use the predict method to predict the results of the model.

4. Evaluation model

# 计算评估指标
from sklearn.metrics import mean_squared_error, r2_score
rmse = np.sqrt(mean_squared_error(y_test, y_pred))
r2 = r2_score(y_test, y_pred)

print('Root Mean Squared Error: ', rmse)
print('R2 Score: ', r2)

The mean_squared_error function and r2_score function in the sklearn library are used to calculate the evaluation indicators, which are the root mean square error (RMSE) and the coefficient of determination respectively. (R2).

Through the above steps, we can use the polynomial regression model to predict the driving distance of the car.

Summary

This article introduces the principle of polynomial regression and its implementation method in Python. Through a prediction example of automobile data, we can see the advantages of polynomial regression in building models and predicting results. Of course, polynomial regression also has some shortcomings, such as overfitting and other problems. Therefore, in practical applications, it is necessary to select appropriate regression methods and parameters according to the actual situation.

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