python实现线性回归之简单回归

python实现线性回归之简单回归

代码来源：https://github.com/eriklindernoren/ML-From-Scratch

首先定义一个基本的回归类，作为各种回归方法的基类：

class Regression(object):

""" Base regression model. Models the relationship between a scalar dependent variable y and the independent 
variables X. 
Parameters:
-----------
n_iterations: float
    The number of training iterations the algorithm will tune the weights for.
learning_rate: float
    The step length that will be used when updating the weights.
"""
def __init__(self, n_iterations, learning_rate):
    self.n_iterations = n_iterations
    self.learning_rate = learning_rate

def initialize_wights(self, n_features):
    """ Initialize weights randomly [-1/N, 1/N] """
    limit = 1 / math.sqrt(n_features)
    self.w = np.random.uniform(-limit, limit, (n_features, ))

def fit(self, X, y):
    # Insert constant ones for bias weights
    X = np.insert(X, 0, 1, axis=1)
    self.training_errors = []
    self.initialize_weights(n_features=X.shape[1])

    # Do gradient descent for n_iterations
    for i in range(self.n_iterations):
        y_pred = X.dot(self.w)
        # Calculate l2 loss
        mse = np.mean(0.5 * (y - y_pred)**2 + self.regularization(self.w))
        self.training_errors.append(mse)
        # Gradient of l2 loss w.r.t w
        grad_w = -(y - y_pred).dot(X) + self.regularization.grad(self.w)
        # Update the weights
        self.w -= self.learning_rate * grad_w

def predict(self, X):
    # Insert constant ones for bias weights
    X = np.insert(X, 0, 1, axis=1)
    y_pred = X.dot(self.w)
    return y_pred

说明：初始化时传入两个参数，一个是迭代次数，另一个是学习率。initialize_weights()用于初始化权重。fit()用于训练。需要注意的是，对于原始的输入X，需要将其最前面添加一项为偏置项。predict()用于输出预测值。

接下来是简单线性回归，继承上面的基类：

class LinearRegression(Regression):

"""Linear model.
Parameters:
-----------
n_iterations: float
    The number of training iterations the algorithm will tune the weights for.
learning_rate: float
    The step length that will be used when updating the weights.
gradient_descent: boolean
    True or false depending if gradient descent should be used when training. If 
    false then we use batch optimization by least squares.
"""
def __init__(self, n_iterations=100, learning_rate=0.001, gradient_descent=True):
    self.gradient_descent = gradient_descent
    # No regularization
    self.regularization = lambda x: 0
    self.regularization.grad = lambda x: 0
    super(LinearRegression, self).__init__(n_iterations=n_iterations,
                                        learning_rate=learning_rate)
def fit(self, X, y):
    # If not gradient descent => Least squares approximation of w
    if not self.gradient_descent:
        # Insert constant ones for bias weights
        X = np.insert(X, 0, 1, axis=1)
        # Calculate weights by least squares (using Moore-Penrose pseudoinverse)
        U, S, V = np.linalg.svd(X.T.dot(X))
        S = np.diag(S)
        X_sq_reg_inv = V.dot(np.linalg.pinv(S)).dot(U.T)
        self.w = X_sq_reg_inv.dot(X.T).dot(y)
    else:
        super(LinearRegression, self).fit(X, y)

这里使用两种方式进行计算。如果规定gradient_descent=True，那么使用随机梯度下降算法进行训练，否则使用标准方程法进行训练。

最后是使用：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import make_regression
import sys
sys.path.append("/content/drive/My Drive/learn/ML-From-Scratch/")

from mlfromscratch.utils import train_test_split, polynomial_features
from mlfromscratch.utils import mean_squared_error, Plot
from mlfromscratch.supervised_learning import LinearRegression

def main():

X, y = make_regression(n_samples=100, n_features=1, noise=20)

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.4)

n_samples, n_features = np.shape(X)

model = LinearRegression(n_iterations=100)

model.fit(X_train, y_train)

# Training error plot
n = len(model.training_errors)
training, = plt.plot(range(n), model.training_errors, label="Training Error")
plt.legend(handles=[training])
plt.title("Error Plot")
plt.ylabel('Mean Squared Error')
plt.xlabel('Iterations')
plt.savefig("test1.png")
plt.show()

y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print ("Mean squared error: %s" % (mse))

y_pred_line = model.predict(X)

# Color map
cmap = plt.get_cmap('viridis')

# Plot the results
m1 = plt.scatter(366 * X_train, y_train, color=cmap(0.9), s=10)
m2 = plt.scatter(366 * X_test, y_test, color=cmap(0.5), s=10)
plt.plot(366 * X, y_pred_line, color='black', linewidth=2, label="Prediction")
plt.suptitle("Linear Regression")
plt.title("MSE: %.2f" % mse, fontsize=10)
plt.xlabel('Day')
plt.ylabel('Temperature in Celcius')
plt.legend((m1, m2), ("Training data", "Test data"), loc='lower right')
plt.savefig("test2.png")
plt.show()

if name == "__main__":

main()

利用sklearn库生成线性回归数据，然后将其拆分为训练集和测试集。

utils下的mean_squared_error()：

def mean_squared_error(y_true, y_pred):

""" Returns the mean squared error between y_true and y_pred """
mse = np.mean(np.power(y_true - y_pred, 2))
return mse

结果：

Mean squared error: 532.3321383700828

原文地址https://www.cnblogs.com/xiximayou/p/12802118.html