python實現線性迴歸之簡單迴歸
代碼來源：https://github.com/eriklindernoren/ML-From-Scratch
首先定義一個基本的迴歸類，作爲各種迴歸方法的基類：
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
說明：初始化時傳入兩個參數，一個是迭代次數，另一個是學習率。initialize_weights()用於初始化權重。fit()用於訓練。需要注意的是，對於原始的輸入X，需要將其最前面添加一項爲偏置項。predict()用於輸出預測值。
接下來是簡單線性迴歸，繼承上面的基類：
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)
這裏使用兩種方式進行計算。如果規定gradient_descent=True，那麼使用隨機梯度下降算法進行訓練，否則使用標準方程法進行訓練。
最後是使用：
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庫生成線性迴歸數據，然後將其拆分爲訓練集和測試集。
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結果：
Mean squared error: 532.3321383700828