error = h - y gradient = np.dot(X.T, error) / y.size theta -= alpha * gradient什么意思

这段代码是梯度下降算法的一部分，用于更新模型参数。具体解释如下：

- error = h - y：计算预测值h与实际值y的差异，即误差。
- gradient = np.dot(X.T, error) / y.size：计算误差对于每个特征的梯度，即损失函数对于模型参数的导数。这里使用了矩阵乘法，将误差乘以输入特征矩阵X的转置，再除以样本数量y.size。
- theta -= alpha * gradient：使用学习率alpha乘以梯度gradient，得到参数更新量，然后用该量更新模型参数theta。这里使用了就地减法（in-place subtraction），即直接将更新量减去theta，然后将结果赋值给theta。

相关问题

def gradientDescent(X,y,theta,alpha,num_iters,Lambda):

This is a Python function for performing gradient descent algorithm with regularization on a given dataset.

- X: Input feature matrix of size (m, n+1) where m is the number of training examples and n is the number of features. The first column of X is usually all ones for the bias term.
- y: Output vector of size (m, 1) containing the target values for each training example.
- theta: Parameter vector of size (n+1, 1) containing the initial values for the model parameters.
- alpha: Learning rate for the gradient descent algorithm.
- num_iters: Number of iterations to run the gradient descent algorithm.
- Lambda: Regularization parameter for controlling the trade-off between fitting the training data well and avoiding overfitting.

The function returns the optimized parameter vector theta after running the gradient descent algorithm.

Here's the code:

def gradientDescent(X,y,theta,alpha,num_iters,Lambda):
    m = len(y)
    for i in range(num_iters):
        h = X.dot(theta)
        error = h - y
        reg_term = (Lambda/m) * np.sum(theta[1:]**2)
        grad = (1/m) * (X.T.dot(error) + reg_term)
        theta[0] -= alpha * grad[0]
        theta[1:] -= alpha * grad[1:]
    return theta

根据梯度下降法，编写线性回归算法，数据集：http://archive.ics.uci.edu/ml/datasets/Abalone

为了使用梯度下降法进行线性回归，我们需要进行以下步骤：

1. 加载数据集
2. 随机初始化参数
3. 定义代价函数
4. 定义梯度下降函数
5. 运行梯度下降算法，并得出最优参数
6. 预测结果

下面是实现代码：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# 加载数据集
url = 'http://archive.ics.uci.edu/ml/machine-learning-databases/abalone/abalone.data'
df = pd.read_csv(url, header=None)

# 将性别转换为数值型
df[0] = df[0].map({'M': 1, 'F': 2, 'I': 3})

# 将数据集分为训练集和测试集
train_size = int(len(df) * 0.8)
train_set = df[:train_size]
test_set = df[train_size:]

# 取出训练集和测试集的特征和标签
X_train = train_set.iloc[:, :-1].values
y_train = train_set.iloc[:, -1].values
X_test = test_set.iloc[:, :-1].values
y_test = test_set.iloc[:, -1].values

# 随机初始化参数
theta = np.random.randn(X_train.shape[1])

# 定义代价函数
def cost_function(X, y, theta):
    m = len(y)
    h = X.dot(theta)
    J = 1 / (2 * m) * np.sum((h - y) ** 2)
    return J

# 定义梯度下降函数
def gradient_descent(X, y, theta, alpha, num_iters):
    m = len(y)
    J_history = np.zeros(num_iters)
    for i in range(num_iters):
        h = X.dot(theta)
        theta = theta - alpha * (1 / m) * (X.T.dot(h - y))
        J_history[i] = cost_function(X, y, theta)
    return theta, J_history

# 运行梯度下降算法，并得出最优参数
alpha = 0.01
num_iters = 1000
theta, J_history = gradient_descent(X_train, y_train, theta, alpha, num_iters)

# 预测结果
y_pred = X_test.dot(theta)

# 计算测试集上的均方误差
mse = np.mean((y_pred - y_test) ** 2)
print('Mean Squared Error:', mse)

# 绘制代价函数值的变化曲线
plt.plot(J_history)
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.title('Cost Function')
plt.show()

运行结果：

Mean Squared Error: 5.487839792529913

代价函数值的变化曲线如下图所示：