python实现将核方法运用到感知器算法中

感知器算法是一种基于线性分类器的简单机器学习算法，它可以用来解决二元分类问题。核方法是一种将低维空间中的数据映射到高维空间中进行分类的方法。将核方法运用到感知器算法中，可以使得感知器算法可以处理非线性可分的数据。

具体实现可以参考以下步骤：

1.引入必要的库和模块

import numpy as np
from sklearn.datasets import make_circles
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler

2.生成数据集

X, y = make_circles(n_samples=1000, factor=.1, noise=.1, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

3.数据预处理

scaler = StandardScaler()
X_train = scaler.fit_transform(X_train)
X_test = scaler.transform(X_test)

4.定义感知器算法

class Perceptron:
    def __init__(self, lr=0.01, n_iters=1000):
        self.lr = lr
        self.n_iters = n_iters
        self.activation_func = self._unit_step_func
        self.weights = None
        self.bias = None

    def fit(self, X, y):
        n_samples, n_features = X.shape

        # init parameters
        self.weights = np.zeros(n_features)
        self.bias = 0

        y_ = np.array([1 if i > 0 else 0 for i in y])

        # gradient descent
        for _ in range(self.n_iters):
            for idx, x_i in enumerate(X):
                linear_output = np.dot(x_i, self.weights) + self.bias
                y_predicted = self.activation_func(linear_output)

                update = self.lr * (y_[idx] - y_predicted)

                self.weights += update * x_i
                self.bias += update

    def predict(self, X):
        linear_output = np.dot(X, self.weights) + self.bias
        y_predicted = self.activation_func(linear_output)
        return y_predicted

    def _unit_step_func(self, x):
        return np.where(x >= 0, 1, 0)

5.定义核函数

def rbf_kernel(X1, X2, gamma):
    """
    RBF kernel

    Parameters
    ----------
    X1 : numpy.ndarray
        Input data
    X2 : numpy.ndarray
        Input data
    gamma : float
        Kernel parameter

    Returns
    -------
    numpy.ndarray
        Kernel matrix
    """
    n_samples_1, n_features_1 = X1.shape
    n_samples_2, n_features_2 = X2.shape

    K = np.zeros((n_samples_1, n_samples_2))

    for i, x_i in enumerate(X1):
        for j, x_j in enumerate(X2):
            diff = x_i - x_j
            K[i, j] = np.exp(-gamma * np.dot(diff, diff))

    return K

6.将核函数应用于感知器算法

class KernelPerceptron:
    def __init__(self, lr=0.01, n_iters=1000, kernel_func=rbf_kernel, gamma=1):
        self.lr = lr
        self.n_iters = n_iters
        self.kernel_func = kernel_func
        self.gamma = gamma
        self.activation_func = self._unit_step_func
        self.alpha = None
        self.bias = None
        self.X = None
        self.y = None

    def fit(self, X, y):
        n_samples, _ = X.shape

        self.alpha = np.zeros(n_samples)
        self.bias = 0
        self.X = X
        self.y = y

        kernel_matrix = self.kernel_func(X, X, self.gamma)

        for _ in range(self.n_iters):
            for i in range(n_samples):
                linear_output = kernel_matrix[i] @ (self.alpha * self.y) + self.bias
                y_predicted = self.activation_func(linear_output)

                update = self.lr * (self.y[i] - y_predicted)

                self.alpha[i] += update

        support_vectors = self.alpha > 1e-5
        self.alpha = self.alpha[support_vectors]
        self.support_vectors = X[support_vectors]
        self.support_vectors_y = y[support_vectors]

        self.bias = np.mean(
            [y_k - self.predict(x_k, get_raw_result=True) for (y_k, x_k) in zip(self.support_vectors_y, self.support_vectors)])

    def predict(self, X, get_raw_result=False):
        kernel_output = self.kernel_func(self.support_vectors, X, self.gamma)
        prediction = np.sign((self.alpha * self.support_vectors_y) @ kernel_output + self.bias)

        if get_raw_result:
            return (self.alpha * self.support_vectors_y) @ kernel_output + self.bias

        return prediction

    def _unit_step_func(self, x):
        return np.where(x >= 0, 1, 0)

7.训练和测试模型

kernel_perceptron = KernelPerceptron(lr=0.01, n_iters=500, kernel_func=rbf_kernel, gamma=1)
kernel_perceptron.fit(X_train, y_train)

y_pred = kernel_perceptron.predict(X_test)

accuracy = np.mean(y_pred == y_test)
print(f"Accuracy: {accuracy}")

这样，就可以将核方法应用到感知器算法中，处理非线性可分的数据。