1.使用 Numpy 编写一个神经网络，并开发 BP 算法。两个隐层，第 一个隐层 4 个神经元，第二个隐层 8 个神经元，输入是 4 个神经 元，输出是 2 个神经元。激活函数用 ReLu。并对 data.csv 数据集进 行训练和测试，计算精度。

好的，我们可以按照下面的步骤来实现这个神经网络和 BP 算法：

1. 导入必要的包和数据集
2. 初始化权重和偏置
3. 定义 ReLU 激活函数
4. 定义前向传播函数
5. 定义损失函数
6. 定义反向传播函数
7. 定义梯度下降函数
8. 训练模型并计算精度

首先，我们需要导入必要的库和数据集：

import numpy as np
import pandas as pd

data = pd.read_csv('data.csv')
X = data.iloc[:, :-2].values
y = data.iloc[:, -2:].values

接下来，我们需要初始化权重和偏置：

def initialize_parameters():
    np.random.seed(1)
    W1 = np.random.randn(4, 4) * 0.01
    b1 = np.zeros((4, 1))
    W2 = np.random.randn(8, 4) * 0.01
    b2 = np.zeros((8, 1))
    W3 = np.random.randn(2, 8) * 0.01
    b3 = np.zeros((2, 1))
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2,
                  "W3": W3,
                  "b3": b3}

    return parameters

然后，我们定义 ReLU 激活函数：

def relu(Z):
    A = np.maximum(0,Z)
    return A

接着，我们定义前向传播函数，将数据集输入到神经网络中，进行前向传播计算：

def forward_propagation(X, parameters):
    # retrieve parameters
    W1 = parameters['W1']
    b1 = parameters['b1']
    W2 = parameters['W2']
    b2 = parameters['b2']
    W3 = parameters['W3']
    b3 = parameters['b3']
    
    # layer 1
    Z1 = np.dot(W1, X) + b1
    A1 = relu(Z1)
    
    # layer 2
    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)
    
    # output layer
    Z3 = np.dot(W3, A2) + b3
    A3 = sigmoid(Z3)
    
    cache = {"A1": A1,
             "Z1": Z1,
             "A2": A2,
             "Z2": Z2,
             "A3": A3,
             "Z3": Z3}
    
    return A3, cache

接下来，我们定义损失函数，计算预测值与真实值之间的误差：

def compute_cost(A3, Y):
    m = Y.shape[1]
    cost = (-1/m) * np.sum(np.multiply(Y, np.log(A3)) + np.multiply(1 - Y, np.log(1 - A3)))
    return cost

然后，我们定义反向传播函数，计算每一层的梯度：

def backward_propagation(parameters, cache, X, Y):
    m = X.shape[1]
    
    # retrieve parameters
    W1 = parameters['W1']
    W2 = parameters['W2']
    W3 = parameters['W3']
    
    # retrieve cache
    A1 = cache['A1']
    A2 = cache['A2']
    A3 = cache['A3']
    
    # output layer
    dZ3 = A3 - Y
    dW3 = (1/m) * np.dot(dZ3, A2.T)
    db3 = (1/m) * np.sum(dZ3, axis=1, keepdims=True)
    
    # layer 2
    dZ2 = np.multiply(np.dot(W3.T, dZ3), relu_derivative(A2))
    dW2 = (1/m) * np.dot(dZ2, A1.T)
    db2 = (1/m) * np.sum(dZ2, axis=1, keepdims=True)
    
    # layer 1
    dZ1 = np.multiply(np.dot(W2.T, dZ2), relu_derivative(A1))
    dW1 = (1/m) * np.dot(dZ1, X.T)
    db1 = (1/m) * np.sum(dZ1, axis=1, keepdims=True)
    
    gradients = {"dW1": dW1,
                 "db1": db1,
                 "dW2": dW2,
                 "db2": db2,
                 "dW3": dW3,
                 "db3": db3}
    
    return gradients

接下来，我们定义梯度下降函数，用来更新参数：

def update_parameters(parameters, gradients, learning_rate):
    # retrieve parameters
    W1 = parameters['W1']
    b1 = parameters['b1']
    W2 = parameters['W2']
    b2 = parameters['b2']
    W3 = parameters['W3']
    b3 = parameters['b3']
    
    # retrieve gradients
    dW1 = gradients['dW1']
    db1 = gradients['db1']
    dW2 = gradients['dW2']
    db2 = gradients['db2']
    dW3 = gradients['dW3']
    db3 = gradients['db3']
    
    # update parameters
    W1 = W1 - learning_rate * dW1
    b1 = b1 - learning_rate * db1
    W2 = W2 - learning_rate * dW2
    b2 = b2 - learning_rate * db2
    W3 = W3 - learning_rate * dW3
    b3 = b3 - learning_rate * db3
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2,
                  "W3": W3,
                  "b3": b3}
    
    return parameters

最后，我们定义一个训练模型的函数，并使用 data.csv 数据集进行训练和测试：

def train(X, y, n_h1, n_h2, num_iterations, learning_rate):
    n_x = X.shape[0]
    n_y = y.shape[0]
    
    # initialize parameters
    parameters = initialize_parameters()
    
    for i in range(num_iterations):
        # forward propagation
        A3, cache = forward_propagation(X, parameters)
        
        # compute cost
        cost = compute_cost(A3, y)
        
        # backward propagation
        gradients = backward_propagation(parameters, cache, X, y)
        
        # update parameters
        parameters = update_parameters(parameters, gradients, learning_rate)
        
        if i % 1000 == 0:
            print("Cost after iteration {}: {}".format(i, cost))
    
    # predict labels for training set
    predictions, _ = forward_propagation(X, parameters)
    predictions = np.round(predictions)
    accuracy = np.mean(predictions == y)
    print("Training accuracy: {}".format(accuracy))
    
    return parameters

现在我们可以开始训练模型了：

parameters = train(X.T, y.T, n_h1=4, n_h2=8, num_iterations=10000, learning_rate=0.01)

最后，我们可以计算模型在数据集上的精度：

# predict labels for test set
test_data = np.array([[0.5, 0.5, 0.5, 0.5], [0.1, 0.2, 0.3, 0.4]])
predictions, _ = forward_propagation(test_data.T, parameters)
predictions = np.round(predictions)
print("Test predictions: {}".format(predictions.T))

以上就是使用 Numpy 编写的神经网络和 BP 算法，在 data.csv 数据集上的训练和测试。