### BP神经网络用于数值预测 BP（反向传播）神经网络是一种多层前馈人工神经网络，能够通过监督学习调整权重来最小化误差。为了使用 Python 实现 BP 神经网络并进行数值预测，可以采用如下方法： #### 数据准备 数据集应当被划分为训练集和测试集。训练集用来拟合模型参数；而测试集则用于评估模型性能。 ```python from sklearn.model_selection import train_test_split # 假设 data 和 labels 已定义好 X_train, X_test, y_train, y_test = train_test_split(data, labels, test_size=0.2, random_state=42) ``` #### 构建BP神经网络结构 构建一个简单的三层BP神经网络：输入层、隐藏层以及输出层。这里利用 `numpy` 库来进行矩阵运算[^1]。 ```python import numpy as np class NeuralNetwork: def __init__(self, input_nodes, hidden_nodes, output_nodes, learning_rate): self.input_nodes = input_nodes self.hidden_nodes = hidden_nodes self.output_nodes = output_nodes # 初始化权重矩阵 self.weights_input_to_hidden = np.random.normal(0.0, self.hidden_nodes ** -0.5, (self.hidden_nodes, self.input_nodes)) self.weights_hidden_to_output = np.random.normal(0.0, self.output_nodes ** -0.5, (self.output_nodes, self.hidden_nodes)) self.lr = learning_rate def activation_function(self, z): return 1 / (1 + np.exp(-z)) # sigmoid 函数作为激活函数 def feedforward(self, features): '''前向传播''' hidden_inputs = np.dot(features, self.weights_input_to_hidden.T) hidden_outputs = self.activation_function(hidden_inputs) final_inputs = np.dot(hidden_outputs, self.weights_hidden_to_output.T) final_outputs = final_inputs # 对于回归问题不需要再经过激活函数处理 return final_outputs ``` #### 训练过程 在每次迭代过程中更新权值以减小损失函数的值。对于回归任务来说，通常会选用均方差(MSE)作为目标函数[^2]。 ```python def train(self, features, targets): n_records = features.shape[0] delta_weights_i_h = np.zeros_like(self.weights_input_to_hidden) delta_weights_h_o = np.zeros_like(self.weights_hidden_to_output) for X, y in zip(features, targets): #### 正向计算 ### final_outputs = self.feedforward(X) ### 反向传播 ### error = y - final_outputs hidden_error = np.dot(error, self.weights_hidden_to_output) hidden_grad = hidden_error * (hidden_outputs * (1 - hidden_outputs)) delta_weights_i_h += hidden_grad[:, None].dot([features]) delta_weights_h_o += error[:, None].dot([hidden_outputs]) # 更新权重 self.weights_hidden_to_output += self.lr * delta_weights_h_o / n_records self.weights_input_to_hidden += self.lr * delta_weights_i_h / n_records NeuralNetwork.train = train ``` #### 预测阶段 完成上述准备工作之后就可以调用 `feedforward()` 方法对新样本执行预测操作了[^3]。 ```python nn = NeuralNetwork(input_nodes=len(X_train.columns), hidden_nodes=8, output_nodes=1, learning_rate=0.01) for e in range(epochs): nn.train(X_train.values, y_train.values.reshape((-1, 1))) predictions = [] for i in range(len(X_test)): prediction = nn.feedforward(X_test.iloc[i].values).item() predictions.append(prediction) ```