612 lines
20 KiB
Markdown
612 lines
20 KiB
Markdown
# 药店单品销售预测系统 - 模型架构与设计
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## 1. 模型概述
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药店单品销售预测系统采用了多种先进的深度学习模型，包括Transformer、矩阵LSTM(mLSTM)和Kolmogorov-Arnold网络(KAN)。这些模型针对时间序列预测任务进行了专门优化，能够有效捕捉销售数据中的各种模式，如季节性、趋势性和突发性变化。
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## 2. 数据预处理流程
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### 2.1 特征工程
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系统使用以下特征进行预测：
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- **销售量**: 目标变量，也作为历史特征输入
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- **价格**: 产品价格
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- **星期几**: 0-6表示周一至周日
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- **月份**: 1-12表示一年中的月份
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- **节假日标志**: 是否为节假日(0或1)
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- **周末标志**: 是否为周末(0或1)
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- **促销标志**: 是否有促销活动(0或1)
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- **气温**: 当日温度
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### 2.2 数据标准化
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所有特征使用MinMaxScaler进行标准化，将值映射到[0,1]区间：
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```python
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scaler_X = MinMaxScaler(feature_range=(0, 1))
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scaler_y = MinMaxScaler(feature_range=(0, 1))
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X_scaled = scaler_X.fit_transform(X)
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y_scaled = scaler_y.fit_transform(y)
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```
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### 2.3 时间序列数据集创建
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系统将原始数据转换为滑动窗口格式的时间序列数据：
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```python
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def create_dataset(X, y, look_back, future_days):
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"""
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创建时间序列数据集
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参数:
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X: 输入特征
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y: 目标变量
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look_back: 使用过去多少天的数据进行预测
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future_days: 预测未来多少天
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返回:
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X_out: 形状为 (samples, look_back, features)
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y_out: 形状为 (samples, future_days)
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"""
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X_out, y_out = [], []
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for i in range(len(X) - look_back - future_days + 1):
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X_out.append(X[i:(i + look_back)])
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y_out.append(y[i + look_back:i + look_back + future_days, 0])
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return np.array(X_out), np.array(y_out)
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```
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## 3. Transformer模型
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### 3.1 架构设计
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Transformer模型基于"Attention is All You Need"论文提出的架构，但针对时间序列预测进行了优化。
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```
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输出序列
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↑
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线性层+激活
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↑
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Transformer解码器
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↑
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Transformer编码器
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↑
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位置编码+嵌入
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↑
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输入序列
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```
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### 3.2 核心组件
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#### 3.2.1 位置编码
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位置编码使模型能够感知序列中的位置信息：
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```python
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def positional_encoding(seq_len, d_model):
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positions = np.arange(seq_len)[:, np.newaxis]
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angles = np.arange(d_model)[np.newaxis, :] / np.power(10000, 2 * (np.arange(d_model)[np.newaxis, :] // 2) / d_model)
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# 应用sin函数到偶数位置
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sines = np.sin(positions * angles[:, 0::2])
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# 应用cos函数到奇数位置
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cosines = np.cos(positions * angles[:, 1::2])
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# 合并sin和cos
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pos_encoding = np.zeros((seq_len, d_model))
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pos_encoding[:, 0::2] = sines
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pos_encoding[:, 1::2] = cosines
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return torch.FloatTensor(pos_encoding)
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```
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#### 3.2.2 多头自注意力机制
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多头自注意力机制允许模型同时关注序列中的不同位置：
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```python
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class MultiHeadAttention(nn.Module):
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def __init__(self, d_model, num_heads):
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super().__init__()
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self.d_model = d_model
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self.num_heads = num_heads
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self.head_dim = d_model // num_heads
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self.query = nn.Linear(d_model, d_model)
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self.key = nn.Linear(d_model, d_model)
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self.value = nn.Linear(d_model, d_model)
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self.fc_out = nn.Linear(d_model, d_model)
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def forward(self, query, key, value, mask=None):
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batch_size = query.shape[0]
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# 线性变换
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Q = self.query(query)
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K = self.key(key)
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V = self.value(value)
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# 分割成多头
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Q = Q.view(batch_size, -1, self.num_heads, self.head_dim).permute(0, 2, 1, 3)
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K = K.view(batch_size, -1, self.num_heads, self.head_dim).permute(0, 2, 1, 3)
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V = V.view(batch_size, -1, self.num_heads, self.head_dim).permute(0, 2, 1, 3)
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# 计算注意力得分
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energy = torch.matmul(Q, K.permute(0, 1, 3, 2)) / (self.head_dim ** 0.5)
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# 应用mask
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if mask is not None:
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energy = energy.masked_fill(mask == 0, -1e10)
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# 应用softmax
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attention = torch.softmax(energy, dim=-1)
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# 应用注意力
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out = torch.matmul(attention, V)
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out = out.permute(0, 2, 1, 3).contiguous()
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out = out.view(batch_size, -1, self.d_model)
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out = self.fc_out(out)
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return out
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```
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#### 3.2.3 前馈网络
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每个Transformer块中的前馈网络包含两个线性变换，中间有ReLU激活函数：
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```python
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class PositionwiseFeedforward(nn.Module):
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def __init__(self, d_model, d_ff):
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super().__init__()
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self.fc1 = nn.Linear(d_model, d_ff)
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self.fc2 = nn.Linear(d_ff, d_model)
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def forward(self, x):
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return self.fc2(F.relu(self.fc1(x)))
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```
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### 3.3 完整的Transformer模型
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```python
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class TimeSeriesTransformer(nn.Module):
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def __init__(self, num_features, embed_dim, num_heads, dense_dim, dropout_rate, num_blocks, output_sequence_length):
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super().__init__()
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self.embedding = nn.Linear(num_features, embed_dim)
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self.pos_encoding = positional_encoding(look_back, embed_dim)
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self.encoder_layers = nn.ModuleList([
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TransformerEncoderLayer(embed_dim, num_heads, dense_dim, dropout_rate)
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for _ in range(num_blocks)
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])
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self.decoder_layers = nn.ModuleList([
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TransformerDecoderLayer(embed_dim, num_heads, dense_dim, dropout_rate)
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for _ in range(num_blocks)
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])
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self.output_layer = nn.Linear(embed_dim, 1)
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self.output_sequence_length = output_sequence_length
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def forward(self, x):
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# x shape: [batch, seq_len, features]
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batch_size = x.shape[0]
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# 嵌入输入序列
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x = self.embedding(x) # [batch, seq_len, embed_dim]
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# 添加位置编码
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x = x + self.pos_encoding.to(x.device)
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# 编码器
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for layer in self.encoder_layers:
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x = layer(x)
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# 解码器初始输入
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decoder_input = x[:, -1:, :] # 使用编码器的最后一个时间步作为初始输入
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outputs = []
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# 自回归解码
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for _ in range(self.output_sequence_length):
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for layer in self.decoder_layers:
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decoder_input = layer(decoder_input, x)
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output = self.output_layer(decoder_input)
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outputs.append(output)
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# 更新解码器输入
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decoder_input = torch.cat([decoder_input, output.unsqueeze(-1)], dim=1)
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decoder_input = decoder_input[:, -1:, :]
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# 合并所有预测结果
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outputs = torch.cat(outputs, dim=1) # [batch, output_seq_len, 1]
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return outputs.squeeze(-1) # [batch, output_seq_len]
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```
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## 4. 矩阵LSTM模型 (mLSTM)
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### 4.1 标准LSTM限制
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标准LSTM使用标量门控单元和隐藏状态，这限制了其表达能力，特别是在处理具有复杂依赖关系的数据时。
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### 4.2 mLSTM创新点
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矩阵LSTM (mLSTM) 使用矩阵代替标量来表示门控单元和隐藏状态，从而提高了模型的表达能力。主要创新点：
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- 使用矩阵运算代替标量运算
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- 隐藏状态表示为矩阵而非向量
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- 使用矩阵乘法而非向量点积
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### 4.3 mLSTM单元实现
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```python
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class MLSTMCell(nn.Module):
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def __init__(self, input_size, hidden_size, matrix_size):
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super().__init__()
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self.input_size = input_size
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self.hidden_size = hidden_size
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self.matrix_size = matrix_size
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# 输入门参数
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self.W_xi = nn.Parameter(torch.Tensor(input_size, hidden_size * matrix_size))
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self.W_hi = nn.Parameter(torch.Tensor(hidden_size * matrix_size, hidden_size * matrix_size))
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self.b_i = nn.Parameter(torch.Tensor(hidden_size * matrix_size))
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# 遗忘门参数
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self.W_xf = nn.Parameter(torch.Tensor(input_size, hidden_size * matrix_size))
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self.W_hf = nn.Parameter(torch.Tensor(hidden_size * matrix_size, hidden_size * matrix_size))
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self.b_f = nn.Parameter(torch.Tensor(hidden_size * matrix_size))
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# 单元状态参数
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self.W_xc = nn.Parameter(torch.Tensor(input_size, hidden_size * matrix_size))
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self.W_hc = nn.Parameter(torch.Tensor(hidden_size * matrix_size, hidden_size * matrix_size))
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self.b_c = nn.Parameter(torch.Tensor(hidden_size * matrix_size))
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# 输出门参数
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self.W_xo = nn.Parameter(torch.Tensor(input_size, hidden_size * matrix_size))
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self.W_ho = nn.Parameter(torch.Tensor(hidden_size * matrix_size, hidden_size * matrix_size))
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self.b_o = nn.Parameter(torch.Tensor(hidden_size * matrix_size))
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self.init_weights()
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def init_weights(self):
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for p in self.parameters():
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if p.data.ndimension() >= 2:
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nn.init.xavier_uniform_(p.data)
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else:
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nn.init.zeros_(p.data)
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def forward(self, x, hidden):
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h, c = hidden
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# 将隐藏状态矩阵展平为向量
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h_flat = h.view(h.size(0), -1)
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# 计算门控值
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i = torch.sigmoid(x @ self.W_xi + h_flat @ self.W_hi + self.b_i)
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f = torch.sigmoid(x @ self.W_xf + h_flat @ self.W_hf + self.b_f)
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o = torch.sigmoid(x @ self.W_xo + h_flat @ self.W_ho + self.b_o)
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# 计算候选单元状态
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c_tilde = torch.tanh(x @ self.W_xc + h_flat @ self.W_hc + self.b_c)
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# 更新单元状态
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c_flat = f * c.view(c.size(0), -1) + i * c_tilde
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c_new = c_flat.view(c.size(0), self.hidden_size, self.matrix_size)
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# 计算新的隐藏状态
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h_new = o * torch.tanh(c_flat)
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h_new = h_new.view(h.size(0), self.hidden_size, self.matrix_size)
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return h_new, c_new
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```
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### 4.4 mLSTM结合Transformer的混合模型
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系统中的mLSTM模型与Transformer结合，形成强大的混合架构：
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```python
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class MLSTMTransformer(nn.Module):
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def __init__(self, num_features, hidden_size, mlstm_layers, embed_dim, dense_dim, num_heads, dropout_rate, num_blocks, output_sequence_length):
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super().__init__()
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self.matrix_size = 4 # 矩阵维度
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self.mlstm = nn.ModuleList([
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MLSTMLayer(num_features if i == 0 else hidden_size * self.matrix_size,
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hidden_size,
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self.matrix_size)
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for i in range(mlstm_layers)
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])
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# Transformer部分
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self.embedding = nn.Linear(hidden_size * self.matrix_size, embed_dim)
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self.transformer_blocks = nn.ModuleList([
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TransformerBlock(embed_dim, num_heads, dense_dim, dropout_rate)
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for _ in range(num_blocks)
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])
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self.output_layer = nn.Linear(embed_dim, 1)
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self.output_sequence_length = output_sequence_length
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self.dropout = nn.Dropout(dropout_rate)
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def forward(self, x):
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batch_size, seq_len, _ = x.shape
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# mLSTM处理
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h_t = torch.zeros(batch_size, self.hidden_size, self.matrix_size).to(x.device)
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c_t = torch.zeros(batch_size, self.hidden_size, self.matrix_size).to(x.device)
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outputs = []
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for t in range(seq_len):
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x_t = x[:, t, :]
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for layer in self.mlstm:
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h_t, c_t = layer(x_t, (h_t, c_t))
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x_t = h_t.view(batch_size, -1)
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outputs.append(h_t.view(batch_size, 1, -1))
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# 合并所有时间步的输出
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mlstm_output = torch.cat(outputs, dim=1)
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# Transformer处理
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transformer_input = self.embedding(mlstm_output.view(batch_size, seq_len, -1))
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transformer_input = self.dropout(transformer_input)
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for block in self.transformer_blocks:
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transformer_input = block(transformer_input)
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# 输出层
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decoder_input = transformer_input[:, -1:, :]
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predictions = []
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# 自回归生成预测序列
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for _ in range(self.output_sequence_length):
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for block in self.transformer_blocks:
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decoder_input = block(decoder_input)
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pred = self.output_layer(decoder_input)
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predictions.append(pred)
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# 更新解码器输入
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embedding = self.embedding(pred.repeat(1, 1, self.hidden_size * self.matrix_size))
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decoder_input = embedding
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# 合并所有预测
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predictions = torch.cat(predictions, dim=1)
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return predictions.squeeze(-1)
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```
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## 5. Kolmogorov-Arnold网络 (KAN)
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### 5.1 理论基础
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KAN基于柯尔莫哥洛夫-阿诺尔德表示定理，该定理表明任何连续多变量函数都可以表示为单变量函数的有限复合。
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### 5.2 KAN架构
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KAN使用B样条基函数作为激活函数，通过学习这些基函数的组合来近似任意复杂的函数：
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```python
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class KANLayer(nn.Module):
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def __init__(self, in_features, out_features, grid_size=10, spline_order=3):
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super().__init__()
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self.in_features = in_features
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self.out_features = out_features
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self.grid_size = grid_size
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self.spline_order = spline_order
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# 网格点
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self.grid = nn.Parameter(torch.linspace(-1, 1, grid_size))
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# 控制点权重
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self.weights = nn.Parameter(torch.Tensor(in_features, out_features, grid_size))
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# 初始化
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nn.init.xavier_uniform_(self.weights)
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def b_spline(self, x, idx, order):
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"""计算B样条基函数"""
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if order == 0:
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return ((x >= self.grid[idx]) & (x < self.grid[idx+1])).float()
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# 递归计算
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w1 = (x - self.grid[idx]) / (self.grid[idx+order] - self.grid[idx] + 1e-7)
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w2 = (self.grid[idx+order+1] - x) / (self.grid[idx+order+1] - self.grid[idx+1] + 1e-7)
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return w1 * self.b_spline(x, idx, order-1) + w2 * self.b_spline(x, idx+1, order-1)
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def forward(self, x):
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batch_size = x.shape[0]
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out = torch.zeros(batch_size, self.out_features).to(x.device)
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# 对每个输入特征计算B样条
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for i in range(self.in_features):
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for j in range(self.grid_size - self.spline_order - 1):
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# 计算样条基函数
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basis = self.b_spline(x[:, i].unsqueeze(1), j, self.spline_order)
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# 应用权重
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out += basis * self.weights[i, :, j].unsqueeze(0)
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return out
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```
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### 5.3 KAN预测模型
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```python
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class KANForecaster(nn.Module):
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def __init__(self, num_features, hidden_sizes=[64, 32], grid_size=10, spline_order=3, output_sequence_length=7):
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super().__init__()
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self.output_sequence_length = output_sequence_length
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# 输入层
|
||
self.input_layer = nn.Linear(num_features * look_back, hidden_sizes[0])
|
||
|
||
# KAN层
|
||
self.kan_layers = nn.ModuleList()
|
||
for i in range(len(hidden_sizes) - 1):
|
||
self.kan_layers.append(KANLayer(hidden_sizes[i], hidden_sizes[i+1], grid_size, spline_order))
|
||
|
||
# 输出层
|
||
self.output_layer = nn.Linear(hidden_sizes[-1], output_sequence_length)
|
||
|
||
def forward(self, x):
|
||
batch_size = x.shape[0]
|
||
|
||
# 展平输入序列
|
||
x = x.view(batch_size, -1)
|
||
|
||
# 输入层
|
||
x = F.relu(self.input_layer(x))
|
||
|
||
# KAN层
|
||
for layer in self.kan_layers:
|
||
x = layer(x)
|
||
|
||
# 输出层
|
||
output = self.output_layer(x)
|
||
|
||
return output
|
||
```
|
||
|
||
## 6. 模型比较
|
||
|
||
### 6.1 各模型优势
|
||
|
||
| 模型 | 优势 | 适用场景 |
|
||
|------|------|----------|
|
||
| Transformer | 并行计算能力强，可以捕捉长期依赖关系 | 数据量大，有明显的季节性模式 |
|
||
| mLSTM | 增强的记忆能力，能处理复杂序列关系 | 数据有复杂的时序依赖，存在不规则波动 |
|
||
| KAN | 可解释性强，能自适应学习复杂非线性关系 | 数据具有复杂的非线性特征，样本量相对较小 |
|
||
|
||
### 6.2 性能对比
|
||
|
||
根据实验结果，不同模型在不同类型的药品销售预测中表现各异：
|
||
|
||
- **季节性强的药品**：Transformer通常表现最佳
|
||
- **突发性销售的药品**：mLSTM能更好地捕捉突变点
|
||
- **非线性关系明显的药品**：KAN往往有更高的预测精度
|
||
|
||
## 7. 模型评估指标
|
||
|
||
系统使用多种指标评估模型性能：
|
||
|
||
- **MSE** (均方误差): 预测值与实际值差的平方的平均值
|
||
- **RMSE** (均方根误差): MSE的平方根，与原始数据单位相同
|
||
- **MAE** (平均绝对误差): 预测值与实际值绝对差的平均值
|
||
- **R²** (决定系数): 模型解释的方差比例，范围通常为0-1
|
||
- **MAPE** (平均绝对百分比误差): 预测值与实际值绝对差的百分比平均值
|
||
|
||
```python
|
||
def evaluate_model(y_true, y_pred):
|
||
"""
|
||
评估模型性能
|
||
|
||
参数:
|
||
y_true: 真实值
|
||
y_pred: 预测值
|
||
|
||
返回:
|
||
包含各评估指标的字典
|
||
"""
|
||
mse = mean_squared_error(y_true, y_pred)
|
||
rmse = np.sqrt(mse)
|
||
mae = mean_absolute_error(y_true, y_pred)
|
||
r2 = r2_score(y_true, y_pred)
|
||
|
||
# 计算MAPE，避免除零错误
|
||
mask = y_true != 0
|
||
mape = np.mean(np.abs((y_true[mask] - y_pred[mask]) / y_true[mask])) * 100
|
||
|
||
return {
|
||
'mse': mse,
|
||
'rmse': rmse,
|
||
'mae': mae,
|
||
'r2': r2,
|
||
'mape': mape
|
||
}
|
||
```
|
||
|
||
## 8. 模型持久化
|
||
|
||
系统使用PyTorch的序列化机制保存和加载模型：
|
||
|
||
```python
|
||
def save_model(model, product_id, model_type, metrics):
|
||
"""保存模型和相关信息"""
|
||
timestamp = datetime.now().strftime("%Y%m%d%H%M%S")
|
||
version = timestamp
|
||
|
||
# 创建保存目录
|
||
save_dir = f"predictions/{model_type}/{product_id}"
|
||
os.makedirs(save_dir, exist_ok=True)
|
||
|
||
# 保存模型
|
||
model_path = f"{save_dir}/{version}.pt"
|
||
torch.save({
|
||
'model_state_dict': model.state_dict(),
|
||
'metrics': metrics,
|
||
'created_at': timestamp,
|
||
'product_id': product_id,
|
||
'model_type': model_type,
|
||
'version': version
|
||
}, model_path)
|
||
|
||
return model_path
|
||
|
||
def load_model(product_id, model_type, version='latest'):
|
||
"""加载模型"""
|
||
model_dir = f"predictions/{model_type}/{product_id}"
|
||
|
||
if not os.path.exists(model_dir):
|
||
return None
|
||
|
||
if version == 'latest':
|
||
# 获取最新版本
|
||
model_files = glob.glob(f"{model_dir}/*.pt")
|
||
if not model_files:
|
||
return None
|
||
model_path = max(model_files, key=os.path.getctime)
|
||
else:
|
||
model_path = f"{model_dir}/{version}.pt"
|
||
if not os.path.exists(model_path):
|
||
return None
|
||
|
||
# 加载模型
|
||
checkpoint = torch.load(model_path)
|
||
|
||
# 根据模型类型创建相应的模型实例
|
||
if model_type == 'transformer':
|
||
model = TimeSeriesTransformer(...)
|
||
elif model_type == 'mlstm':
|
||
model = MLSTMTransformer(...)
|
||
elif model_type == 'kan':
|
||
model = KANForecaster(...)
|
||
else:
|
||
return None
|
||
|
||
model.load_state_dict(checkpoint['model_state_dict'])
|
||
model.eval()
|
||
|
||
return model, checkpoint
|
||
```
|
||
|
||
## 9. 未来模型优化方向
|
||
|
||
### 9.1 模型改进
|
||
|
||
- **注意力机制优化**: 引入多尺度注意力，更好地捕捉不同时间粒度的模式
|
||
- **集成学习**: 结合多个模型的预测结果，提高整体预测精度
|
||
- **贝叶斯优化**: 自动调整超参数，找到最优模型配置
|
||
|
||
### 9.2 特征工程增强
|
||
|
||
- **时间分解**: 将时间序列分解为趋势、季节性和残差组件
|
||
- **外部特征整合**: 引入更多外部因素，如疫情指数、网络搜索热度等
|
||
- **特征自动选择**: 使用特征重要性评估，自动选择最相关的特征
|
||
|
||
### 9.3 可解释性增强
|
||
|
||
- **注意力可视化**: 展示模型在预测时关注的历史数据点
|
||
- **局部可解释性**: 使用SHAP或LIME等技术解释单个预测
|
||
- **规则提取**: 从训练好的模型中提取简单的决策规则 |