[Reading Notes] TPA-LSTM for time series (including Pytorch code implementation)

This article is a summary and reflection after reading the paper. It does not involve the translation of the paper and the interpretation of the model. It is suitable for everyone to exchange ideas after reading the paper. For the translation of the paper, you can check the references. Paper address: https://arxiv.org/abs/1809.04206

1. Summary of the full text

This paper proposes the use of a set of filters to extract time-invariant temporal patterns (CNN) , similar to converting time series data into its "frequency domain". Then, we propose a new attention mechanism to select relevant time series and utilize their frequency domain information for multivariate prediction. This paper applies the proposed model to several real-world tasks and achieves state-of-the-art performance in almost all cases.

2. Research methods

1. A new attention mechanism is proposed, namely Temporal pattern attention (TPA) , where "Temporal pattern" refers to any time-invariant pattern across multiple time steps .
2. In TPA, instead of selecting relevant time steps like a typical attention mechanism, the machine learns to select relevant time series . A convolutional neural network (CNN) is introduced to extract temporal pattern information from each individual variable .

3. Conclusion

This paper takes MTS prediction as the research object and proposes a new temporal pattern attention mechanism , which eliminates the limitations of typical attention mechanisms on such tasks. The attention dimension is allowed to be characterized so that the model learns interdependencies between multiple variables not only within the same time step but also in all previous times and sequences . Experiments on both toy examples and real-world datasets strongly support this idea and show that the proposed model achieves state-of-the-art results.

4. Innovation points

Typical attention mechanisms usually only focus on a few time steps, making it difficult to identify periodic patterns that span multiple time steps. This paper introduces a new concept of attention, where we select relevant variables instead of relevant time steps . This method is simple, versatile and suitable for RNN.

5. Thinking

After simulation, the model indeed has a relatively obvious effect.

6. References

1. TPA attention mechanism (TPA-LSTM)

7. Pytorch implementation⭐

The following code reference: https://github.com/jingw2/demand_forecast , some errors in the original code have been corrected, and some necessary comments have been added for better understanding.

import torch 
from torch import nn
import torch.nn.functional as F 
from torch.optim import Adam

import numpy as np
import math
import os
import random
import matplotlib.pyplot as plt
import pickle
from tqdm import tqdm
import pandas as pd
from sklearn.preprocessing import StandardScaler
from datetime import date
import argparse
from progressbar import *

util(tool function)

def get_data_path():
    folder = os.path.dirname(__file__)
    return os.path.join(folder, "data")

def RSE(ypred, ytrue):
    rse = np.sqrt(np.square(ypred - ytrue).sum()) / \
            np.sqrt(np.square(ytrue - ytrue.mean()).sum())
    return rse

def quantile_loss(ytrue, ypred, qs):
    '''
    Quantile loss version 2
    Args:
    ytrue (batch_size, output_horizon)
    ypred (batch_size, output_horizon, num_quantiles)
    '''
    L = np.zeros_like(ytrue)
    for i, q in enumerate(qs):
        yq = ypred[:, :, i]
        diff = yq - ytrue
        L += np.max(q * diff, (q - 1) * diff)
    return L.mean()

def SMAPE(ytrue, ypred):
    ytrue = np.array(ytrue).ravel()
    ypred = np.array(ypred).ravel() + 1e-4
    mean_y = (ytrue + ypred) / 2.
    return np.mean(np.abs((ytrue - ypred) \
        / mean_y))

def MAPE(ytrue, ypred):
    ytrue = np.array(ytrue).ravel() + 1e-4
    ypred = np.array(ypred).ravel()
    return np.mean(np.abs((ytrue - ypred) \
        / ytrue))

def train_test_split(X, y, train_ratio=0.7):
    '''
    - X (array like): shape (num_samples, num_periods, num_features)
    - y (array like): shape (num_samples, num_periods)
    '''
    num_ts, num_periods, num_features = X.shape
    train_periods = int(num_periods * train_ratio)
    random.seed(2)
    Xtr = X[:, :train_periods, :]
    ytr = y[:, :train_periods]
    Xte = X[:, train_periods:, :]
    yte = y[:, train_periods:]
    return Xtr, ytr, Xte, yte

class StandardScaler:
    
    def fit_transform(self, y):
        self.mean = np.mean(y)
        self.std = np.std(y) + 1e-4
        return (y - self.mean) / self.std
    
    def inverse_transform(self, y):
        return y * self.std + self.mean

    def transform(self, y):
        return (y - self.mean) / self.std

class MaxScaler:

    def fit_transform(self, y):
        self.max = np.max(y)
        return y / self.max
    
    def inverse_transform(self, y):
        return y * self.max

    def transform(self, y):
        return y / self.max


class MeanScaler:
    
    def fit_transform(self, y):
        self.mean = np.mean(y)
        return y / self.mean
    
    def inverse_transform(self, y):
        return y * self.mean

    def transform(self, y):
        return y / self.mean

class LogScaler:

    def fit_transform(self, y):
        return np.log1p(y)
    
    def inverse_transform(self, y):
        return np.expm1(y)

    def transform(self, y):
        return np.log1p(y)


def gaussian_likelihood_loss(z, mu, sigma):
    '''
    Gaussian Liklihood Loss
    Args:
    z (tensor): true observations, shape (num_ts, num_periods)
    mu (tensor): mean, shape (num_ts, num_periods)
    sigma (tensor): standard deviation, shape (num_ts, num_periods)
    likelihood: 
    (2 pi sigma^2)^(-1/2) exp(-(z - mu)^2 / (2 sigma^2))
    log likelihood:
    -1/2 * (log (2 pi) + 2 * log (sigma)) - (z - mu)^2 / (2 sigma^2)
    '''
    negative_likelihood = torch.log(sigma + 1) + (z - mu) ** 2 / (2 * sigma ** 2) + 6
    return negative_likelihood.mean()

def negative_binomial_loss(ytrue, mu, alpha):
    '''
    Negative Binomial Sample
    Args:
    ytrue (array like)
    mu (array like)
    alpha (array like)
    maximuze log l_{nb} = log Gamma(z + 1/alpha) - log Gamma(z + 1) - log Gamma(1 / alpha)
                - 1 / alpha * log (1 + alpha * mu) + z * log (alpha * mu / (1 + alpha * mu))
    minimize loss = - log l_{nb}
    Note: torch.lgamma: log Gamma function
    '''
    batch_size, seq_len = ytrue.size()
    likelihood = torch.lgamma(ytrue + 1. / alpha) - torch.lgamma(ytrue + 1) - torch.lgamma(1. / alpha) \
        - 1. / alpha * torch.log(1 + alpha * mu) \
        + ytrue * torch.log(alpha * mu / (1 + alpha * mu))
    return - likelihood.mean()

def batch_generator(X, y, num_obs_to_train, seq_len, batch_size):
    '''
    Args:
    X (array like): shape (num_samples, train_periods, num_features)
    y (array like): shape (num_samples, train_periods)
    num_obs_to_train (int): 训练的历史窗口长度
    seq_len (int): sequence/encoder/decoder length
    batch_size (int)
    '''
    num_ts, num_periods, _ = X.shape
    if num_ts < batch_size:
        batch_size = num_ts
    t = random.choice(range(num_obs_to_train, num_periods-seq_len)) # 从num_obs_to_train和num_periods-seq_len-1之间随机选一个整数,作为预测点
    batch = random.sample(range(num_ts), batch_size) # 从num_ts条数据中随机选择batch_size条
    X_train_batch = X[batch, t-num_obs_to_train:t, :] # (batch_size, num_obs_to_train, num_features)
    y_train_batch = y[batch, t-num_obs_to_train:t] # (batch_size, num_obs_to_train)
    Xf = X[batch, t:t+seq_len, :] # (batch_size, seq_len, num_features)
    yf = y[batch, t:t+seq_len] # (batch_size, seq_len)
    return X_train_batch, y_train_batch, Xf, yf

Model

class TemporalPatternAttention(nn.Module):

    def __init__(self, filter_size, filter_num, attn_len, attn_size):
        super(TemporalPatternAttention, self).__init__()
        self.filter_size = filter_size # 1
        self.filter_num = filter_num
        self.feat_size = attn_size - self.filter_size + 1 # hidden_size
        self.conv = nn.Conv2d(1, filter_num, (attn_len, filter_size))
        self.linear1 = nn.Linear(attn_size, filter_num)
        self.linear2 = nn.Linear(attn_size + self.filter_num, attn_size)
        self.relu = nn.ReLU()
    
    def forward(self, H, ht): # H:(batch_size, 1, obs_len-1, hidden_size) ht:(batch_size, hidden_size)       
        _, channels, _, attn_size = H.size()

        conv_vecs = self.conv(H) # (batch_size, filter_num, 1, hidden_size)      
        conv_vecs = conv_vecs.view(-1, self.feat_size, self.filter_num) # (batch_size, hidden_size, filter_num)
        conv_vecs = self.relu(conv_vecs) # (batch_size, hidden_size, filter_num)
        
        # score function
        htt = self.linear1(ht) # (batch_size, filter_num) 
        htt = htt.view(-1, self.filter_num, 1) # (batch_size, filter_num, 1)
        s = torch.bmm(conv_vecs, htt) # (batch_size, hidden_size, 1)
        alpha = torch.sigmoid(s) # (batch_size, hidden_size, 1)
        v = torch.bmm(conv_vecs.view(-1,self.filter_num,attn_size), alpha).view(-1, self.filter_num) # (batch_size, filter_num)
        
        concat = torch.cat([ht, v], dim=1) # (batch_size, hidden_size+filter_num)
        new_ht = self.linear2(concat) # (batch_size, hidden_size)
        return new_ht

class TPALSTM(nn.Module):

    def __init__(self, input_size, output_horizon, hidden_size, obs_len, n_layers):
        super(TPALSTM, self).__init__()
        self.hidden = nn.Linear(input_size, hidden_size)
        self.relu = nn.ReLU()
        self.lstm = nn.LSTM(hidden_size, hidden_size, n_layers, \
                    bias=True, batch_first=True) # output (batch_size, obs_len, hidden_size)
        self.hidden_size = hidden_size
        self.filter_num = 16
        self.filter_size = 1
        self.output_horizon = output_horizon
        self.attention = TemporalPatternAttention(self.filter_size, \
            self.filter_num, obs_len-1, hidden_size)
        self.linear = nn.Linear(hidden_size, output_horizon)
        self.n_layers = n_layers

    def forward(self, x):
        batch_size, obs_len, features_size = x.shape #(batch_size, obs_len, features_size)
        xconcat = self.hidden(x) #(batch_size, obs_len, hidden_size)

        H = torch.zeros(batch_size, obs_len-1, self.hidden_size).to(device) #(batch_size, obs_len-1, hidden_size)
        ht = torch.zeros(self.n_layers, batch_size, self.hidden_size).to(device) # (num_layers, batch_size, hidden_size)
        ct = ht.clone()
        for t in range(obs_len):
            xt = xconcat[:, t, :].view(batch_size, 1, -1) #(batch_size, 1, hidden_size)
            out, (ht, ct) = self.lstm(xt, (ht, ct)) # ht size (num_layers, batch_size, hidden_size)
            htt = ht[-1, :, :] # (batch_size, hidden_size)
            if t != obs_len - 1:
                H[:, t, :] = htt
        H = self.relu(H) #(batch_size, obs_len-1, hidden_size)
        
        # reshape hidden states H
        H = H.view(batch_size, 1, obs_len-1, self.hidden_size) #(batch_size, 1, obs_len-1, hidden_size)
        new_ht = self.attention(H, htt) # (batch_size, hidden_size)
        ypred = self.linear(new_ht) # (batch_size, output_horizon)
        return ypred

Load Data

num_epoches = 100
step_per_epoch = 3 #在一个epoch中，从训练集中提取step_per_epoch次训练数据
lr = 1e-3
n_layers = 1
hidden_size = 24
seq_len = 30 #预测的未来窗口长度
num_obs_to_train = 168  #训练的历史窗口长度
num_results_to_sample = 10
show_plot = True
run_test = True
standard_scaler = True
log_scaler = False
mean_scaler = False
max_scaler = False
batch_size = 128

device=torch.device("cuda:0" if torch.cuda.is_available() else "cpu")

# 读取数据
data = pd.read_csv("LD_MT200_hour.csv", parse_dates=["date"])
data["year"] = data["date"].apply(lambda x: x.year)
data["day_of_week"] = data["date"].apply(lambda x: x.dayofweek)
data = data.loc[(data["date"].dt.date >= date(2014, 1, 1)) & (data["date"].dt.date <= date(2014, 3, 1))]
print(data.shape)
plt.figure(figsize=(16, 4)) 
plt.plot(data['MT_200'])
data.head()

# 数据预处理
features = ["hour", "day_of_week"]
# hours = pd.get_dummies(data["hour"])
# dows = pd.get_dummies(data["day_of_week"])
years = data["year"]
hours = data["hour"]
dows = data["day_of_week"]
MT_200 = np.asarray(data["MT_200"]).reshape(-1,1)
yscaler1 = StandardScaler()
MT_200 = yscaler1.fit_transform(MT_200)
X = np.c_[np.asarray(hours),np.asarray(dows),np.asarray(MT_200)] #X:(len,features)
num_features = X.shape[1]
num_periods = len(data)
X = np.asarray(X).reshape((-1, num_periods, num_features))
y = np.asarray(data["MT_200"]).reshape((-1, num_periods))
print("X_shape=",X.shape) # (series_num,len,features_num)
print("y_shape=",y.shape) # (series_num,len)
# X = np.tile(X, (10, 1, 1))
# y = np.tile(y, (10, 1))

输出：
X_shape= (1, 1440, 3)
y_shape= (1, 1440)

def sliding_window(DataSet, width, multi_vector = True): #DataSet has to be as an Array
    if multi_vector: #三维 (num_samples,length,features)
        num_samples,length,features = DataSet.shape
    else: #二维 (num_samples,length)
        DataSet = DataSet[:,:,np.newaxis] #(num_samples,length,1)
        num_samples,length,features = DataSet.shape

    x = DataSet[:,0:width,:] #(num_samples,width,features)
    x = x[np.newaxis,:,:,:] #(1,num_samples,width,features)
    for i in range(length - width):
        i += 1
        tmp = DataSet[:,i:i + width,:]#(num_samples,width,features)
        tmp = tmp[np.newaxis,:,:,:] #(1,num_samples,width,features)
        x = np.concatenate([x,tmp],0) #(i+1,num_samples,width,features)
    return x
    
width = num_obs_to_train + seq_len 
X_data = sliding_window(X, width, multi_vector = True) #(len-width+1,num_samples,width,features)
Y_data = sliding_window(y, width, multi_vector = False) #(len-width+1,num_samples,width,1)
print("x的维度为：",X_data.shape)
print("y的维度为：",Y_data.shape)
# 取其中一类序列
i = 0
X_data = X_data[:,i,:,:]
Y_data = Y_data[:,i,:,0]
print("x的维度为：",X_data.shape)
print("y的维度为：",Y_data.shape)

输出：
x的维度为： (1243, 1, 198, 3)
y的维度为： (1243, 1, 198, 1)
x的维度为： (1243, 198, 3)
y的维度为： (1243, 198)

###### SPLIT TRAIN TEST
from sklearn.model_selection import train_test_split

Xtr, Xte, ytr, yte = train_test_split(X_data, Y_data, 
                                    test_size=0.2, 
                                    random_state=0,
                                    shuffle=False)
print("X_train:{},y_train:{}".format(Xtr.shape,ytr.shape))
print("X_test:{},y_test:{}".format(Xte.shape,yte.shape))

输出：
X_train:(994, 198, 3),y_train:(994, 198)
X_test:(249, 198, 3),y_test:(249, 198)

# 标准化
yscaler = None
if standard_scaler:
    yscaler = StandardScaler()
elif log_scaler:
    yscaler = LogScaler()
elif mean_scaler:
    yscaler = MeanScaler()
if yscaler is not None:
    ytr = yscaler.fit_transform(ytr.reshape(-1,1)).reshape(-1,seq_len+num_obs_to_train)

Xtr=torch.as_tensor(torch.from_numpy(Xtr), dtype=torch.float32)
ytr=torch.as_tensor(torch.from_numpy(ytr),dtype=torch.float32)     
Xte=torch.as_tensor(torch.from_numpy(Xte), dtype=torch.float32)
yte=torch.as_tensor(torch.from_numpy(yte),dtype=torch.float32)

print("X_train:{},y_train:{}".format(Xtr.shape,ytr.shape))
print("X_test:{},y_test:{}".format(Xte.shape,yte.shape))

train_dataset=torch.utils.data.TensorDataset(Xtr,ytr) #训练集dataset
train_Loader=torch.utils.data.DataLoader(train_dataset,batch_size=batch_size)

输出：
X_train:torch.Size([994, 198, 3]),y_train:torch.Size([994, 198])
X_test:torch.Size([249, 198, 3]),y_test:torch.Size([249, 198])

Train

Args:

• X (array like): shape (num_samples, num_periods, num_features)
• y (array like): shape (num_samples, num_periods)
• epochs (int): number of epochs to run
• step_per_epoch (int): steps per epoch to run
• num_obs_to_train (int): The length of the history window for training
• seq_len (int): output horizon
• likelihood (str): what type of likelihood to use, default is gaussian
• num_skus_to_show (int): how many skus to show in test phase
• num_results_to_sample (int): how many samples in test phase as prediction

# 定义模型和优化器
num_ts, num_periods, num_features = X.shape
model = TPALSTM(input_size=Xtr.shape[2], output_horizon=seq_len, hidden_size=32, obs_len=num_obs_to_train, n_layers=1).to(device)
optimizer = Adam(model.parameters(), lr=lr)
random.seed(2)

losses = []
cnt = 0    
    
# training
print("开启训练")
progress = ProgressBar()
for epoch in progress(range(num_epoches)):
#     print("Epoch {} starts...".format(epoch))
    for x,y in train_Loader:
        x = x.to(device) # (batch_size, num_obs_to_train+seq_len, num_features) 
        y = y.to(device) # (batch_size, num_obs_to_train+seq_len)
        Xtrain = x[:,:num_obs_to_train,:].float() # (batch_size, num_obs_to_train, num_features)
        ytrain = y[:,:num_obs_to_train].float() # (batch_size, num_obs_to_train)
        Xf = x[:,-seq_len:,:].float() # (batch_size, seq_len, num_features)
        yf = y[:,-seq_len:].float() # (batch_size, seq_len)             
               
        ypred = model(Xtrain) # ypred:(batch_size, seq_len)
        
        loss = F.mse_loss(ypred, yf)
        
        losses.append(loss.item())
        optimizer.zero_grad()
        loss.backward()
        optimizer.step()
        cnt += 1
        
# 绘制loss
if show_plot:
    plt.plot(range(len(losses)), losses, "k-")
    plt.xlabel("Period")
    plt.ylabel("Loss")
    plt.show()

# test 
print("开启测试")
X_test_sample = Xte[:,:,:].reshape(-1,num_obs_to_train+seq_len,num_features).to(device) # (num_samples, num_obs_to_train+seq_len, num_features)
y_test_sample = yte[:,:].reshape(-1,num_obs_to_train+seq_len).to(device) # (num_samples, num_obs_to_train+seq_len)

X_test = X_test_sample[:,:num_obs_to_train,:] # (num_samples, num_obs_to_train, num_features)
Xf_test = X_test_sample[:, -seq_len:, :] # (num_samples, seq_len, num_features)
y_test = y_test_sample[:, :num_obs_to_train] # (num_samples, num_obs_to_train)
yf_test = y_test_sample[:, -seq_len:] # (num_samples, seq_len)

ypred = model(X_test)
ypred = ypred.cpu().detach().numpy()
if yscaler is not None:
    ypred = yscaler.inverse_transform(ypred.reshape(-1,1)).reshape(-1,seq_len)
# ypred = ypred.ravel()
yf_test = yf_test.cpu().detach().numpy()
loss = np.sqrt(np.sum(np.square(yf_test - ypred)))
print("losses: ", loss)

输出：
开启测试
losses:  11473.168

i = -1
if show_plot: # 序列总长度为：历史窗口长度(num_obs_to_train)+预测长度(seq_len)
    plt.figure(1, figsize=(20, 5))
    plt.plot([k + seq_len + num_obs_to_train - seq_len for k in range(seq_len)], ypred[i,:], "r-") # 绘制50%分位数曲线
    plt.title('Prediction uncertainty')
    yplot = y_test_sample[i,:].cpu() #真实值 (1, seq_len+num_obs_to_train)
    plt.plot(range(len(yplot)), yplot, "k-")
    plt.legend(["P50 forecast", "P10-P90 quantile", "true"], loc="upper left")
    ymin, ymax = plt.ylim()
    plt.vlines(seq_len + num_obs_to_train - seq_len, ymin, ymax, color="blue", linestyles="dashed", linewidth=2)
    plt.ylim(ymin, ymax)
    plt.xlabel("Periods")
    plt.ylabel("Y")
    plt.show()