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LSTM时间序列预测

February 27, 2020 • Read: 523 • Deep Learning

关于时间序列预测

你可能经常会遇到这样的问题,给你一个数据集,要你预测下一个时刻的值是多少?如下图所示,这种数据往往并没有规律可言,也不可能用一个简单的n阶模型去拟合。老实说,以前我遇到这种问题都是直接上灰色模型,但是用的多了就感觉会有点问题。其它还有一些模型比方说ARAM、ARIRM我没有试过。这篇文章主要讲解用LSTM如何进行时间序列预测

数据

数据直接放在代码里,省去了下载文件并读取的麻烦。这组数据是首都国际机场1949-01~1960-12这12年来的客流量,一共12*12=144个月。我们选取前9年的数据(75%)作为Train Data,后3年的数据(25%)作为Test Data。本来还需要Val Data的,但由于数据量比较少,而且不用搞得那么麻烦,所以就不做Validation了。由于我观察到的客流量变化周期是一年的,因为我为客流数据加上了年、月这两个维度的标记

def load_data():
    # passengers number of international airline, 1949-01~1960-12 per month
    seq_number = np.array(
        [112., 118., 132., 129., 121., 135., 148., 148., 136., 119., 104.,
         118., 115., 126., 141., 135., 125., 149., 170., 170., 158., 133.,
         114., 140., 145., 150., 178., 163., 172., 178., 199., 199., 184.,
         162., 146., 166., 171., 180., 193., 181., 183., 218., 230., 242.,
         209., 191., 172., 194., 196., 196., 236., 235., 229., 243., 264.,
         272., 237., 211., 180., 201., 204., 188., 235., 227., 234., 264.,
         302., 293., 259., 229., 203., 229., 242., 233., 267., 269., 270.,
         315., 364., 347., 312., 274., 237., 278., 284., 277., 317., 313.,
         318., 374., 413., 405., 355., 306., 271., 306., 315., 301., 356.,
         348., 355., 422., 465., 467., 404., 347., 305., 336., 340., 318.,
         362., 348., 363., 435., 491., 505., 404., 359., 310., 337., 360.,
         342., 406., 396., 420., 472., 548., 559., 463., 407., 362., 405.,
         417., 391., 419., 461., 472., 535., 622., 606., 508., 461., 390.,
         432.], dtype=np.float32)
    # plt.plot(seq_number)
    # plt.ion()
    seq_number = seq_number[:, np.newaxis] # add a new dimension
    
    # 12 years
    seq_year = np.arange(12)
    # 12 month
    seq_month = np.arange(12)
    seq_year_month = np.transpose(
        [np.repeat(seq_year, len(seq_month)),
         np.tile(seq_month, len(seq_year))],
    )
    seq = np.concatenate((seq_number, seq_year_month), axis=1)

    # normalization
    seq = (seq - seq.mean(axis=0)) / seq.std(axis=0)
    return seq

总的数据维度是(144, 3),即144个月的[客流量,年份,月份]这3个维度的数据。并且我对数据进行了归一化处理

模型

我们希望输入前9年的数据,让LSTM预测后3年的客流,那么我们可以先用前9年中每个月的数据训练LSTM,让它根据前几个月预测下一个月的客流。等训练完成后,我们让LSTM根据前9年的数据预测出下一个月的客流,把刚刚输出的预测客流作为输入,迭代求得后3年的客流

请注意,通常情况下Tensor的第一个维度是批次大小batch size,但是PyTorch建议我们输入循环神经网络的时候,Tensor的第一个维度是序列长度seq len,第二个维度才是batch size

对于这个客流数据,seq_len指的是时间序列的长度,这里前9年,共108个月,则seq_len=108input_dim指的是输入的维度,这里输入数据由三个维度构成,则input_dim=3

接下来我们就可以确定LSTM的结构了

lstm = nn.LSTM(input_dim, mid_dim, mid_layers)
# input_dim 指的是LSTM输入Tensor的维度,根据我们的数据已经确定了这个值是3
# mid_dim 指的是LSTM三个门(gaee)的网络宽度,也是LSTM输出Tensor的维度
# mid_layers 指的是LSTM内部各个gate使用的全连接层的数量,一般设为1或2

x = torch.randn(seq_len, batch_size, input_dim)
output = lstm(x)
assert output.size() == (seq_len, batch_size, mid_dim)

mid_layers一般设置为1或者2:理论上足够宽(神经元个数足够多),并且至少存在一层具有任何一种"挤压"性质的激活函数的2层全连接层就能拟合任何的连续函数

为了进行时间序列预测,我们在LSTM后面街上两层全连接层(1层也行),用于改变最终LSTM输出Tensor的维度。我们只需要预测客流量这一个值,因此out_dim=1

fc = nn.Sequential(
    nn.Linear(mid_dim, mid_dim)
    nn.ReLU(),
    nn.Linear(mid_dim, out_dim),
)

x = output_of_LSTM
seq_len, batch_size, mid_dim = x.shape
x = x.view(-1, mid_dim)
output_of_fc = fc(x)
output_of_fc = output_of_fc.view(seq_len, batch_size, out_dim)

训练

同一批次中序列长度不同,需要使用from torch.nn.utils.rnn import pad_sequence

我们只有一组训练数据,即前9年的客流量。我们可以在同一批次中,训练LSTM预测不同月份的客流量。1~t月的输入对应了t+1月的客流量。由于同一批次里面训练序列长度不统一,直接在末尾补0的操作不优雅,所以我们需要借助torch 自带的工具 pad_sequence的协助,具体如下

var_x = torch.tensor(train_x, dtype=torch.float32, device=device)
var_y = torch.tensor(train_y, dtype=torch.float32, device=device)

batch_var_x = list()
batch_var_y = list()

for i in range(batch_size):
    j = train_size - i
    batch_var_x.append(var_x[j:])
    batch_var_y.append(var_y[j:])

from torch.nn.utils.rnn import pad_sequence
batch_var_x = pad_sequence(batch_var_x)
batch_var_y = pad_sequence(batch_var_y)

放入pad_sequence的序列必须从长到短放置,随着反向传播的进行,PyTorch会逐步忽略完成梯度计算的短序列,具体解释请看PyTorch官网

criterion = nn.MSELoss()  # L2_loss

for e in range(epochs):
    out = net(batch_var_x)

    loss = criterion(out, batch_var_y)

    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

评估

就像前面所说的:使用前9年的数据作为输入,预测得到下一个与的客流,并将此预测结果加到输入序列中,从而逐步预测后3年的客流。就像修路一样,走在刚刚铺好的路面往前推进

最终完整代码

import numpy as np
import torch
from torch import nn
import matplotlib.pyplot as plt

def run_train_lstm():
    inp_dim = 3
    out_dim = 1
    mid_dim = 8
    mid_layers = 1
    batch_size = 12 * 4
    mod_dir = '.'

    '''load data'''
    data = load_data()
    data_x = data[:-1, :] # 0~142
    data_y = data[1:, 0] # 1~143
    assert data_x.shape[1] == inp_dim

    train_size = int(len(data_x) * 0.75)

    train_x = data_x[:train_size]
    train_y = data_y[:train_size]
    train_x = train_x.reshape((train_size, inp_dim))
    train_y = train_y.reshape((train_size, out_dim))

    '''build model'''
    device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
    net = RegLSTM(inp_dim, out_dim, mid_dim, mid_layers).to(device)
    criterion = nn.MSELoss()
    optimizer = torch.optim.Adam(net.parameters(), lr=1e-2)

    '''train'''
    var_x = torch.tensor(train_x, dtype=torch.float32).to(device)
    var_y = torch.tensor(train_y, dtype=torch.float32).to(device)

    batch_var_x = list()
    batch_var_y = list()

    for i in range(batch_size):
        j = train_size - i
        batch_var_x.append(var_x[j:])
        batch_var_y.append(var_y[j:])
    
    from torch.nn.utils.rnn import pad_sequence
    batch_var_x = pad_sequence(batch_var_x)
    batch_var_y = pad_sequence(batch_var_y)

    print("Training Start")
    for e in range(500):
        out = net(batch_var_x)
    
        loss = criterion(out, batch_var_y)
    
        optimizer.zero_grad()
        loss.backward()
        optimizer.step()
    
        if e % 64 == 0:
            print('Epoch: {:4}, Loss: {:.5f}'.format(e, loss.item()))
    torch.save(net.state_dict(), '{}/net.pth'.format(mod_dir))
    print("Save in:", '{}/net.pth'.format(mod_dir))

    '''eval'''
    net.load_state_dict(torch.load('{}/net.pth'.format(mod_dir), map_location=lambda storage, loc: storage))
    net = net.eval()

    test_x = data_x.copy()
    test_x[train_size:, 0] = 0
    test_x = test_x[:, np.newaxis, :]
    test_x = torch.tensor(test_x, dtype=torch.float32, device=device)

    eval_size = 1
    zero_ten = torch.zeros((mid_layers, eval_size, mid_dim), dtype=torch.float32, device=device)
    test_y, hc = net.output_y_hc(test_x[:train_size], (zero_ten, zero_ten))
    test_x[train_size + 1, 0, 0] = test_y[-1]
    for i in range(train_size + 1, len(data) - 2):
        test_y, hc = net.output_y_hc(test_x[i:i + 1], hc)
        test_x[i + 1, 0, 0] = test_y[-1]
    pred_y = test_x[1:, 0, 0]
    pred_y = pred_y.cpu().data.numpy()

    diff_y = pred_y[train_size:] - data_y[train_size:-1]
    l1_loss = np.mean(np.abs(diff_y))
    l2_loss = np.mean(diff_y ** 2)
    print("L1: {:.3f}    L2: {:.3f}".format(l1_loss, l2_loss))

    plt.plot(pred_y, 'r', label='pred')
    plt.plot(data_y, 'b', label='real', alpha=0.3)
    plt.plot([train_size, train_size], [-1, 2], color='k', label='train | pred')
    plt.legend(loc='best')
    plt.savefig('lstm_reg.png')


class RegLSTM(nn.Module):
    def __init__(self, inp_dim, out_dim, mid_dim, mid_layers):
        super(RegLSTM, self).__init__()

        self.lstm = nn.LSTM(inp_dim, mid_dim, mid_layers)  # lstm
        self.fc = nn.Sequential(
            nn.Linear(mid_dim, mid_dim),
            nn.ReLU(),
            nn.Linear(mid_dim, out_dim),
        )

    def forward(self, x):
        y = self.lstm(x)[0]  # y, (h, c) = self.lstm(x)

        seq_len, batch_size, hid_dim = y.shape
        y = y.view(-1, hid_dim)
        y = self.fc(y)
        y = y.view(seq_len, batch_size, -1)
        return y

    """
    Examples::
        >>> rnn = nn.LSTM(10, 20, 2)
        >>> input = torch.randn(5, 3, 10)
        >>> h0 = torch.randn(2, 3, 20)
        >>> c0 = torch.randn(2, 3, 20)
        >>> output, (hn, cn) = rnn(input, (h0, c0))
    """

    def output_y_hc(self, x, hc):
        y, hc = self.lstm(x, hc)  # y, (h, c) = self.lstm(x)

        seq_len, batch_size, hid_dim = y.shape
        y = y.view(-1, hid_dim)
        y = self.fc(y)
        y = y.view(seq_len, batch_size, -1)
        return y, hc

def load_data():
    # passengers number of international airline, 1949-01~1960-12 per month
    seq_number = np.array(
        [112., 118., 132., 129., 121., 135., 148., 148., 136., 119., 104.,
         118., 115., 126., 141., 135., 125., 149., 170., 170., 158., 133.,
         114., 140., 145., 150., 178., 163., 172., 178., 199., 199., 184.,
         162., 146., 166., 171., 180., 193., 181., 183., 218., 230., 242.,
         209., 191., 172., 194., 196., 196., 236., 235., 229., 243., 264.,
         272., 237., 211., 180., 201., 204., 188., 235., 227., 234., 264.,
         302., 293., 259., 229., 203., 229., 242., 233., 267., 269., 270.,
         315., 364., 347., 312., 274., 237., 278., 284., 277., 317., 313.,
         318., 374., 413., 405., 355., 306., 271., 306., 315., 301., 356.,
         348., 355., 422., 465., 467., 404., 347., 305., 336., 340., 318.,
         362., 348., 363., 435., 491., 505., 404., 359., 310., 337., 360.,
         342., 406., 396., 420., 472., 548., 559., 463., 407., 362., 405.,
         417., 391., 419., 461., 472., 535., 622., 606., 508., 461., 390.,
         432.], dtype=np.float32)
    # plt.plot(seq_number)
    # plt.ion()
    seq_number = seq_number[:, np.newaxis] # add a new dimension
    
    # 12 years
    seq_year = np.arange(12)
    # 12 month
    seq_month = np.arange(12)
    seq_year_month = np.transpose(
        [np.repeat(seq_year, len(seq_month)),
         np.tile(seq_month, len(seq_year))],
    )
    seq = np.concatenate((seq_number, seq_year_month), axis=1)

    # normalization
    seq = (seq - seq.mean(axis=0)) / seq.std(axis=0)
    return seq


if __name__ == '__main__':
    torch.manual_seed(1)
    run_train_lstm()

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