4.2 多层感知机回归（时间序列）

这篇教程中，我们使用多层感知机来预测时间序列，这是回归问题。

操作步骤

导入所需的包。

import tensorflow as tf
import numpy as np
import pandas as pd
import matplotlib as mpl
import matplotlib.pyplot as plt

导入数据，并进行预处理。我们使用国际航班乘客数据集，由于它不存在于任何现有库中，我们需要先下载它。

ts = pd.read_csv('international-airline-passengers.csv', usecols=[1], header=0).dropna().values.ravel()

之后，我们需要将其转换为结构化数据集。我知道时间序列有很多实用的特征，但是这篇教程中，为了展示 MLP 的强大，我仅仅使用最简单的特征，也就是乘客数的历史值，并根据历史值来预测当前值。为此，我们需要一个窗口大小，也就是几个历史值与当前值有关。

wnd_sz = 5
ds = []
for i in range(0, len(ts) - wnd_sz + 1):
    ds.append(ts[i:i + wnd_sz])
ds = np.asarray(ds)

x_ = ds[:, 0:wnd_sz - 1]
y_ = ds[:, [wnd_sz - 1]]

之后是训练集和测试集的划分。为时间序列划分训练集和测试集的时候，绝对不能打乱，而是应该把前一部分当做训练集，后一部分当做测试集。因为在时间序列中，未来值依赖历史值，而历史值不依赖未来值，这样可以尽可能避免在训练中使用测试集的信息。

train_size = int(len(x_) * 0.7)
x_train = x_[:train_size]
y_train = y_[:train_size]
x_test = x_[train_size:]
y_test  = y_[train_size:]

定义超参数。时间序列很容易过拟合，为了避免过拟合，建议不要将迭代数设置太大。

n_input = wnd_sz - 1
n_hidden1 = 8
n_hidden2 = 8
n_epoch = 10000
lr = 0.05

搭建模型。要注意隐层的激活函数使用了目前暂时最优的 ELU。由于这个是回归问题，并且标签的取值是正数，输出层激活函数最好是 ReLU，不过我这里用了f(x)=x。

x = tf.placeholder(tf.float64, [None, n_input])
y = tf.placeholder(tf.float64, [None, 1])
w_l1 = tf.Variable(np.random.rand(n_input, n_hidden1))
b_l1 = tf.Variable(np.random.rand(1, n_hidden1))
w_l2 = tf.Variable(np.random.rand(n_hidden1, n_hidden2))
b_l2 = tf.Variable(np.random.rand(1, n_hidden2))
w_l3 = tf.Variable(np.random.rand(n_hidden2, 1))
b_l3 = tf.Variable(np.random.rand(1, 1))
z_l1 = x @ w_l1 + b_l1
a_l1 = tf.nn.elu(z_l1)
z_l2 = a_l1 @ w_l2 + b_l2
a_l2 = tf.nn.elu(z_l2)
z_l3 = a_l2 @ w_l3 + b_l3
a_l3 = z_l3

定义 MSE 损失、优化操作、和 R 方度量指标。

loss = tf.reduce_mean((a_l3 - y) ** 2)
op = tf.train.AdamOptimizer(lr).minimize(loss)

y_mean = tf.reduce_mean(y)
r_sqr = 1 - tf.reduce_sum((y - z_l3) ** 2) / tf.reduce_sum((y - y_mean) ** 2)

使用训练集训练模型。

losses = []
r_sqrs = []

with tf.Session() as sess:
    sess.run(tf.global_variables_initializer())
    
    for e in range(n_epoch):
        _, loss_ = sess.run([op, loss], feed_dict={x: x_train, y: y_train})
        losses.append(loss_)

使用测试集计算 R 方。

        r_sqr_ = sess.run(r_sqr, feed_dict={x: x_test, y: y_test})
        r_sqrs.append(r_sqr_)

每一百步打印损失和度量值。

        if e % 100 == 0:
            print(f'epoch: {e}, loss: {loss_}, r_sqr: {r_sqr_}')

得到模型对训练特征和测试特征的预测值。

    y_train_pred = sess.run(a_l3, feed_dict={x: x_train})
    y_test_pred = sess.run(a_l3, feed_dict={x: x_test})

输出：

epoch: 0, loss: 59209399.053257026, r_sqr: -17520.903006130215
epoch: 100, loss: 54125.98862726741, r_sqr: -28.30371839204463
epoch: 200, loss: 48165.48221823986, r_sqr: -25.13646606476775
epoch: 300, loss: 25826.1223418781, r_sqr: -12.89535810028511
epoch: 400, loss: 1596.701326728818, r_sqr: -0.2350739792412242
epoch: 500, loss: 1396.8836047513207, r_sqr: -0.19979831972491247
epoch: 600, loss: 1386.2307618333675, r_sqr: -0.18952804825771152
epoch: 700, loss: 1374.6194509485028, r_sqr: -0.17864308160044695
epoch: 800, loss: 1362.1306530753875, r_sqr: -0.1669310907644168
epoch: 900, loss: 1348.837516403113, r_sqr: -0.15445861855695942
epoch: 1000, loss: 1334.8048545363076, r_sqr: -0.14128485137041857
epoch: 1100, loss: 1320.0909505177317, r_sqr: -0.1274628487494167
epoch: 1200, loss: 1304.7487050247062, r_sqr: -0.11304061847816937
epoch: 1300, loss: 1288.8264056578596, r_sqr: -0.09806183013209635
epoch: 1400, loss: 1272.368278888685, r_sqr: -0.08256632457099822
epoch: 1500, loss: 1255.4149176209135, r_sqr: -0.06659050598517702
epoch: 1600, loss: 1238.0036386736738, r_sqr: -0.05016766677562812
epoch: 1700, loss: 1220.1688168734415, r_sqr: -0.033328289031115066
epoch: 1800, loss: 1201.942219268364, r_sqr: -0.016100344309701198
epoch: 1900, loss: 1183.3533635535823, r_sqr: 0.001490385551745188
epoch: 2000, loss: 1164.4299150007857, r_sqr: 0.019419953232036713
epoch: 2100, loss: 1145.1981380968932, r_sqr: 0.037665840800028216
epoch: 2200, loss: 1125.683416643312, r_sqr: 0.056206491416253
epoch: 2300, loss: 1105.9108537794123, r_sqr: 0.07502076650340628
epoch: 2400, loss: 1085.9059630721106, r_sqr: 0.0940873169623373
epoch: 2500, loss: 1065.6954608224203, r_sqr: 0.11338385761808756
epoch: 2600, loss: 1045.3081603639205, r_sqr: 0.13288634237228225
epoch: 2700, loss: 1024.7759657787278, r_sqr: 0.15256803995690105
epoch: 2800, loss: 1004.1349451931811, r_sqr: 0.172398525823234
epoch: 2900, loss: 983.4264457196117, r_sqr: 0.1923426217757903
epoch: 3000, loss: 962.6981814611527, r_sqr: 0.2123593431286731
epoch: 3100, loss: 942.0051856419402, r_sqr: 0.23240095064001043
epoch: 3200, loss: 921.4104707924716, r_sqr: 0.25241224904623527
epoch: 3300, loss: 900.9855546712687, r_sqr: 0.2723303372744077
epoch: 3400, loss: 880.8157626399897, r_sqr: 0.2920819860142483
epoch: 3500, loss: 860.9797390814788, r_sqr: 0.3115862647284954
epoch: 3600, loss: 841.5508166258288, r_sqr: 0.3307714419429332
epoch: 3700, loss: 822.65464452708, r_sqr: 0.34954174287751905
epoch: 3800, loss: 804.3510636509582, r_sqr: 0.36784090179299245
epoch: 3900, loss: 786.698489755282, r_sqr: 0.385608999078319
epoch: 4000, loss: 769.742814493071, r_sqr: 0.4028012203684326
epoch: 4100, loss: 753.5066274222577, r_sqr: 0.41939554512386545
epoch: 4200, loss: 737.987317032155, r_sqr: 0.435393667355777
epoch: 4300, loss: 723.1589061501688, r_sqr: 0.450820383097843
epoch: 4400, loss: 708.9775872199175, r_sqr: 0.4657183502307326
epoch: 4500, loss: 695.3902622132391, r_sqr: 0.48014080374835966
epoch: 4600, loss: 682.3445164530003, r_sqr: 0.49414259666687754
epoch: 4700, loss: 669.7979767738486, r_sqr: 0.5077712354635172
epoch: 4800, loss: 657.7254031658086, r_sqr: 0.5210596011864659
epoch: 4900, loss: 646.1225385082785, r_sqr: 0.5340213253103482
epoch: 5000, loss: 635.0063312881094, r_sqr: 0.5466495174400207
epoch: 5100, loss: 624.413372450103, r_sqr: 0.5589148722286865
epoch: 5200, loss: 614.3949181844811, r_sqr: 0.5707707632698614
epoch: 5300, loss: 605.0111648523978, r_sqr: 0.5821553829261457
epoch: 5400, loss: 596.3251407668536, r_sqr: 0.5929950345771348
epoch: 5500, loss: 588.394934666788, r_sqr: 0.6032110372771016
epoch: 5600, loss: 581.2665165995329, r_sqr: 0.6127246745768582
epoch: 5700, loss: 574.966974465473, r_sqr: 0.6214638760543536
epoch: 5800, loss: 569.4991301790525, r_sqr: 0.6293697644027529
epoch: 5900, loss: 564.8386101443393, r_sqr: 0.6364025085215417
epoch: 6000, loss: 560.9342529396988, r_sqr: 0.6425455981413335
epoch: 6100, loss: 557.7121333454768, r_sqr: 0.6478080528894717
epoch: 6200, loss: 555.082871816442, r_sqr: 0.6522228049088226
epoch: 6300, loss: 552.9504878905169, r_sqr: 0.655844018250897
epoch: 6400, loss: 551.2218559344465, r_sqr: 0.6587415604098865
epoch: 6500, loss: 549.8130443197065, r_sqr: 0.6609962191562182
epoch: 6600, loss: 548.6541928338876, r_sqr: 0.6626926870925417
epoch: 6700, loss: 547.6907665851817, r_sqr: 0.6639154520465458
epoch: 6800, loss: 546.8824436922644, r_sqr: 0.6647451519232287
epoch: 6900, loss: 546.2005216521292, r_sqr: 0.6652561458635722
epoch: 7000, loss: 545.624816174369, r_sqr: 0.6655151008532998
epoch: 7100, loss: 545.1407754175443, r_sqr: 0.6655803536739032
epoch: 7200, loss: 544.737161207175, r_sqr: 0.6655015768007708
epoch: 7300, loss: 544.4043734455902, r_sqr: 0.6653207475362519
epoch: 7400, loss: 544.1341742901118, r_sqr: 0.6650726389557293
epoch: 7500, loss: 543.9183617756419, r_sqr: 0.6647841774748728
epoch: 7600, loss: 543.7491045438804, r_sqr: 0.6644772215677328
epoch: 7700, loss: 543.6189281327097, r_sqr: 0.664168178900878
epoch: 7800, loss: 543.5208538865398, r_sqr: 0.6638692273468367
epoch: 7900, loss: 543.448546441615, r_sqr: 0.6635888904468247
epoch: 8000, loss: 543.3964235871173, r_sqr: 0.6633326568412115
epoch: 8100, loss: 543.3597144375057, r_sqr: 0.6631036869053042
epoch: 8200, loss: 543.334470694699, r_sqr: 0.6629032437625964
epoch: 8300, loss: 543.3175272915516, r_sqr: 0.6627311295544881
epoch: 8400, loss: 543.3064268470205, r_sqr: 0.6625860646477272
epoch: 8500, loss: 543.2993218444616, r_sqr: 0.6624660124009056
epoch: 8600, loss: 543.2948676666707, r_sqr: 0.6623684566822827
epoch: 8700, loss: 543.2921173712779, r_sqr: 0.66229063392118
epoch: 8800, loss: 543.2904259948474, r_sqr: 0.6622297208034926
epoch: 8900, loss: 543.2893689519885, r_sqr: 0.6621829791500562
epoch: 9000, loss: 543.2886762658412, r_sqr: 0.6621478605931548
epoch: 9100, loss: 543.2881822202421, r_sqr: 0.6621220749401975
epoch: 9200, loss: 543.2877886467712, r_sqr: 0.6621036272136598
epoch: 9300, loss: 543.2874393833906, r_sqr: 0.6620908290593257
epoch: 9400, loss: 543.2871033165793, r_sqr: 0.662082291939509
epoch: 9500, loss: 543.2867636136157, r_sqr: 0.6620769002407012
epoch: 9600, loss: 543.2864111990842, r_sqr: 0.6620737858271186
epoch: 9700, loss: 543.2860410035256, r_sqr: 0.6620722889010859
epoch: 9800, loss: 543.285649892272, r_sqr: 0.6620719225552976
epoch: 9900, loss: 543.2852355789513, r_sqr: 0.662072338159642

绘制时间序列及其预测值。

plt.figure()
plt.plot(ts, label='Original')
y_train_pred = np.concatenate([
    [np.nan] * n_input, 
    y_train_pred.ravel()
])
y_test_pred = np.concatenate([
    [np.nan] * (n_input + train_size),
    y_test_pred.ravel()
])
plt.plot(y_train_pred, label='y_train_pred')
plt.plot(y_test_pred, label='y_test_pred')
plt.legend()
plt.show()

image

绘制训练集上的损失。

plt.figure()
plt.plot(losses)
plt.title('Loss on Training Set')
plt.xlabel('#epoch')
plt.ylabel('MSE')
plt.show()

image

绘制测试集上的 R 方。

plt.figure()
plt.plot(r_sqrs)
plt.title('$R^2$ on Testing Set')
plt.xlabel('#epoch')
plt.ylabel('$R^2$')
plt.show()

image

扩展阅读

• DeepLearningAI 笔记：浅层神经网络
• DeepLearningAI 笔记：深层神经网络
• 如何将时间序列问题转化为监督学习问题