1.介紹lookback

# convert an array of values into a dataset matrix
def create_dataset(dataset, look_back=1):
	dataX, dataY = [], []
	for i in range(len(dataset)-look_back-1):
		a = dataset[i:(i+look_back), 0]
		dataX.append(a)
		dataY.append(dataset[i + look_back, 0])
	return numpy.array(dataX), numpy.array(dataY)
 

相當於我們之前寫的shift函式啦

結果這樣（lookback=1）

X        Y
112        118
118        132
132        129
129        121
121        135

當然也可以設定lookback=3

     X1    X2    X3    Y
112    118    132    129
118    132    129    121
132    129    121    135
129    121    135    148
121    135    148    148 

2.原來曲線，預測train曲線，預測test曲線畫在一起

# generate predictions for training
trainPredict = model.predict(trainX)
testPredict = model.predict(testX)

# shift train predictions for plotting
trainPredictPlot = numpy.empty_like(dataset)
trainPredictPlot[:, :] = numpy.nan
trainPredictPlot[look_back:len(trainPredict)+look_back, :] = trainPredict

# shift test predictions for plotting
testPredictPlot = numpy.empty_like(dataset)
testPredictPlot[:, :] = numpy.nan
testPredictPlot[len(trainPredict)+(look_back*2)+1:len(dataset)-1, :] = testPredict

# plot baseline and predictions
plt.plot(dataset)
plt.plot(trainPredictPlot)
plt.plot(testPredictPlot)
plt.show()

效果圖應該是這樣

所有程式碼在一起：

# Multilayer Perceptron to Predict International Airline Passengers (t+1, given t, t-1, t-2)
import numpy
import matplotlib.pyplot as plt
from pandas import read_csv
import math
from keras.models import Sequential
from keras.layers import Dense

# convert an array of values into a dataset matrix
def create_dataset(dataset, look_back=1):
	dataX, dataY = [], []
	for i in range(len(dataset)-look_back-1):
		a = dataset[i:(i+look_back), 0]
		dataX.append(a)
		dataY.append(dataset[i + look_back, 0])
	return numpy.array(dataX), numpy.array(dataY)

# fix random seed for reproducibility
numpy.random.seed(7)
# load the dataset
dataframe = read_csv('international-airline-passengers.csv', usecols=[1], engine='python', skipfooter=3)
dataset = dataframe.values
dataset = dataset.astype('float32')
# split into train and test sets
train_size = int(len(dataset) * 0.67)
test_size = len(dataset) - train_size
train, test = dataset[0:train_size,:], dataset[train_size:len(dataset),:]
# reshape dataset
look_back = 3
trainX, trainY = create_dataset(train, look_back)
testX, testY = create_dataset(test, look_back)
# create and fit Multilayer Perceptron model
model = Sequential()
model.add(Dense(12, input_dim=look_back, activation='relu'))
model.add(Dense(8, activation='relu'))
model.add(Dense(1))
model.compile(loss='mean_squared_error', optimizer='adam')
model.fit(trainX, trainY, epochs=400, batch_size=2, verbose=2)
# Estimate model performance
trainScore = model.evaluate(trainX, trainY, verbose=0)
print('Train Score: %.2f MSE (%.2f RMSE)' % (trainScore, math.sqrt(trainScore)))
testScore = model.evaluate(testX, testY, verbose=0)
print('Test Score: %.2f MSE (%.2f RMSE)' % (testScore, math.sqrt(testScore)))
# generate predictions for training
trainPredict = model.predict(trainX)
testPredict = model.predict(testX)
# shift train predictions for plotting
trainPredictPlot = numpy.empty_like(dataset)
trainPredictPlot[:, :] = numpy.nan
trainPredictPlot[look_back:len(trainPredict)+look_back, :] = trainPredict
# shift test predictions for plotting
testPredictPlot = numpy.empty_like(dataset)
testPredictPlot[:, :] = numpy.nan
testPredictPlot[len(trainPredict)+(look_back*2)+1:len(dataset)-1, :] = testPredict
# plot baseline and predictions
plt.plot(dataset)
plt.plot(trainPredictPlot)
plt.plot(testPredictPlot)
plt.show()

結果如下：

Epoch 395/400
0s - loss: 485.3482
Epoch 396/400
0s - loss: 479.9485
Epoch 397/400
0s - loss: 497.2707
Epoch 398/400
0s - loss: 489.5670
Epoch 399/400
0s - loss: 490.8099
Epoch 400/400
0s - loss: 493.6544
Train Score: 564.03 MSE (23.75 RMSE)
Test Score: 2244.82 MSE (47.38 RMSE)

https://machinelearningmastery.com/time-series-prediction-with-deep-learning-in-python-with-keras/