LSTM: How to deal with nonstationarity when predicting a time series

Question

I want to do one-step-ahead predictions for time series with LSTM. To understand the algorithm, I built myself a toy example: A simple autocorrelated process.

def my_process(n, p, drift=0, displacement=0):
    x = np.zeros(n)

    for i in range(1, n):
        x[i] = drift * i + p * x[i-1] + (1-p) * np.random.randn()
    return x + displacement

Then I built an LSTM model in Keras, following this example. I simulated processes with high autocorrelation p=0.99 of length n=10000, trained the neural network on the first 80% of it and let it do one-step-ahead predictions for the remaning 20%.

If I set drift=0, displacement=0, everything works fine:

Then I set drift=0, displacement=10 and things went pear-shaped (notice the different scale on the y-axis):

This is not terribly surprising: LSTMs should be fed with normalized data! So I normalized the data by rescaling it to the interval $[-1, 1]$. Phew, things are fine again:

Then I set drift=0.00001, displacement=10, normalized the data again and ran the algorithm on it. This does not look good:

Apparently the LSTM cannot deal with a drift. What to do? (Yes, in this toy example I could simply subtract the drift; but for real-world time series, this is much harder). Maybe I could run my LSTM on the difference $X_{t} - X_{t-1}$ instead of the original time series $X_t$. This will remove any constant drift from the time series. But running the LSTM on the differenced time series does not work at all:

My question: Why does my algorithm break down when I use it on the differenced time series? What is a good way to deal with drifts in time series?

Here is the full code for my model:

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(42)

from keras.layers.core import Dense, Activation, Dropout
from keras.layers.recurrent import LSTM
from keras.models import Sequential


# The LSTM model
my_model = Sequential()

my_model.add(LSTM(input_shape=(1, 1), units=50, return_sequences=True))
my_model.add(Dropout(0.2))

my_model.add(LSTM(units=100, return_sequences=False))
my_model.add(Dropout(0.2))

my_model.add(Dense(units=1))
my_model.add(Activation('linear'))

my_model.compile(loss='mse', optimizer='rmsprop')


def my_prediction(x, model, normalize=False, difference=False):
    # Plot the process x
    plt.figure(figsize=(15, 7))
    plt.subplot(121)
    plt.plot(x)
    plt.title('Original data')

    n = len(x)
    thrs = int(0.8 * n)    # Train-test split
    # Save starting values for test set to reverse differencing
    x_test_0 = x[thrs + 1]
    # Save minimum and maximum on test set to reverse normalization
    x_min = min(x[:thrs])  
    x_max = max(x[:thrs])

    if difference:
        x = np.diff(x)   # Take difference to remove drift
    if normalize:
        x = (2*x - x_min - x_max) / (x_max - x_min)   # Normalize to [-1, 1]

    # Split into train and test set. The model will be trained on one-step-ahead predictions.
    x_train, y_train, x_test, y_test = x[0:(thrs-1)], x[1:thrs], x[thrs:(n-1)], x[(thrs+1):n]

    x_train, x_test = x_train.reshape(-1, 1, 1), x_test.reshape(-1, 1, 1)
    y_train, y_test = y_train.reshape(-1, 1), y_test.reshape(-1, 1)

    # Fit the model
    model.fit(x_train, y_train, batch_size=200, epochs=10, validation_split=0.05, verbose=0)

    # Predict the test set
    y_pred = model.predict(x_test)

    # Reverse differencing and normalization
    if normalize:
        y_pred = ((x_max - x_min) * y_pred + x_max + x_min) / 2
        y_test = ((x_max - x_min) * y_test + x_max + x_min) / 2  
    if difference:
        y_pred = x_test_0 + np.cumsum(y_pred)
        y_test = x_test_0 + np.cumsum(y_test)

    # Plot estimation
    plt.subplot(122)
    plt.plot(y_pred[-100:], label='One-step-ahead-predictions')
    plt.plot(y_test[-100:], label='Actual data')
    plt.title('Prediction on test set')
    plt.legend()
    plt.show()

# Make plots
x = my_process(10000, 0.99, drift=0, displacement=0)
my_prediction(x, my_model, normalize=False, difference=False)

x = my_process(10000, 0.99, drift=0, displacement=10)
my_prediction(x, my_model, normalize=False, difference=False)

x = my_process(10000, 0.99, drift=0, displacement=10)
my_prediction(x, my_model, normalize=True, difference=False)

x = my_process(10000, 0.99, drift=0.00001, displacement=10)
my_prediction(x, my_model, normalize=True, difference=False)

x = my_process(10000, 0.99, drift=0.00001, displacement=10)
my_prediction(x, my_model, normalize=True, difference=True)

Answer 1

Looking again at your autocorrelated process:

    def my_process(n, p, drift=0, displacement=0):
        x = np.zeros(n)

        for i in range(1, n):
            x[i] = drift * i + p * x[i-1] + (1-p) * np.random.randn()
    return x + displacement

It looks like things are breaking down when the value of displacement is high. This makes sense, as you say, because LSTMs need normalized data.

The drift parameter is a bit different. When a small amount of drift is included, since p is large, the amount of drift is similar to the amount of random noise being added via np.random.randn().

In the plots for drift=0.00001, displacement=10, it looks like the predictions would be fine except for the y-shift. Because of this, I think the root of the problem is still in the displacement parameter, not the drift parameter. Differencing, as has been done, will not help with the displacement parameter; instead, it corrects for drift.

I can't tell from your code, but it looks like perhaps the displacement was not accounted for in model.predict. That's my best guess.

Answer 2

When you choose x_min and x_max, you are choosing it from 1:threshold alone. Since your series is monotonically increasing (well almost..), the testing values are all values > 1. This, the LSTM model has never seen during training at all.

Is that why you are seeing what you are seeing?

Can you try the same with x_min and x_max coming from the whole dataset instead?