Note for course TF 4: Sequences, TS, Prediction

Anh-Thi Dinh
  • Sequence models: focus on time series (there are others) — stock, weather,...
  • At the end, we wanna model sunspot actitivity cycles which is important to NASA and other space agencies.
  • Using RNN on time series data.

Sequences and prediction

Time Series

📙 Notebook: introduction to time series. + explaining video. → How to create synthetic time series data + plot them.
  • Time series is everywhere: stock prices, weather focasts, historical trends (Moore's law),...
  • Univariate TS and Miltivariate TS.
  • Type of things can we do with ML over TS:
    • Any thing has a time factor can be analysed using TS.
    • Predicting a forecasting (eg. birth & death in Japan -> predict future for retirement, immigration, impacts...).
    • Imputation: project back into the past.
    • Fill holes in the data.
    • Nomalies detecction (website attacks).
    • Spot patterns (eg. speed recognition).
  • Common patterns in TS:
    • Trend: a specific direcion that they're moving in.
    • Seasonality: patterns repeat at predictable intervals (eg. active users for a website).
    • Combinition of both trend and seasonality.
    • Stationary TS
    • Autocorrelated TS: a time series is linearly related to a lagged version of itself.. There is no trend, no seasonality.
    • Multiple auto correlation.
    • May be trend + seasonality + autorrelation + noise.
    • Non-stationary TS
      In this case, we base just on the later data to predict the future (not on the whole data).

Train / Validation / Test

  • Fixed partitioning (this course focuses on) = splitting TS data into training period, validation period and test period.
    • If TS is seasonal, we want each period contains the whole number of seasons.
  • We can split + train + test to get a model and then re-train with the data containing also the test period so that the model is optimized! In that case, the test set comes from the future.
  • Roll-forward partitioning: we start with a short training period and we gradually increase it (1 day at a time or 1 week at a time). At each iteration, we train the model on training period, use it to focast the following day/week in the validation period. = Fixed partitioning in a number of times!


For evaluating models:
1errors = forecasts - actual
3# Mean squared error (square to get rid of negative values)
4# Eg. Used if large errors are potentially dangerous
5mse = np.square(errors).mean()
6# Get back to the same scale to error
7rmse = np.sqrt(mse)
9# Mean absolute error (his favorite)
10# this doesn't penalize large errs as much as mse does,
11# used if loss is proportional to the size of err
12mae = np.abs(errors).mean()
14# Mean abs percentage err
15# idea of the size of err compared to the values
16mape = np.abs(errors / x_valid).mean()
1# MAE with TF
2keras.metrics.mean_absolute_error(x_valid, naive_forecast).numpy()

Moving average and differencing

📙 Notebook: Forecasting. + explaining video.
Moving average: a simple forecasting method. Calculate the average of blue lines within a fixed "averaging windows".
  • This can eliminate noises and doesn't anticipate trend or seasonality.
  • Depend on the "averaging window", it can give worse result than naive forecast.
Take the average on each yellow window. MAE=7.14 (optimal is 4).
1def moving_average_forecast(series, window_size):
2    """Forecasts the mean of the last few values.
3        If window_size=1, then this is equivalent to naive forecast"""
4    forecast = []
5    for time in range(len(series) - window_size):
6    forecast.append(series[time:time + window_size].mean())
7    return np.array(forecast)
Differencing: remove the trend and seasonality from the TS. We study on the differences between points and their previous neighbor in period.
Left image: we find the differencing of original values, then we find the average (orange line). Right image: restore the trend and seasonality. MAE=5.8 (optimal is 4).
Above method still get the noises (because we add the differencing to the previous noise). If we remove past noise using moving average on that.
Smoothing both past and present values. MAE=4.5 (optimal is 4).
Keep in mind before using Deep Learning, sometimes simple approaches just work fine!

Deep NN for Time Series

Preparing features and labels

  • We need to split our TS data into features and labels so that we can use them in ML algos.
  • In this case: features=#values in TS, label=next_value.
    • Feature: window size and train to predict next value.
    • Ex: 30 days of values as features and next value as label.
    • Overtime, train ML to match 30 features to match a single label.
1def windowed_dataset(series, window_size, batch_size, shuffle_buffer):
2    dataset =
3    dataset = dataset.window(window_size + 1, shift=1, drop_remainder=True)
4    dataset = dataset.flat_map(lambda window: window.batch(window_size + 1))
5    dataset = dataset.shuffle(shuffle_buffer).map(lambda window: (window[:-1], window[-1]))
6    dataset = dataset.batch(batch_size).prefetch(1)
7    return dataset

Sequence bias

Sequence bias is when the order of things can impact the selection of things. It's ok to shuffle!

Feeding windowed datasets into NN

1# Simple linear regression (1 layer NN)
2dataset = windowed_dataset(x_train, window_size, batch_size, shuffle_buffer_size)
3l0 = tf.keras.layers.Dense(1, input_shape=[window_size])
4model = tf.keras.models.Sequential([l0])
5model.compile(loss="mse", optimizer=tf.keras.optimizers.SGD(lr=1e-6, momentum=0.9)),epochs=100,verbose=0)
7print("Layer weights {}".format(l0.get_weights()))
9forecast = []
11for time in range(len(series) - window_size):
12    forecast.append(model.predict(series[time:time + window_size][np.newaxis]))
13    # np.newaxis: reshape X to input dimension that used by the model
15forecast = forecast[split_time-window_size:]
16results = np.array(forecast)[:, 0, 0]
📙 Notebook: DNN with TS + video explains it.
1# A way to choose an optimal learning rate
2lr_schedule = tf.keras.callbacks.LearningRateScheduler(
3    lambda epoch: 1e-8 * 10**(epoch / 20))
4optimizer = tf.keras.optimizers.SGD(lr=1e-8, momentum=0.9)
5model.compile(loss="mse", optimizer=optimizer)
6history =, epochs=100, callbacks=[lr_schedule], verbose=0)
1lrs = 1e-8 * (10 ** (np.arange(100) / 20))
2plt.semilogx(lrs, history.history["loss"])
3plt.axis([1e-8, 1e-3, 0, 300])
Loss w.r.t different learning rates. We choose the lowest one, around 8e-6.
📙 Notebook: DNN with synthetic TS.

RNN for TS

  • RRN is a NN containing Recurrent layer.
  • The different from DNN is the input shape is 3 dimensional (batch_size x #time_step x dims_input_at each_timestep).
  • Re-use 1 cell multiple times in different layers (in this course).
Idea of how RNN works with TS data. The current location can be impacted more by the nearby locations.

Shape of input to RNN

  • Suppose: window size of 30 time steps, batch size of 4: Shape will be 4x30x1 and the memory cell input will be 4x1 matrix.
  • If the memory cell comprises 3 neurons then the output matrix will be 4x3. Therefore, the full output of the layer will be 4x30x3.
  • is just a copy of .
  • Below figure: input and also output a sequence.
Dimension of input to RNN.

Sequence to vector RNN

  • Sometimes, we want only input a sequence but not output. This called sequence-to-vector RNN. I.E., ignore all of the outputs except the last one!. In tf.keras, it's default setting!
Sequence to vector RNN.
1# Check the figure below as an illustration
2model = tf.keras.models.Sequential([
3tf.keras.layers.SimpleRNN(20, return_sequences=True, input_shape=[None, 1]),
4    # input_shape:
5    #   TF assumes that 1st dim is batch size -> any size at all -> no need to define
6    #   None -> number of time steps, None means RNN can handle sequence of any length
7    #   1 -> univariate TS
9    # if there is `return_sequences=True` -> sequence-to-sequence RNN
Illustration with keras.

Lambda layer

1model = tf.keras.models.Sequential([
2    tf.keras.layers.Lambda(lambda x: tf.expand_dims(x, axis=-1), # expand to 1 dim (from 2) so that we have 3 dims: batch size x #timesteps x series dim
3                        input_shape=[None]), # can use any size of sequences
4    tf.keras.layers.SimpleRNN(40, return_sequences=True),
5    tf.keras.layers.SimpleRNN(40),
6    tf.keras.layers.Dense(1),
7    tf.keras.layers.Lambda(lambda x: x * 100.0)
8        # default activation in RNN is tanh -> (-1, 1) -> scale to -100, 100

Simple RNN

  • Loss function Huber (wiki): less sensitive to outliers. => we use this because our data in this case get a little bit noisy!


1# clear internal variables
3dataset = windowed_dataset(x_train, window_size, batch_size, shuffle_buffer_size)
5model = tf.keras.models.Sequential([
6    tf.keras.layers.Lambda(lambda x: tf.expand_dims(x, axis=-1),
7                        input_shape=[None]),
8    # LSTM here
9    tf.keras.layers.Bidirectional(tf.keras.layers.LSTM(32, return_sequences=True)),
10    tf.keras.layers.Bidirectional(tf.keras.layers.LSTM(32, return_sequences=True)),
11    tf.keras.layers.Bidirectional(tf.keras.layers.LSTM(32)),
12    #
13    tf.keras.layers.Dense(1),
14    tf.keras.layers.Lambda(lambda x: x * 100.0)
📙 Notebook: LSTM with synthetic TS.

Real-world time series data

  • Combine CNN + LSTM.
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