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時系列予測

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This tutorial is an introduction to time series forecasting using TensorFlow. It builds a few different styles of models including Convolutional and Recurrent Neural Networks (CNNs and RNNs).

This is covered in two main parts, with subsections:

  • Forecast for a single timestep:
    • A single feature.
    • All features.
  • Forecast multiple steps:
    • Single-shot: Make the predictions all at once.
    • Autoregressive: Make one prediction at a time and feed the output back to the model.

Setup

import os
import datetime

import IPython
import IPython.display
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns
import tensorflow as tf

mpl.rcParams['figure.figsize'] = (8, 6)
mpl.rcParams['axes.grid'] = False

The weather dataset

This tutorial uses a weather time series dataset recorded by the Max Planck Institute for Biogeochemistry.

This dataset contains 14 different features such as air temperature, atmospheric pressure, and humidity. These were collected every 10 minutes, beginning in 2003. For efficiency, you will use only the data collected between 2009 and 2016. This section of the dataset was prepared by François Chollet for his book Deep Learning with Python.

zip_path = tf.keras.utils.get_file(
    origin='https://storage.googleapis.com/tensorflow/tf-keras-datasets/jena_climate_2009_2016.csv.zip',
    fname='jena_climate_2009_2016.csv.zip',
    extract=True)
csv_path, _ = os.path.splitext(zip_path)
Downloading data from https://storage.googleapis.com/tensorflow/tf-keras-datasets/jena_climate_2009_2016.csv.zip
13574144/13568290 [==============================] - 0s 0us/step

This tutorial will just deal with hourly predictions, so start by sub-sampling the data from 10 minute intervals to 1h:

df = pd.read_csv(csv_path)
# slice [start:stop:step], starting from index 5 take every 6th record.
df = df[5::6]

date_time = pd.to_datetime(df.pop('Date Time'), format='%d.%m.%Y %H:%M:%S')

Let's take a glance at the data. Here are the first few rows:

df.head()

Here is the evolution of a few features over time.

plot_cols = ['T (degC)', 'p (mbar)', 'rho (g/m**3)']
plot_features = df[plot_cols]
plot_features.index = date_time
_ = plot_features.plot(subplots=True)

plot_features = df[plot_cols][:480]
plot_features.index = date_time[:480]
_ = plot_features.plot(subplots=True)

png

png

Inspect and cleanup

Next look at the statistics of the dataset:

df.describe().transpose()

Wind velocity

One thing that should stand out is the min value of the wind velocity, wv (m/s) and max. wv (m/s) columns. This -9999 is likely erroneous. There's a separate wind direction column, so the velocity should be >=0. Replace it with zeros:

wv = df['wv (m/s)']
bad_wv = wv == -9999.0
wv[bad_wv] = 0.0

max_wv = df['max. wv (m/s)']
bad_max_wv = max_wv == -9999.0
max_wv[bad_max_wv] = 0.0

# The above inplace edits are reflected in the DataFrame
df['wv (m/s)'].min()
0.0

Feature engineering

Before diving in to build a model it's important to understand your data, and be sure that you're passing the model appropriately formatted data.

Wind

The last column of the data, wd (deg), gives the wind direction in units of degrees. Angles do not make good model inputs, 360° and 0° should be close to each other, and wrap around smoothly. Direction shouldn't matter if the wind is not blowing.

Right now the distribution of wind data looks like this:

plt.hist2d(df['wd (deg)'], df['wv (m/s)'], bins=(50, 50), vmax=400)
plt.colorbar()
plt.xlabel('Wind Direction [deg]')
plt.ylabel('Wind Velocity [m/s]')
Text(0, 0.5, 'Wind Velocity [m/s]')

png

But this will be easier for the model to interpret if you convert the wind direction and velocity columns to a wind vector:

wv = df.pop('wv (m/s)')
max_wv = df.pop('max. wv (m/s)')

# Convert to radians.
wd_rad = df.pop('wd (deg)')*np.pi / 180

# Calculate the wind x and y components.
df['Wx'] = wv*np.cos(wd_rad)
df['Wy'] = wv*np.sin(wd_rad)

# Calculate the max wind x and y components.
df['max Wx'] = max_wv*np.cos(wd_rad)
df['max Wy'] = max_wv*np.sin(wd_rad)

The distribution of wind vectors is much simpler for the model to correctly interpret.

plt.hist2d(df['Wx'], df['Wy'], bins=(50, 50), vmax=400)
plt.colorbar()
plt.xlabel('Wind X [m/s]')
plt.ylabel('Wind Y [m/s]')
ax = plt.gca()
ax.axis('tight')
(-11.305513973134667, 8.24469928549079, -8.27438540335515, 7.7338312955467785)

png

Time

Similarly the Date Time column is very useful, but not in this string form. Start by converting it to seconds:

timestamp_s = date_time.map(datetime.datetime.timestamp)

Similar to the wind direction the time in seconds is not a useful model input. Being weather data it has clear daily and yearly periodicity. There are many ways you could deal with periodicity.

A simple approach to convert it to a usable signal is to use sin and cos to convert the time to clear "Time of day" and "Time of year" signals:

day = 24*60*60
year = (365.2425)*day

df['Day sin'] = np.sin(timestamp_s * (2 * np.pi / day))
df['Day cos'] = np.cos(timestamp_s * (2 * np.pi / day))
df['Year sin'] = np.sin(timestamp_s * (2 * np.pi / year))
df['Year cos'] = np.cos(timestamp_s * (2 * np.pi / year))
plt.plot(np.array(df['Day sin'])[:25])
plt.plot(np.array(df['Day cos'])[:25])
plt.xlabel('Time [h]')
plt.title('Time of day signal')
Text(0.5, 1.0, 'Time of day signal')

png

This gives the model access to the most important frequency features. In this case you knew ahead of time which frequencies were important.

If you didn't know, you can determine which frequencies are important using an fft. To check our assumptions, here is the tf.signal.rfft of the temperature over time. Note the obvious peaks at frequencies near 1/year and 1/day:

fft = tf.signal.rfft(df['T (degC)'])
f_per_dataset = np.arange(0, len(fft))

n_samples_h = len(df['T (degC)'])
hours_per_year = 24*365.2524
years_per_dataset = n_samples_h/(hours_per_year)

f_per_year = f_per_dataset/years_per_dataset
plt.step(f_per_year, np.abs(fft))
plt.xscale('log')
plt.ylim(0, 400000)
plt.xlim([0.1, max(plt.xlim())])
plt.xticks([1, 365.2524], labels=['1/Year', '1/day'])
_ = plt.xlabel('Frequency (log scale)')

png

Split the data

We'll use a (70%, 20%, 10%) split for the training, validation, and test sets. Note the data is not being randomly shuffled before splitting. This is for two reasons.

  1. It ensures that chopping the data into windows of consecutive samples is still possible.
  2. It ensures that the validation/test results are more realistic, being evaluated on data collected after the model was trained.
column_indices = {name: i for i, name in enumerate(df.columns)}

n = len(df)
train_df = df[0:int(n*0.7)]
val_df = df[int(n*0.7):int(n*0.9)]
test_df = df[int(n*0.9):]

num_features = df.shape[1]

Normalize the data

It is important to scale features before training a neural network. Normalization is a common way of doing this scaling. Subtract the mean and divide by the standard deviation of each feature.

The mean and standard deviation should only be computed using the training data so that the models have no access to the values in the validation and test sets.

It's also arguable that the model shouldn't have access to future values in the training set when training, and that this normalization should be done using moving averages. That's not the focus of this tutorial, and the validation and test sets ensure that you get (somewhat) honest metrics. So in the interest of simplicity this tutorial uses a simple average.

train_mean = train_df.mean()
train_std = train_df.std()

train_df = (train_df - train_mean) / train_std
val_df = (val_df - train_mean) / train_std
test_df = (test_df - train_mean) / train_std

Now peek at the distribution of the features. Some features do have long tails, but there are no obvious errors like the -9999 wind velocity value.

df_std = (df - train_mean) / train_std
df_std = df_std.melt(var_name='Column', value_name='Normalized')
plt.figure(figsize=(12, 6))
ax = sns.violinplot(x='Column', y='Normalized', data=df_std)
_ = ax.set_xticklabels(df.keys(), rotation=90)

png

Data windowing

The models in this tutorial will make a set of predictions based on a window of consecutive samples from the data.

The main features of the input windows are:

  • The width (number of time steps) of the input and label windows
  • The time offset between them.
  • Which features are used as inputs, labels, or both.

This tutorial builds a variety of models (including Linear, DNN, CNN and RNN models), and uses them for both:

  • Single-output, and multi-output predictions.
  • Single-time-step and multi-time-step predictions.

This section focuses on implementing the data windowing so that it can be reused for all of those models.

Depending on the task and type of model you may want to generate a variety of data windows. Here are some examples:

  1. For example, to make a single prediction 24h into the future, given 24h of history you might define a window like this:

    One prediction 24h into the future.

  2. A model that makes a prediction 1h into the future, given 6h of history would need a window like this:

    One prediction 1h into the future.

The rest of this section defines a WindowGenerator class. This class can:

  1. Handle the indexes and offsets as shown in the diagrams above.
  2. Split windows of features into a (features, labels) pairs.
  3. Plot the content of the resulting windows.
  4. Efficiently generate batches of these windows from the training, evaluation, and test data, using tf.data.Datasets.

1. Indexes and offsets

Start by creating the WindowGenerator class. The __init__ method includes all the necessary logic for the input and label indices.

It also takes the train, eval, and test dataframes as input. These will be converted to tf.data.Datasets of windows later.

class WindowGenerator():
  def __init__(self, input_width, label_width, shift,
               train_df=train_df, val_df=val_df, test_df=test_df,
               label_columns=None):
    # Store the raw data.
    self.train_df = train_df
    self.val_df = val_df
    self.test_df = test_df

    # Work out the label column indices.
    self.label_columns = label_columns
    if label_columns is not None:
      self.label_columns_indices = {name: i for i, name in
                                    enumerate(label_columns)}
    self.column_indices = {name: i for i, name in
                           enumerate(train_df.columns)}

    # Work out the window parameters.
    self.input_width = input_width
    self.label_width = label_width
    self.shift = shift

    self.total_window_size = input_width + shift

    self.input_slice = slice(0, input_width)
    self.input_indices = np.arange(self.total_window_size)[self.input_slice]

    self.label_start = self.total_window_size - self.label_width
    self.labels_slice = slice(self.label_start, None)
    self.label_indices = np.arange(self.total_window_size)[self.labels_slice]

  def __repr__(self):
    return '\n'.join([
        f'Total window size: {self.total_window_size}',
        f'Input indices: {self.input_indices}',
        f'Label indices: {self.label_indices}',
        f'Label column name(s): {self.label_columns}'])

Here is code to create the 2 windows shown in the diagrams at the start of this section:

w1 = WindowGenerator(input_width=24, label_width=1, shift=24,
                     label_columns=['T (degC)'])
w1
Total window size: 48
Input indices: [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23]
Label indices: [47]
Label column name(s): ['T (degC)']
w2 = WindowGenerator(input_width=6, label_width=1, shift=1,
                     label_columns=['T (degC)'])
w2
Total window size: 7
Input indices: [0 1 2 3 4 5]
Label indices: [6]
Label column name(s): ['T (degC)']

2. Split

Given a list consecutive inputs, the split_window method will convert them to a window of inputs and a window of labels.

The example w2, above, will be split like this:

The initial window is all consecuitive samples, this splits it into an (inputs, labels) pairs

This diagram doesn't show the features axis of the data, but this split_window function also handles the label_columns so it can be used for both the single output and multi-output examples.

def split_window(self, features):
  inputs = features[:, self.input_slice, :]
  labels = features[:, self.labels_slice, :]
  if self.label_columns is not None:
    labels = tf.stack(
        [labels[:, :, self.column_indices[name]] for name in self.label_columns],
        axis=-1)

  # Slicing doesn't preserve static shape information, so set the shapes
  # manually. This way the `tf.data.Datasets` are easier to inspect.
  inputs.set_shape([None, self.input_width, None])
  labels.set_shape([None, self.label_width, None])

  return inputs, labels

WindowGenerator.split_window = split_window

Try it out:

# Stack three slices, the length of the total window:
example_window = tf.stack([np.array(train_df[:w2.total_window_size]),
                           np.array(train_df[100:100+w2.total_window_size]),
                           np.array(train_df[200:200+w2.total_window_size])])


example_inputs, example_labels = w2.split_window(example_window)

print('All shapes are: (batch, time, features)')
print(f'Window shape: {example_window.shape}')
print(f'Inputs shape: {example_inputs.shape}')
print(f'labels shape: {example_labels.shape}')
All shapes are: (batch, time, features)
Window shape: (3, 7, 19)
Inputs shape: (3, 6, 19)
labels shape: (3, 1, 1)

Typically data in TensorFlow is packed into arrays where the outermost index is across examples (the "batch" dimension). The middle indices are the "time" or "space" (width, height) dimension(s). The innermost indices are the features.

The code above took a batch of 3, 7-timestep windows, with 19 features at each time step. It split them into a batch of 6-timestep, 19 feature inputs, and a 1-timestep 1-feature label. The label only has one feature because the WindowGenerator was initialized with label_columns=['T (degC)']. Initially this tutorial will build models that predict single output labels.

3. Plot

Here is a plot method that allows a simple visualization of the split window:

w2.example = example_inputs, example_labels
def plot(self, model=None, plot_col='T (degC)', max_subplots=3):
  inputs, labels = self.example
  plt.figure(figsize=(12, 8))
  plot_col_index = self.column_indices[plot_col]
  max_n = min(max_subplots, len(inputs))
  for n in range(max_n):
    plt.subplot(3, 1, n+1)
    plt.ylabel(f'{plot_col} [normed]')
    plt.plot(self.input_indices, inputs[n, :, plot_col_index],
             label='Inputs', marker='.', zorder=-10)

    if self.label_columns:
      label_col_index = self.label_columns_indices.get(plot_col, None)
    else:
      label_col_index = plot_col_index

    if label_col_index is None:
      continue

    plt.scatter(self.label_indices, labels[n, :, label_col_index],
                edgecolors='k', label='Labels', c='#2ca02c', s=64)
    if model is not None:
      predictions = model(inputs)
      plt.scatter(self.label_indices, predictions[n, :, label_col_index],
                  marker='X', edgecolors='k', label='Predictions',
                  c='#ff7f0e', s=64)

    if n == 0:
      plt.legend()

  plt.xlabel('Time [h]')

WindowGenerator.plot = plot

This plot aligns inputs, labels, and (later) predictions based on the time that the item refers to:

w2.plot()

png

You can plot the other columns, but the example window w2 configuration only has labels for the T (degC) column.

w2.plot(plot_col='p (mbar)')

png

4. Create tf.data.Datasets

Finally this make_dataset method will take a time series DataFrame and convert it to a tf.data.Dataset of (input_window, label_window) pairs using the preprocessing.timeseries_dataset_from_array function.

def make_dataset(self, data):
  data = np.array(data, dtype=np.float32)
  ds = tf.keras.preprocessing.timeseries_dataset_from_array(
      data=data,
      targets=None,
      sequence_length=self.total_window_size,
      sequence_stride=1,
      shuffle=True,
      batch_size=32,)

  ds = ds.map(self.split_window)

  return ds

WindowGenerator.make_dataset = make_dataset

The WindowGenerator object holds training, validation and test data. Add properties for accessing them as tf.data.Datasets using the above make_dataset method. Also add a standard example batch for easy access and plotting:

@property
def train(self):
  return self.make_dataset(self.train_df)

@property
def val(self):
  return self.make_dataset(self.val_df)

@property
def test(self):
  return self.make_dataset(self.test_df)

@property
def example(self):
  """Get and cache an example batch of `inputs, labels` for plotting."""
  result = getattr(self, '_example', None)
  if result is None:
    # No example batch was found, so get one from the `.train` dataset
    result = next(iter(self.train))
    # And cache it for next time
    self._example = result
  return result

WindowGenerator.train = train
WindowGenerator.val = val
WindowGenerator.test = test
WindowGenerator.example = example

Now the WindowGenerator object gives you access to the tf.data.Dataset objects, so you can easily iterate over the data.

The Dataset.element_spec property tells you the structure, dtypes and shapes of the dataset elements.

# Each element is an (inputs, label) pair
w2.train.element_spec
(TensorSpec(shape=(None, 6, 19), dtype=tf.float32, name=None),
 TensorSpec(shape=(None, 1, 1), dtype=tf.float32, name=None))

Iterating over a Dataset yields concrete batches:

for example_inputs, example_labels in w2.train.take(1):
  print(f'Inputs shape (batch, time, features): {example_inputs.shape}')
  print(f'Labels shape (batch, time, features): {example_labels.shape}')
Inputs shape (batch, time, features): (32, 6, 19)
Labels shape (batch, time, features): (32, 1, 1)

Single step models

The simplest model you can build on this sort of data is one that predicts a single feature's value, 1 timestep (1h) in the future based only on the current conditions.

So start by building models to predict the T (degC) value 1h into the future.

Predict the next time step

Configure a WindowGenerator object to produce these single-step (input, label) pairs:

single_step_window = WindowGenerator(
    input_width=1, label_width=1, shift=1,
    label_columns=['T (degC)'])
single_step_window
Total window size: 2
Input indices: [0]
Label indices: [1]
Label column name(s): ['T (degC)']

The window object creates tf.data.Datasets from the training, validation, and test sets, allowing you to easily iterate over batches of data.

for example_inputs, example_labels in single_step_window.train.take(1):
  print(f'Inputs shape (batch, time, features): {example_inputs.shape}')
  print(f'Labels shape (batch, time, features): {example_labels.shape}')
Inputs shape (batch, time, features): (32, 1, 19)
Labels shape (batch, time, features): (32, 1, 1)

Baseline

Before building a trainable model it would be good to have a performance baseline as a point for comparison with the later more complicated models.

This first task is to predict temperature 1h in the future given the current value of all features. The current values include the current temperature.

So start with a model that just returns the current temperature as the prediction, predicting "No change". This is a reasonable baseline since temperature changes slowly. Of course, this baseline will work less well if you make a prediction further in the future.

Send the input to the output

class Baseline(tf.keras.Model):
  def __init__(self, label_index=None):
    super().__init__()
    self.label_index = label_index

  def call(self, inputs):
    if self.label_index is None:
      return inputs
    result = inputs[:, :, self.label_index]
    return result[:, :, tf.newaxis]

Instantiate and evaluate this model:

baseline = Baseline(label_index=column_indices['T (degC)'])

baseline.compile(loss=tf.losses.MeanSquaredError(),
                 metrics=[tf.metrics.MeanAbsoluteError()])

val_performance = {}
performance = {}
val_performance['Baseline'] = baseline.evaluate(single_step_window.val)
performance['Baseline'] = baseline.evaluate(single_step_window.test, verbose=0)
439/439 [==============================] - 1s 2ms/step - loss: 0.0128 - mean_absolute_error: 0.0785

That printed some performance metrics, but those don't give you a feeling for how well the model is doing.

The WindowGenerator has a plot method, but the plots won't be very interesting with only a single sample. So, create a wider WindowGenerator that generates windows 24h of consecutive inputs and labels at a time.

The wide_window doesn't change the way the model operates. The model still makes predictions 1h into the future based on a single input time step. Here the time axis acts like the batch axis: Each prediction is made independently with no interaction between time steps.

wide_window = WindowGenerator(
    input_width=24, label_width=24, shift=1,
    label_columns=['T (degC)'])

wide_window
Total window size: 25
Input indices: [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23]
Label indices: [ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24]
Label column name(s): ['T (degC)']

This expanded window can be passed directly to the same baseline model without any code changes. This is possible because the inputs and labels have the same number of timesteps, and the baseline just forwards the input to the output:

One prediction 1h into the future, ever hour.

print('Input shape:', wide_window.example[0].shape)
print('Output shape:', baseline(wide_window.example[0]).shape)
Input shape: (32, 24, 19)
Output shape: (32, 24, 1)

Plotting the baseline model's predictions you can see that it is simply the labels, shifted right by 1h.

wide_window.plot(baseline)

png

In the above plots of three examples the single step model is run over the course of 24h. This deserves some explaination:

  • The blue "Inputs" line shows the input temperature at each time step. The model recieves all features, this plot only shows the temperature.
  • The green "Labels" dots show the target prediction value. These dots are shown at the prediction time, not the input time. That is why the range of labels is shifted 1 step relative to the inputs.
  • The orange "Predictions" crosses are the model's prediction's for each output time step. If the model were predicting perfectly the predictions would land directly on the "labels".

Linear model

The simplest trainable model you can apply to this task is to insert linear transformation between the input and output. In this case the output from a time step only depends on that step:

A single step prediction

A layers.Dense with no activation set is a linear model. The layer only transforms the last axis of the data from (batch, time, inputs) to (batch, time, units), it is applied independently to every item across the batch and time axes.

linear = tf.keras.Sequential([
    tf.keras.layers.Dense(units=1)
])
print('Input shape:', single_step_window.example[0].shape)
print('Output shape:', linear(single_step_window.example[0]).shape)
Input shape: (32, 1, 19)
Output shape: (32, 1, 1)

This tutorial trains many models, so package the training procedure into a function:

MAX_EPOCHS = 20

def compile_and_fit(model, window, patience=2):
  early_stopping = tf.keras.callbacks.EarlyStopping(monitor='val_loss',
                                                    patience=patience,
                                                    mode='min')

  model.compile(loss=tf.losses.MeanSquaredError(),
                optimizer=tf.optimizers.Adam(),
                metrics=[tf.metrics.MeanAbsoluteError()])

  history = model.fit(window.train, epochs=MAX_EPOCHS,
                      validation_data=window.val,
                      callbacks=[early_stopping])
  return history

Train the model and evaluate its performance:

history = compile_and_fit(linear, single_step_window)

val_performance['Linear'] = linear.evaluate(single_step_window.val)
performance['Linear'] = linear.evaluate(single_step_window.test, verbose=0)
Epoch 1/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.2864 - mean_absolute_error: 0.2848 - val_loss: 0.0163 - val_mean_absolute_error: 0.0975
Epoch 2/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0121 - mean_absolute_error: 0.0817 - val_loss: 0.0103 - val_mean_absolute_error: 0.0752
Epoch 3/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0103 - mean_absolute_error: 0.0749 - val_loss: 0.0098 - val_mean_absolute_error: 0.0738
Epoch 4/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0099 - mean_absolute_error: 0.0733 - val_loss: 0.0095 - val_mean_absolute_error: 0.0731
Epoch 5/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0096 - mean_absolute_error: 0.0721 - val_loss: 0.0092 - val_mean_absolute_error: 0.0719
Epoch 6/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0095 - mean_absolute_error: 0.0715 - val_loss: 0.0091 - val_mean_absolute_error: 0.0716
Epoch 7/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0094 - mean_absolute_error: 0.0710 - val_loss: 0.0091 - val_mean_absolute_error: 0.0716
Epoch 8/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0093 - mean_absolute_error: 0.0707 - val_loss: 0.0090 - val_mean_absolute_error: 0.0706
Epoch 9/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0092 - mean_absolute_error: 0.0704 - val_loss: 0.0090 - val_mean_absolute_error: 0.0712
Epoch 10/20
1534/1534 [==============================] - 5s 3ms/step - loss: 0.0092 - mean_absolute_error: 0.0703 - val_loss: 0.0091 - val_mean_absolute_error: 0.0715
439/439 [==============================] - 1s 2ms/step - loss: 0.0091 - mean_absolute_error: 0.0715

Like the baseline model, the linear model can be called on batches of wide windows. Used this way the model makes a set of independent predictions on consecuitive time steps. The time axis acts like another batch axis. There are no interactions between the predictions at each time step.

A single step prediction

print('Input shape:', wide_window.example[0].shape)
print('Output shape:', baseline(wide_window.example[0]).shape)
Input shape: (32, 24, 19)
Output shape: (32, 24, 1)

Here is the plot of its example predictions on the wide_window, note how in many cases the prediction is clearly better than just returning the input temperature, but in a few cases it's worse:

wide_window.plot(linear)

png

One advantage to linear models is that they're relatively simple to interpret. You can pull out the layer's weights, and see the weight assigned to each input:

plt.bar(x = range(len(train_df.columns)),
        height=linear.layers[0].kernel[:,0].numpy())
axis = plt.gca()
axis.set_xticks(range(len(train_df.columns)))
_ = axis.set_xticklabels(train_df.columns, rotation=90)

png

Sometimes the model doesn't even place the most weight on the input T (degC). This is one of the risks of random initialization.

Dense

Before applying models that actually operate on multiple time-steps, it's worth checking the performance of deeper, more powerful, single input step models.

Here's a model similar to the linear model, except it stacks several a few Dense layers between the input and the output:

dense = tf.keras.Sequential([
    tf.keras.layers.Dense(units=64, activation='relu'),
    tf.keras.layers.Dense(units=64, activation='relu'),
    tf.keras.layers.Dense(units=1)
])

history = compile_and_fit(dense, single_step_window)

val_performance['Dense'] = dense.evaluate(single_step_window.val)
performance['Dense'] = dense.evaluate(single_step_window.test, verbose=0)
Epoch 1/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0159 - mean_absolute_error: 0.0814 - val_loss: 0.0086 - val_mean_absolute_error: 0.0693
Epoch 2/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0079 - mean_absolute_error: 0.0645 - val_loss: 0.0076 - val_mean_absolute_error: 0.0629
Epoch 3/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0074 - mean_absolute_error: 0.0622 - val_loss: 0.0085 - val_mean_absolute_error: 0.0666
Epoch 4/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0072 - mean_absolute_error: 0.0608 - val_loss: 0.0071 - val_mean_absolute_error: 0.0593
Epoch 5/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0070 - mean_absolute_error: 0.0595 - val_loss: 0.0067 - val_mean_absolute_error: 0.0579
Epoch 6/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0068 - mean_absolute_error: 0.0588 - val_loss: 0.0072 - val_mean_absolute_error: 0.0594
Epoch 7/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0068 - mean_absolute_error: 0.0583 - val_loss: 0.0066 - val_mean_absolute_error: 0.0564
Epoch 8/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0066 - mean_absolute_error: 0.0576 - val_loss: 0.0078 - val_mean_absolute_error: 0.0637
Epoch 9/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0066 - mean_absolute_error: 0.0576 - val_loss: 0.0066 - val_mean_absolute_error: 0.0564
Epoch 10/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0065 - mean_absolute_error: 0.0569 - val_loss: 0.0062 - val_mean_absolute_error: 0.0551
Epoch 11/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0065 - mean_absolute_error: 0.0571 - val_loss: 0.0070 - val_mean_absolute_error: 0.0596
Epoch 12/20
1534/1534 [==============================] - 6s 4ms/step - loss: 0.0064 - mean_absolute_error: 0.0565 - val_loss: 0.0063 - val_mean_absolute_error: 0.0551
439/439 [==============================] - 1s 3ms/step - loss: 0.0063 - mean_absolute_error: 0.0551

Multi-step dense

A single-time-step model has no context for the current values of its inputs. It can't see how the input features are changing over time. To address this issue the model needs access to multiple time steps when making predictions:

Three time steps are used for each prediction.

The baseline, linear and dense models handled each time step independently. Here the model will take multiple time steps as input to produce a single output.

Create a WindowGenerator that will produce batches of the 3h of inputs and, 1h of labels:

Note that the Window's shift parameter is relative to the end of the two windows.

CONV_WIDTH = 3
conv_window = WindowGenerator(
    input_width=CONV_WIDTH,
    label_width=1,
    shift=1,
    label_columns=['T (degC)'])

conv_window
Total window size: 4
Input indices: [0 1 2]
Label indices: [3]
Label column name(s): ['T (degC)']
conv_window.plot()
plt.title("Given 3h as input, predict 1h into the future.")
Text(0.5, 1.0, 'Given 3h as input, predict 1h into the future.')

png

You could train a dense model on a multiple-input-step window by adding a layers.Flatten as the first layer of the model:

multi_step_dense = tf.keras.Sequential([
    # Shape: (time, features) => (time*features)
    tf.keras.layers.Flatten(),
    tf.keras.layers.Dense(units=32, activation='relu'),
    tf.keras.layers.Dense(units=32, activation='relu'),
    tf.keras.layers.Dense(units=1),
    # Add back the time dimension.
    # Shape: (outputs) => (1, outputs)
    tf.keras.layers.Reshape([1, -1]),
])
print('Input shape:', conv_window.example[0].shape)
print('Output shape:', multi_step_dense(conv_window.example[0]).shape)
Input shape: (32, 3, 19)
Output shape: (32, 1, 1)

history = compile_and_fit(multi_step_dense, conv_window)

IPython.display.clear_output()
val_performance['Multi step dense'] = multi_step_dense.evaluate(conv_window.val)
performance['Multi step dense'] = multi_step_dense.evaluate(conv_window.test, verbose=0)
438/438 [==============================] - 1s 2ms/step - loss: 0.0078 - mean_absolute_error: 0.0637

conv_window.plot(multi_step_dense)

png

The main down-side of this approach is that the resulting model can only be executed on input windows of exactly this shape.

print('Input shape:', wide_window.example[0].shape)
try:
  print('Output shape:', multi_step_dense(wide_window.example[0]).shape)
except Exception as e:
  print(f'\n{type(e).__name__}:{e}')
Input shape: (32, 24, 19)

InvalidArgumentError:Matrix size-incompatible: In[0]: [32,456], In[1]: [57,32] [Op:MatMul]

The convolutional models in the next section fix this problem.

Convolution neural network

A convolution layer (layers.Conv1D) also takes multiple time steps as input to each prediction.

Below is the same model as multi_step_dense, re-written with a convolution.

Note the changes:

conv_model = tf.keras.Sequential([
    tf.keras.layers.Conv1D(filters=32,
                           kernel_size=(CONV_WIDTH,),
                           activation='relu'),
    tf.keras.layers.Dense(units=32, activation='relu'),
    tf.keras.layers.Dense(units=1),
])

Run it on an example batch to see that the model produces outputs with the expected shape:

print("Conv model on `conv_window`")
print('Input shape:', conv_window.example[0].shape)
print('Output shape:', conv_model(conv_window.example[0]).shape)
Conv model on `conv_window`
Input shape: (32, 3, 19)
Output shape: (32, 1, 1)

Train and evaluate it on the conv_window and it should give performance similar to the multi_step_dense model.

history = compile_and_fit(conv_model, conv_window)

IPython.display.clear_output()
val_performance['Conv'] = conv_model.evaluate(conv_window.val)
performance['Conv'] = conv_model.evaluate(conv_window.test, verbose=0)
438/438 [==============================] - 1s 3ms/step - loss: 0.0063 - mean_absolute_error: 0.0556

The difference between this conv_model and the multi_step_dense model is that the conv_model can be run on inputs of any length. The convolutional layer is applied to a sliding window of inputs:

Executing a convolutional model on a sequence

If you run it on wider input, it produces wider output:

print("Wide window")
print('Input shape:', wide_window.example[0].shape)
print('Labels shape:', wide_window.example[1].shape)
print('Output shape:', conv_model(wide_window.example[0]).shape)
Wide window
Input shape: (32, 24, 19)
Labels shape: (32, 24, 1)
Output shape: (32, 22, 1)

Note that the output is shorter than the input. To make training or plotting work, you need the labels, and prediction to have the same length. So build a WindowGenerator to produce wide windows with a few extra input time steps so the label and prediction lengths match:

LABEL_WIDTH = 24
INPUT_WIDTH = LABEL_WIDTH + (CONV_WIDTH - 1)
wide_conv_window = WindowGenerator(
    input_width=INPUT_WIDTH,
    label_width=LABEL_WIDTH,
    shift=1,
    label_columns=['T (degC)'])

wide_conv_window
Total window size: 27
Input indices: [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
 24 25]
Label indices: [ 3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26]
Label column name(s): ['T (degC)']
print("Wide conv window")
print('Input shape:', wide_conv_window.example[0].shape)
print('Labels shape:', wide_conv_window.example[1].shape)
print('Output shape:', conv_model(wide_conv_window.example[0]).shape)
Wide conv window
Input shape: (32, 26, 19)
Labels shape: (32, 24, 1)
Output shape: (32, 24, 1)

Now you can plot the model's predictions on a wider window. Note the 3 input time steps before the first prediction. Every prediction here is based on the 3 preceding timesteps:

wide_conv_window.plot(conv_model)

png

Recurrent neural network

A Recurrent Neural Network (RNN) is a type of neural network well-suited to time series data. RNNs process a time series step-by-step, maintaining an internal state from time-step to time-step.

For more details, read the text generation tutorial or the RNN guide.

In this tutorial, you will use an RNN layer called Long Short Term Memory (LSTM).

An important constructor argument for all keras RNN layers is the return_sequences argument. This setting can configure the layer in one of two ways.

  1. If False, the default, the layer only returns the output of the final timestep, giving the model time to warm up its internal state before making a single prediction:

An lstm warming up and making a single prediction

  1. If True the layer returns an output for each input. This is useful for:
    • Stacking RNN layers.
    • Training a model on multiple timesteps simultaneously.

An lstm making a prediction after every timestep

lstm_model = tf.keras.models.Sequential([
    # Shape [batch, time, features] => [batch, time, lstm_units]
    tf.keras.layers.LSTM(32, return_sequences=True),
    # Shape => [batch, time, features]
    tf.keras.layers.Dense(units=1)
])

With return_sequences=True the model can be trained on 24h of data at a time.

print('Input shape:', wide_window.example[0].shape)
print('Output shape:', lstm_model(wide_window.example[0]).shape)
Input shape: (32, 24, 19)
Output shape: (32, 24, 1)

history = compile_and_fit(lstm_model, wide_window)

IPython.display.clear_output()
val_performance['LSTM'] = lstm_model.evaluate(wide_window.val)
performance['LSTM'] = lstm_model.evaluate(wide_window.test, verbose=0)
438/438 [==============================] - 1s 3ms/step - loss: 0.0057 - mean_absolute_error: 0.0523

wide_window.plot(lstm_model)

png

Performance

With this dataset typically each of the models does slightly better than the one before it.

x = np.arange(len(performance))
width = 0.3
metric_name = 'mean_absolute_error'
metric_index = lstm_model.metrics_names.index('mean_absolute_error')
val_mae = [v[metric_index] for v in val_performance.values()]
test_mae = [v[metric_index] for v in performance.values()]

plt.ylabel('mean_absolute_error [T (degC), normalized]')
plt.bar(x - 0.17, val_mae, width, label='Validation')
plt.bar(x + 0.17, test_mae, width, label='Test')
plt.xticks(ticks=x, labels=performance.keys(),
           rotation=45)
_ = plt.legend()

png

for name, value in performance.items():
  print(f'{name:12s}: {value[1]:0.4f}')
Baseline    : 0.0852
Linear      : 0.0694
Dense       : 0.0566
Multi step dense: 0.0667
Conv        : 0.0572
LSTM        : 0.0528

Multi-output models

The models so far all predicted a single output feature, T (degC), for a single time step.

All of these models can be converted to predict multiple features just by changing the number of units in the output layer and adjusting the training windows to include all features in the labels.

single_step_window = WindowGenerator(
    # `WindowGenerator` returns all features as labels if you 
    # don't set the `label_columns` argument.
    input_width=1, label_width=1, shift=1)

wide_window = WindowGenerator(
    input_width=24, label_width=24, shift=1)

for example_inputs, example_labels in wide_window.train.take(1):
  print(f'Inputs shape (batch, time, features): {example_inputs.shape}')
  print(f'Labels shape (batch, time, features): {example_labels.shape}')
Inputs shape (batch, time, features): (32, 24, 19)
Labels shape (batch, time, features): (32, 24, 19)

Note above that the features axis of the labels now has the same depth as the inputs, instead of 1.

Baseline

The same baseline model can be used here, but this time repeating all features instead of selecting a specific label_index.

baseline = Baseline()
baseline.compile(loss=tf.losses.MeanSquaredError(),
                 metrics=[tf.metrics.MeanAbsoluteError()])
val_performance = {}
performance = {}
val_performance['Baseline'] = baseline.evaluate(wide_window.val)
performance['Baseline'] = baseline.evaluate(wide_window.test, verbose=0)
438/438 [==============================] - 1s 2ms/step - loss: 0.0886 - mean_absolute_error: 0.1589

Dense

dense = tf.keras.Sequential([
    tf.keras.layers.Dense(units=64, activation='relu'),
    tf.keras.layers.Dense(units=64, activation='relu'),
    tf.keras.layers.Dense(units=num_features)
])
history = compile_and_fit(dense, single_step_window)

IPython.display.clear_output()
val_performance['Dense'] = dense.evaluate(single_step_window.val)
performance['Dense'] = dense.evaluate(single_step_window.test, verbose=0)
439/439 [==============================] - 1s 3ms/step - loss: 0.0706 - mean_absolute_error: 0.1362

RNN

%%time
wide_window = WindowGenerator(
    input_width=24, label_width=24, shift=1)

lstm_model = tf.keras.models.Sequential([
    # Shape [batch, time, features] => [batch, time, lstm_units]
    tf.keras.layers.LSTM(32, return_sequences=True),
    # Shape => [batch, time, features]
    tf.keras.layers.Dense(units=num_features)
])

history = compile_and_fit(lstm_model, wide_window)

IPython.display.clear_output()
val_performance['LSTM'] = lstm_model.evaluate( wide_window.val)
performance['LSTM'] = lstm_model.evaluate( wide_window.test, verbose=0)

print()
438/438 [==============================] - 1s 3ms/step - loss: 0.0613 - mean_absolute_error: 0.1192

CPU times: user 6min 18s, sys: 1min 36s, total: 7min 55s
Wall time: 2min 53s

Advanced: Residual connections

The Baseline model from earlier took advantage of the fact that the sequence doesn't change drastically from time step to time step. Every model trained in this tutorial so far was randomly initialized, and then had to learn that the output is a a small change from the previous time step.

While you can get around this issue with careful initialization, it's simpler to build this into the model structure.

It's common in time series analysis to build models that instead of predicting the next value, predict how the value will change in the next timestep. Similarly, "Residual networks" or "ResNets" in deep learning refer to architectures where each layer adds to the model's accumulating result.

That is how you take advantage of the knowledge that the change should be small.

A model with a residual connection

Essentially this initializes the model to match the Baseline. For this task it helps models converge faster, with slightly better performance.

This approach can be used in conjunction with any model discussed in this tutorial.

Here it is being applied to the LSTM model, note the use of the tf.initializers.zeros to ensure that the initial predicted changes are small, and don't overpower the residual connection. There are no symmetry-breaking concerns for the gradients here, since the zeros are only used on the last layer.

class ResidualWrapper(tf.keras.Model):
  def __init__(self, model):
    super().__init__()
    self.model = model

  def call(self, inputs, *args, **kwargs):
    delta = self.model(inputs, *args, **kwargs)

    # The prediction for each timestep is the input
    # from the previous time step plus the delta
    # calculated by the model.
    return inputs + delta
%%time
residual_lstm = ResidualWrapper(
    tf.keras.Sequential([
    tf.keras.layers.LSTM(32, return_sequences=True),
    tf.keras.layers.Dense(
        num_features,
        # The predicted deltas should start small
        # So initialize the output layer with zeros
        kernel_initializer=tf.initializers.zeros)
]))

history = compile_and_fit(residual_lstm, wide_window)

IPython.display.clear_output()
val_performance['Residual LSTM'] = residual_lstm.evaluate(wide_window.val)
performance['Residual LSTM'] = residual_lstm.evaluate(wide_window.test, verbose=0)
print()
438/438 [==============================] - 1s 3ms/step - loss: 0.0619 - mean_absolute_error: 0.1181

CPU times: user 2min 2s, sys: 31.2 s, total: 2min 33s
Wall time: 56.9 s

Performance

Here is the overall performance for these multi-output models.

x = np.arange(len(performance))
width = 0.3

metric_name = 'mean_absolute_error'
metric_index = lstm_model.metrics_names.index('mean_absolute_error')
val_mae = [v[metric_index] for v in val_performance.values()]
test_mae = [v[metric_index] for v in performance.values()]

plt.bar(x - 0.17, val_mae, width, label='Validation')
plt.bar(x + 0.17, test_mae, width, label='Test')
plt.xticks(ticks=x, labels=performance.keys(),
           rotation=45)
plt.ylabel('MAE (average over all outputs)')
_ = plt.legend()

png

for name, value in performance.items():
  print(f'{name:15s}: {value[1]:0.4f}')
Baseline       : 0.1638
Dense          : 0.1367
LSTM           : 0.1208
Residual LSTM  : 0.1197

The above performances are averaged across all model outputs.

Multi-step models

Both the single-output and multiple-output models in the previous sections made single time step predictions, 1h into the future.

This section looks at how to expand these models to make multiple time step predictions.

In a multi-step prediction, the model needs to learn to predict a range of future values. Thus, unlike a single step model, where only a single future point is predicted, a multi-step model predicts a sequence of the future values.

There are two rough approaches to this:

  1. Single shot predictions where the entire time series is predicted at once.
  2. Autoregressive predictions where the model only makes single step predictions and its output is fed back as its input.

In this section all the models will predict all the features across all output time steps.

For the multi-step model, the training data again consists of hourly samples. However, here, the models will learn to predict 24h of the future, given 24h of the past.

Here is a Window object that generates these slices from the dataset:

OUT_STEPS = 24
multi_window = WindowGenerator(input_width=24,
                               label_width=OUT_STEPS,
                               shift=OUT_STEPS)

multi_window.plot()
multi_window
Total window size: 48
Input indices: [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23]
Label indices: [24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47]
Label column name(s): None

png

Baselines

A simple baseline for this task is to repeat the last input time step for the required number of output timesteps:

Repeat the last input, for each output step

class MultiStepLastBaseline(tf.keras.Model):
  def call(self, inputs):
    return tf.tile(inputs[:, -1:, :], [1, OUT_STEPS, 1])

last_baseline = MultiStepLastBaseline()
last_baseline.compile(loss=tf.losses.MeanSquaredError(),
                      metrics=[tf.metrics.MeanAbsoluteError()])

multi_val_performance = {}
multi_performance = {}

multi_val_performance['Last'] = last_baseline.evaluate(multi_window.val)
multi_performance['Last'] = last_baseline.evaluate(multi_window.test, verbose=0)
multi_window.plot(last_baseline)
437/437 [==============================] - 1s 2ms/step - loss: 0.6285 - mean_absolute_error: 0.5007

png

Since this task is to predict 24h given 24h another simple approach is to repeat the previous day, assuming tomorrow will be similar:

Repeat the previous day

class RepeatBaseline(tf.keras.Model):
  def call(self, inputs):
    return inputs

repeat_baseline = RepeatBaseline()
repeat_baseline.compile(loss=tf.losses.MeanSquaredError(),
                        metrics=[tf.metrics.MeanAbsoluteError()])

multi_val_performance['Repeat'] = repeat_baseline.evaluate(multi_window.val)
multi_performance['Repeat'] = repeat_baseline.evaluate(multi_window.test, verbose=0)
multi_window.plot(repeat_baseline)
437/437 [==============================] - 1s 2ms/step - loss: 0.4270 - mean_absolute_error: 0.3959

png

Single-shot models

One high level approach to this problem is use a "single-shot" model, where the model makes the entire sequence prediction in a single step.

This can be implemented efficiently as a layers.Dense with OUT_STEPS*features output units. The model just needs to reshape that output to the required (OUTPUT_STEPS, features).

Linear

A simple linear model based on the last input time step does better than either baseline, but is underpowered. The model needs to predict OUTPUT_STEPS time steps, from a single input time step with a linear projection. It can only capture a low-dimensional slice of the behavior, likely based mainly on the time of day and time of year.

Predct all timesteps from the last time-step

multi_linear_model = tf.keras.Sequential([
    # Take the last time-step.
    # Shape [batch, time, features] => [batch, 1, features]
    tf.keras.layers.Lambda(lambda x: x[:, -1:, :]),
    # Shape => [batch, 1, out_steps*features]
    tf.keras.layers.Dense(OUT_STEPS*num_features,
                          kernel_initializer=tf.initializers.zeros),
    # Shape => [batch, out_steps, features]
    tf.keras.layers.Reshape([OUT_STEPS, num_features])
])

history = compile_and_fit(multi_linear_model, multi_window)

IPython.display.clear_output()
multi_val_performance['Linear'] = multi_linear_model.evaluate(multi_window.val)
multi_performance['Linear'] = multi_linear_model.evaluate(multi_window.test, verbose=0)
multi_window.plot(multi_linear_model)
437/437 [==============================] - 1s 2ms/step - loss: 0.2556 - mean_absolute_error: 0.3050

png

Dense

Adding a layers.Dense between the input and output gives the linear model more power, but is still only based on a single input timestep.

multi_dense_model = tf.keras.Sequential([
    # Take the last time step.
    # Shape [batch, time, features] => [batch, 1, features]
    tf.keras.layers.Lambda(lambda x: x[:, -1:, :]),
    # Shape => [batch, 1, dense_units]
    tf.keras.layers.Dense(512, activation='relu'),
    # Shape => [batch, out_steps*features]
    tf.keras.layers.Dense(OUT_STEPS*num_features,
                          kernel_initializer=tf.initializers.zeros),
    # Shape => [batch, out_steps, features]
    tf.keras.layers.Reshape([OUT_STEPS, num_features])
])

history = compile_and_fit(multi_dense_model, multi_window)

IPython.display.clear_output()
multi_val_performance['Dense'] = multi_dense_model.evaluate(multi_window.val)
multi_performance['Dense'] = multi_dense_model.evaluate(multi_window.test, verbose=0)
multi_window.plot(multi_dense_model)
437/437 [==============================] - 1s 3ms/step - loss: 0.2192 - mean_absolute_error: 0.2807

png

CNN

A convolutional model makes predictions based on a fixed-width history, which may lead to better performance than the dense model since it can see how things are changing over time:

A convolutional model sees how things change over time

CONV_WIDTH = 3
multi_conv_model = tf.keras.Sequential([
    # Shape [batch, time, features] => [batch, CONV_WIDTH, features]
    tf.keras.layers.Lambda(lambda x: x[:, -CONV_WIDTH:, :]),
    # Shape => [batch, 1, conv_units]
    tf.keras.layers.Conv1D(256, activation='relu', kernel_size=(CONV_WIDTH)),
    # Shape => [batch, 1,  out_steps*features]
    tf.keras.layers.Dense(OUT_STEPS*num_features,
                          kernel_initializer=tf.initializers.zeros),
    # Shape => [batch, out_steps, features]
    tf.keras.layers.Reshape([OUT_STEPS, num_features])
])

history = compile_and_fit(multi_conv_model, multi_window)

IPython.display.clear_output()

multi_val_performance['Conv'] = multi_conv_model.evaluate(multi_window.val)
multi_performance['Conv'] = multi_conv_model.evaluate(multi_window.test, verbose=0)
multi_window.plot(multi_conv_model)
437/437 [==============================] - 1s 3ms/step - loss: 0.2142 - mean_absolute_error: 0.2805

png

RNN

A recurrent model can learn to use a long history of inputs, if it's relevant to the predictions the model is making. Here the model will accumulate internal state for 24h, before making a single prediction for the next 24h.

In this single-shot format, the LSTM only needs to produce an output at the last time step, so set return_sequences=False.

The lstm accumulates state over the input window, and makes a single prediction for the next 24h

multi_lstm_model = tf.keras.Sequential([
    # Shape [batch, time, features] => [batch, lstm_units]
    # Adding more `lstm_units` just overfits more quickly.
    tf.keras.layers.LSTM(32, return_sequences=False),
    # Shape => [batch, out_steps*features]
    tf.keras.layers.Dense(OUT_STEPS*num_features,
                          kernel_initializer=tf.initializers.zeros),
    # Shape => [batch, out_steps, features]
    tf.keras.layers.Reshape([OUT_STEPS, num_features])
])

history = compile_and_fit(multi_lstm_model, multi_window)

IPython.display.clear_output()

multi_val_performance['LSTM'] = multi_lstm_model.evaluate(multi_window.val)
multi_performance['LSTM'] = multi_lstm_model.evaluate(multi_window.test, verbose=0)
multi_window.plot(multi_lstm_model)
437/437 [==============================] - 1s 3ms/step - loss: 0.2146 - mean_absolute_error: 0.2829

png

Advanced: Autoregressive model

The above models all predict the entire output sequence as a in a single step.

In some cases it may be helpful for the model to decompose this prediction into individual time steps. Then each model's output can be fed back into itself at each step and predictions can be made conditioned on the previous one, like in the classic Generating Sequences With Recurrent Neural Networks.

One clear advantage to this style of model is that it can be set up to produce output with a varying length.

You could take any of single single-step multi-output models trained in the first half of this tutorial and run in an autoregressive feedback loop, but here you'll focus on building a model that's been explicitly trained to do that.

Feedback a model's output to its input

RNN

This tutorial only builds an autoregressive RNN model, but this pattern could be applied to any model that was designed to output a single timestep.

The model will have the same basic form as the single-step LSTM models: An LSTM followed by a layers.Dense that converts the LSTM outputs to model predictions.

A layers.LSTM is a layers.LSTMCell wrapped in the higher level layers.RNN that manages the state and sequence results for you (See Keras RNNs for details).

In this case the model has to manually manage the inputs for each step so it uses layers.LSTMCell directly for the lower level, single time step interface.

class FeedBack(tf.keras.Model):
  def __init__(self, units, out_steps):
    super().__init__()
    self.out_steps = out_steps
    self.units = units
    self.lstm_cell = tf.keras.layers.LSTMCell(units)
    # Also wrap the LSTMCell in an RNN to simplify the `warmup` method.
    self.lstm_rnn = tf.keras.layers.RNN(self.lstm_cell, return_state=True)
    self.dense = tf.keras.layers.Dense(num_features)
feedback_model = FeedBack(units=32, out_steps=OUT_STEPS)

The first method this model needs is a warmup method to initialize its internal state based on the inputs. Once trained this state will capture the relevant parts of the input history. This is equivalent to the single-step LSTM model from earlier:

def warmup(self, inputs):
  # inputs.shape => (batch, time, features)
  # x.shape => (batch, lstm_units)
  x, *state = self.lstm_rnn(inputs)

  # predictions.shape => (batch, features)
  prediction = self.dense(x)
  return prediction, state

FeedBack.warmup = warmup

This method returns a single time-step prediction, and the internal state of the LSTM:

prediction, state = feedback_model.warmup(multi_window.example[0])
prediction.shape
TensorShape([32, 19])

With the RNN's state, and an initial prediction you can now continue iterating the model feeding the predictions at each step back as the input.

The simplest approach to collecting the output predictions is to use a python list, and tf.stack after the loop.

def call(self, inputs, training=None):
  # Use a TensorArray to capture dynamically unrolled outputs.
  predictions = []
  # Initialize the lstm state
  prediction, state = self.warmup(inputs)

  # Insert the first prediction
  predictions.append(prediction)

  # Run the rest of the prediction steps
  for n in range(1, self.out_steps):
    # Use the last prediction as input.
    x = prediction
    # Execute one lstm step.
    x, state = self.lstm_cell(x, states=state,
                              training=training)
    # Convert the lstm output to a prediction.
    prediction = self.dense(x)
    # Add the prediction to the output
    predictions.append(prediction)

  # predictions.shape => (time, batch, features)
  predictions = tf.stack(predictions)
  # predictions.shape => (batch, time, features)
  predictions = tf.transpose(predictions, [1, 0, 2])
  return predictions

FeedBack.call = call

Test run this model on the example inputs:

print('Output shape (batch, time, features): ', feedback_model(multi_window.example[0]).shape)
Output shape (batch, time, features):  (32, 24, 19)

Now train the model:

history = compile_and_fit(feedback_model, multi_window)

IPython.display.clear_output()

multi_val_performance['AR LSTM'] = feedback_model.evaluate(multi_window.val)
multi_performance['AR LSTM'] = feedback_model.evaluate(multi_window.test, verbose=0)
multi_window.plot(feedback_model)
437/437 [==============================] - 3s 8ms/step - loss: 0.2352 - mean_absolute_error: 0.3116

png

Performance

There are clearly diminishing returns as a function of model complexity on this problem.

x = np.arange(len(multi_performance))
width = 0.3


metric_name = 'mean_absolute_error'
metric_index = lstm_model.metrics_names.index('mean_absolute_error')
val_mae = [v[metric_index] for v in multi_val_performance.values()]
test_mae = [v[metric_index] for v in multi_performance.values()]

plt.bar(x - 0.17, val_mae, width, label='Validation')
plt.bar(x + 0.17, test_mae, width, label='Test')
plt.xticks(ticks=x, labels=multi_performance.keys(),
           rotation=45)
plt.ylabel(f'MAE (average over all times and outputs)')
_ = plt.legend()

png

The metrics for the multi-output models in the first half of this tutorial show the performance averaged across all output features. These performances similar but also averaged across output timesteps.

for name, value in multi_performance.items():
  print(f'{name:8s}: {value[1]:0.4f}')
Last    : 0.5157
Repeat  : 0.3774
Linear  : 0.2980
Dense   : 0.2754
Conv    : 0.2724
LSTM    : 0.2770
AR LSTM : 0.3026

The gains achieved going from a dense model to convolutional and recurrent models are only a few percent (if any), and the autoregressive model performed clearly worse. So these more complex approaches may not be worth while on this problem, but there was no way to know without trying, and these models could be helpful for your problem.

Next steps

This tutorial was a quick introduction to time series forecasting using TensorFlow.