Sequence to Sequence - for time series prediction

Question

I\'ve tried to build a sequence to sequence model to predict a sensor signal over time based on its first few inputs (see figure below)

The model works OK, but I wa

Answer 1

THIS IS THE ANSWER TO THE EDITED QUESTION

first of all, when you call fit, decoder_inputs is a tensor and you can't use it to fit your model. the author of the code you cited, use an array of zeros and so you have to do the same (I do it in the dummy example below)

secondly, look at your output layer in the model summary... it is 3D so you have to manage your target as 3D array

thirdly, the decoder input must be 1 feature dimension and not 20 as you reported

set initial parameters

layers = [35, 35]
learning_rate = 0.01
decay = 0 
optimiser = keras.optimizers.Adam(lr=learning_rate, decay=decay)

num_input_features = 20
num_output_features = 1
loss = "mse"

lambda_regulariser = 0.000001
regulariser = None

batch_size = 128
steps_per_epoch = 200
epochs = 100

define encoder

encoder_inputs = keras.layers.Input(shape=(None, num_input_features), name='encoder_input')

encoder_cells = []
for hidden_neurons in layers:
    encoder_cells.append(keras.layers.GRUCell(hidden_neurons,
                                              kernel_regularizer=regulariser,
                                              recurrent_regularizer=regulariser,
                                              bias_regularizer=regulariser))

encoder = keras.layers.RNN(encoder_cells, return_state=True, name='encoder_layer')
encoder_outputs_and_states = encoder(encoder_inputs)
encoder_states = encoder_outputs_and_states[1:] # only keep the states

define decoder (1 feature dimension input!)

decoder_inputs = keras.layers.Input(shape=(None, 1), name='decoder_input') #### <=== must be 1

decoder_cells = []
for hidden_neurons in layers:
    decoder_cells.append(keras.layers.GRUCell(hidden_neurons,
                                              kernel_regularizer=regulariser,
                                              recurrent_regularizer=regulariser,
                                              bias_regularizer=regulariser))

decoder = keras.layers.RNN(decoder_cells, return_sequences=True, return_state=True, name='decoder_layer')
decoder_outputs_and_states = decoder(decoder_inputs, initial_state=encoder_states)

decoder_outputs = decoder_outputs_and_states[0] # only keep the output sequence
decoder_dense = keras.layers.Dense(num_output_features,
                                   activation='linear',
                                   kernel_regularizer=regulariser,
                                   bias_regularizer=regulariser)

decoder_outputs = decoder_dense(decoder_outputs)

define model

model = keras.models.Model(inputs=[encoder_inputs, decoder_inputs], outputs=decoder_outputs)
model.compile(optimizer=optimiser, loss=loss)
model.summary()

Layer (type)                    Output Shape         Param #     Connected to                     
==================================================================================================
encoder_input (InputLayer)      (None, None, 20)     0                                            
__________________________________________________________________________________________________
decoder_input (InputLayer)      (None, None, 1)      0                                            
__________________________________________________________________________________________________
encoder_layer (RNN)             [(None, 35), (None,  13335       encoder_input[0][0]              
__________________________________________________________________________________________________
decoder_layer (RNN)             [(None, None, 35), ( 11340       decoder_input[0][0]              
                                                                 encoder_layer[0][1]              
                                                                 encoder_layer[0][2]              
__________________________________________________________________________________________________
dense_4 (Dense)                 (None, None, 1)      36          decoder_layer[0][0]              
==================================================================================================

this is my dummy data. the same as yours in shapes. pay attention to decoder_zero_inputs it has the same dimension of your y but is an array of zeros

train_x = np.random.uniform(0,1, (439, 5, 20))
train_y = np.random.uniform(0,1, (439, 56, 1))
validation_x = np.random.uniform(0,1, (10, 5, 20))
validation_y = np.random.uniform(0,1, (10, 56, 1))
decoder_zero_inputs = np.zeros((439, 56, 1)) ### <=== attention

fitting

history = model.fit([train_x, decoder_zero_inputs],train_y, epochs=epochs,
                     validation_split=0.3, verbose=1)

Epoch 1/100
307/307 [==============================] - 2s 8ms/step - loss: 0.1038 - val_loss: 0.0845
Epoch 2/100
307/307 [==============================] - 1s 2ms/step - loss: 0.0851 - val_loss: 0.0832
Epoch 3/100
307/307 [==============================] - 1s 2ms/step - loss: 0.0842 - val_loss: 0.0828

prediction on validation

pred_validation = model.predict([validation_x, np.zeros((10,56,1))])

Answer 2

the attention layer in Keras is not a trainable layer (unless we use the scale parameter). it only computes matrix operation. In my opinion, this layer can result in some mistakes if applied directly on time series, but let proceed with order...

the most natural choice to replicate the attention mechanism on our time-series problem is to adopt the solution presented here and explained again here. It's the classical application of attention in enc-dec structure in NLP

following TF implementation, for our attention layer, we need query, value, key tensor in 3d format. we obtain these values directly from our recurrent layer. more specifically we utilize the sequence output and the hidden state. these are all we need to build an attention mechanism.

query is the output sequence [batch_dim, time_step, features]

value is the hidden state [batch_dim, features] where we add a temporal dimension for matrix operation [batch_dim, 1, features]

as the key, we utilize as before the hidden state so key = value

In the above definition and implementation I found 2 problems:

• the scores are calculated with softmax(dot(sequence, hidden)). the dot is ok but the softmax following Keras implementation is calculated on the last dimension and not on the temporal dimension. this implies the scores to be all 1 so they are useless
• the output attention is dot(scores, hidden) and not dot(scores, sequences) as we need

the example:

def attention_keras(query_value):

    query, value = query_value # key == value
    score = tf.matmul(query, value, transpose_b=True) # (batch, timestamp, 1)
    score = tf.nn.softmax(score) # softmax on -1 axis ==> score always = 1 !!!
    print((score.numpy()!=1).any()) # False ==> score always = 1 !!!
    score = tf.matmul(score, value) # (batch, timestamp, feat)
    return score

np.random.seed(33)
time_steps = 20
features = 50
sample = 5

X = np.random.uniform(0,5, (sample,time_steps,features))
state = np.random.uniform(0,5, (sample,features))
attention_keras([X,tf.expand_dims(state,1)]) # ==> the same as Attention(dtype='float64')([X,tf.expand_dims(state,1)])

so for this reason, for time series attention I propose this solution

def attention_seq(query_value, scale):

    query, value = query_value
    score = tf.matmul(query, value, transpose_b=True) # (batch, timestamp, 1)
    score = scale*score # scale with a fixed number (it can be finetuned or learned during train)
    score = tf.nn.softmax(score, axis=1) # softmax on timestamp axis
    score = score*query # (batch, timestamp, feat)
    return score

np.random.seed(33)
time_steps = 20
features = 50
sample = 5

X = np.random.uniform(0,5, (sample,time_steps,features))
state = np.random.uniform(0,5, (sample,features))
attention_seq([X,tf.expand_dims(state,1)], scale=0.05)

query is the output sequence [batch_dim, time_step, features]

value is the hidden state [batch_dim, features] where we add a temporal dimension for matrix operation [batch_dim, 1, features]

the weights are calculated with softmax(scale*dot(sequence, hidden)). the scale parameter is a scalar value that can be used to scale the weights before applying the softmax operation. the softmax is calculated correctly on the time dimension. the attention output is the weighted product of input sequence and scores. I use the scalar parameter as a fixed value, but it can be tuned or insert as a learnable weight in a custom layer (as scale parameter in Keras attention).

In term of network implementation these are the two possibilities available:

######### KERAS #########
inp = Input((time_steps,features))
seq, state = GRU(32, return_state=True, return_sequences=True)(inp)
att = Attention()([seq, tf.expand_dims(state,1)])

######### CUSTOM #########
inp = Input((time_steps,features))
seq, state = GRU(32, return_state=True, return_sequences=True)(inp)
att = Lambda(attention_seq, arguments={'scale': 0.05})([seq, tf.expand_dims(state,1)])

CONCLUSION

I don't know how much added-value an introduction of an attention layer in simple problems can have. If you have short sequences, I suggest you leave all as is. What I reported here is an answer where I express my considerations, I'll accept comment or consideration about possible mistakes or misunderstandings

---

In your model, these solutions can be embedded in this way

######### KERAS #########
inp = Input((n_features, n_steps))
seq, state = GRU(n_units, activation='relu',
                 return_state=True, return_sequences=True)(inp)
att = Attention()([seq, tf.expand_dims(state,1)])
x = GRU(n_units, activation='relu')(att)
x = Dense(64, activation='relu')(x)
x = Dropout(0.5)(x)
out = Dense(n_steps_out)(x)

model = Model(inp, out)
model.compile(optimizer='adam', loss='mse', metrics=['mse'])
model.summary()

######### CUSTOM #########
inp = Input((n_features, n_steps))
seq, state = GRU(n_units, activation='relu',
                 return_state=True, return_sequences=True)(inp)
att = Lambda(attention_seq, arguments={'scale': 0.05})([seq, tf.expand_dims(state,1)])
x = GRU(n_units, activation='relu')(att)
x = Dense(64, activation='relu')(x)
x = Dropout(0.5)(x)
out = Dense(n_steps_out)(x)

model = Model(inp, out)
model.compile(optimizer='adam', loss='mse', metrics=['mse'])
model.summary()