That is the ultimate publish in a four-part introduction to time-series forecasting with `torch`

. These posts have been the story of a quest for multiple-step prediction, and by now, we’ve seen three completely different approaches: forecasting in a loop, incorporating a multi-layer perceptron (MLP), and sequence-to-sequence fashions. Right here’s a fast recap.

As one ought to when one units out for an adventurous journey, we began with an in-depth examine of the instruments at our disposal: recurrent neural networks (RNNs). We educated a mannequin to foretell the very subsequent statement in line, after which, considered a intelligent hack: How about we use this for multi-step prediction, feeding again particular person predictions in a loop? The end result , it turned out, was fairly acceptable.

Then, the journey actually began. We constructed our first mannequin “natively” for multi-step prediction, relieving the RNN a little bit of its workload and involving a second participant, a tiny-ish MLP. Now, it was the MLP’s activity to venture RNN output to a number of time factors sooner or later. Though outcomes have been fairly passable, we didn’t cease there.

As a substitute, we utilized to numerical time collection a way generally utilized in pure language processing (NLP): sequence-to-sequence (

*seq2seq*) prediction. Whereas forecast efficiency was not a lot completely different from the earlier case, we discovered the method to be extra intuitively interesting, because it displays the*causal*relationship between successive forecasts.

At this time we’ll enrich the seq2seq strategy by including a brand new element: the *consideration* module. Initially launched round 2014, consideration mechanisms have gained monumental traction, a lot so {that a} current paper title begins out “Consideration is Not All You Want”.

The thought is the next.

Within the basic encoder-decoder setup, the decoder will get “primed” with an encoder abstract only a single time: the time it begins its forecasting loop. From then on, it’s by itself. With consideration, nevertheless, it will get to see the whole sequence of encoder outputs once more each time it forecasts a brand new worth. What’s extra, each time, it will get to zoom in on *these* outputs that appear *related for the present prediction step*.

This can be a notably helpful technique in translation: In producing the subsequent phrase, a mannequin might want to know what a part of the supply sentence to give attention to. How a lot the method helps with numerical sequences, in distinction, will doubtless rely upon the options of the collection in query.

As earlier than, we work with `vic_elec`

, however this time, we partly deviate from the way in which we used to make use of it. With the unique, bi-hourly dataset, coaching the present mannequin takes a very long time, longer than readers will need to wait when experimenting. So as an alternative, we mixture observations by day. With a view to have sufficient knowledge, we practice on years 2012 and 2013, reserving 2014 for validation in addition to post-training inspection.

We’ll try and forecast demand as much as fourteen days forward. How lengthy, then, ought to be the enter sequences? This can be a matter of experimentation; all of the extra so now that we’re including within the consideration mechanism. (I think that it may not deal with very lengthy sequences so properly).

Under, we go along with fourteen days for enter size, too, however that won’t essentially be the absolute best selection for this collection.

```
n_timesteps <- 7 * 2
n_forecast <- 7 * 2
elec_dataset <- dataset(
identify = "elec_dataset",
initialize = perform(x, n_timesteps, sample_frac = 1) {
self$n_timesteps <- n_timesteps
self$x <- torch_tensor((x - train_mean) / train_sd)
n <- size(self$x) - self$n_timesteps - 1
self$begins <- kind(pattern.int(
n = n,
measurement = n * sample_frac
))
},
.getitem = perform(i) {
begin <- self$begins[i]
finish <- begin + self$n_timesteps - 1
lag <- 1
listing(
x = self$x[start:end],
y = self$x[(start+lag):(end+lag)]$squeeze(2)
)
},
.size = perform() {
size(self$begins)
}
)
batch_size <- 32
train_ds <- elec_dataset(elec_train, n_timesteps)
train_dl <- train_ds %>% dataloader(batch_size = batch_size, shuffle = TRUE)
valid_ds <- elec_dataset(elec_valid, n_timesteps)
valid_dl <- valid_ds %>% dataloader(batch_size = batch_size)
test_ds <- elec_dataset(elec_test, n_timesteps)
test_dl <- test_ds %>% dataloader(batch_size = 1)
```

Mannequin-wise, we once more encounter the three *modules* acquainted from the earlier publish: encoder, decoder, and top-level seq2seq module. Nonetheless, there’s an extra element: the *consideration* module, utilized by the decoder to acquire *consideration weights*.

### Encoder

The encoder nonetheless works the identical method. It wraps an RNN, and returns the ultimate state.

```
encoder_module <- nn_module(
initialize = perform(kind, input_size, hidden_size, num_layers = 1, dropout = 0) {
self$kind <- kind
self$rnn <- if (self$kind == "gru") {
nn_gru(
input_size = input_size,
hidden_size = hidden_size,
num_layers = num_layers,
dropout = dropout,
batch_first = TRUE
)
} else {
nn_lstm(
input_size = input_size,
hidden_size = hidden_size,
num_layers = num_layers,
dropout = dropout,
batch_first = TRUE
)
}
},
ahead = perform(x) {
# return outputs for all timesteps, in addition to last-timestep states for all layers
x %>% self$rnn()
}
)
```

### Consideration module

In fundamental seq2seq, each time it needed to generate a brand new worth, the decoder took under consideration two issues: its prior state, and the earlier output generated. In an attention-enriched setup, the decoder moreover receives the whole output from the encoder. In deciding what subset of that output ought to matter, it will get assist from a brand new agent, the eye module.

This, then, is the eye module’s raison d’être: Given present decoder state and properly as full encoder outputs, get hold of a weighting of these outputs indicative of how related they’re to what the decoder is at the moment as much as. This process ends in the so-called *consideration weights*: a normalized rating, for every time step within the encoding, that quantify their respective significance.

Consideration could also be carried out in quite a few other ways. Right here, we present two implementation choices, one additive, and one multiplicative.

#### Additive consideration

In additive consideration, encoder outputs and decoder state are generally both added or concatenated (we select to do the latter, beneath). The ensuing tensor is run by way of a linear layer, and a softmax is utilized for normalization.

```
attention_module_additive <- nn_module(
initialize = perform(hidden_dim, attention_size) {
self$consideration <- nn_linear(2 * hidden_dim, attention_size)
},
ahead = perform(state, encoder_outputs) {
# perform argument shapes
# encoder_outputs: (bs, timesteps, hidden_dim)
# state: (1, bs, hidden_dim)
# multiplex state to permit for concatenation (dimensions 1 and a pair of should agree)
seq_len <- dim(encoder_outputs)[2]
# ensuing form: (bs, timesteps, hidden_dim)
state_rep <- state$permute(c(2, 1, 3))$repeat_interleave(seq_len, 2)
# concatenate alongside function dimension
concat <- torch_cat(listing(state_rep, encoder_outputs), dim = 3)
# run by way of linear layer with tanh
# ensuing form: (bs, timesteps, attention_size)
scores <- self$consideration(concat) %>%
torch_tanh()
# sum over consideration dimension and normalize
# ensuing form: (bs, timesteps)
attention_weights <- scores %>%
torch_sum(dim = 3) %>%
nnf_softmax(dim = 2)
# a normalized rating for each supply token
attention_weights
}
)
```

#### Multiplicative consideration

In multiplicative consideration, scores are obtained by computing dot merchandise between decoder state and the entire encoder outputs. Right here too, a softmax is then used for normalization.

```
attention_module_multiplicative <- nn_module(
initialize = perform() {
NULL
},
ahead = perform(state, encoder_outputs) {
# perform argument shapes
# encoder_outputs: (bs, timesteps, hidden_dim)
# state: (1, bs, hidden_dim)
# enable for matrix multiplication with encoder_outputs
state <- state$permute(c(2, 3, 1))
# put together for scaling by variety of options
d <- torch_tensor(dim(encoder_outputs)[3], dtype = torch_float())
# scaled dot merchandise between state and outputs
# ensuing form: (bs, timesteps, 1)
scores <- torch_bmm(encoder_outputs, state) %>%
torch_div(torch_sqrt(d))
# normalize
# ensuing form: (bs, timesteps)
attention_weights <- scores$squeeze(3) %>%
nnf_softmax(dim = 2)
# a normalized rating for each supply token
attention_weights
}
)
```

### Decoder

As soon as consideration weights have been computed, their precise software is dealt with by the decoder. Concretely, the strategy in query, `weighted_encoder_outputs()`

, computes a product of weights and encoder outputs, ensuring that every output can have applicable affect.

The remainder of the motion then occurs in `ahead()`

. A concatenation of weighted encoder outputs (usually known as “context”) and present enter is run by way of an RNN. Then, an ensemble of RNN output, context, and enter is handed to an MLP. Lastly, each RNN state and present prediction are returned.

```
decoder_module <- nn_module(
initialize = perform(kind, input_size, hidden_size, attention_type, attention_size = 8, num_layers = 1) {
self$kind <- kind
self$rnn <- if (self$kind == "gru") {
nn_gru(
input_size = input_size,
hidden_size = hidden_size,
num_layers = num_layers,
batch_first = TRUE
)
} else {
nn_lstm(
input_size = input_size,
hidden_size = hidden_size,
num_layers = num_layers,
batch_first = TRUE
)
}
self$linear <- nn_linear(2 * hidden_size + 1, 1)
self$consideration <- if (attention_type == "multiplicative") attention_module_multiplicative()
else attention_module_additive(hidden_size, attention_size)
},
weighted_encoder_outputs = perform(state, encoder_outputs) {
# encoder_outputs is (bs, timesteps, hidden_dim)
# state is (1, bs, hidden_dim)
# ensuing form: (bs * timesteps)
attention_weights <- self$consideration(state, encoder_outputs)
# ensuing form: (bs, 1, seq_len)
attention_weights <- attention_weights$unsqueeze(2)
# ensuing form: (bs, 1, hidden_size)
weighted_encoder_outputs <- torch_bmm(attention_weights, encoder_outputs)
weighted_encoder_outputs
},
ahead = perform(x, state, encoder_outputs) {
# encoder_outputs is (bs, timesteps, hidden_dim)
# state is (1, bs, hidden_dim)
# ensuing form: (bs, 1, hidden_size)
context <- self$weighted_encoder_outputs(state, encoder_outputs)
# concatenate enter and context
# NOTE: this repeating is finished to compensate for the absence of an embedding module
# that, in NLP, would give x the next proportion within the concatenation
x_rep <- x$repeat_interleave(dim(context)[3], 3)
rnn_input <- torch_cat(listing(x_rep, context), dim = 3)
# ensuing shapes: (bs, 1, hidden_size) and (1, bs, hidden_size)
rnn_out <- self$rnn(rnn_input, state)
rnn_output <- rnn_out[[1]]
next_hidden <- rnn_out[[2]]
mlp_input <- torch_cat(listing(rnn_output$squeeze(2), context$squeeze(2), x$squeeze(2)), dim = 2)
output <- self$linear(mlp_input)
# shapes: (bs, 1) and (1, bs, hidden_size)
listing(output, next_hidden)
}
)
```

`seq2seq`

module

The `seq2seq`

module is principally unchanged (aside from the truth that now, it permits for consideration module configuration). For an in depth clarification of what occurs right here, please seek the advice of the earlier publish.

```
seq2seq_module <- nn_module(
initialize = perform(kind, input_size, hidden_size, attention_type, attention_size, n_forecast,
num_layers = 1, encoder_dropout = 0) {
self$encoder <- encoder_module(kind = kind, input_size = input_size, hidden_size = hidden_size,
num_layers, encoder_dropout)
self$decoder <- decoder_module(kind = kind, input_size = 2 * hidden_size, hidden_size = hidden_size,
attention_type = attention_type, attention_size = attention_size, num_layers)
self$n_forecast <- n_forecast
},
ahead = perform(x, y, teacher_forcing_ratio) {
outputs <- torch_zeros(dim(x)[1], self$n_forecast)
encoded <- self$encoder(x)
encoder_outputs <- encoded[[1]]
hidden <- encoded[[2]]
# listing of (batch_size, 1), (1, batch_size, hidden_size)
out <- self$decoder(x[ , n_timesteps, , drop = FALSE], hidden, encoder_outputs)
# (batch_size, 1)
pred <- out[[1]]
# (1, batch_size, hidden_size)
state <- out[[2]]
outputs[ , 1] <- pred$squeeze(2)
for (t in 2:self$n_forecast) {
teacher_forcing <- runif(1) < teacher_forcing_ratio
enter <- if (teacher_forcing == TRUE) y[ , t - 1, drop = FALSE] else pred
enter <- enter$unsqueeze(3)
out <- self$decoder(enter, state, encoder_outputs)
pred <- out[[1]]
state <- out[[2]]
outputs[ , t] <- pred$squeeze(2)
}
outputs
}
)
```

When instantiating the top-level mannequin, we now have an extra selection: that between additive and multiplicative consideration. Within the “accuracy” sense of efficiency, my exams didn’t present any variations. Nonetheless, the multiplicative variant is rather a lot sooner.

```
web <- seq2seq_module("gru", input_size = 1, hidden_size = 32, attention_type = "multiplicative",
attention_size = 8, n_forecast = n_forecast)
```

Similar to final time, in mannequin coaching, we get to decide on the diploma of trainer forcing. Under, we go along with a fraction of 0.0, that’s, no forcing in any respect.

```
optimizer <- optim_adam(web$parameters, lr = 0.001)
num_epochs <- 1000
train_batch <- perform(b, teacher_forcing_ratio) {
optimizer$zero_grad()
output <- web(b$x, b$y, teacher_forcing_ratio)
goal <- b$y
loss <- nnf_mse_loss(output, goal[ , 1:(dim(output)[2])])
loss$backward()
optimizer$step()
loss$merchandise()
}
valid_batch <- perform(b, teacher_forcing_ratio = 0) {
output <- web(b$x, b$y, teacher_forcing_ratio)
goal <- b$y
loss <- nnf_mse_loss(output, goal[ , 1:(dim(output)[2])])
loss$merchandise()
}
for (epoch in 1:num_epochs) {
web$practice()
train_loss <- c()
coro::loop(for (b in train_dl) {
loss <-train_batch(b, teacher_forcing_ratio = 0.0)
train_loss <- c(train_loss, loss)
})
cat(sprintf("nEpoch %d, coaching: loss: %3.5f n", epoch, imply(train_loss)))
web$eval()
valid_loss <- c()
coro::loop(for (b in valid_dl) {
loss <- valid_batch(b)
valid_loss <- c(valid_loss, loss)
})
cat(sprintf("nEpoch %d, validation: loss: %3.5f n", epoch, imply(valid_loss)))
}
```

```
# Epoch 1, coaching: loss: 0.83752
# Epoch 1, validation: loss: 0.83167
# Epoch 2, coaching: loss: 0.72803
# Epoch 2, validation: loss: 0.80804
# ...
# ...
# Epoch 99, coaching: loss: 0.10385
# Epoch 99, validation: loss: 0.21259
# Epoch 100, coaching: loss: 0.10396
# Epoch 100, validation: loss: 0.20975
```

For visible inspection, we choose a couple of forecasts from the check set.

```
web$eval()
test_preds <- vector(mode = "listing", size = size(test_dl))
i <- 1
vic_elec_test <- vic_elec_daily %>%
filter(yr(Date) == 2014, month(Date) %in% 1:4)
coro::loop(for (b in test_dl) {
output <- web(b$x, b$y, teacher_forcing_ratio = 0)
preds <- as.numeric(output)
test_preds[[i]] <- preds
i <<- i + 1
})
test_pred1 <- test_preds[[1]]
test_pred1 <- c(rep(NA, n_timesteps), test_pred1, rep(NA, nrow(vic_elec_test) - n_timesteps - n_forecast))
test_pred2 <- test_preds[[21]]
test_pred2 <- c(rep(NA, n_timesteps + 20), test_pred2, rep(NA, nrow(vic_elec_test) - 20 - n_timesteps - n_forecast))
test_pred3 <- test_preds[[41]]
test_pred3 <- c(rep(NA, n_timesteps + 40), test_pred3, rep(NA, nrow(vic_elec_test) - 40 - n_timesteps - n_forecast))
test_pred4 <- test_preds[[61]]
test_pred4 <- c(rep(NA, n_timesteps + 60), test_pred4, rep(NA, nrow(vic_elec_test) - 60 - n_timesteps - n_forecast))
test_pred5 <- test_preds[[81]]
test_pred5 <- c(rep(NA, n_timesteps + 80), test_pred5, rep(NA, nrow(vic_elec_test) - 80 - n_timesteps - n_forecast))
preds_ts <- vic_elec_test %>%
choose(Demand, Date) %>%
add_column(
ex_1 = test_pred1 * train_sd + train_mean,
ex_2 = test_pred2 * train_sd + train_mean,
ex_3 = test_pred3 * train_sd + train_mean,
ex_4 = test_pred4 * train_sd + train_mean,
ex_5 = test_pred5 * train_sd + train_mean) %>%
pivot_longer(-Date) %>%
update_tsibble(key = identify)
preds_ts %>%
autoplot() +
scale_color_hue(h = c(80, 300), l = 70) +
theme_minimal()
```

We are able to’t instantly examine efficiency right here to that of earlier fashions in our collection, as we’ve pragmatically redefined the duty. The primary aim, nevertheless, has been to introduce the idea of consideration. Particularly, find out how to *manually* implement the method – one thing that, when you’ve understood the idea, chances are you’ll by no means must do in apply. As a substitute, you’d doubtless make use of present instruments that include `torch`

(multi-head consideration and transformer modules), instruments we might introduce in a future “season” of this collection.

Thanks for studying!

Photograph by David Clode on Unsplash

*CoRR*abs/1409.0473. http://arxiv.org/abs/1409.0473.

*arXiv e-Prints*, March, arXiv:2103.03404. https://arxiv.org/abs/2103.03404.

*arXiv e-Prints*, June, arXiv:1706.03762. https://arxiv.org/abs/1706.03762.

*CoRR*abs/1412.7449. http://arxiv.org/abs/1412.7449.

*CoRR*abs/1502.03044. http://arxiv.org/abs/1502.03044.