This is the ultimate submit 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 totally different approaches: forecasting in a loop, incorporating a multi-layer perceptron (MLP), and sequence-to-sequence fashions. Here’s a fast recap.
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As one ought to when one units out for an adventurous journey, we began with an in-depth research of the instruments at our disposal: recurrent neural networks (RNNs). We educated a mannequin to foretell the very subsequent remark 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 consequence , it turned out, was fairly acceptable.
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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 process to challenge RNN output to a number of time factors sooner or later. Although outcomes have been fairly passable, we didn’t cease there.
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Instead, we utilized to numerical time collection a method generally utilized in pure language processing (NLP): sequence-to-sequence (seq2seq) prediction. While forecast efficiency was not a lot totally different from the earlier case, we discovered the approach to be extra intuitively interesting, because it displays the causal relationship between successive forecasts.
Today we’ll enrich the seq2seq method by including a brand new part: the consideration module. Originally launched round 2014, consideration mechanisms have gained huge traction, a lot so {that a} current paper title begins out “Attention is Not All You Need”.
The thought is the next.
In the traditional 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 entire 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 is a very helpful technique in translation: In producing the following phrase, a mannequin might want to know what a part of the supply sentence to concentrate on. How a lot the approach helps with numerical sequences, in distinction, will possible rely on 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 a substitute, we mixture observations by day. In order to have sufficient information, we prepare on years 2012 and 2013, reserving 2014 for validation in addition to post-training inspection.
We’ll try to forecast demand as much as fourteen days forward. How lengthy, then, ought to be the enter sequences? This is 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 nicely).
Below, we go together 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 = operate(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$begins <- type(sample.int(
n = n,
dimension = n * sample_frac
))
},
.getitem = operate(i) {
begin <- self$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 = operate() {
size(self$begins)
}
)
batch_size <- 32
train_ds <- elec_dataset(elec_train, n_timesteps)
train_dl <- train_ds %>% dataloader(batch_size = batch_size, shuffle = TRUE)
valid_ds <- elec_dataset(elec_valid, n_timesteps)
valid_dl <- valid_ds %>% dataloader(batch_size = batch_size)
test_ds <- elec_dataset(elec_test, n_timesteps)
test_dl <- test_ds %>% dataloader(batch_size = 1)Model-wise, we once more encounter the three modules acquainted from the earlier submit: encoder, decoder, and top-level seq2seq module. However, there may be a further part: the consideration module, utilized by the decoder to acquire consideration weights.
Encoder
The encoder nonetheless works the identical means. It wraps an RNN, and returns the ultimate state.
encoder_module <- nn_module(
initialize = operate(sort, input_size, hidden_size, num_layers = 1, dropout = 0) {
self$sort <- sort
self$rnn <- if (self$sort == "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 = operate(x) {
# return outputs for all timesteps, in addition to last-timestep states for all layers
x %>% self$rnn()
}
)Attention module
In primary seq2seq, each time it needed to generate a brand new worth, the decoder took into consideration two issues: its prior state, and the earlier output generated. In an attention-enriched setup, the decoder moreover receives the entire output from the encoder. In deciding what subset of that output ought to matter, it will get assist from a brand new agent, the eye module.
This, then, is the eye module’s raison d’être: Given present decoder state and nicely as full encoder outputs, get hold of a weighting of these outputs indicative of how related they’re to what the decoder is at the moment as much as. This process leads to the so-called consideration weights: a normalized rating, for every time step within the encoding, that quantify their respective significance.
Attention could also be applied in various alternative ways. Here, we present two implementation choices, one additive, and one multiplicative.
Additive consideration
In additive consideration, encoder outputs and decoder state are generally both added or concatenated (we select to do the latter, beneath). The ensuing tensor is run via a linear layer, and a softmax is utilized for normalization.
attention_module_additive <- nn_module(
initialize = operate(hidden_dim, attention_size) {
self$consideration <- nn_linear(2 * hidden_dim, attention_size)
},
ahead = operate(state, encoder_outputs) {
# operate 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 characteristic dimension
concat <- torch_cat(listing(state_rep, encoder_outputs), dim = 3)
# run via linear layer with tanh
# ensuing form: (bs, timesteps, attention_size)
scores <- self$consideration(concat) %>%
torch_tanh()
# sum over consideration dimension and normalize
# ensuing form: (bs, timesteps)
attention_weights <- scores %>%
torch_sum(dim = 3) %>%
nnf_softmax(dim = 2)
# a normalized rating for each supply token
attention_weights
}
)Multiplicative consideration
In multiplicative consideration, scores are obtained by computing dot merchandise between decoder state and all the encoder outputs. Here too, a softmax is then used for normalization.
attention_module_multiplicative <- nn_module(
initialize = operate() {
NULL
},
ahead = operate(state, encoder_outputs) {
# operate argument shapes
# encoder_outputs: (bs, timesteps, hidden_dim)
# state: (1, bs, hidden_dim)
# enable for matrix multiplication with encoder_outputs
state <- state$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$squeeze(3) %>%
nnf_softmax(dim = 2)
# a normalized rating for each supply token
attention_weights
}
)Decoder
Once consideration weights have been computed, their precise software is dealt with by the decoder. Concretely, the strategy in query, weighted_encoder_outputs(), computes a product of weights and encoder outputs, ensuring that every output can have acceptable impression.
The remainder of the motion then occurs in ahead(). A concatenation of weighted encoder outputs (typically known as “context”) and present enter is run via an RNN. Then, an ensemble of RNN output, context, and enter is handed to an MLP. Finally, each RNN state and present prediction are returned.
decoder_module <- nn_module(
initialize = operate(sort, input_size, hidden_size, attention_type, attention_size = 8, num_layers = 1) {
self$sort <- sort
self$rnn <- if (self$sort == "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 = operate(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 = operate(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$weighted_encoder_outputs(state, encoder_outputs)
# concatenate enter and context
# NOTE: this repeating is finished to compensate for the absence of an embedding module
# that, in NLP, would give x a better proportion within the concatenation
x_rep <- x$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$linear(mlp_input)
# shapes: (bs, 1) and (1, bs, hidden_size)
listing(output, next_hidden)
}
)seq2seq module
The seq2seq module is principally unchanged (aside from the truth that now, it permits for consideration module configuration). For an in depth rationalization of what occurs right here, please seek the advice of the earlier submit.
seq2seq_module <- nn_module(
initialize = operate(sort, input_size, hidden_size, attention_type, attention_size, n_forecast,
num_layers = 1, encoder_dropout = 0) {
self$encoder <- encoder_module(sort = sort, input_size = input_size, hidden_size = hidden_size,
num_layers, encoder_dropout)
self$decoder <- decoder_module(sort = sort, input_size = 2 * hidden_size, hidden_size = hidden_size,
attention_type = attention_type, attention_size = attention_size, num_layers)
self$n_forecast <- n_forecast
},
ahead = operate(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 a further selection: that between additive and multiplicative consideration. In the “accuracy” sense of efficiency, my assessments didn’t present any variations. However, the multiplicative variant is so much quicker.
internet <- seq2seq_module("gru", input_size = 1, hidden_size = 32, attention_type = "multiplicative",
attention_size = 8, n_forecast = n_forecast)Just like final time, in mannequin coaching, we get to decide on the diploma of instructor forcing. Below, we go together with a fraction of 0.0, that’s, no forcing in any respect.
optimizer <- optim_adam(internet$parameters, lr = 0.001)
num_epochs <- 1000
train_batch <- operate(b, teacher_forcing_ratio) {
optimizer$zero_grad()
output <- internet(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 <- operate(b, teacher_forcing_ratio = 0) {
output <- internet(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) {
internet$prepare()
train_loss <- c()
coro::loop(for (b in train_dl) {
loss <-train_batch(b, teacher_forcing_ratio = 0.0)
train_loss <- c(train_loss, loss)
})
cat(sprintf("nEpoch %d, coaching: loss: %3.5f n", epoch, imply(train_loss)))
internet$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 just a few forecasts from the take a look at set.
internet$eval()
test_preds <- vector(mode = "listing", size = size(test_dl))
i <- 1
vic_elec_test <- vic_elec_daily %>%
filter(12 months(Date) == 2014, month(Date) %in% 1:4)
coro::loop(for (b in test_dl) {
output <- internet(b$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()
Figure 1: A pattern of two-weeks-ahead predictions for the take a look at set, 2014.
We can’t immediately examine efficiency right here to that of earlier fashions in our collection, as we’ve pragmatically redefined the duty. The major objective, nevertheless, has been to introduce the idea of consideration. Specifically, methods to manually implement the approach – one thing that, when you’ve understood the idea, it’s possible you’ll by no means must do in follow. Instead, you’ll possible 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!
Photo by David Clode on Unsplash