# A primary go at multi-step prediction

We choose up the place the first post in this series left us: confronting the duty of multi-step time-series forecasting.

Our first try was a workaround of types. The mannequin had been educated to ship a single prediction, similar to the very subsequent cut-off date. Thus, if we would have liked an extended forecast, all we might do is use that prediction and feed it again to the mannequin, shifting the enter sequence by one worth (from ([x_{t-n}, …, x_t]) to ([x_{t-n-1}, …, x_{t+1}]), say).

In distinction, the brand new mannequin might be designed – and educated – to forecast a configurable variety of observations without delay. The structure will nonetheless be primary – about as primary as doable, given the duty – and thus, can function a baseline for later makes an attempt.

We work with the identical knowledge as earlier than, `vic_elec`

from `tsibbledata`

.

In comparison with final time although, the `dataset`

class has to alter. Whereas, beforehand, for every batch merchandise the goal (`y`

) was a single worth, it now’s a vector, identical to the enter, `x`

. And identical to `n_timesteps`

was (and nonetheless is) used to specify the size of the enter sequence, there may be now a second parameter, `n_forecast`

, to configure goal measurement.

In our instance, `n_timesteps`

and `n_forecast`

are set to the identical worth, however there isn’t a want for this to be the case. You can equally effectively practice on week-long sequences after which forecast developments over a single day, or a month.

Other than the truth that `.getitem()`

now returns a vector for `y`

in addition to `x`

, there may be not a lot to be mentioned about dataset creation. Right here is the whole code to arrange the info enter pipeline:

```
n_timesteps <- 7 * 24 * 2
n_forecast <- 7 * 24 * 2
batch_size <- 32
vic_elec_get_year <- operate(yr, month = NULL) {
vic_elec %>%
filter(yr(Date) == yr, month(Date) == if (is.null(month)) month(Date) else month) %>%
as_tibble() %>%
choose(Demand)
}
elec_train <- vic_elec_get_year(2012) %>% as.matrix()
elec_valid <- vic_elec_get_year(2013) %>% as.matrix()
elec_test <- vic_elec_get_year(2014, 1) %>% as.matrix()
train_mean <- mean(elec_train)
train_sd <- sd(elec_train)
elec_dataset <- dataset(
identify = "elec_dataset",
initialize = operate(x, n_timesteps, n_forecast, sample_frac = 1) {
self$n_timesteps <- n_timesteps
self$n_forecast <- n_forecast
self$x <- torch_tensor((x - train_mean) / train_sd)
n <- length(self$x) - self$n_timesteps - self$n_forecast + 1
self$begins <- sort(sample.int(
n = n,
measurement = n * sample_frac
))
},
.getitem = operate(i) {
begin <- self$begins[i]
finish <- begin + self$n_timesteps - 1
pred_length <- self$n_forecast
list(
x = self$x[start:end],
y = self$x[(end + 1):(end + pred_length)]$squeeze(2)
)
},
.size = operate() {
length(self$begins)
}
)
train_ds <- elec_dataset(elec_train, n_timesteps, n_forecast, sample_frac = 0.5)
train_dl <- train_ds %>% dataloader(batch_size = batch_size, shuffle = TRUE)
valid_ds <- elec_dataset(elec_valid, n_timesteps, n_forecast, sample_frac = 0.5)
valid_dl <- valid_ds %>% dataloader(batch_size = batch_size)
test_ds <- elec_dataset(elec_test, n_timesteps, n_forecast)
test_dl <- test_ds %>% dataloader(batch_size = 1)
```

The mannequin replaces the only linear layer that, within the earlier publish, had been tasked with outputting the ultimate prediction, with a small community, full with two linear layers and – non-obligatory – dropout.

In `ahead()`

, we first apply the RNN, and identical to within the earlier publish, we make use of the `outputs`

solely; or extra particularly, the `output`

similar to the ultimate time step. (See that earlier publish for a detailed discussion of what a `torch`

RNN returns.)

```
mannequin <- nn_module(
initialize = operate(kind, input_size, hidden_size, linear_size, output_size,
num_layers = 1, dropout = 0, linear_dropout = 0) {
self$kind <- kind
self$num_layers <- num_layers
self$linear_dropout <- linear_dropout
self$rnn <- if (self$kind == "gru") {
nn_gru(
input_size = input_size,
hidden_size = hidden_size,
num_layers = num_layers,
dropout = dropout,
batch_first = TRUE
)
} else {
nn_lstm(
input_size = input_size,
hidden_size = hidden_size,
num_layers = num_layers,
dropout = dropout,
batch_first = TRUE
)
}
self$mlp <- nn_sequential(
nn_linear(hidden_size, linear_size),
nn_relu(),
nn_dropout(linear_dropout),
nn_linear(linear_size, output_size)
)
},
ahead = operate(x) {
x <- self$rnn(x)
x[[1]][ ,-1, ..] %>%
self$mlp()
}
)
```

For mannequin instantiation, we now have an extra configuration parameter, associated to the quantity of dropout between the 2 linear layers.

```
internet <- mannequin(
"gru", input_size = 1, hidden_size = 32, linear_size = 512, output_size = n_forecast, linear_dropout = 0
)
# coaching RNNs on the GPU at present prints a warning which will muddle
# the console
# see https://github.com/mlverse/torch/points/461
# alternatively, use
# machine <- "cpu"
machine <- torch_device(if (cuda_is_available()) "cuda" else "cpu")
internet <- internet$to(machine = machine)
```

The coaching process is totally unchanged.

```
optimizer <- optim_adam(internet$parameters, lr = 0.001)
num_epochs <- 30
train_batch <- operate(b) {
optimizer$zero_grad()
output <- internet(b$x$to(machine = machine))
goal <- b$y$to(machine = machine)
loss <- nnf_mse_loss(output, goal)
loss$backward()
optimizer$step()
loss$merchandise()
}
valid_batch <- operate(b) {
output <- internet(b$x$to(machine = machine))
goal <- b$y$to(machine = machine)
loss <- nnf_mse_loss(output, goal)
loss$merchandise()
}
for (epoch in 1:num_epochs) {
internet$practice()
train_loss <- c()
coro::loop(for (b in train_dl) {
loss <-train_batch(b)
train_loss <- c(train_loss, loss)
})
cat(sprintf("nEpoch %d, coaching: loss: %3.5f n", epoch, mean(train_loss)))
internet$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, mean(valid_loss)))
}
```

```
# Epoch 1, coaching: loss: 0.65737
#
# Epoch 1, validation: loss: 0.54586
#
# Epoch 2, coaching: loss: 0.43991
#
# Epoch 2, validation: loss: 0.50588
#
# Epoch 3, coaching: loss: 0.42161
#
# Epoch 3, validation: loss: 0.50031
#
# Epoch 4, coaching: loss: 0.41718
#
# Epoch 4, validation: loss: 0.48703
#
# Epoch 5, coaching: loss: 0.39498
#
# Epoch 5, validation: loss: 0.49572
#
# Epoch 6, coaching: loss: 0.38073
#
# Epoch 6, validation: loss: 0.46813
#
# Epoch 7, coaching: loss: 0.36472
#
# Epoch 7, validation: loss: 0.44957
#
# Epoch 8, coaching: loss: 0.35058
#
# Epoch 8, validation: loss: 0.44440
#
# Epoch 9, coaching: loss: 0.33880
#
# Epoch 9, validation: loss: 0.41995
#
# Epoch 10, coaching: loss: 0.32545
#
# Epoch 10, validation: loss: 0.42021
#
# Epoch 11, coaching: loss: 0.31347
#
# Epoch 11, validation: loss: 0.39514
#
# Epoch 12, coaching: loss: 0.29622
#
# Epoch 12, validation: loss: 0.38146
#
# Epoch 13, coaching: loss: 0.28006
#
# Epoch 13, validation: loss: 0.37754
#
# Epoch 14, coaching: loss: 0.27001
#
# Epoch 14, validation: loss: 0.36636
#
# Epoch 15, coaching: loss: 0.26191
#
# Epoch 15, validation: loss: 0.35338
#
# Epoch 16, coaching: loss: 0.25533
#
# Epoch 16, validation: loss: 0.35453
#
# Epoch 17, coaching: loss: 0.25085
#
# Epoch 17, validation: loss: 0.34521
#
# Epoch 18, coaching: loss: 0.24686
#
# Epoch 18, validation: loss: 0.35094
#
# Epoch 19, coaching: loss: 0.24159
#
# Epoch 19, validation: loss: 0.33776
#
# Epoch 20, coaching: loss: 0.23680
#
# Epoch 20, validation: loss: 0.33974
#
# Epoch 21, coaching: loss: 0.23070
#
# Epoch 21, validation: loss: 0.34069
#
# Epoch 22, coaching: loss: 0.22761
#
# Epoch 22, validation: loss: 0.33724
#
# Epoch 23, coaching: loss: 0.22390
#
# Epoch 23, validation: loss: 0.34013
#
# Epoch 24, coaching: loss: 0.22155
#
# Epoch 24, validation: loss: 0.33460
#
# Epoch 25, coaching: loss: 0.21820
#
# Epoch 25, validation: loss: 0.33755
#
# Epoch 26, coaching: loss: 0.22134
#
# Epoch 26, validation: loss: 0.33678
#
# Epoch 27, coaching: loss: 0.21061
#
# Epoch 27, validation: loss: 0.33108
#
# Epoch 28, coaching: loss: 0.20496
#
# Epoch 28, validation: loss: 0.32769
#
# Epoch 29, coaching: loss: 0.20223
#
# Epoch 29, validation: loss: 0.32969
#
# Epoch 30, coaching: loss: 0.20022
#
# Epoch 30, validation: loss: 0.33331
```

From the best way loss decreases on the coaching set, we conclude that, sure, the mannequin is studying one thing. It in all probability would proceed bettering for fairly some epochs nonetheless. We do, nevertheless, see much less of an enchancment on the validation set.

Naturally, now we’re inquisitive about test-set predictions. (Keep in mind, for testing we’re selecting the “notably arduous” month of January, 2014 – notably arduous due to a heatwave that resulted in exceptionally excessive demand.)

With no loop to be coded, analysis now turns into fairly simple:

```
internet$eval()
test_preds <- vector(mode = "record", size = length(test_dl))
i <- 1
coro::loop(for (b in test_dl) {
enter <- b$x
output <- internet(enter$to(machine = machine))
preds <- as.numeric(output)
test_preds[[i]] <- preds
i <<- i + 1
})
vic_elec_jan_2014 <- vic_elec %>%
filter(yr(Date) == 2014, month(Date) == 1)
test_pred1 <- test_preds[[1]]
test_pred1 <- c(rep(NA, n_timesteps), test_pred1, rep(NA, nrow(vic_elec_jan_2014) - n_timesteps - n_forecast))
test_pred2 <- test_preds[[408]]
test_pred2 <- c(rep(NA, n_timesteps + 407), test_pred2, rep(NA, nrow(vic_elec_jan_2014) - 407 - n_timesteps - n_forecast))
test_pred3 <- test_preds[[817]]
test_pred3 <- c(rep(NA, nrow(vic_elec_jan_2014) - n_forecast), test_pred3)
preds_ts <- vic_elec_jan_2014 %>%
choose(Demand) %>%
add_column(
mlp_ex_1 = test_pred1 * train_sd + train_mean,
mlp_ex_2 = test_pred2 * train_sd + train_mean,
mlp_ex_3 = test_pred3 * train_sd + train_mean) %>%
pivot_longer(-Time) %>%
update_tsibble(key = identify)
preds_ts %>%
autoplot() +
scale_colour_manual(values = c("#08c5d1", "#00353f", "#ffbf66", "#d46f4d")) +
theme_minimal()
```

Evaluate this to the forecast obtained by feeding again predictions. The demand profiles over the day look much more sensible now. How in regards to the phases of maximum demand? Evidently, these should not mirrored within the forecast, not any greater than within the “loop method”. In truth, the forecast permits for fascinating insights into this mannequin’s character: Apparently, it actually likes fluctuating across the imply – “prime” it with inputs that oscillate round a considerably larger degree, and it’ll shortly shift again to its consolation zone.

Seeing how, above, we supplied an choice to make use of dropout contained in the MLP, chances are you’ll be questioning if this is able to assist with forecasts on the check set. Seems it didn’t, in my experiments. Possibly this isn’t so unusual both: How, absent exterior cues (temperature), ought to the community know that top demand is developing?

In our evaluation, we are able to make an extra distinction. With the primary week of predictions, what we see is a failure to anticipate one thing that *couldn’t* moderately have been anticipated (two, or two-and-a-half, say, days of exceptionally excessive demand). Within the second, all of the community would have needed to do was keep on the present, elevated degree. It will likely be fascinating to see how that is dealt with by the architectures we talk about subsequent.

Lastly, an extra concept you’ll have had is – what if we used temperature as a second enter variable? As a matter of truth, coaching efficiency certainly improved, however no efficiency impression was noticed on the validation and check units. Nonetheless, chances are you’ll discover the code helpful – it’s simply prolonged to datasets with extra predictors. Due to this fact, we reproduce it within the appendix.

Thanks for studying!

```
# Information enter code modified to accommodate two predictors
n_timesteps <- 7 * 24 * 2
n_forecast <- 7 * 24 * 2
vic_elec_get_year <- operate(yr, month = NULL) {
vic_elec %>%
filter(yr(Date) == yr, month(Date) == if (is.null(month)) month(Date) else month) %>%
as_tibble() %>%
choose(Demand, Temperature)
}
elec_train <- vic_elec_get_year(2012) %>% as.matrix()
elec_valid <- vic_elec_get_year(2013) %>% as.matrix()
elec_test <- vic_elec_get_year(2014, 1) %>% as.matrix()
train_mean_demand <- mean(elec_train[ , 1])
train_sd_demand <- sd(elec_train[ , 1])
train_mean_temp <- mean(elec_train[ , 2])
train_sd_temp <- sd(elec_train[ , 2])
elec_dataset <- dataset(
identify = "elec_dataset",
initialize = operate(knowledge, n_timesteps, n_forecast, sample_frac = 1) {
demand <- (knowledge[ , 1] - train_mean_demand) / train_sd_demand
temp <- (knowledge[ , 2] - train_mean_temp) / train_sd_temp
self$x <- cbind(demand, temp) %>% torch_tensor()
self$n_timesteps <- n_timesteps
self$n_forecast <- n_forecast
n <- nrow(self$x) - self$n_timesteps - self$n_forecast + 1
self$begins <- sort(sample.int(
n = n,
measurement = n * sample_frac
))
},
.getitem = operate(i) {
begin <- self$begins[i]
finish <- begin + self$n_timesteps - 1
pred_length <- self$n_forecast
list(
x = self$x[start:end, ],
y = self$x[(end + 1):(end + pred_length), 1]
)
},
.size = operate() {
length(self$begins)
}
)
### relaxation equivalent to single-predictor code above
```

Picture by Monica Bourgeau on Unsplash