Introductory time-series forecasting with torch

That is the primary submit in a sequence introducing time-series forecasting with torch. It does assume some prior expertise with torch and/or deep studying. However so far as time sequence are involved, it begins proper from the start, utilizing recurrent neural networks (GRU or LSTM) to foretell how one thing develops in time.

On this submit, we construct a community that makes use of a sequence of observations to foretell a price for the very subsequent time limit. What if we’d wish to forecast a sequence of values, akin to, say, per week or a month of measurements?

One factor we might do is feed again into the system the beforehand forecasted worth; that is one thing we’ll attempt on the finish of this submit. Subsequent posts will discover different choices, a few of them involving considerably extra complicated architectures. It is going to be attention-grabbing to check their performances; however the important aim is to introduce some torch “recipes” which you can apply to your individual information.

We begin by analyzing the dataset used. It’s a low-dimensional, however fairly polyvalent and complicated one.

The vic_elec dataset, out there by package deal tsibbledata, offers three years of half-hourly electrical energy demand for Victoria, Australia, augmented by same-resolution temperature info and a every day vacation indicator.

Rows: 52,608
Columns: 5
$ Time        <dttm> 2012-01-01 00:00:00, 2012-01-01 00:30:00, 2012-01-01 01:00:00,…
$ Demand      <dbl> 4382.825, 4263.366, 4048.966, 3877.563, 4036.230, 3865.597, 369…
$ Temperature <dbl> 21.40, 21.05, 20.70, 20.55, 20.40, 20.25, 20.10, 19.60, 19.10, …
$ Date        <date> 2012-01-01, 2012-01-01, 2012-01-01, 2012-01-01, 2012-01-01, 20…
$ Vacation     <lgl> TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRU…

Relying on what subset of variables is used, and whether or not and the way information is temporally aggregated, these information might serve as an example a wide range of totally different methods. For instance, within the third version of Forecasting: Principles and Practice every day averages are used to show quadratic regression with ARMA errors. On this first introductory submit although, in addition to in most of its successors, we’ll try to forecast Demand with out counting on further info, and we preserve the unique decision.

To get an impression of how electrical energy demand varies over totally different timescales. Let’s examine information for 2 months that properly illustrate the U-shaped relationship between temperature and demand: January, 2014 and July, 2014.

First, right here is July.

vic_elec_2014 <-  vic_elec %>%
  filter(year(Date) == 2014) %>%
  choose(-c(Date, Vacation)) %>%
  mutate(Demand = scale(Demand), Temperature = scale(Temperature)) %>%
  pivot_longer(-Time, names_to = "variable") %>%
  update_tsibble(key = variable)

vic_elec_2014 %>% filter(month(Time) == 7) %>% 
  autoplot() + 
  scale_colour_manual(values = c("#08c5d1", "#00353f")) +
  theme_minimal()

Determine 1: Temperature and electrical energy demand (normalized). Victoria, Australia, 07/2014.

It’s winter; temperature fluctuates beneath common, whereas electrical energy demand is above common (heating). There’s robust variation over the course of the day; we see troughs within the demand curve akin to ridges within the temperature graph, and vice versa. Whereas diurnal variation dominates, there is also variation over the times of the week. Between weeks although, we don’t see a lot distinction.

Examine this with the information for January:

vic_elec_2014 %>% filter(month(Time) == 1) %>% 
  autoplot() + 
  scale_colour_manual(values = c("#08c5d1", "#00353f")) +
  theme_minimal()

Determine 2: Temperature and electrical energy demand (normalized). Victoria, Australia, 01/2014.

We nonetheless see the robust circadian variation. We nonetheless see some day-of-week variation. However now it’s excessive temperatures that trigger elevated demand (cooling). Additionally, there are two intervals of unusually excessive temperatures, accompanied by distinctive demand. We anticipate that in a univariate forecast, not considering temperature, this might be exhausting – and even, unimaginable – to forecast.

Let’s see a concise portrait of how Demand behaves utilizing feasts::STL(). First, right here is the decomposition for July:

vic_elec_2014 <-  vic_elec %>%
  filter(year(Date) == 2014) %>%
  choose(-c(Date, Vacation))

cmp <- vic_elec_2014 %>% filter(month(Time) == 7) %>%
  mannequin(STL(Demand)) %>% 
  elements()

cmp %>% autoplot()

Determine 3: STL decomposition of electrical energy demand. Victoria, Australia, 07/2014.

And right here, for January:

Determine 4: STL decomposition of electrical energy demand. Victoria, Australia, 01/2014.

Each properly illustrate the robust circadian and weekly seasonalities (with diurnal variation considerably stronger in January). If we glance intently, we are able to even see how the pattern element is extra influential in January than in July. This once more hints at a lot stronger difficulties predicting the January than the July developments.

Now that we’ve got an concept what awaits us, let’s start by making a torch dataset.

Here’s what we intend to do. We wish to begin our journey into forecasting by utilizing a sequence of observations to foretell their quick successor. In different phrases, the enter (x) for every batch merchandise is a vector, whereas the goal (y) is a single worth. The size of the enter sequence, x, is parameterized as n_timesteps, the variety of consecutive observations to extrapolate from.

The dataset will replicate this in its .getitem() methodology. When requested for the observations at index i, it’ll return tensors like so:

list(
      x = self$x[start:end],
      y = self$x[end+1]
)

the place begin:finish is a vector of indices, of size n_timesteps, and finish+1 is a single index.

Now, if the dataset simply iterated over its enter so as, advancing the index one after the other, these traces might merely learn

list(
      x = self$x[i:(i + self$n_timesteps - 1)],
      y = self$x[self$n_timesteps + i]
)

Since many sequences within the information are comparable, we are able to scale back coaching time by making use of a fraction of the information in each epoch. This may be completed by (optionally) passing a sample_frac smaller than 1. In initialize(), a random set of begin indices is ready; .getitem() then simply does what it usually does: search for the (x,y) pair at a given index.

Right here is the whole dataset code:

elec_dataset <- dataset(
  title = "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 <- length(self$x) - self$n_timesteps 
    
    self$begins <- sort(sample.int(
      n = n,
      measurement = n * sample_frac
    ))

  },
  
  .getitem = operate(i) {
    
    begin <- self$begins[i]
    finish <- begin + self$n_timesteps - 1
    
    list(
      x = self$x[start:end],
      y = self$x[end + 1]
    )

  },
  
  .size = operate() {
    length(self$begins) 
  }
)

You’ll have seen that we normalize the information by globally outlined train_mean and train_sd. We but need to calculate these.

The way in which we break up the information is easy. We use the entire of 2012 for coaching, and all of 2013 for validation. For testing, we take the “tough” month of January, 2014. You’re invited to check testing outcomes for July that very same yr, and examine performances.

vic_elec_get_year <- operate(yr, month = NULL) {
  vic_elec %>%
    filter(year(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() # or 2014, 7, alternatively

train_mean <- mean(elec_train)
train_sd <- sd(elec_train)

Now, to instantiate a dataset, we nonetheless want to choose sequence size. From prior inspection, per week looks as if a good choice.

n_timesteps <- 7 * 24 * 2 # days * hours * half-hours

Now we are able to go forward and create a dataset for the coaching information. Let’s say we’ll make use of fifty% of the information in every epoch:

train_ds <- elec_dataset(elec_train, n_timesteps, sample_frac = 0.5)
length(train_ds)

 8615

Fast verify: Are the shapes right?

$x
torch_tensor
-0.4141
-0.5541
[...]       ### traces eliminated by me
 0.8204
 0.9399
... [the output was truncated (use n=-1 to disable)]
[ CPUFloatType{336,1} ]

$y
torch_tensor
-0.6771
[ CPUFloatType{1} ]

Sure: That is what we wished to see. The enter sequence has n_timesteps values within the first dimension, and a single one within the second, akin to the one function current, Demand. As supposed, the prediction tensor holds a single worth, corresponding– as we all know – to n_timesteps+1.

That takes care of a single input-output pair. As standard, batching is organized for by torch’s dataloader class. We instantiate one for the coaching information, and instantly once more confirm the end result:

batch_size <- 32
train_dl <- train_ds %>% dataloader(batch_size = batch_size, shuffle = TRUE)
length(train_dl)

b <- train_dl %>% dataloader_make_iter() %>% dataloader_next()
b

$x
torch_tensor
(1,.,.) = 
  0.4805
  0.3125
[...]       ### traces eliminated by me
 -1.1756
 -0.9981
... [the output was truncated (use n=-1 to disable)]
[ CPUFloatType{32,336,1} ]

$y
torch_tensor
 0.1890
 0.5405
[...]       ### traces eliminated by me
 2.4015
 0.7891
... [the output was truncated (use n=-1 to disable)]
[ CPUFloatType{32,1} ]

We see the added batch dimension in entrance, leading to total form (batch_size, n_timesteps, num_features). That is the format anticipated by the mannequin, or extra exactly, by its preliminary RNN layer.

Earlier than we go on, let’s rapidly create datasets and dataloaders for validation and take a look at information, as nicely.

valid_ds <- elec_dataset(elec_valid, n_timesteps, sample_frac = 0.5)
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)

The mannequin consists of an RNN – of sort GRU or LSTM, as per the person’s alternative – and an output layer. The RNN does a lot of the work; the single-neuron linear layer that outputs the prediction compresses its vector enter to a single worth.

Right here, first, is the mannequin definition.

mannequin <- nn_module(
  
  initialize = operate(sort, input_size, hidden_size, num_layers = 1, dropout = 0) {
    
    self$sort <- sort
    self$num_layers <- num_layers
    
    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
      )
    }
    
    self$output <- nn_linear(hidden_size, 1)
    
  },
  
  ahead = operate(x) {
    
    # record of [output, hidden]
    # we use the output, which is of measurement (batch_size, n_timesteps, hidden_size)
    x <- self$rnn(x)[[1]]
    
    # from the output, we solely need the ultimate timestep
    # form now's (batch_size, hidden_size)
    x <- x[ , dim(x)[2], ]
    
    # feed this to a single output neuron
    # remaining form then is (batch_size, 1)
    x %>% self$output() 
  }
  
)

Most significantly, that is what occurs in ahead().

1. The RNN returns a listing. The record holds two tensors, an output, and a synopsis of hidden states. We discard the state tensor, and preserve the output solely. The excellence between state and output, or quite, the way in which it’s mirrored in what a torch RNN returns, deserves to be inspected extra intently. We’ll do this in a second.

2. Of the output tensor, we’re involved in solely the ultimate time-step, although.

3. Solely this one, thus, is handed to the output layer.

4. Lastly, the mentioned output layer’s output is returned.

Now, a bit extra on states vs. outputs. Contemplate Fig. 1, from Goodfellow, Bengio, and Courville (2016).

Let’s faux there are three time steps solely, akin to (t-1), (t), and (t+1). The enter sequence, accordingly, consists of (x_{t-1}), (x_{t}), and (x_{t+1}).

At every (t), a hidden state is generated, and so is an output. Usually, if our aim is to foretell (y_{t+2}), that’s, the very subsequent remark, we wish to consider the whole enter sequence. Put in another way, we wish to have run by the whole equipment of state updates. The logical factor to do would thus be to decide on (o_{t+1}), for both direct return from ahead() or for additional processing.

Certainly, return (o_{t+1}) is what a Keras LSTM or GRU would do by default. Not so its torch counterparts. In torch, the output tensor contains all of (o). For this reason, in step two above, we choose the only time step we’re involved in – particularly, the final one.

In later posts, we are going to make use of greater than the final time step. Typically, we’ll use the sequence of hidden states (the (h)s) as an alternative of the outputs (the (o)s). So you might really feel like asking, what if we used (h_{t+1}) right here as an alternative of (o_{t+1})? The reply is: With a GRU, this may not make a distinction, as these two are equivalent. With LSTM although, it will, as LSTM retains a second, particularly, the “cell,” state.

On to initialize(). For ease of experimentation, we instantiate both a GRU or an LSTM based mostly on person enter. Two issues are price noting:

• We move batch_first = TRUE when creating the RNNs. That is required with torch RNNs once we wish to persistently have batch gadgets stacked within the first dimension. And we do need that; it’s arguably much less complicated than a change of dimension semantics for one sub-type of module.

• num_layers can be utilized to construct a stacked RNN, akin to what you’d get in Keras when chaining two GRUs/LSTMs (the primary one created with return_sequences = TRUE). This parameter, too, we’ve included for fast experimentation.

Let’s instantiate a mannequin for coaching. It is going to be a single-layer GRU with thirty-two models.

# coaching RNNs on the GPU presently prints a warning which will muddle 
# the console
# see https://github.com/mlverse/torch/points/461
# alternatively, use 
# system <- "cpu"
system <- torch_device(if (cuda_is_available()) "cuda" else "cpu")

internet <- mannequin("gru", 1, 32)
internet <- internet$to(system = system)

In spite of everything these RNN specifics, the coaching course of is totally customary.

optimizer <- optim_adam(internet$parameters, lr = 0.001)

num_epochs <- 30

train_batch <- operate(b) {
  
  optimizer$zero_grad()
  output <- internet(b$x$to(system = system))
  goal <- b$y$to(system = system)
  
  loss <- nnf_mse_loss(output, goal)
  loss$backward()
  optimizer$step()
  
  loss$merchandise()
}

valid_batch <- operate(b) {
  
  output <- internet(b$x$to(system = system))
  goal <- b$y$to(system = system)
  
  loss <- nnf_mse_loss(output, goal)
  loss$merchandise()
  
}

for (epoch in 1:num_epochs) {
  
  internet$prepare()
  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: 0.21908 

Epoch 1, validation: loss: 0.05125 

Epoch 2, coaching: loss: 0.03245 

Epoch 2, validation: loss: 0.03391 

Epoch 3, coaching: loss: 0.02346 

Epoch 3, validation: loss: 0.02321 

Epoch 4, coaching: loss: 0.01823 

Epoch 4, validation: loss: 0.01838 

Epoch 5, coaching: loss: 0.01522 

Epoch 5, validation: loss: 0.01560 

Epoch 6, coaching: loss: 0.01315 

Epoch 6, validation: loss: 0.01374 

Epoch 7, coaching: loss: 0.01205 

Epoch 7, validation: loss: 0.01200 

Epoch 8, coaching: loss: 0.01155 

Epoch 8, validation: loss: 0.01157 

Epoch 9, coaching: loss: 0.01118 

Epoch 9, validation: loss: 0.01096 

Epoch 10, coaching: loss: 0.01070 

Epoch 10, validation: loss: 0.01132 

Epoch 11, coaching: loss: 0.01003 

Epoch 11, validation: loss: 0.01150 

Epoch 12, coaching: loss: 0.00943 

Epoch 12, validation: loss: 0.01106 

Epoch 13, coaching: loss: 0.00922 

Epoch 13, validation: loss: 0.01069 

Epoch 14, coaching: loss: 0.00862 

Epoch 14, validation: loss: 0.01125 

Epoch 15, coaching: loss: 0.00842 

Epoch 15, validation: loss: 0.01095 

Epoch 16, coaching: loss: 0.00820 

Epoch 16, validation: loss: 0.00975 

Epoch 17, coaching: loss: 0.00802 

Epoch 17, validation: loss: 0.01120 

Epoch 18, coaching: loss: 0.00781 

Epoch 18, validation: loss: 0.00990 

Epoch 19, coaching: loss: 0.00757 

Epoch 19, validation: loss: 0.01017 

Epoch 20, coaching: loss: 0.00735 

Epoch 20, validation: loss: 0.00932 

Epoch 21, coaching: loss: 0.00723 

Epoch 21, validation: loss: 0.00901 

Epoch 22, coaching: loss: 0.00708 

Epoch 22, validation: loss: 0.00890 

Epoch 23, coaching: loss: 0.00676 

Epoch 23, validation: loss: 0.00914 

Epoch 24, coaching: loss: 0.00666 

Epoch 24, validation: loss: 0.00922 

Epoch 25, coaching: loss: 0.00644 

Epoch 25, validation: loss: 0.00869 

Epoch 26, coaching: loss: 0.00620 

Epoch 26, validation: loss: 0.00902 

Epoch 27, coaching: loss: 0.00588 

Epoch 27, validation: loss: 0.00896 

Epoch 28, coaching: loss: 0.00563 

Epoch 28, validation: loss: 0.00886 

Epoch 29, coaching: loss: 0.00547 

Epoch 29, validation: loss: 0.00895 

Epoch 30, coaching: loss: 0.00523 

Epoch 30, validation: loss: 0.00935 

Loss decreases rapidly, and we don’t appear to be overfitting on the validation set.

Numbers are fairly summary, although. So, we’ll use the take a look at set to see how the forecast truly appears.

Right here is the forecast for January, 2014, thirty minutes at a time.

internet$eval()

preds <- rep(NA, n_timesteps)

coro::loop(for (b in test_dl) {
  output <- internet(b$x$to(system = system))
  preds <- c(preds, output %>% as.numeric())
})

vic_elec_jan_2014 <-  vic_elec %>%
  filter(year(Date) == 2014, month(Date) == 1) %>%
  choose(Demand)

preds_ts <- vic_elec_jan_2014 %>%
  add_column(forecast = preds * train_sd + train_mean) %>%
  pivot_longer(-Time) %>%
  update_tsibble(key = title)

preds_ts %>%
  autoplot() +
  scale_colour_manual(values = c("#08c5d1", "#00353f")) +
  theme_minimal()

Determine 6: One-step-ahead predictions for January, 2014.

General, the forecast is great, however it’s attention-grabbing to see how the forecast “regularizes” essentially the most excessive peaks. This type of “regression to the imply” might be seen far more strongly in later setups, once we attempt to forecast additional into the longer term.

Can we use our present structure for multi-step prediction? We will.

One factor we are able to do is feed again the present prediction, that’s, append it to the enter sequence as quickly as it’s out there. Successfully thus, for every batch merchandise, we get hold of a sequence of predictions in a loop.

We’ll attempt to forecast 336 time steps, that’s, a whole week.

n_forecast <- 2 * 24 * 7

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(system = system))
  preds <- as.numeric(output)
  
  for(j in 2:n_forecast) {
    enter <- torch_cat(list(enter[ , 2:length(input), ], output$view(c(1, 1, 1))), dim = 2)
    output <- internet(enter$to(system = system))
    preds <- c(preds, as.numeric(output))
  }
  
  test_preds[[i]] <- preds
  i <<- i + 1
  
})

For visualization, let’s choose three non-overlapping sequences.

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 %>%
  filter(year(Date) == 2014, month(Date) == 1) %>%
  choose(Demand) %>%
  add_column(
    iterative_ex_1 = test_pred1 * train_sd + train_mean,
    iterative_ex_2 = test_pred2 * train_sd + train_mean,
    iterative_ex_3 = test_pred3 * train_sd + train_mean) %>%
  pivot_longer(-Time) %>%
  update_tsibble(key = title)

preds_ts %>%
  autoplot() +
  scale_colour_manual(values = c("#08c5d1", "#00353f", "#ffbf66", "#d46f4d")) +
  theme_minimal()

Determine 7: Multi-step predictions for January, 2014, obtained in a loop.

Even with this very fundamental forecasting approach, the diurnal rhythm is preserved, albeit in a strongly smoothed kind. There even is an obvious day-of-week periodicity within the forecast. We do see, nevertheless, very robust regression to the imply, even in loop situations the place the community was “primed” with the next enter sequence.

Hopefully this submit supplied a helpful introduction to time sequence forecasting with torch. Evidently, we picked a difficult time sequence – difficult, that’s, for a minimum of two causes:

• To accurately issue within the pattern, exterior info is required: exterior info in type of a temperature forecast, which, “in actuality,” can be simply obtainable.

• Along with the extremely essential pattern element, the information are characterised by a number of ranges of seasonality.

Of those, the latter is much less of an issue for the methods we’re working with right here. If we discovered that some stage of seasonality went undetected, we might attempt to adapt the present configuration in plenty of uncomplicated methods:

• Use an LSTM as an alternative of a GRU. In concept, LSTM ought to higher have the ability to seize further lower-frequency elements because of its secondary storage, the cell state.

• Stack a number of layers of GRU/LSTM. In concept, this could permit for studying a hierarchy of temporal options, analogously to what we see in a convolutional neural community.

To deal with the previous impediment, greater modifications to the structure can be wanted. We might try to try this in a later, “bonus,” submit. However within the upcoming installments, we’ll first dive into often-used methods for sequence prediction, additionally porting to numerical time sequence issues which are generally achieved in pure language processing.

Thanks for studying!

Picture by Nick Dunn on Unsplash