Predicting Sunspot Frequency with Keras

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Forecasting sunspots with deep studying

On this publish we are going to look at making time sequence predictions utilizing the sunspots dataset that ships with base R. Sunspots are darkish spots on the solar, related to decrease temperature. Right here’s a picture from NASA displaying the photo voltaic phenomenon.

We’re utilizing the month-to-month model of the dataset, sunspots.month (there’s a yearly model, too). It accommodates 265 years price of knowledge (from 1749 by means of 2013) on the variety of sunspots per thirty days.

Forecasting this dataset is difficult due to excessive brief time period variability in addition to long-term irregularities evident within the cycles. For instance, most amplitudes reached by the low frequency cycle differ quite a bit, as does the variety of excessive frequency cycle steps wanted to succeed in that most low frequency cycle top.

Our publish will concentrate on two dominant elements: the way to apply deep studying to time sequence forecasting, and the way to correctly apply cross validation on this area. For the latter, we are going to use the rsample bundle that enables to do resampling on time sequence information. As to the previous, our purpose is to not attain utmost efficiency however to point out the overall plan of action when utilizing recurrent neural networks to mannequin this sort of information.

Recurrent neural networks

When our information has a sequential construction, it’s recurrent neural networks (RNNs) we use to mannequin it.

As of right this moment, amongst RNNs, the perfect established architectures are the GRU (Gated Recurrent Unit) and the LSTM (Lengthy Brief Time period Reminiscence). For right this moment, let’s not zoom in on what makes them particular, however on what they’ve in frequent with essentially the most stripped-down RNN: the essential recurrence construction.

In distinction to the prototype of a neural community, usually known as Multilayer Perceptron (MLP), the RNN has a state that’s carried on over time. That is properly seen on this diagram from Goodfellow et al., a.okay.a. the “bible of deep studying”:

At every time, the state is a mix of the present enter and the earlier hidden state. That is harking back to autoregressive fashions, however with neural networks, there needs to be some level the place we halt the dependence.

That’s as a result of with the intention to decide the weights, we maintain calculating how our loss adjustments because the enter adjustments. Now if the enter we’ve to contemplate, at an arbitrary timestep, ranges again indefinitely – then we won’t be able to calculate all these gradients. In follow, then, our hidden state will, at each iteration, be carried ahead by means of a hard and fast variety of steps.

We’ll come again to that as quickly as we’ve loaded and pre-processed the information.

Setup, pre-processing, and exploration

Libraries

Right here, first, are the libraries wanted for this tutorial.

If in case you have not beforehand run Keras in R, you will want to put in Keras utilizing the install_keras() operate.

# Set up Keras if in case you have not put in earlier than
install_keras()

Knowledge

sunspot.month is a ts class (not tidy), so we’ll convert to a tidy information set utilizing the tk_tbl() operate from timetk. We use this as an alternative of as.tibble() from tibble to routinely protect the time sequence index as a zoo yearmon index. Final, we’ll convert the zoo index so far utilizing lubridate::as_date() (loaded with tidyquant) after which change to a tbl_time object to make time sequence operations simpler.

sun_spots <- datasets::sunspot.month %>%
    tk_tbl() %>%
    mutate(index = as_date(index)) %>%
    as_tbl_time(index = index)

sun_spots
# A time tibble: 3,177 x 2
# Index: index
   index      worth
   <date>     <dbl>
 1 1749-01-01  58  
 2 1749-02-01  62.6
 3 1749-03-01  70  
 4 1749-04-01  55.7
 5 1749-05-01  85  
 6 1749-06-01  83.5
 7 1749-07-01  94.8
 8 1749-08-01  66.3
 9 1749-09-01  75.9
10 1749-10-01  75.5
# ... with 3,167 extra rows

Exploratory information evaluation

The time sequence is lengthy (265 years!). We will visualize the time sequence each in full, and zoomed in on the primary 10 years to get a really feel for the sequence.

Visualizing sunspot information with cowplot

We’ll make two ggplots and mix them utilizing cowplot::plot_grid(). Notice that for the zoomed in plot, we make use of tibbletime::time_filter(), which is a simple approach to carry out time-based filtering.

p1 <- sun_spots %>%
    ggplot(aes(index, worth)) +
    geom_point(shade = palette_light()[[1]], alpha = 0.5) +
    theme_tq() +
    labs(
        title = "From 1749 to 2013 (Full Knowledge Set)"
    )

p2 <- sun_spots %>%
    filter_time("begin" ~ "1800") %>%
    ggplot(aes(index, worth)) +
    geom_line(shade = palette_light()[[1]], alpha = 0.5) +
    geom_point(shade = palette_light()[[1]]) +
    geom_smooth(methodology = "loess", span = 0.2, se = FALSE) +
    theme_tq() +
    labs(
        title = "1749 to 1759 (Zoomed In To Present Modifications over the 12 months)",
        caption = "datasets::sunspot.month"
    )

p_title <- ggdraw() + 
  draw_label("Sunspots", measurement = 18, fontface = "daring", 
             color = palette_light()[[1]])

plot_grid(p_title, p1, p2, ncol = 1, rel_heights = c(0.1, 1, 1))

Backtesting: time sequence cross validation

When doing cross validation on sequential information, the time dependencies on previous samples should be preserved. We will create a cross validation sampling plan by offsetting the window used to pick sequential sub-samples. In essence, we’re creatively coping with the truth that there’s no future check information out there by creating a number of artificial “futures” – a course of usually, esp. in finance, known as “backtesting”.

As talked about within the introduction, the rsample bundle consists of facitlities for backtesting on time sequence. The vignette, “Time Sequence Evaluation Instance”, describes a process that makes use of the rolling_origin() operate to create samples designed for time sequence cross validation. We’ll use this method.

Creating a backtesting technique

The sampling plan we create makes use of 100 years (preliminary = 12 x 100 samples) for the coaching set and 50 years (assess = 12 x 50) for the testing (validation) set. We choose a skip span of about 22 years (skip = 12 x 22 – 1) to roughly evenly distribute the samples into 6 units that span your entire 265 years of sunspots historical past. Final, we choose cumulative = FALSE to permit the origin to shift which ensures that fashions on more moderen information should not given an unfair benefit (extra observations) over these working on much less current information. The tibble return accommodates the rolling_origin_resamples.

periods_train <- 12 * 100
periods_test  <- 12 * 50
skip_span     <- 12 * 22 - 1

rolling_origin_resamples <- rolling_origin(
  sun_spots,
  preliminary    = periods_train,
  assess     = periods_test,
  cumulative = FALSE,
  skip       = skip_span
)

rolling_origin_resamples
# Rolling origin forecast resampling 
# A tibble: 6 x 2
  splits       id    
  <checklist>       <chr> 
1 <S3: rsplit> Slice1
2 <S3: rsplit> Slice2
3 <S3: rsplit> Slice3
4 <S3: rsplit> Slice4
5 <S3: rsplit> Slice5
6 <S3: rsplit> Slice6

Visualizing the backtesting technique

We will visualize the resamples with two customized capabilities. The primary, plot_split(), plots one of many resampling splits utilizing ggplot2. Notice that an expand_y_axis argument is added to develop the date vary to the complete sun_spots dataset date vary. It will grow to be helpful after we visualize all plots collectively.

# Plotting operate for a single break up
plot_split <- operate(break up, expand_y_axis = TRUE, 
                       alpha = 1, measurement = 1, base_size = 14) {
    
    # Manipulate information
    train_tbl <- coaching(break up) %>%
        add_column(key = "coaching") 
    
    test_tbl  <- testing(break up) %>%
        add_column(key = "testing") 
    
    data_manipulated <- bind_rows(train_tbl, test_tbl) %>%
        as_tbl_time(index = index) %>%
        mutate(key = fct_relevel(key, "coaching", "testing"))
        
    # Acquire attributes
    train_time_summary <- train_tbl %>%
        tk_index() %>%
        tk_get_timeseries_summary()
    
    test_time_summary <- test_tbl %>%
        tk_index() %>%
        tk_get_timeseries_summary()
    
    # Visualize
    g <- data_manipulated %>%
        ggplot(aes(x = index, y = worth, shade = key)) +
        geom_line(measurement = measurement, alpha = alpha) +
        theme_tq(base_size = base_size) +
        scale_color_tq() +
        labs(
          title    = glue("Cut up: {break up$id}"),
          subtitle = glue("{train_time_summary$begin} to ", 
                          "{test_time_summary$finish}"),
            y = "", x = ""
        ) +
        theme(legend.place = "none") 
    
    if (expand_y_axis) {
        
        sun_spots_time_summary <- sun_spots %>% 
            tk_index() %>% 
            tk_get_timeseries_summary()
        
        g <- g +
            scale_x_date(limits = c(sun_spots_time_summary$begin, 
                                    sun_spots_time_summary$finish))
    }
    
    g
}

The plot_split() operate takes one break up (on this case Slice01), and returns a visible of the sampling technique. We develop the axis to the vary for the complete dataset utilizing expand_y_axis = TRUE.

rolling_origin_resamples$splits[[1]] %>%
    plot_split(expand_y_axis = TRUE) +
    theme(legend.place = "backside")

The second operate, plot_sampling_plan(), scales the plot_split() operate to all the samples utilizing purrr and cowplot.

# Plotting operate that scales to all splits 
plot_sampling_plan <- operate(sampling_tbl, expand_y_axis = TRUE, 
                               ncol = 3, alpha = 1, measurement = 1, base_size = 14, 
                               title = "Sampling Plan") {
    
    # Map plot_split() to sampling_tbl
    sampling_tbl_with_plots <- sampling_tbl %>%
        mutate(gg_plots = map(splits, plot_split, 
                              expand_y_axis = expand_y_axis,
                              alpha = alpha, base_size = base_size))
    
    # Make plots with cowplot
    plot_list <- sampling_tbl_with_plots$gg_plots 
    
    p_temp <- plot_list[[1]] + theme(legend.place = "backside")
    legend <- get_legend(p_temp)
    
    p_body  <- plot_grid(plotlist = plot_list, ncol = ncol)
    
    p_title <- ggdraw() + 
        draw_label(title, measurement = 14, fontface = "daring", 
                   color = palette_light()[[1]])
    
    g <- plot_grid(p_title, p_body, legend, ncol = 1, 
                   rel_heights = c(0.05, 1, 0.05))
    
    g
    
}

We will now visualize your entire backtesting technique with plot_sampling_plan(). We will see how the sampling plan shifts the sampling window with every progressive slice of the practice/check splits.

rolling_origin_resamples %>%
    plot_sampling_plan(expand_y_axis = T, ncol = 3, alpha = 1, measurement = 1, base_size = 10, 
                       title = "Backtesting Technique: Rolling Origin Sampling Plan")

And, we are able to set expand_y_axis = FALSE to zoom in on the samples.

rolling_origin_resamples %>%
    plot_sampling_plan(expand_y_axis = F, ncol = 3, alpha = 1, measurement = 1, base_size = 10, 
                       title = "Backtesting Technique: Zoomed In")

We’ll use this backtesting technique (6 samples from one time sequence every with 50/10 break up in years and a ~20 yr offset) when testing the veracity of the LSTM mannequin on the sunspots dataset.

The LSTM mannequin

To start, we’ll develop an LSTM mannequin on a single pattern from the backtesting technique, particularly, the newest slice. We’ll then apply the mannequin to all samples to analyze modeling efficiency.

example_split    <- rolling_origin_resamples$splits[[6]]
example_split_id <- rolling_origin_resamples$id[[6]]

We will reuse the plot_split() operate to visualise the break up. Set expand_y_axis = FALSE to zoom in on the subsample.

plot_split(example_split, expand_y_axis = FALSE, measurement = 0.5) +
    theme(legend.place = "backside") +
    ggtitle(glue("Cut up: {example_split_id}"))

Knowledge setup

To assist hyperparameter tuning, moreover the coaching set we additionally want a validation set. For instance, we are going to use a callback, callback_early_stopping, that stops coaching when no important efficiency is seen on the validation set (what’s thought-about important is as much as you).

We are going to dedicate 2 thirds of the evaluation set to coaching, and 1 third to validation.

df_trn <- evaluation(example_split)[1:800, , drop = FALSE]
df_val <- evaluation(example_split)[801:1200, , drop = FALSE]
df_tst <- evaluation(example_split)

First, let’s mix the coaching and testing information units right into a single information set with a column key that specifies the place they got here from (both “coaching” or “testing)”. Notice that the tbl_time object might want to have the index respecified through the bind_rows() step, however this situation must be corrected in dplyr quickly.

df <- bind_rows(
  df_trn %>% add_column(key = "coaching"),
  df_val %>% add_column(key = "validation"),
  df_tst %>% add_column(key = "testing")
) %>%
  as_tbl_time(index = index)

df
# A time tibble: 1,800 x 3
# Index: index
   index      worth key     
   <date>     <dbl> <chr>   
 1 1849-06-01  81.1 coaching
 2 1849-07-01  78   coaching
 3 1849-08-01  67.7 coaching
 4 1849-09-01  93.7 coaching
 5 1849-10-01  71.5 coaching
 6 1849-11-01  99   coaching
 7 1849-12-01  97   coaching
 8 1850-01-01  78   coaching
 9 1850-02-01  89.4 coaching
10 1850-03-01  82.6 coaching
# ... with 1,790 extra rows

Preprocessing with recipes

The LSTM algorithm will normally work higher if the enter information has been centered and scaled. We will conveniently accomplish this utilizing the recipes bundle. Along with step_center and step_scale, we’re utilizing step_sqrt to cut back variance and remov outliers. The precise transformations are executed after we bake the information in response to the recipe:

rec_obj <- recipe(worth ~ ., df) %>%
    step_sqrt(worth) %>%
    step_center(worth) %>%
    step_scale(worth) %>%
    prep()

df_processed_tbl <- bake(rec_obj, df)

df_processed_tbl
# A tibble: 1,800 x 3
   index      worth key     
   <date>     <dbl> <fct>   
 1 1849-06-01 0.714 coaching
 2 1849-07-01 0.660 coaching
 3 1849-08-01 0.473 coaching
 4 1849-09-01 0.922 coaching
 5 1849-10-01 0.544 coaching
 6 1849-11-01 1.01  coaching
 7 1849-12-01 0.974 coaching
 8 1850-01-01 0.660 coaching
 9 1850-02-01 0.852 coaching
10 1850-03-01 0.739 coaching
# ... with 1,790 extra rows

Subsequent, let’s seize the unique heart and scale so we are able to invert the steps after modeling. The sq. root step can then merely be undone by squaring the back-transformed information.

center_history <- rec_obj$steps[[2]]$means["value"]
scale_history  <- rec_obj$steps[[3]]$sds["value"]

c("heart" = center_history, "scale" = scale_history)
heart.worth  scale.worth 
    6.694468     3.238935 

Reshaping the information

Keras LSTM expects the enter in addition to the goal information to be in a particular form. The enter needs to be a three-D array of measurement num_samples, num_timesteps, num_features.

Right here, num_samples is the variety of observations within the set. It will get fed to the mannequin in parts of batch_size. The second dimension, num_timesteps, is the size of the hidden state we have been speaking about above. Lastly, the third dimension is the variety of predictors we’re utilizing. For univariate time sequence, that is 1.

How lengthy ought to we select the hidden state to be? This typically depends upon the dataset and our purpose. If we did one-step-ahead forecasts – thus, forecasting the next month solely – our predominant concern can be selecting a state size that enables to be taught any patterns current within the information.

Now say we wished to forecast 12 months as an alternative, as does SILSO, the World Knowledge Heart for the manufacturing, preservation and dissemination of the worldwide sunspot quantity. The way in which we are able to do that, with Keras, is by wiring the LSTM hidden states to units of consecutive outputs of the identical size. Thus, if we need to produce predictions for 12 months, our LSTM ought to have a hidden state size of 12.

These 12 time steps will then get wired to 12 linear predictor items utilizing a time_distributed() wrapper. That wrapper’s activity is to use the identical calculation (i.e., the identical weight matrix) to each state enter it receives.

Now, what’s the goal array’s format imagined to be? As we’re forecasting a number of timesteps right here, the goal information once more must be third-dimensional. Dimension 1 once more is the batch dimension, dimension 2 once more corresponds to the variety of timesteps (the forecasted ones), and dimension 3 is the scale of the wrapped layer. In our case, the wrapped layer is a layer_dense() of a single unit, as we wish precisely one prediction per time limit.

So, let’s reshape the information. The primary motion right here is creating the sliding home windows of 12 steps of enter, adopted by 12 steps of output every. That is best to know with a shorter and easier instance. Say our enter have been the numbers from 1 to 10, and our chosen sequence size (state measurement) have been 4. Tthis is how we might need our coaching enter to look:

1,2,3,4
2,3,4,5
3,4,5,6

And our goal information, correspondingly:

5,6,7,8
6,7,8,9
7,8,9,10

We’ll outline a brief operate that does this reshaping on a given dataset. Then lastly, we add the third axis that’s formally wanted (regardless that that axis is of measurement 1 in our case).

# these variables are being outlined simply due to the order through which
# we current issues on this publish (first the information, then the mannequin)
# they are going to be outdated by FLAGS$n_timesteps, FLAGS$batch_size and n_predictions
# within the following snippet
n_timesteps <- 12
n_predictions <- n_timesteps
batch_size <- 10

# capabilities used
build_matrix <- operate(tseries, overall_timesteps) {
  t(sapply(1:(size(tseries) - overall_timesteps + 1), operate(x) 
    tseries[x:(x + overall_timesteps - 1)]))
}

reshape_X_3d <- operate(X) {
  dim(X) <- c(dim(X)[1], dim(X)[2], 1)
  X
}

# extract values from information body
train_vals <- df_processed_tbl %>%
  filter(key == "coaching") %>%
  choose(worth) %>%
  pull()
valid_vals <- df_processed_tbl %>%
  filter(key == "validation") %>%
  choose(worth) %>%
  pull()
test_vals <- df_processed_tbl %>%
  filter(key == "testing") %>%
  choose(worth) %>%
  pull()


# construct the windowed matrices
train_matrix <-
  build_matrix(train_vals, n_timesteps + n_predictions)
valid_matrix <-
  build_matrix(valid_vals, n_timesteps + n_predictions)
test_matrix <- build_matrix(test_vals, n_timesteps + n_predictions)

# separate matrices into coaching and testing components
# additionally, discard final batch if there are fewer than batch_size samples
# (a purely technical requirement)
X_train <- train_matrix[, 1:n_timesteps]
y_train <- train_matrix[, (n_timesteps + 1):(n_timesteps * 2)]
X_train <- X_train[1:(nrow(X_train) %/% batch_size * batch_size), ]
y_train <- y_train[1:(nrow(y_train) %/% batch_size * batch_size), ]

X_valid <- valid_matrix[, 1:n_timesteps]
y_valid <- valid_matrix[, (n_timesteps + 1):(n_timesteps * 2)]
X_valid <- X_valid[1:(nrow(X_valid) %/% batch_size * batch_size), ]
y_valid <- y_valid[1:(nrow(y_valid) %/% batch_size * batch_size), ]

X_test <- test_matrix[, 1:n_timesteps]
y_test <- test_matrix[, (n_timesteps + 1):(n_timesteps * 2)]
X_test <- X_test[1:(nrow(X_test) %/% batch_size * batch_size), ]
y_test <- y_test[1:(nrow(y_test) %/% batch_size * batch_size), ]
# add on the required third axis
X_train <- reshape_X_3d(X_train)
X_valid <- reshape_X_3d(X_valid)
X_test <- reshape_X_3d(X_test)

y_train <- reshape_X_3d(y_train)
y_valid <- reshape_X_3d(y_valid)
y_test <- reshape_X_3d(y_test)

Constructing the LSTM mannequin

Now that we’ve our information within the required kind, let’s lastly construct the mannequin. As at all times in deep studying, an essential, and infrequently time-consuming, a part of the job is tuning hyperparameters. To maintain this publish self-contained, and contemplating that is primarily a tutorial on the way to use LSTM in R, let’s assume the next settings have been discovered after in depth experimentation (in actuality experimentation did happen, however to not a level that efficiency couldn’t probably be improved).

As a substitute of exhausting coding the hyperparameters, we’ll use tfruns to arrange an setting the place we might simply carry out grid search.

We’ll rapidly touch upon what these parameters do however primarily go away these matters to additional posts.

FLAGS <- flags(
  # There's a so-called "stateful LSTM" in Keras. Whereas LSTM is stateful
  # per se, this provides an extra tweak the place the hidden states get 
  # initialized with values from the merchandise at similar place within the earlier
  # batch. That is useful just below particular circumstances, or if you need
  # to create an "infinite stream" of states, through which case you'd use 1 as 
  # the batch measurement. Beneath, we present how the code must be modified to
  # use this, however it will not be additional mentioned right here.
  flag_boolean("stateful", FALSE),
  # Ought to we use a number of layers of LSTM?
  # Once more, simply included for completeness, it didn't yield any superior 
  # efficiency on this activity.
  # It will truly stack precisely one extra layer of LSTM items.
  flag_boolean("stack_layers", FALSE),
  # variety of samples fed to the mannequin in a single go
  flag_integer("batch_size", 10),
  # measurement of the hidden state, equals measurement of predictions
  flag_integer("n_timesteps", 12),
  # what number of epochs to coach for
  flag_integer("n_epochs", 100),
  # fraction of the items to drop for the linear transformation of the inputs
  flag_numeric("dropout", 0.2),
  # fraction of the items to drop for the linear transformation of the 
  # recurrent state
  flag_numeric("recurrent_dropout", 0.2),
  # loss operate. Discovered to work higher for this particular case than imply
  # squared error
  flag_string("loss", "logcosh"),
  # optimizer = stochastic gradient descent. Appeared to work higher than adam 
  # or rmsprop right here (as indicated by restricted testing)
  flag_string("optimizer_type", "sgd"),
  # measurement of the LSTM layer
  flag_integer("n_units", 128),
  # studying price
  flag_numeric("lr", 0.003),
  # momentum, an extra parameter to the SGD optimizer
  flag_numeric("momentum", 0.9),
  # parameter to the early stopping callback
  flag_integer("endurance", 10)
)

# the variety of predictions we'll make equals the size of the hidden state
n_predictions <- FLAGS$n_timesteps
# what number of options = predictors we've
n_features <- 1
# simply in case we wished to attempt totally different optimizers, we might add right here
optimizer <- swap(FLAGS$optimizer_type,
                    sgd = optimizer_sgd(lr = FLAGS$lr, 
                                        momentum = FLAGS$momentum)
                    )

# callbacks to be handed to the match() operate
# We simply use one right here: we could cease earlier than n_epochs if the loss on the
# validation set doesn't lower (by a configurable quantity, over a 
# configurable time)
callbacks <- checklist(
  callback_early_stopping(endurance = FLAGS$endurance)
)

In spite of everything these preparations, the code for setting up and coaching the mannequin is quite brief! Let’s first rapidly view the “lengthy model”, that might help you check stacking a number of LSTMs or use a stateful LSTM, then undergo the ultimate brief model (that does neither) and touch upon it.

This, only for reference, is the whole code.

mannequin <- keras_model_sequential()

mannequin %>%
  layer_lstm(
    items = FLAGS$n_units,
    batch_input_shape = c(FLAGS$batch_size, FLAGS$n_timesteps, n_features),
    dropout = FLAGS$dropout,
    recurrent_dropout = FLAGS$recurrent_dropout,
    return_sequences = TRUE,
    stateful = FLAGS$stateful
  )

if (FLAGS$stack_layers) {
  mannequin %>%
    layer_lstm(
      items = FLAGS$n_units,
      dropout = FLAGS$dropout,
      recurrent_dropout = FLAGS$recurrent_dropout,
      return_sequences = TRUE,
      stateful = FLAGS$stateful
    )
}
mannequin %>% time_distributed(layer_dense(items = 1))

mannequin %>%
  compile(
    loss = FLAGS$loss,
    optimizer = optimizer,
    metrics = checklist("mean_squared_error")
  )

if (!FLAGS$stateful) {
  mannequin %>% match(
    x          = X_train,
    y          = y_train,
    validation_data = checklist(X_valid, y_valid),
    batch_size = FLAGS$batch_size,
    epochs     = FLAGS$n_epochs,
    callbacks = callbacks
  )
  
} else {
  for (i in 1:FLAGS$n_epochs) {
    mannequin %>% match(
      x          = X_train,
      y          = y_train,
      validation_data = checklist(X_valid, y_valid),
      callbacks = callbacks,
      batch_size = FLAGS$batch_size,
      epochs     = 1,
      shuffle    = FALSE
    )
    mannequin %>% reset_states()
  }
}

if (FLAGS$stateful)
  mannequin %>% reset_states()

Now let’s step by means of the easier, but higher (or equally) performing configuration under.

# create the mannequin
mannequin <- keras_model_sequential()

# add layers
# we've simply two, the LSTM and the time_distributed 
mannequin %>%
  layer_lstm(
    items = FLAGS$n_units, 
    # the primary layer in a mannequin must know the form of the enter information
    batch_input_shape  = c(FLAGS$batch_size, FLAGS$n_timesteps, n_features),
    dropout = FLAGS$dropout,
    recurrent_dropout = FLAGS$recurrent_dropout,
    # by default, an LSTM simply returns the ultimate state
    return_sequences = TRUE
  ) %>% time_distributed(layer_dense(items = 1))

mannequin %>%
  compile(
    loss = FLAGS$loss,
    optimizer = optimizer,
    # along with the loss, Keras will inform us about present 
    # MSE whereas coaching
    metrics = checklist("mean_squared_error")
  )

historical past <- mannequin %>% match(
  x          = X_train,
  y          = y_train,
  validation_data = checklist(X_valid, y_valid),
  batch_size = FLAGS$batch_size,
  epochs     = FLAGS$n_epochs,
  callbacks = callbacks
)

As we see, coaching was stopped after ~55 epochs as validation loss didn’t lower any extra. We additionally see that efficiency on the validation set is manner worse than efficiency on the coaching set – usually indicating overfitting.

This subject too, we’ll go away to a separate dialogue one other time, however curiously regularization utilizing greater values of dropout and recurrent_dropout (mixed with rising mannequin capability) didn’t yield higher generalization efficiency. That is most likely associated to the traits of this particular time sequence we talked about within the introduction.

plot(historical past, metrics = "loss")

Now let’s see how nicely the mannequin was in a position to seize the traits of the coaching set.

pred_train <- mannequin %>%
  predict(X_train, batch_size = FLAGS$batch_size) %>%
  .[, , 1]

# Retransform values to unique scale
pred_train <- (pred_train * scale_history + center_history) ^2
compare_train <- df %>% filter(key == "coaching")

# construct a dataframe that has each precise and predicted values
for (i in 1:nrow(pred_train)) {
  varname <- paste0("pred_train", i)
  compare_train <-
    mutate(compare_train,!!varname := c(
      rep(NA, FLAGS$n_timesteps + i - 1),
      pred_train[i,],
      rep(NA, nrow(compare_train) - FLAGS$n_timesteps * 2 - i + 1)
    ))
}

We compute the common RSME over all sequences of predictions.

coln <- colnames(compare_train)[4:ncol(compare_train)]
cols <- map(coln, quo(sym(.)))
rsme_train <-
  map_dbl(cols, operate(col)
    rmse(
      compare_train,
      reality = worth,
      estimate = !!col,
      na.rm = TRUE
    )) %>% imply()

rsme_train
21.01495

How do these predictions actually look? As a visualization of all predicted sequences would look fairly crowded, we arbitrarily choose begin factors at common intervals.

ggplot(compare_train, aes(x = index, y = worth)) + geom_line() +
  geom_line(aes(y = pred_train1), shade = "cyan") +
  geom_line(aes(y = pred_train50), shade = "crimson") +
  geom_line(aes(y = pred_train100), shade = "inexperienced") +
  geom_line(aes(y = pred_train150), shade = "violet") +
  geom_line(aes(y = pred_train200), shade = "cyan") +
  geom_line(aes(y = pred_train250), shade = "crimson") +
  geom_line(aes(y = pred_train300), shade = "crimson") +
  geom_line(aes(y = pred_train350), shade = "inexperienced") +
  geom_line(aes(y = pred_train400), shade = "cyan") +
  geom_line(aes(y = pred_train450), shade = "crimson") +
  geom_line(aes(y = pred_train500), shade = "inexperienced") +
  geom_line(aes(y = pred_train550), shade = "violet") +
  geom_line(aes(y = pred_train600), shade = "cyan") +
  geom_line(aes(y = pred_train650), shade = "crimson") +
  geom_line(aes(y = pred_train700), shade = "crimson") +
  geom_line(aes(y = pred_train750), shade = "inexperienced") +
  ggtitle("Predictions on the coaching set")

This seems fairly good. From the validation loss, we don’t fairly anticipate the identical from the check set, although.

Let’s see.

pred_test <- mannequin %>%
  predict(X_test, batch_size = FLAGS$batch_size) %>%
  .[, , 1]

# Retransform values to unique scale
pred_test <- (pred_test * scale_history + center_history) ^2
pred_test[1:10, 1:5] %>% print()
compare_test <- df %>% filter(key == "testing")

# construct a dataframe that has each precise and predicted values
for (i in 1:nrow(pred_test)) {
  varname <- paste0("pred_test", i)
  compare_test <-
    mutate(compare_test,!!varname := c(
      rep(NA, FLAGS$n_timesteps + i - 1),
      pred_test[i,],
      rep(NA, nrow(compare_test) - FLAGS$n_timesteps * 2 - i + 1)
    ))
}

compare_test %>% write_csv(str_replace(model_path, ".hdf5", ".check.csv"))
compare_test[FLAGS$n_timesteps:(FLAGS$n_timesteps + 10), c(2, 4:8)] %>% print()

coln <- colnames(compare_test)[4:ncol(compare_test)]
cols <- map(coln, quo(sym(.)))
rsme_test <-
  map_dbl(cols, operate(col)
    rmse(
      compare_test,
      reality = worth,
      estimate = !!col,
      na.rm = TRUE
    )) %>% imply()

rsme_test
31.31616
ggplot(compare_test, aes(x = index, y = worth)) + geom_line() +
  geom_line(aes(y = pred_test1), shade = "cyan") +
  geom_line(aes(y = pred_test50), shade = "crimson") +
  geom_line(aes(y = pred_test100), shade = "inexperienced") +
  geom_line(aes(y = pred_test150), shade = "violet") +
  geom_line(aes(y = pred_test200), shade = "cyan") +
  geom_line(aes(y = pred_test250), shade = "crimson") +
  geom_line(aes(y = pred_test300), shade = "inexperienced") +
  geom_line(aes(y = pred_test350), shade = "cyan") +
  geom_line(aes(y = pred_test400), shade = "crimson") +
  geom_line(aes(y = pred_test450), shade = "inexperienced") +  
  geom_line(aes(y = pred_test500), shade = "cyan") +
  geom_line(aes(y = pred_test550), shade = "violet") +
  ggtitle("Predictions on check set")

That’s inferior to on the coaching set, however not dangerous both, given this time sequence is kind of difficult.

Having outlined and run our mannequin on a manually chosen instance break up, let’s now revert to our general re-sampling body.

Backtesting the mannequin on all splits

To acquire predictions on all splits, we transfer the above code right into a operate and apply it to all splits. First, right here’s the operate. It returns a listing of two dataframes, one for the coaching and check units every, that include the mannequin’s predictions along with the precise values.

obtain_predictions <- operate(break up) {
  df_trn <- evaluation(break up)[1:800, , drop = FALSE]
  df_val <- evaluation(break up)[801:1200, , drop = FALSE]
  df_tst <- evaluation(break up)
  
  df <- bind_rows(
    df_trn %>% add_column(key = "coaching"),
    df_val %>% add_column(key = "validation"),
    df_tst %>% add_column(key = "testing")
  ) %>%
    as_tbl_time(index = index)
  
  rec_obj <- recipe(worth ~ ., df) %>%
    step_sqrt(worth) %>%
    step_center(worth) %>%
    step_scale(worth) %>%
    prep()
  
  df_processed_tbl <- bake(rec_obj, df)
  
  center_history <- rec_obj$steps[[2]]$means["value"]
  scale_history  <- rec_obj$steps[[3]]$sds["value"]
  
  FLAGS <- flags(
    flag_boolean("stateful", FALSE),
    flag_boolean("stack_layers", FALSE),
    flag_integer("batch_size", 10),
    flag_integer("n_timesteps", 12),
    flag_integer("n_epochs", 100),
    flag_numeric("dropout", 0.2),
    flag_numeric("recurrent_dropout", 0.2),
    flag_string("loss", "logcosh"),
    flag_string("optimizer_type", "sgd"),
    flag_integer("n_units", 128),
    flag_numeric("lr", 0.003),
    flag_numeric("momentum", 0.9),
    flag_integer("endurance", 10)
  )
  
  n_predictions <- FLAGS$n_timesteps
  n_features <- 1
  
  optimizer <- swap(FLAGS$optimizer_type,
                      sgd = optimizer_sgd(lr = FLAGS$lr, momentum = FLAGS$momentum))
  callbacks <- checklist(
    callback_early_stopping(endurance = FLAGS$endurance)
  )
  
  train_vals <- df_processed_tbl %>%
    filter(key == "coaching") %>%
    choose(worth) %>%
    pull()
  valid_vals <- df_processed_tbl %>%
    filter(key == "validation") %>%
    choose(worth) %>%
    pull()
  test_vals <- df_processed_tbl %>%
    filter(key == "testing") %>%
    choose(worth) %>%
    pull()
  
  train_matrix <-
    build_matrix(train_vals, FLAGS$n_timesteps + n_predictions)
  valid_matrix <-
    build_matrix(valid_vals, FLAGS$n_timesteps + n_predictions)
  test_matrix <-
    build_matrix(test_vals, FLAGS$n_timesteps + n_predictions)
  
  X_train <- train_matrix[, 1:FLAGS$n_timesteps]
  y_train <-
    train_matrix[, (FLAGS$n_timesteps + 1):(FLAGS$n_timesteps * 2)]
  X_train <-
    X_train[1:(nrow(X_train) %/% FLAGS$batch_size * FLAGS$batch_size),]
  y_train <-
    y_train[1:(nrow(y_train) %/% FLAGS$batch_size * FLAGS$batch_size),]
  
  X_valid <- valid_matrix[, 1:FLAGS$n_timesteps]
  y_valid <-
    valid_matrix[, (FLAGS$n_timesteps + 1):(FLAGS$n_timesteps * 2)]
  X_valid <-
    X_valid[1:(nrow(X_valid) %/% FLAGS$batch_size * FLAGS$batch_size),]
  y_valid <-
    y_valid[1:(nrow(y_valid) %/% FLAGS$batch_size * FLAGS$batch_size),]
  
  X_test <- test_matrix[, 1:FLAGS$n_timesteps]
  y_test <-
    test_matrix[, (FLAGS$n_timesteps + 1):(FLAGS$n_timesteps * 2)]
  X_test <-
    X_test[1:(nrow(X_test) %/% FLAGS$batch_size * FLAGS$batch_size),]
  y_test <-
    y_test[1:(nrow(y_test) %/% FLAGS$batch_size * FLAGS$batch_size),]
  
  X_train <- reshape_X_3d(X_train)
  X_valid <- reshape_X_3d(X_valid)
  X_test <- reshape_X_3d(X_test)
  
  y_train <- reshape_X_3d(y_train)
  y_valid <- reshape_X_3d(y_valid)
  y_test <- reshape_X_3d(y_test)
  
  mannequin <- keras_model_sequential()
  
  mannequin %>%
    layer_lstm(
      items            = FLAGS$n_units,
      batch_input_shape  = c(FLAGS$batch_size, FLAGS$n_timesteps, n_features),
      dropout = FLAGS$dropout,
      recurrent_dropout = FLAGS$recurrent_dropout,
      return_sequences = TRUE
    )     %>% time_distributed(layer_dense(items = 1))
  
  mannequin %>%
    compile(
      loss = FLAGS$loss,
      optimizer = optimizer,
      metrics = checklist("mean_squared_error")
    )
  
  mannequin %>% match(
    x          = X_train,
    y          = y_train,
    validation_data = checklist(X_valid, y_valid),
    batch_size = FLAGS$batch_size,
    epochs     = FLAGS$n_epochs,
    callbacks = callbacks
  )
  
  
  pred_train <- mannequin %>%
    predict(X_train, batch_size = FLAGS$batch_size) %>%
    .[, , 1]
  
  # Retransform values
  pred_train <- (pred_train * scale_history + center_history) ^ 2
  compare_train <- df %>% filter(key == "coaching")
  
  for (i in 1:nrow(pred_train)) {
    varname <- paste0("pred_train", i)
    compare_train <-
      mutate(compare_train, !!varname := c(
        rep(NA, FLAGS$n_timesteps + i - 1),
        pred_train[i, ],
        rep(NA, nrow(compare_train) - FLAGS$n_timesteps * 2 - i + 1)
      ))
  }
  
  pred_test <- mannequin %>%
    predict(X_test, batch_size = FLAGS$batch_size) %>%
    .[, , 1]
  
  # Retransform values
  pred_test <- (pred_test * scale_history + center_history) ^ 2
  compare_test <- df %>% filter(key == "testing")
  
  for (i in 1:nrow(pred_test)) {
    varname <- paste0("pred_test", i)
    compare_test <-
      mutate(compare_test, !!varname := c(
        rep(NA, FLAGS$n_timesteps + i - 1),
        pred_test[i, ],
        rep(NA, nrow(compare_test) - FLAGS$n_timesteps * 2 - i + 1)
      ))
  }
  checklist(practice = compare_train, check = compare_test)
  
}

Mapping the operate over all splits yields a listing of predictions.

all_split_preds <- rolling_origin_resamples %>%
     mutate(predict = map(splits, obtain_predictions))

Calculate RMSE on all splits:

calc_rmse <- operate(df) {
  coln <- colnames(df)[4:ncol(df)]
  cols <- map(coln, quo(sym(.)))
  map_dbl(cols, operate(col)
    rmse(
      df,
      reality = worth,
      estimate = !!col,
      na.rm = TRUE
    )) %>% imply()
}

all_split_preds <- all_split_preds %>% unnest(predict)
all_split_preds_train <- all_split_preds[seq(1, 11, by = 2), ]
all_split_preds_test <- all_split_preds[seq(2, 12, by = 2), ]

all_split_rmses_train <- all_split_preds_train %>%
  mutate(rmse = map_dbl(predict, calc_rmse)) %>%
  choose(id, rmse)

all_split_rmses_test <- all_split_preds_test %>%
  mutate(rmse = map_dbl(predict, calc_rmse)) %>%
  choose(id, rmse)

How does it look? Right here’s RMSE on the coaching set for the 6 splits.

# A tibble: 6 x 2
  id      rmse
  <chr>  <dbl>
1 Slice1  22.2
2 Slice2  20.9
3 Slice3  18.8
4 Slice4  23.5
5 Slice5  22.1
6 Slice6  21.1
# A tibble: 6 x 2
  id      rmse
  <chr>  <dbl>
1 Slice1  21.6
2 Slice2  20.6
3 Slice3  21.3
4 Slice4  31.4
5 Slice5  35.2
6 Slice6  31.4

Taking a look at these numbers, we see one thing fascinating: Generalization efficiency is significantly better for the primary three slices of the time sequence than for the latter ones. This confirms our impression, said above, that there appears to be some hidden growth happening, rendering forecasting tougher.

And listed below are visualizations of the predictions on the respective coaching and check units.

First, the coaching units:

plot_train <- operate(slice, title) {
  ggplot(slice, aes(x = index, y = worth)) + geom_line() +
    geom_line(aes(y = pred_train1), shade = "cyan") +
    geom_line(aes(y = pred_train50), shade = "crimson") +
    geom_line(aes(y = pred_train100), shade = "inexperienced") +
    geom_line(aes(y = pred_train150), shade = "violet") +
    geom_line(aes(y = pred_train200), shade = "cyan") +
    geom_line(aes(y = pred_train250), shade = "crimson") +
    geom_line(aes(y = pred_train300), shade = "crimson") +
    geom_line(aes(y = pred_train350), shade = "inexperienced") +
    geom_line(aes(y = pred_train400), shade = "cyan") +
    geom_line(aes(y = pred_train450), shade = "crimson") +
    geom_line(aes(y = pred_train500), shade = "inexperienced") +
    geom_line(aes(y = pred_train550), shade = "violet") +
    geom_line(aes(y = pred_train600), shade = "cyan") +
    geom_line(aes(y = pred_train650), shade = "crimson") +
    geom_line(aes(y = pred_train700), shade = "crimson") +
    geom_line(aes(y = pred_train750), shade = "inexperienced") +
    ggtitle(title)
}

train_plots <- map2(all_split_preds_train$predict, all_split_preds_train$id, plot_train)
p_body_train  <- plot_grid(plotlist = train_plots, ncol = 3)
p_title_train <- ggdraw() + 
  draw_label("Backtested Predictions: Coaching Units", measurement = 18, fontface = "daring")

plot_grid(p_title_train, p_body_train, ncol = 1, rel_heights = c(0.05, 1, 0.05))

And the check units:

plot_test <- operate(slice, title) {
  ggplot(slice, aes(x = index, y = worth)) + geom_line() +
    geom_line(aes(y = pred_test1), shade = "cyan") +
    geom_line(aes(y = pred_test50), shade = "crimson") +
    geom_line(aes(y = pred_test100), shade = "inexperienced") +
    geom_line(aes(y = pred_test150), shade = "violet") +
    geom_line(aes(y = pred_test200), shade = "cyan") +
    geom_line(aes(y = pred_test250), shade = "crimson") +
    geom_line(aes(y = pred_test300), shade = "inexperienced") +
    geom_line(aes(y = pred_test350), shade = "cyan") +
    geom_line(aes(y = pred_test400), shade = "crimson") +
    geom_line(aes(y = pred_test450), shade = "inexperienced") +  
    geom_line(aes(y = pred_test500), shade = "cyan") +
    geom_line(aes(y = pred_test550), shade = "violet") +
    ggtitle(title)
}

test_plots <- map2(all_split_preds_test$predict, all_split_preds_test$id, plot_test)

p_body_test  <- plot_grid(plotlist = test_plots, ncol = 3)
p_title_test <- ggdraw() + 
  draw_label("Backtested Predictions: Take a look at Units", measurement = 18, fontface = "daring")

plot_grid(p_title_test, p_body_test, ncol = 1, rel_heights = c(0.05, 1, 0.05))

This has been an extended publish, and essentially could have left plenty of questions open, at first: How can we get hold of good settings for the hyperparameters (studying price, variety of epochs, dropout)? How can we select the size of the hidden state? And even, can we’ve an instinct how nicely LSTM will carry out on a given dataset (with its particular traits)? We are going to sort out questions just like the above in upcoming posts.

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