Artificial Intelligence 

Time collection prediction with FNN-LSTM

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As we speak, we choose up on the plan alluded to within the conclusion of the latest Deep attractors: The place deep studying meets chaos: make use of that very same approach to generate forecasts for empirical time collection information.

“That very same approach,” which for conciseness, I’ll take the freedom of referring to as FNN-LSTM, is because of William Gilpin’s 2020 paper “Deep reconstruction of unusual attractors from time collection” (Gilpin 2020).

In a nutshell, the issue addressed is as follows: A system, recognized or assumed to be nonlinear and extremely depending on preliminary situations, is noticed, leading to a scalar collection of measurements. The measurements aren’t simply – inevitably – noisy, however as well as, they’re – at finest – a projection of a multidimensional state area onto a line.

Classically in nonlinear time collection evaluation, such scalar collection of observations are augmented by supplementing, at each time limit, delayed measurements of that very same collection – a method known as delay coordinate embedding (Sauer, Yorke, and Casdagli 1991). For instance, as an alternative of only a single vector X1, we may have a matrix of vectors X1, X2, and X3, with X2 containing the identical values as X1, however ranging from the third commentary, and X3, from the fifth. On this case, the delay could be 2, and the embedding dimension, 3. Numerous theorems state that if these parameters are chosen adequately, it’s attainable to reconstruct the entire state area. There’s a downside although: The theorems assume that the dimensionality of the true state area is thought, which in lots of real-world functions, received’t be the case.

That is the place Gilpin’s thought is available in: Prepare an autoencoder, whose intermediate illustration encapsulates the system’s attractor. Not simply any MSE-optimized autoencoder although. The latent illustration is regularized by false nearest neighbors (FNN) loss, a method generally used with delay coordinate embedding to find out an ample embedding dimension. False neighbors are those that are shut in n-dimensional area, however considerably farther aside in n+1-dimensional area. Within the aforementioned introductory put up, we confirmed how this method allowed to reconstruct the attractor of the (artificial) Lorenz system. Now, we wish to transfer on to prediction.

We first describe the setup, together with mannequin definitions, coaching procedures, and information preparation. Then, we let you know the way it went.

Setup

From reconstruction to forecasting, and branching out into the actual world

Within the earlier put up, we educated an LSTM autoencoder to generate a compressed code, representing the attractor of the system. As ordinary with autoencoders, the goal when coaching is similar because the enter, that means that general loss consisted of two parts: The FNN loss, computed on the latent illustration solely, and the mean-squared-error loss between enter and output. Now for prediction, the goal consists of future values, as many as we want to predict. Put in a different way: The structure stays the identical, however as an alternative of reconstruction we carry out prediction, in the usual RNN manner. The place the standard RNN setup would simply immediately chain the specified variety of LSTMs, we’ve got an LSTM encoder that outputs a (timestep-less) latent code, and an LSTM decoder that ranging from that code, repeated as many instances as required, forecasts the required variety of future values.

This in fact implies that to judge forecast efficiency, we have to examine towards an LSTM-only setup. That is precisely what we’ll do, and comparability will turn into attention-grabbing not simply quantitatively, however qualitatively as properly.

We carry out these comparisons on the 4 datasets Gilpin selected to display attractor reconstruction on observational information. Whereas all of those, as is obvious from the pictures in that pocket book, exhibit good attractors, we’ll see that not all of them are equally suited to forecasting utilizing easy RNN-based architectures – with or with out FNN regularization. However even people who clearly demand a special strategy enable for attention-grabbing observations as to the impression of FNN loss.

Mannequin definitions and coaching setup

In all 4 experiments, we use the identical mannequin definitions and coaching procedures, the one differing parameter being the variety of timesteps used within the LSTMs (for causes that can change into evident once we introduce the person datasets).

Each architectures had been chosen to be easy, and about comparable in variety of parameters – each mainly encompass two LSTMs with 32 models (n_recurrent will likely be set to 32 for all experiments).

FNN-LSTM

FNN-LSTM appears practically like within the earlier put up, other than the truth that we cut up up the encoder LSTM into two, to uncouple capability (n_recurrent) from maximal latent state dimensionality (n_latent, stored at 10 identical to earlier than).

# DL-related packages
library(tensorflow)
library(keras)
library(tfdatasets)
library(tfautograph)
library(reticulate)

# going to want these later
library(tidyverse)
library(cowplot)

encoder_model <- operate(n_timesteps,
                          n_features,
                          n_recurrent,
                          n_latent,
                          title = NULL) {
  
  keras_model_custom(title = title, operate(self) {
    
    self$noise <- layer_gaussian_noise(stddev = 0.5)
    self$lstm1 <-  layer_lstm(
      models = n_recurrent,
      input_shape = c(n_timesteps, n_features),
      return_sequences = TRUE
    ) 
    self$batchnorm1 <- layer_batch_normalization()
    self$lstm2 <-  layer_lstm(
      models = n_latent,
      return_sequences = FALSE
    ) 
    self$batchnorm2 <- layer_batch_normalization()
    
    operate (x, masks = NULL) {
      x %>%
        self$noise() %>%
        self$lstm1() %>%
        self$batchnorm1() %>%
        self$lstm2() %>%
        self$batchnorm2() 
    }
  })
}

decoder_model <- operate(n_timesteps,
                          n_features,
                          n_recurrent,
                          n_latent,
                          title = NULL) {
  
  keras_model_custom(title = title, operate(self) {
    
    self$repeat_vector <- layer_repeat_vector(n = n_timesteps)
    self$noise <- layer_gaussian_noise(stddev = 0.5)
    self$lstm <- layer_lstm(
      models = n_recurrent,
      return_sequences = TRUE,
      go_backwards = TRUE
    ) 
    self$batchnorm <- layer_batch_normalization()
    self$elu <- layer_activation_elu() 
    self$time_distributed <- time_distributed(layer = layer_dense(models = n_features))
    
    operate (x, masks = NULL) {
      x %>%
        self$repeat_vector() %>%
        self$noise() %>%
        self$lstm() %>%
        self$batchnorm() %>%
        self$elu() %>%
        self$time_distributed()
    }
  })
}

n_latent <- 10L
n_features <- 1
n_hidden <- 32

encoder <- encoder_model(n_timesteps,
                         n_features,
                         n_hidden,
                         n_latent)

decoder <- decoder_model(n_timesteps,
                         n_features,
                         n_hidden,
                         n_latent)

The regularizer, FNN loss, is unchanged:

loss_false_nn <- operate(x) {
  
  # altering these parameters is equal to
  # altering the power of the regularizer, so we hold these fastened (these values
  # correspond to the unique values utilized in Kennel et al 1992).
  rtol <- 10 
  atol <- 2
  k_frac <- 0.01
  
  okay <- max(1, ground(k_frac * batch_size))
  
  ## Vectorized model of distance matrix calculation
  tri_mask <-
    tf$linalg$band_part(
      tf$ones(
        form = c(tf$forged(n_latent, tf$int32), tf$forged(n_latent, tf$int32)),
        dtype = tf$float32
      ),
      num_lower = -1L,
      num_upper = 0L
    )
  
  # latent x batch_size x latent
  batch_masked <-
    tf$multiply(tri_mask[, tf$newaxis,], x[tf$newaxis, reticulate::py_ellipsis()])
  
  # latent x batch_size x 1
  x_squared <-
    tf$reduce_sum(batch_masked * batch_masked,
                  axis = 2L,
                  keepdims = TRUE)
  
  # latent x batch_size x batch_size
  pdist_vector <- x_squared + tf$transpose(x_squared, perm = c(0L, 2L, 1L)) -
    2 * tf$matmul(batch_masked, tf$transpose(batch_masked, perm = c(0L, 2L, 1L)))
  
  #(latent, batch_size, batch_size)
  all_dists <- pdist_vector
  # latent
  all_ra <-
    tf$sqrt((1 / (
      batch_size * tf$vary(1, 1 + n_latent, dtype = tf$float32)
    )) *
      tf$reduce_sum(tf$sq.(
        batch_masked - tf$reduce_mean(batch_masked, axis = 1L, keepdims = TRUE)
      ), axis = c(1L, 2L)))
  
  # Keep away from singularity within the case of zeros
  #(latent, batch_size, batch_size)
  all_dists <-
    tf$clip_by_value(all_dists, 1e-14, tf$reduce_max(all_dists))
  
  #inds = tf.argsort(all_dists, axis=-1)
  top_k <- tf$math$top_k(-all_dists, tf$forged(okay + 1, tf$int32))
  # (#(latent, batch_size, batch_size)
  top_indices <- top_k[[1]]
  
  #(latent, batch_size, batch_size)
  neighbor_dists_d <-
    tf$collect(all_dists, top_indices, batch_dims = -1L)
  #(latent - 1, batch_size, batch_size)
  neighbor_new_dists <-
    tf$collect(all_dists[2:-1, , ],
              top_indices[1:-2, , ],
              batch_dims = -1L)
  
  # Eq. 4 of Kennel et al.
  #(latent - 1, batch_size, batch_size)
  scaled_dist <- tf$sqrt((
    tf$sq.(neighbor_new_dists) -
      # (9, 8, 2)
      tf$sq.(neighbor_dists_d[1:-2, , ])) /
      # (9, 8, 2)
      tf$sq.(neighbor_dists_d[1:-2, , ])
  )
  
  # Kennel situation #1
  #(latent - 1, batch_size, batch_size)
  is_false_change <- (scaled_dist > rtol)
  # Kennel situation 2
  #(latent - 1, batch_size, batch_size)
  is_large_jump <-
    (neighbor_new_dists > atol * all_ra[1:-2, tf$newaxis, tf$newaxis])
  
  is_false_neighbor <-
    tf$math$logical_or(is_false_change, is_large_jump)
  #(latent - 1, batch_size, 1)
  total_false_neighbors <-
    tf$forged(is_false_neighbor, tf$int32)[reticulate::py_ellipsis(), 2:(k + 2)]
  
  # Pad zero to match dimensionality of latent area
  # (latent - 1)
  reg_weights <-
    1 - tf$reduce_mean(tf$forged(total_false_neighbors, tf$float32), axis = c(1L, 2L))
  # (latent,)
  reg_weights <- tf$pad(reg_weights, record(record(1L, 0L)))
  
  # Discover batch common exercise
  
  # L2 Exercise regularization
  activations_batch_averaged <-
    tf$sqrt(tf$reduce_mean(tf$sq.(x), axis = 0L))
  
  loss <- tf$reduce_sum(tf$multiply(reg_weights, activations_batch_averaged))
  loss
  
}

Coaching is unchanged as properly, other than the truth that now, we regularly output latent variable variances along with the losses. It’s because with FNN-LSTM, we’ve got to decide on an ample weight for the FNN loss element. An “ample weight” is one the place the variance drops sharply after the primary n variables, with n thought to correspond to attractor dimensionality. For the Lorenz system mentioned within the earlier put up, that is how these variances regarded:

     V1       V2        V3        V4        V5        V6        V7        V8        V9       V10
 0.0739   0.0582   1.12e-6   3.13e-4   1.43e-5   1.52e-8   1.35e-6   1.86e-4   1.67e-4   4.39e-5

If we take variance as an indicator of significance, the primary two variables are clearly extra essential than the remaining. This discovering properly corresponds to “official” estimates of Lorenz attractor dimensionality. For instance, the correlation dimension is estimated to lie round 2.05 (Grassberger and Procaccia 1983).

Thus, right here we’ve got the coaching routine:

train_step <- operate(batch) {
  with (tf$GradientTape(persistent = TRUE) %as% tape, {
    code <- encoder(batch[[1]])
    prediction <- decoder(code)
    
    l_mse <- mse_loss(batch[[2]], prediction)
    l_fnn <- loss_false_nn(code)
    loss <- l_mse + fnn_weight * l_fnn
  })
  
  encoder_gradients <-
    tape$gradient(loss, encoder$trainable_variables)
  decoder_gradients <-
    tape$gradient(loss, decoder$trainable_variables)
  
  optimizer$apply_gradients(purrr::transpose(record(
    encoder_gradients, encoder$trainable_variables
  )))
  optimizer$apply_gradients(purrr::transpose(record(
    decoder_gradients, decoder$trainable_variables
  )))
  
  train_loss(loss)
  train_mse(l_mse)
  train_fnn(l_fnn)
  
  
}

training_loop <- tf_function(autograph(operate(ds_train) {
  for (batch in ds_train) {
    train_step(batch)
  }
  
  tf$print("Loss: ", train_loss$consequence())
  tf$print("MSE: ", train_mse$consequence())
  tf$print("FNN loss: ", train_fnn$consequence())
  
  train_loss$reset_states()
  train_mse$reset_states()
  train_fnn$reset_states()
  
}))


mse_loss <-
  tf$keras$losses$MeanSquaredError(discount = tf$keras$losses$Discount$SUM)

train_loss <- tf$keras$metrics$Imply(title = 'train_loss')
train_fnn <- tf$keras$metrics$Imply(title = 'train_fnn')
train_mse <-  tf$keras$metrics$Imply(title = 'train_mse')

# fnn_multiplier needs to be chosen individually per dataset
# that is the worth we used on the geyser dataset
fnn_multiplier <- 0.7
fnn_weight <- fnn_multiplier * nrow(x_train)/batch_size

# studying price may want adjustment
optimizer <- optimizer_adam(lr = 1e-3)

for (epoch in 1:200) {
 cat("Epoch: ", epoch, " -----------n")
 training_loop(ds_train)
 
 test_batch <- as_iterator(ds_test) %>% iter_next()
 encoded <- encoder(test_batch[[1]]) 
 test_var <- tf$math$reduce_variance(encoded, axis = 0L)
 print(test_var %>% as.numeric() %>% spherical(5))
}

On to what we’ll use as a baseline for comparability.

Vanilla LSTM

Right here is the vanilla LSTM, stacking two layers, every, once more, of measurement 32. Dropout and recurrent dropout had been chosen individually per dataset, as was the training price.

lstm <- operate(n_latent, n_timesteps, n_features, n_recurrent, dropout, recurrent_dropout,
                 optimizer = optimizer_adam(lr =  1e-3)) {
  
  mannequin <- keras_model_sequential() %>%
    layer_lstm(
      models = n_recurrent,
      input_shape = c(n_timesteps, n_features),
      dropout = dropout, 
      recurrent_dropout = recurrent_dropout,
      return_sequences = TRUE
    ) %>% 
    layer_lstm(
      models = n_recurrent,
      dropout = dropout,
      recurrent_dropout = recurrent_dropout,
      return_sequences = TRUE
    ) %>% 
    time_distributed(layer_dense(models = 1))
  
  mannequin %>%
    compile(
      loss = "mse",
      optimizer = optimizer
    )
  mannequin
  
}

mannequin <- lstm(n_latent, n_timesteps, n_features, n_hidden, dropout = 0.2, recurrent_dropout = 0.2)

Information preparation

For all experiments, information had been ready in the identical manner.

In each case, we used the primary 10000 measurements accessible within the respective .pkl recordsdata supplied by Gilpin in his GitHub repository. To avoid wasting on file measurement and never rely upon an exterior information supply, we extracted these first 10000 entries to .csv recordsdata downloadable immediately from this weblog’s repo:

geyser <- obtain.file(
  "https://uncooked.githubusercontent.com/rstudio/ai-blog/grasp/docs/posts/2020-07-20-fnn-lstm/information/geyser.csv",
  "information/geyser.csv")

electrical energy <- obtain.file(
  "https://uncooked.githubusercontent.com/rstudio/ai-blog/grasp/docs/posts/2020-07-20-fnn-lstm/information/electrical energy.csv",
  "information/electrical energy.csv")

ecg <- obtain.file(
  "https://uncooked.githubusercontent.com/rstudio/ai-blog/grasp/docs/posts/2020-07-20-fnn-lstm/information/ecg.csv",
  "information/ecg.csv")

mouse <- obtain.file(
  "https://uncooked.githubusercontent.com/rstudio/ai-blog/grasp/docs/posts/2020-07-20-fnn-lstm/information/mouse.csv",
  "information/mouse.csv")

Must you wish to entry the entire time collection (of significantly better lengths), simply obtain them from Gilpin’s repo and cargo them utilizing reticulate:

Right here is the information preparation code for the primary dataset, geyser – all different datasets had been handled the identical manner.

# the primary 10000 measurements from the compilation supplied by Gilpin
geyser <- read_csv("geyser.csv", col_names = FALSE) %>% choose(X1) %>% pull() %>% unclass()

# standardize
geyser <- scale(geyser)

# varies per dataset; see under 
n_timesteps <- 60
batch_size <- 32

# remodel into [batch_size, timesteps, features] format required by RNNs
gen_timesteps <- operate(x, n_timesteps) {
  do.name(rbind,
          purrr::map(seq_along(x),
                     operate(i) {
                       begin <- i
                       finish <- i + n_timesteps - 1
                       out <- x[start:end]
                       out
                     })
  ) %>%
    na.omit()
}

n <- 10000
prepare <- gen_timesteps(geyser[1:(n/2)], 2 * n_timesteps)
take a look at <- gen_timesteps(geyser[(n/2):n], 2 * n_timesteps) 

dim(prepare) <- c(dim(prepare), 1)
dim(take a look at) <- c(dim(take a look at), 1)

# cut up into enter and goal  
x_train <- prepare[ , 1:n_timesteps, , drop = FALSE]
y_train <- prepare[ , (n_timesteps + 1):(2*n_timesteps), , drop = FALSE]

x_test <- take a look at[ , 1:n_timesteps, , drop = FALSE]
y_test <- take a look at[ , (n_timesteps + 1):(2*n_timesteps), , drop = FALSE]

# create tfdatasets
ds_train <- tensor_slices_dataset(record(x_train, y_train)) %>%
  dataset_shuffle(nrow(x_train)) %>%
  dataset_batch(batch_size)

ds_test <- tensor_slices_dataset(record(x_test, y_test)) %>%
  dataset_batch(nrow(x_test))

Now we’re prepared to take a look at how forecasting goes on our 4 datasets.

Experiments

Geyser dataset

Folks working with time collection could have heard of Outdated Trustworthy, a geyser in Wyoming, US that has regularly been erupting each 44 minutes to 2 hours because the yr 2004. For the subset of knowledge Gilpin extracted,

geyser_train_test.pkl corresponds to detrended temperature readings from the principle runoff pool of the Outdated Trustworthy geyser in Yellowstone Nationwide Park, downloaded from the GeyserTimes database. Temperature measurements begin on April 13, 2015 and happen in one-minute increments.

Like we mentioned above, geyser.csv is a subset of those measurements, comprising the primary 10000 information factors. To decide on an ample timestep for the LSTMs, we examine the collection at numerous resolutions:


Geyer dataset. Top: First 1000 observations. Bottom: Zooming in on the first 200.

Determine 1: Geyer dataset. Prime: First 1000 observations. Backside: Zooming in on the primary 200.

It looks like the habits is periodic with a interval of about 40-50; a timestep of 60 thus appeared like a superb attempt.

Having educated each FNN-LSTM and the vanilla LSTM for 200 epochs, we first examine the variances of the latent variables on the take a look at set. The worth of fnn_multiplier comparable to this run was 0.7.

test_batch <- as_iterator(ds_test) %>% iter_next()
encoded <- encoder(test_batch[[1]]) %>%
  as.array() %>%
  as_tibble()

encoded %>% summarise_all(var)

   V1     V2        V3          V4       V5       V6       V7       V8       V9      V10
0.258 0.0262 0.0000627 0.000000600 0.000533 0.000362 0.000238 0.000121 0.000518 0.000365

There’s a drop in significance between the primary two variables and the remaining; nevertheless, in contrast to within the Lorenz system, V1 and V2 variances additionally differ by an order of magnitude.

Now, it’s attention-grabbing to check prediction errors for each fashions. We’re going to make a remark that can carry by means of to all three datasets to come back.

Maintaining the suspense for some time, right here is the code used to compute per-timestep prediction errors from each fashions. The identical code will likely be used for all different datasets.

calc_mse <- operate(df, y_true, y_pred) {
  (sum((df[[y_true]] - df[[y_pred]])^2))/nrow(df)
}

get_mse <- operate(test_batch, prediction) {
  
  comp_df <- 
    information.body(
      test_batch[[2]][, , 1] %>%
        as.array()) %>%
        rename_with(operate(title) paste0(title, "_true")) %>%
    bind_cols(
      information.body(
        prediction[, , 1] %>%
          as.array()) %>%
          rename_with(operate(title) paste0(title, "_pred")))
  
  mse <- purrr::map(1:dim(prediction)[2],
                        operate(varno)
                          calc_mse(comp_df,
                                   paste0("X", varno, "_true"),
                                   paste0("X", varno, "_pred"))) %>%
    unlist()
  
  mse
}

prediction_fnn <- decoder(encoder(test_batch[[1]]))
mse_fnn <- get_mse(test_batch, prediction_fnn)

prediction_lstm <- mannequin %>% predict(ds_test)
mse_lstm <- get_mse(test_batch, prediction_lstm)

mses <- information.body(timestep = 1:n_timesteps, fnn = mse_fnn, lstm = mse_lstm) %>%
  collect(key = "kind", worth = "mse", -timestep)

ggplot(mses, aes(timestep, mse, shade = kind)) +
  geom_point() +
  scale_color_manual(values = c("#00008B", "#3CB371")) +
  theme_classic() +
  theme(legend.place = "none")

And right here is the precise comparability. One factor particularly jumps to the attention: FNN-LSTM forecast error is considerably decrease for preliminary timesteps, before everything, for the very first prediction, which from this graph we count on to be fairly good!


Per-timestep prediction error as obtained by FNN-LSTM and a vanilla stacked LSTM. Green: LSTM. Blue: FNN-LSTM.

Determine 2: Per-timestep prediction error as obtained by FNN-LSTM and a vanilla stacked LSTM. Inexperienced: LSTM. Blue: FNN-LSTM.

Curiously, we see “jumps” in prediction error, for FNN-LSTM, between the very first forecast and the second, after which between the second and the following ones, reminding of the same jumps in variable significance for the latent code! After the primary ten timesteps, vanilla LSTM has caught up with FNN-LSTM, and we received’t interpret additional growth of the losses primarily based on only a single run’s output.

As a substitute, let’s examine precise predictions. We randomly choose sequences from the take a look at set, and ask each FNN-LSTM and vanilla LSTM for a forecast. The identical process will likely be adopted for the opposite datasets.

given <- information.body(as.array(tf$concat(record(
  test_batch[[1]][, , 1], test_batch[[2]][, , 1]
),
axis = 1L)) %>% t()) %>%
  add_column(kind = "given") %>%
  add_column(num = 1:(2 * n_timesteps))

fnn <- information.body(as.array(prediction_fnn[, , 1]) %>%
                    t()) %>%
  add_column(kind = "fnn") %>%
  add_column(num = (n_timesteps  + 1):(2 * n_timesteps))

lstm <- information.body(as.array(prediction_lstm[, , 1]) %>%
                     t()) %>%
  add_column(kind = "lstm") %>%
  add_column(num = (n_timesteps + 1):(2 * n_timesteps))

compare_preds_df <- bind_rows(given, lstm, fnn)

plots <- 
  purrr::map(pattern(1:dim(compare_preds_df)[2], 16),
             operate(v) {
               ggplot(compare_preds_df, aes(num, .information[[paste0("X", v)]], shade = kind)) +
                 geom_line() +
                 theme_classic() +
                 theme(legend.place = "none", axis.title = element_blank()) +
                 scale_color_manual(values = c("#00008B", "#DB7093", "#3CB371"))
             })

plot_grid(plotlist = plots, ncol = 4)

Listed here are sixteen random picks of predictions on the take a look at set. The bottom fact is displayed in pink; blue forecasts are from FNN-LSTM, inexperienced ones from vanilla LSTM.


60-step ahead predictions from FNN-LSTM (blue) and vanilla LSTM (green) on randomly selected sequences from the test set. Pink: the ground truth.

Determine 3: 60-step forward predictions from FNN-LSTM (blue) and vanilla LSTM (inexperienced) on randomly chosen sequences from the take a look at set. Pink: the bottom fact.

What we count on from the error inspection comes true: FNN-LSTM yields considerably higher predictions for quick continuations of a given sequence.

Let’s transfer on to the second dataset on our record.

Electrical energy dataset

It is a dataset on energy consumption, aggregated over 321 totally different households and fifteen-minute-intervals.

electricity_train_test.pkl corresponds to common energy consumption by 321 Portuguese households between 2012 and 2014, in models of kilowatts consumed in fifteen minute increments. This dataset is from the UCI machine studying database.

Right here, we see a really common sample:


Electricity dataset. Top: First 2000 observations. Bottom: Zooming in on 500 observations, skipping the very beginning of the series.

Determine 4: Electrical energy dataset. Prime: First 2000 observations. Backside: Zooming in on 500 observations, skipping the very starting of the collection.

With such common habits, we instantly tried to foretell the next variety of timesteps (120) – and didn’t should retract behind that aspiration.

For an fnn_multiplier of 0.5, latent variable variances appear like this:

V1          V2            V3       V4       V5            V6       V7         V8      V9     V10
0.390 0.000637 0.00000000288 1.48e-10 2.10e-11 0.00000000119 6.61e-11 0.00000115 1.11e-4 1.40e-4

We positively see a pointy drop already after the primary variable.

How do prediction errors examine on the 2 architectures?


Per-timestep prediction error as obtained by FNN-LSTM and a vanilla stacked LSTM. Green: LSTM. Blue: FNN-LSTM.

Determine 5: Per-timestep prediction error as obtained by FNN-LSTM and a vanilla stacked LSTM. Inexperienced: LSTM. Blue: FNN-LSTM.

Right here, FNN-LSTM performs higher over a protracted vary of timesteps, however once more, the distinction is most seen for quick predictions. Will an inspection of precise predictions affirm this view?


60-step ahead predictions from FNN-LSTM (blue) and vanilla LSTM (green) on randomly selected sequences from the test set. Pink: the ground truth.

Determine 6: 60-step forward predictions from FNN-LSTM (blue) and vanilla LSTM (inexperienced) on randomly chosen sequences from the take a look at set. Pink: the bottom fact.

It does! In truth, forecasts from FNN-LSTM are very spectacular on all time scales.

Now that we’ve seen the simple and predictable, let’s strategy the bizarre and tough.

ECG dataset

Says Gilpin,

ecg_train.pkl and ecg_test.pkl correspond to ECG measurements for 2 totally different sufferers, taken from the PhysioNet QT database.

How do these look?


ECG dataset. Top: First 1000 observations. Bottom: Zooming in on the first 400 observations.

Determine 7: ECG dataset. Prime: First 1000 observations. Backside: Zooming in on the primary 400 observations.

To the layperson that I’m, these don’t look practically as common as anticipated. First experiments confirmed that each architectures aren’t able to coping with a excessive variety of timesteps. In each attempt, FNN-LSTM carried out higher for the very first timestep.

That is additionally the case for n_timesteps = 12, the ultimate attempt (after 120, 60 and 30). With an fnn_multiplier of 1, the latent variances obtained amounted to the next:

     V1        V2          V3        V4         V5       V6       V7         V8         V9       V10
  0.110  1.16e-11     3.78e-9 0.0000992    9.63e-9  4.65e-5  1.21e-4    9.91e-9    3.81e-9   2.71e-8

There is a niche between the primary variable and all different ones; however not a lot variance is defined by V1 both.

Aside from the very first prediction, vanilla LSTM reveals decrease forecast errors this time; nevertheless, we’ve got so as to add that this was not persistently noticed when experimenting with different timestep settings.


Per-timestep prediction error as obtained by FNN-LSTM and a vanilla stacked LSTM. Green: LSTM. Blue: FNN-LSTM.

Determine 8: Per-timestep prediction error as obtained by FNN-LSTM and a vanilla stacked LSTM. Inexperienced: LSTM. Blue: FNN-LSTM.

precise predictions, each architectures carry out finest when a persistence forecast is ample – in actual fact, they produce one even when it’s not.


60-step ahead predictions from FNN-LSTM (blue) and vanilla LSTM (green) on randomly selected sequences from the test set. Pink: the ground truth.

Determine 9: 60-step forward predictions from FNN-LSTM (blue) and vanilla LSTM (inexperienced) on randomly chosen sequences from the take a look at set. Pink: the bottom fact.

On this dataset, we definitely would wish to discover different architectures higher in a position to seize the presence of excessive and low frequencies within the information, comparable to combination fashions. However – had been we compelled to stick with one in every of these, and will do a one-step-ahead, rolling forecast, we’d go together with FNN-LSTM.

Talking of blended frequencies – we haven’t seen the extremes but …

Mouse dataset

“Mouse,” that’s spike charges recorded from a mouse thalamus.

mouse.pkl A time collection of spiking charges for a neuron in a mouse thalamus. Uncooked spike information was obtained from CRCNS and processed with the authors’ code with the intention to generate a spike price time collection.


Mouse dataset. Top: First 2000 observations. Bottom: Zooming in on the first 500 observations.

Determine 10: Mouse dataset. Prime: First 2000 observations. Backside: Zooming in on the primary 500 observations.

Clearly, this dataset will likely be very onerous to foretell. How, after “lengthy” silence, have you learnt {that a} neuron goes to fireside?

As ordinary, we examine latent code variances (fnn_multiplier was set to 0.4):

Whereas it’s straightforward to acquire these estimates, utilizing, as an illustration, the nonlinearTseries package deal explicitly modeled after practices described in Kantz & Schreiber’s basic (Kantz and Schreiber 2004), we don’t wish to extrapolate from our tiny pattern of datasets, and go away such explorations and analyses to additional posts, and/or the reader’s ventures :-). In any case, we hope you loved the demonstration of sensible usability of an strategy that within the previous put up, was primarily launched when it comes to its conceptual attractivity.

Thanks for studying!

Gilpin, William. 2020. “Deep Reconstruction of Unusual Attractors from Time Sequence.” https://arxiv.org/abs/2002.05909.
Grassberger, Peter, and Itamar Procaccia. 1983. “Measuring the Strangeness of Unusual Attractors.” Physica D: Nonlinear Phenomena 9 (1): 189–208. https://doi.org/https://doi.org/10.1016/0167-2789(83)90298-1.

Kantz, Holger, and Thomas Schreiber. 2004. Nonlinear Time Sequence Evaluation. Cambridge College Press.

Sauer, Tim, James A. Yorke, and Martin Casdagli. 1991. “Embedology.” Journal of Statistical Physics 65 (3-4): 579–616. https://doi.org/10.1007/BF01053745.

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