This submit didn’t find yourself fairly the best way I’d imagined. A fast follow-up on the latest Time sequence prediction with
FNN-LSTM, it was imagined to reveal how noisy time sequence (so widespread in
apply) might revenue from a change in structure: As an alternative of FNN-LSTM, an LSTM autoencoder regularized by false nearest
neighbors (FNN) loss, use FNN-VAE, a variational autoencoder constrained by the identical. Nevertheless, FNN-VAE didn’t appear to deal with
noise higher than FNN-LSTM. No plot, no submit, then?

Then again – this isn’t a scientific research, with speculation and experimental setup all preregistered; all that actually
issues is that if there’s one thing helpful to report. And it seems like there may be.

Firstly, FNN-VAE, whereas on par performance-wise with FNN-LSTM, is way superior in that different which means of “efficiency”:
Coaching goes a lot quicker for FNN-VAE.

Secondly, whereas we don’t see a lot distinction between FNN-LSTM and FNN-VAE, we do see a transparent impression of utilizing FNN loss. Including in FNN loss strongly reduces imply squared error with respect to the underlying (denoised) sequence – particularly within the case of VAE, however for LSTM as effectively. That is of specific curiosity with VAE, because it comes with a regularizer
out-of-the-box – specifically, Kullback-Leibler (KL) divergence.

After all, we don’t declare that related outcomes will at all times be obtained on different noisy sequence; nor did we tune any of
the fashions “to dying.” For what could possibly be the intent of such a submit however to indicate our readers attention-grabbing (and promising) concepts
to pursue in their very own experimentation?

The context

This submit is the third in a mini-series.

In Deep attractors: The place deep studying meets chaos, we
defined, with a considerable detour into chaos idea, the thought of FNN loss, launched in (Gilpin 2020). Please seek the advice of
that first submit for theoretical background and intuitions behind the method.

The following submit, Time sequence prediction with FNN-LSTM, confirmed
tips on how to use an LSTM autoencoder, constrained by FNN loss, for forecasting (versus reconstructing an attractor). The outcomes had been beautiful: In multi-step prediction (12-120 steps, with that quantity various by
dataset), the short-term forecasts had been drastically improved by including in FNN regularization. See that second submit for
experimental setup and outcomes on 4 very totally different, non-synthetic datasets.

As we speak, we present tips on how to exchange the LSTM autoencoder by a – convolutional – VAE. In mild of the experimentation outcomes,
already hinted at above, it’s fully believable that the “variational” half isn’t even so necessary right here – {that a}
convolutional autoencoder with simply MSE loss would have carried out simply as effectively on these knowledge. In actual fact, to seek out out, it’s
sufficient to take away the decision to reparameterize() and multiply the KL element of the loss by 0. (We depart this to the
reader, to maintain the submit at cheap size.)

One final piece of context, in case you haven’t learn the 2 earlier posts and wish to soar in right here instantly. We’re
doing time sequence forecasting; so why this speak of autoencoders? Shouldn’t we simply be evaluating an LSTM (or another sort of
RNN, for that matter) to a convnet? In actual fact, the need of a latent illustration is because of the very concept of FNN: The
latent code is meant to mirror the true attractor of a dynamical system. That’s, if the attractor of the underlying
system is roughly two-dimensional, we hope to seek out that simply two of the latent variables have appreciable variance. (This
reasoning is defined in a whole lot of element within the earlier posts.)

FNN-VAE

So, let’s begin with the code for our new mannequin.

The encoder takes the time sequence, of format batch_size x num_timesteps x num_features similar to within the LSTM case, and
produces a flat, 10-dimensional output: the latent code, which FNN loss is computed on.

library(tensorflow)
library(keras)
library(tfdatasets)
library(tfautograph)
library(reticulate)
library(purrr)

vae_encoder_model <- perform(n_timesteps,
                               n_features,
                               n_latent,
                               title = NULL) {
  keras_model_custom(title = title, perform(self) {
    self$conv1 <- layer_conv_1d(kernel_size = 3,
                                filters = 16,
                                strides = 2)
    self$act1 <- layer_activation_leaky_relu()
    self$batchnorm1 <- layer_batch_normalization()
    self$conv2 <- layer_conv_1d(kernel_size = 7,
                                filters = 32,
                                strides = 2)
    self$act2 <- layer_activation_leaky_relu()
    self$batchnorm2 <- layer_batch_normalization()
    self$conv3 <- layer_conv_1d(kernel_size = 9,
                                filters = 64,
                                strides = 2)
    self$act3 <- layer_activation_leaky_relu()
    self$batchnorm3 <- layer_batch_normalization()
    self$conv4 <- layer_conv_1d(
      kernel_size = 9,
      filters = n_latent,
      strides = 2,
      activation = "linear" 
    )
    self$batchnorm4 <- layer_batch_normalization()
    self$flat <- layer_flatten()
    
    perform (x, masks = NULL) {
      x %>%
        self$conv1() %>%
        self$act1() %>%
        self$batchnorm1() %>%
        self$conv2() %>%
        self$act2() %>%
        self$batchnorm2() %>%
        self$conv3() %>%
        self$act3() %>%
        self$batchnorm3() %>%
        self$conv4() %>%
        self$batchnorm4() %>%
        self$flat()
    }
  })
}

The decoder begins from this – flat – illustration and decompresses it right into a time sequence. In each encoder and decoder
(de-)conv layers, parameters are chosen to deal with a sequence size (num_timesteps) of 120, which is what we’ll use for
prediction under.

vae_decoder_model <- perform(n_timesteps,
                               n_features,
                               n_latent,
                               title = NULL) {
  keras_model_custom(title = title, perform(self) {
    self$reshape <- layer_reshape(target_shape = c(1, n_latent))
    self$conv1 <- layer_conv_1d_transpose(kernel_size = 15,
                                          filters = 64,
                                          strides = 3)
    self$act1 <- layer_activation_leaky_relu()
    self$batchnorm1 <- layer_batch_normalization()
    self$conv2 <- layer_conv_1d_transpose(kernel_size = 11,
                                          filters = 32,
                                          strides = 3)
    self$act2 <- layer_activation_leaky_relu()
    self$batchnorm2 <- layer_batch_normalization()
    self$conv3 <- layer_conv_1d_transpose(
      kernel_size = 9,
      filters = 16,
      strides = 2,
      output_padding = 1
    )
    self$act3 <- layer_activation_leaky_relu()
    self$batchnorm3 <- layer_batch_normalization()
    self$conv4 <- layer_conv_1d_transpose(
      kernel_size = 7,
      filters = 1,
      strides = 1,
      activation = "linear"
    )
    self$batchnorm4 <- layer_batch_normalization()
    
    perform (x, masks = NULL) {
      x %>%
        self$reshape() %>%
        self$conv1() %>%
        self$act1() %>%
        self$batchnorm1() %>%
        self$conv2() %>%
        self$act2() %>%
        self$batchnorm2() %>%
        self$conv3() %>%
        self$act3() %>%
        self$batchnorm3() %>%
        self$conv4() %>%
        self$batchnorm4()
    }
  })
}

Notice that though we referred to as these constructors vae_encoder_model() and vae_decoder_model(), there may be nothing
variational to those fashions per se; they’re actually simply an encoder and a decoder, respectively. Metamorphosis right into a VAE will
occur within the coaching process; actually, the one two issues that can make this a VAE are going to be the
reparameterization of the latent layer and the added-in KL loss.

Talking of coaching, these are the routines we’ll name. The perform to compute FNN loss, loss_false_nn(), will be present in
each of the abovementioned predecessor posts; we kindly ask the reader to repeat it from one in every of these locations.

# to reparameterize encoder output earlier than calling decoder
reparameterize <- perform(imply, logvar = 0) {
  eps <- k_random_normal(form = n_latent)
  eps * k_exp(logvar * 0.5) + imply
}

# loss has 3 elements: NLL, KL, and FNN
# in any other case, that is simply regular TF2-style coaching 
train_step_vae <- perform(batch) {
  with (tf$GradientTape(persistent = TRUE) %as% tape, {
    code <- encoder(batch[[1]])
    z <- reparameterize(code)
    prediction <- decoder(z)
    
    l_mse <- mse_loss(batch[[2]], prediction)
    # see loss_false_nn in 2 earlier posts
    l_fnn <- loss_false_nn(code)
    # KL divergence to an ordinary regular
    l_kl <- -0.5 * k_mean(1 - k_square(z))
    # general loss is a weighted sum of all 3 elements
    loss <- l_mse + fnn_weight * l_fnn + kl_weight * l_kl
  })
  
  encoder_gradients <-
    tape$gradient(loss, encoder$trainable_variables)
  decoder_gradients <-
    tape$gradient(loss, decoder$trainable_variables)
  
  optimizer$apply_gradients(purrr::transpose(checklist(
    encoder_gradients, encoder$trainable_variables
  )))
  optimizer$apply_gradients(purrr::transpose(checklist(
    decoder_gradients, decoder$trainable_variables
  )))
  
  train_loss(loss)
  train_mse(l_mse)
  train_fnn(l_fnn)
  train_kl(l_kl)
}

# wrap all of it in autograph
training_loop_vae <- tf_function(autograph(perform(ds_train) {
  
  for (batch in ds_train) {
    train_step_vae(batch) 
  }
  
  tf$print("Loss: ", train_loss$consequence())
  tf$print("MSE: ", train_mse$consequence())
  tf$print("FNN loss: ", train_fnn$consequence())
  tf$print("KL loss: ", train_kl$consequence())
  
  train_loss$reset_states()
  train_mse$reset_states()
  train_fnn$reset_states()
  train_kl$reset_states()
  
}))

To complete up the mannequin part, right here is the precise coaching code. That is practically an identical to what we did for FNN-LSTM earlier than.

n_latent <- 10L
n_features <- 1

encoder <- vae_encoder_model(n_timesteps,
                         n_features,
                         n_latent)

decoder <- vae_decoder_model(n_timesteps,
                         n_features,
                         n_latent)
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$Imply(title = 'train_mse')
train_kl <-  tf$keras$metrics$Imply(title = 'train_kl')

fnn_multiplier <- 1 # default worth utilized in practically all circumstances (see textual content)
fnn_weight <- fnn_multiplier * nrow(x_train)/batch_size

kl_weight <- 1

optimizer <- optimizer_adam(lr = 1e-3)

for (epoch in 1:100) {
  cat("Epoch: ", epoch, " -----------n")
  training_loop_vae(ds_train)
 
  test_batch <- as_iterator(ds_test) %>% iter_next()
  encoded <- encoder(test_batch[[1]][1:1000])
  test_var <- tf$math$reduce_variance(encoded, axis = 0L)
  print(test_var %>% as.numeric() %>% spherical(5))
}

Experimental setup and knowledge

The thought was so as to add white noise to a deterministic sequence. This time, the Roessler
system was chosen, primarily for the prettiness of its attractor, obvious
even in its two-dimensional projections:

Determine 1: Roessler attractor, two-dimensional projections.

Like we did for the Lorenz system within the first a part of this sequence, we use deSolve to generate knowledge from the Roessler
equations.

library(deSolve)

parameters <- c(a = .2,
                b = .2,
                c = 5.7)

initial_state <-
  c(x = 1,
    y = 1,
    z = 1.05)

roessler <- perform(t, state, parameters) {
  with(as.checklist(c(state, parameters)), {
    dx <- -y - z
    dy <- x + a * y
    dz = b + z * (x - c)
    
    checklist(c(dx, dy, dz))
  })
}

instances <- seq(0, 2500, size.out = 20000)

roessler_ts <-
  ode(
    y = initial_state,
    instances = instances,
    func = roessler,
    parms = parameters,
    technique = "lsoda"
  ) %>% unclass() %>% as_tibble()

n <- 10000
roessler <- roessler_ts$x[1:n]

roessler <- scale(roessler)

Then, noise is added, to the specified diploma, by drawing from a standard distribution, centered at zero, with customary deviations
various between 1 and a pair of.5.

# add noise
noise <- 1 # additionally used 1.5, 2, 2.5
roessler <- roessler + rnorm(10000, imply = 0, sd = noise)

Right here you may evaluate results of not including any noise (left), customary deviation-1 (center), and customary deviation-2.5 Gaussian noise:

Determine 2: Roessler sequence with added noise. Prime: none. Center: SD = 1. Backside: SD = 2.5.

In any other case, preprocessing proceeds as within the earlier posts. Within the upcoming outcomes part, we’ll evaluate forecasts not simply
to the “actual,” after noise addition, take a look at cut up of the information, but additionally to the underlying Roessler system – that’s, the factor
we’re actually all in favour of. (Simply that in the actual world, we are able to’t do this test.) This second take a look at set is ready for
forecasting similar to the opposite one; to keep away from duplication we don’t reproduce the code.

n_timesteps <- 120
batch_size <- 32

gen_timesteps <- perform(x, n_timesteps) {
  do.name(rbind,
          purrr::map(seq_along(x),
                     perform(i) {
                       begin <- i
                       finish <- i + n_timesteps - 1
                       out <- x[start:end]
                       out
                     })
  ) %>%
    na.omit()
}

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

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

x_train <- prepare[ , 1:n_timesteps, , drop = FALSE]
y_train <- prepare[ , (n_timesteps + 1):(2*n_timesteps), , drop = FALSE]

ds_train <- tensor_slices_dataset(checklist(x_train, y_train)) %>%
  dataset_shuffle(nrow(x_train)) %>%
  dataset_batch(batch_size)

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]

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

Outcomes

The LSTM used for comparability with the VAE described above is an identical to the structure employed within the earlier submit.
Whereas with the VAE, an fnn_multiplier of 1 yielded adequate regularization for all noise ranges, some extra experimentation
was wanted for the LSTM: At noise ranges 2 and a pair of.5, that multiplier was set to five.

Because of this, in all circumstances, there was one latent variable with excessive variance and a second one in every of minor significance. For all
others, variance was near 0.

In all circumstances right here means: In all circumstances the place FNN regularization was used. As already hinted at within the introduction, the primary
regularizing issue offering robustness to noise right here appears to be FNN loss, not KL divergence. So for all noise ranges,
moreover FNN-regularized LSTM and VAE fashions we additionally examined their non-constrained counterparts.

Low noise

Seeing how all fashions did fantastically on the unique deterministic sequence, a noise degree of 1 can nearly be handled as
a baseline. Right here you see sixteen 120-timestep predictions from each regularized fashions, FNN-VAE (darkish blue), and FNN-LSTM
(orange). The noisy take a look at knowledge, each enter (x, 120 steps) and output (y, 120 steps) are displayed in (blue-ish) gray. In
inexperienced, additionally spanning the entire sequence, we now have the unique Roessler knowledge, the best way they might look had no noise been added.

Determine 3: Roessler sequence with added Gaussian noise of ordinary deviation 1. Gray: precise (noisy) take a look at knowledge. Inexperienced: underlying Roessler system. Orange: Predictions from FNN-LSTM. Darkish blue: Predictions from FNN-VAE.

Regardless of the noise, forecasts from each fashions look wonderful. Is that this because of the FNN regularizer?

forecasts from their unregularized counterparts, we now have to confess these don’t look any worse. (For higher
comparability, the sixteen sequences to forecast had been initiallly picked at random, however used to check all fashions and
circumstances.)

Determine 4: Roessler sequence with added Gaussian noise of ordinary deviation 1. Gray: precise (noisy) take a look at knowledge. Inexperienced: underlying Roessler system. Orange: Predictions from unregularized LSTM. Darkish blue: Predictions from unregularized VAE.

What occurs once we begin to add noise?

Substantial noise

Between noise ranges 1.5 and a pair of, one thing modified, or turned noticeable from visible inspection. Let’s soar on to the
highest-used degree although: 2.5.

Right here first are predictions obtained from the unregularized fashions.

Determine 5: Roessler sequence with added Gaussian noise of ordinary deviation 2.5. Gray: precise (noisy) take a look at knowledge. Inexperienced: underlying Roessler system. Orange: Predictions from unregularized LSTM. Darkish blue: Predictions from unregularized VAE.

Each LSTM and VAE get “distracted” a bit an excessive amount of by the noise, the latter to a fair increased diploma. This results in circumstances
the place predictions strongly “overshoot” the underlying non-noisy rhythm. This isn’t stunning, in fact: They had been educated
on the noisy model; predict fluctuations is what they realized.

Can we see the identical with the FNN fashions?

Determine 6: Roessler sequence with added Gaussian noise of ordinary deviation 2.5. Gray: precise (noisy) take a look at knowledge. Inexperienced: underlying Roessler system. Orange: Predictions from FNN-LSTM. Darkish blue: Predictions from FNN-VAE.

Curiously, we see a significantly better match to the underlying Roessler system now! Particularly the VAE mannequin, FNN-VAE, surprises
with an entire new smoothness of predictions; however FNN-LSTM turns up a lot smoother forecasts as effectively.

“Easy, becoming the system…” – by now you could be questioning, when are we going to provide you with extra quantitative
assertions? If quantitative implies “imply squared error” (MSE), and if MSE is taken to be some divergence between forecasts
and the true goal from the take a look at set, the reply is that this MSE doesn’t differ a lot between any of the 4 architectures.
Put in a different way, it’s principally a perform of noise degree.

Nevertheless, we might argue that what we’re actually all in favour of is how effectively a mannequin forecasts the underlying course of. And there,
we see variations.

Within the following plot, we distinction MSEs obtained for the 4 mannequin varieties (gray: VAE; orange: LSTM; darkish blue: FNN-VAE; inexperienced:
FNN-LSTM). The rows mirror noise ranges (1, 1.5, 2, 2.5); the columns signify MSE in relation to the noisy(“actual”) goal
(left) on the one hand, and in relation to the underlying system on the opposite (proper). For higher visibility of the impact,
MSEs have been normalized as fractions of the utmost MSE in a class.

So, if we wish to predict sign plus noise (left), it’s not extraordinarily vital whether or not we use FNN or not. But when we wish to
predict the sign solely (proper), with rising noise within the knowledge FNN loss turns into more and more efficient. This impact is way
stronger for VAE vs. FNN-VAE than for LSTM vs. FNN-LSTM: The space between the gray line (VAE) and the darkish blue one
(FNN-VAE) turns into bigger and bigger as we add extra noise.

Determine 7: Normalized MSEs obtained for the 4 mannequin varieties (gray: VAE; orange: LSTM; darkish blue: FNN-VAE; inexperienced: FNN-LSTM). Rows are noise ranges (1, 1.5, 2, 2.5); columns are MSE as associated to the actual goal (left) and the underlying system (proper).

Summing up

Our experiments present that when noise is more likely to obscure measurements from an underlying deterministic system, FNN
regularization can strongly enhance forecasts. That is the case particularly for convolutional VAEs, and possibly convolutional
autoencoders normally. And if an FNN-constrained VAE performs as effectively, for time sequence prediction, as an LSTM, there’s a
robust incentive to make use of the convolutional mannequin: It trains considerably quicker.

With that, we conclude our mini-series on FNN-regularized fashions. As at all times, we’d love to listen to from you should you had been in a position to
make use of this in your personal work!

Thanks for studying!