FNN-VAE for noisy time collection forecasting

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

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

Firstly, FNN-VAE, whereas on par performance-wise with FNN-LSTM, is much superior in that different that means of “efficiency”:
Coaching goes a lot sooner 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) collection – 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 – particularly, Kullback-Leibler (KL) divergence.

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

The context

This publish is the third in a mini-series.

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

The next publish, Time collection prediction with FNN-LSTM, confirmed
the best way 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 publish for
experimental setup and outcomes on 4 very totally different, non-synthetic datasets.

Right this moment, we present the best way to exchange the LSTM autoencoder by a – convolutional – VAE. In gentle of the experimentation outcomes,
already hinted at above, it’s fully believable that the “variational” half shouldn’t be even so essential right here – {that a}
convolutional autoencoder with simply MSE loss would have carried out simply as effectively on these knowledge. In actual fact, to search out out, it’s
sufficient to take away the decision to reparameterize() and multiply the KL part of the loss by 0. (We depart this to the
reader, to maintain the publish at cheap size.)

One final piece of context, in case you haven’t learn the 2 earlier posts and wish to bounce in right here immediately. We’re
doing time collection forecasting; so why this speak of autoencoders? Shouldn’t we simply be evaluating an LSTM (or another kind of
RNN, for that matter) to a convnet? In actual fact, the need of a latent illustration is as a result of very thought 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 search out that simply two of the latent variables have appreciable variance. (This
reasoning is defined in a variety of element within the earlier posts.)

FNN-VAE

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

The encoder takes the time collection, of format batch_size x num_timesteps x num_features identical 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,
                               identify = NULL) {
  keras_model_custom(identify = identify, 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 beneath.

vae_decoder_model <- perform(n_timesteps,
                               n_features,
                               n_latent,
                               identify = NULL) {
  keras_model_custom(identify = identify, 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()
    }
  })
}

Observe that regardless that we known as these constructors vae_encoder_model() and vae_decoder_model(), there’s 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; the truth is, 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(), may be present in
each of the abovementioned predecessor posts; we kindly ask the reader to repeat it from one in all 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 parts: 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 a normal regular
    l_kl <- -0.5 * k_mean(1 - k_square(z))
    # general loss is a weighted sum of all 3 parts
    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$apply_gradients(purrr::transpose(listing(
    encoder_gradients, encoder$trainable_variables
  )))
  optimizer$apply_gradients(purrr::transpose(listing(
    decoder_gradients, decoder$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$print("Loss: ", train_loss$outcome())
  tf$print("MSE: ", train_mse$outcome())
  tf$print("FNN loss: ", train_fnn$outcome())
  tf$print("KL loss: ", train_kl$outcome())
  
  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 almost 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$Imply(identify = 'train_loss')
train_fnn <- tf$keras$metrics$Imply(identify = 'train_fnn')
train_mse <-  tf$keras$metrics$Imply(identify = 'train_mse')
train_kl <-  tf$keras$metrics$Imply(identify = 'train_kl')

fnn_multiplier <- 1 # default worth utilized in almost 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 collection. 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 collection, 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.listing(c(state, parameters)), {
    dx <- -y - z
    dy <- x + a * y
    dz = b + z * (x - c)
    
    listing(c(dx, dy, dz))
  })
}

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

roessler_ts <-
  ode(
    y = initial_state,
    occasions = occasions,
    func = roessler,
    parms = parameters,
    methodology = "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 traditional 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 possibly can evaluate results of not including any noise (left), customary deviation-1 (center), and customary deviation-2.5 Gaussian noise:

Determine 2: Roessler collection 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, check cut up of the information, but additionally to the underlying Roessler system – that’s, the factor
we’re actually excited about. (Simply that in the true world, we are able to’t do this verify.) This second check set is ready for
forecasting identical 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()
}

practice <- gen_timesteps(roessler[1:(n/2)], 2 * n_timesteps)
check <- gen_timesteps(roessler[(n/2):n], 2 * n_timesteps) 

dim(practice) <- c(dim(practice), 1)
dim(check) <- c(dim(check), 1)

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

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

x_test <- check[ , 1:n_timesteps, , drop = FALSE]
y_test <- check[ , (n_timesteps + 1):(2*n_timesteps), , drop = FALSE]

ds_test <- tensor_slices_dataset(listing(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 publish.
Whereas with the VAE, an fnn_multiplier of 1 yielded ample 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.

In consequence, in all circumstances, there was one latent variable with excessive variance and a second one in all 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 principle
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 collection, a noise degree of 1 can virtually 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 check 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’ve got the unique Roessler knowledge, the best way they’d look had no noise been added.

Determine 3: Roessler collection with added Gaussian noise of normal deviation 1. Gray: precise (noisy) check 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 glorious. Is that this as a result of FNN regularizer?

Taking a look at forecasts from their unregularized counterparts, we’ve got 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
situations.)

Determine 4: Roessler collection with added Gaussian noise of normal deviation 1. Gray: precise (noisy) check 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 bounce on to the
highest-used degree although: 2.5.

Right here first are predictions obtained from the unregularized fashions.

Determine 5: Roessler collection with added Gaussian noise of normal deviation 2.5. Gray: precise (noisy) check 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 an excellent larger diploma. This results in circumstances
the place predictions strongly “overshoot” the underlying non-noisy rhythm. This isn’t stunning, in fact: They had been skilled
on the noisy model; predict fluctuations is what they realized.

Can we see the identical with the FNN fashions?

Determine 6: Roessler collection with added Gaussian noise of normal deviation 2.5. Gray: precise (noisy) check knowledge. Inexperienced: underlying Roessler system. Orange: Predictions from FNN-LSTM. Darkish blue: Predictions from FNN-VAE.

Apparently, 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.

“Clean, becoming the system…” – by now chances are you’ll be questioning, when are we going to give you 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 check set, the reply is that this MSE doesn’t differ a lot between any of the 4 architectures.
Put otherwise, it’s principally a perform of noise degree.

Nonetheless, we might argue that what we’re actually excited about 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 characterize 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 need to predict sign plus noise (left), it’s not extraordinarily crucial whether or not we use FNN or not. But when we need to
predict the sign solely (proper), with rising noise within the knowledge FNN loss turns into more and more efficient. This impact is much
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 true goal (left) and the underlying system (proper).

Summing up

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

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

Thanks for studying!