This submit didn’t find yourself fairly the way in which I’d imagined. A fast follow-up on the current Time sequence prediction with
FNN-LSTM, it was purported to exhibit how noisy time sequence (so widespread in
observe) may 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 examine, with speculation and experimental setup all preregistered; all that basically
issues is that if there’s one thing helpful to report. And it seems like there’s.

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 affect 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 properly. That is of explicit 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 all the time be obtained on different noisy sequence; nor did we tune any of
the fashions “to demise.” For what could possibly be the intent of such a submit however to point out 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 concept, the concept of FNN loss, launched in (Gilpin 2020). Please seek the advice of
that first submit for theoretical background and intuitions behind the method.

The next submit, Time sequence prediction with FNN-LSTM, confirmed
the right way to use an LSTM autoencoder, constrained by FNN loss, for forecasting (versus reconstructing an attractor). The outcomes have been gorgeous: In multi-step prediction (12-120 steps, with that quantity various by
dataset), the short-term forecasts have 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 the right way to substitute 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 will not be even so vital right here – {that a}
convolutional autoencoder with simply MSE loss would have carried out simply as properly on these information. 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 submit at cheap size.)

One final piece of context, in case you haven’t learn the 2 earlier posts and want to leap in right here immediately. 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 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 quite a 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 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 under.

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()
    }
  })
}

Word that regardless that we referred to 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; 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(), may be present in
each of the abovementioned predecessor posts; we kindly ask the reader to repeat it from considered one 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 a typical 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$apply_gradients(purrr::transpose(listing(
    encoder_gradients, encoder$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$print("Loss: ", train_loss$end result())
  tf$print("MSE: ", train_mse$end result())
  tf$print("FNN loss: ", train_fnn$end result())
  tf$print("KL loss: ", train_kl$end result())
  
  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 similar 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 instances (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 information

The concept 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 information 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))
  })
}

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

roessler_ts <-
  ode(
    y = initial_state,
    instances = instances,
    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 standard distribution, centered at zero, with normal deviations
various between 1 and a couple 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), normal deviation-1 (center), and normal deviation-2.5 Gaussian noise:

Determine 2: Roessler sequence with added noise. High: 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 info, but in addition to the underlying Roessler system – that’s, the factor
we’re actually fascinated by. (Simply that in the actual world, we will’t do this verify.) This second take a look at 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()
}

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(listing(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(listing(x_test, y_test)) %>%
  dataset_batch(nrow(x_test))

Outcomes

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

Consequently, in all instances, there was one latent variable with excessive variance and a second considered one of minor significance. For all
others, variance was near 0.

In all instances right here means: In all instances 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,
apart from 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 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 take a look at information, each enter (x, 120 steps) and output (y, 120 steps) are displayed in (blue-ish) gray. In
inexperienced, additionally spanning the entire sequence, we have now the unique Roessler information, the way in which they might look had no noise been added.

Determine 3: Roessler sequence with added Gaussian noise of normal deviation 1. Gray: precise (noisy) take a look at information. 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?

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

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

What occurs after we begin to add noise?

Substantial noise

Between noise ranges 1.5 and a couple of, one thing modified, or grew to become noticeable from visible inspection. Let’s leap 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 normal deviation 2.5. Gray: precise (noisy) take a look at information. 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 good larger diploma. This results in instances
the place predictions strongly “overshoot” the underlying non-noisy rhythm. This isn’t stunning, after all: They have been skilled
on the noisy model; predict fluctuations is what they realized.

Will we see the identical with the FNN fashions?

Determine 6: Roessler sequence with added Gaussian noise of normal deviation 2.5. Gray: precise (noisy) take a look at information. 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 properly.

“Clean, becoming the system…” – by now you might 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 otherwise, it’s principally a perform of noise degree.

Nevertheless, we may argue that what we’re actually fascinated by is how properly 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 isn’t extraordinarily essential whether or not we use FNN or not. But when we wish to
predict the sign solely (proper), with rising noise within the information 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 gap 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 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 on the whole. And if an FNN-constrained VAE performs as properly, for time sequence prediction, as an LSTM, there’s a
sturdy incentive to make use of the convolutional mannequin: It trains considerably quicker.

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

Thanks for studying!