FNN-VAE for noisy time sequence forecasting

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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 display how noisy time sequence (so frequent in follow) 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?

However – 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 to be like there may be.

Firstly, FNN-VAE, whereas on par performance-wise with FNN-LSTM, is way superior in that different that 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 influence 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 – particularly, Kullback-Leibler (KL) divergence.

After all, we don’t declare that comparable outcomes will all the time be obtained on different noisy sequence; nor did we tune any of the fashions “to loss of life.” 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 concept, the thought of FNN loss, launched in (Gilpin 2020). Please seek the advice of that first submit for theoretical background and intuitions behind the approach.

The following submit, Time sequence prediction with FNN-LSTM, confirmed the way to use an LSTM autoencoder, constrained by FNN loss, for forecasting (versus reconstructing an attractor). The outcomes have been beautiful: 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 completely different, non-synthetic datasets.

In the present day, we present the way to substitute 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 is just not even so necessary right here – {that a} convolutional autoencoder with simply MSE loss would have carried out simply as effectively on these information. In actual fact, to seek 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 bounce 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 seek 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 <- operate(n_timesteps,
                               n_features,
                               n_latent,
                               title = NULL) {
  keras_model_custom(title = title, operate(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()
    
    operate (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 <- operate(n_timesteps,
                               n_features,
                               n_latent,
                               title = NULL) {
  keras_model_custom(title = title, operate(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()
    
    operate (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 regardless that we known 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; in reality, 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 operate to compute FNN loss, loss_false_nn(), might 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 <- operate(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 <- operate(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$apply_gradients(purrr::transpose(checklist(
    encoder_gradients, encoder$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(operate(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 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 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 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:


Roessler attractor, 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 <- operate(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))
  })
}

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

roessler_ts <-
  ode(
    y = initial_state,
    occasions = occasions,
    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 traditional distribution, centered at zero, with commonplace 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 possibly can examine results of not including any noise (left), commonplace deviation-1 (center), and commonplace deviation-2.5 Gaussian noise:


Roessler series with added noise. Top: none. Middle: SD = 1. Bottom: SD = 2.5.

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 examine forecasts not simply to the “actual,” after noise addition, take a look at cut up of the info, but additionally to the underlying Roessler system – that’s, the factor we’re actually occupied with. (Simply that in the actual world, we will’t do this test.) 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 <- 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()
}

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

dim(practice) <- c(dim(practice), 1)
dim(take a look at) <- c(dim(take a look at), 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(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 couple of.5, that multiplier was set to five.

Consequently, 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 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 stage 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’ve got the unique Roessler information, the way in which they’d look had no noise been added.


Roessler series with added Gaussian noise of standard deviation 1. Grey: actual (noisy) test data. Green: underlying Roessler system. Orange: Predictions from FNN-LSTM. Dark blue: Predictions from FNN-VAE.

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’ve got 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 circumstances.)


Roessler series with added Gaussian noise of standard deviation 1. Grey: actual (noisy) test data. Green: underlying Roessler system. Orange: Predictions from unregularized LSTM. Dark blue: Predictions from unregularized VAE.

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 bounce on to the highest-used stage although: 2.5.

Right here first are predictions obtained from the unregularized fashions.


Roessler series with added Gaussian noise of standard deviation 2.5. Grey: actual (noisy) test data. Green: underlying Roessler system. Orange: Predictions from unregularized LSTM. Dark blue: Predictions from unregularized VAE.

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 fair increased diploma. This results in circumstances the place predictions strongly “overshoot” the underlying non-noisy rhythm. This isn’t shocking, after all: They have been educated on the noisy model; predict fluctuations is what they discovered.

Can we see the identical with the FNN fashions?


Roessler series with added Gaussian noise of standard deviation 2.5. Grey: actual (noisy) test data. Green: underlying Roessler system. Orange: Predictions from FNN-LSTM. Dark blue: Predictions from FNN-VAE.

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.

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.

“Easy, becoming the system…” – by now chances are you’ll 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 largely a operate of noise stage.

Nevertheless, we may argue that what we’re actually occupied with 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 sorts (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 isn’t extraordinarily important whether or not we use FNN or not. But when we need to predict the sign solely (proper), with growing 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 space between the gray line (VAE) and the darkish blue one (FNN-VAE) turns into bigger and bigger as we add extra noise.


Normalized MSEs obtained for the four model types (grey: VAE; orange: LSTM; dark blue: FNN-VAE; green: FNN-LSTM). Rows are noise levels (1, 1.5, 2, 2.5); columns are MSE as related to the real target (left) and the underlying system (right).

Determine 7: Normalized MSEs obtained for the 4 mannequin sorts (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 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 all the time, we’d love to listen to from you when you have been capable of make use of this in your individual work!

Thanks for studying!

Gilpin, William. 2020. “Deep Reconstruction of Unusual Attractors from Time Sequence.” https://arxiv.org/abs/2002.05909.

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