For us deep studying practitioners, the world is – not flat, however – linear, largely. Or piecewise linear. Like different

linear approximations, or perhaps much more so, deep studying may be extremely profitable at making predictions. However let’s

admit it – typically we simply miss the fun of the nonlinear, of excellent, outdated, deterministic-yet-unpredictable chaos. Can we

have each? It appears like we are able to. On this submit, we’ll see an software of deep studying (DL) to nonlinear time collection

prediction – or fairly, the important step that predates it: reconstructing the attractor underlying its dynamics. Whereas this

submit is an introduction, presenting the subject from scratch, additional posts will construct on this and extrapolate to observational

datasets.

### What to anticipate from this submit

In his 2020 paper *Deep reconstruction of unusual attractors from time collection* (Gilpin 2020), William Gilpin makes use of an

autoencoder structure, mixed with a regularizer implementing the *false nearest neighbors* statistic

(Kennel, Brown, and Abarbanel 1992), to reconstruct attractors from univariate observations of multivariate, nonlinear dynamical programs. If

you’re feeling you fully perceive the sentence you simply learn, you could as properly instantly soar to the paper – come again for the

code although. If, however, you’re extra acquainted with the chaos in your desk (extrapolating … apologies) than

*chaos idea chaos*, learn on. Right here, we’ll first go into what it’s all about, after which, present an instance software,

that includes Edward Lorenz’s well-known butterfly attractor. Whereas this preliminary submit is primarily purported to be a enjoyable introduction

to an interesting matter, we hope to observe up with purposes to real-world datasets sooner or later.

## Rabbits, butterflies, and low-dimensional projections: Our downside assertion in context

In curious misalignment with how we use “chaos” in day-to-day language, chaos, the technical idea, could be very totally different from

stochasticity, or randomness. Chaos might emerge from purely deterministic processes – very simplistic ones, even. Let’s see

how; with rabbits.

### Rabbits, or: Delicate dependence on preliminary circumstances

Chances are you’ll be acquainted with the *logistic* equation, used as a toy mannequin for inhabitants development. Typically it’s written like this –

with (x) being the dimensions of the inhabitants, expressed as a fraction of the maximal dimension (a fraction of potential rabbits, thus),

and (r) being the expansion charge (the speed at which rabbits reproduce):

[

x_{n + 1} = r x_n (1 – x_n)

]

This equation describes an *iterated map* over discrete timesteps (n). Its repeated software leads to a *trajectory*

describing how the inhabitants of rabbits evolves. Maps can have *fastened factors*, states the place additional perform software goes

on producing the identical end result eternally. Instance-wise, say the expansion charge quantities to (2.1), and we begin at two (fairly

totally different!) preliminary values, (0.3) and (0.8). Each trajectories arrive at a hard and fast level – the identical fastened level – in fewer

than 10 iterations. Had been we requested to foretell the inhabitants dimension after 100 iterations, we might make a really assured

guess, regardless of the of beginning worth. (If the preliminary worth is (0), we keep at (0), however we may be fairly sure of that as

properly.)

What if the expansion charge had been considerably greater, at (3.3), say? Once more, we instantly evaluate trajectories ensuing from preliminary

values (0.3) and (0.9):

This time, don’t see a single fastened level, however a *two-cycle*: Because the trajectories stabilize, inhabitants dimension inevitably is at

considered one of two potential values – both too many rabbits or too few, you could possibly say. The 2 trajectories are phase-shifted, however

once more, the attracting values – the *attractor* – is shared by each preliminary circumstances. So nonetheless, predictability is fairly

excessive. However we haven’t seen the whole lot but.

Let’s once more improve the expansion charge some. Now *this* (actually) is chaos:

Even after 100 iterations, there isn’t any set of values the trajectories recur to. We are able to’t be assured about any

prediction we would make.

Or can we? In spite of everything, we have now the governing equation, which is deterministic. So we must always have the ability to calculate the dimensions of

the inhabitants at, say, time (150)? In precept, sure; however this presupposes we have now an correct measurement for the beginning

state.

How correct? Let’s evaluate trajectories for preliminary values (0.3) and (0.301):

At first, trajectories appear to leap round in unison; however throughout the second dozen iterations already, they dissociate extra and

extra, and more and more, all bets are off. What if preliminary values are *actually* shut, as in, (0.3) vs. (0.30000001)?

It simply takes a bit longer for the disassociation to floor.

What we’re seeing right here is *delicate dependence on preliminary circumstances*, a vital precondition for a system to be chaotic.

In an nutshell: Chaos arises when a *deterministic* system reveals *delicate dependence on preliminary circumstances*. Or as Edward

Lorenz is said to have put it,

When the current determines the long run, however the approximate current doesn’t roughly decide the long run.

Now if these unstructured, random-looking level clouds represent chaos, what with the all-but-amorphous butterfly (to be

displayed very quickly)?

### Butterflies, or: Attractors and unusual attractors

Truly, within the context of chaos idea, the time period butterfly could also be encountered in several contexts.

Firstly, as so-called “butterfly impact,” it’s an instantiation of the templatic phrase “the flap of a butterfly’s wing in

_________ profoundly impacts the course of the climate in _________.” On this utilization, it’s largely a

metaphor for delicate dependence on preliminary circumstances.

Secondly, the existence of this metaphor led to a Rorschach-test-like identification with two-dimensional visualizations of

attractors of the Lorenz system. The Lorenz system is a set of three first-order differential equations designed to explain

atmospheric convection:

[

begin{aligned}

& frac{dx}{dt} = sigma (y – x)

& frac{dy}{dt} = rho x – x z – y

& frac{dz}{dt} = x y – beta z

end{aligned}

]

This set of equations is nonlinear, as required for chaotic habits to look. It additionally has the required dimensionality, which

for clean, steady programs, is a minimum of 3. Whether or not we really see chaotic attractors – amongst which, the butterfly –

is determined by the settings of the parameters (sigma), (rho) and (beta). For the values conventionally chosen, (sigma=10),

(rho=28), and (beta=8/3) , we see it when projecting the trajectory on the (x) and (z) axes:

The butterfly is an *attractor* (as are the opposite two projections), however it’s neither some extent nor a cycle. It’s an attractor

within the sense that ranging from a wide range of totally different preliminary values, we find yourself in some sub-region of the state house, and we

don’t get to flee no extra. That is simpler to see when watching evolution over time, as on this animation:

Now, to plot the attractor in two dimensions, we threw away the third. However in “actual life,” we don’t normally have too *a lot*

info (though it might typically look like we had). We’d have a number of measurements, however these don’t normally mirror

the precise state variables we’re concerned with. In these instances, we might wish to really *add* info.

### Embeddings (as a non-DL time period), or: Undoing the projection

Assume that as an alternative of all three variables of the Lorenz system, we had measured only one: (x), the speed of convection. Typically

in nonlinear dynamics, the strategy of delay coordinate embedding (Sauer, Yorke, and Casdagli 1991) is used to reinforce a collection of univariate

measurements.

On this technique – or household of strategies – the univariate collection is augmented by time-shifted copies of itself. There are two

selections to be made: What number of copies so as to add, and the way huge the delay needs to be. For example, if we had a scalar collection,

`1 2 3 4 5 6 7 8 9 10 11 ...`

a three-dimensional embedding with time delay 2 would seem like this:

```
1 3 5
2 4 6
3 5 7
4 6 8
5 7 9
6 8 10
7 9 11
...
```

Of the 2 selections to be made – variety of shifted collection and time lag – the primary is a choice on the dimensionality of

the reconstruction house. Numerous theorems, comparable to Taken’s theorem,

point out bounds on the variety of dimensions required, offered the dimensionality of the true state house is understood – which,

in real-world purposes, typically will not be the case.The second has been of little curiosity to mathematicians, however is vital

in follow. The truth is, Kantz and Schreiber (Kantz and Schreiber 2004) argue that in follow, it’s the product of each parameters that issues,

because it signifies the time span represented by an embedding vector.

How are these parameters chosen? Relating to reconstruction dimensionality, the reasoning goes that even in chaotic programs,

factors which can be shut in state house at time (t) ought to nonetheless be shut at time (t + Delta t), offered (Delta t) could be very

small. So say we have now two factors which can be shut, by some metric, when represented in two-dimensional house. However in three

dimensions, that’s, if we don’t “mission away” the third dimension, they’re much more distant. As illustrated in

(Gilpin 2020):

If this occurs, then projecting down has eradicated some important info. In 2nd, the factors had been *false neighbors*. The

*false nearest neighbors* (FNN) statistic can be utilized to find out an ample embedding dimension, like this:

For every level, take its closest neighbor in (m) dimensions, and compute the ratio of their distances in (m) and (m+1)

dimensions. If the ratio is bigger than some threshold (t), the neighbor was false. Sum the variety of false neighbors over all

factors. Do that for various (m) and (t), and examine the ensuing curves.

At this level, let’s look forward on the autoencoder method. The autoencoder will use that very same FNN statistic as a

regularizer, along with the same old autoencoder reconstruction loss. This may lead to a brand new heuristic concerning embedding

dimensionality that entails fewer selections.

Going again to the traditional technique for an immediate, the second parameter, the time lag, is much more tough to type out

(Kantz and Schreiber 2004). Normally, mutual info is plotted for various delays after which, the primary delay the place it falls under some

threshold is chosen. We don’t additional elaborate on this query as it’s rendered out of date within the neural community method.

Which we’ll see now.

## Studying the Lorenz attractor

Our code intently follows the structure, parameter settings, and knowledge setup used within the reference

implementation William offered. The loss perform, particularly, has been ported

one-to-one.

The final concept is the next. An autoencoder – for instance, an LSTM autoencoder as introduced right here – is used to compress

the univariate time collection right into a latent illustration of some dimensionality, which is able to represent an higher certain on the

dimensionality of the realized attractor. Along with imply squared error between enter and reconstructions, there will probably be a

second loss time period, making use of the FNN regularizer. This leads to the latent models being roughly ordered by *significance*, as

measured by their variance. It’s anticipated that someplace within the itemizing of variances, a pointy drop will seem. The models

earlier than the drop are then assumed to encode the *attractor* of the system in query.

On this setup, there’s nonetheless a option to be made: the right way to weight the FNN loss. One would run coaching for various weights

(lambda) and search for the drop. Certainly, this might in precept be automated, however given the novelty of the strategy – the

paper was printed this 12 months – it is smart to give attention to thorough evaluation first.

### Knowledge technology

We use the `deSolve`

bundle to generate knowledge from the Lorenz equations.

```
library(deSolve)
library(tidyverse)
parameters <- c(sigma = 10,
rho = 28,
beta = 8/3)
initial_state <-
c(x = -8.60632853,
y = -14.85273055,
z = 15.53352487)
lorenz <- perform(t, state, parameters) {
with(as.list(c(state, parameters)), {
dx <- sigma * (y - x)
dy <- x * (rho - z) - y
dz <- x * y - beta * z
list(c(dx, dy, dz))
})
}
instances <- seq(0, 500, size.out = 125000)
lorenz_ts <-
ode(
y = initial_state,
instances = instances,
func = lorenz,
parms = parameters,
technique = "lsoda"
) %>% as_tibble()
lorenz_ts[1:10,]
```

```
# A tibble: 10 x 4
time x y z
<dbl> <dbl> <dbl> <dbl>
1 0 -8.61 -14.9 15.5
2 0.00400 -8.86 -15.2 15.9
3 0.00800 -9.12 -15.6 16.3
4 0.0120 -9.38 -16.0 16.7
5 0.0160 -9.64 -16.3 17.1
6 0.0200 -9.91 -16.7 17.6
7 0.0240 -10.2 -17.0 18.1
8 0.0280 -10.5 -17.3 18.6
9 0.0320 -10.7 -17.7 19.1
10 0.0360 -11.0 -18.0 19.7
```

We’ve already seen the attractor, or fairly, its three two-dimensional projections, in determine 6 above. However now our situation is

totally different. We solely have entry to (x), a univariate time collection. Because the time interval used to numerically combine the

differential equations was fairly tiny, we simply use each tenth statement.

### Preprocessing

The primary half of the collection is used for coaching. The information is scaled and reworked into the three-dimensional kind anticipated

by recurrent layers.

```
library(keras)
library(tfdatasets)
library(tfautograph)
library(reticulate)
library(purrr)
# scale observations
obs <- obs %>% mutate(
x = scale(x)
)
# generate timesteps
n <- nrow(obs)
n_timesteps <- 10
gen_timesteps <- perform(x, n_timesteps) {
do.call(rbind,
purrr::map(seq_along(x),
perform(i) {
begin <- i
finish <- i + n_timesteps - 1
out <- x[start:end]
out
})
) %>%
na.omit()
}
# practice with begin of time collection, take a look at with finish of time collection
x_train <- gen_timesteps(as.matrix(obs$x)[1:(n/2)], n_timesteps)
x_test <- gen_timesteps(as.matrix(obs$x)[(n/2):n], n_timesteps)
# add required dimension for options (we have now one)
dim(x_train) <- c(dim(x_train), 1)
dim(x_test) <- c(dim(x_test), 1)
# some batch dimension (worth not essential)
batch_size <- 100
# remodel to datasets so we are able to use customized coaching
ds_train <- tensor_slices_dataset(x_train) %>%
dataset_batch(batch_size)
ds_test <- tensor_slices_dataset(x_test) %>%
dataset_batch(nrow(x_test))
```

### Autoencoder

With newer variations of TensorFlow (>= 2.0, actually if >= 2.2), autoencoder-like fashions are greatest coded as customized fashions,

and skilled in an “autographed” loop.

The encoder is centered round a single LSTM layer, whose dimension determines the utmost dimensionality of the attractor. The

decoder then undoes the compression – once more, primarily utilizing a single LSTM.

```
# dimension of the latent code
n_latent <- 10L
n_features <- 1
encoder_model <- perform(n_timesteps,
n_features,
n_latent,
identify = NULL) {
keras_model_custom(identify = identify, perform(self) {
self$noise <- layer_gaussian_noise(stddev = 0.5)
self$lstm <- layer_lstm(
models = n_latent,
input_shape = c(n_timesteps, n_features),
return_sequences = FALSE
)
self$batchnorm <- layer_batch_normalization()
perform (x, masks = NULL) {
x %>%
self$noise() %>%
self$lstm() %>%
self$batchnorm()
}
})
}
decoder_model <- perform(n_timesteps,
n_features,
n_latent,
identify = NULL) {
keras_model_custom(identify = identify, perform(self) {
self$repeat_vector <- layer_repeat_vector(n = n_timesteps)
self$noise <- layer_gaussian_noise(stddev = 0.5)
self$lstm <- layer_lstm(
models = n_latent,
return_sequences = TRUE,
go_backwards = TRUE
)
self$batchnorm <- layer_batch_normalization()
self$elu <- layer_activation_elu()
self$time_distributed <- time_distributed(layer = layer_dense(models = n_features))
perform (x, masks = NULL) {
x %>%
self$repeat_vector() %>%
self$noise() %>%
self$lstm() %>%
self$batchnorm() %>%
self$elu() %>%
self$time_distributed()
}
})
}
encoder <- encoder_model(n_timesteps, n_features, n_latent)
decoder <- decoder_model(n_timesteps, n_features, n_latent)
```

### Loss

As already defined above, the loss perform we practice with is twofold. On the one hand, we evaluate the unique inputs with

the decoder outputs (the reconstruction), utilizing imply squared error:

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

As well as, we attempt to hold the variety of false neighbors small, by the use of the next regularizer.

```
loss_false_nn <- perform(x) {
# authentic values utilized in Kennel et al. (1992)
rtol <- 10
atol <- 2
k_frac <- 0.01
ok <- max(1, floor(k_frac * batch_size))
tri_mask <-
tf$linalg$band_part(
tf$ones(
form = c(n_latent, n_latent),
dtype = tf$float32
),
num_lower = -1L,
num_upper = 0L
)
batch_masked <- tf$multiply(
tri_mask[, tf$newaxis,], x[tf$newaxis, reticulate::py_ellipsis()]
)
x_squared <- tf$reduce_sum(
batch_masked * batch_masked,
axis = 2L,
keepdims = TRUE
)
pdist_vector <- x_squared +
tf$transpose(
x_squared, perm = c(0L, 2L, 1L)
) -
2 * tf$matmul(
batch_masked,
tf$transpose(batch_masked, perm = c(0L, 2L, 1L))
)
all_dists <- pdist_vector
all_ra <-
tf$sqrt((1 / (
batch_size * tf$vary(1, 1 + n_latent, dtype = tf$float32)
)) *
tf$reduce_sum(tf$sq.(
batch_masked - tf$reduce_mean(batch_masked, axis = 1L, keepdims = TRUE)
), axis = c(1L, 2L)))
all_dists <- tf$clip_by_value(all_dists, 1e-14, tf$reduce_max(all_dists))
top_k <- tf$math$top_k(-all_dists, tf$forged(ok + 1, tf$int32))
top_indices <- top_k[[1]]
neighbor_dists_d <- tf$collect(all_dists, top_indices, batch_dims = -1L)
neighbor_new_dists <- tf$collect(
all_dists[2:-1, , ],
top_indices[1:-2, , ],
batch_dims = -1L
)
# Eq. 4 of Kennel et al. (1992)
scaled_dist <- tf$sqrt((
tf$sq.(neighbor_new_dists) -
tf$sq.(neighbor_dists_d[1:-2, , ])) /
tf$sq.(neighbor_dists_d[1:-2, , ])
)
# Kennel situation #1
is_false_change <- (scaled_dist > rtol)
# Kennel situation #2
is_large_jump <-
(neighbor_new_dists > atol * all_ra[1:-2, tf$newaxis, tf$newaxis])
is_false_neighbor <-
tf$math$logical_or(is_false_change, is_large_jump)
total_false_neighbors <-
tf$forged(is_false_neighbor, tf$int32)[reticulate::py_ellipsis(), 2:(k + 2)]
reg_weights <- 1 -
tf$reduce_mean(tf$forged(total_false_neighbors, tf$float32), axis = c(1L, 2L))
reg_weights <- tf$pad(reg_weights, list(list(1L, 0L)))
activations_batch_averaged <-
tf$sqrt(tf$reduce_mean(tf$sq.(x), axis = 0L))
loss <- tf$reduce_sum(tf$multiply(reg_weights, activations_batch_averaged))
loss
}
```

MSE and FNN are added , with FNN loss weighted in accordance with the important hyperparameter of this mannequin:

This worth was experimentally chosen because the one greatest conforming to our *look-for-the-highest-drop* heuristic.

### Mannequin coaching

The coaching loop intently follows the aforementioned recipe on the right way to

practice with customized fashions and `tfautograph`

.

```
train_loss <- tf$keras$metrics$Imply(identify='train_loss')
train_fnn <- tf$keras$metrics$Imply(identify='train_fnn')
train_mse <- tf$keras$metrics$Imply(identify='train_mse')
train_step <- perform(batch) {
with (tf$GradientTape(persistent = TRUE) %as% tape, {
code <- encoder(batch)
reconstructed <- decoder(code)
l_mse <- mse_loss(batch, reconstructed)
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$apply_gradients(
purrr::transpose(list(encoder_gradients, encoder$trainable_variables))
)
optimizer$apply_gradients(
purrr::transpose(list(decoder_gradients, decoder$trainable_variables))
)
train_loss(loss)
train_mse(l_mse)
train_fnn(l_fnn)
}
training_loop <- tf_function(autograph(perform(ds_train) {
for (batch in ds_train) {
train_step(batch)
}
tf$print("Loss: ", train_loss$end result())
tf$print("MSE: ", train_mse$end result())
tf$print("FNN loss: ", train_fnn$end result())
train_loss$reset_states()
train_mse$reset_states()
train_fnn$reset_states()
}))
optimizer <- optimizer_adam(lr = 1e-3)
for (epoch in 1:200) {
cat("Epoch: ", epoch, " -----------n")
training_loop(ds_train)
}
```

After 2 hundred epochs, general loss is at 2.67, with the MSE element at 1.8 and FNN at 0.09.

### Acquiring the attractor from the take a look at set

We use the take a look at set to examine the latent code:

```
# A tibble: 6,242 x 10
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.439 0.401 -0.000614 -0.0258 -0.00176 -0.0000276 0.000276 0.00677 -0.0239 0.00906
2 0.415 0.504 0.0000481 -0.0279 -0.00435 -0.0000970 0.000921 0.00509 -0.0214 0.00921
3 0.389 0.619 0.000848 -0.0240 -0.00661 -0.000171 0.00106 0.00454 -0.0150 0.00794
4 0.363 0.729 0.00137 -0.0143 -0.00652 -0.000244 0.000523 0.00450 -0.00594 0.00476
5 0.335 0.809 0.00128 -0.000450 -0.00338 -0.000307 -0.000561 0.00407 0.00394 -0.000127
6 0.304 0.828 0.000631 0.0126 0.000889 -0.000351 -0.00167 0.00250 0.0115 -0.00487
7 0.274 0.769 -0.000202 0.0195 0.00403 -0.000367 -0.00220 -0.000308 0.0145 -0.00726
8 0.246 0.657 -0.000865 0.0196 0.00558 -0.000359 -0.00208 -0.00376 0.0134 -0.00709
9 0.224 0.535 -0.00121 0.0162 0.00608 -0.000335 -0.00169 -0.00697 0.0106 -0.00576
10 0.211 0.434 -0.00129 0.0129 0.00606 -0.000306 -0.00134 -0.00927 0.00820 -0.00447
# … with 6,232 extra rows
```

Because of the FNN regularizer, the latent code models needs to be ordered roughly by reducing variance, with a pointy drop

showing some place (if the FNN weight has been chosen adequately).

For an `fnn_weight`

of 10, we do see a drop after the primary two models:

`predicted %>% summarise_all(var)`

```
# A tibble: 1 x 10
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 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
```

So the mannequin signifies that the Lorenz attractor may be represented in two dimensions. If we nonetheless wish to plot the

full (reconstructed) state house of three dimensions, we must always reorder the remaining variables by magnitude of

variance. Right here, this leads to three projections of the set `V1`

, `V2`

and `V4`

:

## Wrapping up (for this time)

At this level, we’ve seen the right way to reconstruct the Lorenz attractor from knowledge we didn’t practice on (the take a look at set), utilizing an

autoencoder regularized by a customized *false nearest neighbors* loss. You will need to stress that at no level was the community

introduced with the anticipated resolution (attractor) – coaching was purely unsupervised.

This can be a fascinating end result. After all, pondering virtually, the following step is to acquire predictions on heldout knowledge. Given

how lengthy this textual content has turn into already, we reserve that for a follow-up submit. And once more *after all*, we’re serious about different

datasets, particularly ones the place the true state house will not be recognized beforehand. What about measurement noise? What about

datasets that aren’t fully deterministic? There’s a lot to discover, keep tuned – and as all the time, thanks for

studying!

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

*Phys. Rev. A*45 (March): 3403–11. https://doi.org/10.1103/PhysRevA.45.3403.

*Journal of Statistical Physics*65 (3-4): 579–616. https://doi.org/10.1007/BF01053745.

Strang, Gilbert. 2019. *Linear Algebra and Studying from Knowledge*. Wellesley Cambridge Press.

Strogatz, Steven. 2015. *Nonlinear Dynamics and Chaos: With Functions to Physics, Biology, Chemistry, and Engineering*. Westview Press.