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admit it – generally we simply miss the joys of the nonlinear, of excellent, outdated, deterministic-yet-unpredictable chaos.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

have each? It appears to be like like we will. On this submit, we’ll see an utility of deep studying (DL) to nonlinear time sequence

prediction – or quite, 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 sequence* (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 techniques. If

you are feeling you utterly perceive the sentence you simply learn, you could as properly immediately soar to the paper – come again for the

code although. If, alternatively, you’re extra aware of the chaos in your desk (extrapolating … apologies) than

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

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

to an enchanting 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

You could be aware of the *logistic* equation, used as a toy mannequin for inhabitants development. Usually it’s written like this –

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

and (r) being the expansion price (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 utility leads to a *trajectory*

describing how the inhabitants of rabbits evolves. Maps can have *mounted factors*, states the place additional operate utility goes

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

totally different!) preliminary values, (0.3) and (0.8). Each trajectories arrive at a set level – the identical mounted 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 price have been considerably increased, 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 mounted level, however a *two-cycle*: Because the trajectories stabilize, inhabitants dimension inevitably is at

certainly one of two doable values – both too many rabbits or too few, you may 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 price some. Now *this* (actually) is chaos:

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

prediction we would make.

Or can we? In spite of everything, we’ve got the governing equation, which is deterministic. So we should always have the ability to calculate the dimensions of

the inhabitants at, say, time (150)? In precept, sure; however this presupposes we’ve got 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 in the course of 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*, an important 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 claimed to have put it,

When the current determines the longer term, however the approximate current doesn’t roughly decide the longer term.

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 principle, 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 techniques, is not less than 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 a degree 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 generally appear to be we had). We would have lots of measurements, however these don’t normally mirror

the precise state variables we’re keen on. In these instances, we might need to really *add* info.

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

Assume that as a substitute of all three variables of the Lorenz system, we had measured only one: (x), the speed of convection. Usually

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

measurements.

On this methodology – or household of strategies – the univariate sequence 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 must be. As an instance, if we had a scalar sequence,

`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 sequence and time lag – the primary is a choice on the dimensionality of

the reconstruction house. Varied theorems, akin 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 is just not the case.The second has been of little curiosity to mathematicians, however is necessary

in apply. The truth is, Kantz and Schreiber (Kantz and Schreiber 2004) argue that in apply, 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 techniques,

factors which might 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’ve got two factors which might 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 have been *false neighbors*. The

*false nearest neighbors* (FNN) statistic can be utilized to find out an enough 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 strategy. The autoencoder will use that very same FNN statistic as a

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

dimensionality that entails fewer selections.

Going again to the basic methodology for an immediate, the second parameter, the time lag, is much more tough to kind 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 strategy.

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 operate, particularly, has been ported

one-to-one.

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

the univariate time sequence right into a latent illustration of some dimensionality, which is able to represent an higher sure 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: how you can 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 tactic – the

paper was printed this 12 months – it is smart to concentrate on thorough evaluation first.

### Information 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 <- operate(t, state, parameters) {
with(as.listing(c(state, parameters)), {
dx <- sigma * (y - x)
dy <- x * (rho - z) - y
dz <- x * y - beta * z
listing(c(dx, dy, dz))
})
}
occasions <- seq(0, 500, size.out = 125000)
lorenz_ts <-
ode(
y = initial_state,
occasions = occasions,
func = lorenz,
parms = parameters,
methodology = "lsoda"
) %>% as_tibble()
lorenz_ts[1:10,]
```

```
# A tibble: 10 x 4
time x y z
```
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 quite, its three two-dimensional projections, in determine 6 above. However now our state of affairs is

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

differential equations was quite tiny, we simply use each tenth commentary.

### Preprocessing

The primary half of the sequence is used for coaching. The info is scaled and reworked into the three-dimensional type 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 <- 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()
}
# prepare with begin of time sequence, take a look at with finish of time sequence
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've got 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
# rework to datasets so we will 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 finest coded as customized fashions,

and educated 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 <- operate(n_timesteps,
n_features,
n_latent,
identify = NULL) {
keras_model_custom(identify = identify, operate(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$batchnorm <- layer_batch_normalization()
operate (x, masks = NULL) {
x %>%
self$noise() %>%
self$lstm() %>%
self$batchnorm()
}
})
}
decoder_model <- operate(n_timesteps,
n_features,
n_latent,
identify = NULL) {
keras_model_custom(identify = identify, operate(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))
operate (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 operate we prepare 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 preserve the variety of false neighbors small, via the next regularizer.

```
loss_false_nn <- operate(x) {
# authentic values utilized in Kennel et al. (1992)
rtol <- 10
atol <- 2
k_frac <- 0.01
ok <- max(1, ground(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$top_k(-all_dists, tf$solid(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$logical_or(is_false_change, is_large_jump)
total_false_neighbors <-
tf$solid(is_false_neighbor, tf$int32)[reticulate::py_ellipsis(), 2:(k + 2)]
reg_weights <- 1 -
tf$reduce_mean(tf$solid(total_false_neighbors, tf$float32), axis = c(1L, 2L))
reg_weights <- tf$pad(reg_weights, listing(listing(1L, 0L)))
activations_batch_averaged <-
tf$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 finest conforming to our *look-for-the-highest-drop* heuristic.

### Mannequin coaching

The coaching loop intently follows the aforementioned recipe on how you can

prepare 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$Imply(identify='train_mse')
train_step <- operate(batch) {
with (tf$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$apply_gradients(
purrr::transpose(listing(encoder_gradients, encoder$trainable_variables))
)
optimizer$apply_gradients(
purrr::transpose(listing(decoder_gradients, decoder$trainable_variables))
)
train_loss(loss)
train_mse(l_mse)
train_fnn(l_fnn)
}
training_loop <- tf_function(autograph(operate(ds_train) {
for (batch in ds_train) {
train_step(batch)
}
tf$print("Loss: ", train_loss$consequence())
tf$print("MSE: ", train_mse$consequence())
tf$print("FNN loss: ", train_fnn$consequence())
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, total loss is at 2.67, with the MSE part 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
```
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 must be ordered roughly by lowering 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
```
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 need to plot the

full (reconstructed) state house of three dimensions, we should 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 how you can reconstruct the Lorenz attractor from knowledge we didn’t prepare 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

offered with the anticipated answer (attractor) – coaching was purely unsupervised.

It is a fascinating consequence. In fact, considering 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 *in fact*, we’re excited about different

datasets, particularly ones the place the true state house is just not identified beforehand. What about measurement noise? What about

datasets that aren’t utterly deterministic? There’s a lot to discover, keep tuned – and as at all times, thanks for

studying!

Kantz, Holger, and Thomas Schreiber. 2004. *Nonlinear Time Collection 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 Information*. Wellesley Cambridge Press.

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