For us deep studying practitioners, the world is – not flat, however – linear, principally. Or piecewise linear. Like different
linear approximations, or possibly much more so, deep studying might be extremely profitable at making predictions. However let’s
admit it – typically we simply miss the joys of the nonlinear, of fine, outdated, deterministic-yet-unpredictable chaos. Can we
have each? It appears like we are able to. On this publish, we’ll see an utility of deep studying (DL) to nonlinear time collection
prediction – or relatively, the important step that predates it: reconstructing the attractor underlying its dynamics. Whereas this
publish is an introduction, presenting the subject from scratch, additional posts will construct on this and extrapolate to observational
datasets.
What to anticipate from this publish
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 techniques. If
you are feeling you fully perceive the sentence you simply learn, you might as nicely immediately soar to the paper – come again for the
code although. If, however, you’re extra accustomed to 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 publish is primarily presupposed to be a enjoyable introduction
to an interesting subject, 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 completely different from
stochasticity, or randomness. Chaos could 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 accustomed to the logistic equation, used as a toy mannequin for inhabitants progress. 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 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 utility leads to a trajectory
describing how the inhabitants of rabbits evolves. Maps can have fastened factors, states the place additional perform utility goes
on producing the identical end result ceaselessly. Instance-wise, say the expansion charge quantities to (2.1), and we begin at two (fairly
completely different!) preliminary values, (0.3) and (0.8). Each trajectories arrive at a set level – the identical fastened level – in fewer
than 10 iterations. Had been we requested to foretell the inhabitants dimension after 100 iterations, we may make a really assured
guess, regardless of the of beginning worth. (If the preliminary worth is (0), we keep at (0), however we might be fairly sure of that as
nicely.)
Determine 1: Trajectory of the logistic map for r = 2.1 and two completely different preliminary values.
What if the expansion charge have been considerably greater, at (3.3), say? Once more, we instantly examine trajectories ensuing from preliminary
values (0.3) and (0.9):
Determine 2: Trajectory of the logistic map for r = 3.3 and two completely different preliminary values.
This time, don’t see a single fastened level, however a two-cycle: Because the trajectories stabilize, inhabitants dimension inevitably is at
one in every of two doable values – both too many rabbits or too few, you would 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 every part but.
Let’s once more improve the expansion charge some. Now this (actually) is chaos:
Determine 3: Trajectory of the logistic map for r = 3.6 and two completely different preliminary values, 0.3 and 0.9.
Even after 100 iterations, there isn’t any set of values the trajectories recur to. We will’t be assured about any
prediction we’d make.
Or can we? In any case, we’ve got the governing equation, which is deterministic. So we should always be capable of 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 examine trajectories for preliminary values (0.3) and (0.301):
Determine 4: Trajectory of the logistic map for r = 3.6 and two completely different 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.
Determine 5: Trajectory of the logistic map for r = 3.6 and two completely different preliminary values, 0.3 and 0.30000001.
What we’re seeing right here is delicate dependence on preliminary circumstances, a necessary precondition for a system to be chaotic.
In an nutshell: Chaos arises when a deterministic system exhibits delicate dependence on preliminary circumstances. Or as Edward
Lorenz is alleged 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 principally 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 seem. It additionally has the required dimensionality, which
for clean, steady techniques, is a minimum of 3. Whether or not we really see chaotic attractors – amongst which, the butterfly –
will depend on 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:
Determine 6: Two-dimensional projections of the Lorenz attractor for sigma = 10, rho = 28, beta = 8 / 3. On the suitable: the butterfly.
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 quite a lot of completely 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:
Determine 7: How the Lorenz attractor traces out the well-known “butterfly” form.
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 could typically look like we had). We would have plenty of measurements, however these don’t normally replicate
the precise state variables we’re thinking about. In these instances, we could need 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. Usually
in nonlinear dynamics, the strategy of delay coordinate embedding (Sauer, Yorke, and Casdagli 1991) is used to boost 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
choices to be made: What number of copies so as to add, and the way massive the delay must 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 appear to be 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 choices to be made – variety of shifted collection and time lag – the primary is a call on the dimensionality of
the reconstruction house. Varied theorems, resembling Taken’s theorem,
point out bounds on the variety of dimensions required, supplied the dimensionality of the true state house is thought – which,
in real-world purposes, usually just isn’t the case.The second has been of little curiosity to mathematicians, however is essential
in follow. In actual fact, 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 techniques,
factors which might be shut in state house at time (t) ought to nonetheless be shut at time (t + Delta t), supplied (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 “challenge away” the third dimension, they’re much more distant. As illustrated in
(Gilpin 2020):
Determine 8: Within the two-dimensional projection on axes x and y, the pink factors are shut neighbors. In 3d, nonetheless, they’re separate. Evaluate with the blue factors, which keep shut even in higher-dimensional house. Determine from Gilpin (2020).
If this occurs, then projecting down has eradicated some important info. In second, the factors have 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 standard autoencoder reconstruction loss. This can end in a brand new heuristic relating to embedding
dimensionality that includes fewer choices.
Going again to the basic technique for an on the spot, the second parameter, the time lag, is much more troublesome 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 method.
Which we’ll see now.
Studying the Lorenz attractor
Our code carefully follows the structure, parameter settings, and knowledge setup used within the reference
implementation William supplied. The loss perform, particularly, has been ported
one-to-one.
The final thought is the next. An autoencoder – for instance, an LSTM autoencoder as offered 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 shall 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: methods to weight the FNN loss. One would run coaching for various weights
(lambda) and search for the drop. Absolutely, this might in precept be automated, however given the novelty of the tactic – the
paper was revealed this 12 months – it is sensible to deal with thorough evaluation first.
Knowledge technology
We use the deSolve package deal 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.listing(c(state, parameters)), {
dx <- sigma * (y - x)
dy <- x * (rho - z) - y
dz <- x * y - beta * z
listing(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 19.7We’ve already seen the attractor, or relatively, its three two-dimensional projections, in determine 6 above. However now our state of affairs is
completely different. We solely have entry to (x), a univariate time collection. Because the time interval used to numerically combine the
differential equations was relatively tiny, we simply use each tenth commentary.
Determine 9: Convection charges as a univariate time collection.
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.name(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'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
# 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 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 <- 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 examine the unique inputs with
the decoder outputs (the reconstruction), utilizing imply squared error:
mse_loss <- tf$keras$losses$MeanSquaredError(
discount = tf$keras$losses$Discount$SUM)As well as, we attempt to preserve the variety of false neighbors small, by way of the next regularizer.
loss_false_nn <- perform(x) {
# unique values utilized in Kennel et al. (1992)
rtol <- 10
atol <- 2
k_frac <- 0.01
ok <- max(1, ground(k_frac * batch_size))
tri_mask <-
tf$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$multiply(reg_weights, activations_batch_averaged))
loss
}MSE and FNN are added , with FNN loss weighted based on the important hyperparameter of this mannequin:
This worth was experimentally chosen because the one greatest conforming to our look-for-the-highest-drop heuristic.
Mannequin coaching
The coaching loop carefully follows the aforementioned recipe on methods to
practice with customized fashions and tfautograph.
train_loss <- tf$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(listing(encoder_gradients, encoder$trainable_variables))
)
optimizer$apply_gradients(
purrr::transpose(listing(decoder_gradients, decoder$trainable_variables))
)
train_loss(loss)
train_mse(l_mse)
train_fnn(l_fnn)
}
training_loop <- tf_function(autograph(perform(ds_train) {
for (batch in ds_train) {
train_step(batch)
}
tf$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 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
<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 rowsBecause 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
<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-5So the mannequin signifies that the Lorenz attractor might 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:
Determine 10: Attractors as predicted from the latent code (take a look at set). The three highest-variance variables have been chosen.
Wrapping up (for this time)
At this level, we’ve seen methods 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
offered with the anticipated resolution (attractor) – coaching was purely unsupervised.
It is a fascinating end result. After all, pondering virtually, the subsequent step is to acquire predictions on heldout knowledge. Given
how lengthy this textual content has grow to be already, we reserve that for a follow-up publish. And once more after all, we’re serious about different
datasets, particularly ones the place the true state house just isn’t identified 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.
Strang, Gilbert. 2019. Linear Algebra and Studying from Knowledge. Wellesley Cambridge Press.
Strogatz, Steven. 2015. Nonlinear Dynamics and Chaos: With Purposes to Physics, Biology, Chemistry, and Engineering. Westview Press.