Posit AI Weblog: Wavelet Remodel

Be aware: Like a number of prior ones, this submit is an excerpt from the forthcoming e book, Deep Studying and Scientific Computing with R torch. And like many excerpts, it’s a product of arduous trade-offs. For extra depth and extra examples, I’ve to ask you to please seek the advice of the e book.

Wavelets and the Wavelet Remodel

What are wavelets? Just like the Fourier foundation, they’re capabilities; however they don’t prolong infinitely. As an alternative, they’re localized in time: Away from the middle, they shortly decay to zero. Along with a location parameter, in addition they have a scale: At completely different scales, they seem squished or stretched. Squished, they are going to do higher at detecting excessive frequencies; the converse applies after they’re stretched out in time.

The essential operation concerned within the Wavelet Remodel is convolution – have the (flipped) wavelet slide over the information, computing a sequence of dot merchandise. This manner, the wavelet is mainly on the lookout for similarity.

As to the wavelet capabilities themselves, there are lots of of them. In a sensible software, we’d need to experiment and choose the one which works greatest for the given information. In comparison with the DFT and spectrograms, extra experimentation tends to be concerned in wavelet evaluation.

The subject of wavelets may be very completely different from that of Fourier transforms in different respects, as properly. Notably, there’s a lot much less standardization in terminology, use of symbols, and precise practices. On this introduction, I’m leaning closely on one particular exposition, the one in Arnt Vistnes’ very good e book on waves (Vistnes 2018). In different phrases, each terminology and examples replicate the alternatives made in that e book.

Introducing the Morlet wavelet

The Morlet, also referred to as Gabor, wavelet is outlined like so:

[
Psi_{omega_{a},K,t_{k}}(t_n) = (e^{-i omega_{a} (t_n – t_k)} – e^{-K^2}) e^{- omega_a^2 (t_n – t_k )^2 /(2K )^2}
]

This formulation pertains to discretized information, the sorts of information we work with in follow. Thus, (t_k) and (t_n) designate cut-off dates, or equivalently, particular person time-series samples.

This equation appears daunting at first, however we will “tame” it a bit by analyzing its construction, and pointing to the primary actors. For concreteness, although, we first have a look at an instance wavelet.

We begin by implementing the above equation:

Evaluating code and mathematical formulation, we discover a distinction. The operate itself takes one argument, (t_n); its realization, 4 (omega, Ok, t_k, and t). It’s because the torch code is vectorized: On the one hand, omega, Ok, and t_k, which, within the components, correspond to (omega_{a}), (Ok), and (t_k) , are scalars. (Within the equation, they’re assumed to be fastened.) t, then again, is a vector; it should maintain the measurement occasions of the collection to be analyzed.

We choose instance values for omega, Ok, and t_k, in addition to a variety of occasions to judge the wavelet on, and plot its values:

omega <- 6 * pi
Ok <- 6
t_k <- 5
 
sample_time <- torch_arange(3, 7, 0.0001)

create_wavelet_plot <- operate(omega, Ok, t_k, sample_time) {
  morlet <- morlet(omega, Ok, t_k, sample_time)
  df <- information.body(
    x = as.numeric(sample_time),
    actual = as.numeric(morlet$actual),
    imag = as.numeric(morlet$imag)
  ) %>%
    pivot_longer(-x, names_to = "half", values_to = "worth")
  ggplot(df, aes(x = x, y = worth, colour = half)) +
    geom_line() +
    scale_colour_grey(begin = 0.8, finish = 0.4) +
    xlab("time") +
    ylab("wavelet worth") +
    ggtitle("Morlet wavelet",
      subtitle = paste0("ω_a = ", omega / pi, "π , Ok = ", Ok)
    ) +
    theme_minimal()
}

create_wavelet_plot(omega, Ok, t_k, sample_time)

What we see here’s a complicated sine curve – notice the actual and imaginary elements, separated by a part shift of (pi/2) – that decays on each side of the middle. Trying again on the equation, we will establish the components liable for each options. The primary time period within the equation, (e^{-i omega_{a} (t_n – t_k)}), generates the oscillation; the third, (e^{- omega_a^2 (t_n – t_k )^2 /(2K )^2}), causes the exponential decay away from the middle. (In case you’re questioning in regards to the second time period, (e^{-Ok^2}): For given (Ok), it’s only a fixed.)

The third time period really is a Gaussian, with location parameter (t_k) and scale (Ok). We’ll discuss (Ok) in nice element quickly, however what’s with (t_k)? (t_k) is the middle of the wavelet; for the Morlet wavelet, that is additionally the placement of most amplitude. As distance from the middle will increase, values shortly method zero. That is what is supposed by wavelets being localized: They’re “lively” solely on a brief vary of time.

The roles of (Ok) and (omega_a)

Now, we already mentioned that (Ok) is the dimensions of the Gaussian; it thus determines how far the curve spreads out in time. However there may be additionally (omega_a). Trying again on the Gaussian time period, it, too, will affect the unfold.

First although, what’s (omega_a)? The subscript (a) stands for “evaluation”; thus, (omega_a) denotes a single frequency being probed.

Now, let’s first examine visually the respective impacts of (omega_a) and (Ok).

p1 <- create_wavelet_plot(6 * pi, 4, 5, sample_time)
p2 <- create_wavelet_plot(6 * pi, 6, 5, sample_time)
p3 <- create_wavelet_plot(6 * pi, 8, 5, sample_time)
p4 <- create_wavelet_plot(4 * pi, 6, 5, sample_time)
p5 <- create_wavelet_plot(6 * pi, 6, 5, sample_time)
p6 <- create_wavelet_plot(8 * pi, 6, 5, sample_time)

(p1 | p4) /
  (p2 | p5) /
  (p3 | p6)

Within the left column, we maintain (omega_a) fixed, and differ (Ok). On the best, (omega_a) adjustments, and (Ok) stays the identical.

Firstly, we observe that the upper (Ok), the extra the curve will get unfold out. In a wavelet evaluation, which means that extra cut-off dates will contribute to the rework’s output, leading to excessive precision as to frequency content material, however lack of decision in time. (We’ll return to this – central – trade-off quickly.)

As to (omega_a), its affect is twofold. On the one hand, within the Gaussian time period, it counteracts – precisely, even – the dimensions parameter, (Ok). On the opposite, it determines the frequency, or equivalently, the interval, of the wave. To see this, check out the best column. Similar to the completely different frequencies, we’ve, within the interval between 4 and 6, 4, six, or eight peaks, respectively.

This double position of (omega_a) is the explanation why, all-in-all, it does make a distinction whether or not we shrink (Ok), protecting (omega_a) fixed, or enhance (omega_a), holding (Ok) fastened.

This state of issues sounds sophisticated, however is much less problematic than it may appear. In follow, understanding the position of (Ok) is necessary, since we have to choose wise (Ok) values to strive. As to the (omega_a), then again, there might be a large number of them, equivalent to the vary of frequencies we analyze.

So we will perceive the affect of (Ok) in additional element, we have to take a primary have a look at the Wavelet Remodel.

Wavelet Remodel: An easy implementation

Whereas total, the subject of wavelets is extra multifaceted, and thus, could appear extra enigmatic than Fourier evaluation, the rework itself is simpler to know. It’s a sequence of native convolutions between wavelet and sign. Right here is the components for particular scale parameter (Ok), evaluation frequency (omega_a), and wavelet location (t_k):

[
W_{K, omega_a, t_k} = sum_n x_n Psi_{omega_{a},K,t_{k}}^*(t_n)
]

That is only a dot product, computed between sign and complex-conjugated wavelet. (Right here complicated conjugation flips the wavelet in time, making this convolution, not correlation – a undeniable fact that issues rather a lot, as you’ll see quickly.)

Correspondingly, easy implementation leads to a sequence of dot merchandise, every equivalent to a unique alignment of wavelet and sign. Beneath, in wavelet_transform(), arguments omega and Ok are scalars, whereas x, the sign, is a vector. The result’s the wavelet-transformed sign, for some particular Ok and omega of curiosity.

wavelet_transform <- operate(x, omega, Ok) {
  n_samples <- dim(x)[1]
  W <- torch_complex(
    torch_zeros(n_samples), torch_zeros(n_samples)
  )
  for (i in 1:n_samples) {
    # transfer heart of wavelet
    t_k <- x[i, 1]
    m <- morlet(omega, Ok, t_k, x[, 1])
    # compute native dot product
    # notice wavelet is conjugated
    dot <- torch_matmul(
      m$conj()$unsqueeze(1),
      x[, 2]$to(dtype = torch_cfloat())
    )
    W[i] <- dot
  }
  W
}

To check this, we generate a easy sine wave that has a frequency of 100 Hertz in its first half, and double that within the second.

gencos <- operate(amp, freq, part, fs, period) {
  x <- torch_arange(0, period, 1 / fs)[1:-2]$unsqueeze(2)
  y <- amp * torch_cos(2 * pi * freq * x + part)
  torch_cat(checklist(x, y), dim = 2)
}

# sampling frequency
fs <- 8000

f1 <- 100
f2 <- 200
part <- 0
period <- 0.25

s1 <- gencos(1, f1, part, fs, period)
s2 <- gencos(1, f2, part, fs, period)

s3 <- torch_cat(checklist(s1, s2), dim = 1)
s3[(dim(s1)[1] + 1):(dim(s1)[1] * 2), 1] <-
  s3[(dim(s1)[1] + 1):(dim(s1)[1] * 2), 1] + period

df <- information.body(
  x = as.numeric(s3[, 1]),
  y = as.numeric(s3[, 2])
)
ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("amplitude") +
  theme_minimal()

Now, we run the Wavelet Remodel on this sign, for an evaluation frequency of 100 Hertz, and with a Ok parameter of two, discovered by fast experimentation:

Ok <- 2
omega <- 2 * pi * f1

res <- wavelet_transform(x = s3, omega, Ok)
df <- information.body(
  x = as.numeric(s3[, 1]),
  y = as.numeric(res$abs())
)

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("Wavelet Remodel") +
  theme_minimal()

The rework appropriately picks out the a part of the sign that matches the evaluation frequency. When you really feel like, you may need to double-check what occurs for an evaluation frequency of 200 Hertz.

Now, in actuality we’ll need to run this evaluation not for a single frequency, however a variety of frequencies we’re inquisitive about. And we’ll need to strive completely different scales Ok. Now, in case you executed the code above, you may be nervous that this might take a lot of time.

Properly, it by necessity takes longer to compute than its Fourier analogue, the spectrogram. For one, that’s as a result of with spectrograms, the evaluation is “simply” two-dimensional, the axes being time and frequency. With wavelets there are, as well as, completely different scales to be explored. And secondly, spectrograms function on complete home windows (with configurable overlap); a wavelet, then again, slides over the sign in unit steps.

Nonetheless, the state of affairs shouldn’t be as grave because it sounds. The Wavelet Remodel being a convolution, we will implement it within the Fourier area as a substitute. We’ll try this very quickly, however first, as promised, let’s revisit the subject of various Ok.

Decision in time versus in frequency

We already noticed that the upper Ok, the extra spread-out the wavelet. We will use our first, maximally easy, instance, to analyze one speedy consequence. What, for instance, occurs for Ok set to twenty?

Ok <- 20

res <- wavelet_transform(x = s3, omega, Ok)
df <- information.body(
  x = as.numeric(s3[, 1]),
  y = as.numeric(res$abs())
)

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("Wavelet Remodel") +
  theme_minimal()

The Wavelet Remodel nonetheless picks out the proper area of the sign – however now, as a substitute of a rectangle-like end result, we get a considerably smoothed model that doesn’t sharply separate the 2 areas.

Notably, the primary 0.05 seconds, too, present appreciable smoothing. The bigger a wavelet, the extra element-wise merchandise might be misplaced on the finish and the start. It’s because transforms are computed aligning the wavelet in any respect sign positions, from the very first to the final. Concretely, after we compute the dot product at location t_k = 1, only a single pattern of the sign is taken into account.

Other than probably introducing unreliability on the boundaries, how does wavelet scale have an effect on the evaluation? Properly, since we’re correlating (convolving, technically; however on this case, the impact, in the long run, is identical) the wavelet with the sign, point-wise similarity is what issues. Concretely, assume the sign is a pure sine wave, the wavelet we’re utilizing is a windowed sinusoid just like the Morlet, and that we’ve discovered an optimum Ok that properly captures the sign’s frequency. Then every other Ok, be it bigger or smaller, will lead to much less point-wise overlap.

Performing the Wavelet Remodel within the Fourier area

Quickly, we’ll run the Wavelet Remodel on an extended sign. Thus, it’s time to pace up computation. We already mentioned that right here, we profit from time-domain convolution being equal to multiplication within the Fourier area. The general course of then is that this: First, compute the DFT of each sign and wavelet; second, multiply the outcomes; third, inverse-transform again to the time area.

The DFT of the sign is shortly computed:

F <- torch_fft_fft(s3[ , 2])

With the Morlet wavelet, we don’t even should run the FFT: Its Fourier-domain illustration will be said in closed kind. We’ll simply make use of that formulation from the outset. Right here it’s:

morlet_fourier <- operate(Ok, omega_a, omega) {
  2 * (torch_exp(-torch_square(
    Ok * (omega - omega_a) / omega_a
  )) -
    torch_exp(-torch_square(Ok)) *
      torch_exp(-torch_square(Ok * omega / omega_a)))
}

Evaluating this assertion of the wavelet to the time-domain one, we see that – as anticipated – as a substitute of parameters t and t_k it now takes omega and omega_a. The latter, omega_a, is the evaluation frequency, the one we’re probing for, a scalar; the previous, omega, the vary of frequencies that seem within the DFT of the sign.

In instantiating the wavelet, there may be one factor we have to pay particular consideration to. In FFT-think, the frequencies are bins; their quantity is set by the size of the sign (a size that, for its half, straight is determined by sampling frequency). Our wavelet, then again, works with frequencies in Hertz (properly, from a consumer’s perspective; since this unit is significant to us). What this implies is that to morlet_fourier, as omega_a we have to go not the worth in Hertz, however the corresponding FFT bin. Conversion is finished relating the variety of bins, dim(x)[1], to the sampling frequency of the sign, fs:

# once more search for 100Hz elements
omega <- 2 * pi * f1

# want the bin equivalent to some frequency in Hz
omega_bin <- f1/fs * dim(s3)[1]

We instantiate the wavelet, carry out the Fourier-domain multiplication, and inverse-transform the end result:

Ok <- 3

m <- morlet_fourier(Ok, omega_bin, 1:dim(s3)[1])
prod <- F * m
reworked <- torch_fft_ifft(prod)

Placing collectively wavelet instantiation and the steps concerned within the evaluation, we’ve the next. (Be aware the way to wavelet_transform_fourier, we now, conveniently, go within the frequency worth in Hertz.)

wavelet_transform_fourier <- operate(x, omega_a, Ok, fs) {
  N <- dim(x)[1]
  omega_bin <- omega_a / fs * N
  m <- morlet_fourier(Ok, omega_bin, 1:N)
  x_fft <- torch_fft_fft(x)
  prod <- x_fft * m
  w <- torch_fft_ifft(prod)
  w
}

We’ve already made vital progress. We’re prepared for the ultimate step: automating evaluation over a variety of frequencies of curiosity. This may lead to a three-dimensional illustration, the wavelet diagram.

Creating the wavelet diagram

Within the Fourier Remodel, the variety of coefficients we get hold of is determined by sign size, and successfully reduces to half the sampling frequency. With its wavelet analogue, since anyway we’re doing a loop over frequencies, we would as properly resolve which frequencies to research.

Firstly, the vary of frequencies of curiosity will be decided working the DFT. The subsequent query, then, is about granularity. Right here, I’ll be following the advice given in Vistnes’ e book, which is predicated on the relation between present frequency worth and wavelet scale, Ok.

Iteration over frequencies is then applied as a loop:

wavelet_grid <- operate(x, Ok, f_start, f_end, fs) {
  # downsample evaluation frequency vary
  # as per Vistnes, eq. 14.17
  num_freqs <- 1 + log(f_end / f_start)/ log(1 + 1/(8 * Ok))
  freqs <- seq(f_start, f_end, size.out = ground(num_freqs))
  
  reworked <- torch_zeros(
    num_freqs, dim(x)[1],
    dtype = torch_cfloat()
    )
  for(i in 1:num_freqs) {
    w <- wavelet_transform_fourier(x, freqs[i], Ok, fs)
    reworked[i, ] <- w
  }
  checklist(reworked, freqs)
}

Calling wavelet_grid() will give us the evaluation frequencies used, along with the respective outputs from the Wavelet Remodel.

Subsequent, we create a utility operate that visualizes the end result. By default, plot_wavelet_diagram() shows the magnitude of the wavelet-transformed collection; it might probably, nevertheless, plot the squared magnitudes, too, in addition to their sq. root, a way a lot really useful by Vistnes whose effectiveness we’ll quickly have alternative to witness.

The operate deserves a number of additional feedback.

Firstly, identical as we did with the evaluation frequencies, we down-sample the sign itself, avoiding to recommend a decision that isn’t really current. The components, once more, is taken from Vistnes’ e book.

Then, we use interpolation to acquire a brand new time-frequency grid. This step might even be essential if we maintain the unique grid, since when distances between grid factors are very small, R’s picture() might refuse to just accept axes as evenly spaced.

Lastly, notice how frequencies are organized on a log scale. This results in rather more helpful visualizations.

plot_wavelet_diagram <- operate(x,
                                 freqs,
                                 grid,
                                 Ok,
                                 fs,
                                 f_end,
                                 sort = "magnitude") {
  grid <- swap(sort,
    magnitude = grid$abs(),
    magnitude_squared = torch_square(grid$abs()),
    magnitude_sqrt = torch_sqrt(grid$abs())
  )

  # downsample time collection
  # as per Vistnes, eq. 14.9
  new_x_take_every <- max(Ok / 24 * fs / f_end, 1)
  new_x_length <- ground(dim(grid)[2] / new_x_take_every)
  new_x <- torch_arange(
    x[1],
    x[dim(x)[1]],
    step = x[dim(x)[1]] / new_x_length
  )
  
  # interpolate grid
  new_grid <- nnf_interpolate(
    grid$view(c(1, 1, dim(grid)[1], dim(grid)[2])),
    c(dim(grid)[1], new_x_length)
  )$squeeze()
  out <- as.matrix(new_grid)

  # plot log frequencies
  freqs <- log10(freqs)
  
  picture(
    x = as.numeric(new_x),
    y = freqs,
    z = t(out),
    ylab = "log frequency [Hz]",
    xlab = "time [s]",
    col = hcl.colours(12, palette = "Gentle grays")
  )
  most important <- paste0("Wavelet Remodel, Ok = ", Ok)
  sub <- swap(sort,
    magnitude = "Magnitude",
    magnitude_squared = "Magnitude squared",
    magnitude_sqrt = "Magnitude (sq. root)"
  )

  mtext(aspect = 3, line = 2, at = 0, adj = 0, cex = 1.3, most important)
  mtext(aspect = 3, line = 1, at = 0, adj = 0, cex = 1, sub)
}

Let’s use this on a real-world instance.

An actual-world instance: Chaffinch’s music

For the case research, I’ve chosen what, to me, was probably the most spectacular wavelet evaluation proven in Vistnes’ e book. It’s a pattern of a chaffinch’s singing, and it’s accessible on Vistnes’ web site.

url <- "http://www.physics.uio.no/pow/wavbirds/chaffinch.wav"

obtain.file(
 file.path(url),
 destfile = "/tmp/chaffinch.wav"
)

We use torchaudio to load the file, and convert from stereo to mono utilizing tuneR’s appropriately named mono(). (For the sort of evaluation we’re doing, there isn’t any level in protecting two channels round.)

library(torchaudio)
library(tuneR)

wav <- tuneR_loader("/tmp/chaffinch.wav")
wav <- mono(wav, "each")
wav

Wave Object
    Variety of Samples:      1864548
    Length (seconds):     42.28
    Samplingrate (Hertz):   44100
    Channels (Mono/Stereo): Mono
    PCM (integer format):   TRUE
    Bit (8/16/24/32/64):    16 

For evaluation, we don’t want the entire sequence. Helpfully, Vistnes additionally printed a advice as to which vary of samples to research.

waveform_and_sample_rate <- transform_to_tensor(wav)
x <- waveform_and_sample_rate[[1]]$squeeze()
fs <- waveform_and_sample_rate[[2]]

# http://www.physics.uio.no/pow/wavbirds/chaffinchInfo.txt
begin <- 34000
N <- 1024 * 128
finish <- begin + N - 1
x <- x[start:end]

dim(x)

[1] 131072

How does this look within the time area? (Don’t miss out on the event to truly pay attention to it, in your laptop computer.)

df <- information.body(x = 1:dim(x)[1], y = as.numeric(x))
ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("pattern") +
  ylab("amplitude") +
  theme_minimal()

Now, we have to decide an inexpensive vary of research frequencies. To that finish, we run the FFT:

On the x-axis, we plot frequencies, not pattern numbers, and for higher visibility, we zoom in a bit.

bins <- 1:dim(F)[1]
freqs <- bins / N * fs

# the bin, not the frequency
cutoff <- N/4

df <- information.body(
  x = freqs[1:cutoff],
  y = as.numeric(F$abs())[1:cutoff]
)
ggplot(df, aes(x = x, y = y)) +
  geom_col() +
  xlab("frequency (Hz)") +
  ylab("magnitude") +
  theme_minimal()

Primarily based on this distribution, we will safely limit the vary of research frequencies to between, roughly, 1800 and 8500 Hertz. (That is additionally the vary really useful by Vistnes.)

First, although, let’s anchor expectations by making a spectrogram for this sign. Appropriate values for FFT measurement and window measurement have been discovered experimentally. And although, in spectrograms, you don’t see this performed typically, I discovered that displaying sq. roots of coefficient magnitudes yielded probably the most informative output.

fft_size <- 1024
window_size <- 1024
energy <- 0.5

spectrogram <- transform_spectrogram(
  n_fft = fft_size,
  win_length = window_size,
  normalized = TRUE,
  energy = energy
)

spec <- spectrogram(x)
dim(spec)

[1] 513 257

Like we do with wavelet diagrams, we plot frequencies on a log scale.

bins <- 1:dim(spec)[1]
freqs <- bins * fs / fft_size
log_freqs <- log10(freqs)

frames <- 1:(dim(spec)[2])
seconds <- (frames / dim(spec)[2])  * (dim(x)[1] / fs)

picture(x = seconds,
      y = log_freqs,
      z = t(as.matrix(spec)),
      ylab = 'log frequency [Hz]',
      xlab = 'time [s]',
      col = hcl.colours(12, palette = "Gentle grays")
)
most important <- paste0("Spectrogram, window measurement = ", window_size)
sub <- "Magnitude (sq. root)"
mtext(aspect = 3, line = 2, at = 0, adj = 0, cex = 1.3, most important)
mtext(aspect = 3, line = 1, at = 0, adj = 0, cex = 1, sub)

The spectrogram already exhibits a particular sample. Let’s see what will be performed with wavelet evaluation. Having experimented with a number of completely different Ok, I agree with Vistnes that Ok = 48 makes for a superb selection:

f_start <- 1800
f_end <- 8500

Ok <- 48
c(grid, freqs) %<-% wavelet_grid(x, Ok, f_start, f_end, fs)
plot_wavelet_diagram(
  torch_tensor(1:dim(grid)[2]),
  freqs, grid, Ok, fs, f_end,
  sort = "magnitude_sqrt"
)

The acquire in decision, on each the time and the frequency axis, is totally spectacular.

Thanks for studying!

Picture by Vlad Panov on Unsplash