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 added 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 features; however they don’t lengthen infinitely. As an alternative, they’re localized in time: Away from the middle, they rapidly decay to zero. Along with a location parameter, additionally they have a scale: At totally different scales, they seem squished or stretched. Squished, they’ll 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 info, computing a sequence of dot merchandise. This manner, the wavelet is mainly on the lookout for similarity.

As to the wavelet features themselves, there are numerous of them. In a sensible utility, we’d need to experiment and decide 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 totally different from that of Fourier transforms in different respects, as nicely. 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 called 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 varieties of information we work with in apply. Thus, (t_k) and (t_n) designate closing dates, or equivalently, particular person time-series samples.

This equation seems to be 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, Okay, t_k, and t). It’s because the torch code is vectorized: On the one hand, omega, Okay, and t_k, which, within the method, correspond to (omega_{a}), (Okay), and (t_k) , are scalars. (Within the equation, they’re assumed to be fastened.) t, alternatively, is a vector; it’ll maintain the measurement occasions of the sequence to be analyzed.

We decide instance values for omega, Okay, and t_k, in addition to a variety of occasions to guage the wavelet on, and plot its values:

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

create_wavelet_plot <- operate(omega, Okay, t_k, sample_time) {
  morlet <- morlet(omega, Okay, 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 , Okay = ", Okay)
    ) +
    theme_minimal()
}

create_wavelet_plot(omega, Okay, t_k, sample_time)

What we see here’s a advanced sine curve – observe the true and imaginary elements, separated by a section shift of (pi/2) – that decays on each side of the middle. Trying again on the equation, we will determine 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 concerning the second time period, (e^{-Okay^2}): For given (Okay), it’s only a fixed.)

The third time period really is a Gaussian, with location parameter (t_k) and scale (Okay). We’ll speak about (Okay) 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 situation of most amplitude. As distance from the middle will increase, values rapidly method zero. That is what is supposed by wavelets being localized: They’re “lively” solely on a brief vary of time.

The roles of (Okay) and (omega_a)

Now, we already mentioned that (Okay) 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 influence 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 (Okay).

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 hold (omega_a) fixed, and fluctuate (Okay). On the proper, (omega_a) adjustments, and (Okay) stays the identical.

Firstly, we observe that the upper (Okay), the extra the curve will get unfold out. In a wavelet evaluation, which means extra closing dates will contribute to the remodel’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 influence is twofold. On the one hand, within the Gaussian time period, it counteracts – precisely, even – the dimensions parameter, (Okay). On the opposite, it determines the frequency, or equivalently, the interval, of the wave. To see this, check out the proper column. Similar to the totally different frequencies, now we have, within the interval between 4 and 6, 4, six, or eight peaks, respectively.

This double function of (omega_a) is the explanation why, all-in-all, it does make a distinction whether or not we shrink (Okay), conserving (omega_a) fixed, or improve (omega_a), holding (Okay) fastened.

This state of issues sounds difficult, however is much less problematic than it might sound. In apply, understanding the function of (Okay) is essential, since we have to decide smart (Okay) values to strive. As to the (omega_a), alternatively, there might be a large number of them, comparable to the vary of frequencies we analyze.

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

Wavelet Remodel: An easy implementation

Whereas general, the subject of wavelets is extra multifaceted, and thus, could seem extra enigmatic than Fourier evaluation, the remodel itself is less complicated to understand. It’s a sequence of native convolutions between wavelet and sign. Right here is the method for particular scale parameter (Okay), 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 advanced conjugation flips the wavelet in time, making this convolution, not correlation – a indisputable fact that issues loads, as you’ll see quickly.)

Correspondingly, simple implementation leads to a sequence of dot merchandise, every comparable to a special alignment of wavelet and sign. Under, in wavelet_transform(), arguments omega and Okay are scalars, whereas x, the sign, is a vector. The result’s the wavelet-transformed sign, for some particular Okay and omega of curiosity.

wavelet_transform <- operate(x, omega, Okay) {
  n_samples <- dim(x)[1]
  W <- torch_complex(
    torch_zeros(n_samples), torch_zeros(n_samples)
  )
  for (i in 1:n_samples) {
    # transfer middle of wavelet
    t_k <- x[i, 1]
    m <- morlet(omega, Okay, t_k, x[, 1])
    # compute native dot product
    # observe 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, section, fs, length) {
  x <- torch_arange(0, length, 1 / fs)[1:-2]$unsqueeze(2)
  y <- amp * torch_cos(2 * pi * freq * x + section)
  torch_cat(listing(x, y), dim = 2)
}

# sampling frequency
fs <- 8000

f1 <- 100
f2 <- 200
section <- 0
length <- 0.25

s1 <- gencos(1, f1, section, fs, length)
s2 <- gencos(1, f2, section, fs, length)

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

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 Okay parameter of two, discovered via fast experimentation:

Okay <- 2
omega <- 2 * pi * f1

res <- wavelet_transform(x = s3, omega, Okay)
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 remodel appropriately picks out the a part of the sign that matches the evaluation frequency. In case 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 fascinated about. And we’ll need to strive totally different scales Okay. Now, in the event you executed the code above, you is perhaps apprehensive that this might take a lot of time.

Effectively, 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, totally different scales to be explored. And secondly, spectrograms function on complete home windows (with configurable overlap); a wavelet, alternatively, slides over the sign in unit steps.

Nonetheless, the state of affairs just isn’t 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 Okay.

Decision in time versus in frequency

We already noticed that the upper Okay, the extra spread-out the wavelet. We are able to use our first, maximally simple, instance, to analyze one rapid consequence. What, for instance, occurs for Okay set to twenty?

Okay <- 20

res <- wavelet_transform(x = s3, omega, Okay)
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, once we compute the dot product at location t_k = 1, only a single pattern of the sign is taken into account.

Aside from probably introducing unreliability on the boundaries, how does wavelet scale have an effect on the evaluation? Effectively, since we’re correlating (convolving, technically; however on this case, the impact, ultimately, 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 Okay that properly captures the sign’s frequency. Then some other Okay, 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 velocity 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 rapidly computed:

F <- torch_fft_fft(s3[ , 2])

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

morlet_fourier <- operate(Okay, omega_a, omega) {
  2 * (torch_exp(-torch_square(
    Okay * (omega - omega_a) / omega_a
  )) -
    torch_exp(-torch_square(Okay)) *
      torch_exp(-torch_square(Okay * 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 dependent upon sampling frequency). Our wavelet, alternatively, 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 comparable 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:

Okay <- 3

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

Placing collectively wavelet instantiation and the steps concerned within the evaluation, now we have the next. (Be aware how you can wavelet_transform_fourier, we now, conveniently, go within the frequency worth in Hertz.)

wavelet_transform_fourier <- operate(x, omega_a, Okay, fs) {
  N <- dim(x)[1]
  omega_bin <- omega_a / fs * N
  m <- morlet_fourier(Okay, 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 acquire is dependent upon 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 nicely resolve which frequencies to research.

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

Iteration over frequencies is then applied as a loop:

wavelet_grid <- operate(x, Okay, 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 * Okay))
  freqs <- seq(f_start, f_end, size.out = ground(num_freqs))
  
  remodeled <- torch_zeros(
    num_freqs, dim(x)[1],
    dtype = torch_cfloat()
    )
  for(i in 1:num_freqs) {
    w <- wavelet_transform_fourier(x, freqs[i], Okay, fs)
    remodeled[i, ] <- w
  }
  listing(remodeled, 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 sequence; it will possibly, nonetheless, plot the squared magnitudes, too, in addition to their sq. root, a technique a lot really helpful by Vistnes whose effectiveness we’ll quickly have alternative to witness.

The operate deserves just a few additional feedback.

Firstly, similar as we did with the evaluation frequencies, we down-sample the sign itself, avoiding to recommend a decision that’s not really current. The method, 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 hold 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, observe how frequencies are organized on a log scale. This results in way more helpful visualizations.

plot_wavelet_diagram <- operate(x,
                                 freqs,
                                 grid,
                                 Okay,
                                 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 sequence
  # as per Vistnes, eq. 14.9
  new_x_take_every <- max(Okay / 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")
  )
  important <- paste0("Wavelet Remodel, Okay = ", Okay)
  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, 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 tune

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 out there 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 a level in conserving 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 study 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 prohibit the vary of study frequencies to between, roughly, 1800 and 8500 Hertz. (That is additionally the vary really helpful 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 executed 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")
)
important <- paste0("Spectrogram, window measurement = ", window_size)
sub <- "Magnitude (sq. root)"
mtext(aspect = 3, line = 2, at = 0, adj = 0, cex = 1.3, important)
mtext(aspect = 3, line = 1, at = 0, adj = 0, cex = 1, sub)

The spectrogram already reveals a particular sample. Let’s see what might be executed with wavelet evaluation. Having experimented with just a few totally different Okay, I agree with Vistnes that Okay = 48 makes for a wonderful selection:

f_start <- 1800
f_end <- 8500

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

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

Thanks for studying!

Photograph by Vlad Panov on Unsplash