Posit AI Weblog: AO, NAO, ENSO: A wavelet evaluation instance

Lately, we confirmed the best way to use torch for wavelet evaluation. A member of the household of spectral evaluation strategies, wavelet evaluation bears some similarity to the Fourier Remodel, and particularly, to its well-liked two-dimensional utility, the spectrogram.

As defined in that e-book excerpt, although, there are important variations. For the needs of the present submit, it suffices to know that frequency-domain patterns are found by having a bit “wave” (that, actually, will be of any form) “slide” over the info, computing diploma of match (or mismatch) within the neighborhood of each pattern.

With this submit, then, my purpose is two-fold.

First, to introduce torchwavelets, a tiny, but helpful package deal that automates all the important steps concerned. In comparison with the Fourier Remodel and its functions, the subject of wavelets is quite “chaotic” – that means, it enjoys a lot much less shared terminology, and far much less shared observe. Consequently, it is sensible for implementations to observe established, community-embraced approaches, each time such can be found and nicely documented. With torchwavelets, we offer an implementation of Torrence and Compo’s 1998 “Sensible Information to Wavelet Evaluation” (Torrence and Compo (1998)), an oft-cited paper that proved influential throughout a variety of utility domains. Code-wise, our package deal is usually a port of Tom Runia’s PyTorch implementation, itself primarily based on a previous implementation by Aaron O’Leary.

Second, to point out a lovely use case of wavelet evaluation in an space of nice scientific curiosity and super social significance (meteorology/climatology). Being not at all an professional myself, I’d hope this may very well be inspiring to individuals working in these fields, in addition to to scientists and analysts in different areas the place temporal information come up.

Concretely, what we’ll do is take three completely different atmospheric phenomena – El Niño–Southern Oscillation (ENSO), North Atlantic Oscillation (NAO), and Arctic Oscillation (AO) – and examine them utilizing wavelet evaluation. In every case, we additionally have a look at the general frequency spectrum, given by the Discrete Fourier Remodel (DFT), in addition to a basic time-series decomposition into development, seasonal parts, and the rest.

Three oscillations

By far the best-known – essentially the most notorious, I ought to say – among the many three is El Niño–Southern Oscillation (ENSO), a.ok.a. El Niño/La Niña. The time period refers to a altering sample of sea floor temperatures and sea-level pressures occurring within the equatorial Pacific. Each El Niño and La Niña can and do have catastrophic influence on individuals’s lives, most notably, for individuals in creating nations west and east of the Pacific.

El Niño happens when floor water temperatures within the japanese Pacific are greater than regular, and the robust winds that usually blow from east to west are unusually weak. From April to October, this results in sizzling, extraordinarily moist climate situations alongside the coasts of northern Peru and Ecuador, frequently leading to main floods. La Niña, however, causes a drop in sea floor temperatures over Southeast Asia in addition to heavy rains over Malaysia, the Philippines, and Indonesia. Whereas these are the areas most gravely impacted, adjustments in ENSO reverberate throughout the globe.

Much less well-known than ENSO, however extremely influential as nicely, is the North Atlantic Oscillation (NAO). It strongly impacts winter climate in Europe, Greenland, and North America. Its two states relate to the dimensions of the stress distinction between the Icelandic Excessive and the Azores Low. When the stress distinction is excessive, the jet stream – these robust westerly winds that blow between North America and Northern Europe – is but stronger than regular, resulting in heat, moist European winters and calmer-than-normal situations in Japanese North America. With a lower-than-normal stress distinction, nonetheless, the American East tends to incur extra heavy storms and cold-air outbreaks, whereas winters in Northern Europe are colder and extra dry.

Lastly, the Arctic Oscillation (AO) is a ring-like sample of sea-level stress anomalies centered on the North Pole. (Its Southern-hemisphere equal is the Antarctic Oscillation.) AO’s affect extends past the Arctic Circle, nonetheless; it’s indicative of whether or not and the way a lot Arctic air flows down into the center latitudes. AO and NAO are strongly associated, and may designate the identical bodily phenomenon at a elementary stage.

Now, let’s make these characterizations extra concrete by precise information.

Evaluation: ENSO

We start with the best-known of those phenomena: ENSO. Knowledge can be found from 1854 onwards; nonetheless, for comparability with AO, we discard all data previous to January, 1950. For evaluation, we decide NINO34_MEAN, the month-to-month common sea floor temperature within the Niño 3.4 area (i.e., the world between 5° South, 5° North, 190° East, and 240° East). Lastly, we convert to a tsibble, the format anticipated by feasts::STL().

library(tidyverse)
library(tsibble)

obtain.file(
  "https://bmcnoldy.rsmas.miami.edu/tropics/oni/ONI_NINO34_1854-2022.txt",
  destfile = "ONI_NINO34_1854-2022.txt"
)

enso <- read_table("ONI_NINO34_1854-2022.txt", skip = 9) %>%
  mutate(x = yearmonth(as.Date(paste0(YEAR, "-", `MON/MMM`, "-01")))) %>%
  choose(x, enso = NINO34_MEAN) %>%
  filter(x >= yearmonth("1950-01"), x <= yearmonth("2022-09")) %>%
  as_tsibble(index = x)

enso

# A tsibble: 873 x 2 [1M]
          x  enso
      <mth> <dbl>
 1 1950 Jan  24.6
 2 1950 Feb  25.1
 3 1950 Mar  25.9
 4 1950 Apr  26.3
 5 1950 Might  26.2
 6 1950 Jun  26.5
 7 1950 Jul  26.3
 8 1950 Aug  25.9
 9 1950 Sep  25.7
10 1950 Oct  25.7
# … with 863 extra rows

As already introduced, we need to have a look at seasonal decomposition, as nicely. By way of seasonal periodicity, what can we anticipate? Until advised in any other case, feasts::STL() will fortunately decide a window dimension for us. Nevertheless, there’ll probably be a number of vital frequencies within the information. (Not eager to spoil the suspense, however for AO and NAO, this may undoubtedly be the case!). In addition to, we need to compute the Fourier Remodel anyway, so why not try this first?

Right here is the ability spectrum:

Within the beneath plot, the x axis corresponds to frequencies, expressed as “variety of occasions per 12 months.” We solely show frequencies as much as and together with the Nyquist frequency, i.e., half the sampling charge, which in our case is 12 (per 12 months).

num_samples <- nrow(enso)
nyquist_cutoff <- ceiling(num_samples / 2) # highest discernible frequency
bins_below_nyquist <- 0:nyquist_cutoff

sampling_rate <- 12 # per 12 months
frequencies_per_bin <- sampling_rate / num_samples
frequencies <- frequencies_per_bin * bins_below_nyquist

df <- information.body(f = frequencies, y = as.numeric(fft[1:(nyquist_cutoff + 1)]$abs()))
df %>% ggplot(aes(f, y)) +
  geom_line() +
  xlab("frequency (per 12 months)") +
  ylab("magnitude") +
  ggtitle("Spectrum of Niño 3.4 information")

There’s one dominant frequency, equivalent to about yearly. From this part alone, we’d anticipate one El Niño occasion – or equivalently, one La Niña – per 12 months. However let’s find vital frequencies extra exactly. With not many different periodicities standing out, we could as nicely limit ourselves to a few:

strongest <- torch_topk(fft[1:(nyquist_cutoff/2)]$abs(), 3)
strongest

[[1]]
torch_tensor
233.9855
172.2784
142.3784
[ CPUFloatType{3} ]

[[2]]
torch_tensor
74
21
7
[ CPULongType{3} ]

What we’ve got listed below are the magnitudes of the dominant parts, in addition to their respective bins within the spectrum. Let’s see which precise frequencies these correspond to:

important_freqs <- frequencies[as.numeric(strongest[[2]])]
important_freqs

[1] 1.00343643 0.27491409 0.08247423 

That’s as soon as per 12 months, as soon as per quarter, and as soon as each twelve years, roughly. Or, expressed as periodicity, when it comes to months (i.e., what number of months are there in a interval):

num_observations_in_season <- 12/important_freqs  
num_observations_in_season

[1] 11.95890  43.65000 145.50000  

We now move these to feasts::STL(), to acquire a five-fold decomposition into development, seasonal parts, and the rest.

library(feasts)
enso %>%
  mannequin(STL(enso ~ season(interval = 12) + season(interval = 44) +
              season(interval = 145))) %>%
  parts() %>%
  autoplot()

In keeping with Loess decomposition, there nonetheless is critical noise within the information – the rest remaining excessive regardless of our hinting at vital seasonalities. In truth, there isn’t any huge shock in that: Wanting again on the DFT output, not solely are there many, shut to at least one one other, low- and lowish-frequency parts, however as well as, high-frequency parts simply received’t stop to contribute. And actually, as of at this time, ENSO forecasting – tremendously vital when it comes to human influence – is concentrated on predicting oscillation state only a 12 months upfront. This shall be fascinating to bear in mind for after we proceed to the opposite sequence – as you’ll see, it’ll solely worsen.

By now, we’re nicely knowledgeable about how dominant temporal rhythms decide, or fail to find out, what really occurs in ambiance and ocean. However we don’t know something about whether or not, and the way, these rhythms could have different in power over the time span thought of. That is the place wavelet evaluation is available in.

In torchwavelets, the central operation is a name to wavelet_transform(), to instantiate an object that takes care of all required operations. One argument is required: signal_length, the variety of information factors within the sequence. And one of many defaults we want to override: dt, the time between samples, expressed within the unit we’re working with. In our case, that’s 12 months, and, having month-to-month samples, we have to move a worth of 1/12. With all different defaults untouched, evaluation shall be accomplished utilizing the Morlet wavelet (accessible options are Mexican Hat and Paul), and the rework shall be computed within the Fourier area (the quickest means, until you will have a GPU).

library(torchwavelets)
enso_idx <- enso$enso %>% as.numeric() %>% torch_tensor()
dt <- 1/12
wtf <- wavelet_transform(size(enso_idx), dt = dt)

A name to energy() will then compute the wavelet rework:

power_spectrum <- wtf$energy(enso_idx)
power_spectrum$form

[1]  71 873

The result’s two-dimensional. The second dimension holds measurement occasions, i.e., the months between January, 1950 and September, 2022. The primary dimension warrants some extra clarification.

Specifically, we’ve got right here the set of scales the rework has been computed for. In the event you’re conversant in the Fourier Remodel and its analogue, the spectrogram, you’ll most likely assume when it comes to time versus frequency. With wavelets, there may be an extra parameter, the size, that determines the unfold of the evaluation sample.

Some wavelets have each a scale and a frequency, by which case these can work together in complicated methods. Others are outlined such that no separate frequency seems. Within the latter case, you instantly find yourself with the time vs. scale structure we see in wavelet diagrams (scaleograms). Within the former, most software program hides the complexity by merging scale and frequency into one, leaving simply scale as a user-visible parameter. In torchwavelets, too, the wavelet frequency (if existent) has been “streamlined away.” Consequently, we’ll find yourself plotting time versus scale, as nicely. I’ll say extra after we really see such a scaleogram.

For visualization, we transpose the info and put it right into a ggplot-friendly format:

occasions <- lubridate::12 months(enso$x) + lubridate::month(enso$x) / 12
scales <- as.numeric(wtf$scales)

df <- as_tibble(as.matrix(power_spectrum$t()), .name_repair = "common") %>%
  mutate(time = occasions) %>%
  pivot_longer(!time, names_to = "scale", values_to = "energy") %>%
  mutate(scale = scales[scale %>%
    str_remove("[.]{3}") %>%
    as.numeric()])
df %>% glimpse()

Rows: 61,983
Columns: 3
$ time  <dbl> 1950.083, 1950.083, 1950.083, 1950.083, 195…
$ scale <dbl> 0.1613356, 0.1759377, 0.1918614, 0.2092263,…
$ energy <dbl> 0.03617507, 0.05985500, 0.07948010, 0.09819…

There’s one further piece of data to be integrated, nonetheless: the so-called “cone of affect” (COI). Visually, it is a shading that tells us which a part of the plot displays incomplete, and thus, unreliable and to-be-disregarded, information. Specifically, the larger the size, the extra spread-out the evaluation wavelet, and the extra incomplete the overlap on the borders of the sequence when the wavelet slides over the info. You’ll see what I imply in a second.

The COI will get its personal information body:

And now we’re able to create the scaleogram:

labeled_scales <- c(0.25, 0.5, 1, 2, 4, 8, 16, 32, 64)
labeled_frequencies <- spherical(as.numeric(wtf$fourier_period(labeled_scales)), 1)

ggplot(df) +
  scale_y_continuous(
    trans = scales::compose_trans(scales::log2_trans(), scales::reverse_trans()),
    breaks = c(0.25, 0.5, 1, 2, 4, 8, 16, 32, 64),
    limits = c(max(scales), min(scales)),
    broaden = c(0, 0),
    sec.axis = dup_axis(
      labels = scales::label_number(labeled_frequencies),
      title = "Fourier interval (years)"
    )
  ) +
  ylab("scale (years)") +
  scale_x_continuous(breaks = seq(1950, 2020, by = 5), broaden = c(0, 0)) +
  xlab("12 months") +
  geom_contour_filled(aes(time, scale, z = energy), present.legend = FALSE) +
  scale_fill_viridis_d(possibility = "turbo") +
  geom_ribbon(information = coi_df, aes(x = x, ymin = y, ymax = max(scales)),
              fill = "black", alpha = 0.6) +
  theme(legend.place = "none")

What we see right here is how, in ENSO, completely different rhythms have prevailed over time. As an alternative of “rhythms,” I may have stated “scales,” or “frequencies,” or “durations” – all these translate into each other. Since, to us people, wavelet scales don’t imply that a lot, the interval (in years) is displayed on an extra y axis on the correct.

So, we see that within the eighties, an (roughly) four-year interval had distinctive affect. Thereafter, but longer periodicities gained in dominance. And, in accordance with what we anticipate from prior evaluation, there’s a basso continuo of annual similarity.

Additionally, notice how, at first sight, there appears to have been a decade the place a six-year interval stood out: proper firstly of the place (for us) measurement begins, within the fifties. Nevertheless, the darkish shading – the COI – tells us that, on this area, the info is to not be trusted.

Summing up, the two-dimensional evaluation properly enhances the extra compressed characterization we obtained from the DFT. Earlier than we transfer on to the following sequence, nonetheless, let me simply rapidly handle one query, in case you have been questioning (if not, simply learn on, since I received’t be going into particulars anyway): How is that this completely different from a spectrogram?

In a nutshell, the spectrogram splits the info into a number of “home windows,” and computes the DFT independently on all of them. To compute the scaleogram, however, the evaluation wavelet slides repeatedly over the info, leading to a spectrum-equivalent for the neighborhood of every pattern within the sequence. With the spectrogram, a hard and fast window dimension implies that not all frequencies are resolved equally nicely: The upper frequencies seem extra ceaselessly within the interval than the decrease ones, and thus, will enable for higher decision. Wavelet evaluation, in distinction, is finished on a set of scales intentionally organized in order to seize a broad vary of frequencies theoretically seen in a sequence of given size.

Evaluation: NAO

The information file for NAO is in fixed-table format. After conversion to a tsibble, we’ve got:

obtain.file(
 "https://crudata.uea.ac.uk/cru/information//nao/nao.dat",
 destfile = "nao.dat"
)

# wanted for AO, as nicely
use_months <- seq.Date(
  from = as.Date("1950-01-01"),
  to = as.Date("2022-09-01"),
  by = "months"
)

nao <-
  read_table(
    "nao.dat",
    col_names = FALSE,
    na = "-99.99",
    skip = 3
  ) %>%
  choose(-X1, -X14) %>%
  as.matrix() %>%
  t() %>%
  as.vector() %>%
  .[1:length(use_months)] %>%
  tibble(
    x = use_months,
    nao = .
  ) %>%
  mutate(x = yearmonth(x)) %>%
  fill(nao) %>%
  as_tsibble(index = x)

nao

# A tsibble: 873 x 2 [1M]
          x   nao
      <mth> <dbl>
 1 1950 Jan -0.16
 2 1950 Feb  0.25
 3 1950 Mar -1.44
 4 1950 Apr  1.46
 5 1950 Might  1.34
 6 1950 Jun -3.94
 7 1950 Jul -2.75
 8 1950 Aug -0.08
 9 1950 Sep  0.19
10 1950 Oct  0.19
# … with 863 extra rows

Like earlier than, we begin with the spectrum:

fft <- torch_fft_fft(as.numeric(scale(nao$nao)))

num_samples <- nrow(nao)
nyquist_cutoff <- ceiling(num_samples / 2)
bins_below_nyquist <- 0:nyquist_cutoff

sampling_rate <- 12 
frequencies_per_bin <- sampling_rate / num_samples
frequencies <- frequencies_per_bin * bins_below_nyquist

df <- information.body(f = frequencies, y = as.numeric(fft[1:(nyquist_cutoff + 1)]$abs()))
df %>% ggplot(aes(f, y)) +
  geom_line() +
  xlab("frequency (per 12 months)") +
  ylab("magnitude") +
  ggtitle("Spectrum of NAO information")

Have you ever been questioning for a tiny second whether or not this was time-domain information – not spectral? It does look much more noisy than the ENSO spectrum for certain. And actually, with NAO, predictability is way worse – forecast lead time often quantities to simply one or two weeks.

Continuing as earlier than, we decide dominant seasonalities (not less than this nonetheless is feasible!) to move to feasts::STL().

strongest <- torch_topk(fft[1:(nyquist_cutoff/2)]$abs(), 6)
strongest

[[1]]
torch_tensor
102.7191
80.5129
76.1179
75.9949
72.9086
60.8281
[ CPUFloatType{6} ]

[[2]]
torch_tensor
147
99
146
59
33
78
[ CPULongType{6} ]

important_freqs <- frequencies[as.numeric(strongest[[2]])]
important_freqs

[1] 2.0068729 1.3470790 1.9931271 0.7972509 0.4398625 1.0584192

num_observations_in_season <- 12/important_freqs  
num_observations_in_season

[1]  5.979452  8.908163  6.020690 15.051724 27.281250 11.337662

Necessary seasonal durations are of size six, 9, eleven, fifteen, and twenty-seven months, roughly – fairly shut collectively certainly! No surprise that, in STL decomposition, the rest is much more important than with ENSO:

nao %>%
  mannequin(STL(nao ~ season(interval = 6) + season(interval = 9) +
              season(interval = 15) + season(interval = 27) +
              season(interval = 12))) %>%
  parts() %>%
  autoplot()

Now, what is going to we see when it comes to temporal evolution? A lot of the code that follows is similar as for ENSO, repeated right here for the reader’s comfort:

nao_idx <- nao$nao %>% as.numeric() %>% torch_tensor()
dt <- 1/12 # identical interval as for ENSO
wtf <- wavelet_transform(size(nao_idx), dt = dt)
power_spectrum <- wtf$energy(nao_idx)

occasions <- lubridate::12 months(nao$x) + lubridate::month(nao$x)/12 # additionally identical
scales <- as.numeric(wtf$scales) # shall be identical as a result of each sequence have identical size

df <- as_tibble(as.matrix(power_spectrum$t()), .name_repair = "common") %>%
  mutate(time = occasions) %>%
  pivot_longer(!time, names_to = "scale", values_to = "energy") %>%
  mutate(scale = scales[scale %>%
    str_remove("[.]{3}") %>%
    as.numeric()])

coi <- wtf$coi(occasions[1], occasions[length(nao_idx)])
coi_df <- information.body(x = as.numeric(coi[[1]]), y = as.numeric(coi[[2]]))

labeled_scales <- c(0.25, 0.5, 1, 2, 4, 8, 16, 32, 64) # identical since scales are identical 
labeled_frequencies <- spherical(as.numeric(wtf$fourier_period(labeled_scales)), 1)

ggplot(df) +
  scale_y_continuous(
    trans = scales::compose_trans(scales::log2_trans(), scales::reverse_trans()),
    breaks = c(0.25, 0.5, 1, 2, 4, 8, 16, 32, 64),
    limits = c(max(scales), min(scales)),
    broaden = c(0, 0),
    sec.axis = dup_axis(
      labels = scales::label_number(labeled_frequencies),
      title = "Fourier interval (years)"
    )
  ) +
  ylab("scale (years)") +
  scale_x_continuous(breaks = seq(1950, 2020, by = 5), broaden = c(0, 0)) +
  xlab("12 months") +
  geom_contour_filled(aes(time, scale, z = energy), present.legend = FALSE) +
  scale_fill_viridis_d(possibility = "turbo") +
  geom_ribbon(information = coi_df, aes(x = x, ymin = y, ymax = max(scales)),
              fill = "black", alpha = 0.6) +
  theme(legend.place = "none")

That, actually, is a way more colourful image than with ENSO! Excessive frequencies are current, and frequently dominant, over the entire time interval.

Apparently, although, we see similarities to ENSO, as nicely: In each, there is a crucial sample, of periodicity 4 or barely extra years, that exerces affect through the eighties, nineties, and early two-thousands – solely with ENSO, it reveals peak influence through the nineties, whereas with NAO, its dominance is most seen within the first decade of this century. Additionally, each phenomena exhibit a strongly seen peak, of interval two years, round 1970. So, is there an in depth(-ish) connection between each oscillations? This query, in fact, is for the area specialists to reply. At the least I discovered a current research (Scaife et al. (2014)) that not solely suggests there may be, however makes use of one (ENSO, the extra predictable one) to tell forecasts of the opposite:

Earlier research have proven that the El Niño–Southern Oscillation can drive interannual variations within the NAO [Brönnimann et al., 2007] and therefore Atlantic and European winter local weather by way of the stratosphere [Bell et al., 2009]. […] this teleconnection to the tropical Pacific is lively in our experiments, with forecasts initialized in El Niño/La Niña situations in November tending to be adopted by detrimental/constructive NAO situations in winter.

Will we see an analogous relationship for AO, our third sequence underneath investigation? We would anticipate so, since AO and NAO are intently associated (and even, two sides of the identical coin).

Evaluation: AO

First, the info:

obtain.file(
 "https://www.cpc.ncep.noaa.gov/merchandise/precip/CWlink/daily_ao_index/month-to-month.ao.index.b50.present.ascii.desk",
 destfile = "ao.dat"
)

ao <-
  read_table(
    "ao.dat",
    col_names = FALSE,
    skip = 1
  ) %>%
  choose(-X1) %>%
  as.matrix() %>% 
  t() %>%
  as.vector() %>%
  .[1:length(use_months)] %>%
  tibble(x = use_months,
         ao = .) %>%
  mutate(x = yearmonth(x)) %>%
  fill(ao) %>%
  as_tsibble(index = x) 

ao

# A tsibble: 873 x 2 [1M]
          x     ao
      <mth>  <dbl>
 1 1950 Jan -0.06 
 2 1950 Feb  0.627
 3 1950 Mar -0.008
 4 1950 Apr  0.555
 5 1950 Might  0.072
 6 1950 Jun  0.539
 7 1950 Jul -0.802
 8 1950 Aug -0.851
 9 1950 Sep  0.358
10 1950 Oct -0.379
# … with 863 extra rows

And the spectrum:

fft <- torch_fft_fft(as.numeric(scale(ao$ao)))

num_samples <- nrow(ao)
nyquist_cutoff <- ceiling(num_samples / 2)
bins_below_nyquist <- 0:nyquist_cutoff

sampling_rate <- 12 # per 12 months
frequencies_per_bin <- sampling_rate / num_samples
frequencies <- frequencies_per_bin * bins_below_nyquist

df <- information.body(f = frequencies, y = as.numeric(fft[1:(nyquist_cutoff + 1)]$abs()))
df %>% ggplot(aes(f, y)) +
  geom_line() +
  xlab("frequency (per 12 months)") +
  ylab("magnitude") +
  ggtitle("Spectrum of AO information")

Nicely, this spectrum appears to be like much more random than NAO’s, in that not even a single frequency stands out. For completeness, right here is the STL decomposition:

strongest <- torch_topk(fft[1:(nyquist_cutoff/2)]$abs(), 5)

important_freqs <- frequencies[as.numeric(strongest[[2]])]
important_freqs
# [1] 0.01374570 0.35738832 1.77319588 1.27835052 0.06872852

num_observations_in_season <- 12/important_freqs  
num_observations_in_season
# [1] 873.000000  33.576923   6.767442   9.387097 174.600000 

ao %>%
  mannequin(STL(ao ~ season(interval = 33) + season(interval = 7) +
              season(interval = 9) + season(interval = 174))) %>%
  parts() %>%
  autoplot()

Lastly, what can the scaleogram inform us about dominant patterns?

ao_idx <- ao$ao %>% as.numeric() %>% torch_tensor()
dt <- 1/12 # identical interval as for ENSO and NAO
wtf <- wavelet_transform(size(ao_idx), dt = dt)
power_spectrum <- wtf$energy(ao_idx)

occasions <- lubridate::12 months(ao$x) + lubridate::month(ao$x)/12 # additionally identical
scales <- as.numeric(wtf$scales) # shall be identical as a result of all sequence have identical size

df <- as_tibble(as.matrix(power_spectrum$t()), .name_repair = "common") %>%
  mutate(time = occasions) %>%
  pivot_longer(!time, names_to = "scale", values_to = "energy") %>%
  mutate(scale = scales[scale %>%
    str_remove("[.]{3}") %>%
    as.numeric()])

coi <- wtf$coi(occasions[1], occasions[length(ao_idx)])
coi_df <- information.body(x = as.numeric(coi[[1]]), y = as.numeric(coi[[2]]))

labeled_scales <- c(0.25, 0.5, 1, 2, 4, 8, 16, 32, 64) # identical since scales are identical 
labeled_frequencies <- spherical(as.numeric(wtf$fourier_period(labeled_scales)), 1)

ggplot(df) +
  scale_y_continuous(
    trans = scales::compose_trans(scales::log2_trans(), scales::reverse_trans()),
    breaks = c(0.25, 0.5, 1, 2, 4, 8, 16, 32, 64),
    limits = c(max(scales), min(scales)),
    broaden = c(0, 0),
    sec.axis = dup_axis(
      labels = scales::label_number(labeled_frequencies),
      title = "Fourier interval (years)"
    )
  ) +
  ylab("scale (years)") +
  scale_x_continuous(breaks = seq(1950, 2020, by = 5), broaden = c(0, 0)) +
  xlab("12 months") +
  geom_contour_filled(aes(time, scale, z = energy), present.legend = FALSE) +
  scale_fill_viridis_d(possibility = "turbo") +
  geom_ribbon(information = coi_df, aes(x = x, ymin = y, ymax = max(scales)),
              fill = "black", alpha = 0.6) +
  theme(legend.place = "none")

Having seen the general spectrum, the dearth of strongly dominant patterns within the scaleogram doesn’t come as an enormous shock. It’s tempting – for me, not less than – to see a mirrored image of ENSO round 1970, all of the extra since by transitivity, AO and ENSO ought to be associated not directly. However right here, certified judgment actually is reserved to the specialists.

Conclusion

Like I stated to start with, this submit can be about inspiration, not technical element or reportable outcomes. And I hope that inspirational it has been, not less than a bit bit. In the event you’re experimenting with wavelets your self, or plan to – or when you work within the atmospheric sciences, and want to present some perception on the above information/phenomena – we’d love to listen to from you!

As all the time, thanks for studying!

Photograph by ActionVance on Unsplash