Recently, we have successfully validated methodologies for utilization. torch for . While a member of the household of spectral evaluation strategies, wavelet analysis shares similarities with the Fourier transform, particularly in its 2D counterpart, the spectrogram.
Defined in that eBook excerpt, despite notable disparities exist. To identify patterns in the frequency domain for this submission, it is sufficient to have a minimal wave-like structure “glide” over the data, calculating the degree of matching (or mismatching) in the vicinity of each pattern.
With this submission, I intend to achieve a twofold objective.
To begin with, consider introducing a small yet invaluable package deal that streamlines the crucial process, thereby simplifying and accelerating all essential steps involved. Compared to the well-established Fourier Transform and its established conventions, the field of wavelets appears relatively “chaotic” – with significantly less common vocabulary and far fewer standardized practices. In light of this, it is advisable for implementations to adhere to widely accepted, community-validated methodologies whenever possible, ensuring their clear documentation for easy reference. With torchwavelets
Our solution implements the seminal work of Torrence and Compo (1998), “Sensible Information to Wavelet Evaluation”, a groundbreaking paper whose impact has been felt across multiple software landscapes. Our proprietary code, essentially a port of, draws heavily from an earlier version developed by.
In addition, a striking example of the application of wavelet analysis emerges in the realm of meteorology and climatology, highlighting its potential for impactful scientific exploration. While lacking expertise myself, I hope this effort may spark inspiration for those working in these domains, as well as for scientists and analysts from diverse fields where temporal data arise.
We will specifically investigate the interplay between three distinct atmospheric patterns – ENSO, NAO, and AO – by applying wavelet analysis to their respective dynamics. We also examine the overall frequency spectrum provided by the Discrete Fourier Transform (DFT), as well as conducting a fundamental time-series decomposition into trend, seasonal components, and residuals.
Three oscillations
One of the most infamous and widely recognized among the trio is, colloquially known as. El Niño/La Niña. The targeted time interval pertains to a shifting pattern of seafloor temperature fluctuations and pressure variations in the equatorial Pacific region, with implications for ocean dynamics. El Niño and La Niña events can have devastating impacts on individuals’ daily lives, with particularly severe effects felt by people in developing regions surrounding the Pacific Ocean, both to its west and east.
El Niño occurs when sea surface temperatures in the equatorial Pacific Ocean exceed normal levels, accompanied by an unusual weakening of the trade winds that typically blow from east to west. Along the coasts of northern Peru and Ecuador, a unique combination of scorching hot and exceptionally humid climate conditions prevails from April to October, frequently giving rise to severe flooding episodes. La Niña’s impact is characterized by a decline in sea floor temperatures across Southeast Asia, concurrent with heavy rainfall events over Malaysia, the Philippines, and Indonesia. While these regions are disproportionately affected, shifts in ENSO have far-reaching consequences globally?
While ENSO may garner more attention, its lesser-known counterpart, the Madden-Julian Oscillation (MJO), has a profound impact on global climate patterns. The North Atlantic Oscillation has a profound influence on winter climates across Europe, Greenland, and North America? The relationship between these two weather systems lies in their distinct dimensional properties, specifically regarding the stress dynamics that characterize the Icelandic High and the Azores Low. When stress in the jet stream becomes excessively pronounced, its robust westerly winds blowing between North America and Northern Europe intensify, leading to milder, moister European winters and unusually calm conditions prevailing over northern Japan. While exhibiting a relatively low stress difference, the American East is prone to more intense storm systems and frequent cold air outbreaks; conversely, Northern European winters tend to be colder and drier than usual.
Lastly, the dataset consists of a ring-like sample of sea-level stress anomalies centered on the North Pole. While its counterpart in the Southern Hemisphere exists, AO’s impact doesn’t stop at the Arctic Circle; instead, it serves as an indicator of the extent to which Arctic air masses flow southward into mid-latitudes. The Arctic Oscillation (AO) and North Atlantic Oscillation (NAO) exhibit a strong correlation, potentially signifying the same fundamental atmospheric process at its core.
Let’s quantify these characterizations with specific data to make them more concrete.
Evaluation: ENSO
The first phenomenon to consider is El Niño Southern Oscillation (ENSO), arguably the most well-known of these events. Data prior to January 1950 is excluded for consistency with the Archives Office, making information available from 1854 onwards. For evaluation, we decide NINO34_MEANThe average monthly sea floor temperature within the Niño 3.4 region, spanning from 5° South to 5° North latitude and 190° to 240° East longitude. Lastly, we convert to a tsibble, the format anticipated by feasts::STL().
# 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 rowsAs previously discussed, let’s examine seasonal decomposition in closer detail. As the calendar turns, we can expect a predictable sequence of events to unfold. Informed of circumstances surrounding the situation feasts::STL() Will we be fortunate enough to have a decision made regarding our desired window dimensions? While it may seem that nothing significant will emerge from this data, there are likely to be several crucial frequencies hidden within the information. For those keenly anticipating a dramatic turn of events, rest assured that the fate of the protagonists will undoubtedly unfold in a thrilling manner! The anticipation is palpable, and the stakes are high, setting the stage for an unforgettable climax. Let’s perform the Fourier transform upfront to get a head start on our analysis. This will give us valuable insights into the frequency domain.
The human ability spectrum is a wide-ranging array that encompasses diverse cognitive and non-cognitive skills.
The plot displays frequencies along the x-axis, measured in “number of occurrences per year”. Frequencies are capped at the Nyquist frequency, or half the sampling rate, which in this instance is 12 per year.
The dominant frequency appears to be consistent, roughly approximating a yearly periodicity. Based on historical records, we would expect to see approximately one El Niño event and one La Niña event each year. Let’s pinpoint those crucial frequencies with precision. With limited distinct periodicities, we can realistically confine our analysis to just three.
[[1]]
torch_tensor
233.9855
172.2784
142.3784
[ CPUFloatType{3} ]
[[2]]
torch_tensor
74
21
7
[ CPULongType{3} ]Below, we present the magnitudes of the prominent spectral components, along with their corresponding frequency ranges. Which precise frequencies do these signals correspond to?
[1] 1.00343643 0.27491409 0.08247423 About once a year, approximately once every three months, and roughly once every 12 years. There are 12 months in a year, so the concept of periodicity in this context is often expressed in terms of annual frequency.
[1] 11.95890 43.65000 145.50000 We now go these to feasts::STL()Can we obtain a five-fold stratification by partitioning the data into development, seasonal components, and residuals?
Despite aligning with Loess decomposition principles, a significant amount of noise persists in the data, rendering seasonal patterns ineffective despite attempts to highlight their importance? Truthfully, this revelation comes as no surprise: A reexamination of the DFT output reveals that numerous, nearly indistinguishable, low- and low-to-moderate-frequency components abound, while high-frequency contributors persistently add to the mix. Currently, the focus of ENSO forecasting, which has profound implications for humanity, is limited to accurately predicting the oscillation state just 12 months ahead. As we venture forward, this realization will prove significant to keep in mind once we transition to the alternative dataset – an unsettling prospect that will only continue to deteriorate further.
With our current understanding, we’re well-versed on how prominent temporal cycles influence the dynamics between atmosphere and ocean, determining – or sometimes failing to reveal – what’s actually happening in this intricate system? Whether there is a correlation between these rhythms and their varying levels of influence across different periods remains unknown. Where wavelet analysis is readily accessible.
In torchwaveletsWhat drives the core of our organization? wavelet_transform()To create an instance of a class that handles all necessary actions? One argument is required: signal_lengthThe diverse array of data elements within this dataset. One of many default settings we often need to override: dtThe interval between successive measurements, denoted in the relevant unit of measurement. Given the limited data, we need to adjust for the month-to-month variability by normalizing it over a one-twelfth period. Evaluation shall be completed using default settings unless otherwise specified. The computation will be performed in the Fourier domain, which is the fastest approach available, assuming you do not have access to a Graphics Processing Unit (GPU).
A name to energy() Will subsequently calculate the wavelet decomposition.
[1] 71 873The result’s two-dimensional. The second dimension captures measurement occasions – a span of time from January 1950 to September 2022. What specifically requires further elaboration in this context is the primary dimension?
In particular, the reworked computations are based on this specific set of scales. If you’re familiar with the Fourier Transform and its analog, the spectrogram, you likely consider the relationship between time and frequency. The incorporation of wavelets into this context introduces an additional consideration: the dimensionality parameter, which governs the extent to which the evaluation sample is unfolded in its evaluation.
Some wavelet types feature both scale and frequency attributes, allowing them to interact in intricate ways. No apparent pattern exists among others. Instantly situated within the paradigm of time vs scale, a format reminiscent of wavelet diagrams’ characteristic scaleograms emerges. Within many software programs, underlying complexity is effectively masked by integrating both scale and frequency into a single, user-facing parameter that simplifies the process to just selecting a scale. In torchwaveletsAs a result, when the wavelet frequency is present, it is effectively eliminated from the analysis; thus, our graph will display time plotted against scale in a straightforward manner. We’ll discuss further options once we’ve visually assessed this level of complexity.
The data is then reorganized and presented in its entirety. ggplot-friendly format:
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…Despite being nearly complete, there is still one additional dataset that needs to be incorporated: the conceptually significant “cone of influence.” The narrative employs subtle visual cues, such as shading, to indicate sections of the story that are fragmented or untrustworthy, thereby prompting the reader to approach these portions with skepticism. As dimensionality increases, the evaluation wavelet becomes increasingly dispersed, resulting in a proportionally greater extent of incomplete overlap between successive wavelet slides across the data’s boundaries. As I soon will.
The Company’s Interest (COI) is set to receive a comprehensive personal information framework.
Now that we have the necessary data in place, we’re finally able to generate the scaleogram.
As evidenced by ENSO’s complex history, distinct rhythmic patterns have dominated at various intervals. Rather than suggesting alternatives to “rhythms,” the text could be reframed as: To enhance understanding of wavelet scales for non-experts, we overlay the time interval in years on an additional y-axis beside it.
During the 1980s, a roughly four-year period exhibited a distinct impact. Subsequently, oscillations with longer intervals between peaks became more prevalent. As anticipated, the observed annual similarity remains consistent with our previous expectations.
Initially, it seems that roughly a decade-long gap emerged, where a six-year span stood out; this is particularly evident at the start of our era’s measurement, which began in the 1950s. Despite the ominous dark shading, the Conflict of Interest indicator warns us that the data presented here may be unreliable, and we should exercise caution when interpreting its findings.
By summarizing the two-dimensional assessment effectively, we can gain a more comprehensive understanding of the additional compressed data provided by the DFT. Before moving forward with the next series, though, I’ll just quickly address one question: how is this different from a spectrogram?
The spectrogram breaks down data into multiple ‘home windows’ and calculates the discrete Fourier transform (DFT) for each, processing the information independently. To generate the scaleogram, the wavelet subsequently scans the data in repeated iterations, thereby yielding a frequency-domain representation of local patterns throughout the dataset. With the spectrogram, a specified window dimension introduces an inherent trade-off: while it allows for higher resolution in the upper frequency range, which seems to be more densely populated within the interval, it does so at the expense of reduced resolution in the lower frequency range, thereby enabling greater precision in the former. In contrast, wavelet analysis involves a hierarchy of scales deliberately designed to capture a wide range of frequencies typically encountered in a dataset of a specified size.
Evaluation: NAO
The information file for NAO is formatted in a fixed-table structure. After conversion to a tsibble, now we have:
# 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 rowsPrior to commencing our exploration, we delve into the concept of a spectrum.
Have you ever paused briefly to wonder if this was, in fact, time-domain data rather than its frequency-domain equivalent? The comparison suggests that the noise level appears significantly higher in this spectrum compared to the ENSO counterpart, particularly when focusing on the positive frequency range. In reality, however, predictability takes a significant hit when using NAO, as the forecast lead time often dwindles down to a mere few days.
Let’s focus on identifying and capitalizing on dominant seasonalities in our strategy, rather than trying to chase fleeting market trends. feasts::STL().
[[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} ]
2.0068729 1.3470790 1.9931271, the numbers flow smoothly with a gentle pace. The sequence unfolds steadily, like a well-paced narrative, guiding the reader through a series of precise values.
5.98, 8.91, 6.02, 15.05, 27.28, 11.34Seasonal patterns are characterized by intervals of approximately six, nine, eleven, fifteen, and twenty-seven months, with some overlap. It’s no surprise that, in STL decomposition, the remainder is significantly more crucial than with ENSO.
As the concept of time itself evolves, what will become apparent is that our understanding of its fabric will continue to unravel. To facilitate comprehension, this section duplicates code patterns commonly found in ENSO-related analyses.
This vibrant representation surpasses even the most vivid depictions of ENSO’s impact. Excessive frequency phenomena persistently predominate throughout the entire duration.
Although similarities to ENSO are evident, a crucial aspect in each phenomenon is a periodicity of approximately four years, which exerts significant influence throughout the 1980s, 1990s, and early 2000s; however, while ENSO’s peak impact occurs primarily during the 1990s, NAO’s dominance is most pronounced in the first decade of this century. Each phenomenon exhibits a pronounced peak approximately every two years, centered around 1970. Are the subtle ties between successive oscillations a subject of fascination for you? The Area Consultants are requested to respond. Recent research reveals that ENSO, a more predictable climate phenomenon, can not only suggest but also inform predictions about another.
Research has consistently shown that the El Niño–Southern Oscillation drives interannual variations in the North Atlantic Oscillation (NAO), subsequently influencing Atlantic and European winter weather patterns through stratospheric pathways [Brönnimann et al., 2007; Bell et al., 2009]. The teleconnection between our tropical Pacific-based experiments and North Atlantic Oscillation (NAO) patterns is strikingly evident, with predictions initialized during El Niño/La Niña conditions in November consistently aligning with negative/positive NAO phases in winter.
Will we witness a parallel trend with AO, our third dataset under scrutiny? Given their inherent correlation, we naturally expect this connection.
Evaluation: AO
First, the info:
# 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 rowsAnd the spectrum:
In stark contrast to NAO’s distinct spectral pattern, this spectrum appears fundamentally more erratic, with no individual frequency noticeably dominating the overall profile. Here are the top-level components of the Standard Template Library (STL): containers, algorithms, and iterators.
The scaleogram, by revealing relationships between variables on different scales, informs us that dominant patterns emerge when there is a strong correlation between highly correlated features.
Given the overall range, the scarcity of prominent patterns on the scaleogram is hardly surprising. Given the historical context around 1970, it’s plausible to infer that any amplified ENSO patterns would mirror themselves; conversely, since an excess in ENSO can be linked to both AO and ENSO, it appears that these phenomena are intertwined in their ultimate causes. However, certification for making judgments is reserved solely to the consultants.
Conclusion
The spark within us that drives creativity and innovation is often rooted in a sense of purpose, passion, and perseverance. I hope this small spark of inspiration ignites a fire within you, even if it’s just a tiny flame. If you’re exploring wavelet applications yourself or planning to do so, or if your expertise lies in the atmospheric sciences and you’d like to share insights on this topic, we invite your thoughts.
Thank you for your time spent studying.
Picture by on
Torrence, C., and G. P. Compo. 1998. 79 (1): 61–78.
