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Element analysis: a wavelet-based method for analyzing time-localized events in noisy time series

J. M. Lilly

arXiv:1703.04882v2stat.ME

TL;DR

The paper addresses quantitative detection and characterization of isolated, time-localized events in noisy signals. It develops element analysis using generalized Morse wavelets, significance testing under power-law noise, and an isolation criterion, then applies the method to satellite-altimetry eddy detection. The method identifies coherent eddy events with encouraging results, while event shielding and limited observational records remain scope boundaries.

  • Problem

    Existing wavelet methods emphasize nearly sinusoidal or nearly singular signals, whereas isolated, self-similar events require a model that estimates their amplitude, phase, time, and scale.

  • Method

    Element analysis models signals as isolated rescaled and phase-shifted copies of generalized Morse wavelets, detects transform maxima, evaluates significance under power-law noise, and tests event isolation.

  • Results

    The method detects coherent eddy events in along-track satellite altimetry with encouraging results.

  • Takeaways & Limitations

    Element analysis is also suitable for strongly modulated wave packets and other impulse-like events.

  • Takeaways & Limitations

    Smaller events may be obscured when too close to larger events, and extending eddy statistics across a longer record would require additional data processing.

Abstract

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A method is derived for the quantitative analysis of signals that are composed of superpositions of isolated, time-localized "events". Here these events are taken to be well represented as rescaled and phase-rotated versions of generalized Morse wavelets, a broad family of continuous analytic functions. Analyzing a signal composed of replicates of such a function using another Morse wavelet allows one to directly estimate the properties of events from the values of the wavelet transform at its own maxima. The distribution of events in general power-law noise is determined in order to establish significance based on an expected false detection rate. Finally, an expression for an event's "region of influence" within the wavelet transform permits the formation of a criterion for rejecting spurious maxima due to numerical artifacts or other unsuitable events. Signals can then be reconstructed based on a small number of isolated points on the time/scale plane. This method, termed element analysis, is applied to the identification of long-lived eddy structures in ocean currents as observed by along-track measurements of sea surface elevation from satellite altimetry

1 Introduction

The paper introduces element analysis for nonsinusoidal signals composed of isolated, time-localized events. It models events with flexible generalized Morse functions and estimates their properties from significant, isolated wavelet-transform maxima.

  • Motivation: Nonsinusoidal signals can consist of isolated, self-similar events that are sparsely distributed on the time/scale plane.This provides a third signal category beyond nearly sinusoidal and nearly singular signals.
  • Application: Applied to along-track altimetry, element analysis targets Gaussian-like sea-surface-height anomalies associated with coherent ocean eddies amid noise and other variability.The paper motivates the application using narrow satellite ground tracks that observe eddy-related bumps or depressions.
  • Model: The element model represents a real signal as time-offset, phase-shifted, rescaled copies of a known localized function plus stationary Gaussian measurement noise.Each event is characterized by amplitude, phase, temporal location, and scale.
  • Method: Element analysis detects transform maxima, assesses their significance against power-law noise, and rejects events that are insufficiently isolated.Unlike wavelet thresholding, it seeks significant features that also match the specified element function.
  • Element family: Generalized Morse wavelets provide a flexible element family encompassing commonly used analytic wavelets and limiting cases such as analytic delta functions and complex exponentials.Their simple frequency-domain form permits analytic treatment of key properties.
  • Method: The method assumes event form is known but duration is unknown, contrasting with sliding-window methods that assume duration while leaving form unspecified.This model-based formulation supports inference with few adjustable parameters and enables automation and scalability.

2 Background

The background develops generalized Morse wavelets and the continuous wavelet transform as the basis for element analysis. It explains how wavelet parameters control form and why amplitude normalization supports localized event analysis, while noting limitations of energy-normalized maxima.

  • Morse wavelets: Generalized Morse wavelets are a two-parameter family defined by order β and family γ, with analytic, complex-valued frequency-domain representations.Their frequency-domain construction restricts support to nonnegative frequencies and naturally captures phase variability.
  • Continuous wavelet transform: The wavelet transform compares a signal with shifted, rescaled wavelets, while scale controls temporal stretching or compression and determines the scale frequency.The frequency-domain wavelet peaks at ω_s = ω_β,γ/s.
  • Normalization: The 1/s normalization preserves transform magnitude under temporal rescaling and makes transform values reflect signal amplitude rather than constant energy.This normalization is therefore suited to describing time-localized signals by amplitude.
  • Morse wavelets: The zeroth-order Morse functions are not strictly wavelets because they are not zero mean, whereas β > 0 functions have vanishing mean.For β = 0, the functions are treated as Morse functions and require a separate reference-frequency definition.
  • Morse wavelet properties: Increasing β makes Morse functions more oscillatory, whereas decreasing β and γ produces increasingly localized, impulse-like forms.For fixed γ, increasing β packs more oscillations into the envelope; changing γ alters the family and shape.
  • Optimization principle: Energy-normalized transform maxima provide local least-squares fits, but on real-world data they are overly influenced by variability at adjacent times.The paper therefore motivates amplitude normalization as a better basis for localized event analysis.

3 Element analysis

Element analysis models signals as isolated, rescaled Morse-wavelet events and uses wavelet-transform maxima to infer their properties. Statistical significance and isolation criteria reject spurious maxima, enabling reconstruction from a small set of time/scale points.

  • 3.1 Generalized Morse functions as signal elements: Element analysis represents signals as isolated events modeled by Morse functions plus zero-mean noise, then analyzes them with Morse wavelets from the same γ family.The element and analyzing wavelets may have different orders, while sharing γ.
  • 3.1 Generalized Morse functions as signal elements: The analysis identifies maxima by local modulus conditions, then thresholds noise-generated maxima and enforces separation using a region-of-influence criterion.The procedure has three stages: detection, significance assessment from noise, and isolation testing.
  • 3.2 The Morse transform of another Morse function: The wavelet transform of a rescaled Morse element rescales both time and transform scale without changing transform amplitude.Its transform remains expressible as a modified Morse wavelet.
  • 3.3 Transform values at transform maxima: When events are sufficiently separated and the transform profile decays monotonically, each event produces one primary maximum, while interactions and sidelobes can create weak spurious maxima.These minor maxima may be rejected with an amplitude cutoff when they fall below the noise level.
  • 3.4 Inferring element properties from maxima points: Transform maxima provide estimates of each event’s temporal location, scale, and complex coefficient when the element function is known.The method works backward from observed time/scale maxima and their transform values to infer event properties.
  • 3.5 Examples of transform maxima: In a synthetic six-event signal obscured by noise, selecting statistically significant and isolated maxima produced a reconstruction virtually identical to the original.The reconstruction used estimates formed from the transform values at the selected maxima.

4 Significance and isolation

The method assesses transform maxima using noise-calibrated statistical significance and an isolation criterion, enabling reconstruction from a small set of meaningful time/scale points. In the example, these criteria reject a duplicate maximum and recover the underlying events despite substantial noise.

  • Significance: The analysis derives the expected wavelet spectrum for stationary Gaussian power-law noise and uses it to calibrate transform-maxima significance.The noise framework includes both white noise and red noise with power-law spectra.
  • Significance: A covariance-based Monte Carlo method simulates a five-component noise vector instead of repeatedly transforming noise realizations.The simulated vector preserves the local covariance structure governing maxima probabilities and normalized maxima amplitudes.
  • Significance: Normalizing maxima amplitudes by σα,β,γ(s) and densities by Lα,β,γ(s) nearly collapses distributions across scales, supporting a common false-detection calibration.Residual scale dependence appears at larger scales because neighboring components become more correlated, reducing the effective sample size.
  • Significance: A normalized maximum above about 1.7 σα,β,γ(s) corresponds to roughly one event per 100 wavelet footprints in the white-noise example.The threshold provides an amplitude cutoff for a tolerable false-detection rate.
  • Isolation: At the 1/1000 cutoff, seven significant maxima were detected: six associated with true events and one duplicate near the second event.The duplicate caused an overshoot in reconstruction, motivating the separate isolation criterion.
  • Isolation: The region-of-influence test rejects the poorly isolated duplicate, leaving six maxima and a reconstruction virtually identical to the original signal.The example demonstrates accurate extraction even with relatively large noise levels.
  • Isolation: Smaller events can be obscured by nearby larger events, with the larger event’s region of influence indicating the resulting shielding region.A fuller investigation of nearby-event effects is outside the paper’s scope.

5 Application

The method is applied to along-track satellite altimetry in the Labrador Sea, detecting statistically significant and isolated eddy-like events. Reconstructions capture the meaningful structure with strong compression and resolve coherent eddy properties at O(10) km scales.

  • The application analyzes one year of along-track altimetry from the Labrador Sea using a Gaussian element function and a Morse wavelet.The dataset contains 5216 valid data points from repeated satellite ground tracks.
  • Events with a false detection rate above one per 1000 dataset realizations are rejected, along with maxima lacking isolation or containing more than 10% missing data.
  • 67 detected events, or fewer than two per track, are highly significant and isolated from one another and missing-data segments.
  • Reconstructions explain virtually all meaningful structure, while using only 5% of the number of datapoint coefficients.The residuals appear virtually devoid of meaningful structure.
  • The present paper develops the detection method rather than extending physical analysis of the detected events or assessing interannual eddy variability.Such follow-up analysis is left to a sequel, and longer records would be needed for interannual variability.
  • The application extracts coherent eddy properties as small as O(10) km from the along-track dataset.The analyzed segment represents only one-thousandth of one percent of the complete along-track altimeter dataset.

6 Conclusions and Discussion

The conclusions position element analysis as a wavelet-based method for isolated, time-localized events, with applications to eddies and other impulse-like signals. Its specificity depends on events being isolated and well matched to the proposed model, while several theoretical extensions remain open.

  • Element analysis models signals as isolated, rescaled, phase-shifted replicates of a specified time-localized function and uses generalized Morse wavelets.
  • The method complements wavelet ridge and modulus maxima approaches by representing signals at isolated points on the time/scale plane.
  • Its innovations include efficient simulation of maxima statistics and an approximate region-of-influence criterion.
  • The method produces encouraging results for coherent eddy detection and is also described as suitable for strongly modulated wave packets and other impulse-like events.
  • Compared with wavelet thresholding, element analysis is more specific because it requires isolated support points and filters out poorly matched events.
  • Future work includes explaining variation in total noise detection rates, deriving estimator bias and variance, selecting element wavelets, and extending the model with higher-order Morse-wavelet components.

Appendix A. A freely available software package

The paper's freely available Matlab toolbox jLab implements the element-analysis workflow, including wavelet transforms, maxima statistics, regions of influence, and isolation checks. Monte Carlo simulations dominate runtime, while application analysis is comparatively faster.

  • The jLab toolbox implements Morse-wavelet frequency selection, wavelet transforms, and Morse-wavelet generation through morsespace, wavetrans, and morsewave.
  • The transmaxdist routine simulates transform-maxima histograms and false detection rates using the five-vector method.
  • The toolbox computes regions of influence and localization regions analytically, then uses isomax to verify that maxima are isolated.
  • Running the complete analysis and figure-generation script takes about nine minutes on a 12-core Mac Pro with 2.7 GHz Intel Xeon E5 processors.
  • Most runtime is spent on Monte Carlo noise-distribution simulations, whereas applying the method to a larger dataset excerpt takes about two minutes.

Appendix B. The wavelet footprint vs. standard deviation

The appendix relates the wavelet footprint to the wavelet's time-domain standard deviation. The footprint is scaled to be comparable to approximately four standard deviations of the scale-s wavelet.

  • The wavelet footprint Lβ,γ(s) is related to the time-domain standard deviation of the scale-s wavelet.
  • The dimensional time-domain standard deviation is sσt;β,γ/ωβ,γ, while Pβ,γ has the simpler expression √βγ.
  • Numerical calculations show that Pβ,γ is roughly equal to 2σt;β,γ over a large range of β and γ values.
  • The factor 2√2 in Lβ,γ(s) makes the footprint comparable to four times the scale-s wavelet's dimensional time-domain standard deviation.
  • The relationship between Pβ,γ and σt;β,γ supports interpreting Pβ,γ as an inverse-bandwidth measure under the Heisenberg uncertainty principle.

Appendix C: Wavelet transform implementation details

The implementation specifies logarithmically spaced wavelet scale frequencies using high- and low-frequency cutoffs, density, and packing parameters. Mirror boundary conditions are used to reduce edge effects.

  • Frequency limits: The high-frequency cutoff ensures that the wavelet value at the Nyquist frequency is no greater than η times its peak value.Scale frequencies above the cutoff would extend substantially beyond the Nyquist frequency.
  • Frequency discretization: Scale frequencies are logarithmically spaced from a high-frequency cutoff, with density parameter D controlling the frequency resolution.D can be interpreted as the number of scale-frequency bands fitting within a bandwidth interval; D = 4 or D = 8 is generally suitable.
  • Frequency discretization: The lowest scale frequency is chosen so that p wavelet footprints span the time series, where p is the packing number.A typical choice is p = 5, placing five wavelet footprints across the lowest-frequency band.
  • Boundary conditions: The transform uses mirror boundary conditions because they generally minimize edge effects better than zero-padding or periodic boundaries.This condition extends the time series by flipping it about both endpoints.
  • Applied settings: The synthetic example uses 59 logarithmically spaced scale levels, whereas the application uses 38 frequency bands with different η, D, and p settings.The synthetic settings are η = 0.05, D = 4, p = 3; the application settings are η = 0.1, D = 8, p = 2.

Appendix D. A scale range for the region of influence

The appendix determines the scale range where region-of-influence curves have real-valued solutions by comparing two monotonically increasing functions. Their relative growth implies a bounded interval of possible crossings.

  • Scale-range determination: The scale range is found by locating where the first monotonically increasing function rises above a second one and later falls below it.For small scales, f1 is below f2; for sufficiently large scales, f2 grows faster and again exceeds f1.
  • Scale-range determination: The first crossing cannot occur below ˜sa ≡ (λϑβ,µ,γ)1/β, where f1(˜s) equals λϑβ,µ,γ.This lower bound follows because f2 remains above λϑβ,µ,γ at small scales.

Appendix E. Regions of influence vs. localization regions

The appendix compares numerically computed Morse-wavelet regions of influence with cumulant-based approximations and localization regions. The two region types differ substantially, especially in their dependence on γ.

  • Figure 7 comparison: The actual λ-contours and second-order cumulant approximations are generally visually identical except for the outermost λ = 0.25 curve.Solid black lines show numerical contours, while dotted black lines show the approximations.
  • Figure 7 comparison: Figure 7 compares regions of influence across four γ values and five analyzing-wavelet β values using five λ levels.The plotted λ values are 0.25, 0.5, 0.75, 0.85, and 0.95; higher λ contours lie closer to each plot’s center.
  • Regions of influence: The influence curves form upward-pointing wedges or arrowheads that become more oval as λ increases and the contours shrink.Increasing β shifts the aspect ratio from frequency-elongated toward time-elongated when the element function is fixed.
  • Localization regions: Localization regions arise from limiting reconstruction integrals to regions in the time/frequency plane, producing an eigenvalue problem.Figure 7 compares these regions for area parameters Aβ,γ = 1, 10, 20, 40, and 100.
  • Localization regions: Localization regions depend strongly on γ: they point upward for γ < 2, downward for γ > 2, and are nearly symmetric for γ = 2.This behavior contrasts with the regions of influence and makes localization regions inappropriate indicators of influence regions.

Section S1. Moments and cumulants of generalized Morse wavelets

This section develops generalized Morse-wavelet moments and cumulants, derives transform expressions for Morse functions, and relates power-law-noise spectra and autocovariances to wavelet moments. It also states the integrability condition required for the noise-spectrum expressions.

  • Moments and cumulants: Generalized Morse-wavelet frequency moments provide the basis for expressions used throughout the analysis.A change of variables and the gamma-function definition yield the relevant moment forms.
  • Moments and cumulants: The zeroth-order moment equals the wavelet value at its temporal center and is used to set element-model coefficients.In the synthetic example, this relation gives |cn| = 5.39 from the chosen center value.
  • Moments and cumulants: Wavelet cumulants are obtained by equating the moment and cumulant expansions of the generalized Morse wavelet.The derivation retains terms through third order when relating the expansions.
  • Morse-function transforms: The transform of one rescaled Morse function by another is derived using the wavelet definition, rescaling, and a change of variables.The resulting expressions define ζβ,µ,γ(τ, s) and lead to the stated Morse-wavelet transform relation.
  • Power-law noise: For power-law noise, the expected wavelet spectrum is expressed through a function fα,β,γ and corresponding energy moments.The integral is finite only when β > α − 1/2, ensuring the wavelet decay overcomes the low-frequency noise singularity.
  • Autocovariance: The normalized wavelet-transform autocovariance is reduced to a wavelet expression using the noise spectrum, scaling law, and the wavelet-spectrum variance.The derivation concludes with Ξα,β,γ(0, s, 1) = σ²α,β,γ(s).

Supplementary figures for “Element analysis” by J. M. Lilly

The supplementary figures test element detection under red noise and in signals containing closely spaced weaker events. They compare normalized noise behavior, detection rates, and regions of influence around detected maxima.

  • α = 1 red noise is generated by cumulatively summing discrete Gaussian white noise, then standardizing and removing the mean.
  • Normalized red-noise curves differ only marginally from white-noise curves, mainly in event-density and detection-rate magnitudes rather than shapes.The y-axes differ from those in the corresponding white-noise figure.
  • Red-noise event magnitudes increase with scale or period, while wavelet-spectrum normalization makes red- and white-noise distributions qualitatively similar.For fixed-magnitude events, significance decreases toward larger scales in red noise.
  • The closely spaced-event test adds 110 weaker events at one-tenth the peak magnitude of six major events, with random phases and two frequencies.One hundred weaker events use ωρ = 2π/100 and ten use ωρ = 2π/1000, both uniformly spaced in time.
  • Detected maxima are distinguished by marker style, while λ = 1/2 and λ = 1/10 regions of influence assess neighborhoods around weaker and major detections.Gray curves mark λ = 1/2 regions; black dotted lines mark λ = 1/10 regions around large-amplitude maxima.
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