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Wavelet Selection Guide

This guide helps you choose the right wavelet for your specific application.

Quick Selection Matrix​

ApplicationRecommended WaveletsKey Considerations
General Signal AnalysisMorlet, DB4-DB8Balance between time and frequency resolution
Financial Time SeriesPaul, Morlet, Mexican HatAsymmetric patterns, volatility clustering
Image ProcessingDB2-DB4, Symlets, BiorCompact support, symmetry
Audio ProcessingMorlet, Shannon, DB8-DB16Frequency resolution, phase information
DenoisingDB4-DB8, Symlets, CoifletsSmoothness, vanishing moments
Edge DetectionMexican Hat (DOG2), DB2Zero crossings, sharp transitions
CompressionDB4-DB8, BiorEnergy compaction, perfect reconstruction
Scientific DataMorlet, Paul, DOGPhysical interpretation, continuous analysis
Smooth Signal ApproximationBLEM3-BLEM4, Coiflets, DB8+High regularity, polynomial approximation

Continuous Wavelets (CWT)​

Morlet Wavelet​

MorletWavelet wavelet = new MorletWavelet(); // Default: ω₀=6, σ=1
MorletWavelet custom = new MorletWavelet(5.0, 1.5); // Custom parameters

Best for:

  • General time-frequency analysis
  • Musical signal analysis
  • Geophysical data processing
  • Neurological signal analysis (EEG, MEG)

Characteristics:

  • Complex-valued (provides phase information)
  • Gaussian envelope with sinusoidal modulation
  • Excellent frequency localization
  • No perfect time localization due to infinite support

Parameters:

  • ω₀ (omega0): Center frequency (typically 5-6)
    • Higher values → better frequency resolution
    • Lower values → better time resolution
  • σ (sigma): Gaussian width (typically 1.0)

Paul Wavelet​

PaulWavelet wavelet = new PaulWavelet(4); // Order 4 (recommended)

Best for:

  • Financial market crash detection
  • Asymmetric pattern recognition
  • Sharp transient detection
  • Directional price movements

Characteristics:

  • Complex-valued
  • Asymmetric shape
  • Good for capturing one-sided features
  • Order parameter controls smoothness

Parameters:

  • m (order): Typically 1-20
    • m=4: Best balance for financial analysis
    • Higher m → smoother wavelet, better frequency localization

Mexican Hat (DOG2)​

DOGWavelet dog2 = new DOGWavelet(2); // Second derivative
MATLABMexicanHat mexhat = new MATLABMexicanHat(); // MATLAB-compatible

Best for:

  • Edge detection
  • Volatility analysis
  • Peak detection
  • Zero-crossing analysis

Characteristics:

  • Real-valued
  • Second derivative of Gaussian
  • Zero crossings at inflection points
  • Symmetric shape

MATLAB Compatibility:

  • Use MATLABMexicanHat for exact MATLAB reproduction
  • Different normalization (σ = 5/√8 instead of 1)

Derivative of Gaussian (DOG)​

DOGWavelet dog = new DOGWavelet(n); // n-th derivative

Best for:

  • Smooth feature extraction
  • Multi-scale edge detection
  • Pattern matching at different smoothness levels

Parameters:

  • n (order): Derivative order
    • n=1: First derivative (edge detection)
    • n=2: Mexican Hat (ridge detection)
    • n=3+: Smoother patterns

Shannon Wavelet​

ShannonWavelet wavelet = new ShannonWavelet();
ShannonWavelet custom = new ShannonWavelet(0.5, 1.5); // Custom band

Best for:

  • Band-limited signal analysis
  • Theoretical studies
  • Signals with known frequency bands
  • Communication systems

Characteristics:

  • Perfect frequency localization (rectangular in frequency)
  • Poor time localization (sinc function in time)
  • Ideal for strictly band-limited signals

Discrete Wavelets (DWT)​

Daubechies Wavelets​

OrthogonalWavelet db4 = Daubechies.DB4;
OrthogonalWavelet db8 = Daubechies.DB8;

Selection Guide:

  • DB2-DB4: Sharp features, edge detection, fast computation
  • DB4-DB8: General purpose, good balance
  • DB10-DB20: Smooth signals, high accuracy requirements

Vanishing Moments:

  • DBN has N vanishing moments
  • Removes polynomial trends up to degree N-1
  • Higher N → better frequency localization, longer filters

Symlets​

OrthogonalWavelet sym4 = Symlet.SYM4;
OrthogonalWavelet sym8 = Symlet.SYM8;

Best for:

  • Symmetric/near-symmetric signals
  • Image processing
  • Reduced phase distortion requirements

Characteristics:

  • Near-symmetric (least asymmetric)
  • Same vanishing moments as Daubechies
  • Better phase properties than Daubechies

Coiflets​

OrthogonalWavelet coif2 = Coiflet.COIF2;

Best for:

  • Numerical analysis
  • Signals with polynomial components
  • Applications requiring scaling function moments

Characteristics:

  • Both wavelet and scaling function have vanishing moments
  • More symmetric than Daubechies
  • Longer support for same number of vanishing moments

Biorthogonal Wavelets​

BiorthogonalWavelet bior = BiorthogonalSpline.BIOR2_2;

Best for:

  • Image compression (JPEG2000 uses BIOR4_4)
  • Perfect reconstruction with symmetric filters
  • Applications where phase linearity matters

Characteristics:

  • Symmetric (linear phase)
  • Perfect reconstruction
  • Different analysis and synthesis wavelets

Battle-Lemarié Wavelets (B-Spline)​

BattleLemarieWavelet blem3 = BattleLemarieWavelet.BLEM3; // Cubic (recommended)
BattleLemarieWavelet blem2 = BattleLemarieWavelet.BLEM2; // Quadratic

Best for:

  • Very smooth signal approximation
  • Numerical analysis with spline methods
  • Signals with polynomial-like behavior
  • Applications requiring high regularity

Characteristics:

  • Constructed from orthogonalized B-splines
  • m-1 continuous derivatives for order m
  • Excellent smoothness properties
  • Exponential decay in time domain

⚠️ Implementation Note: Current implementation uses approximations with:

  • Up to 6% reconstruction error (particularly BLEM3)
  • Relaxed orthogonality conditions
  • Good practical performance for smooth signals

Selection Guide:

  • BLEM1: Linear splines, piecewise linear approximation
  • BLEM2: Quadratic splines, C¹ continuity
  • BLEM3: Cubic splines, C² continuity (most common)
  • BLEM4: Quartic splines, C³ continuity
  • BLEM5: Quintic splines, C⁴ continuity (may have normalization issues)

Selection Criteria​

1. Time-Frequency Resolution Trade-off​

  • Need precise frequency? → Morlet with high ω₀, Shannon
  • Need precise timing? → Paul, DB2, Mexican Hat
  • Need balance? → Morlet (ω₀=6), DB4-DB8

2. Signal Characteristics​

  • Smooth signals → Battle-Lemarié (BLEM3-4), Higher-order wavelets (DB8+, Coif)
  • Sharp transients → Low-order wavelets (DB2-DB4, Paul)
  • Oscillatory → Morlet, Shannon
  • Polynomial trends → Battle-Lemarié, Coiflets, high-order Daubechies

3. Application Requirements​

  • Real-time → Orthogonal wavelets (Daubechies, Symlets)
  • Phase information needed → Complex wavelets (Morlet, Paul)
  • Perfect reconstruction → Any orthogonal or biorthogonal
  • Compression → Wavelets with good energy compaction (DB4-DB8)

4. Computational Constraints​

  • Fast computation → Low-order wavelets, dyadic scales
  • Memory limited → Compact support wavelets
  • FFT available → CWT with any wavelet
  • Streaming → Orthogonal wavelets with small support

Financial Applications Special Considerations​

Market Crash Detection​

// Paul wavelet for asymmetric patterns in financial data
PaulWavelet paul = new PaulWavelet(4);
CWTTransform cwt = new CWTTransform(paul);
double[] scales = {1.0, 2.0, 4.0, 8.0};
CWTResult result = cwt.analyze(priceData, scales);

Volatility Analysis​

// Mexican Hat for volatility clustering
DOGWavelet dog2 = new DOGWavelet(2);
CWTTransform cwt = new CWTTransform(dog2);
// Focus on scales corresponding to daily/weekly patterns

Trend Analysis​

// Morlet for multi-scale trends
MorletWavelet morlet = new MorletWavelet(6.0, 1.0);
// Use logarithmic scales for multi-timeframe analysis

Performance Tips​

  1. For large signals (>10k samples):

    • Use FFT-accelerated CWT
    • Consider dyadic scales for efficiency
    • Use signal-adaptive scale selection

  2. For real-time processing:

    • Use orthogonal wavelets (DWT)
    • Keep wavelet support small (DB4-DB8)
    • Pre-compute wavelet values if possible

  3. For highest accuracy:

    • Use higher-order wavelets
    • Increase scale density
    • Consider complex wavelets for phase

Common Pitfalls​

  1. Using Shannon wavelet on non-band-limited signals

    • Causes severe artifacts
    • Use Morlet instead for general signals

  2. Too few scales in CWT

    • Misses important features
    • Use adaptive scale selection

  3. Wrong boundary handling

    • Periodic for non-periodic signals causes artifacts
    • Use symmetric or zero-padding for most signals

  4. Ignoring phase information

    • Missing important signal relationships
    • Use complex wavelets when phase matters

Code Examples​

Automatic Wavelet Selection​

// For unknown signal characteristics
SignalAdaptiveScaleSelector selector = new SignalAdaptiveScaleSelector();
MorletWavelet wavelet = new MorletWavelet(); // Good general choice
double[] scales = selector.selectScales(signal, wavelet, samplingRate);

Multi-Wavelet Analysis​

// Compare different wavelets
ContinuousWavelet[] wavelets = {
    new MorletWavelet(),
    new PaulWavelet(4),
    new DOGWavelet(2)
};

for (ContinuousWavelet w : wavelets) {
    CWTTransform cwt = new CWTTransform(w);
    CWTResult result = cwt.analyze(signal, scales);
    // Compare results...
}

Optimal Parameter Selection​

// Grid search for best Morlet parameters
double bestOmega = 6.0;
double bestSigma = 1.0;
double minError = Double.MAX_VALUE;

for (double omega = 4.0; omega <= 8.0; omega += 0.5) {
    for (double sigma = 0.5; sigma <= 2.0; sigma += 0.25) {
        MorletWavelet wavelet = new MorletWavelet(omega, sigma);
        // Evaluate performance...
    }
}

See Also​