Multi-scale variance

dwt_multilevel separates a signal into one approximation plus N detail bands, each concentrating variance from a distinct frequency octave. The band variances sum to the signal's total variance (Parseval), giving a clean attribution by scale.

When to use it

• You want to know how much of your realized volatility lives at intraday vs daily vs weekly scale.
• You want a compact, interpretable multi-scale feature vector per window for a downstream model.
• You want an exact attribution — band variances that sum to the total by Parseval, not an approximation.

Example

use ferro_wave::{Signal, Daubechies, DaubechiesType,
    BoundaryMode, dwt_multilevel};
 
let signal  = Signal::from_slice(&log_returns);
let wavelet = Daubechies::new(DaubechiesType::Db4);
 
let decomp = dwt_multilevel(
    &signal, &wavelet, 5,
    BoundaryMode::Periodic,
)?;
 
// Variance per band — sums to signal.variance()
let bands: Vec<f64> = decomp.details.iter()
    .map(|d| variance(d))
    .collect();
 
// D1 ≈ intraday · D2-3 ≈ daily · D4-5 ≈ weekly+
# Ok::<(), ferro_wave::WaveletError>(())

Notes

• Each detail band Dj concentrates variance from one dyadic frequency octave; the approximation A5 holds the residual trend.
• For a shift-invariant version whose levels stay the input length, swap in modwt_multilevel and use the wavelet-variance energy identity.