SignalY (version 1.1.1)

filter_wavelet: Wavelet Multiresolution Analysis Filter

Description

Performs wavelet-based signal decomposition using the Maximal Overlap Discrete Wavelet Transform (MODWT) to extract trend components at specified frequency scales. This method decomposes the signal into detail coefficients (D1, D2, ..., DJ) capturing progressively lower frequencies and a smooth coefficient (SJ) representing the underlying trend.

Usage

filter_wavelet(
  y,
  wf = "la8",
  J = 4,
  boundary = "periodic",
  levels_to_combine = c(3, 4),
  first_difference = FALSE,
  verbose = FALSE
)

Value

A list of class "signaly_wavelet" containing:

trend

Numeric vector of the extracted trend component

mra

Full multiresolution analysis object from waveslim::mra

detail_levels

Data frame with all detail level coefficients

smooth_level

Vector of the smooth (SJ) coefficients

combined_levels

Character string indicating which levels were combined

settings

List of parameters used in the analysis

diagnostics

List with variance decomposition and energy distribution

Arguments

y

Numeric vector of the time series to decompose. Length must be at least 2^J.

wf

Character string specifying the wavelet filter. Options include "la8" (least asymmetric with 8 vanishing moments, 16 coefficients), "la16", "la20", "haar", "d4", "d6", "d8", etc. Default is "la8".

J

Integer specifying the decomposition depth (number of levels). Default is 4, yielding D1-D4 detail levels plus S4 smooth level.

boundary

Character string specifying boundary handling: "periodic" (default) or "reflection".

levels_to_combine

Integer vector specifying which detail levels to combine for the trend estimate. Default is c(3, 4) for D3+D4.

first_difference

Logical. If TRUE, applies wavelet to first differences and reconstructs via cumulative sum. Default is FALSE.

verbose

Logical indicating whether to print diagnostic messages.

Frequency Interpretation

For a series with unit sampling interval, the detail levels correspond to approximate frequency bands:

• D1: periods 2-4 (highest frequency noise)

• D2: periods 4-8 (short-term fluctuations)

• D3: periods 8-16 (medium-term cycles)

• D4: periods 16-32 (longer cycles)

• S4: periods > 32 (smooth trend)

For annual economic data, D3+D4 typically captures business cycle dynamics (8-32 year periods), while D1+D2 captures short-term noise.

Details

The MODWT (Maximal Overlap Discrete Wavelet Transform) is preferred over the classical DWT for several reasons relevant to signal extraction:

1. Translation invariance: Unlike DWT, MODWT does not depend on the starting point of the series, producing consistent results regardless of circular shifts.

2. Any sample size: MODWT can be applied to series of any length, not just powers of 2.

3. Additive decomposition: The MRA (multiresolution analysis) coefficients sum exactly to the original series.

The choice of wavelet filter affects the trade-off between time and frequency localization:

• la8 (Daubechies least asymmetric, 8 vanishing moments): Good balance of smoothness and localization, recommended for economic data.

• Higher order (la16, la20): Better frequency resolution at cost of temporal smearing.

• haar: Maximum time localization but poor frequency resolution.

References

Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM.

Percival, D. B., & Walden, A. T. (2000). Wavelet Methods for Time Series Analysis. Cambridge University Press.

Gencay, R., Selcuk, F., & Whitcher, B. (2002). An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Academic Press.

See Also

mra, filter_emd, filter_hpgc

Examples

set.seed(123)
y <- cumsum(rnorm(100)) + sin(seq(0, 4*pi, length.out = 100))
result <- filter_wavelet(y, wf = "la8", J = 4)
plot(y, type = "l", col = "gray")
lines(result$trend, col = "red", lwd = 2)