Package 'WSwavelet'

WSwavelet: Bayesian Wavelet Denoising with Wendland-Semicircle Slab Mixture

Description

Bayesian wavelet denoising with a resolution-adaptive spike-and-slab prior. The continuous slab is a mixture of a compactly supported Wendland-type density and the semicircle density. The package provides Gaussian and Laplace working-likelihood versions, empirical-Bayes fitting of resolution-specific slab weights, robust noise-scale estimation, posterior-mean coefficient shrinkage, inverse-DWT reconstruction, and diagnostic tools.

Details

The primary user-facing function is wswavelet. It requires a finite numeric vector whose length is a dyadic integer, such as 128 or 1024. The function applies an orthogonal discrete wavelet transform using wavethresh, retains the scaling coefficients, shrinks detail coefficients, and reconstructs the denoised signal.

The prior for a detail coefficient at level $j$ is

$\theta_{j,k} \sim \pi_j\delta_0 + (1-\pi_j)\{\omega_j g_W(\theta;\beta_j) +(1-\omega_j)g_S(\theta;\beta_j)\}.$

Here $\pi_j$ is the spike probability, $\omega_j$ is the conditional Wendland weight among nonzero coefficients, and $\beta_j$ is the common support scale.

See wswavelet for the empirical-Bayes fitting details and level_posterior for the coefficient-level posterior calculation.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

wswavelet, level_posterior, evaluate_shrinkage_curve

---

Robust Wavelet-Based Noise-Scale Estimation

Description

Estimates the observational noise standard deviation from high-frequency wavelet coefficients using a robust median absolute deviation rule.

Usage

estimate_noise_sd(
  wavelet_object,
  detail_levels,
  supplied_sd = NULL,
  mad_levels = 1L
)

Arguments

wavelet_object	A wavelet object returned by wd.
detail_levels	A nonempty numeric or integer vector of detail levels available in wavelet_object. The finest levels are selected.
supplied_sd	Optional known positive noise standard deviation. If supplied, it is returned directly and the MAD calculation is skipped.
mad_levels	The positive integer number of finest detail levels to pool when estimating the scale. The default is 1.

Details

Unless supplied_sd is provided, the function computes

$\widehat{\sigma} = \frac{\operatorname{median}(|d-\operatorname{median}(d)|)}{0.6745}$

using the finest mad_levels detail levels. If the resulting value is not positive and finite, the function falls back to pooling up to the three finest available detail levels. A strictly positive machine-scale value is returned in the remaining degenerate case.

Value

A single positive numeric value giving the supplied or estimated noise standard deviation.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

wswavelet, wd

Examples

set.seed(1)
y <- rnorm(128)
wd <- wavethresh::wd(y, filter.number = 6L, bc = "periodic")
lev <- 0:(wavethresh::nlevelsWT(wd) - 1L)
estimate_noise_sd(wd, lev)

---

Evaluate a Fitted Wavelet Shrinkage Curve

Description

Evaluates the fitted posterior-mean shrinkage rule and posterior component probabilities over a grid of hypothetical observed coefficients at one detail level.

Usage

evaluate_shrinkage_curve(
  fit,
  level_index = 1L,
  d_grid = seq(-4, 4, length.out = 401L)
)

Arguments

fit	A fitted object returned by wswavelet.
level_index	The position of the detail level in fit$detail, beginning at 1. This is the position in the fitted list, not necessarily the numerical wavelet level.
d_grid	A numeric vector of hypothetical observed coefficient values at which the shrinkage rule is evaluated.

Details

For each value in d_grid, the function recomputes the posterior for the selected level using the fitted $\pi_j$, $\omega_j$, $\beta_j$, and likelihood. The resulting curve is the empirical-Bayes version of the coefficientwise posterior-mean rule

$\delta_j(d)=p_{W,j}(d)\mu_{W,j}(d) +p_{S,j}(d)\mu_{S,j}(d).$

The output can be used to plot attenuation, compare the rule with the identity map $\delta(d)=d$, or inspect how posterior evidence for the spike changes with the observed coefficient.

Value

A data frame with columns d, estimate, posterior_spike, posterior_wendland, and posterior_semicircle.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

wswavelet, level_posterior

Examples

set.seed(1)
n <- 128
x <- seq(0, 1, length.out = n)
y <- sin(6 * pi * x) + rnorm(n, sd = 0.5)
fit <- wswavelet(
  y, filter.number = 6L, quadrature_n = 24L
)
curve <- evaluate_shrinkage_curve(fit, level_index = 1L)
head(curve)

---

Posterior Probabilities and Means for One Wavelet Level

Description

Computes the complete three-component posterior for a vector of observed wavelet coefficients at one resolution level.

Usage

level_posterior(
  d,
  pi_j,
  omega_j,
  beta_j,
  sigma,
  likelihood,
  laplace_rate,
  quadrature_n,
  use_exact_wendland
)

Arguments

d	A numeric vector of observed wavelet coefficients. The calculation is vectorized, with one posterior calculation for each element.
pi_j	The prior probability of the point-mass spike at zero for the resolution level. It must lie in $[0,1]$.
omega_j	The conditional prior probability of the Wendland slab among nonzero coefficients. It must lie in $[0,1]$.
beta_j	The positive common support scale for the two continuous slabs.
sigma	The positive Gaussian noise standard deviation. It is used for the Gaussian likelihood and for Gaussian quadrature calculations.
likelihood	Either "gaussian" or "laplace".
laplace_rate	The positive rate $\lambda$ used by the Laplace working likelihood. It is ignored when likelihood = "gaussian", but should still be supplied.
quadrature_n	The number of Gauss-Legendre nodes used for numerical integration.
use_exact_wendland	Logical value indicating whether the exact finite-sum Wendland formula should be used for Laplace moments.

Details

The continuous slab is

$g_j(\theta)=\omega_j g_W(\theta;\beta_j) +(1-\omega_j)g_S(\theta;\beta_j).$

For component $c\in\{W,S\}$, define

$M_{r,c,j}(d)=\int_{-\beta_j}^{\beta_j} \theta^r g_c(\theta;\beta_j)L(d\mid\theta)\,d\theta, \qquad r\in\{0,1\}.$

The posterior component probabilities are

$p_{0,j}(d)=\frac{\pi_jL(d\mid0)}{D_j(d)},\quad p_{W,j}(d)=\frac{(1-\pi_j)\omega_jM_{0,W,j}(d)}{D_j(d)},\quad p_{S,j}(d)=\frac{(1-\pi_j)(1-\omega_j)M_{0,S,j}(d)}{D_j(d)},$

where

$D_j(d)=\pi_jL(d\mid0)+(1-\pi_j) \{\omega_jM_{0,W,j}(d)+(1-\omega_j)M_{0,S,j}(d)\}.$

The returned posterior mean is

$\delta_j(d)=p_{W,j}(d)\mu_{W,j}(d) +p_{S,j}(d)\mu_{S,j}(d), \qquad \mu_{c,j}(d)=M_{1,c,j}(d)/M_{0,c,j}(d).$

The spike contributes zero to this posterior mean. Computations are performed on the log scale where appropriate to reduce numerical underflow. Endpoint cases such as $\pi_j=1$ or $\omega_j=0$ are handled explicitly.

Value

A list with one row or one entry per value of d:

log_marginal

The log marginal density $\log D_j(d)$.

probability

A matrix with columns spike, wendland, and semicircle, containing posterior component probabilities.

posterior_mean

The posterior-mean shrinkage estimate $\delta_j(d)$.

component_mean

A matrix containing the posterior means conditional on the Wendland and semicircle components.

component_log_m0

A matrix containing the log zeroth moments for the two continuous components.

quadrature_fallbacks

The number of Wendland coefficients for which the exact Laplace calculation used numerical fallback.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

wswavelet, evaluate_shrinkage_curve, wendland_kernel, semicircle_kernel

Examples

d <- c(-1, 0, 1)
post <- level_posterior(
  d = d,
  pi_j = 0.75,
  omega_j = 0.5,
  beta_j = 2,
  sigma = 1,
  likelihood = "gaussian",
  laplace_rate = 0.5,
  quadrature_n = 24L,
  use_exact_wendland = TRUE
)
post$posterior_mean
post$probability

---

Standardized Semicircle Slab Density

Description

Evaluates the normalized semicircle slab density on the standardized interval $[-1,1]$.

Usage

semicircle_kernel(u)

Arguments

u	A numeric vector of standardized coefficient values.

Details

The density is

$K_S(u)=\frac{2}{\pi}\sqrt{1-u^2}I(|u|<1).$

It is symmetric, integrates to one, and is more dispersed than the Wendland component used in the package. For a support scale $\beta>0$, the corresponding density for a coefficient $\theta$ is $g_S(\theta;\beta)=\beta^{-1}K_S(\theta/\beta)$.

Value

A numeric vector of the same length as u, containing the density values.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

wendland_kernel, level_posterior

Examples

u <- seq(-1.2, 1.2, length.out = 7)
semicircle_kernel(u)

---

Standardized Wendland-Type Slab Density

Description

Evaluates the normalized compactly supported Wendland-type slab density on the standardized interval $[-1,1]$.

Usage

wendland_kernel(u)

Arguments

u	A numeric vector of standardized coefficient values.

Details

The density is

$K_W(u)=\frac{3}{2}(1-|u|)^4(1+4|u|) I(|u|<1).$

It is symmetric, integrates to one, is concentrated near zero, and is exactly zero outside the open unit interval. For a support scale $\beta>0$, the corresponding density for a coefficient $\theta$ is $g_W(\theta;\beta)=\beta^{-1}K_W(\theta/\beta)$.

Value

A numeric vector of the same length as u, containing the density values.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

semicircle_kernel, level_posterior

Examples

u <- seq(-1.2, 1.2, length.out = 7)
wendland_kernel(u)

---

Numerical Checks for the WS Posterior-Mean Rule

Description

Runs a standard collection of numerical checks for endpoint handling, symmetry, support bounds, finiteness, and Gaussian monotonicity of the posterior-mean shrinkage rule.

Usage

ws_numerical_checks(quadrature_n = 128L)

Arguments

quadrature_n

The number of Gauss-Legendre nodes used in the numerical integrations. Larger values can improve accuracy but increase computation time.

Details

The routine evaluates Gaussian and Laplace posterior means over a fixed coefficient grid for all combinations of spike probability $\pi\in\{0,0.5,1\}$ and Wendland weight $\omega\in\{0,0.5,1\}$. It checks the numerical versions of

$\delta(-d)=-\delta(d)$

and

$|\delta(d)|\leq\beta$

. For the Gaussian likelihood, min_first_difference checks the expected nondecreasing behavior on the grid. The endpoint columns verify the exact no-spike and all-spike rules.

Value

A data frame with one row for each likelihood, spike probability, and slab weight combination. Important columns are max_oddness_error, min_first_difference, max_support_excess, all_finite, pi0_exact_slab, and pi1_exact_zero.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

level_posterior, wswavelet

Examples

checks <- ws_numerical_checks(quadrature_n = 48L)
checks

---

Resolution-Adaptive Bayesian Wavelet Denoising

Description

Fits the resolution-adaptive Wendland-Semicircle Bayesian wavelet denoiser, shrinks detail coefficients by posterior means, and reconstructs the signal with the inverse discrete wavelet transform.

Usage

wswavelet(
  y,
  filter.number = 10L,
  family = "DaubExPhase",
  bc = "periodic",
  j0 = 0L,
  likelihood = c("gaussian", "laplace"),
  supplied_sd = NULL,
  spike_offset = 2,
  spike_gamma = 2.4,
  beta_quantile = 0.99,
  beta_quantile_type = 8L,
  beta_floor = 1e-8,
  omega_model = c("adaptive", "constant", "fixed"),
  fixed_omega = NULL,
  eta_start = c(0, 0),
  eta_bounds = c(-12, 12),
  laplace_rate = NULL,
  quadrature_n = 48L,
  use_exact_wendland = TRUE,
  mad_levels = 1L,
  return_wavelet = FALSE
)

Arguments

y	A finite numeric signal vector. Its length must be a dyadic integer, such as 128, 256, or 1024.
filter.number	The wavelet filter order passed to wd. The default is 10.
family	The wavelet family passed to wd. Common choices include "DaubExPhase" and "DaubLeAsymm".
bc	The boundary convention passed to wd. The default is "periodic".
j0	The coarsest detail level to shrink. Detail levels below j0 are not shrunk. The default is 0.
likelihood	The coefficientwise working likelihood. Use "gaussian" for the primary version or "laplace" for the robustness-oriented alternative.
supplied_sd	Optional known noise standard deviation. If NULL, the function estimates it from high-frequency wavelet coefficients using a robust MAD rule.
spike_offset	The offset $\ell$ in the level-dependent spike probability. The implementation permits values at least 1; the default is 2.
spike_gamma	The positive resolution-sparsity exponent $\gamma$. Larger values make the spike probability increase more rapidly with resolution.
beta_quantile	The probability $\kappa$ used in the empirical upper-quantile rule for the support scale $\beta_j$. It must lie strictly between 0 and 1; the default is 0.99.
beta_quantile_type	The integer quantile convention passed to quantile. The default is type 8.
beta_floor	A positive numerical floor used to prevent a zero support scale when the noise estimate and coefficients are numerically zero.
omega_model	How the Wendland weight is modeled across resolution levels. "adaptive" estimates an intercept and a resolution trend, "constant" estimates one common weight, and "fixed" uses fixed_omega.
fixed_omega	A fixed Wendland weight in $[0,1]$. Supplying this argument forces omega_model = "fixed"; 0 gives the semicircle-only endpoint and 1 gives the Wendland-only endpoint.
eta_start	Starting values for the empirical-Bayes logistic trend parameters $(\eta_0,\eta_1)$. For a constant mixture, only the first value is used.
eta_bounds	Common finite lower and upper bounds for the optimized logistic parameters. The default is c(-12, 12).
laplace_rate	Optional positive rate parameter for the Laplace working likelihood. If NULL, the function sets $\widehat{\lambda}=1/(2\widehat{\sigma}^2)$ once before optimization and holds it fixed. This argument has no effect for the Gaussian likelihood.
quadrature_n	Number of Gauss-Legendre nodes used for numerical integration. The default is 48. Larger values may improve accuracy but increase computation time.
use_exact_wendland	Logical value. For the Laplace likelihood, TRUE uses the finite-sum Wendland moment formula with a numerical fallback. If FALSE, numerical quadrature is used. For the Gaussian likelihood, numerical quadrature is used.
mad_levels	Number of finest detail levels pooled for the robust noise-scale estimate. The default is 1.
return_wavelet	Logical value. If TRUE, also returns the original and thresholded wavethresh wavelet objects.

Details

The prior for a detail coefficient at resolution level $j$ is

$\theta_{j,k} \sim \pi_j\delta_0 + (1-\pi_j)\{\omega_j g_W(\theta;\beta_j) +(1-\omega_j)g_S(\theta;\beta_j)\}.$

The spike probability is

$\pi_j = 1-(j-j_0+\ell)^{-\gamma}.$

For the adaptive model, the Wendland weight is

$\omega_j(\eta)=\operatorname{logit}^{-1}(\eta_0+\eta_1 t_j), \qquad t_j=(j-j_0)/(J-1-j_0).$

The support scale is selected from the observed detail coefficients using

$\widehat{\beta}_j = \max\{\widehat{\sigma},Q_{\kappa}(|d_{j,k}|)\},$

followed by the positive numerical floor

$\beta_{\min}=\epsilon_\beta s_\beta,\qquad s_\beta=\max\{1,\widehat{\sigma},\max_{j,k}|d_{j,k}|\},\qquad \widehat{\beta}_j\leftarrow\max\{\widehat{\beta}_j,\beta_{\min}\},$

where $\epsilon_\beta$ is beta_floor.

When omega_model = "constant", the fitted Wendland weight is $\operatorname{logit}^{-1}(\eta_0)$ at every level. When omega_model = "fixed", no weight optimization is performed.

For Gaussian errors, the coefficientwise likelihood is

$L_N(d\mid\theta,\sigma^2)= (2\pi\sigma^2)^{-1/2} \exp\{-(d-\theta)^2/(2\sigma^2)\}.$

For the Laplace working likelihood, the implementation uses

$L_L(d\mid\theta,\lambda)=(a/2)\exp\{-a|d-\theta|\}, \qquad a=\sqrt{2\lambda}.$

The empirical-Bayes parameters are fitted by bounded L-BFGS-B optimization of the joint log marginal likelihood. The posterior mean replaces each detail coefficient, while the scaling coefficients are retained without shrinkage.

Value

A list with the following principal components:

estimate

The reconstructed denoised signal.

sigma_hat

The estimated or supplied noise standard deviation.

laplace_rate_hat

The rate used for the Laplace working likelihood.

likelihood

The selected likelihood.

omega_model

The selected mixture-weight model.

eta_hat

The fitted logistic-trend parameters, padded to length two for the constant and fixed cases.

optimization

The optimization result or fixed-weight record.

level_summary

A data frame containing level-specific support scales, prior probabilities, fitted weights, average posterior component probabilities, and numerical diagnostics.

detail

A list containing the observed and fitted coefficients and posterior quantities for each detail level.

original

The input signal.

diagnostics

Quadrature, fallback, non-finite-integral, quantile-type, and MAD-level information.

settings

The principal fitting settings.

If return_wavelet = TRUE, the list also contains wavelet_object and thresholded_wavelet.

Author(s)

Nilotpal Sanyal

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

See Also

level_posterior, evaluate_shrinkage_curve, estimate_noise_sd, wd

Examples

set.seed(1)
n <- 128
x <- seq(0, 1, length.out = n)
y <- sin(6 * pi * x) + rnorm(n, sd = 0.5)

fit <- wswavelet(
  y,
  likelihood = "gaussian",
  filter.number = 6L,
  quadrature_n = 24L
)
fit$estimate
fit$level_summary

---