Package 'NLPwavelet'

Bayesian Wavelet Analysis Using Non-local Priors

Description

Performs Bayesian wavelet analysis using individual non-local priors as described in Sanyal & Ferreira (2017) <DOI:10.1007/s13571-016-0129-3> and non-local prior mixtures as described in Sanyal (2025) <DOI:10.48550/arXiv.2501.18134>.

Details

The main function is BNLPWA, which has arguments for specifying analysis using individual non-local priors or non-local prior mixtures and various hyperparameter specifications for the wavelet coefficients and scale parameters of the non-local priors. See the manual of BNLPWA for examples.

Author(s)

Nilotpal Sanyal <[email protected]>

Maintainer: Nilotpal Sanyal <[email protected]>

References

Sanyal, Nilotpal. "Nonlocal prior mixture-based Bayesian wavelet regression." arXiv preprint arXiv:2501.18134 (2025).

Sanyal, Nilotpal, and Marco AR Ferreira. "Bayesian wavelet analysis using nonlocal priors with an application to FMRI analysis." Sankhya B 79.2 (2017): 361-388.

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Bayesian Non-Local Prior-Based Wavelet Analysis

Description

BNLPWA is the main function of this package that performs Bayesian wavelet analysis using individual non-local priors as described in Sanyal & Ferreira (2017) and non-local prior mixtures as described in Sanyal (2025). It currently works with one-dimensional data. The usage is described below.

Usage

BNLPWA(
  data, 
  func=NULL, 
  method=c("mixture","mom","imom"), 
  mixprob_dist=c("logit","genlogit","hypsec","gennormal"), 
  scale_dist=c("polynom","doubleexp"),
  r=1, 
  nu=1, 
  wave.family="DaubLeAsymm", 
  filter.number=6, 
  bc="periodic"
)

Arguments

data	Vector of noisy data.
func	Vector of true functional values. NULL by default. If available, they are used to compute and return mean squared error (MSE) of the estimates.
method	"mixture" for non-local prior mixture-based analysis, "mom" or "imom" for individual non-local prior-based analysis.
mixprob_dist	Specification for the mixture probabilities of the spike-and-slab prior. Equations given in the Details.
scale_dist	Specification for the scale parameters of the non-local priors. Equations given in the Details.
r	Integer specifying a) the order of the MOM prior or the shape parameter of the IMOM prior for individual non-local prior-based analysis, or b) the order of the MOM prior for non-local prior mixture-based analysis.
nu	Integer specifying the shape parameter of the IMOM prior for non-local prior mixture-based analysis. Not used for individual non-local prior-based analysis.
wave.family	The family of wavelets to use - "DaubExPhase" or "DaubLeAsymm". Default is "DaubLeAsymm".
filter.number	The number of vanishing moments of the wavelet. Default is 6.
bc	The boundary condition to use - "periodic" or "symmetric". Default is "periodic".

Details

Spike-and-slab prior for the wavelet coefficients

For individual MOM prior-based analysis, the spike-and-slab prior for the wavelet coefficient $d_{lj}$ is given by

$d_{lj} \mid \gamma_l, \tau_l, \sigma^2, r \sim \gamma_l \; \text{MOM}(\tau_l, \sigma^2, r) + (1-\gamma_l) \; \delta(0),$

for individual IMOM prior-based analysis, the spike-and-slab prior for the wavelet coefficient $d_{lj}$ is given by

$d_{lj} \mid \gamma_l, \tau_l, \sigma^2, r \sim \gamma_l \; \text{IMOM}(\tau_l, \sigma^2, r) + (1-\gamma_l) \; \delta(0),$

and for non-local prior mixture-based analysis, the spike-and-slab prior for the wavelet coefficient $d_{lj}$ is given by

$d_{lj} \mid \gamma_l^{(1)}, \gamma_l^{(2)}, \tau_l^{(1)}, \tau_l^{(2)}, \sigma^2, r, \nu \sim \gamma_l^{(1)} \; \text{MOM}(\tau_l^{(1)}, r, \sigma^2) + (1-\gamma_l^{(1)})\gamma_l^{(2)} \;\text{IMOM}(\tau_l^{(2)}, \nu, \sigma^2) + (1-\gamma_l^{(1)})(1-\gamma_l^{(2)}) \; \delta(0),$

where the density of the MOM prior is

$mom(d_{lj} | \tau_{l}^{(1)},r,\sigma^{2}) = \widetilde{M}_{r} \left(\tau_{l}^{(1)}\sigma^{2}\right)^{-r-1/2} d_{lj}^{2r} \exp\left(-\frac{d_{lj}^{2}}{2\tau_{l}^{(1)}\sigma^{2}}\right), \quad r>1, \tau_{l}^{(1)}>0, \widetilde{M}_{r} = \frac{(2\pi)^{-1/2}}{(2r-1)!!}$

and the density of the IMOM prior is

$imom(d_{lj} | \tau_{l}^{(2)},\nu,\sigma^{2}) = \frac{\left(\tau_{l}^{(2)}\sigma^{2}\right)^{\nu/2}}{\Gamma(\nu/2)} |d_{lj}|^{-\nu-1} \exp\left( -\frac{\tau_{l}^{(2)}\sigma^{2}}{d_{lj}^{2}} \right),\quad \nu>1, \tau_{l}^{(2)}>0.$

Hyperparameter specifications

For non-local prior mixture-based analysis, the available specifications for the mixture probabilities are

1. Logit:

   $\gamma_l^{(1)} = \frac{\exp(\theta^{\gamma}_{1} - \theta^{\gamma}_{2}l)} {1 + \exp(\theta^{\gamma}_{1} - \theta^{\gamma}_{2}l)}, \quad \theta^{\gamma}_{1} \in \mathbb{R}, \; \theta^{\gamma}_{2} > 0$

   $\gamma_l^{(2)} = \frac{\exp(\theta^{\gamma}_{3} - \theta^{\gamma}_{4}l)} {1 + \exp(\theta^{\gamma}_{3} - \theta^{\gamma}_{4}l)}, \quad \theta^{\gamma}_{3} \in \mathbb{R}, \; \theta^{\gamma}_{4} > 0$

2. Generalized logit or Richards:

   $\gamma_l^{(1)} = \frac{1}{[1 + \exp\{-(\theta^{\gamma}_{1} - \theta^{\gamma}_{2}l)\}]^{\theta^{\gamma}_{3}}}, \quad \theta^{\gamma}_{1} \in \mathbb{R}, \; \theta^{\gamma}_{2},\theta^{\gamma}_{3} > 0$

   $\gamma_l^{(2)} = \frac{1}{[1 + \exp\{-(\theta^{\gamma}_{4} - \theta^{\gamma}_{5}l)\}]^{\theta^{\gamma}_{6}}}, \quad \theta^{\gamma}_{4} \in \mathbb{R}, \; \theta^{\gamma}_{5},\theta^{\gamma}_{6} > 0;$

3. Hyperbolic secant:

   $\gamma_l^{(1)} = \frac{2}{\pi} \arctan\left[\exp\left(\frac{\pi}{2} \left(\theta^{\gamma}_{1} - \theta^{\gamma}_{2}l\right)\right)\right], \quad \theta^{\gamma}_{1} \in \mathbb{R}, \; \theta^{\gamma}_{2} > 0$

   $\gamma_l^{(2)} = \frac{2}{\pi} \arctan\left[\exp\left(\frac{\pi}{2} \left(\theta^{\gamma}_{3} - \theta^{\gamma}_{4}l\right)\right)\right], \quad \theta^{\gamma}_{3} \in \mathbb{R}, \; \theta^{\gamma}_{4} > 0$

4. Generalized normal:

   $\gamma_l^{(1)} = \frac{1}{2} + \text{sign}(\theta^{\gamma}_{1}-l) \frac{1}{2\Gamma(1/\theta^{\gamma}_{2})} \gamma\left(1/\theta^{\gamma}_{2} ,\left|\frac{\theta^{\gamma}_{1}-l}{\theta^{\gamma}_{3}}\right|^{\theta^{\gamma}_{2}}\right), \quad \theta^{\gamma}_{1} \in \mathbb{R}, \; \theta^{\gamma}_{2},\theta^{\gamma}_{3} > 0$

   $\gamma_l^{(2)} = \frac{1}{2} + \text{sign}(\theta^{\gamma}_{4}-l) \frac{1}{2\Gamma(1/\theta^{\gamma}_{5})} \gamma\left(1/\theta^{\gamma}_{5} ,\left|\frac{\theta^{\gamma}_{4}-l}{\theta^{\gamma}_{6}}\right|^{\theta^{\gamma}_{5}}\right), \quad \theta^{\gamma}_{4} \in \mathbb{R}, \; \theta^{\gamma}_{5},\theta^{\gamma}_{6} > 0$

For individual non-local prior based analysis, $gamma_l$ is defined likewise.

For non-local prior mixture-based analysis, the available specifications for the scale parameters are

1. Polynomial decay:

   $\tau_{l}^{(1)} = \theta^{\tau}_{1} l^{-\theta^{\tau}_{2}}, \quad \theta^{\tau}_{1},\theta^{\tau}_{2} > 0$

   $\tau_{l}^{(2)} = \theta^{\tau}_{3} l^{-\theta^{\tau}_{4}}, \quad \theta^{\tau}_{3},\theta^{\tau}_{4} > 0$

2. Double-exponential decay:

   $\tau_{l}^{(1)} = \theta^{\tau}_{1} \exp(-\theta^{\tau}_{2} l) + \theta^{\tau}_{3} \exp(-\theta^{\tau}_{4} l), \quad \theta^{\tau}_{1},\theta^{\tau}_{2},\theta^{\tau}_{3},\theta^{\tau}_{4} > 0$

   $\tau_{l}^{(2)} = \theta^{\tau}_{5} \exp(-\theta^{\tau}_{6} l) + \theta^{\tau}_{7} \exp(-\theta^{\tau}_{8} l), \quad \theta^{\tau}_{5},\theta^{\tau}_{6},\theta^{\tau}_{7},\theta^{\tau}_{8} > 0$

For individual non-local prior based analysis, $tau_l$ is defined likewise.

Note: The wavelet computations are performed by using the R package wavethresh.

Value

A list containing the following.

data	The data vector.
func.post.mean	Posterior estimate (mean) of the function that generated the data.
wavelet.empirical	Empirical wavelet coefficients obtained by wavelet transformation of the data.
wavelet.post.mean	Posterior estimate (mean) of the true wavelet coefficients obtained by wavelet transformation of the underlying function.
hyperparam	Estimates of the hyperparameters that specify the spike-and-slab prior for the wavelet coefficients.
sigma	Estimate of the standard deviation of the error.
MSE.mean	Mean squared error of the estimates, computable only if true functional values are supplied in the input argument func.
runtime	System run-time of the function.

Author(s)

Nilotpal Sanyal <[email protected]>

Maintainer: Nilotpal Sanyal <[email protected]>

References

Sanyal, Nilotpal. "Nonlocal prior mixture-based Bayesian wavelet regression." arXiv preprint arXiv:2501.18134 (2025).

Sanyal, Nilotpal, and Marco AR Ferreira. "Bayesian wavelet analysis using nonlocal priors with an application to FMRI analysis." Sankhya B 79.2 (2017): 361-388.

See Also

wd, wr

Examples

# Using the well-known Doppler function to 
  # illustrate the use of the function BNLPWA

  # set seed for reproducibility
  set.seed(1)

  # Define the doppler function
  doppler <- function(x) { 
    sqrt(x * (1 - x)) * sin((2 * pi * 1.05) / (x + 0.05)) 
  }

  # Generate true values over a grid of length an integer power of 2 
  n <- 128  # Number of points
  x <- seq(0, 1, length.out = n)
  true_signal <- doppler(x) 

  # Add noise to generate data
  sigma <- 0.2  # Noise level
  y <- true_signal + rnorm(n, mean = 0, sd = sigma)

  # BNLPWA analysis based on MOM prior using logit specification
  # for the mixture probabilities and polynomial decay
  # specification for the scale parameter
  fit_mom <- BNLPWA(data=y, func=true_signal, r=1, wave.family=
    "DaubLeAsymm", filter.number=6, bc="periodic", method="mom", 
    mixprob_dist="logit", scale_dist="polynom")

  plot(y,type="l",col="grey") # plot of data
  lines(fit_mom$func.post.mean,col="blue") # plot of posterior 
  # smoothed estimates
  fit_mom$MSE.mean

  
    # BNLPWA analysis using non-local prior mixtures using generalized 
    # logit (Richard's) specification for the mixture probabilities and 
    # double exponential decay specification for the scale parameter
    fit_mixture <- BNLPWA(data=y, func=true_signal, r=1, nu=1, wave.family=
      "DaubLeAsymm", filter.number=6, bc="periodic", method="mixture", 
      mixprob_dist="genlogit", scale_dist="doubleexp")

    plot(y,type="l",col="grey") # plot of data
    lines(fit_mixture$func.post.mean,col="blue") # plot of posterior 
    # smoothed estimates
    fit_mixture$MSE.mean
  
  
  # Compare with other wavelet methods
  library(wavethresh)
  wd <- wd(y, family="DaubLeAsymm", filter.number=6, bc="periodic")  # Wavelet decomposition  
  
  wd_thresh_universal <- threshold(wd, policy="universal", type="hard")
  fit_universal <- wr(wd_thresh_universal)
  MSE_universal <- mean((true_signal-fit_universal)^2)
  MSE_universal

  wd_thresh_sure <- threshold(wd, policy="sure", type="soft")
  fit_sure <- wr(wd_thresh_sure)
  MSE_sure <- mean((true_signal-fit_sure)^2)
  MSE_sure

  wd_thresh_BayesThresh <- threshold(wd, policy="BayesThresh", type="hard")
  fit_BayesThresh <- wr(wd_thresh_BayesThresh)
  MSE_BayesThresh <- mean((true_signal-fit_BayesThresh)^2)
  MSE_BayesThresh

  wd_thresh_cv <- threshold(wd, policy="cv", type="hard")
  fit_cv <- wr(wd_thresh_cv)
  MSE_cv <- mean((true_signal-fit_cv)^2)
  MSE_cv

  wd_thresh_fdr <- threshold(wd, policy="fdr", type="hard")
  fit_fdr <- wr(wd_thresh_fdr)
  MSE_fdr <- mean((true_signal-fit_fdr)^2)
  MSE_fdr

  # Compare with non-wavelet methods
      # Kernel smoothing
  fit_ksmooth <- ksmooth(x, y, kernel="normal", bandwidth=0.05)
  MSE_ksmooth <- mean((true_signal-fit_ksmooth$y)^2)
  MSE_ksmooth
      # LOESS smoothing
  fit_loess <- loess(y ~ x, span=0.1)  # Adjust span for more or less smoothing
  MSE_loess <- mean((true_signal-predict(fit_loess))^2)
  MSE_loess
      # Cubic spline smoothing
  spline_fit <- smooth.spline(x, y, spar=0.5)  # Adjust spar for smoothness
  MSE_spline <- mean((true_signal-spline_fit$y)^2)
  MSE_spline

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