Simulation studies to compare bayesian wavelet shrinkage methods in aggregated functional data Alex Rodrigo dos S. Sousa

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Simulation studies to compare bayesian wavelet shrinkage methods

in aggregated functional data

Alex Rodrigo dos S. Sousa

State University of Campinas, Brazil

Abstract

The present work describes simulation studies to compare the performances of bayesian wavelet shrink-

age methods in estimating component curves from aggregated functional data. To do so, ﬁve methods

were considered: the bayesian shrinkage rule under logistic prior by Sousa (2020), bayesian shrinkage

rule under beta prior by Sousa et al. (2020), Large Posterior Mode method by Cutillo et al. (2008),

Amplitude-scale invariant Bayes Estimator by Figueiredo and Nowak (2001) and Bayesian Adaptive Mul-

tiresolution Smoother by Vidakovic and Ruggeri (2001). Further, the so called Donoho-Johnstone test

functions, Logit and SpaHet functions were considered as component functions. It was observed that the

signal to noise ratio of the data had impact on the performances of the methods.

1 Introduction

The statistical problem of estimating component curves from aggregated curves, also known as calibration

problem, has been widely studied in recent years in several areas of science. In Chemometrics, for example,

there is an interest in estimating absorbance curves of the constituents of a substance from samples of

absorbance curves of the substance itself. In this case, the substance’s absorbance curve is formed by the

linear combination of the absorbance curves of its constituents, according to the Beer-Lambert Law (Brereton,

2003). Another example appears in the study of electricity consumption in a given region, in which the energy

consumption curve over a period of time is composed of the aggregations of the individual consumption curves

of households and establishments (Dias et al., 2013).

The ﬁrst proposed methods to estimate component curves from aggregated data approach this problem in a

multivariate way, since, in practice, we observe such curves in a ﬁnite number of locations. Thus, it is possible

to see the observations as a random vector with a certain correlation structure between nearby locations.

Methods based on Principal Components Regression (PCR) by Cowe and McNicol (1985) and Partial Least

Squares Regression (PLS) by Wold et al. (1983) have been successfully proposed in applications. Bayesian

multivariate approaches was also proposed by Brown et al. (1998a,b) and Brown et al. (2001).

Methods based on functional data analysis to the calibration problem were proposed later by Dias et al.

arXiv:2210.04966v1 [stat.ME] 10 Oct 2022

(2009) and Dias et al. (2013), taking into account the functional structure of the observations. In this way,

each component curve can be represented in terms of some convenient function basis, such as splines basis for

example, so that the problem of estimating the curve becomes a ﬁnite-dimensional problem of estimating the

coeﬃcients of the basis expansion. In this functional approach, one possibility is to expand the functions in

wavelet basis, recently proposed by Sousa (2022). An advantage of such an approach is the well localization

property of the wavelet coeﬃcients and the possibility of estimating important characteristics of a function,

such as peaks, oscillations, discontinuities, among others, through wavelet coeﬃcients. Further, the sparsity of

the wavelet coeﬃcients allow to identify these function features by the magnitude of the nonzero coeﬃcients

at their localizations. See Vidakovic (1999) for details and properties of wavelets and its application on

statistical modelling.

Sousa (2022) proposes the expansion of the component curves by wavelet basis and, to estimate the

wavelet coeﬃcients, the use of a bayesian approach by considering, for each wavelet coeﬃcient, a prior

distribution composed of a point mass function at zero and the logistic distribution. Thus, the Bayesian rule

for estimating coeﬃcients under quadratic loss function assumption is the posterior expected value of the

wavelet coeﬃcient. In fact, the associated rule acts by shrinking empirical wavelet coeﬃcients and reducing

the present noise eﬀects on the coeﬃcients. This kind of estimator is also called shrinkage rules, see Donoho

and Johnstone (1994a,b and 1995) and Donoho (1993a,b and 1995a,b) for details about wavelet shrinkage

procedures. Further, in the work by Sousa (2022), a simulation study was carried out comparing the proposed

method with the expansion of the component curves by B-splines basis. It was concluded that the proposed

method performed better in terms of mean squared error than the method based on B-splines for curves with

characteristics such as discontinuities, peaks and oscillations.

Although the study by Sousa (2022) indicates the feasibility of applying the wavelet basis expansion and

the use of the Bayesian shrinkage rule under logistic prior to estimate the wavelet coeﬃcients of component

curves from aggregated curves, it would be interesting a study that compares diﬀerent wavelet shrinkage

methods in estimating component curves. In this sense, this work intends to compare bayesian methods of

wavelet shrinkage in the problem of estimating component curves from aggregated curves. For that, ﬁve

methods were considered, the rule based on the logistic prior of Sousa (2020 and 2022), the rule based on the

symmetric around zero beta prior proposed by Sousa et al. (2020), the Large Posterior Mode (LPM) method

by Cutillo et al. (2008), Amplitude-scale invariant Bayes Estimator (ABE) by Figueiredo and Nowak (2001)

and Bayesian Adaptive Multiresolution Smoother (BAMS) by Vidakovic and Ruggeri (2001).

This paper is organized as follows: the statistical model for de aggregated curves and the estimation

procedure of the component curves are deﬁned in Section 2. The descriptions of the considered bayesian

methods in the simulation studies are in Section 3. Section 4 is dedicated to the simulation studies results

and analysis. We ﬁnish with further considerations and discussions in Section 5.

2 Statistical model and estimation procedure

We consider a univariate function A(t)∈L2(R) = {f:R→R|Rf2<∞} that can be written as

A(t) =

l=1

ylαl(t) + e(t),(2.1)

where αl(t)∈L2(R) are unknown component functions, ylare known real valued weights, l= 1,··· , L,

and {e(t), t ∈R}is a zero mean gaussian process with unknown variance σ2,σ > 0. The estimation of the

functions αl(t) is considered in this work. In fact, this estimation process is usually done by multivariate

methods (Brown et al., 1998a,b) or functional data analysis (FDA) approach (Dias et al., 2009, 2013). In this

last point of view, each function αlof (2.1) is represented in terms of some functional basis such as splines,

B-splines or Fourier basis, for example. Here, we expand each component function by wavelet basis,

αl(t) = X

j,k∈Z

γ(l)

jk ψjk(t), l = 1,··· , L, (2.2)

where {ψjk(x) = 2j/2ψ(2jx−k), j, k ∈Z}is an orthonormal wavelet basis for L2(R) constructed by dilations

jand translations kof a function ψcalled wavelet or mother wavelet and γ(l)

jk ’s are unknown wavelet

coeﬃcients of the expansion of the component function αl. Note that we consider the same wavelet family for

all component functions expansion. Thus, the problem of estimating the functions αlbecomes a problem of

estimating the ﬁnite number of wavelet coeﬃcients γ(l)

jk ’s of the representation (2.2). Further, the magnitude

of the wavelet coeﬃcients allow to recover local features of the component functions, such as discontinuities

points, spikes and oscillations, due the well localization in time and frequency domain of wavelets. This

characteristic does not occur in spline-based and Fourier basis representations.

In practice, one observes Isamples of the aggregated curve A(t) at M= 2Jequally spaced locations

t1,··· , tM, i.e, our dataset is {(tm, Ai(tm)), m= 1,··· , M and i= 1,··· , I}. Thus, the discrete version of

(2.1) is

Ai(tm) =

l=1

yilαl(tm) + ei(tm), i = 1,··· , I, m = 1,··· , M = 2J,(2.3)

where ei(tm) are independent and identically normal distributed random noise with zero mean and variance

σ2,∀i, m. Further, the Isamples are obtained at the same locations t1,··· , tMbut the weights of the linear

combinations are allowed to be diﬀerent from one sample to another. We can rewrite (2.3) in matrix notation

A=αy +e,(2.4)

where A= (Ami =Ai(tm))1≤m≤M,1≤i≤I,α= (αml =αl(tm))1≤m≤M,1≤l≤L,y= (yli)1≤l≤L,1≤i≤Iand

e= (emi =ei(tm))1≤m≤M,1≤i≤I.

The wavelet shrinkage procedure will be made in the wavelet domain to estimate the coeﬃcients γ’s of

(2.2). For this reason, we apply a discrete wavelet transform (DWT) on the original aggregated data, which

can be represented by a M×Mwavelet transformation matrix W, and applied on both sides of (2.4), i.e,

W A =W(αy +e)

W A =W αy +W e

D=Γy+ε,(2.5)

where D=W A = (dmi)1≤m≤M,1≤i≤Iis the matrix with the empirical wavelet coeﬃcients of the aggregated

curves, Γ=W α = (γml)1≤m≤M,1≤l≤Lis the matrix with the unknown wavelet coeﬃcients of the component

curves and ε=W e = (εmi)1≤m≤M,1≤i≤Iis the matrix with the random errors on the wavelet domain, which

remain zero mean normal distributed with variance σ2due the orthogonality property of wavelet transforms.

Thus, for a particular empirical wavelet coeﬃcient dmi of D, one has the additive model

dmi =

l=1

yliγml +εmi =θmi +εmi,(2.6)

where θmi =PL

l=1 yliγml and εmi is zero mean normal with variance σ2, i.e, a single empirical wavelet

coeﬃcient of the aggregated curve is also a linear combination of the unknown wavelet coeﬃcients of the

component curves plus a random error. Moreover, the weights of this linear combination on wavelet domain

remain the same that the original combination of the curves at time domain.

The estimation of the wavelet coeﬃcients matrix Γin (2.5) is done by applying a wavelet shrinkage rule

δon each single empirical wavelet coeﬃcient d, obtaining the matrix δ(D)such that

δ(D)= (δ(dmi))1≤m≤M,1≤i≤I.(2.7)

We can see the matrix δ(D)as a denoising version of D, i.e, the shrinkage rule δ(d) acts by denoising

the empirical coeﬃcient din order to estimate θin (2.6), δ(d) = ˆ

θ. Thus, the estimation ˆ

Γof the wavelet

coeﬃcients matrix Γis given by least squares method,

Γ=δ(D)yt(yyt)−1,(2.8)

and ﬁnally αcan be estimated at locations t1,··· , tMby the inverse discrete wavelet transformation (IDWT),

ˆα=Wtˆ

Γ.(2.9)

For more details about wavelet shrinkage in aggregated curves, see Sousa (2022).

3 Bayesian methods

There are several available wavelet shrinkage methods in the literature. Most of them are thresholding,

i.e, the rule shrinks suﬃcient small empirical wavelet coeﬃcients as exactly zero. Two extremely applied

thresholding rules are the so called hard and soft rules proposed by Donoho and Johnstone (1994b). In this

work will be considered bayesian wavelet shrinkage methods that have been proposed in recent years. In

general, these methods propose a prior distribution to the wavelet coeﬃcients and estimate them according

to a loss function, as quadratic loss function for example, by the Bayes rule. The main advantage of bayesian

methods is the ability to incorporate prior information about the wavelet coeﬃcients, as sparsity, boundedness,

self similarity and others by convenient choices of prior distributions and their hyperparameters values.

In the next subsections, we brieﬂy describe the considered bayesian shrinkage rules in this work.

3.1 Shrinkage rule under logistic prior

The shrinkage rule under logistic prior was proposed by Sousa (2020). It assumes a mixture of a point

mass function at zero and a symmetric logistic distribution as prior to a single linear combination of wavelet

coeﬃcients θ,

π(θ;p, τ) = pδ0(θ) + (1 −p)g(θ;τ),(3.1)

where p∈(0,1), δ0(θ) is the point mass function at zero and g(θ;τ) is the logistic density function symmetric

around zero, for τ > 0,

g(θ;τ) = exp{−θ

τ}

τ(1 + exp{−θ

τ})2IR(θ).(3.2)

Under squared loss function, the associated bayesian shrinkage rule is the posterior expected value of θ,

Eπ(θ|d), that under the model prior (3.1), is given by (Sousa, 2020),

δ(d) = Eπ(θ|d) = (1 −p)RR(σu +d)g(σu +d;τ)φ(u)du

σφ(d

σ) + (1 −p)RRg(σu +d;τ)φ(u)du,(3.3)

where φ(·) is the standard normal density function. The shrinkage rule (3.3) under the model (3.2) is called

logistic shrinkage rule and has interesting features under estimation point of view. First, its hyperparameters

pand τcontrol the degree of shrinkage of the rule. Higher values of τor pimply higuer shrinkage level, i.e, the

rule will reduce severely the magnitudes of the empirical coeﬃcients. Further, as described in Sousa (2020),

the logistic shrinkage rule had good performances in terms of averaged mean squared error in simulation

studies against standard shrinkage or thresholding procedures.

3.2 Shrinkage rule under beta prior

Sousa et al. (2020) proposed the use of a mixture of a point mass function at zero and the beta distribution

with symmetric support around zero as a prior distribution to the wavelet coeﬃcients,

π(θ;p, a, m) = pδ0(θ) + (1 −p)g(θ;a, m),(3.4)

and

g(θ;a, m) = (m2−θ2)(a−1)

(2m)(2a−1)B(a, a)I[−m,m](θ),(3.5)

where B(·,·) is the standard beta function, a > 0 and m > 0 are the parameters of the distribution, and

I[−m,m](·) is an indicator function equal to 1 for its argument in the interval [−m, m] and 0 else.

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SimulationstudiestocomparebayesianwaveletshrinkagemethodsinaggregatedfunctionaldataAlexRodrigodosS.SousaStateUniversityofCampinas,BrazilAbstractThepresentworkdescribessimulationstudiestocomparetheperformancesofbayesianwaveletshrink-agemethodsinestimatingcomponentcurvesfromaggregatedfunctionaldata.To...

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