Multivariate denoising using wavelets and principal component analysis
Introduction
On one hand, denoising algorithms based on wavelet decompositions are a popular method for one-dimensional statistical signal extraction and filtering. On the other hand, principal component analysis (PCA) is among the most notorious data-analysis tools designed to simplify multidimensional data by tracking new factors supposed to capture the main features.
This paper proposes a multivariate extension of wavelet denoising procedures, combining a straightforward multivariate generalization of the classical one for scalar valued signals and principal component analysis. This proposal takes place among the various recent approaches combining wavelet strategies and data analytic tools to cope with the problem of feature extraction in regression models. Numerous applied situations strongly motivate this interest. Let us mention some of them together with some references focusing on a wavelet-data analysis approach: spectral calibration problems (Vannucci et al., 2003), multivariate statistical process control (see Bakshi, 1998; Teppola and Minkkinen, 2000), blind source separation (Roberts et al., 2004), functional magnetic resonance imaging (fMRI) analysis (Meyer and Chinrungrueng, 2003), spike detection and sorting (Oweiss and Anderson, 2001).
This paper focuses on multivariate wavelet denoising and deals with regression models of the form , where the observation X is p-dimensional, F is deterministic and is the signal to be recovered and is a spatially correlated noise. This kind of model is well suited for situations for which such an additive spatially correlated noise is realistic. For example, a longitudinal study on p subjects, the analysis of a part a fMRI region (involving p voxels) or the noise reduction in multichannel neural recordings (using p channels). Let us be a little bit more precise on this last example which is chosen to illustrate the behavior of the proposed procedure on a real world data set, at the end of the paper. Following Oweiss and Anderson (2001), extra-cellular neural recordings can be modeled as an invariant deterministic signal and an additive noise which obscures neural discharges from cells of interest. This noise contains a component exhibiting spatial correlation coming from background activity caused by neural cells.
To close this introduction, let us recall some facts about classical univariate wavelet denoising dealing both with signal processing and functional estimation in statistics and which is of interest in various applied fields. Valuable references are the books (Mallat, 1998, Percival and Walden, 2000, Vidakovic, 1999) and the survey paper (Antoniadis, 1997). For basics on wavelets, we refer the reader to Mallat (1998) or Misiti et al. (2003) for example.
The simplest considered model is of the following form:where is observed, is a centered Gaussian white noise of unknown variance and f is an unknown function to be recovered through the observations.
For a given orthogonal wavelet basis denoted by where is a wavelet, the associated scaling function, J a suitably chosen decomposition level and where , wavelet denoising proceeds in three steps:
Step 1: Compute the wavelet decomposition of the observed signal up to level J;
Step 2: Threshold conveniently the wavelet detail coefficients;
Step 3: Reconstruct a denoised version of the original signal, from the thresholded detail coefficients and the approximation coefficients, using the inverse wavelet transform.
This paper dedicated to a multivariate denoising procedure that takes into account the correlation structure of the noise, is organized in two main sections. Section 2 proposes a first denoising procedure which is a direct generalization of the one-dimensional strategy. The method is based on a change of basis followed by a classical one-dimensional soft-thresholding. This new procedure exhibits promising behavior on classical test signals and the associated estimator is found to be near minimax in the one-dimensional sense, for Besov balls. The change of basis is obtained from the diagonalization of a robust estimate of the noise covariance matrix given by the minimum covariance determinant estimator based on the matrix of finest details.
Section 3 first recalls the multiscale PCA proposed by Bakshi (1998) for statistical process control purposes. This scheme is discussed and then a second denoising procedure combining wavelets and PCA is proposed. The introduction of a PCA step try to take advantage of the deterministic relationships between the signals, leading to an additional denoising effect. It is then illustrated by some simulation examples and by an application to multichannel neural recordings.
Section snippets
Procedure
Let us consider the following p-dimensional model:where are of size and is a centered Gaussian white noise with unknown covariance matrix . Each component of is of the previous form (1), for : where belongs to some functional space (typically or Besov spaces).
The covariance matrix , supposed to be positive definite, captures the stochastic link between the components of and models
Multivariate wavelet denoising using PCA
The multivariate procedure previously examined can be generalized by looking at the deterministic relationships between the p signals. The idea is to use principal component analysis, not to discover new variables which could be of interest, but to kill unsignificant principal components to obtain an additional denoising effect.
In this section, we first recall the multiscale PCA proposed by Bakshi (1998) in another context and we discuss it from the denoising perspective. Next, a second
Conclusion
We have proposed a multivariate denoising procedure combining wavelets and PCA, that takes into account the correlation structure of the noise. This new procedure exhibits promising behavior on classical bench signals and seems to perform well when it is applied to multichannel neural recordings, the real world example which illustrates the method.
This work could be extended in various directions, let us mention some of them for future work. First, the way to select the parameters of the
Acknowledgements
The authors thank Anestis Antoniadis for valuable discussions, Karim Oweiss and Yasir Suhail for making available to us the multichannel neural recordings that we used to illustrate our method here and the three anonymous referees for helpful comments and suggestions.
References (22)
- et al.
Multi-channel spike detection and sorting using an array processing technique
Neurocomputing
(1999) - et al.
Noise reduction in multichannel neural recordings using a new array wavelet denoising algorithm
Neurocomputing
(2001) - et al.
Hierarchy, priors and waveletsstructure and signal modelling using ICA
Signal Process.
(2004) - et al.
A decision theoretical approach to wavelet regression on curves with a high number of regressors
J. Statist. Plann. Inference
(2003) Wavelet in statisticsa review
J. Ital. Statist. Soc.
(1997)- et al.
Wavelet estimators in nonparametric regressiona comparative simulation study
J. Statist. Software
(2001) Multiscale PCA with application to MSPC monitoring
AIChE J.
(1998)Ten Lectures on Wavelets
(1992)- et al.
Constructive Approximation
(1993) De-noising by soft-thresholding
IEEE Trans. Inform. Theory
(1995)
Ideal spatial adaptation by wavelet shrinkage
Biometrika
Cited by (169)
The Msegram: A useful multichannel feature synchronous extraction tool for detecting rolling bearing faults
2023, Mechanical Systems and Signal ProcessingCitation Excerpt :However, with the continuous growth of big data, an urgent task turns out to be the powerful methods to multichannel signal simultaneous processing for the high efficiency and effectiveness of data cleaning and digging. Hereinto, Aminghafari studied multivariate wavelet denoising (MWD) with principal component analysis for multichannel denoising [19]. Rehman proposed multivariate empirical mode decomposition (MEMD) for synchronized processing of multichannel signals and adaptive decompositions [20].
Recognition of lower limb movements using empirical mode decomposition and k-nearest neighbor entropy estimator with surface electromyogram signals
2022, Biomedical Signal Processing and ControlAutomated interface detection in liquid-liquid systems using self-calibrating ultrasonic sensor
2021, Chemical Engineering Science