Covariance Structure of Wavelet Coefficients: Theory and Models in a Bayesian Perspective

Vannucci, Marina; Corradi, Fabio

doi:10.1111/1467-9868.00214

Cited by 74 publications

(52 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Vannucci and Corradi (1999) proposed a fast recursive algorithm for computing quantities such as WHW 0 . Their algorithm has a useful link to the twodimensional DWT (DWT2), making computations simple.…”

Section: Transformation To Waveletsmentioning

confidence: 99%

Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem

Brown

Fearn

Vannucci

2001

Journal of the American Statistical Association

157

117

View full text Add to dashboard Cite

Motivated by calibration problems in near-infrared (N IR) spectroscopy, we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands. We tackle this regression problem by calculating the wavelet transforms of the discretized curves, then applying a Bayesian variable selection method using mixture priors to the multivariate regression of predictands on wavelet coef cients. For prediction purposes, we average over a set of likely models. Applied to a particular problem in N IR spectroscopy, this approach was able to nd subsets of the wavelet coef cients with overall better predictive performance than the more usual approaches. In the application, the available predictors are measurements of the N IR re ectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths. The aim is to predict the composition (i.e., the fat, our, sugar, and water content) of the dough pieces using the spectral variables. Thus we have a multivariate regression of four predictands on 256 predictors with quite high intercorrelation among the predictors. A training set of 39 samples is available to t this regression. Applying a wavelet transform replaces the 256 measurements on each spectrum with 256 wavelet coef cients that carry the same information. The variable selection method could use subsets of these coef cients that gave good predictions for all four compositional variables on a separate test set of samples. Selecting in the wavelet domain rather than from the original spectral variables is appealing in this application, because a single wavelet coef cient can carry information from a band of wavelengths in the original spectrum. This band can be narrow or wide, depending on the scale of the wavelet selected.

show abstract

Section: Transformation To Waveletsmentioning

confidence: 99%

Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem

Brown

Fearn

Vannucci

2001

Journal of the American Statistical Association

157

117

View full text Add to dashboard Cite

show abstract

“…(13). In Vannucci and Corradi (1999), the authors provide an algorithm to do so for a 1-D wavelet basis. We developed a 2-D version of this algorithm to perform the experiments presented in Section 5.4.…”

Section: Correlated Gaussian White Noisementioning

confidence: 99%

Accounting for observation errors in image data assimilation

Chabot¹,

Nodet²,

Papadakis

et al. 2015

Tellus A: Dynamic Meteorology and Oceanography

View full text Add to dashboard Cite

A B S T R A C T This paper deals with the assimilation of image-type data. Such kinds of data, such as satellite images, have good properties (dense coverage in space and time), but also one crucial problem for data assimilation: they are affected by spatially correlated errors. Classical approaches in data assimilation assume uncorrelated noise, because the proper description and numerical manipulation of non-diagonal error covariance matrices is complex. This paper proposes a simple way to provide observation error covariance matrices adapted to spatially correlated errors. This is done using various image transformations: multiscale (Wavelets, Fourier, Curvelets), gradients, and gradient orientations. These transformations are described and compared to classical approaches, such as pixel-to-pixel comparison and observation thinning. We provide simple yet effective covariance matrices for each of these transformations, which take into account the observation error correlations and improve the results. The effectiveness of the proposed approach is demonstrated on twin experiments performed on a 2-D shallow-water model.

show abstract

“…However, it is presented in different literature that by only keeping few wavelet coefficients with approximate sub-band coefficients can improve the quality of the reconstructed image significantly (Donoho et al 1996;Vannucci and Corradi 1999). Therefore, in this example, the distance calculation was performed by using few wavelet coefficients along with approximate sub-band coefficients when the conditioning data event is fully informed.…”

Section: Sensitivity To the Number Of Wavelet Coefficientsmentioning

confidence: 99%

Dimensional Reduction of Pattern-Based Simulation Using Wavelet Analysis

2012

View full text Add to dashboard Cite

A pattern-based simulation technique using wavelet analysis is proposed for the simulation (wavesim) of categorical and continuous variables. Patterns are extracted by scanning a training image with a template and then storing them in a pattern database. The dimension reduction of patterns in the pattern database is performed by wavelet decomposition at certain scale and the approximate sub-band is used for pattern database classification. The pattern database classification is performed by the k-means clustering algorithm and classes are represented by a class prototype. For the simulation of categorical variables, the conditional cumulative density function (ccdf ) for each class is generated based on the frequency of the individual categories at the central node of the template. During the simulation process, the similarity of the conditioning data event with the class prototypes is measured using the L 2 -norm. When simulating categorical variables, the ccdf of the best matched class is used to draw a pattern from a class. When continuous variables are simulated, a random pattern is drawn from the best matched class. Several examples of conditional and unconditional simulation with two-and three-dimensional data sets show that the spatial continuity of geometric features and shapes is well reproduced. A comparative study with the filtersim algorithm shows that the wavesim performs better than filtersim in all examples. A full-field case study at the Olympic Dam base metals de-

show abstract

Covariance Structure of Wavelet Coefficients: Theory and Models in a Bayesian Perspective

Cited by 74 publications

References 29 publications

Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem

Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem

Accounting for observation errors in image data assimilation

Dimensional Reduction of Pattern-Based Simulation Using Wavelet Analysis

Contact Info

Product

Resources

About