Bayesian Wavelet Estimation of Long Memory Parameter

A Bayesian wavelet estimation method for estimating parameters of a stationary I(d) process is represented as an useful alternative to the existing frequentist wavelet estimation methods. The effectiveness of the proposed method is demonstrated through Monte Carlo simulations. The sampling from the posterior distribution is through the Markov Chain Monte Carlo (MCMC) easily implemented in the WinBUGS software package

Copula Density Estimation by Total Variation Penalized Likelihood with Linear Equality Constraints

A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on a maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by log-barrier method for second order cone program. A data-driven selection of the regularization parameter is through K-fold cross-validation (CV). Simulation and real data application show the effectiveness of the proposed approach. The MATLAB code implementing the methodology is available online

Change Point Estimation of Bilevel Functions

Reconstruction of a bilevel function such as a bar code signal in a partially blind deconvolution problem is an important task in industrial processes. Existing methods are based on either the local approach or the regularization approach with a total variation penalty. This article reformulated the problem explicitly in terms of change points of the 0-1 step function. The bilevel function is then reconstructed by solving the nonlinear least squares problem subject to linear inequality constraints, with starting values provided by the local extremas of the derivative of the convolved signal from discrete noisy data. Simulation results show a considerable improvement of the quality of the bilevel function using the proposed hybrid approach over the local approach. The hybrid approach extends the workable range of the standard deviation of the Gaussian kernel significantly

Wavelet-Based Bayesian Estimation of Partially Linear Regression Models with Long Memory Errors

In this paper we focus on partially linear regression models with long memory errors, and propose a wavelet-based Bayesian procedure that allows the simultaneous estimation of the model parameters and the nonparametric part of the model. Employing discrete wavelet transforms is crucial in order to simplify the dense variance-covariance matrix of the long memory error. We achieve a fully Bayesian inference by adopting a Metropolis algorithm within a Gibbs sampler. We evaluate the performances of the proposed method on simulated data. In addition, we present an application to Northern hemisphere temperature data, a benchmark in the long memory literature

Wavelet Deconvolution in a Periodic Setting Using Cross-Validation

The wavelet deconvolution method WaveD using band-limited wavelets offers both theoretical and computational advantages over traditional compactly supported wavelets. The translation-invariant WaveD with a fast algorithm improves further. The twofold cross-validation method for choosing the threshold parameter and the finest resolution level in WaveD is introduced. The algorithm’s performance is compared with the fixed constant tuning and the default tuning in WaveD

A Semiparametric Approach for Analyzing Nonignorable Missing Data

In missing data analysis, there is often a need to assess the sensitivity of key inferences to departures from untestable assumptions regarding the missing data process. Such sensitivity analysis often requires specifying a missing data model which commonly assumes parametric functional forms for the predictors of missingness. In this paper, we relax the parametric assumption and investigate the use of a generalized additive missing data model. We also consider the possibility of a non-linear relationship between missingness and the potentially missing outcome, whereas the existing literature commonly assumes a more restricted linear relationship. To avoid the computational complexity, we adopt an index approach for local sensitivity. We derive explicit formulas for the resulting semiparametric sensitivity index. The computation of the index is simple and completely avoids the need to repeatedly fit the semiparametric nonignorable model. Only estimates from the standard software analysis are required with a moderate amount of additional computation. Thus, the semiparametric index provides a fast and robust method to adjust the standard estimates for nonignorable missingness. An extensive simulation study is conducted to evaluate the effects of misspecifying the missing data model and to compare the performance of the proposed approach with the commonly used parametric approaches. The simulation study shows that the proposed method helps reduce bias that might arise from the misspecification of the functional forms of predictors in the missing data model. We illustrate the method in a Wage Offer dataset.

Wavelet Reconstruction of Nonuniformly Sampled Signals

For the reconstruction of a nonuniformly sampled signal based on its noisy observations, we propose a level dependent l1 penalized wavelet reconstruction method. The LARS/Lasso algorithm is applied to solve the Lasso problem. The data adaptive choice of the regularization parameters is based on the AIC and the degrees of freedom is estimated by the number of nonzero elements in the Lasso solution. Simulation results conducted on some commonly used 1_D test signals illustrate that the proposed method possesses good empirical properties

Inversion for Non-Smooth Models with Physical Bounds

Geological processes produce structures at multiple scales. A discontinuity in the subsurface can occur due to layering, tectonic activities such as faulting, folding and fractures. Traditional approaches to invert geophysical data employ smoothness constraints. Such methods produce smooth models and thefore sharp contrasts in the medium such as lithological boundaries are not easily discernible. The methods that are able to produce non-smooth models, can help interpret the geological discontinuity. In this paper we examine various approaches to obtain non-smooth models from a finite set of noisy data. Broadly they can be categorized into approaches: (1) imposing non-smooth regularization in the inverse problem and (2) solve the inverse problem in a domain that provides multi-scale resolution, such as wavelet domain. In addition to applying non-smooth constraints, we further constrain the inverse problem to obtain models within prescribed physical bounds. The optimization with non-smooth regularization and physical bounds is solved using an interior point method. We demonstrate the applicability and usefulness of these methods with realistic synthetic examples and provide a field example from crosswell radar data

Using Decision Forest to Classify Prostate Cancer Samples on the Basis of SELDI-TOF MS Data: Assessing Chance Correlation and Prediction Confidence

Class prediction using “omics” data is playing an increasing role in toxicogenomics, diagnosis/prognosis, and risk assessment. These data are usually noisy and represented by relatively few samples and a very large number of predictor variables (e.g., genes of DNA microarray data or m/z peaks of mass spectrometry data). These characteristics manifest the importance of assessing potential random correlation and overfitting of noise for a classification model based on omics data. We present a novel classification method, decision forest (DF), for class prediction using omics data. DF combines the results of multiple heterogeneous but comparable decision tree (DT) models to produce a consensus prediction. The method is less prone to overfitting of noise and chance correlation. A DF model was developed to predict presence of prostate cancer using a proteomic data set generated from surface-enhanced laser deposition/ionization time-of-flight mass spectrometry (SELDI-TOF MS). The degree of chance correlation and prediction confidence of the model was rigorously assessed by extensive cross-validation and randomization testing. Comparison of model prediction with imposed random correlation demonstrated biologic relevance of the model and the reduction of overfitting in DF. Furthermore, two confidence levels (high and low confidences) were assigned to each prediction, where most misclassifications were associated with the low-confidence region. For the high-confidence prediction, the model achieved 99.2% sensitivity and 98.2% specificity. The model also identified a list of significant peaks that could be useful for biomarker identification. DF should be equally applicable to other omics data such as gene expression data or metabolomic data. The DF algorithm is available upon request