Penalized wavelet estimation and robust denoising for irregular spaced data

Nonparametric univariate regression via wavelets is usually implemented under the assumptions of dyadic sample size, equally spaced fixed sample points, and i.i.d. normal errors. In this work, we propose, study and compare some wavelet based nonparametric estimation methods designed to recover a one-dimensional regression function for data that not necessary possess the above requirements. These methods use appropriate regularizations by penalizing the decomposition of the unknown regression function on a wavelet basis of functions evaluated on the sampling design. Exploiting the sparsity of wavelet decompositions for signals belonging to homogeneous Besov spaces, we use some efficient proximal gradient descent algorithms, available in recent literature, for computing the estimates with fast computation times. Our wavelet based procedures, in both the standard and the robust regression case have favorable theoretical properties, thanks in large part to the separability nature of the (non convex) regularization they are based on. We establish asymptotic global optimal rates of convergence under weak conditions. It is known that such rates are, in general, unattainable by smoothing splines or other linear nonparametric smoothers. Lastly, we present several experiments to examine the empirical performance of our procedures and their comparisons with other proposals available in the literature. An interesting regression analysis of some real data applications using these procedures unambiguously demonstrate their effectiveness.

A Jackson-type inequality associated with wavelet bases decomposition

Although wavelet decompositions of functions in Besov spaces have been extensively investigated, those involved with mild decay bases are relatively unexplored. In this paper, we study wavelet bases of Besov spaces and the relation between norms and wavelet coefficients. We establish the $l^{p}$ l p -stability as a measure of how effectively the Besov norm of a function is evaluated by its wavelet coefficients and the $L^{p}$ L p -completeness of wavelet bases. We also discuss wavelets with decay conditions and establish the Jackson inequality.

A wavelet approach of investing behaviors and their effects on risk exposures

Exposure to market risk is a core objective of the Capital Asset Pricing Model (CAPM) with a focus on systematic risk. However, traditional OLS Beta model estimations (Ordinary Least Squares) are plagued with several statistical issues. Moreover, the CAPM considers only one source of risk and supposes that investors only engage in similar behaviors. In order to analyze short and long exposures to different sources of risk, we developed a Time–Frequency Multi-Betas Model with ARMA-EGARCH errors (Auto Regressive Moving Average Exponential AutoRegressive Conditional Heteroskedasticity). Our model considers gold, oil, and Fama–French factors as supplementary sources of risk and wavelets decompositions. We used 30 French stocks listed on the CAC40 (Cotations Assistées Continues 40) within a daily period from 2005 to 2015. The conjugation of the wavelet decompositions and the parameters estimates constitutes decision-making support for managers by multiplying the interpretive possibilities. In the short-run, (“Noise Trader” and “High-Frequency Trader”) only a few equities are insensitive to Oil and Gold fluctuations, and the estimated Market Betas parameters are scant different compared to the Model without wavelets. Oppositely, in the long-run, (fundamentalists investors), Oil and Gold affect all stocks but their impact varies according to the Beta (sensitivity to the market). We also observed significant differences between parameters estimated with and without wavelets.

Sensitivity Analysis of Acoustic Emission Detection Using Fiber Bragg Gratings with Different Optical Fiber Diameters

Acoustic Emission (AE) detection and, in particular, ultrasound detection are excellent tools for structural health monitoring or medical diagnosis. Despite the technological maturity of the well-received piezoelectric transducer, optical fiber AE detection sensors are attracting increasing attention due to their small size, and electromagnetic and chemical immunity as well as the broad frequency response of Fiber Bragg Grating (FBG) sensors in these fibers. Due to the merits of their small size, FBGs were inscribed in optical fibers with diameters of 50 and 80 μm in this work. The manufactured FBGs were used for the detection of reproducible acoustic waves using the edge filter detection method. The acquired acoustic signals were compared to the ones captured by a standard 125 μm-diameter optical fiber FBG. Result analysis was performed by utilizing fast Fourier and wavelet decompositions. Both analyses reveal a higher sensitivity and dynamic range for the 50 μm-diameter optical fiber, despite it being more prone to noise than the other two, due to non-standard splicing methods and mode field mismatch losses. Consequently, the use of smaller-diameter optical fibers for AE detection is favorable for both the sensor sensitivity as well as physical footprint.

3D Textural Analysis of Fatigue Fracture Surfaces

Three CT specimens from stainless steel AISI 304L were subjected to constant amplitude cyclic loadings with various asymmetries. Crack growth was recorded in detail. Fracture surfaces were documented by 3D maps in about 110 locations in the crack growth direction. 3D maps and their local gradients were represented by 2D wavelet decompositions in 10 levels resulting in 60 textural features. Statistical models expressing crack growth rate as a function of textural features were optimized. Training and testing approach, a high ratio of overfitting, and testing of significance of components ensured model's robustness. Quality of results is documented by graphs confronting model outputs with real data known from experiment. Results are acceptable in all cases.

Fast wavelet decomposition of linear operators through product-convolution expansions

Wavelet decompositions of integral operators have proven their efficiency in reducing computing times for many problems, ranging from the simulation of waves or fluids to the resolution of inverse problems in imaging. Unfortunately, computing the decomposition is itself a hard problem which is oftentimes out of reach for large-scale problems. The objective of this work is to design fast decomposition algorithms based on another representation called product-convolution expansion. This decomposition can be evaluated efficiently, assuming that a few impulse responses of the operator are available, but it is usually less efficient than the wavelet decomposition when incorporated in iterative methods. The proposed decomposition algorithms, run in quasi-linear time and we provide some numerical experiments to assess its performance for an imaging problem involving space-varying blurs.

Reducing image search time by improved BOVW with wavelet decomposition

<p>In the last decade, the bag of visual words (BOVW) has been used widely in image classification, image retrieval and has significantly improved the performance of CBIR system. In this paper we propose a new method to enhance BOVW using features obtained from wavelet decomposition in order to reduce computational costs in vocabulary construction and training time. We apply several level of wavelet decompositions and evaluate their impact on accuracy of the BOVW. We apply our method on MURA-v1.1 dataset and the experiments results confirm the performance of our approach.</p>

From subgaussianity to stochastic approximation and modelling

The modern theory of subgaussian random variables and processes was created by independent efforts of several research schools from France, USA and Ukraine. Professor Yu.Kozachenko was a founder and leading figure of this research direction of the Ukrainian probability school. An outline of Professor Yu.Kozachenko's contribution to the theory of sub-Gaussian random variables and processes is presented. The class of $\varphi$-subgaussian random variables is introduced and its key property is discussed. Then it is demonstrated how these results can be used in stochastic approximation and modeling. In particular, applications to approximation of trajectories of $\varphi$-subgaussian random processes with given accuracy and reliability are discussed. Two important clases of algorithms from the signal processing theory, the Shannon sampling method and wavelet decompositions, are used as examples. Some personal memories of the author about Yu. Kozachenko are included at the end of the paper.