Functional data analysis is ubiquitous in most areas of sciences and engineering. Several paradigms are proposed to deal with the dimensionality problem which is inherent to this type of data. Sparseness, penalization, thresholding, among other principles, have been used to tackle this issue. We discuss here a solution based on a finite-dimensional functional space. We employ wavelet representation of the functionals to estimate this finite dimension, and successfully model a time series of curves. The proposed method is shown to have nice asymptotic properties. Moreover, the wavelet representation permits the use of several bootstrap procedures, and it results in faster computing algorithms. Besides the theoretical and computational properties, some simulation studies and an application to real data are provided. MSC2010 Classification: 62G05; 62G20; 62G99.Keywords Aggregate data · bootstrap testing · finite dimension · functional data analysis 1 Introduction Many phenomena, natural or anthropogenic, can be appropriately modeled by a function on a suitable domain. Examples on the literature have been around for several decades, but the last three have made them ubiquitous in
We address the issue of performing inference on the parameters that index a bimodal extension of the Birnbaum-Saunders distribution (BS). We show that maximum likelihood point estimation can be problematic since the standard nonlinear optimization algorithms may fail to converge. To deal with this problem, we penalize the log-likelihood function. The numerical evidence we present shows that maximum likelihood estimation based on such penalized function is made considerably more reliable. We also consider hypothesis testing inference based on the penalized log-likelihood function. In particular, we consider likelihood ratio, signed likelihood ratio, score and Wald tests. Bootstrap-based testing inference is also considered. We use a nonnested hypothesis test to distinguish between two bimodal BS laws. We derive analytical corrections to some tests. Monte Carlo simulation results and empirical applications are presented and discussed.
This paper focuses on wavelet analysis of variability for heavy‐tailed time series. Under the assumption that time‐series values have finite second but infinite fourth moments, stable asymptotics are derived for wavelet variances across different time scales. These stable asymptotics have a slower rate of convergence than the square root of the sample size and are markedly different from conventional normal asymptotics. Furthermore, the asymptotic results apply even when the time series exhibits long‐range dependence. Wavelet variances and stable asymptotics are then used to analyze three streamflows: one in Arizona, one in Connecticut, and one in Illinois. These analyses provide a deeper understanding of streamflow variability at different time scales (e.g., extreme variation at short time scales that are characteristic of heavy rainfall, presence of seasonal variations, and, in one case, some quasi‐biennial fluctuations). Furthermore, this paper includes evaluations of local characteristic time scales, a discussion of tail heaviness, computation of Hurst exponents, and some future directions of research.
The log-linear Birnbaum-Saunders model has been widely used in empirical applications. We introduce an extension of this model based on a recently proposed version of the Birnbaum-Saunders distribution which is more flexible than the standard Birnbaum-Saunders law since its density may assume both unimodal and bimodal shapes. We show how to perform point estimation, interval estimation and hypothesis testing inferences on the parameters that index the regression model we propose. We also present a number of diagnostic tools, such as residual analysis, local influence, generalized leverage, generalized Cook's distance and model misspecification tests. We investigate the usefulness of model selection criteria and the accuracy of prediction intervals for the proposed model. Results of Monte Carlo simulations are presented. Finally, we also present and discuss an empirical application.
We introduce the WECS (Wavelet Energies Correlation Screening), an unsupervised method to detect spatio-temporal changes on multitemporal SAR images. The procedure is based on wavelet approximation for the multitemporal images, wavelet energy apportionment, and ultra-high dimensional correlation screening for the wavelet coefficients. We show WECS's performance on simulated multitemporal image data. We also evaluate the proposed method on a time series of 85 Sentinel-1 images of a forest region at the border of Brazil and French Guiana. Comparisons with well-known change detection methods found in the literature highlight the proposal's superiority in terms of change detection accuracy. Additionally, the introduced method has simple architecture and low computational cost.
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