arXiv subjects

A. Alexandre Trindade

Publications and source records attributed to A. Alexandre Trindade.

7 recordsLinked to original sources

arXiv2026

Modeling Brain MRI Using Persistent Homology and Multilevel Functional Data Analysis

Persistent homology provides a multiscale representation of biomedical images by capturing higher-order topological features that reflect their underlying structural organization. However, the resulting topological summaries are typically used as predictors or features for classification and group comparisons rather than treated as primary variables of interest. We construct a generalized multilevel functional framework for analyzing persistent-homology summaries as longitudinal functional responses in repeated three-dimensional structural magnetic resonance imaging (MRI). Specifically, we represent topological features using Betti curves and model these curves as count-valued functional responses. A negative-binomial distribution accommodates the discrete and potentially overdispersed nature of Betti counts, while the multilevel formulation accounts for the dependence induced by repeated measurements and separates between-subject and within-subject sources of functional variation. Functional principal component analysis further evaluates the dominant modes of variation at each level. A Bayesian approach is used for joint estimation of the functional regression and multilevel functional principal components. We apply this modeling framework to longitudinal structural MRI data from the OASIS-2 study to investigate associations between brain topology and demographic and clinical characteristics, including age, gender, follow-up time, and dementia severity. The results demonstrate that the framework can capture covariate-associated variation across the filtration continuum while evaluating distinct sources and patterns of longitudinal variation across homology dimensions.

stat.ME↗

arXiv2025

The Three-Dimensional Decomposition of Volatility Memory

This paper develops a three-dimensional decomposition of volatility memory into orthogonal components of level, shape, and tempo. The framework unifies regime-switching, fractional-integration, and business-time approaches within a single canonical representation that identifies how each dimension governs persistence strength, long-memory form, and temporal speed. We establish conditions for existence, uniqueness, and ergodicity of this decomposition and show that all GARCH-type processes arise as special cases. Empirically, applications to SPY and EURUSD (2005--2024) reveal that volatility memory is state-dependent: regime and tempo gates dominate in equities, while fractional-memory gates prevail in foreign exchange. The unified tri-gate model jointly captures these effects. By formalizing volatility dynamics through a level--shape--tempo structure, the paper provides a coherent link between information flow, market activity, and the evolving memory of financial volatility.

q-fin.MF↗

arXiv2024

On the use of the M-quantiles for outlier detection in multivariate data

Defining a successful notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of [8] and [25] leading to M-quantiles, is very appealing for its mathematical elegance combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of R^d and the unit ball in R^d. By analogy with the d=1 case where the K-transform is a cumulative distribution function-like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d>=1. Over the past twenty years the resulting M-quantiles have seen applications in a variety of fields, primarily for the purpose of detecting outliers in multidimensional spaces. In this article we prove that for odd d>=3, it is not the gradient but a poly-Laplacian of the K-function that is (almost everywhere) proportional to the density function. For d even one cannot establish a differential equation connecting the K-function with the density. These results show that usage of the K-transform for outlier detection in higher odd-dimensions is in principle flawed, as the K-transform does not originate from inversion of a true M-distribution. We demonstrate these conclusions in two dimensions through examples from non-standard asymmetric distributions. Our examples illustrate a feature of the K-transform whereby regions in the domain with higher density map to larger volumes in the co-domain, thereby producing a magnification effect that moves inliers closer to the boundary of the co-domain than outliers. This feature obviously disrupts any outlier detection mechanism that relies on the inverse K-transform.

math.ST↗

arXiv2021

Global and Tail Dependence: A Differential Geometry Approach

Measures of tail dependence between random variables aim to numerically quantify the degree of association between their extreme realizations. Existing tail dependence coefficients (TDCs) are based on an asymptotic analysis of relevant conditional probabilities, and do not provide a complete framework in which to compare extreme dependence between two random variables. In fact, for many important classes of bivariate distributions, these coefficients take on non-informative boundary values. We propose a new approach by first considering global measures based on the surface area of the conditional cumulative probability in copula space, normalized with respect to departures from independence and scaled by the difference between the two boundary copulas of co-monotonicity and counter-monotonicity. The measures could be approached by cumulating probability on either the lower left or upper right domain of the copula space, and offer the novel perspective of being able to differentiate asymmetric dependence with respect to direction of conditioning. The resulting TDCs produce a smoother and more refined taxonomy of tail dependence. The empirical performance of the measures is examined in a simulated data context, and illustrated through a case study examining tail dependence between stock indices.

stat.AP↗

arXiv2020

A Socioeconomic Well-Being Index

An annual well-being index constructed from thirteen socioeconomic factors is proposed in order to dynamically measure the mood of the US citizenry. Econometric models are fitted to the log-returns of the index in order to quantify its tail risk and perform option pricing and risk budgeting. By providing a statistically sound assessment of socioeconomic content, the index is consistent with rational finance theory, enabling the construction and valuation of insurance-type financial instruments to serve as contracts written against it. Endogenously, the VXO volatility measure of the stock market appears to be the greatest contributor to tail risk. Exogenously, "stress-testing" the index against the politically important factors of trade imbalance and legal immigration, quantify the systemic risk. For probability levels in the range of 5% to 10%, values of trade below these thresholds are associated with larger downward movements of the index than for immigration at the same level. The main intent of the index is to provide early-warning for negative changes in the mood of citizens, thus alerting policy makers and private agents to potential future market downturns.

econ.GN↗

arXiv2017

Improved Inference for the Signal Significance

We study the properties of several likelihood-based statistics commonly used in testing for the presence of a known signal under a mixture model with known background, but unknown signal fraction. Under the null hypothesis of no signal, all statistics follow a standard normal distribution in large samples, but substantial deviations can occur at low sample sizes. Approximations for respective $p$-values are derived to various orders of accuracy using the methodology of Edgeworth expansions. Adherence to normality is studied, and the magnitude of deviations is quantified according to resulting inflation or deflation. We find that approximations to third-order accuracy are generally sufficient to guarantee $p$-values with nominal false positive error rates in the five sigma range ($p$-value $= 2.87 \times 10^{-7}$) for the classic Wald, score, and likelihood ratio (LR) statistics at relatively low samples. Not only does LR have better adherence to normality, but it also consistently outperforms all other statistics in terms of false negative error rates. The reasons for this are shown to be connected with high-order cumulant behavior gleaned from fourth order Edgeworth expansions. Finally, a conservative procedure is suggested for making finite sample adjustments while accounting for the look elsewhere effect with the theory of random fields (a.k.a. the Gross-Vitells method).

physics.data-an↗

arXiv2015

Local Orthogonal Polynomial Expansion for Density Estimation

A Local Orthogonal Polynomial Expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localized expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalizes straightforwardly to multivariate settings. By manner of construction, it is similar to Local Likelihood Density Estimation (LLDE). In the limit of small bandwidths, LOrPE functions as Kernel Density Estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Faster asymptotic convergence rates follow. In the limit of large bandwidths, LOrPE is equivalent to Orthogonal Series Density Estimation (OSDE) with Legendre polynomials. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.

stat.AP↗