arXiv subjects

Gustavo Didier

Publications and source records attributed to Gustavo Didier.

29 records · Page 2Linked to original sources

arXiv2016

Non-Linear Wavelet Regression and Branch & Bound Optimization for the Full Identification of Bivariate Operator Fractional Brownian Motion

Self-similarity is widely considered the reference framework for modeling the scaling properties of real-world data. However, most theoretical studies and their practical use have remained univariate. Operator Fractional Brownian Motion (OfBm) was recently proposed as a multivariate model for self-similarity. Yet it has remained seldom used in applications because of serious issues that appear in the joint estimation of its numerous parameters. While the univariate fractional Brownian motion requires the estimation of two parameters only, its mere bivariate extension already involves 7 parameters which are very different in nature. The present contribution proposes a method for the full identification of bivariate OfBm (i.e., the joint estimation of all parameters) through an original formulation as a non-linear wavelet regression coupled with a custom-made Branch & Bound numerical scheme. The estimation performance (consistency and asymptotic normality) is mathematically established and numerically assessed by means of Monte Carlo experiments. The impact of the parameters defining OfBm on the estimation performance as well as the associated computational costs are also thoroughly investigated.

math.PR↗

arXiv2016

Exponents of operator self-similar random fields

If X(c^E t) and c^H X(t) have the same finite-dimensional distributions for some linear operators E and H, we say that the random vector field X(t) is operator self-similar. The exponents E and H are not unique in general, due to symmetry. This paper characterizes the possible set of range exponents H for a given domain exponent, and conversely, the set of domain exponents E for a given range exponent.

math.PR↗

arXiv2016

The asymptotic distribution of the pathwise mean squared displacement in single particle tracking experiments

Recent advances in light microscopy have spawned new research frontiers in microbiology by working around the diffraction barrier and allowing for the observation of nanometric biological structures. Microrheology is the study of the properties of complex fluids, such as those found in biology, through the dynamics of small embedded particles, typically latex beads. Statistics based on the recorded sample paths are then used by biophysicists to infer rheological properties of the fluid. In the biophysical literature, the main statistic for characterizing diffusivity is the so-named mean squared displacement (MSD) of the tracer particles. Notwithstanding the central role played by the MSD, its asymptotic distribution in different cases has not yet been established. In this paper, we tackle this problem. We take a pathwise approach and assume that the particle movement undergoes a Gaussian, stationary-increment stochastic process. We show that as the sample and the increment lag sizes go to infinity, the MSD displays Gaussian or non-Gaussian limiting distributions, as well as distinct convergence rates, depending on the diffusion exponent parameter. We illustrate our results analytically and computationally based on fractional Brownian motion and the (integrated) fractional Ornstein-Uhlenbeck process.

math.PR↗

arXiv2015

Wavelet estimation for operator fractional Brownian motion

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. The proposed approach consists of considering the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the coordinate system itself under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

math.PR↗

arXiv2014

On integral representations of operator fractional Brownian fields

Operator fractional Brownian fields (OFBFs) are Gaussian, stationary-increment vector random fields that satisfy the operator self-similarity relation {X(c^{E}t)}_{t in R^m} L= {c^{H}X(t)}_{t in R^m}. We establish a general harmonizable representation (Fourier domain stochastic integral) for OFBFs. Under additional assumptions, we also show how the harmonizable representation can be reexpressed as a moving average stochastic integral, thus answering an open problem described in Bierme et al.(2007), "Operator scaling stable random fields", Stochastic Processes and their Applications 117, 312--332.

math.PR↗

arXiv2013

On the vaguelet and Riesz properties of L^2-unbounded transformations of orthogonal wavelet bases

In this work, we prove that certain L^2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L^2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L^2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.

math.FA↗

arXiv2012

On the wavelet-based simulation of anomalous diffusion

The characterization of particle diffusion is a classical problem in physics and probability theory. The field of microrheology is based on experiments in which microscopic tracer beads are placed into a non-Newtonian fluid and tracked using high speed video capture. The modeling of the behavior of these beads is now an active scientific area which demands multiple stochastic and statistical methods. We propose an approximate wavelet-based simulation technique for two classes of continuous time anomalous diffusion models, the fractional Ornstein-Uhlenbeck process and the fractional generalized Langevin equation. The proposed algorithm is an iterative method that provides approximate discretizations that converge quickly and in an appropriate sense to the continuous time target process. As compared to previous works, it covers cases where the natural discretization of the target process does not have closed form in the time domain. Moreover, we propose smoothing procedures as to speed the time domain decay of the filters.

stat.ME↗

arXiv2012

Statistical Challenges in Microrheology

Microrheology is the study of the properties of a complex fluid through the diffusion dynamics of small particles, typically latex beads, moving through that material. Currently, it is the dominant technique in the study of the physical properties of biological fluids, of the material properties of membranes or the cytoplasm of cells, or of the entire cell. The theoretical underpinning of microrheology was given in Mason and Weitz (Physical Review Letters; 1995), who introduced a framework for the use of path data of diffusing particles to infer viscoelastic properties of its fluid environment. The multi-particle tracking techniques that were subsequently developed have presented numerous challenges for experimentalists and theoreticians. This paper describes some specific challenges that await the attention of statisticians and applied probabilists. We describe relevant aspects of the physical theory, current inferential efforts and simulation aspects of a central model for the dynamics of nano-scale particles in viscoelastic fluids, the generalized Langevin equation.

stat.ME↗

arXiv2011

Integral representations and properties of operator fractional Brownian motions

Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)--(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.

math.ST↗

arXiv2011

Exponents, symmetry groups and classification of operator fractional Brownian motions

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.), Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group. In this paper, we revisit and study both the symmetry groups and exponent sets for the class of OFBMs based on their spectral domain integral representations. A general description of the symmetry groups of OFBMs in terms of subsets of centralizers of the spectral domain parameters is provided. OFBMs with symmetry groups of maximal and minimal types are studied in any dimension. In particular, it is shown that OFBMs have minimal symmetry groups (as thus, unique exponents) in general, in the topological sense. Finer classification results of OFBMs, based on the explicit construction of their symmetry groups, are given in the lower dimensions 2 and 3. It is also shown that the parametrization of spectral domain integral representations are, in a suitable sense, not affected by the multiplicity of exponents, whereas the same is not true for time domain integral representations.

math.PR↗

arXiv2008

On the Behrens--Fisher problem: A globally convergent algorithm and a finite-sample study of the Wald, LR and LM Tests

In this paper we provide a provably convergent algorithm for the multivariate Gaussian Maximum Likelihood version of the Behrens--Fisher Problem. Our work builds upon a formulation of the log-likelihood function proposed by Buot and Richards \citeBR. Instead of focusing on the first order optimality conditions, the algorithm aims directly for the maximization of the log-likelihood function itself to achieve a global solution. Convergence proof and complexity estimates are provided for the algorithm. Computational experiments illustrate the applicability of such methods to high-dimensional data. We also discuss how to extend the proposed methodology to a broader class of problems. We establish a systematic algebraic relation between the Wald, Likelihood Ratio and Lagrangian Multiplier Test ($W\geq \mathit{LR}\geq \mathit{LM}$) in the context of the Behrens--Fisher Problem. Moreover, we use our algorithm to computationally investigate the finite-sample size and power of the Wald, Likelihood Ratio and Lagrange Multiplier Tests, which previously were only available through asymptotic results. The methods developed here are applicable to much higher dimensional settings than the ones available in the literature. This allows us to better capture the role of high dimensionality on the actual size and power of the tests for finite samples.

math.ST↗