Joaquin Ortega

Mathematics Research Center | CIMAT · Department of Probability and Statistics

In this work, we propose a robust cluster analysis methodology based on cellwise trimming as an extension to a robust version of Principal Component Analysis. This new approach is more reasonable than traditional casewise trimming when the dimension is not small. This type of trimming avoids an unnecessary loss of information when only a few cells...

A new time series clustering procedure, based on Functional Data Analysis techniques applied to spectral densities, is employed in this work for the detection of stationary intervals in random waves. Long records of wave data are divided into 30-minute or one-hour segments and the spectral density of each interval is estimated by one of the standar...

In this work, a robust clustering algorithm for stationary time series is proposed. The algorithm is based on the use of estimated spectral densities, which are considered as functional data, as the basic characteristic of stationary time series for clustering purposes. A robust algorithm for functional data is then applied to the set of spectral d...

In this work a robust clustering algorithm for stationary time series is proposed. The algorithm is based on the use of estimated spectral densities, which are considered as functional data, as the basic characteristic of stationary time series for clustering purposes. A robust algorithm for functional data is then applied to the set of spectral de...

In this work a robust clustering algorithm for stationary time series is proposed. The algorithm is based on the use of estimated spectral densities, which are considered as functional data, as the basic characteristic of stationary time series for clustering purposes. A robust algorithm for functional data is then applied to the set of spectral de...

Many clustering algorithms when the data are curves or functions have been recently proposed. However, the presence of contamination in the sample of curves can influence the performance of most of them. In this work we propose a robust, model-based clustering method based on an approximation to the "density function" for functional data. The robus...

Many clustering algorithms when the data are curves or functions have been recently proposed. However, the presence of contamination in the sample of curves can influence the performance of most of them. In this work we propose a robust, model-based clustering method based on an approximation to the "density function" for functional data. The robus...

We present a new method for time series clustering which we call the Hierarchical Spectral Merger (HSM) method. This procedure is based on the spectral theory of time series and identifies series that share similar oscillations or waveforms. The extent of similarity between a pair of time series is measured using the total variation distance betwee...

We present a new method for time series clustering which we call the Hierarchical Spectral Merger (HSM) method. This procedure is based on the spectral theory of time series and identifies series that share similar oscillations or waveforms. The extent of similarity between a pair of time series is measured using the total variation distance betwee...

Brain activity following stimulus presentation and during resting state are often the result of highly coordinated responses of large numbers of neurons both locally and globally. Coordinated activity of neurons can give rise to oscillations which are captured by electroencephalograms (EEG). In this paper, we examine EEGs as this is the primary dat...

The use of quadratic forms of the empirical process for the two-sample problem in the context of functional data is considered. The convergence of the family of statistics proposed to a Gaussian limit is established under metric entropy conditions for smooth functional data. The applicability of the proposed methodology is evaluated in examples.

A time series clustering algorithm based on the use of the total variation distance between normalized spectra as a measure of dissimilarity is proposed in this work. The oscillatory behavior of the series is thus considered the central characteristic for classification purposes. The proposed algorithm is compared to several other methods which are...

Large quantiles of extreme value distributions are useful to assess the risk of environmental disasters. Profile likelihood intervals of quantiles are shown here to be optimal for samples of sizes of n ≥ 50. However, they are seldom used, notwithstanding their reasonable coverage frequencies. In contrast, asymptotic maximum likelihood confidence in...

The problem of detecting changes in the state of the sea is very important for the analysis and determination of wave climate in a given location. Wave measurements are frequently statistically analyzed as a time series, and segmentation algorithms developed in this context are used to determine change-points. However, most methods found in the lit...

Functional data analysis (FDA) is a set of tools developed to perform statistical analysis on data having a functional form. In our case we consider the one-dimensional wave surface profiles registered during a North-Sea storm as functional data. The data is split into 20 min intervals within which an individual wave is defined as the profile betwe...

Random sea waves are often modeled as stationary processes for short or moderately long periods of time and therefore the problem of detecting changes in the sea state is very important. We look at this problem from the spectral point of view, proposing a method based on the total variation distance. The method considers processes normalized to hav...

In this work we perform a comparative study of storm wave spectra using the Hilbert-Huang Transform and the Smooth Localized complex EXponen-cial (SLEX) algorithm. This last algorithm divides the data set into approxi-mately stationary sections by detecting changes in their spectra. We compare the spectra produced by both algorithms and also look a...

: In this work we look at the spectral evolution of waves from the point of view of the total variation (TV) distance. There are several methods for determining changes in the variance of a random process, which correspond to changes in the total energy of the waves. We look instead at changes in the distribution of the energy as given by the energ...

Functional Data Analysis is a set of statistical tools developed to perform statistical analysis on data having a functional form. In our case we consider the one-dimensional wave profiles registered during a North-Sea storm as functional data. The waves are defined as the surface height between two consecutive downcrossings. Data is split into 20-...

Profile likelihood intervals of large quantiles in Extreme Value distributions provide a good way to estimate these parameters of interest since they take into account the asymmetry of the likelihood surface in the case of small and moderate sample sizes; however they are seldom used in practice. In contrast, maximum likelihood asymptotic (mla) int...

We use the Hilbert–Huang transform (HHT) for the spectral analysis of a North Sea storm that took place in 1997. We look at the contribution of the different Intrinsic Mode Functions (IMF) obtained using the Empirical Mode Decomposition algorithm, and also compare the Hilbert Marginal Spectra and the classical Fourier Spectra for the data set and t...

The Hilbert-Huang Transform (HHT) was proposed by Huang et al. [2] as a method for the analysis of non-linear, non-stationary time series. This procedure requires the decomposition of the signal into intrinsic mode functions using a method called empirical mode decomposition. These functions represent the essential oscillatory modes contained in th...

In this work we consider the evolution of power spectra of waves during a period of one year. Soukissian and Samalekos (2005) have proposed a segmentation method for significant wave height based on determining periods of stability, increase and decrease using time-series techniques. The second segmentation method is based on the mean value over a...

We consider the evolution of spectra of random waves over periods of three days. Two segmentation methods are used: Detection of Changes by Penalized Contrasts (DCPC) proposed and developed by Lavielle (1998, 1999) and Smooth Localized complex EXponentials (SLEX) proposed in Ombao et al. (2001). We compare the results obtained with both methods. In...

In this work, we study some geometrical properties of a stationary Gaussian field modeling the sea surface, using the energy spectrum. We consider the length of a crest and the mean speed of contours, which can be expressed as integrals over level sets. We also give central limit theorems for some of these quantities, using chaos expansions.

In this work, we study some geometrical properties of a stationary Gaussian field modeling the sea surface, using the energy spectrum. We consider the length of a crest and the mean speed of contours, which can be expressed as integrals over level sets. We also give central limit theorems for some of these quantities, using chaos expansions.

We consider regularizations by convolution of the empirical process and study the asymptotic behaviour of non-linear functionals of this process. Using a result for the same type of non-linear functionals of the Brownian bridge, shown in a previous paper [4], and a strong approximation theorem, we prove several results for the p-deviation in estima...

Let {bF(t),t[set membership, variant][0,1]} be an F-Brownian bridge process. We study the asymptotic behaviour of non-linear functionals of regularizations by convolution of this process and apply these results to the estimation of the variance of a non-homogeneous diffusion and to the convergence of the number of crossings of a level by the regula...

We study the asymptotic behaviour of several non-linear functionals of the empirical bridge and obtain some applications to G-deviation in density estimation and to Kullback deviation convergence.

Let bF(t),t ε [0,1] be an F-Brownian bridge process. We study the asymptotic behaviour of non-linear functionals of regularizations by convolution of this process and obtain several applications. Résumé Soit bF(t),t ε [0,1] un F-pont brownien. Nous étudions le comportement asymptotique de fonctionnelles non linéaires des régularisations de ce proce...

Let {X t ,t∈[0,1]} be a centred stationary Gaussian process defined on (Ω,A,P) with covariance function satisfying r(t)∼1-C|t| 2α , 0<α<1, as t→0. Define the regularized processes X ε =φ ε *X and Y ε =X ε /σ ε , where σ ε 2 =varX t ε , φ ε is a kernel which approaches the Dirac delta function as ε→0 and * denotes the convolution. We study the conve...

Let {X t ,t∈[0,1]} be a centred stationary Gaussian process defined on (Ω,A,P) with covariance function satisfying r(t)∼1-C|t| 2α , 0<α<1, as t→0. We define the regularized processes X ε =φ ε *X and Y ε =X ε /σ ε , where σ ε 2 =varX t ε , with φ ε is a kernel that approaches Dirac's delta function. We study the convergence of Z ε (f)=ε -a(α) ∫ -∞ ∞...

We study the weak convergence of the number of level crossings to the local time of a stationary Gaussian process whose covariance does not have a second order derivative at the origin, and which has been regularized by convolution with a kernel that approaches Dirac’s delta function as the regularization parameter ε goes to zero. We consider the d...

Let (X(t), t≧0) be a centred Gaussian process with stationary increments andEX 2(t)=C 0t 2α for someC 0>0, 0<α<1, and let 0<a t ≦t be a nondecreasing function oft witha t /t nonincreasing. The asymptotic behaviour of several increment processes constructed fromX anda t is studied in terms of their upper classes.

Let {W(t), t supo \leqq t \leqq T supo < s \leqq at Ù(T - t) (W(t + s) - W(t))at - 1/2\mathop {\sup }\limits_{o \leqq t \leqq T} \mathop {\sup }\limits_{o < s \leqq a_t \wedge (T - t)} (W(t + s) - W(t))a_t^{ - 1/2}

Let {X(t), t[greater-or-equal, slanted]0} be a centred nonstationary Gaussian process with EX2(t) = C0t2[alpha] for some C0 > 0, 0<[alpha]<1, and [beta]T = 1/[sigma](aT)(2(log T/aT+log log T)1/2). In this paper the a.s. asymptotic behaviour of I(T,aT[beta]T as T-->[infinity] is studied where I(T, aT) = sup{X(t')-X(t): 0[less-than-or-equals, slant]t...

Let {Xn, n [greater-or-equal, slanted] 1} be a sequence of identically distributed random variables, Zn = max {X1,..., Xn} and {un, n [greater-or-equal, slanted] 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossi...

Let X={X(t), t∈ℝ N} be a centred Gaussian random field with covariance ℰX(t)X(s)=r(t−s) continuous on ℝN×ℝN and r(0)=1. Let σ(t,s)=(ℰ(X(t)−X(s)) 2)1/2; σ(t,s) is a pseudometric on ℝN. Assume X is σ-separable. Let D 1 be the unit cube in ℝN and for 0<k∈ℝ, D k= {x∈ℝN: k −1x∈D1}, Z(k)=sup{X(t),t∈D k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦)...

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