arXiv subjects

Alexander Aue

Publications and source records attributed to Alexander Aue.

34 records · Page 2Linked to original sources

arXiv2017

Functional data analysis in the Banach space of continuous functions

Functional data analysis is typically conducted within the $L^2$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same time, there have recently been a number of publications that sidestep dimension reduction steps and focus on a fully functional $L^2$-methodology. This paper goes one step further and develops data analysis methodology for functional time series in the space of all continuous functions. The work is motivated by the fact that objects with rather different shapes may still have a small $L^2$-distance and are therefore identified as similar when using an $L^2$-metric. However, in applications it is often desirable to use metrics reflecting the visualization of the curves in the statistical analysis. The methodological contributions are focused on developing two-sample and change-point tests as well as confidence bands, as these procedures appear do be conducive to the proposed setting. Particular interest is put on relevant differences; that is, on not trying to test for exact equality, but rather for pre-specified deviations under the null hypothesis. The procedures are justified through large-sample theory. To ensure practicability, non-standard bootstrap procedures are developed and investigated addressing particular features that arise in the problem of testing relevant hypotheses. The finite sample properties are explored through a simulation study and an application to annual temperature profiles.

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arXiv2017

Detecting and dating structural breaks in functional data without dimension reduction

Methodology is proposed to uncover structural breaks in functional data that is "fully functional" in the sense that it does not rely on dimension reduction techniques. A thorough asymptotic theory is developed for a fully functional break detection procedure as well as for a break date estimator, assuming a fixed break size and a shrinking break size. The latter result is utilized to derive confidence intervals for the unknown break date. The main results highlight that the fully functional procedures perform best under conditions when analogous fPCA based estimators are at their worst, namely when the feature of interest is orthogonal to the leading principal components of the data. The theoretical findings are confirmed by means of a Monte Carlo simulation study in finite samples. An application to annual temperature curves illustrates the practical relevance of the proposed procedures.

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arXiv2017

Estimating functional time series by moving average model fitting

Functional time series have become an integral part of both functional data and time series analysis. Important contributions to methodology, theory and application for the prediction of future trajectories and the estimation of functional time series parameters have been made in the recent past. This paper continues this line of research by proposing a first principled approach to estimate invertible functional time series by fitting functional moving average processes. The idea is to estimate the coefficient operators in a functional linear filter. To do this a functional Innovations Algorithm is utilized as a starting point to estimate the corresponding moving average operators via suitable projections into principal directions. In order to establish consistency of the proposed estimators, asymptotic theory is developed for increasing subspaces of these principal directions. For practical purposes, several strategies to select the number of principal directions to include in the estimation procedure as well as the choice of order of the functional moving average process are discussed. Their empirical performance is evaluated through simulations and an application to vehicle traffic data.

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arXiv2016

Piecewise quantile autoregressive modeling for nonstationary time series

We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is defined in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is done with the use of genetic algorithms. If the data generating process follows indeed a piecewise quantile autoregression structure, we show that our method is consistent for estimating the break points and the autoregressive parameters. Empirical work suggests that the proposed method performs well in finite samples.

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arXiv2016

Consistent Estimation for Partition-wise Regression and Classification Models

Partition-wise models offer a flexible approach for modeling complex and multidimensional data that are capable of producing interpretable results. They are based on partitioning the observed data into regions, each of which is modeled with a simple submodel. The success of this approach highly depends on the quality of the partition, as too large a region could lead to a non-simple submodel, while too small a region could inflate estimation variance. This paper proposes an automatic procedure for choosing the partition (i.e., the number of regions and the boundaries between regions) as well as the submodels for the regions. It is shown that, under the assumption of the existence of a true partition, the proposed partition estimator is statistically consistent. The methodology is demonstrated for both regression and classification problems.

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arXiv2015

Functional generalized autoregressive conditional heteroskedasticity

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of ARCH and GARCH processes. More recently multivariate variants of these processes have been in the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Due to the need for estimation of many parameters, however, these models may not be suitable for modeling now prevalent high-frequency volatility data. One potentially useful way to bypass these issues is to take a functional approach. In this paper, theory is developed for a new functional version of the generalized autoregressive conditionally heteroskedastic process, termed fGARCH. The main results are concerned with the structure of the fGARCH(1,1) process, providing criteria for the existence of a strictly stationary solutions both in the space of square-integrable and continuous functions. An estimation procedure is introduced and its consistency verified. A small empirical study highlights potential applications to intraday volatility estimation.

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arXiv2015

Reaction times of monitoring schemes for ARMA time series

This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples.

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arXiv2015

Spectral analysis of linear time series in moderately high dimensions

This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $p\times p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.

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arXiv2015

On the Marčenko-Pastur law for linear time series

This paper is concerned with extensions of the classical Marčenko-Pastur law to time series. Specifically, $p$-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed (real- or complex-valued) entries possessing zero mean, unit variance and finite fourth moments. The coefficient matrices of the linear process are assumed to be simultaneously diagonalizable. In this setting, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting $p/n\to c\in (0,\infty)$ for which dimension $p$ and sample size $n$ diverge to infinity at the same rate. The results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case.

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arXiv2014

Automatic estimation of flux distributions of astrophysical source populations

In astrophysics a common goal is to infer the flux distribution of populations of scientifically interesting objects such as pulsars or supernovae. In practice, inference for the flux distribution is often conducted using the cumulative distribution of the number of sources detected at a given sensitivity. The resulting "$\log(N>S)$-$\log (S)$" relationship can be used to compare and evaluate theoretical models for source populations and their evolution. Under restrictive assumptions the relationship should be linear. In practice, however, when simple theoretical models fail, it is common for astrophysicists to use prespecified piecewise linear models. This paper proposes a methodology for estimating both the number and locations of "breakpoints" in astrophysical source populations that extends beyond existing work in this field. An important component of the proposed methodology is a new interwoven EM algorithm that computes parameter estimates. It is shown that in simple settings such estimates are asymptotically consistent despite the complex nature of the parameter space. Through simulation studies it is demonstrated that the proposed methodology is capable of accurately detecting structural breaks in a variety of parameter configurations. This paper concludes with an application of our methodology to the Chandra Deep Field North (CDFN) data set.

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arXiv2014

On the prediction of stationary functional time series

This paper addresses the prediction of stationary functional time series. Existing contributions to this problem have largely focused on the special case of first-order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology. It is shown here how standard multivariate prediction techniques can be utilized in this context. The connection between functional and multivariate predictions is made precise for the important case of vector and functional autoregressions. The proposed method is easy to implement, making use of existing statistical software packages, and may therefore be attractive to a broader, possibly non-academic, audience. Its practical applicability is enhanced through the introduction of a novel functional final prediction error model selection criterion that allows for an automatic determination of the lag structure and the dimensionality of the model. The usefulness of the proposed methodology is demonstrated in a simulation study and an application to environmental data, namely the prediction of daily pollution curves describing the concentration of particulate matter in ambient air. It is found that the proposed prediction method often significantly outperforms existing methods.

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arXiv2013

Limit Laws in Transaction-Level Asset Price Models

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility, and non-trading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. We also obtain the asymptotic distribution of the ordinary least-squares estimator of the cointegrating parameter based on data sampled from an equally-spaced discretization of calendar time, in the case of weak fractional cointegration. For this same case, we obtain the asymptotic distribution for a tapered estimator under more

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arXiv2012

On image segmentation using information theoretic criteria

Image segmentation is a long-studied and important problem in image processing. Different solutions have been proposed, many of which follow the information theoretic paradigm. While these information theoretic segmentation methods often produce excellent empirical results, their theoretical properties are still largely unknown. The main goal of this paper is to conduct a rigorous theoretical study into the statistical consistency properties of such methods. To be more specific, this paper investigates if these methods can accurately recover the true number of segments together with their true boundaries in the image as the number of pixels tends to infinity. Our theoretical results show that both the Bayesian information criterion (BIC) and the minimum description length (MDL) principle can be applied to derive statistically consistent segmentation methods, while the same is not true for the Akaike information criterion (AIC). Numerical experiments were conducted to illustrate and support our theoretical findings.

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arXiv2009

Break detection in the covariance structure of multivariate time series models

In this paper, we introduce an asymptotic test procedure to assess the stability of volatilities and cross-volatilites of linear and nonlinear multivariate time series models. The test is very flexible as it can be applied, for example, to many of the multivariate GARCH models established in the literature, and also works well in the case of high dimensionality of the underlying data. Since it is nonparametric, the procedure avoids the difficulties associated with parametric model selection, model fitting and parameter estimation. We provide the theoretical foundation for the test and demonstrate its applicability via a simulation study and an analysis of financial data. Extensions to multiple changes and the case of infinite fourth moments are also discussed.

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arXiv2008

Testing for changes in polynomial regression

We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value distribution of a maximum-type test statistic which is asymptotically equivalent to the maximally selected likelihood ratio. The resulting test is easy to apply and has good size and power, even in small samples.

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arXiv2008

Selection from a stable box

Let $\{X_j\}$ be independent, identically distributed random variables. It is well known that the functional CUSUM statistic and its randomly permuted version both converge weakly to a Brownian bridge if second moments exist. Surprisingly, an infinite-variance counterpart does not hold true. In the present paper, we let $\{X_j\}$ be in the domain of attraction of a strictly $α$-stable law, $α\in(0,2)$. While the functional CUSUM statistics itself converges to an $α$-stable bridge and so does the permuted version, provided both the $\{X_j\}$ and the permutation are random, the situation turns out to be more delicate if a realization of the $\{X_j\}$ is fixed and randomness is restricted to the permutation. Here, the conditional distribution function of the permuted CUSUM statistics converges in probability to a random and nondegenerate limit.

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