arXiv subjects

Ansgar Steland

Publications and source records attributed to Ansgar Steland.

34 records · Page 2Linked to original sources

arXiv2018

Automatic Processing and Solar Cell Detection in Photovoltaic Electroluminescence Images

Electroluminescence (EL) imaging is a powerful and established technique for assessing the quality of photovoltaic (PV) modules, which consist of many electrically connected solar cells arranged in a grid. The analysis of imperfect real-world images requires reliable methods for preprocessing, detection and extraction of the cells. We propose several methods for those tasks, which, however, can be modified to related imaging problems where similar geometric objects need to be detected accurately. Allowing for images taken under difficult outdoor conditions, we present methods to correct for rotation and perspective distortions. The next important step is the extraction of the solar cells of a PV module, for instance to pass them to a procedure to detect and analyze defects on their surface. We propose a method based on specialized Hough transforms, which allows to extract the cells even when the module is surrounded by disturbing background and a fast method based on cumulated sums (CUSUM) change detection to extract the cell area of single-cell mini-module, where the correction of perspective distortion is implicitly done. The methods are highly automatized to allow for big data analyses. Their application to a large database of EL images substantiates that the methods work reliably on a large scale for real-world images. Simulations show that the approach achieves high accuracy, reliability and robustness. This even holds for low contrast images as evaluated by comparing the simulated accuracy for a low and a high contrast image.

cs.CV↗

arXiv2018

On Convergence of Moments for Approximating Processes and Applications to Surrogate Models

We study critera for a pair $ (\{ X_n \} $, $ \{ Y_n \}) $ of approximating processes which guarantee closeness of moments by generalizing known results for the special case that $ Y_n = Y $ for all $n$ and $ X_n $ converges to $Y$ in probability. This problem especially arises when working with surrogate models, e.g. to enrich observed data by simulated data, where the surrogates $Y_n$'s are constructed to justify that they approximate the $ X_n $'s. The results of this paper deal with sequences of random variables. Since this framework does not cover many applications where surrogate models such as deep neural networks are used to approximate more general stochastic processes, we extend the results to the more general framework of random fields of stochastic processes. This framework especially covers image data and sequences of images. We show that uniform integrability is sufficient, and this holds even for the case of processes provided they satisfy a weak stationarity condition.

stat.ML↗

arXiv2018

Estimation of the asymptotic variance of univariate and multivariate random fields and statistical inference

Correlated random fields are a common way to model dependence struc- tures in high-dimensional data, especially for data collected in imaging. One important parameter characterizing the degree of dependence is the asymp- totic variance which adds up all autocovariances in the temporal and spatial domain. Especially, it arises in the standardization of test statistics based on partial sums of random fields and thus the construction of tests requires its estimation. In this paper we propose consistent estimators for this parameter for strictly stationary ϕ-mixing random fields with arbitrary dimension of the domain and taking values in a Euclidean space of arbitrary dimension, thus allowing for multivariate random fields. We establish consistency, provide cen- tral limit theorems and show that distributional approximations of related test statistics based on sample autocovariances of random fields can be obtained by the subsampling approach. As in applications the spatial-temporal correlations are often quite local, such that a large number of autocovariances vanish or are negligible, we also investigate a thresholding approach where sample autocovariances of small magnitude are omitted. Extensive simulation studies show that the proposed estimators work well in practice and, when used to standardize image test statistics, can provide highly accurate image testing procedures.

math.ST↗

arXiv2017

Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage

We establish large sample approximations for an arbitray number of bilinear forms of the sample variance-covariance matrix of a high-dimensional vector time series using $ \ell_1$-bounded and small $\ell_2$-bounded weighting vectors. Estimation of the asymptotic covariance structure is also discussed. The results hold true without any constraint on the dimension, the number of forms and the sample size or their ratios. Concrete and potential applications are widespread and cover high-dimensional data science problems such as tests for large numbers of covariances, sparse portfolio optimization and projections onto sparse principal components or more general spanning sets as frequently considered, e.g. in classification and dictionary learning. As two specific applications of our results, we study in greater detail the asymptotics of the trace functional and shrinkage estimation of covariance matrices. In shrinkage estimation, it turns out that the asymptotics differs for weighting vectors bounded away from orthogonaliy and nearly orthogonal ones in the sense that their inner product converges to 0.

math.PR↗

arXiv2017

Large-sample approximations for variance-covariance matrices of high-dimensional time series

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly $ \ell_1 $-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high--dimensional time series, which matter for the analysis of big data and are discussed in greater detail.

math.PR↗

arXiv2017

Sequential Detection of Three-Dimensional Signals under Dependent Noise

We study detection methods for multivariable signals under dependent noise. The main focus is on three-dimensional signals, i.e. on signals in the space-time domain. Examples for such signals are multifaceted. They include geographic and climatic data as well as image data, that are observed over a fixed time horizon. We assume that the signal is observed as a finite block of noisy samples whereby we are interested in detecting changes from a given reference signal. Our detector statistic is based on a sequential partial sum process, related to classical signal decomposition and reconstruction approaches applied to the sampled signal. We show that this detector process converges weakly under the no change null hypothesis that the signal coincides with the reference signal, provided that the spatial-temporal partial sum process associated to the random field of the noise terms disturbing the sampled signal con- verges to a Brownian motion. More generally, we also establish the limiting distribution under a wide class of local alternatives that allows for smooth as well as discontinuous changes. Our results also cover extensions to the case that the reference signal is unknown. We conclude with an extensive simulation study of the detection algorithm.

math.PR↗

arXiv2016

On the Accuracy of Fixed Sampled and Fixed Width Confidence Intervals Based on the Vertically Weighted Averages

Vertically weighted averages perform a bilateral filtering of data, in order to preserve fine details of the underlying signal, especially discontinuities such as jumps (in dimension one) or edges (in dimension two). In homogeneous regions of the domain the procedure smoothes the data by averaging nearby data points to reduce the noise, whereas in inhomogenous regions the neighboring points are only taken into account when their value is close to the current one. This results in a denoised reconstruction or estimate of the true signal without blurring finer details. This paper addresses the lack of results about the construction and evaluation of confidence intervals based on the vertically weighted average, which is required for a proper statistical evaluation of its estimation accuracy. Based on recent results we discuss and investigate in greater detail fixed sample as well as fixed width (conditional) confidence intervals constructed from this estimator. The fixed width approach allows to specify explicitly the estimator's accuracy and determines a random sample size to ensure the required coverage probability. This also fixes to some extent the inherent property of the vertically weighted average that its variability is higher in low-density regions than in high-density regions. To estimate the variances required to construct the procedures, we rely on resampling techniques, especially the bootstrap and the jackknife. Extensive Monte Carlo simulations show that the proposed confidence intervals are reliable in terms of their coverage probabilities for a wide range of parameter settings.

stat.ME↗

arXiv2014

Sampling Plans for Control-Inspection Schemes Under Independent and Dependent Sampling Designs With Applications to Photovoltaics

The evaluation of produced items at the time of delivery is, in practice, usually amended by at least one inspection at later time points. We extend the methodology of acceptance sampling for variables for arbitrary unknown distributions when additional sampling infor- mation is available to such settings. Based on appropriate approximations of the operating characteristic, we derive new acceptance sampling plans that control the overall operating characteristic. The results cover the case of independent sampling as well as the case of dependent sampling. In particular, we study a modified panel sampling design and the case of spatial batch sampling. The latter is advisable in photovoltaic field monitoring studies, since it allows to detect and analyze local clusters of degraded or damaged modules. Some finite sample properties are examined by a simulation study, focusing on the accuracy of estimation.

math.ST↗

arXiv2012

Sequential Cross-Validated Bandwidth Selection Under Dependence and Anscombe-Type Extensions to Random Time Horizons

To detect changes in the mean of a time series, one may use previsible detection procedures based on nonparametric kernel prediction smoothers which cover various classic detection statistics as special cases. Bandwidth selection, particularly in a data-adaptive way, is a serious issue and not well studied for detection problems. To ensure data adaptation, we select the bandwidth by cross-validation, but in a sequential way leading to a functional estimation approach. This article provides the asymptotic theory for the method under fairly weak assumptions on the dependence structure of the error terms, which cover, e.g., GARCH($p,q$) processes, by establishing (sequential) functional central limit theorems for the cross-validation objective function and the associated bandwidth selector. It turns out that the proof can be based in a neat way on \cite{KurtzProtter1996}'s results on the weak convergence of \ito integrals and a diagonal argument. Our gradual change-point model covers multiple change-points in that it allows for a nonlinear regression function after the first change-point possibly with further jumps and Lipschitz continuous between those discontinuities. In applications, the time horizon where monitoring stops latest is often determined by a random experiment, e.g. a first-exit stopping time applied to a cumulated cost process or a risk measure, possibly stochastically dependent from the monitored time series. Thus, we also study that case and establish related limit theorems in the spirit of \citet{Anscombe1952}'s result. The result has various applications including statistical parameter estimation and monitoring financial investment strategies with risk-controlled early termination, which are briefly discussed.

math.PR↗

arXiv2010

Sequential Data-Adaptive Bandwidth Selection by Cross-Validation for Nonparametric Prediction

We consider the problem of bandwidth selection by cross-validation from a sequential point of view in a nonparametric regression model. Having in mind that in applications one often aims at estimation, prediction and change detection simultaneously, we investigate that approach for sequential kernel smoothers in order to base these tasks on a single statistic. We provide uniform weak laws of large numbers and weak consistency results for the cross-validated bandwidth. Extensions to weakly dependent error terms are discussed as well. The errors may be α-mixing or L2-near epoch dependent, which guarantees that the uniform convergence of the cross validation sum and the consistency of the cross-validated bandwidth hold true for a large class of time series. The method is illustrated by analyzing photovoltaic data.

math.ST↗

arXiv2010

Optimal Sequential Kernel Detection for Dependent Processes

In many applications one is interested to detect certain (known) patterns in the mean of a process with smallest delay. Using an asymptotic framework which allows to capture that feature, we study a class of appropriate sequential nonparametric kernel procedures under local nonparametric alternatives. We prove a new theorem on the con- vergence of the normed delay of the associated sequential detection procedure which holds for dependent time series under a weak mixing condition. The result suggests a simple procedure to select a kernel from a finite set of candidate kernels, and therefore may also be of interest from a practical point of view. Further, we provide two new theorems about the existence and an explicit representation of optimal kernels minimizing the asymptotic normed delay. The results are illustrated by some examples.

math.ST↗

arXiv2010

Monitoring Procedures to Detect Unit Roots and Stationarity

When analysing time series an important issue is to decide whether the time series is stationary or a random walk. Relaxing these notions, we consider the problem to decide in favor of the I(0)- or I(1)-property. Fixed-sample statistical tests for that problem are well studied in the literature. In this paper we provide first results for the problem to monitor sequentially a time series. Our stopping times are based on a sequential version of a kernel-weighted variance-ratio statistic. The asymptotic distributions are established for I(1) processes, a rich class of stationary processes, possibly affected by local nonpara- metric alternatives, and the local-to-unity model. Further, we consider the two interesting change-point models where the time series changes its behaviour after a certain fraction of the observations and derive the associated limiting laws. Our Monte-Carlo studies show that the proposed detection procedures have high power when interpreted as a hypothesis test, and that the decision can often be made very early.

math.ST↗

arXiv2010

Weighted Dickey-Fuller Processes for Detecting Stationarity

Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey-Fuller (DF) processes, and related stopping times, which may be called weighted Dickey-Fuller control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For gen- eral dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance pa- rameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation.

math.PR↗

arXiv2010

A Binary Control Chart to Detect Small Jumps

The classic N p chart gives a signal if the number of successes in a sequence of inde- pendent binary variables exceeds a control limit. Motivated by engineering applications in industrial image processing and, to some extent, financial statistics, we study a simple modification of this chart, which uses only the most recent observations. Our aim is to construct a control chart for detecting a shift of an unknown size, allowing for an unknown distribution of the error terms. Simulation studies indicate that the proposed chart is su- perior in terms of out-of-control average run length, when one is interest in the detection of very small shifts. We provide a (functional) central limit theorem under a change-point model with local alternatives which explains that unexpected and interesting behavior. Since real observations are often not independent, the question arises whether these re- sults still hold true for the dependent case. Indeed, our asymptotic results work under the fairly general condition that the observations form a martingale difference array. This enlarges the applicability of our results considerably, firstly, to a large class time series models, and, secondly, to locally dependent image data, as we demonstrate by an example.

stat.ME↗

arXiv2010

Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models

The question whether a time series behaves as a random walk or as a station- ary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This paper studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a monitoring procedure given by a control chart, i.e., a stopping time, which is related to a well known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process which implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.

math.PR↗

arXiv2010

Random walks - a sequential approach

In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative.

math.ST↗