On excursion sets, tube formulas and maxima of random fields

Adler, Robert J.

doi:10.1214/aoap/1019737664

Cited by 189 publications

(196 citation statements)

References 87 publications

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“…Given the disparity maps, it is straightforward to compute the depths of the corresponding 3-D points and their heights with respect to an average plane via triangulation (see Hartley and Sturm, 1997). In the following, we will estimate directional spectra from the reconstructed wave surface and the observed wave extremes in space-time will be compared against theoretical models based on Euler Characteristics (Adler, 1981;Adler, 2000;Adler and Taylor, 2007).…”

Section: Stereo Processingmentioning

confidence: 99%

Space–time measurements of oceanic sea states

et al. 2013

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Section: Stereo Processingmentioning

confidence: 99%

Space–time measurements of oceanic sea states

et al. 2013

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“…In particular, Theorem 1 underscores the intimate connection between model selection and the distribution of the maxima of random fields. In general, the maximum of random processes and fields is a well studied challenging mathematical problem with a rich history, see for example [2], [3] and references therein. For our purposes, we shall use the following result (see Theorem 2 and Lemma 1 below): For large n,…”

Section: B a New Model Selection Criterionmentioning

confidence: 99%

Model Selection for Sinusoids in Noise: Statistical Analysis and a New Penalty Term

Nadler

Kontorovich

2011

IEEE Trans. Signal Process.

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Abstract-Detection of the number of sinusoids embedded in noise is a fundamental problem in statistical signal processing. Most parametric methods minimize the sum of a data fit (likelihood) term and a complexity penalty term. The latter is often derived via information theoretic criteria, such as minimum description length (MDL), or via Bayesian approaches including Bayesian information criterion (BIC) or maximum a-posteriori (MAP). While the resulting estimators are asymptotically consistent, empirically their finite sample performance is strongly dependent on the specific penalty term chosen. In this paper we elucidate the source of this behavior, by relating the detection performance to the extreme value distribution of the maximum of the periodogram and of related random fields. Based on this relation, we propose a combined detection-estimation algorithm with a new penalty term. Our proposed penalty term is sharp in the sense that the resulting estimator achieves a nearly constant false alarm rate. A series of simulations support our theoretical analysis and show the superior detection performance of the suggested estimator.Index Terms-sinusoids in noise, maxima of random fields, extreme value theory, periodogram, statistical hypothesis tests.

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“…However, under quite general assumptions, there are accurate asymptotic approximations for the probability of exceedance ( [11], [4], [32] for example). For our purposes, we will use the following well known result for the case of a stationary Gaussian process (proposition 4.1 in [32]).…”

Section: A Proof Of Theoremmentioning

confidence: 99%

Parametric joint detection-estimation of the number of sources in array processing

Arkind

Nadler

2010

2010 IEEE Sensor Array and Multichannel Signal Processing Workshop

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Detection of the number of signals and estimation of their directions of arrival (DOAs) are fundamental problems in signal array processing. In this paper we present three main contributions to this problem.First, we show that there is an explicit relation between model selection and the breakdown phenomena of the Maximum Likelihood estimator (MLE). Second, we provide a simple formula for the location of the breakdown of the MLE, using tools from extreme value theory. This gives an explicit formula for the signal strength required for reliable detection. Third, we apply these results and propose a new joint detection-estimation algorithm with state-of-the-art performance. We present computer simulations that demonstrate the performance of our algorithm, compared to other popular methods. Index Termsjoint detection-estimation, statistical hypothesis tests, extreme value theory, MLE breakdown phenomena.

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On excursion sets, tube formulas and maxima of random fields

Cited by 189 publications

References 87 publications

Space–time measurements of oceanic sea states

Space–time measurements of oceanic sea states

Model Selection for Sinusoids in Noise: Statistical Analysis and a New Penalty Term

Parametric joint detection-estimation of the number of sources in array processing

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