The Geometry of Random Fields

Adler, Robert J.

doi:10.1137/1.9780898718980

Cited by 805 publications

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“…In the following, we will show that WASS measurements allow for a complete statistical analysis of wave extremes exploiting Adler's Euler Characteristics (Adler, 1981; see also Piterbarg, 1995). Drawing from Fedele et al (2011a,b) and Fedele (2012) we first consider wave surface maxima that occur over a given area at a fixed instant of time (space extremes) and then extend the analysis to global maxima over an area during the sea state duration (space-time extremes).…”

Section: Space-time Extreme Value Analysismentioning

confidence: 98%

“…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%

“…If the threshold is high, then all the holes tend to disappear and the EC counts the number of connected components, or local maxima of the wave surface. For weakly nonlinear wave surfaces, the expected value of the EC is given by (Adler, 1981;Worsley, 1995;Adler and Tay also Fedele et al 2011a;Fedele et al 2011b;Fedele, 2012) (12) where lor, 2007, see…”

Section: Space Extremesmentioning

confidence: 99%

“…The EC b term in (12) will be referred to as a boundary correction because it is important when the excursion set touches the boundary/perimeter of the region A (Worsley 1995). Adler (1981) and Adler and Taylor (2007) have shown that the probability that the global maximum of a random field η exceeds a threshold h is well approximated by the expected EC of the excursion set U η,h , provided the threshold is high. Indeed, as the threshold h increases, the holes in the excursion set disappear until each of its connected components includes just one local maximum, and the EC counts the number of local maxima.…”

Section: Space Extremesmentioning

confidence: 99%

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Space–time measurements of oceanic sea states

et al. 2013

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Section: Space-time Extreme Value Analysismentioning

confidence: 98%

Section: Stereo Processingmentioning

confidence: 99%

Section: Space Extremesmentioning

confidence: 99%

Section: Space Extremesmentioning

confidence: 99%

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Space–time measurements of oceanic sea states

et al. 2013

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“…Euler characteristic derivations of Adler (1981) and Worsley (1994) at a significance level of 5%. The methodologies developed in this paper were applied to these maps.…”

Section: Experimental and Imaging Setupmentioning

confidence: 99%

A re-defined and generalized percent-overlap-of-activation measure for studies of fMRI reproducibility and its use in identifying outlier activation maps

Maitra

2010

NeuroImage

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Functional Magnetic Resonance Imaging (fMRI) is a popular noninvasive modality to investigate activation in the human brain. The end result of most fMRI experiments is an activation map corresponding to the given paradigm. These maps can vary greatly from one study to the next, so quantifying the reliability of identified activation over several fMRI studies is important. The percent overlap of activation (Rombouts et al., 1998 andMachielsen et al., 2000) is a global reliability measure between activation maps drawn from any two fMRI studies. A slightly modified but more intuitive measure is provided by the Jaccard (1901) coefficient of similarity, whose use we study in this paper. A generalization of these measures is also proposed to comprehensively summarize the reliability of multiple fMRI studies. Finally, a testing mechanism to flag potentially anomalous studies is developed. The methodology is illustrated on studies involving left-and righthand motor task paradigms performed by a right-hand dominant male subject several times over a period of two months, with excellent results. KeywordsEigenvalues, Reliability, Intra-class correlation coefficient, Outlier detection, Percent overlap, Principal components, Finger-thumb opposition experiment, Summarized multiple Jaccard similarity coefficient, Dice coefficient, Perron-Frobenius theorem Disciplines Statistics and ProbabilityComments NOTICE:this is the author's version of a work that was accepted for publication in NeuroImage. Changes resulting from the publishing process, such as peer review, editing, corrections structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in NeuroImage, [v.50, iss.1,(2010) AbstractFunctional Magnetic Resonance Imaging (fMRI) is a popular non-invasive modality to investigate activation in the human brain. The end result of most fMRI experiments is an activation map corresponding to the given paradigm. These maps can vary greatly from one study to the next, so quantifying the reliability of identified activation over several fMRI studies is important. The percent overlap of activation (Rombouts et al., 1998;Machielsen et al., 2000) is a global reliability measure between activation maps drawn from any two fMRI studies. A slightly modified but more intuitive measure is provided by the Jaccard (1901) coefficient of similarity, whose use we study in this paper. A generalization of these measures is also proposed to comprehensively summarize the reliability of multiple fMRI studies. Finally, a testing mechanism to flag potentially anomalous studies is developed. The methodology is illustrated on studies involving left-and right-hand motor task paradigms performed by a right-hand dominant male subject several times over a period of two months, with excellent results.

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Conditions for successful data assimilation

Chorin

Morzfeld

2013

JGR Atmospheres

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[1] We show, using idealized models, that numerical data assimilation can be successful only if an effective dimension of the problem is not excessive. This effective dimension depends on the noise in the model and the data, and in physically reasonable problems, it can be moderate even when the number of variables is huge. We then analyze several data assimilation algorithms, including particle filters and variational methods. We show that well-designed particle filters can solve most of those data assimilation problems that can be solved in principle and compare the conditions under which variational methods can succeed to the conditions required of particle filters. We also discuss the limitations of our analysis.

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The Geometry of Random Fields

Cited by 805 publications

References 0 publications

Space–time measurements of oceanic sea states

Space–time measurements of oceanic sea states

A re-defined and generalized percent-overlap-of-activation measure for studies of fMRI reproducibility and its use in identifying outlier activation maps

Conditions for successful data assimilation

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