On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

Takemura, Akimichi; Kuriki, Shinya

doi:10.1214/aoap/1026915624

Cited by 46 publications

(65 citation statements)

References 19 publications

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“…Although neither the tube method nor the expected EC is guaranteed to be a good approximation to the probability of interest, in geometrically well behaved problems they seem to agree with each other [Siegmund and Worsley (1995) and Takemura and Kuriki (2002)] and to be reasonably accurate as judged by numerical comparisons with simulations. The present problem is poorly behaved in the sense that the rotation parameter is not identifiable when the scale parameters are equal.…”

Section: The Tubes Approachmentioning

confidence: 93%

“…The evidence for this comes from Takemura and Kuriki (2002) who show that if the random field has a terminating Karhunen-Loève expansion then the EC approach and the tubes approach always give the same answer. In our case (and most interesting cases), the expansion does not terminate, so the agreement between the two approaches is still an open question.…”

Section: Introductionmentioning

confidence: 99%

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Rotation space random fields with an application to fMRI data

Shafie¹,

Sigal²,

Siegmund³

et al. 2003

Ann. Statist.

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Siegmund and Worsley considered the problem of testing for a signal with unknown location and scale in a Gaussian random field defined on R N . The test statistic was the maximum of a Gaussian random field in an (N + 1)-dimensional "scale space," N dimensions for location and one dimension for the scale of a smoothing kernel. Siegmund and Worsley used two methods, one involving the expected Euler characteristic of the excursion set and the other involving the volume of tubes, to derive an approximate null distribution. The purpose of this paper is to extend the scale space result to the rotation space random field when N = 2, where the maximum is taken over all rotations of the filter as well as scales. We apply this result to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

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Section: The Tubes Approachmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Rotation space random fields with an application to fMRI data

Shafie¹,

Sigal²,

Siegmund³

et al. 2003

Ann. Statist.

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“…There is also significant literature in geometric combinatorics concerning Euler characteristic as a measure [5,23,24]. More recently, integrals involving Euler characteristic have arisen in analyses of the geometry of Gaussian random fields, as in [1,2,32,29]. Many of these papers appear to use integration with respect to Euler characteristic without the formal machinery.…”

Section: Results and Related Workmentioning

confidence: 99%

Target Enumeration via Euler Characteristic Integrals

Baryshnikov¹,

Ghrist²

2009

SIAM J. Appl. Math.

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We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a network of sensors, each of which measures the number of targets nearby but neither their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological sheaf integration theory -integration with respect to Euler characteristic -to yield complete solutions to these problems. KeywordsEuler characteristic, sensor network, enumeration, integration Abstract.We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a network of sensors, each of which measures the number of targets nearby but neither their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological sheaf integration theory-integration with respect to Euler characteristic-to yield complete solutions to these problems.

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“…Here the random field is approximated by a finite Karhunen-Loève expansion, and the P -value for its maximum is then the volume of a particular tube about the search region, which can be evaluated using Weyl's (1939) tube formula. Takemura and Kuriki (2002) 15 have proved this conjecture when the expansion is finite. Using results of Piterbarg, Robert Adler has shown that the expected EC is an even more precise P -value approximation than previously thought: the error is exponentially (not just polynomially) smaller than the smallest term in the expected EC expansion.…”

Section: The Roughness Of the Random Fieldmentioning

confidence: 88%

Detecting activation in fMRI data

Worsley

2003

Stat Methods Med Res

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1We present a simple approach to the analysis of fMRI data collected from several runs, sessions and subjects. We take advantage of the spatial nature of the data to reduce the noise in certain key parameters, achieving an increase in degrees of freedom for a mixed effects analysis. Our main interest is the analysis of the resulting images of test statistics using the geometry of random fields. We show how the Euler characteristic of the excursion set plays a key role in setting the threshold of the image to detect regions of the brain activated by a stimulus.

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On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

Cited by 46 publications

References 19 publications

Rotation space random fields with an application to fMRI data

Rotation space random fields with an application to fMRI data

Target Enumeration via Euler Characteristic Integrals

Detecting activation in fMRI data

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