Model fields in crossing theory: a weak convergence perspective

1988 ◽  
Vol 20 (4) ◽  
pp. 756-774
Author(s):  
Richard J. Wilson
Keyword(s):  
Weak Convergence ◽  
Random Field ◽  
Lebesgue Measure ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Asymptotic Results ◽  
Model Field ◽  
Fixed Level ◽  
Excursion Set ◽  
Crossing Theory

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

Model fields in crossing theory: a weak convergence perspective

1988 ◽  
Vol 20 (04) ◽  
pp. 756-774
Author(s):  
Richard J. Wilson
Keyword(s):  
Weak Convergence ◽  
Random Field ◽  
Lebesgue Measure ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Asymptotic Results ◽  
Model Field ◽  
Fixed Level ◽  
Excursion Set ◽  
Crossing Theory

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

The structure of Gaussian fields near a level crossing

1982 ◽  
Vol 14 (03) ◽  
pp. 543-565 ◽  
Author(s):  
Richard J. Wilson ◽  
Robert J. Adler
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Final Section ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Gaussian Fields ◽  
Level Crossing ◽  
Model Field ◽  
Definition Of

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

The structure of Gaussian fields near a level crossing

1982 ◽  
Vol 14 (3) ◽  
pp. 543-565 ◽  
Author(s):  
Richard J. Wilson ◽  
Robert J. Adler
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Final Section ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Gaussian Fields ◽  
Level Crossing ◽  
Model Field ◽  
Definition Of

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI

2001 ◽  
Vol 33 (4) ◽  
pp. 773-793 ◽  
Author(s):  
Keith J. Worsley
Keyword(s):  
Random Field ◽  
Gaussian Random Field ◽  
Null Distribution ◽  
Scale Space ◽  
Functional Magnetic Resonance ◽  
Test Statistic ◽  
Brain Images ◽  
Smoothing Filter ◽  
Excursion Set ◽  
Special Case

Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN. The test statistic was the maximum of a Gaussian random field in an N+1 dimensional ‘scale space’, N dimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution for N=2 and N=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2 fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI

2001 ◽  
Vol 33 (04) ◽  
pp. 773-793 ◽  
Author(s):  
Keith J. Worsley
Keyword(s):  
Random Field ◽  
Gaussian Random Field ◽  
Null Distribution ◽  
Scale Space ◽  
Functional Magnetic Resonance ◽  
Test Statistic ◽  
Brain Images ◽  
Smoothing Filter ◽  
Excursion Set ◽  
Special Case

Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN. The test statistic was the maximum of a Gaussian random field in anN+1 dimensional ‘scale space’,Ndimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution forN=2 andN=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

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