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).

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.

A NONPARAMETRIC TEST OF SIGNIFICANT VARIABLES IN GRADIENTS

2020 ◽  
pp. 1-45
Author(s):  
Feng Yao ◽  
Taining Wang
Keyword(s):  
Null Distribution ◽  
Monte Carlo Study ◽  
Nonparametric Test ◽  
Hedonic Price ◽  
Finite Sample ◽  
Test Statistic ◽  
Bootstrap Test ◽  
Local Polynomial Estimation ◽  
Polynomial Estimation ◽  
Mean Function

We propose a nonparametric test of significant variables in the partial derivative of a regression mean function. The derivative is estimated by local polynomial estimation and the test statistic is constructed through a variation-based measure of the derivative in the direction of variables of interest. We establish the asymptotic null distribution of the test statistic and demonstrate that it is consistent. Motivated by the null distribution, we propose a wild bootstrap test, and show that it exhibits the same null distribution, whether the null is valid or not. We perform a Monte Carlo study to demonstrate its encouraging finite sample performance. An empirical application is conducted showing how the test can be applied to infer certain aspects of regression structures in a hedonic price model.

scholarly journals ULTRAMETRIC RANDOM FIELD

2006 ◽  
Vol 09 (02) ◽  
pp. 199-213 ◽  
Author(s):  
A. YU. KHRENNIKOV ◽  
S. V. KOZYREV
Keyword(s):  
Wavelet Analysis ◽  
Random Field ◽  
Stochastic Equation ◽  
Gaussian Random Field ◽  
Ultrametric Space ◽  
Ultrametric Spaces

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property.

scholarly journals Monitoring Persistence Change in Heavy-Tailed Observations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 936
Author(s):  
Dan Wang
Keyword(s):  
Kernel Method ◽  
Alternative Hypothesis ◽  
Null Distribution ◽  
Real Data ◽  
Ratio Test ◽  
Finite Sample ◽  
Test Statistic ◽  
Bootstrap Approximation ◽  
Heavy Tailed ◽  
Better Than

In this paper, a ratio test based on bootstrap approximation is proposed to detect the persistence change in heavy-tailed observations. This paper focuses on the symmetry testing problems of I(1)-to-I(0) and I(0)-to-I(1). On the basis of residual CUSUM, the test statistic is constructed in a ratio form. I prove the null distribution of the test statistic. The consistency under alternative hypothesis is also discussed. However, the null distribution of the test statistic contains an unknown tail index. To address this challenge, I present a bootstrap approximation method for determining the rejection region of this test. Simulation studies of artificial data are conducted to assess the finite sample performance, which shows that our method is better than the kernel method in all listed cases. The analysis of real data also demonstrates the excellent performance of this method.

scholarly journals The Transparent Dead Leaves Model

2012 ◽  
Vol 44 (01) ◽  
pp. 1-20 ◽  
Author(s):  
B. Galerne ◽  
Y. Gousseau
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Gaussian Random Field ◽  
Natural Images ◽  
Simulation Algorithm ◽  
New Model ◽  
Grain Model ◽  
Dead Leaves Model

In this paper we introduce the transparent dead leaves (TDL) random field, a new germ-grain model in which the grains are combined according to a transparency principle. Informally, this model may be seen as the superposition of infinitely many semitransparent objects. It is therefore of interest in view of the modeling of natural images. Properties of this new model are established and a simulation algorithm is proposed. The main contribution of the paper is to establish a central limit theorem, showing that, when varying the transparency of the grain from opacity to total transparency, the TDL model ranges from the dead leaves model to a Gaussian random field.

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