Conjunction Probability of Smooth Centered Gaussian Processes
- PDF / 305,202 Bytes
- 10 Pages / 439.642 x 666.49 pts Page_size
- 55 Downloads / 225 Views
Conjunction Probability of Smooth Centered Gaussian Processes Viet-Hung Pham1 Received: 14 December 2018 / Revised: 13 June 2019 / Accepted: 28 June 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper we provide an upper bound for the conjunction probability of independent Gaussian smooth processes, and then, we prove that this bound is a good approximation with exponentially smaller error. Our result confirms the heuristic approximation by Euler characteristic method of Worsley and Friston and also implies the exact value of generalized Pickands constant in a special case. Some results for conjunction probability of correlated processes are also discussed. Keywords Conjunction probability · Gaussian processes · Pickands constant · Euler characteristic method · Rice formula Mathematics Subject Classification (2010) 60G15 · 60G60 · 62G09
1 Introduction In this paper, we investigate the conjunction probability of independent Gaussian processes, that is P
sup min Xi (t) ≥ u ,
t∈[0,T ]1≤i≤n
(1)
where u is a fixed threshold, and Xi ’s are the independent smooth centered Gaussian processes with unit variance. In a more general setting where Xi ’s are random fields defined on Rd , this problem has been addressed by Worsley and Friston in the seminal contribution [9] with the statistical application to test whether the functional organization of the brain for language differs according to sex. With the same application to fRMI data, Alodat [1] was interested in the distribution of the duration of the conjunction time.
Viet-Hung Pham
[email protected]; [email protected] 1
Institute of Mathematics, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam
Viet-Hung Pham
Most published papers [4, 5, 9] assumed more that the processes Xi ’s are stationary with the covariance functions ri (.), 1 ≤ i ≤ n, satisfying ri (t) = 1 − Ci t 2 + o(t 2 ) as t → 0, and ri (t) < 1, ∀t ∈ (0, T ], where Ci ’s are positive constants. In this case, Debicki et al. [4] introduced the generalized Pickands constant 1 HC1 ,...,Cn = lim P maxZ(ak) ≤ 0 , a↓0 a k≥1 where √ Z(t) = min 2Bi ( Ci t) − Ci t 2 + Ei , 1≤i≤n
with Bi ’s being independent copies of a centered Gaussian process B(t) with the covariance function Cov(B(t), B(s)) = |ts|, ∀t, s ≥ 0, and Ei ’s being mutually independent unit mean exponential random variables and also independent of Bi ’s. Using the double-sum method, they proved the asymptotic formula ϕ n (u) P sup min Xi (t) > u = HC1 ,...,Cn T n−1 (1 + o(1)), u t∈[0,T ]1≤i≤n where ϕ(.) is the density function of the standard normal distribution. However, one main disadvantage in statistical application of this result is the difficulty to estimate the exact value of the generalized Pickands constant HC1 ,...,Cn . Worsley and Friston [9] followed an heuristic argument that as the threshold u is large enough, then the Euler characteristic χ (Cu ) of the excursion set Cu =
Data Loading...
