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Approximations for a conditional two-dimensional scan statistic
Statistics & Probability Letters 58 (2002) 287–296
Approximations for a conditional two-dimensional scan statistic Jie Chena , Joseph Glazb; ∗ a
Department of Computing Services, University of Massachusetts Boston, Boston, MA 02125, USA b Department of Statistics, University of Connecticut, Storrs, CT 06269, USA Received May 2001
Abstract Let Xi; j ; 1 6 i 6 n1 ; 1 6 j 6 n2 , be a sequence of independent and identically distributed nonnegative integer valued random variables. The observation Xi; j denotes the number of events that have occurred in the i; jth location in a two dimensional rectangular region R. For 2 6 mi 6 ni − 1; i = 1; 2, the two dimensional discrete scan statistic is de4ned as the maximum number of events in any of the m1 by m2 consecutive rectangular windows in that region. Conditional on the total number of events that have occurred in R, we refer to this scan statistic as the conditional two-dimensional scan statistic. Two-dimensional scan statistics have been extensively used in many areas of science to analyze the occurrence of observed clusters of events in space. Since this scan statistic is based on highly dependent consecutive subsequences of observed data, accurate approximations for its distributions are of great value. In this article, based on the scanning window representation of the scan statistic, accurate product-type, Poisson and a compound Poisson approximations are investigated. Moreover, accurate approximations for the expected size of the scan statistic are derived. Numerical results are presented to evaluate the performance of the approximations c 2002 Elsevier Science B.V. All rights reserved. discussed in this article. Keywords: Compound Poisson approximation; Moving window detection; Poisson approximations; Product-type approximations; Testing for randomness in two dimensions
1. Introduction Let [0; T1 ] × [0; T2 ] be rectangular region. Let hi = Ti =ni ¿ 0; where ni are positive integers, i = 1; 2. In many applications the exact locations of the observed events in the region are unknown. What is Research was supported in part by the O@ce of Naval Research Contract No. N00014-94-1-0061 and the University of Connecticut Research Foundation. ∗ Corresponding author. E-mail address: [email protected] (J. Glaz).
c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 2 2 - 0
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usually available are the counts in small rectangular subregions. For 1 6 i 6 n1 and 1 6 j 6 n2 , let Xij be the number of events that have been observed in the rectangular subregion [(i − 1)h1 ; ih1 ] × [(j − 1)h2 ; jh2 ]. We are interested in detecting unusual clustering of these events under the null hypothesis that Xij are independent and identically distributed nonnegative integer valued random variables from a speci4ed distribution. For 2 6 m1 6 n1 − 1; 2 6 m2 6 n2 − 1; 1 6 i1 6 n1 − m1 + 1 and 1 6 i2 6 n2 − m2 + 1 de4ne Yi1 ;i2 =
i2 +m 2 −1 i1 +m 1 −1 j=i2
Xij
(1)
j=i1
to be the number of events in a rectangular region comprised of m1 by m2 adjacent rectangular subregions with area h1 h2 and the south west corner located at the point ((i1 − 1)h1 ; (i2 − 1)h2 ). If Yi1 ;i2 equals or exceeds a preassigned value of k, we will say that k events are clustered within the inspected region. A two-dimensional discrete scan statistic (Chen and Glaz, 1996) is de4ned as the largest number of events in any m1 by m2 adjacent rectangular subregions with area h1 h2 and the south west corner located at the point ((i1 − 1)h1 ; (i2 − 1)h2 ): Sm1 ; m2 = max{Yi1 ; i2 ; 1 6 i1 6 n1 − m1 + 1; 1 6 i2 6 n2 − m2 + 1}:
(2)
Sm1 ; m2 can be viewed as an extension of the one-dimensional discrete scan statistic discussed in Glaz and Naus (1991). It is used for testing the null hypothesis of randomness that assumes the Xij ’s are iid binomial random variables with parameters L and 0 ¡ p ¡ 1 or iid Poisson random variables with mean ¿ 0; respectively. For the alternative hypothesis of clustering we specify a rectangular subregion R(i1 ; i2 ) = [(i1 − 1)h1 ; (i1 + m1 − 1)h1 ] × [(i2 − 1)h2 ; (i2 + m2 − 1)h2 ] such that for any i1 6 i 6 i1 + m1 − 1 and i2 6 j 6 i2 + m2 − 1; Xij has a binomial distribution with parameters L and p1 , where p1 ¿ p or a Poisson distribution with mean 1 where 1 ¿ , respectively. For ij ∈ [i1 ; i1 + m1 − 1] × [i2 ; i2 + m2 − 1]; Xij is distributed according to the distribution speci4ed by the null hypothesis. It is routine to verify that the generalized likelihood ratio test rejects the null hypothesis in favor of the alternative hypothesis whenever Sm1 ;m2 exceeds the value k, where k is determined from a speci4ed signi4cance level of the testing procedure. Approximations for P(Sm1 ; m2 ¿ k) have been investigated in Boutsikas and Koutras (2000a, b), Darling and Waterman (1986), Chen and Glaz (1996) and Sheng and Naus (1995). The use of Sm1 ; m2 for testing the null hypothesis of randomness speci4ed above is of interest in the many areas of applications including: astronomy (Darling and Waterman, 1986), ecology (Cressie, 1991), epidemiology (Cressie, 1991; KulldorL, 1999), mine4eld detection via remote sensing (Glaz, 1996) and reliability theory (Barbour et al., 1996; Boutsikas and Koutras, 2000a, b; Fu and Koutras, 1994; Koutras et al., 1993; Salvia and Lasher, 1990; Yamamoto and Miyakawa, 1995; Zao, 1993). In this article we are interested in approximations for the distribution of the two-dimensional discrete scan statistic, conditioned on the number of events that have occurred in the rectangular region [0; T1 ] × [0; T2 ]. In Section 2 we derive product-type, Poisson and compound Poisson approximations for the distribution of the conditional scan statistic. These approximations yield approximations for the expected size and standard deviation of the conditional scan statistic. These approximations are based on simulation algorithms, described in Section 3. Numerical results are presented in Section 4. Conclusions are stated in Section 5.
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2. Approximations 2.1. Product-type approximations Let Xi; j ; 1 6 i 6 n1 ; 1 6 j 6 n2 , be iid nonnegative integer valued random variables. If the total number of events that have occurred in the two-dimensional rectangular region is known to be a, then the scan statistic de4ned in the Introduction, Eq. (2), is referred to as the conditional two-dimensional discrete scan statistic. In this article we are interested in approximations for n2 n1 P(Sm1 ; m2 (a) ¿ k) = P Sm1 ; m2 ¿ k | Xi; j = a : (3) j=1 i=1
To simplify the presentation of the results, we assume that n1 = n2 = n and m1 = m2 = m. For 1 6 i1 ; i2 6 n − m + 1, de4ne Ai1 ; i2 = (Yi1 ; i2 ¿ k) and
B=
n n
(4)
Xi; j = a :
(5)
j=1 i=1
Then, P(Sm; m (a) ¿ k) = P
n−m+1 n−m+1 i1 =1
Ai1 ; i2 |B :
(6)
i2 =1
The following product-type approximation for P(Sm; m ¿ k) has been derived in Chen and Glaz (1996):
q2m (n−2m+1)(n−m+1) P(Sm; m ¿ k) ≈ 1 − q2m−1 ; (7) q2m−1 where for 1 6 t 6 m + 1, qm+t −1 = P(Ac1; 1 ∩ Ac1; 2 · · · ∩ Ac1; t ): This approximation was at times not accurate, since we were not able to obtain an accurate approximation for P(Ei1 ∩ Ei1 +1 ); where P(Ei1 ) = P
n−m+1
Aci1 ; i2
i2 =1
and 1 6 i1 6 n − m. Therefore, we will not pursue this approximation here.
(8)
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In this section we develop a simulation-based method for approximating the distribution of the conditional scan statistic. This method is based on the representation: n−m+1 P(Sm; m (a) 6 k − 1) = P Ei1 |B =P
i1 =1
n−m+1 i r 1 Ei |B) P( j=1 Ei |B ; i1 − 1 P( j=1 Ei |B) i=1 i1 =r+1
where r = 1 or 2. We propose to employ a Markov-like approximation of order r = 1; 2: 1 1 Ei |B) Ei |B) P( r+1 P( ij=i P( ij=1 j=1 Ei |B) 1 −r : ≈ i1 − 1 = r i1 − 1 P( j=1 Ei |B) P( j=1 Ei |B) P( j=i1 −r Ei |B) This yields product-type approximations: P(Sm; m (a) 6 k − 1) ≈
[P(E1 ∩ E2 |B)]n−m [P(E1 |B)]n−m−1
P(Sm; m (a) 6 k − 1) ≈
[P(E1 ∩ E2 ∩ E3 |B)]n−m−1 ; [P(E1 ∩ E2 |B)]n−m−2
(9)
and (10)
respectively. To implement these approximations, e@cient algorithms for simulating P(E1 ∩ E2 ∩ E3 |B); P(E1 ∩ E2 |B) and P(E1 |B) are needed: If m and n are not too large, it is a feasible task. Moreover, one gains signi4cant savings over simulating P(Sm; m 6 k − 1) on the entire rectangular region. The algorithms for simulating P(E1 ∩ E2 ∩ E3 |B); P(E1 ∩ E2 |B) and P(E1 |B) are given in Section 3. 2.2. Poisson approximations For 1 6 j 6 n − m + 1; let Ij be a Bernoulli random variable with P(Ij = 1) = P(Ejc |B) = 1 − P(Ij = 0);
(11)
where Ej and B are de4ned in Eqs. (8) and (5), respectively. Then n− m+1 P(Sm; m (a) ¿ k) = 1 − P Ij : j=1
−m+1 The distribution of nj=1 Ij can be approximated by a Poisson distribution with mean , where n− m+1 =E Ij = (n − m + 1)(1 − P(E1 |B)): j=1
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Poisson approximations for the unconditional case have been discussed in Barbour et al. (1996), Chen and Glaz (1996), Darling and Waterman (1986), Koutras et al. (1993), and Roos (1993). In Section 4, we evaluate the performance of the following Poisson approximation: P(Sm; m (a) ¿ k) ≈ 1 − exp(− )
(12)
for the binomial and Poisson models for selected values of n; m; k and a. Poisson approximation (12) is not expected to perform well when k ¡ m2 , since the events {(Ij = 1); 1 6 j 6 n−m+1} tend to clump. We propose to employ a local declumping approach discussed in Chen and Glaz (1996), for the unconditional case, to obtain a more accurate Poisson approximation. For 1 6 j 6 n − m, let Ij∗ be a Bernoulli random variable with P(Ij∗ = 1) = P(Ejc ∩ Ej+1 |B) = 1 − P(Ij∗ = 0)
(13)
and P(In∗−m+1 = 1) = P(Enc−m+1 |B) = 1 − P(In∗−m+1 = 0): The improved Poisson approximation is given by P(Sm; m (a) ¿ k) ≈ 1 − exp(− ∗ );
(14)
where ∗ = 1 − P(E1 |B) + (n − m)[P(E1 |B) − P(E1 ∩ E2 |B)]: In Section 4, we evaluate the performance of the improved Poisson approximation (14) for selected values of n; m k and a, for the binomial and Poisson models. 2.3. A compound Poisson approximation The following compound Poisson approximation for P(Sm; m (a) ¿ k) is based on the approach in Roos (1993, 1994): 3 P(Sm; m (a) ¿ k) ≈ 1 − exp − (15) i : i=1
The constants i are evaluated by extending the method in Roos (1993): 1 i = {2 1; i + (n − m − 1) 2; i }; i = 1; 2; 3; i where 1; 1 = P{I1 = 1; I2 = 0} = P(E1 |B) − P(E1 ∩ E2 |B); 1; 2 = P{I1 = 1; I2 = 1} = 1 − 2P(E1 |B) + P(E1 ∩ E2 |B); 2; 1 = P{I1 + I3 = 0; I2 = 1} = P(E1 ∩ E3 |B) − P(E1 ∩ E2 ∩ E3 |B);
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2; 2 = P{I1 + I3 = 1; I2 = 1} = 2[P(E1 |B) − P(E1 ∩ E2 |B) − P(E1 ∩ E3 |B) + P(E1 ∩ E2 ∩ E3 |B)] and 2; 3 = P{I1 + I3 = 2; I2 = 1}; = 1 − 3P(E1 |B) + 2P(E1 ∩ E2 |B) + P(E1 ∩ E3 |B) − P(E1 ∩ E2 ∩ E3 |B); where Ij is de4ned in Eq. (11). The algorithms for evaluating these terms are presented in Section 3. We evaluate this compound Poisson approximation for selected values of n; m; k and a, for the binomial and Poisson models. 2.4. A Bonferroni-type upper bound Eqs. (6) and (8) imply: n−m+1 P(Sm; m (a) ¿ k) = P Eic1 |B :
(16)
i1 =1
It follows from Hoover (1990) and Glaz and Ravishanker (1991) that a third order Bonferroni-type upper bound for P(Sm; m (a) ¿ k) is given by P(Sm; m (a) ¿ k) 6 1 + (n − m − 2)P(E1 ∩ E2 |B) − (n − m − 1)P(E1 ∩ E2 ∩ E3 |B):
(2.17)
In Tables 1 and 2 we evaluate the performance of this Bonferroni-type inequality for selected values of n; m; k and a, for binomial and Poisson models. 2.5. Approximations for the expected size Since Sm; m (a) is a discrete random variable 2
E(Sm; m (a)) =
m
P(Sm; m (a) ¿ k):
(17)
k=1
Approximations for P(Sm; m (a) ¿ k) yield approximations for E(Sm; m (a)). In Section 5, we present approximations for E(Sm; m (a)) based on the product-type approximation (10), for selected values of the parameters for the Poisson model. To evaluate the performance of these approximations we present simulated values for E(Sm; m (a)) based on 10,000 trials. 3. Simulation algorithms 3.1. Binomial models We proceed to describe an algorithm for evaluating P(E1 |B); P(E1 ∩ E2 |B); P(E1 ∩ E3 |B) and P(E1 ∩ E2 ∩ E3 |B), for the binomial model. Let Xi; j ; 1 6 i; j 6 n, be iid binomial random variables
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293
with parameters L ¿ 1 and 0 ¡ p ¡ 1. The joint distribution of Xi; j ; 1 6 i; j 6 n, conditional on B, is a multivariate hypergeometric distribution given by:
L L ··· x n; n x1; 1 2
; (18) P(X1; 1 = x1; 1 ; : : : ; Xn; n = x n; n |B) = nL a n−1 n where 0 6 xi; j and 0 6 x n; n = a − j=1 i=1 xi; j are integers. Let V1; 1 ; : : : ; Vn; n be a sequence of random variables with the multivariate hypergeometric distribution given in Eq. (18). For 1 6 t 6 3, it follows that t t n−m+1 s+m−1 i +m−1 2 P Es |B = P Vi; j 6 k − 1 (19) s=1
and
s=1 i2 =1
P(E1 ∩ E3 |B) = P
m+1 n−
t=1;3 i2 =1
i=s
j=i2
m+t −1 i +m−1 2 i=t
Vi; j 6 k − 1
:
(20)
j=i2
To evaluate the probabilities given in Eqs. (19) and (20) we use the algorithm in Pate4eld (1981) for simulating the multivariate hypergeometric distribution for V1; 1 ; : : : ; Vm+2; n ; given by
n(n − m − 2)L L L m+2 n ··· v1; 1 vm+2; n vi; j a− i=1 j=1 2
: P(V1; 1 = v1; 1 ; : : : ; Vm+2; n = vm+2; n ) = nL a The results of this simulation are arranged in an (m + 2) × n matrix. This process is repeated 10,000 times to obtain simulated values of the probabilities given in Eqs. (19) and (20). 3.2. Poisson model Let Xi; j ; 1 6 i; j 6 n, be iid Poisson random variables with mean . The joint distribution of Xi; j ; 1 6 i; j 6 n, conditional on B, is a multinomial distribution given by
a 1 a P(X1; 1 = x1; 1 ; : : : ; Xn; n = x n; n |B) = ; (21) x1; 1 ; : : : ; x n; n n2 n−1 n where 0 6 xi; j and 0 6 x n; n = a − j=1 i=1 xi; j are integers. Let W1; 1 ; : : : ; Wn; n be a sequence of random variables with the multinomial distribution given in Eq. (21). For 1 6 t 6 3, it follows that t n−m+1 m+t −1 i +m−1 t 2 Es |B = P Wi; j 6 k − 1 (22) P s=1
and
s=1 i2 =1
P(E1 ∩ E3 |B) = P
n− m+1
t=1;3 i2 =1
i=t
j=i2
m+t −1 i +m−1 2 i=t
j=i2
Wi; j 6 k − 1
:
(23)
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Table 1 Comparison of 4ve approximations to P(Sm; m (a) ¿ k) for a binomial model, L = 5 n 25
m 5
a 25 50
50
10
150
5
100
10
150
k
(10)
(12)
(14)
(15)
(16)
Simulation
5 6 8 9 10 37 38 39 40 6 7 16 17 18 19
0.3109 0.0620 0.1188 0.0274 0.0051 0.1858 0.1146 0.0668 0.0374 0.3162 0.0622 0.1295 0.0493 0.0156 0.0060
0.4795 0.0936 0.1786 0.0384 0.0063 0.3406 0.2038 0.1154 0.0626 0.4491 0.0808 0.2302 0.0835 0.0247 0.0074
0.3055 0.0552 0.1318 0.0291 0.0054 0.1930 0.1172 0.0688 0.0402 0.3228 0.0624 0.1302 0.0438 0.0160 0.0051
0.3313 0.0611 0.1341 0.0296 0.0052 0.2212 0.1317 0.0762 0.0430 0.3322 0.0617 0.1478 0.0513 0.0171 0.0053
0.3531 0.0635 0.1242 0.0276 0.0051 0.1974 0.1190 0.0682 0.0378 0.3708 0.0640 0.1372 0.0504 0.0157 0.0060
0.3159 0.0631 0.1177 0.0271 0.0040 0.1790 0.1053 0.0619 0.0336 0.3009 0.0533 0.1375 0.0548 0.0184 0.0060
To evaluate the probabilities given in Eqs. (22) and (23) we use the Fortran IMSL library for simulating the multinomial distribution for W1; 1 ; : : : ; Wm+2; n , given by P(W1; 1 = w1; 1 ; : : : ; Wm+2; n = wm+2; n ) n
m+2
∗ i=1 j=1 wi; j 1 (m + 2)n w a 1 − ; = w1; 1 ; : : : ; wm+2; n ; w∗ n2 n2 n where w∗ = a − m+2 i=1 j=1 wi; j . The results of this simulation are arranged in an (m + 2) × n matrix. This process is repeated 10,000 times to obtain simulated values of the probabilities given in Eqs. (22) and (23). 4. Numerical results In this section, for selected values of n; m; k and a, we evaluate in Tables 1 and 2 approximations for P(Sm; m (a) ¿ k) for the binomial and Poisson models, respectively, developed in Sections 2 and 3. The simulation results presented in Tables 1 and 2 are based on 10,000 trials. 5. Conclusions In this article accurate approximations were evaluated for the distribution and the expected value of a conditional two dimensional scan statistic. This is of interest for this particular problem, as there are no exact results available for conditional two dimensional scan statistics. From the numerical results in Section 4, Tables 1–2, it is evident that the product-type approximation given in Eq. (10) is the most accurate one. The frequently used Poisson approximation, given in Eq. (12) is inaccurate
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Table 2 Comparison of 4ve approximations to P(Sm; m (a) ¿ k) for a Poisson model n
m
a
25
5
300
100
5
1000
10
1000
k
(10)
(12)
(14)
(15)
(16)
Simulation
23 24 25 26 27 28 11 12 13 24 25 26 27 28
0.3747 0.2137 0.1097 0.0521 0.0252 0.0104 0.2506 0.0626 0.0172 0.2235 0.1097 0.0456 0.0169 0.0075
0.5103 0.2974 0.1514 0.0703 0.0311 0.0126 0.3411 0.0833 0.0200 0.3842 0.1802 0.0758 0.0295 0.0131
0.3726 0.2173 0.1172 0.0579 0.0264 0.0108 0.2379 0.0683 0.0134 0.2332 0.1118 0.0491 0.0199 0.0057
0.3898 0.2251 0.1187 0.0574 0.0260 0.0107 0.2544 0.0682 0.0146 0.2615 0.1225 0.0527 0.0215 0.0075
0.4408 0.2329 0.1144 0.0532 0.0255 0.0104 0.2860 0.0646 0.0174 0.2506 0.1157 0.0466 0.0170 0.0076
0.3790 0.2128 0.1087 0.0500 0.0239 0.0103 0.2662 0.0768 0.0211 0.2192 0.1028 0.0438 0.0188 0.0079
and can be signi4cantly improved by the modi4ed Poisson approximation and a compound Poisson approximation, given in Eqs. (14) and (15), respectively. Accurate approximations for P(Sm; m (a) ¿ k) lead to quite accurate approximations for E(Sm; m (a)). For example, for the Poisson model with n = 100; m = 5; a = 250 and 1000, respectively, based on Eq. (10), E(S5; 5 (250)) ≈ 4:94 and E(S5; 5 (1000)) ≈ 9:98, while the simulated values for E(S5; 5 (250)) and E(S5; 5 (1000)), based on 10,000 trials, are 4:95 and 10:03, respectively. Acknowledgements The authors thank the referee for the suggestions to improve the presentation of the results. References Barbour, A.D., Chryssaphinou, O., Roos, M., 1996. Compound Poisson approximation in system reliability. Naval Res. Logist. 43, 251–264. Boutsikas, M.V., Koutras, M.V., 2000a. Generalized reliability bounds for coherent structures. J. Appl. Probab. 37, 778–794. Boutsikas, M.V., Koutras, M.V., 2000b. Reliability approximations for Markov chain imbeddable systems. Methodol. Comput. Appl. Probab. 2, 393–412. Chen, J., Glaz, J., 1996. Two dimensional discrete scan statistics. Statist. Probab. Lett. 31, 59–68. Cressie, N., 1991. Statistics for Spatial Data. Wiley, New York. Darling, R.W.R., Waterman, M.S., 1986. Extreme value distribution for the largest cube in a random lattice. SIAM J. Appl. Math. 46, 118–132. Fu, J.C., Koutras, M.V., 1994. Poisson approximation for 2-dimensional patterns. Ann. Inst. Statist. Math. 46, 179–192. Glaz, J., 1996. Discrete scan statistics with applications to mine4elds detection. In: Proceedings of Conference SPIE, Vol. 2765, Orlando, FL, pp. 420 – 429. Glaz, J., Naus, J.I., 1991. Tight bounds and approximations for scan statistic probabilities for discrete data. Ann. Appl. Probab. 1, 306–318.
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Koutras, M.V., Papadopoulos, G.K., Papastavridis, S.G., 1993. Reliability of 2-dimensional consecutive-k-out-of n: F systems IEEE Trans. Reliab. R-42, 658–661. KulldorL, M., 1999. Spatial scan statistics: models, calculations and applications. In: Glaz, J., Balakrishnan, N. (Eds.), Scan Statistics and Applications. Birkhauser, Boston, pp. 303–322. Pate4eld, W.M., 1981. An e@cient method of generating Random R × C tables with given row and column totals. Appl. Statist. 30, 91–97. Roos, M., 1993. Stein-Chen method for compound Poisson approximation. Ph.D. Dissertation, University of Zurich, Zurich. Roos, M., 1994. Stein’s method for compound Poisson approximation. Ann. Appl. Probab. 4, 1177–1187. Salvia, A.A., Lasher, W.C., 1990. 2-dimensional consecutive-k-out-of n: F models IEEE, Trans. Reliab. R-39, 382–385. Sheng, K.N., Naus, J.I., 1995. Matching 4xed rectangles in 2-dimensions. Statist. Probab. Lett., in press. Yamamoto, H., Miyakawa, M., 1995. Reliability of a linear connected-(r,s)-out-of-(m,n): F lattice system IEEE Trans. Reliab. 44, 333–336. Zao, M.J., 1993. Reliability and design of a 2-dimensional consecutive-k-out-of-n systems. IEEE Trans. Reliab. 42, 488–490.
Approximations for a conditional two-dimensional scan statistic Jie Chena , Joseph Glazb; ∗ a
Department of Computing Services, University of Massachusetts Boston, Boston, MA 02125, USA b Department of Statistics, University of Connecticut, Storrs, CT 06269, USA Received May 2001
Abstract Let Xi; j ; 1 6 i 6 n1 ; 1 6 j 6 n2 , be a sequence of independent and identically distributed nonnegative integer valued random variables. The observation Xi; j denotes the number of events that have occurred in the i; jth location in a two dimensional rectangular region R. For 2 6 mi 6 ni − 1; i = 1; 2, the two dimensional discrete scan statistic is de4ned as the maximum number of events in any of the m1 by m2 consecutive rectangular windows in that region. Conditional on the total number of events that have occurred in R, we refer to this scan statistic as the conditional two-dimensional scan statistic. Two-dimensional scan statistics have been extensively used in many areas of science to analyze the occurrence of observed clusters of events in space. Since this scan statistic is based on highly dependent consecutive subsequences of observed data, accurate approximations for its distributions are of great value. In this article, based on the scanning window representation of the scan statistic, accurate product-type, Poisson and a compound Poisson approximations are investigated. Moreover, accurate approximations for the expected size of the scan statistic are derived. Numerical results are presented to evaluate the performance of the approximations c 2002 Elsevier Science B.V. All rights reserved. discussed in this article. Keywords: Compound Poisson approximation; Moving window detection; Poisson approximations; Product-type approximations; Testing for randomness in two dimensions
1. Introduction Let [0; T1 ] × [0; T2 ] be rectangular region. Let hi = Ti =ni ¿ 0; where ni are positive integers, i = 1; 2. In many applications the exact locations of the observed events in the region are unknown. What is Research was supported in part by the O@ce of Naval Research Contract No. N00014-94-1-0061 and the University of Connecticut Research Foundation. ∗ Corresponding author. E-mail address: [email protected] (J. Glaz).
c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 2 2 - 0
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usually available are the counts in small rectangular subregions. For 1 6 i 6 n1 and 1 6 j 6 n2 , let Xij be the number of events that have been observed in the rectangular subregion [(i − 1)h1 ; ih1 ] × [(j − 1)h2 ; jh2 ]. We are interested in detecting unusual clustering of these events under the null hypothesis that Xij are independent and identically distributed nonnegative integer valued random variables from a speci4ed distribution. For 2 6 m1 6 n1 − 1; 2 6 m2 6 n2 − 1; 1 6 i1 6 n1 − m1 + 1 and 1 6 i2 6 n2 − m2 + 1 de4ne Yi1 ;i2 =
i2 +m 2 −1 i1 +m 1 −1 j=i2
Xij
(1)
j=i1
to be the number of events in a rectangular region comprised of m1 by m2 adjacent rectangular subregions with area h1 h2 and the south west corner located at the point ((i1 − 1)h1 ; (i2 − 1)h2 ). If Yi1 ;i2 equals or exceeds a preassigned value of k, we will say that k events are clustered within the inspected region. A two-dimensional discrete scan statistic (Chen and Glaz, 1996) is de4ned as the largest number of events in any m1 by m2 adjacent rectangular subregions with area h1 h2 and the south west corner located at the point ((i1 − 1)h1 ; (i2 − 1)h2 ): Sm1 ; m2 = max{Yi1 ; i2 ; 1 6 i1 6 n1 − m1 + 1; 1 6 i2 6 n2 − m2 + 1}:
(2)
Sm1 ; m2 can be viewed as an extension of the one-dimensional discrete scan statistic discussed in Glaz and Naus (1991). It is used for testing the null hypothesis of randomness that assumes the Xij ’s are iid binomial random variables with parameters L and 0 ¡ p ¡ 1 or iid Poisson random variables with mean ¿ 0; respectively. For the alternative hypothesis of clustering we specify a rectangular subregion R(i1 ; i2 ) = [(i1 − 1)h1 ; (i1 + m1 − 1)h1 ] × [(i2 − 1)h2 ; (i2 + m2 − 1)h2 ] such that for any i1 6 i 6 i1 + m1 − 1 and i2 6 j 6 i2 + m2 − 1; Xij has a binomial distribution with parameters L and p1 , where p1 ¿ p or a Poisson distribution with mean 1 where 1 ¿ , respectively. For ij ∈ [i1 ; i1 + m1 − 1] × [i2 ; i2 + m2 − 1]; Xij is distributed according to the distribution speci4ed by the null hypothesis. It is routine to verify that the generalized likelihood ratio test rejects the null hypothesis in favor of the alternative hypothesis whenever Sm1 ;m2 exceeds the value k, where k is determined from a speci4ed signi4cance level of the testing procedure. Approximations for P(Sm1 ; m2 ¿ k) have been investigated in Boutsikas and Koutras (2000a, b), Darling and Waterman (1986), Chen and Glaz (1996) and Sheng and Naus (1995). The use of Sm1 ; m2 for testing the null hypothesis of randomness speci4ed above is of interest in the many areas of applications including: astronomy (Darling and Waterman, 1986), ecology (Cressie, 1991), epidemiology (Cressie, 1991; KulldorL, 1999), mine4eld detection via remote sensing (Glaz, 1996) and reliability theory (Barbour et al., 1996; Boutsikas and Koutras, 2000a, b; Fu and Koutras, 1994; Koutras et al., 1993; Salvia and Lasher, 1990; Yamamoto and Miyakawa, 1995; Zao, 1993). In this article we are interested in approximations for the distribution of the two-dimensional discrete scan statistic, conditioned on the number of events that have occurred in the rectangular region [0; T1 ] × [0; T2 ]. In Section 2 we derive product-type, Poisson and compound Poisson approximations for the distribution of the conditional scan statistic. These approximations yield approximations for the expected size and standard deviation of the conditional scan statistic. These approximations are based on simulation algorithms, described in Section 3. Numerical results are presented in Section 4. Conclusions are stated in Section 5.
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2. Approximations 2.1. Product-type approximations Let Xi; j ; 1 6 i 6 n1 ; 1 6 j 6 n2 , be iid nonnegative integer valued random variables. If the total number of events that have occurred in the two-dimensional rectangular region is known to be a, then the scan statistic de4ned in the Introduction, Eq. (2), is referred to as the conditional two-dimensional discrete scan statistic. In this article we are interested in approximations for n2 n1 P(Sm1 ; m2 (a) ¿ k) = P Sm1 ; m2 ¿ k | Xi; j = a : (3) j=1 i=1
To simplify the presentation of the results, we assume that n1 = n2 = n and m1 = m2 = m. For 1 6 i1 ; i2 6 n − m + 1, de4ne Ai1 ; i2 = (Yi1 ; i2 ¿ k) and
B=
n n
(4)
Xi; j = a :
(5)
j=1 i=1
Then, P(Sm; m (a) ¿ k) = P
n−m+1 n−m+1 i1 =1
Ai1 ; i2 |B :
(6)
i2 =1
The following product-type approximation for P(Sm; m ¿ k) has been derived in Chen and Glaz (1996):
q2m (n−2m+1)(n−m+1) P(Sm; m ¿ k) ≈ 1 − q2m−1 ; (7) q2m−1 where for 1 6 t 6 m + 1, qm+t −1 = P(Ac1; 1 ∩ Ac1; 2 · · · ∩ Ac1; t ): This approximation was at times not accurate, since we were not able to obtain an accurate approximation for P(Ei1 ∩ Ei1 +1 ); where P(Ei1 ) = P
n−m+1
Aci1 ; i2
i2 =1
and 1 6 i1 6 n − m. Therefore, we will not pursue this approximation here.
(8)
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In this section we develop a simulation-based method for approximating the distribution of the conditional scan statistic. This method is based on the representation: n−m+1 P(Sm; m (a) 6 k − 1) = P Ei1 |B =P
i1 =1
n−m+1 i r 1 Ei |B) P( j=1 Ei |B ; i1 − 1 P( j=1 Ei |B) i=1 i1 =r+1
where r = 1 or 2. We propose to employ a Markov-like approximation of order r = 1; 2: 1 1 Ei |B) Ei |B) P( r+1 P( ij=i P( ij=1 j=1 Ei |B) 1 −r : ≈ i1 − 1 = r i1 − 1 P( j=1 Ei |B) P( j=1 Ei |B) P( j=i1 −r Ei |B) This yields product-type approximations: P(Sm; m (a) 6 k − 1) ≈
[P(E1 ∩ E2 |B)]n−m [P(E1 |B)]n−m−1
P(Sm; m (a) 6 k − 1) ≈
[P(E1 ∩ E2 ∩ E3 |B)]n−m−1 ; [P(E1 ∩ E2 |B)]n−m−2
(9)
and (10)
respectively. To implement these approximations, e@cient algorithms for simulating P(E1 ∩ E2 ∩ E3 |B); P(E1 ∩ E2 |B) and P(E1 |B) are needed: If m and n are not too large, it is a feasible task. Moreover, one gains signi4cant savings over simulating P(Sm; m 6 k − 1) on the entire rectangular region. The algorithms for simulating P(E1 ∩ E2 ∩ E3 |B); P(E1 ∩ E2 |B) and P(E1 |B) are given in Section 3. 2.2. Poisson approximations For 1 6 j 6 n − m + 1; let Ij be a Bernoulli random variable with P(Ij = 1) = P(Ejc |B) = 1 − P(Ij = 0);
(11)
where Ej and B are de4ned in Eqs. (8) and (5), respectively. Then n− m+1 P(Sm; m (a) ¿ k) = 1 − P Ij : j=1
−m+1 The distribution of nj=1 Ij can be approximated by a Poisson distribution with mean , where n− m+1 =E Ij = (n − m + 1)(1 − P(E1 |B)): j=1
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Poisson approximations for the unconditional case have been discussed in Barbour et al. (1996), Chen and Glaz (1996), Darling and Waterman (1986), Koutras et al. (1993), and Roos (1993). In Section 4, we evaluate the performance of the following Poisson approximation: P(Sm; m (a) ¿ k) ≈ 1 − exp(− )
(12)
for the binomial and Poisson models for selected values of n; m; k and a. Poisson approximation (12) is not expected to perform well when k ¡ m2 , since the events {(Ij = 1); 1 6 j 6 n−m+1} tend to clump. We propose to employ a local declumping approach discussed in Chen and Glaz (1996), for the unconditional case, to obtain a more accurate Poisson approximation. For 1 6 j 6 n − m, let Ij∗ be a Bernoulli random variable with P(Ij∗ = 1) = P(Ejc ∩ Ej+1 |B) = 1 − P(Ij∗ = 0)
(13)
and P(In∗−m+1 = 1) = P(Enc−m+1 |B) = 1 − P(In∗−m+1 = 0): The improved Poisson approximation is given by P(Sm; m (a) ¿ k) ≈ 1 − exp(− ∗ );
(14)
where ∗ = 1 − P(E1 |B) + (n − m)[P(E1 |B) − P(E1 ∩ E2 |B)]: In Section 4, we evaluate the performance of the improved Poisson approximation (14) for selected values of n; m k and a, for the binomial and Poisson models. 2.3. A compound Poisson approximation The following compound Poisson approximation for P(Sm; m (a) ¿ k) is based on the approach in Roos (1993, 1994): 3 P(Sm; m (a) ¿ k) ≈ 1 − exp − (15) i : i=1
The constants i are evaluated by extending the method in Roos (1993): 1 i = {2 1; i + (n − m − 1) 2; i }; i = 1; 2; 3; i where 1; 1 = P{I1 = 1; I2 = 0} = P(E1 |B) − P(E1 ∩ E2 |B); 1; 2 = P{I1 = 1; I2 = 1} = 1 − 2P(E1 |B) + P(E1 ∩ E2 |B); 2; 1 = P{I1 + I3 = 0; I2 = 1} = P(E1 ∩ E3 |B) − P(E1 ∩ E2 ∩ E3 |B);
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2; 2 = P{I1 + I3 = 1; I2 = 1} = 2[P(E1 |B) − P(E1 ∩ E2 |B) − P(E1 ∩ E3 |B) + P(E1 ∩ E2 ∩ E3 |B)] and 2; 3 = P{I1 + I3 = 2; I2 = 1}; = 1 − 3P(E1 |B) + 2P(E1 ∩ E2 |B) + P(E1 ∩ E3 |B) − P(E1 ∩ E2 ∩ E3 |B); where Ij is de4ned in Eq. (11). The algorithms for evaluating these terms are presented in Section 3. We evaluate this compound Poisson approximation for selected values of n; m; k and a, for the binomial and Poisson models. 2.4. A Bonferroni-type upper bound Eqs. (6) and (8) imply: n−m+1 P(Sm; m (a) ¿ k) = P Eic1 |B :
(16)
i1 =1
It follows from Hoover (1990) and Glaz and Ravishanker (1991) that a third order Bonferroni-type upper bound for P(Sm; m (a) ¿ k) is given by P(Sm; m (a) ¿ k) 6 1 + (n − m − 2)P(E1 ∩ E2 |B) − (n − m − 1)P(E1 ∩ E2 ∩ E3 |B):
(2.17)
In Tables 1 and 2 we evaluate the performance of this Bonferroni-type inequality for selected values of n; m; k and a, for binomial and Poisson models. 2.5. Approximations for the expected size Since Sm; m (a) is a discrete random variable 2
E(Sm; m (a)) =
m
P(Sm; m (a) ¿ k):
(17)
k=1
Approximations for P(Sm; m (a) ¿ k) yield approximations for E(Sm; m (a)). In Section 5, we present approximations for E(Sm; m (a)) based on the product-type approximation (10), for selected values of the parameters for the Poisson model. To evaluate the performance of these approximations we present simulated values for E(Sm; m (a)) based on 10,000 trials. 3. Simulation algorithms 3.1. Binomial models We proceed to describe an algorithm for evaluating P(E1 |B); P(E1 ∩ E2 |B); P(E1 ∩ E3 |B) and P(E1 ∩ E2 ∩ E3 |B), for the binomial model. Let Xi; j ; 1 6 i; j 6 n, be iid binomial random variables
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293
with parameters L ¿ 1 and 0 ¡ p ¡ 1. The joint distribution of Xi; j ; 1 6 i; j 6 n, conditional on B, is a multivariate hypergeometric distribution given by:
L L ··· x n; n x1; 1 2
; (18) P(X1; 1 = x1; 1 ; : : : ; Xn; n = x n; n |B) = nL a n−1 n where 0 6 xi; j and 0 6 x n; n = a − j=1 i=1 xi; j are integers. Let V1; 1 ; : : : ; Vn; n be a sequence of random variables with the multivariate hypergeometric distribution given in Eq. (18). For 1 6 t 6 3, it follows that t t n−m+1 s+m−1 i +m−1 2 P Es |B = P Vi; j 6 k − 1 (19) s=1
and
s=1 i2 =1
P(E1 ∩ E3 |B) = P
m+1 n−
t=1;3 i2 =1
i=s
j=i2
m+t −1 i +m−1 2 i=t
Vi; j 6 k − 1
:
(20)
j=i2
To evaluate the probabilities given in Eqs. (19) and (20) we use the algorithm in Pate4eld (1981) for simulating the multivariate hypergeometric distribution for V1; 1 ; : : : ; Vm+2; n ; given by
n(n − m − 2)L L L m+2 n ··· v1; 1 vm+2; n vi; j a− i=1 j=1 2
: P(V1; 1 = v1; 1 ; : : : ; Vm+2; n = vm+2; n ) = nL a The results of this simulation are arranged in an (m + 2) × n matrix. This process is repeated 10,000 times to obtain simulated values of the probabilities given in Eqs. (19) and (20). 3.2. Poisson model Let Xi; j ; 1 6 i; j 6 n, be iid Poisson random variables with mean . The joint distribution of Xi; j ; 1 6 i; j 6 n, conditional on B, is a multinomial distribution given by
a 1 a P(X1; 1 = x1; 1 ; : : : ; Xn; n = x n; n |B) = ; (21) x1; 1 ; : : : ; x n; n n2 n−1 n where 0 6 xi; j and 0 6 x n; n = a − j=1 i=1 xi; j are integers. Let W1; 1 ; : : : ; Wn; n be a sequence of random variables with the multinomial distribution given in Eq. (21). For 1 6 t 6 3, it follows that t n−m+1 m+t −1 i +m−1 t 2 Es |B = P Wi; j 6 k − 1 (22) P s=1
and
s=1 i2 =1
P(E1 ∩ E3 |B) = P
n− m+1
t=1;3 i2 =1
i=t
j=i2
m+t −1 i +m−1 2 i=t
j=i2
Wi; j 6 k − 1
:
(23)
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Table 1 Comparison of 4ve approximations to P(Sm; m (a) ¿ k) for a binomial model, L = 5 n 25
m 5
a 25 50
50
10
150
5
100
10
150
k
(10)
(12)
(14)
(15)
(16)
Simulation
5 6 8 9 10 37 38 39 40 6 7 16 17 18 19
0.3109 0.0620 0.1188 0.0274 0.0051 0.1858 0.1146 0.0668 0.0374 0.3162 0.0622 0.1295 0.0493 0.0156 0.0060
0.4795 0.0936 0.1786 0.0384 0.0063 0.3406 0.2038 0.1154 0.0626 0.4491 0.0808 0.2302 0.0835 0.0247 0.0074
0.3055 0.0552 0.1318 0.0291 0.0054 0.1930 0.1172 0.0688 0.0402 0.3228 0.0624 0.1302 0.0438 0.0160 0.0051
0.3313 0.0611 0.1341 0.0296 0.0052 0.2212 0.1317 0.0762 0.0430 0.3322 0.0617 0.1478 0.0513 0.0171 0.0053
0.3531 0.0635 0.1242 0.0276 0.0051 0.1974 0.1190 0.0682 0.0378 0.3708 0.0640 0.1372 0.0504 0.0157 0.0060
0.3159 0.0631 0.1177 0.0271 0.0040 0.1790 0.1053 0.0619 0.0336 0.3009 0.0533 0.1375 0.0548 0.0184 0.0060
To evaluate the probabilities given in Eqs. (22) and (23) we use the Fortran IMSL library for simulating the multinomial distribution for W1; 1 ; : : : ; Wm+2; n , given by P(W1; 1 = w1; 1 ; : : : ; Wm+2; n = wm+2; n ) n
m+2
∗ i=1 j=1 wi; j 1 (m + 2)n w a 1 − ; = w1; 1 ; : : : ; wm+2; n ; w∗ n2 n2 n where w∗ = a − m+2 i=1 j=1 wi; j . The results of this simulation are arranged in an (m + 2) × n matrix. This process is repeated 10,000 times to obtain simulated values of the probabilities given in Eqs. (22) and (23). 4. Numerical results In this section, for selected values of n; m; k and a, we evaluate in Tables 1 and 2 approximations for P(Sm; m (a) ¿ k) for the binomial and Poisson models, respectively, developed in Sections 2 and 3. The simulation results presented in Tables 1 and 2 are based on 10,000 trials. 5. Conclusions In this article accurate approximations were evaluated for the distribution and the expected value of a conditional two dimensional scan statistic. This is of interest for this particular problem, as there are no exact results available for conditional two dimensional scan statistics. From the numerical results in Section 4, Tables 1–2, it is evident that the product-type approximation given in Eq. (10) is the most accurate one. The frequently used Poisson approximation, given in Eq. (12) is inaccurate
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295
Table 2 Comparison of 4ve approximations to P(Sm; m (a) ¿ k) for a Poisson model n
m
a
25
5
300
100
5
1000
10
1000
k
(10)
(12)
(14)
(15)
(16)
Simulation
23 24 25 26 27 28 11 12 13 24 25 26 27 28
0.3747 0.2137 0.1097 0.0521 0.0252 0.0104 0.2506 0.0626 0.0172 0.2235 0.1097 0.0456 0.0169 0.0075
0.5103 0.2974 0.1514 0.0703 0.0311 0.0126 0.3411 0.0833 0.0200 0.3842 0.1802 0.0758 0.0295 0.0131
0.3726 0.2173 0.1172 0.0579 0.0264 0.0108 0.2379 0.0683 0.0134 0.2332 0.1118 0.0491 0.0199 0.0057
0.3898 0.2251 0.1187 0.0574 0.0260 0.0107 0.2544 0.0682 0.0146 0.2615 0.1225 0.0527 0.0215 0.0075
0.4408 0.2329 0.1144 0.0532 0.0255 0.0104 0.2860 0.0646 0.0174 0.2506 0.1157 0.0466 0.0170 0.0076
0.3790 0.2128 0.1087 0.0500 0.0239 0.0103 0.2662 0.0768 0.0211 0.2192 0.1028 0.0438 0.0188 0.0079
and can be signi4cantly improved by the modi4ed Poisson approximation and a compound Poisson approximation, given in Eqs. (14) and (15), respectively. Accurate approximations for P(Sm; m (a) ¿ k) lead to quite accurate approximations for E(Sm; m (a)). For example, for the Poisson model with n = 100; m = 5; a = 250 and 1000, respectively, based on Eq. (10), E(S5; 5 (250)) ≈ 4:94 and E(S5; 5 (1000)) ≈ 9:98, while the simulated values for E(S5; 5 (250)) and E(S5; 5 (1000)), based on 10,000 trials, are 4:95 and 10:03, respectively. Acknowledgements The authors thank the referee for the suggestions to improve the presentation of the results. References Barbour, A.D., Chryssaphinou, O., Roos, M., 1996. Compound Poisson approximation in system reliability. Naval Res. Logist. 43, 251–264. Boutsikas, M.V., Koutras, M.V., 2000a. Generalized reliability bounds for coherent structures. J. Appl. Probab. 37, 778–794. Boutsikas, M.V., Koutras, M.V., 2000b. Reliability approximations for Markov chain imbeddable systems. Methodol. Comput. Appl. Probab. 2, 393–412. Chen, J., Glaz, J., 1996. Two dimensional discrete scan statistics. Statist. Probab. Lett. 31, 59–68. Cressie, N., 1991. Statistics for Spatial Data. Wiley, New York. Darling, R.W.R., Waterman, M.S., 1986. Extreme value distribution for the largest cube in a random lattice. SIAM J. Appl. Math. 46, 118–132. Fu, J.C., Koutras, M.V., 1994. Poisson approximation for 2-dimensional patterns. Ann. Inst. Statist. Math. 46, 179–192. Glaz, J., 1996. Discrete scan statistics with applications to mine4elds detection. In: Proceedings of Conference SPIE, Vol. 2765, Orlando, FL, pp. 420 – 429. Glaz, J., Naus, J.I., 1991. Tight bounds and approximations for scan statistic probabilities for discrete data. Ann. Appl. Probab. 1, 306–318.
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Koutras, M.V., Papadopoulos, G.K., Papastavridis, S.G., 1993. Reliability of 2-dimensional consecutive-k-out-of n: F systems IEEE Trans. Reliab. R-42, 658–661. KulldorL, M., 1999. Spatial scan statistics: models, calculations and applications. In: Glaz, J., Balakrishnan, N. (Eds.), Scan Statistics and Applications. Birkhauser, Boston, pp. 303–322. Pate4eld, W.M., 1981. An e@cient method of generating Random R × C tables with given row and column totals. Appl. Statist. 30, 91–97. Roos, M., 1993. Stein-Chen method for compound Poisson approximation. Ph.D. Dissertation, University of Zurich, Zurich. Roos, M., 1994. Stein’s method for compound Poisson approximation. Ann. Appl. Probab. 4, 1177–1187. Salvia, A.A., Lasher, W.C., 1990. 2-dimensional consecutive-k-out-of n: F models IEEE, Trans. Reliab. R-39, 382–385. Sheng, K.N., Naus, J.I., 1995. Matching 4xed rectangles in 2-dimensions. Statist. Probab. Lett., in press. Yamamoto, H., Miyakawa, M., 1995. Reliability of a linear connected-(r,s)-out-of-(m,n): F lattice system IEEE Trans. Reliab. 44, 333–336. Zao, M.J., 1993. Reliability and design of a 2-dimensional consecutive-k-out-of-n systems. IEEE Trans. Reliab. 42, 488–490.