Approximations for the distribution of the discrete scan statistics when the scanning window has arbitrary shape

Approximations for the distribution of the discrete scan statistics when the scanning window has arbitrary shape Alexandru Am rioarei National Institute of Research and Development for Biological Sciences The 9 th Conference of the Romanian Society of Probability and Statistics 27-28 May, 206, Bucure³ti, România A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 / 34

A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 2 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Introduction Framework A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 3 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Introduction Framework A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 4 / 34 Denitions and notations

Introduction Framework A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 5 / 34 Preliminary notations Let T, T 2 be positive integers.. s. T s 2 T 2,s 2 T 2 T Rectangular region R 2 = [0, T ] [0, T 2 ] (,s 2 ) s T s 2 T 2 i.i.d. r.v.'s Bernoulli(B(, p)) Binomial(B(n, p)) Poisson(P(λ)) Normal(N (µ, σ 2 )),s 2 number of observed events in the elementary subregion r s,s 2 = [s, s ] [s 2, s 2 ]

Introduction Framework A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 6 / 34 Two dimensional scan statistic Let 2 m s T s, s {, 2} be positive integers Dene for i s T s m s + and j s m s the 2-way tensor X i,i 2 R m m 2, X i,i 2 (j, j 2 ) = X i +j,i 2 +j 2 Take S : R m m 2 R to be a measurable real valued function (score function) and dene Y i,i 2 (S) = S (X i,i 2 ) Definition The two dimensional scan statistic with score function S is dened by S m,m 2 (T, T 2 ; S) = max i T m + i 2 T 2 m 2 + Y i,i 2 (S)

Introduction Framework A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 7 / 34 Shape of the scanning window Let G be the geometrical shape of the scanning window (rectangle, quadrilateral, ellipse, etc.) and G be its corresponding discrete form. Rasterization algorithms (omputer vision): continuous shape discrete shape Line - Bresenham line algorithm ([Bresenham, 965]) Circle - Bresenham circle algorithm ([Bresenham, 977]) Bezier curves - [Foley, 995]

Introduction Framework Shape of the scanning window To each discrete shape G it corresponds an unique matrix (2-way tensor) A (G) = A ( G ) (of smallest size) with 0 and entries ( if there is a point and 0 otherwise): G A ( G ) A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 8 / 34

Introduction Framework Arbitrary window scan statistic Let G be a geometric shape and A = A(G) its corresponding {0, } matrix of size m m 2. Dene the score function S associated to the shape G by S (X i,i 2 ) = A X i,i 2 = Remark i +m s =i i 2 +m 2 s 2 =i 2 A(s i +, s 2 i 2 + ),s 2 If, in particular, the shape G is a rectangle of size m m 2 than its corresponding {0, } matrix of the same size has all the entries equal to so the score function S (X i,i 2 ) = i +m s =i i 2 +m 2 s 2 =i 2,s 2 is the classical rectangular window of the two dimensional scan statistics. A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 9 / 34

Introduction Problem A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 0 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Introduction Problem A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 / 34 Problem and related work

Introduction Problem Objective Find a good estimate for the distribution of the two dimensional discrete scan statistic with score function S Q m (T; S) = P (S m (T; S) τ) with m = (m, m 2 ) and T = (T, T 2 ) Previous work: Continuous scan statistics Rectangles: [Loader, 99], [Glaz et al., 200], [Glaz et al., 2009] Circles: [Anderson and Titterington, 997] Triangles, ellipses and other convex shapes: [Alm, 983, Alm, 997, Alm, 998], [Tango and Takahashi, 2005], [Assunção et al., 2006] Discrete scan statistics No results! A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 2 / 34

Methodology Approximation A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 3 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Methodology Approximation A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 4 / 34 Approximation methodology for the general scan statistic

Methodology Approximation A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 5 / 34 Approximation and error bounds Theorem (Generalization of [Am rioarei, 204]) Let t, t 2 {2, 3}, Q t,t2 = P (Sm (t (m ), t 2 (m 2 ); S) τ) and L s = Ts, m s s {, 2}. If ˆQt,t2 is an estimate of Qt,t2, ˆQt,t2 Qt,t2 β t,t2 and τ is such that ˆQ2,2 (τ) 0. then ( ) [ ] L P (Sm(T; S) τ) 2 ˆQ2 ˆQ3 + ˆQ2 ˆQ3 + 2( ˆQ2 ˆQ3 ) 2 E sf + E sapp, where, for t {2, 3} ) ˆQ t = (2 ˆQt,2 ˆQt,3 [ + ˆQt,2 ˆQt,3 + 2( ˆQt,2 ˆQt,3 ) 2] L2 E sf = (L )(L 2 ) (β 2,2 + β 2,3 + β 3,2 + β 3,3 ) ( ) ] 2 E sapp = (L ) [F ˆQ2 + A 2 + C 2 + (L2 )(F 2 C 2 + F 3 C 3 ) A 2 = (L 2 ) (β 2,2 + β 2,3 ) C t = (L 2 )F t ( ˆQt,2 + β t,2 ) 2.

Methodology Approximation A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 6 / 34 Illustration of the approximation process T 2 T R 2 Find Approximation m m 2 T T T 2 Q 2 T 2 Q 3 m 2 m 2 2(m ) 3(m ) T T T T T 2 T 2 T 2 T 2 Q 22 Q 3(m 2 ) 2(m 2 ) 23 Q 32 Q 3(m 2 ) 2(m 2 ) 33 2(m ) 2(m ) 3(m ) 3(m )

Methodology Approximation A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 6 / 34 Illustration of the approximation process T 2 T R 2 First Step Approximation m m 2 T 2 T 2 T Q 2 T Q 3 m m 2(m 2 ) 3(m 2 ) T T T T T 2 T 2 T 2 T 2 Q 22 Q 3(m 2 ) 2(m ) 23 Q 32 Q 3(m 2 ) 2(m 2 ) 33 2(m ) 2(m ) 3(m ) 3(m )

Methodology Approximation A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 6 / 34 Illustration of the approximation process T 2 T R 2 Second Step Approximation m m 2 T 2 T 2 T Q 2 T Q 3 m m 2(m 2 ) 3(m 2 ) T 2 T 2 T 2 T 2 T T T T Q 22 Q 3(m ) 2(m ) 23 Q 32 Q 3(m ) 2(m ) 33 2(m 2 ) 2(m 2 ) 3(m 2 ) 3(m 2 )

Methodology Simulation methods: Normal data A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 7 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Methodology Simulation methods: Normal data A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 8 / 34 Simulation methods for Normal data

Methodology Simulation methods: Normal data A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 9 / 34 Importance sampling algorithm Test the null hypothesis of randomness against an alternative of clustering H 0 : The r.v.'s,s 2 are i.i.d. N (µ, σ 2 ) H : There exists R(i, i 2 ) = [i, i + m ] [i 2, i 2 + m 2 ] R 2 where the r.v.'s,s 2 N (µ, σ 2 ), µ > µ and,s 2 N (µ, σ 2 ) outside R(i, i 2 ) Objective Find a good estimate for P H0 (Sm(T; S) τ). We are interested in evaluating the probability P H0 (Sm(T; S) τ) = P where E i,i 2 (S) = {Y i,i 2 (S) τ}. ( T m + i = T 2 m 2+ i 2= E i,i 2 (S) )

Methodology Simulation methods: Normal data A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 20 / 34 Importance sampling algorithm Algorithm Importance Sampling Algorithm for Scan Statistics Begin Repeat for each k from to ITER (iterations number) : Generate uniformly the couple (i (k), i (k) 2 ) from the set {,..., T m + } {,..., T 2 m 2 + }. { } 2: Given the couple (i (k), i (k) 2 ), generate a sample of the random eld X ~ (k) (k) =,s2, with s j { } {,..., T j } and j {, 2}, from the conditional distribution of X given Y (k) (k) (S) τ. i,i 2 3: Take c k = C( X ~ (k) ) the number of all couples (i, i 2 ) for which Ỹ i,i2 (S) τ and put ρ k (2) =. c k End Repeat Return ρ(2) = ITER ρ k (2), ITER End k= Var [ ρ(2)] ITER ITER k= ( ρ k (2) ITER ρ k (2) ITER k= ) 2

Methodology Simulation methods: Normal data A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 2 / 34 Importance sampling algorithm: N (µ, σ 2 ) Step 2 requires to sample: ( (S) from the tail distribution P Y i (k) (k),i 2 Y i (k) (k),i 2 for the other indices, from the conditional distribution given ) (S) τ ([Devroye, 986]) { Y i (k) (k),i 2 } (S) τ Lemma (Generalization of [Am rioarei, 204, Lemma 3.4.4]) Let N be a positive integer, X = (X, X 2,..., X N ) be a vector of i.i.d. N (µ, σ 2 ) and a = (a,..., a N ) R N a non zero constant vector ( a j 0 for some particular j). Then conditionally given a, X = t, the r.v.'s with s j are jointly distributed as = a s a t µa j a a a j i j a i ( Z i µ a j a ) + Z s where Z s are i.i.d. N (µ, σ 2 ) r.v.s.

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 22 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 23 / 34 Numerical examples

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 24 / 34 Scanning a region of size T T 2 = 250 250 Table : Numerical results for P(S τ): Triangle Window's shape Triangle (m = 4, m2 = 8, Nt = 33, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.9675 0.99820 0.00823 59 0.86250 0.89485 0.0202 4 0.99747 0.997507 0.000257 60 0.93563 0.948980 0.00556 5 0.999946 0.999946 0.000005 6 0.972665 0.975087 0.002637 6 0.999999 0.999999 0 62 0.986982 0.988558 0.00273 7 0.999999 0.999999 0 63 0.994493 0.994676 0.000599 8.000000.000000 0 64 0.997532 0.997604 0.000280 9.000000.000000 0 65 0.998943 0.998978 0.00026 0.000000.000000 0 66 0.999556 0.999562 0.000057.000000.000000 0 67 0.999824 0.99989 0.000025,s 2 P(0.25),s 2 N (0, ) 59 0.737933 0.822902 0.024424 50 0.9285 0.93266 0.007379 60 0.87787 0.903293 0.003 5 0.94649 0.957850 0.004762 6 0.93894 0.95589 0.005389 52 0.96694 0.974052 0.003052 62 0.97229 0.975560 0.002639 53 0.980486 0.982878 0.00962 63 0.987388 0.988468 0.00309 54 0.98829 0.989444 0.00243 64 0.99428 0.994332 0.000642 55 0.993024 0.993292 0.000805 65 0.997330 0.997386 0.000307 56 0.995643 0.99603 0.000505 66 0.998794 0.998844 0.00047 57 0.99753 0.997494 0.00038 67 0.999474 0.999490 0.000068 58 0.998482 0.998597 0.00098

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 24 / 34 Scanning a region of size T T 2 = 250 250 Table : Numerical results for P(S τ): Triangle Window's shape Triangle (m = 4, m2 = 8, Nt = 33, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.9675 0.99820 0.00823 59 0.86250 0.89485 0.0202 4 0.99747 0.997507 0.000257 60 0.93563 0.948980 0.00556 5 0.999946 0.999946 0.000005 6 0.972665 0.975087 0.002637 6 0.999999 0.999999 0 62 0.986982 0.988558 0.00273 7 0.999999 0.999999 0 63 0.994493 0.994676 0.000599 8.000000.000000 0 64 0.997532 0.997604 0.000280 9.000000.000000 0 65 0.998943 0.998978 0.00026 0.000000.000000 0 66 0.999556 0.999562 0.000057.000000.000000 0 67 0.999824 0.99989 0.000025,s 2 P(0.25),s 2 N (0, ) 59 0.737933 0.822902 0.024424 50 0.9285 0.93266 0.007379 60 0.87787 0.903293 0.003 5 0.94649 0.957850 0.004762 6 0.93894 0.95589 0.005389 52 0.96694 0.974052 0.003052 62 0.97229 0.975560 0.002639 53 0.980486 0.982878 0.00962 63 0.987388 0.988468 0.00309 54 0.98829 0.989444 0.00243 64 0.99428 0.994332 0.000642 55 0.993024 0.993292 0.000805 65 0.997330 0.997386 0.000307 56 0.995643 0.99603 0.000505 66 0.998794 0.998844 0.00047 57 0.99753 0.997494 0.00038 67 0.999474 0.999490 0.000068 58 0.998482 0.998597 0.00098

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 25 / 34 Scanning a region of size T T 2 = 250 250 Table 2 : Numerical results for P(S τ): Rectangle Window's shape Rectangle (m =, m2 = 2, Nt = 32, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.947673 0.947757 0.005972 59 0.857404 0.856832 0.0264 4 0.997963 0.997979 0.00080 60 0.920590 0.99603 0.005960 5 0.999942 0.999942 0.000004 6 0.956954 0.956530 0.002886 6 0.999998 0.999998 0 62 0.97755 0.97779 0.00395 7 0.999999 0.999999 0 63 0.9886 0.988263 0.000669 8.000000.000000 0 64 0.994206 0.994089 0.00039 9.000000.000000 0 65 0.9974 0.9974 0.00049 0.000000.000000 0 66 0.998593 0.99863 0.000069.000000.000000 0 67 0.99934 0.99989 0.00003,s 2 P(0.25),s 2 N (0, ) 59 0.763228 0.763426 0.02599 50 0.864066 0.86608 0.00457 60 0.8565 0.85872 0.02264 5 0.90488 0.9036 0.00665 6 0.9830 0.98395 0.00594 52 0.93252 0.932663 0.00428 62 0.95483 0.95457 0.002958 53 0.95403 0.953874 0.00275 63 0.975084 0.975426 0.00483 54 0.968702 0.968700 0.00775 64 0.986979 0.98707 0.000735 55 0.97875 0.978566 0.0045 65 0.993240 0.993240 0.00036 56 0.98577 0.98582 0.00073 66 0.9965 0.996547 0.00076 57 0.990663 0.990575 0.000468 67 0.998276 0.998279 0.000084 58 0.993763 0.993827 0.000296

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 25 / 34 Scanning a region of size T T 2 = 250 250 Table 2 : Numerical results for P(S τ): Rectangle Window's shape Rectangle (m =, m2 = 2, Nt = 32, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.947673 0.947757 0.005972 59 0.857404 0.856832 0.0264 4 0.997963 0.997979 0.00080 60 0.920590 0.99603 0.005960 5 0.999942 0.999942 0.000004 6 0.956954 0.956530 0.002886 6 0.999998 0.999998 0 62 0.97755 0.97779 0.00395 7 0.999999 0.999999 0 63 0.9886 0.988263 0.000669 8.000000.000000 0 64 0.994206 0.994089 0.00039 9.000000.000000 0 65 0.9974 0.9974 0.00049 0.000000.000000 0 66 0.998593 0.99863 0.000069.000000.000000 0 67 0.99934 0.99989 0.00003,s 2 P(0.25),s 2 N (0, ) 59 0.763228 0.763426 0.02599 50 0.864066 0.86608 0.00457 60 0.8565 0.85872 0.02264 5 0.90488 0.9036 0.00665 6 0.9830 0.98395 0.00594 52 0.93252 0.932663 0.00428 62 0.95483 0.95457 0.002958 53 0.95403 0.953874 0.00275 63 0.975084 0.975426 0.00483 54 0.968702 0.968700 0.00775 64 0.986979 0.98707 0.000735 55 0.97875 0.978566 0.0045 65 0.993240 0.993240 0.00036 56 0.98577 0.98582 0.00073 66 0.9965 0.996547 0.00076 57 0.990663 0.990575 0.000468 67 0.998276 0.998279 0.000084 58 0.993763 0.993827 0.000296

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 26 / 34 Scanning a region of size T T 2 = 250 250 Table 3 : Numerical results for P(S τ): Quadrilateral Window's shape Quadrilateral (m = 4, m2 = 8, Nt = 3, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.925258 0.92674 0.009722 59 0.93203 0.926709 0.007540 4 0.99765 0.99769 0.000235 60 0.958806 0.963738 0.003582 5 0.999945 0.999945 0.000005 6 0.98433 0.982643 0.00725 6 0.999999 0.999999 0 62 0.99635 0.9989 0.000823 7 0.999999 0.999999 0 63 0.99670 0.996220 0.000387 8.000000.000000 0 64 0.998287 0.998322 0.00080 9.000000.000000 0 65 0.999243 0.999240 0.000082 0.000000.000000 0 66 0.999679 0.999693 0.000037.000000.000000 0 67 0.999866 0.999869 0.00006,s 2 P(0.25),s 2 N (0, ) 59 0.83677 0.876 0.05388 50 0.92057 0.935366 0.006875 60 0.98389 0.93407 0.007235 5 0.949483 0.95763 0.004378 6 0.960554 0.963979 0.003555 52 0.968977 0.972699 0.002844 62 0.98340 0.982444 0.00754 53 0.980772 0.982865 0.00823 63 0.9948 0.99396 0.000863 54 0.98882 0.989097 0.0075 64 0.995798 0.996026 0.000423 55 0.992870 0.992982 0.000758 65 0.99848 0.99868 0.000205 56 0.995739 0.995736 0.00048 66 0.99926 0.99957 0.000097 57 0.997355 0.997429 0.000302 67 0.99968 0.99962 0.000045 58 0.998397 0.998445 0.00090

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 26 / 34 Scanning a region of size T T 2 = 250 250 Table 3 : Numerical results for P(S τ): Quadrilateral Window's shape Quadrilateral (m = 4, m2 = 8, Nt = 3, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.925258 0.92674 0.009722 59 0.93203 0.926709 0.007540 4 0.99765 0.99769 0.000235 60 0.958806 0.963738 0.003582 5 0.999945 0.999945 0.000005 6 0.98433 0.982643 0.00725 6 0.999999 0.999999 0 62 0.99635 0.9989 0.000823 7 0.999999 0.999999 0 63 0.99670 0.996220 0.000387 8.000000.000000 0 64 0.998287 0.998322 0.00080 9.000000.000000 0 65 0.999243 0.999240 0.000082 0.000000.000000 0 66 0.999679 0.999693 0.000037.000000.000000 0 67 0.999866 0.999869 0.00006,s 2 P(0.25),s 2 N (0, ) 59 0.83677 0.876 0.05388 50 0.92057 0.935366 0.006875 60 0.98389 0.93407 0.007235 5 0.949483 0.95763 0.004378 6 0.960554 0.963979 0.003555 52 0.968977 0.972699 0.002844 62 0.98340 0.982444 0.00754 53 0.980772 0.982865 0.00823 63 0.9948 0.99396 0.000863 54 0.98882 0.989097 0.0075 64 0.995798 0.996026 0.000423 55 0.992870 0.992982 0.000758 65 0.99848 0.99868 0.000205 56 0.995739 0.995736 0.00048 66 0.99926 0.99957 0.000097 57 0.997355 0.997429 0.000302 67 0.99968 0.99962 0.000045 58 0.998397 0.998445 0.00090

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 27 / 34 Scanning a region of size T T 2 = 250 250 Table 4 : Numerical results for P(S τ): Circle Window's shape Circle (m = 3, m2 = 3, Nt = 29, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.949902 0.950399 0.006284 59 0.92034 0.920987 0.005962 4 0.9983 0.998099 0.00087 60 0.95687 0.95737 0.002840 5 0.999947 0.999947 0.000004 6 0.977475 0.977539 0.00366 6 0.999998 0.999998 0 62 0.988597 0.988595 0.00065 7 0.999999 0.999999 0 63 0.994265 0.994274 0.000306 8.000000.000000 0 64 0.997227 0.997257 0.00042 9.000000.000000 0 65 0.998698 0.998675 0.000065 0.000000.000000 0 66 0.999377 0.99938 0.000029.000000.000000 0 67 0.999709 0.99975 0.00003,s 2 P(0.25),s 2 N (0, ) 59 0.860740 0.860278 0.02220 50 0.889276 0.889385 0.00824 60 0.98923 0.99454 0.005955 5 0.92070 0.920950 0.005242 6 0.955942 0.955339 0.00295 52 0.945380 0.94554 0.00335 62 0.97565 0.975720 0.00449 53 0.962727 0.962884 0.00242 63 0.987403 0.98734 0.00072 54 0.974988 0.974968 0.00370 64 0.993559 0.993485 0.000347 55 0.983200 0.98388 0.000875 65 0.996698 0.996696 0.00067 56 0.989078 0.988928 0.000556 66 0.998365 0.998353 0.000079 57 0.992649 0.992725 0.000350 67 0.99923 0.999202 0.000037 58 0.995300 0.995278 0.000220

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 27 / 34 Scanning a region of size T T 2 = 250 250 Table 4 : Numerical results for P(S τ): Circle Window's shape Circle (m = 3, m2 = 3, Nt = 29, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.949902 0.950399 0.006284 59 0.92034 0.920987 0.005962 4 0.9983 0.998099 0.00087 60 0.95687 0.95737 0.002840 5 0.999947 0.999947 0.000004 6 0.977475 0.977539 0.00366 6 0.999998 0.999998 0 62 0.988597 0.988595 0.00065 7 0.999999 0.999999 0 63 0.994265 0.994274 0.000306 8.000000.000000 0 64 0.997227 0.997257 0.00042 9.000000.000000 0 65 0.998698 0.998675 0.000065 0.000000.000000 0 66 0.999377 0.99938 0.000029.000000.000000 0 67 0.999709 0.99975 0.00003,s 2 P(0.25),s 2 N (0, ) 59 0.860740 0.860278 0.02220 50 0.889276 0.889385 0.00824 60 0.98923 0.99454 0.005955 5 0.92070 0.920950 0.005242 6 0.955942 0.955339 0.00295 52 0.945380 0.94554 0.00335 62 0.97565 0.975720 0.00449 53 0.962727 0.962884 0.00242 63 0.987403 0.98734 0.00072 54 0.974988 0.974968 0.00370 64 0.993559 0.993485 0.000347 55 0.983200 0.98388 0.000875 65 0.996698 0.996696 0.00067 56 0.989078 0.988928 0.000556 66 0.998365 0.998353 0.000079 57 0.992649 0.992725 0.000350 67 0.99923 0.999202 0.000037 58 0.995300 0.995278 0.000220

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 28 / 34 Scanning a region of size T T 2 = 250 250 Table 5 : Numerical results for P(S τ): Ellipse Window's shape Ellipse (m = 9, m2 = 9, Nt = 35, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.94549 0.94408 0.006856 59 0.76436 0.764423 0.022290 4 0.997692 0.997783 0.00027 60 0.86070 0.858224 0.020 5 0.999934 0.999935 0.000005 6 0.922564 0.920545 0.005593 6 0.999998 0.999998 0 62 0.956994 0.956925 0.002803 7 0.999999 0.999999 0 63 0.97724 0.977 0.0039 8.000000.000000 0 64 0.98864 0.98870 0.000678 9.000000.000000 0 65 0.993942 0.99403 0.000328 0.000000.000000 0 66 0.99706 0.997005 0.00055.000000.000000 0 67 0.99856 0.998565 0.000072,s 2 P(0.25),s 2 N (0, ) 59 0.638328 0.638600 0.04242 50 0.845330 0.842760 0.0460 60 0.76953 0.769090 0.02264 5 0.887757 0.887823 0.007568 6 0.860703 0.859829 0.00885 52 0.92023 0.920804 0.005026 62 0.98782 0.99289 0.005606 53 0.94475 0.944745 0.00332 63 0.954499 0.95497 0.002890 54 0.96667 0.96830 0.00276 64 0.975360 0.97527 0.00468 55 0.973786 0.973964 0.0043 65 0.986788 0.986846 0.000746 56 0.982379 0.982058 0.000929 66 0.993076 0.993056 0.000372 57 0.988009 0.98842 0.000599 67 0.996464 0.996456 0.00083 58 0.99239 0.99260 0.000385

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 28 / 34 Scanning a region of size T T 2 = 250 250 Table 5 : Numerical results for P(S τ): Ellipse Window's shape Ellipse (m = 9, m2 = 9, Nt = 35, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.94549 0.94408 0.006856 59 0.76436 0.764423 0.022290 4 0.997692 0.997783 0.00027 60 0.86070 0.858224 0.020 5 0.999934 0.999935 0.000005 6 0.922564 0.920545 0.005593 6 0.999998 0.999998 0 62 0.956994 0.956925 0.002803 7 0.999999 0.999999 0 63 0.97724 0.977 0.0039 8.000000.000000 0 64 0.98864 0.98870 0.000678 9.000000.000000 0 65 0.993942 0.99403 0.000328 0.000000.000000 0 66 0.99706 0.997005 0.00055.000000.000000 0 67 0.99856 0.998565 0.000072,s 2 P(0.25),s 2 N (0, ) 59 0.638328 0.638600 0.04242 50 0.845330 0.842760 0.0460 60 0.76953 0.769090 0.02264 5 0.887757 0.887823 0.007568 6 0.860703 0.859829 0.00885 52 0.92023 0.920804 0.005026 62 0.98782 0.99289 0.005606 53 0.94475 0.944745 0.00332 63 0.954499 0.95497 0.002890 54 0.96667 0.96830 0.00276 64 0.975360 0.97527 0.00468 55 0.973786 0.973964 0.0043 65 0.986788 0.986846 0.000746 56 0.982379 0.982058 0.000929 66 0.993076 0.993056 0.000372 57 0.988009 0.98842 0.000599 67 0.996464 0.996456 0.00083 58 0.99239 0.99260 0.000385

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 29 / 34 Scanning a region of size T T 2 = 250 250 Table 6 : Numerical results for P(S τ): Annulus Window's shape Annulus (m = 7, m2 = 7, Nt = 24, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.88687 0.882372 0.009897 59 0.9532 0.95283 0.00572 4 0.995456 0.995488 0.00020 60 0.975645 0.975784 0.000654 5 0.999882 0.999883 0.000004 6 0.9889 0.988284 0.000280 6 0.999997 0.999997 0 62 0.99445 0.99445 0.00020 7 0.999999 0.999999 0 63 0.997437 0.997444 0.00005 8.000000.000000 0 64 0.998835 0.998838 0.00002 9.000000.000000 0 65 0.99948 0.999483 0.000009 0.000000.000000 0 66 0.999774 0.999774 0.000003.000000.000000 0 67 0.999903 0.999903 0.00000,s 2 P(0.25),s 2 N (0, ) 59 0.902926 0.903783 0.003972 50 0.860688 0.860266 0.006625 60 0.948792 0.94906 0.00623 5 0.9057 0.904708 0.00357 6 0.973906 0.973904 0.000700 52 0.936387 0.935930 0.0099 62 0.986863 0.98697 0.0003 53 0.957907 0.957630 0.0037 63 0.993560 0.993580 0.00038 54 0.97230 0.972343 0.00066 64 0.996904 0.99690 0.00006 55 0.98234 0.98227 0.000386 65 0.998536 0.998536 0.000027 56 0.988540 0.988555 0.000228 66 0.99938 0.99938 0.0000 57 0.992783 0.992775 0.00035 67 0.999688 0.999689 0.000005 58 0.995458 0.995467 0.000079

Simulation study Numerical examples A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 29 / 34 Scanning a region of size T T 2 = 250 250 Table 6 : Numerical results for P(S τ): Annulus Window's shape Annulus (m = 7, m2 = 7, Nt = 24, IS = e4, IA = e5),s 2 B(, 0.0),s 2 B(5, 0.05) 3 0.88687 0.882372 0.009897 59 0.9532 0.95283 0.00572 4 0.995456 0.995488 0.00020 60 0.975645 0.975784 0.000654 5 0.999882 0.999883 0.000004 6 0.9889 0.988284 0.000280 6 0.999997 0.999997 0 62 0.99445 0.99445 0.00020 7 0.999999 0.999999 0 63 0.997437 0.997444 0.00005 8.000000.000000 0 64 0.998835 0.998838 0.00002 9.000000.000000 0 65 0.99948 0.999483 0.000009 0.000000.000000 0 66 0.999774 0.999774 0.000003.000000.000000 0 67 0.999903 0.999903 0.00000,s 2 P(0.25),s 2 N (0, ) 59 0.902926 0.903783 0.003972 50 0.860688 0.860266 0.006625 60 0.948792 0.94906 0.00623 5 0.9057 0.904708 0.00357 6 0.973906 0.973904 0.000700 52 0.936387 0.935930 0.0099 62 0.986863 0.98697 0.0003 53 0.957907 0.957630 0.0037 63 0.993560 0.993580 0.00038 54 0.97230 0.972343 0.00066 64 0.996904 0.99690 0.00006 55 0.98234 0.98227 0.000386 65 0.998536 0.998536 0.000027 56 0.988540 0.988555 0.000228 66 0.99938 0.99938 0.0000 57 0.992783 0.992775 0.00035 67 0.999688 0.999689 0.000005 58 0.995458 0.995467 0.000079

Simulation study Power A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 30 / 34 Outline Introduction Framework Problem 2 Methodology Approximation Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 References

Simulation study Power A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 3 / 34 Power of the scan statistic test

Simulation study Power Power evaluation for B(, 0.00) model Triangular simulated cluster 0.9 0.8 0.7 Rectangular simulated cluster 0.9 0.8 0.7 Power 0.6 0.5 0.4 0.3 0.2 0. Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 0.05 0. 0.5 p Power 0.6 0.5 0.4 0.3 0.2 0. Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 0.05 0. 0.5 p A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 32 / 34

Simulation study Power Power evaluation for B(, 0.00) model Quadrilateral simulated cluster 0.9 0.8 0.7 Circular simulated cluster 0.9 0.8 0.7 Power 0.6 0.5 0.4 0.3 0.2 0. Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 0.05 0. 0.5 p Power 0.6 0.5 0.4 0.3 0.2 0. Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 0.05 0. 0.5 p A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 33 / 34

Simulation study Power Power evaluation for B(, 0.00) model Ellipsoidal simulated cluster 0.9 0.8 0.7 Annular simulated cluster 0.9 0.8 0.7 Power 0.6 0.5 0.4 0.3 0.2 0. Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 0.05 0. 0.5 p Power 0.6 0.5 0.4 0.3 0.2 0. Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 0.05 0. 0.5 p A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 34 / 34

Simulation study Power A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 34 / 34

References Anderson, N. H. and Titterington, D. M. (997). Some methods for investigating spatial clustering, with epidemiological applications. Journal of the Royal Statistical Society: Series A (Statistics in Society), 60():8705. A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 34 / 34 Alm, S. E. (983). On the distribution of scan statistic of poisson process. In Gut, A. and Helst, L., editors, Probability and Mathematical Statistics, pages 0. Upsalla University Press. Alm, S. E. (997). On the distributions of scan statistics of a two-dimensional poisson process. Advances in Applied Probability, pages 8. Alm, S. E. (998). Approximation and simulation of the distributions of scan statistics for poisson processes in higher dimensions. Extremes, ():26. Am rioarei, A. (204). Approximations for the multidimensional discrete scan statistics. PhD thesis, University of Lille.

References Assunção, R., Costa, M., Tavares, A., and Ferreira, S. (2006). Fast detection of arbitrarily shaped disease clusters. Statistics in Medicine, 25(5):723742. Bresenham, J. (965). Algorithm for computer control of a digital printer. IBM Systems Journal, 4(). Bresenham, J. (977). A linear algorithm for incremental digital display of circular arcs. Communications of the ACM, 20(2):0006. Devroye, L. (986). Non uniform random variate generation. Springer-Verlag, New York. Foley, J. (995). Computer Graphics, Principles and Practice in C. Addison-Wesley Professional. Glaz, J., Naus, J., and Wallenstein, S. (200). Scan Statistics. A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 34 / 34

References Springer. Glaz, J., Pozdnyakov, V., and Wallenstein, S. (2009). Scan Statistics: Methods and Applications. Birkhaüser Boston. Loader, C. R. (99). Large-deviation approximations to the distribution of scan statistics. Advances in Applied Probability, pages 7577. Tango, T. and Takahashi, K. (2005). A exibly shaped spatial scan statistic for detecting clusters. Int J Health Geogr, 4:. Tango, Toshiro Takahashi, Kunihiko England Int J Health Geogr. 2005 May 8;4:. A. Am rioarei (INCDSB) Scan statistics with arbitrary shape SPSR 206 34 / 34