Discrete scan statistics with windows of arbitrary shape

Discrete scan statistics with windows of arbitrary shape Alexandru Am rioarei National Institute of Research and Development for Biological Sciences Bucharest, Romania MΘDAL TEAM - INRIA Lille Nord Europe Lille, France The 8 th International Workshop on Applied Probability 20-23 June, 206, Toronto, Canada A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 / 50

A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 2 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Introduction Framework A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 3 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Introduction Framework A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 4 / 50 Denitions and notations

Introduction Framework A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 5 / 50 Preliminary notations Let T, T 2 be positive integers s T s 2 T 2 X s,s 2 T 2 T Rectangular region R 2 = [0, T ] [0, T 2 ] (X s,s 2 ) s T s 2 T 2 iid rv's Bernoulli(B(, p)) Binomial(B(n, p)) Poisson(P(λ)) Normal(N (µ, σ 2 )) X s,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 IWAP 206 6 / 50 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 IWAP 206 7 / 50 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 : Circle A ( G ) : Circle A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 8 / 50

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 : Annulus A ( G ) : Annulus A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 8 / 50

Introduction Framework A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 9 / 50 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 + )X s,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 X s,s 2 is the classical rectangular window of the two dimensional scan statistics

Introduction Problem A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 0 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Introduction Problem A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 / 50 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 IWAP 206 2 / 50

Methodology A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 3 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Methodology A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 4 / 50 methodology for the general scan statistic

Methodology A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 5 / 50 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 A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 6 / 50 Illustration of the approximation process T 2 T R 2 Find 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 A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 6 / 50 Illustration of the approximation process T 2 T R 2 First Step 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 A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 6 / 50 Illustration of the approximation process T 2 T R 2 Second Step 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 IWAP 206 7 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Methodology Simulation methods: Normal data A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 8 / 50 Simulation methods for Normal data

Methodology Simulation methods: Normal data A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 9 / 50 Importance sampling algorithm Test the null hypothesis of randomness against an alternative of clustering H 0 : The rv's X s,s 2 are iid N (µ, σ 2 ) H : There exists R(i, i 2 ) = [i, i + m ] [i 2, i 2 + m 2 ] R 2 where the rv's X s,s 2 N (µ, σ 2 ), µ > µ and X s,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 IWAP 206 20 / 50 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) = X s,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 k= End 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 IWAP 206 2 / 50 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 344]) Let N be a positive integer, X = (X, X 2,, X N ) be a vector of iid 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 rv's X s with s j are jointly distributed as X s = 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 iid N (µ, σ 2 ) rvs

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 22 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 23 / 50 Numerical examples

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 24 / 50 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 = e5, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 096397 098667 0004333 59 0863336 08970 0004902 4 0997488 0997483 0000082 60 0936684 094860 000200 5 0999947 0999947 0000002 6 097529 0974938 0000894 6 0999999 0999999 0 62 0987094 0988279 000042 7 0999999 0999999 0 63 0994372 0994664 000092 8 000000 000000 0 64 0997563 0997643 0000089 9 000000 000000 0 65 0998946 099897 000004 0 000000 000000 0 66 0999564 0999567 000008 000000 000000 0 67 099987 0999820 0000008 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0739866 0820277 0083 50 0970 0934999 0002737 60 087829 0902957 0004483 5 094529 0957777 000655 6 0939577 0950977 0009 52 0966494 0972997 000026 62 097673 0975649 0000890 53 0979944 0982929 0000644 63 0987075 0988206 0000425 54 0987840 0989469 0000406 64 099404 0994492 0000204 55 0992768 0993477 0000257 65 0997384 0997452 0000098 56 0995667 0996022 000062 66 099882 0998855 0000046 57 099742 0997574 000002 67 0999489 0999490 0000022 58 0998509 0998563 0000063

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 25 / 50 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 = e5, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 0947896 0948226 00020 59 0856993 0857094 0005485 4 0997983 0997986 0000058 60 09946 099589 0002300 5 0999943 0999943 000000 6 0956420 0956569 000024 6 0999999 0999999 0 62 097742 0977065 000047 7 0999999 0999999 0 63 0988228 0988208 0000220 8 000000 000000 0 64 099430 0994095 000003 9 000000 000000 0 65 09972 099707 0000048 0 000000 000000 0 66 099860 0998607 0000022 000000 000000 0 67 0999342 0999342 000000 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0764372 0764495 003285 50 0865858 0865484 0004572 60 0857872 0859289 0005320 5 0904555 0904805 000269 6 098972 098732 0002307 52 0933323 0933206 000620 62 0954682 0954579 000059 53 0953950 0953807 0000993 63 097527 097539 0000502 54 0968340 0968483 000067 64 0986996 0986966 0000242 55 0978632 0978687 0000386 65 0993240 099326 00007 56 098579 0985752 0000242 66 0996557 099655 0000056 57 099062 0990585 000052 67 0998290 0998283 0000027 58 0993837 099385 0000096

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 26 / 50 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 = e5, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 0926068 0927398 0003806 59 094546 092763 0002942 4 0997622 0997627 0000075 60 0959599 0963873 000255 5 0999946 0999946 0000002 6 098235 0982506 000057 6 0999999 0999999 0 62 099423 099796 0000266 7 0999999 0999999 0 63 09963 0996233 000024 8 000000 000000 0 64 0998283 0998337 0000057 9 000000 000000 0 65 0999266 0999266 0000026 0 000000 000000 0 66 0999684 0999684 000002 000000 000000 0 67 0999868 0999869 0000005 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0835054 087035 0006852 50 0920004 0935266 000257 60 097972 093040 0002768 5 0950232 09577 000556 6 0960397 09647 000237 52 0968755 0972594 0000964 62 098228 098245 0000585 53 0980695 0982566 0000606 63 09942 09950 000028 54 09880 0989060 0000383 64 0995855 099597 000036 55 0992626 09930 0000242 65 099808 099824 0000065 56 0995569 099577 000053 66 099935 099953 000003 57 099736 0997394 0000096 67 0999620 0999622 000004 58 0998379 0998435 0000060

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 27 / 50 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 = e54, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 09503 095046 000295 59 0920229 0920388 000238 4 09988 09984 0000059 60 095684 095743 00006 5 0999947 0999947 000000 6 0977460 097764 0000462 6 0999999 0999999 0 62 0988568 0988567 000024 7 0999999 0999999 0 63 099432 0994309 0000099 8 000000 000000 0 64 0997229 0997228 0000046 9 000000 000000 0 65 0998678 0998679 000002 0 000000 000000 0 66 0999380 099938 0000009 000000 000000 0 67 099975 099975 0000004 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0858454 085978 000539 50 088837 088789 0003485 60 09982 099586 000230 5 09273 092549 0002058 6 0955229 0955388 000047 52 094576 0945644 000243 62 0976023 0975987 000049 53 0962790 0962825 0000760 63 098744 0987344 0000234 54 0974848 0974878 0000470 64 0993477 0993502 00002 55 0983235 0983263 0000293 65 0996706 0996703 0000054 56 0988885 0988907 000082 66 0998372 0998365 0000025 57 0992730 0992734 00004 67 0999207 0999203 000002 58 0995269 0995287 000007

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 27 / 50 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 = e54, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 09503 095046 000295 59 0920229 0920388 000238 4 09988 09984 0000059 60 095684 095743 00006 5 0999947 0999947 000000 6 0977460 097764 0000462 6 0999999 0999999 0 62 0988568 0988567 000024 7 0999999 0999999 0 63 099432 0994309 0000099 8 000000 000000 0 64 0997229 0997228 0000046 9 000000 000000 0 65 0998678 0998679 000002 0 000000 000000 0 66 0999380 099938 0000009 000000 000000 0 67 099975 099975 0000004 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0858454 085978 000539 50 088837 088789 0003485 60 09982 099586 000230 5 09273 092549 0002058 6 0955229 0955388 000047 52 094576 0945644 000243 62 0976023 0975987 000049 53 0962790 0962825 0000760 63 098744 0987344 0000234 54 0974848 0974878 0000470 64 0993477 0993502 00002 55 0983235 0983263 0000293 65 0996706 0996703 0000054 56 0988885 0988907 000082 66 0998372 0998365 0000025 57 0992730 0992734 00004 67 0999207 09992032 000002 58 0995269 0995287 000007

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 28 / 50 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 = e5, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 094400 09442 0002297 59 076487 0763482 000928 4 0997757 0997758 0000069 60 0860903 0860548 000427 5 0999935 0999935 0000002 6 092089 0920882 00094 6 0999998 0999998 0 62 0956735 0956866 0000934 7 000000 000000 0 63 09778 0977094 0000452 8 000000 000000 0 64 098882 098852 000028 9 000000 000000 0 65 0994044 099402 000004 0 000000 000000 0 66 0997037 0997037 0000049 000000 000000 0 67 0998554 0998558 0000023 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0638962 063902 009442 50 084356 08443 0004369 60 0768283 0769666 000860 5 0887477 0887692 0002755 6 08664 0860885 000402 52 092060 0920385 000757 62 09944 09930 000948 53 0944398 0944328 00027 63 095494 0954864 0000965 54 096682 096667 0000725 64 0975255 0975369 000048 55 0973797 0973864 0000468 65 0986869 0986900 0000240 56 0982232 0982330 000030 66 09935 099327 00009 57 098883 098832 000093 67 0996485 0996472 0000058 58 099238 099234 000023

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 29 / 50 Scanning a region of size T T 2 = 250 250 Table 6 : Numerical results for P(S τ): Ellipse2 Window's shape Ellipse2 (m = 9, m2 = 9, Nt = 35, IS = e5, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 094480 0943779 0003420 59 0764090 0763753 003670 4 099776 0997767 000007 60 08595 0860602 0029 5 0999934 0999935 0000002 6 09273 0920669 000386 6 0999998 0999998 0 62 0956920 0956693 000440 7 0999999 0999999 0 63 097798 097729 0000586 8 000000 000000 0 64 098862 098877 0000253 9 000000 000000 0 65 0993979 0994008 00003 0 000000 000000 0 66 0997032 0997036 000005 000000 000000 0 67 0998558 0998562 0000023 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0638389 0640805 07483 50 0843792 0844342 003346 60 0770899 0769795 0034030 5 0888070 0887823 0006857 6 0860847 086092 000890 52 0920460 092069 0003626 62 099522 099537 0003909 53 094454 0944368 000974 63 0954873 0954742 00056 54 09659 096748 00009 64 0975326 0975379 0000635 55 097386 0973830 0000640 65 0986856 0986843 0000282 56 0982294 0982269 0000377 66 099354 09939 000030 57 098880 098843 0000226 67 0996482 0996478 000006 58 099254 099223 000038

Simulation study Numerical examples A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 30 / 50 Scanning a region of size T T 2 = 250 250 Table 7 : Numerical results for P(S τ): Annulus Window's shape Annulus (m = 7, m2 = 7, Nt = 24, IS = e5, IA = e6) X s,s 2 B(, 00) X s,s 2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 3 088798 0882489 000482 59 09570 095245 0000699 4 0995434 0995465 0000069 60 0975772 0975727 0000255 5 0999883 0999883 000000 6 0988275 0988270 0000099 6 0999998 0999998 0 62 0994460 0994466 000004 7 0999999 0999999 0 63 0997439 0997440 000007 8 000000 000000 0 64 0998839 0998840 0000007 9 000000 000000 0 65 0999484 0999484 0000003 0 000000 000000 0 66 0999775 0999775 000000 000000 000000 0 67 0999903 0999903 0000000 X s,s 2 P(025) X s,s 2 N (0, ) τ Sim AppH ETotal τ Sim AppH ETotal 59 0903956 0903852 000228 50 0860708 086046 0004097 60 0949083 0949059 0000735 5 090465 0904644 000977 6 097384 0973844 0000277 52 0936077 0935938 0000987 62 0986909 0986876 0000 53 0957564 0957650 0000508 63 0993576 0993570 0000047 54 0972378 09723 0000270 64 099690 0996907 0000020 55 098239 098236 000048 65 0998539 0998539 0000009 56 0988564 0988587 0000082 66 0999320 0999320 0000004 57 0992769 0992769 0000047 67 0999689 0999689 0000002 58 099547 0995466 0000027

Simulation study Power A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 3 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Simulation study Power A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 32 / 50 Power of the scan statistic test

Simulation study Power Power evaluation for B(, 000) model Triangular simulated cluster 09 08 07 Rectangular simulated cluster 09 08 07 Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 005 0 05 p Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 005 0 05 p A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 33 / 50

Simulation study Power Power evaluation for B(, 000) model Quadrilateral simulated cluster 09 08 07 Circular simulated cluster 09 08 07 Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 005 0 05 p Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 005 0 05 p A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 34 / 50

Simulation study Power Power evaluation for B(, 000) model Ellipsoidal simulated cluster 09 08 07 Annular simulated cluster 09 08 07 Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 005 0 05 p Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Annulus (24) 0 0 005 0 05 p A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 35 / 50

Simulation study Power Power evaluation for B(5, 005) model Triangular simulated cluster 09 08 07 Rectangular simulated cluster 09 08 07 Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Ellipse2 (35) Annulus (24) 0 005 0 05 02 025 p Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Ellipse2 (35) Annulus (24) 0 005 0 05 02 025 p A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 36 / 50

Simulation study Power Power evaluation for B(5, 005) model Quadrilateral simulated cluster 09 08 07 Circular simulated cluster 09 08 07 Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Ellipse2 (35) Annulus (24) 0 005 0 05 02 025 p Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Ellipse2 (35) Annulus (24) 0 005 0 05 02 025 p A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 37 / 50

Simulation study Power Power evaluation for B(5, 005) model Ellipsoidal simulated cluster 09 08 07 Annular simulated cluster 09 08 07 Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Ellipse2 (35) Annulus (24) 0 005 0 05 02 025 p Power 06 05 04 03 02 0 Shape of the scanning window Triangle (33) Rectangle (32) Quadrillater (3) Circle (29) Ellipse (35) Ellipse2 (35) Annulus (24) 0 005 0 05 02 025 p A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 38 / 50

Scan on a Cylinder Problem A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 39 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Scan on a Cylinder Problem A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 40 / 50 Scanning the surface of a cylinder

Scan on a Cylinder Problem A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 4 / 50 Scanning the surface of a cylinder Unfolded cylinder of size T T 2 Cylinder

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 42 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 43 / 50 Transformation into a one dimensional problem

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 44 / 50 Transformation of the unfolded cylinder m2 X, X,m2 X,T2 m X m, Xm,m 2 X m,t 2 T X T, X T,m 2 X T,T 2 T2 X, X,m2 X,T2 X,m2 Xm,m 2 XT,m 2 X,T2 X m,t 2 X T,T 2

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 44 / 50 Transformation of the unfolded cylinder m2 X, X,m2 X,T2 m X m, Xm,m 2 X m,t 2 T X T, X T,m 2 X T,T 2 T2 X, Xm, X T, X,m2 Xm,m 2 XT,m 2 X,T2 X m,t 2 X T,T 2

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 45 / 50 Defining the score function m2 X, X,m2 m T X m, Xm,m 2 X T, X T,m 2 The size of the scanning window is m = T m 2 (T m ) The score function is dened by S ( X i ) = A Xi where A is the corresponding {0, } vector 0 0

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 45 / 50 Defining the score function m2 m T 0 0 0 0 The size of the scanning window is m = T m 2 (T m ) The score function is dened by S ( X i ) = A Xi where A is the corresponding {0, } vector 0 0

Scan on a Cylinder A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 46 / 50 and error bounds Theorem [Am rioarei, 204] Let t {2, 3} and Q t = Qt (τ) = P (S m (t ( m ); S) τ) and L = T m If ˆQt is an estimate of Qt with ˆQt Qt β t and τ is such that ˆQ2 (τ) 0 then ) ) L (S P m ( T, S) τ (2 ˆQ2 ˆQ3 [ + ˆQ2 ˆQ3 + 2( ˆQ2 ˆQ3 ) 2] E total (), [ ( ) ) ] 2 E total () = (L ) β 2 + β 3 + F ˆQ2, L ( ˆQ2 + β 2 T X X m X m X2( m ) X 2 m X 3( m ) X T m 2( m ) Q 2 3( m ) Q 3

Scan on a Cylinder Numerical Results A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 47 / 50 Outline Introduction Framework Problem 2 Methodology Simulation methods: Normal data 3 Simulation study Numerical examples Power 4 Scanning the surface of a cylinder Problem Numerical Results 5 References

Scan on a Cylinder Numerical Results A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 48 / 50 Numerical examples

Scan on a Cylinder Numerical Results A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 49 / 50 Scanning a region of size T T 2 = 300 350 Table 8 : Numerical results for P(S τ): Cylinder Cylinder: (m = 0, m 2 = 5, T = 300, T 2 = 350, IS = e4, IA = e5) X s,s2 B(, 0) Xs,s2 B(5, 005) τ Sim AppH ETotal τ Sim AppH ETotal 33 087559 0870200 0003674 68 0955593 095567 0000938 34 094626 0946527 00077 69 0976348 0976285 000046 35 0979458 097938 0000393 70 0987406 0987574 0000227 36 0992379 0992465 00003 7 0993526 0993593 0000 37 0997384 0997367 0000043 72 0996772 099675 0000054 38 09996 0999 000004 73 099840 099839 0000026 39 099973 099974 0000004 74 099924 099922 000002 40 09999 09999 000000 75 0999623 0999626 0000006

Scan on a Cylinder Numerical Results A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 50 / 50 Scanning a region of size T T 2 = 300 350 Table 9 : Numerical results for P(S τ): Cylinder Cylinder: (m = 0, m 2 = 5, T = 300, T 2 = 350, IS = e4, IA = e5) X s,s2 P(025) τ Sim AppH ETotal 68 095283 09569 0002055 69 095447 095723 000023 70 0973445 0973488 000055 7 0985486 0985509 0000263 72 0992349 0992285 000033 73 0995950 0995979 0000066 74 099796 0997920 0000033 75 099895 0998945 000006

Scan on a Cylinder Numerical Results A Am rioarei (INCDSB) Scan statistics with arbitrary shape IWAP 206 50 / 50

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 IWAP 206 50 / 50 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) and simulation of the distributions of scan statistics for poisson processes in higher dimensions Extremes, ():26 Am rioarei, A (204) s 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 IWAP 206 50 / 50

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 IWAP 206 50 / 50