1 computation of E {K(r, n)}
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- Loreen Simmons
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1 1 Anlysis of the sptil orgniztion of molecules with robust sttistics: Supplementry Mteril Thibult Lgche 1,,, Gbriel Lng 3, Nthlie Suvonnet 4,, Jen-Christophe Olivo-Mrin 1,, 1 Unité d Anlyse d Imges Quntittive, Institut Psteur. Pris, Frnce Unité de Recherche Associée 58, Centre Ntionl de l Recherche Scientifique. Pris, Frnce 3 Unité Mixte de Recherche 518 Mthémtiques et Informtique Appliquées, AgroPrisTech nd INRA. Pris, Frnce 4 Unité de Biologie des Interctions Cellulires, Institut Psteur. Pris, Frnce E-mil: thibult.lgche@psteur.fr, jcolivo@psteur.fr In this supplementry mteril, our gol is to compute the four first moment of the Ripley s K-function Kr, n. Computtion of E Kr, n} nd vr Kr, n} cn be found in [1], however we refine here the computtion of vr Kr, n} for smll number of points in section nd we reproduce the computtion of E Kr, n} for ske of clrity in section 1. Finlly, the third nd the fourth moments of Kr, n, E Kr, n E Kr, n} 3} nd E Kr, n E Kr, n} 4}, re computed in sections 3 nd 4. 1 computtion of E Kr, n} Denoting ψx, y = 1 x y r} kx, y, we decompose the symmetric function φx, y s nd we re-write Kr, n s Kr, n = nn 1 φx, y = 1 ψx, y + ψy, x, 1 φx, y = x y Then, ssuming uniform distribution of points in Ω nd denoting 1 ψx, y + ψy, x. nn 1 x y α r = nn 1... n r + 1 r µ r, 3 where µ r is the Lebesgue mesure on R r, we hve [1] E Kr, n} = φx, ydα x, y. 4 nn 1 Ω tht is Denoting we re-write E Kr, n} = ψx, ydα x, y. 5 nn 1 Ω I 0 = ψx, ydx, y = Ω 1 x y r} kx, ydx, y, Ω 6 E Kr, n} = I 0, 7 nd we further compute I 0 by considering locl polr coordintes yr y, θ y round x0, 0 in Ω, with 0 r y r nd Θr y θ y Θr y where Θr y is the prt of the perimeter bx, r y tht is in Ω: π Θr y r y = bx, r y Ω, 8 tht is π Θr y = bx, r y Ω r y = π kx, y. 9
2 We then hve I 0 = Ω r Finlly, reinjecting I 0 Eq. 10 in Eq. 7 we hve 0 πr y dr y dx = πr. 10 E Kr, n} = I 0 = πr. 11 Integrls I j j 0 tht re introduced ll long this supplementry mteril re summrized with their numericl vlue in Supplementry Tble S1. computtion of vr Kr, n} = E Kr, n E Kr, n} } We first introduce the centered E = 0 estimtor K 0 r, n: where K 0 r, n = Kr, n E Kr, n} = nn 1 φ 0 x, y, 1 x y We then hve tht we expnd s φ 0 x, y = φx, y πr. 13 vr Kr, n} = E K0r, n } = nn 1 E φ 0 x, y x y, 14 φ 0 x, y x y = φ 0x, y + 3 x y x y z + 4 φ 0 x, yφ 0 x, z φ 0 x, yφ 0 z, w. 15 Computing the number of terms in ech sum of the eqution bove, we obtin tht nn 1 = nn nn 1n + 4 nn 1n n 3, 16 tht is n 4 n 3 + n = 4 n n n Identifying polynomil coefficients, we obtin tht =, 3 = 4, nd 4 = 1, 18 leding to vr Kr, n} = E nn 1 φ 0x, y x y + 4E φ 0 x, yφ 0 x, y + E x y z φ 0 x, yφ 0 z, w. 19
3 3 Considering tht x nd y re uniformly distributed in Ω, Eq. 19 becomes vr Kr, n} = nn 1 φ 0x, ydα x, y 0 Ω + 4 φ 0 x, yφ 0 x, wdα 3 x, y, w + φ 0 x, yφ 0 w, zdα 4 x, y, w, z Ω 3 where α r is given by Eq. 3. Becuse φ 0 x, ydα x, y = Ω φx, ydα x, y πr = 0, Ω 1 we hve φ 0 x, yφ 0 w, zdα 4 x, y, w, z = α 4 φ 0 x, ydx, y Ω φ 0 w, zdw, z = 0. Ω Then, denoting β = πr, we expnd the two remining integrls of Eq. 46 nd Becuse Ω φx, y = πr = β, we hve nd φ 0 x, y = φx, y βφx, y + β, 3 φ 0 x, yφ 0 x, z = φx, yφx, z β φx, y + φx, z + β, 4 φ 0x, ydx, y = Ω φx, y βφx, y + β dx, y = Ω φx, y β, Ω 5 φ 0 x, yφ 0 x, zdx, y, z = Ω 3 φx, yφx, zdx, y, z 3 β. Ω 6 Consequently, we re now left with the computtions of the integrls I 1 = φx, y dx, y 7 Ω nd I = φx, yφx, zdx, y, z. Ω 3 8 We first observe tht for points x tht re t distnce x Ω > r from the domin boundry, there is no edge correction for ny points y, z inside the domin Ω: kx, y = ky, x = 1 x y <r}. Consequently, integrls I 1 nd I cn be decomposed s follows I 1 = 1 x Ω >r} 1 x y <r} dx, y 9 Ω x Ω <r} 1 x y <r} kx, y + ky, x dx, y 4 Ω nd, I = 1 x Ω >r} 1 x y <r} 1 x z <r} dx, y, z 30 Ω x Ω <r} 1 x y <r} 1 x z <r} kx, y + ky, x kx, z + kz, x dx, y, z. 4 Ω 3
4 4 Becuse Ω 1 x Ω >r}dx = Ω r Ω = ur, where u = Ω is the perimeter of the domin Ω, nd tht for ll x, such tht x Ω > r, Ω 1 x y <r}dy = πr, we hve nd Ω 1 x Ω >r} 1 x y <r} dx, y = urπr, 31 Ω 3 1 x Ω >r} 1 x y <r} 1 x y <r} dx, y, z = ur πr. 3 Consequently, denoting A h = y Ω such tht x y < r given tht x Ω = h}, I 1 nd I reduce to I 1 = urπr + u 4 r 0 A h kh, y + ky, h dydh 33 nd, I = ur πr + u 4 r 0 kh, y + ky, h dy dh. 34 A h Assuming tht the edge of the domin boundry Ω is stright where it intersects bx, x y, kh, y nd ky, h cn be determined nlyticlly [], nd re given by: kh, y min x y, h π rccos, nd, x y ky, h min x y, y Ω π rccos. 35 x y However, using nlyticl expressions 35 in Eq does not led to closed form expressions for I 1 nd I. We thus use finite difference lgorithm with respect to the vrible h n h = r dh steps of size dh = coupled with Monte-Crlo smpling of y in ech A hj=j.dh, 1 j n h n y = 1000 rndom drws y i, 1 i n, nd pproximte r 0 n h kh, y + ky, h dydh A hj 1 n y kh j, y i + ky i, h j dh 36 A h n y i nd with [3] r 0 n h kh, y + ky, h dy dh A hj 1 n y kh j, y i + ky i, h j dh 37 A h n y A hj = πr 1 1 π rccos hj r + h j 1 r i hj r. 38 Finlly, we obtin following numericl pproximtions u r 0 A h kh, y + ky, h dydh ruπr.305, 39 nd u 4 r 0 kh, y + ky, h dy dh ruπr A h
5 5 Reinjecting pproximtions 39 nd 40 in 33 nd 34, we hve I 1 = β ur nd I = 3 β ur, 41 leding to nd Finlly, we obtin tht is vr Kr, n} = Ω φ 0x, ydx, y = I 1 β = β ur φ 0 x, yφ 0 x, zdx, y, z = I 3 β = 3 β Ω 3 = vr Kr, n} = nn 1 nn 1 β β, ur. 43 } φ 0x, ydα x, y + 4 φ 0 x, yφ 0 x, wdα 3 x, y, w Ω Ω ur β + n β ur β ur nn 1 which reduces for n 1 to vr Kr, n} β λ ur where λ = n 44 + β 1 + n ur β ur. 46 is the empiricl density of points. Formul 46 is in greement with [1], pge computtion of E Kr, n E Kr, n} 3} Using φ 0 x, y = φx, y β, we hve E Kr, n E Kr, n} 3} = 3 nn 1 3 E 3 φ 0 x, y x y. 47 We expnd 3 x y φ 0x, y s φ 0 x, y x y where S j is the sum of the terms contining j different points: S = φ 3 0x, y S 3 = 1 3 S 4 = x y x y z S 5 = 5 r nd S 6 = 6 φ 0x, yφ 0 x, z + 3 r u 3 = 6 S j 48 j= x y z φ 0x, yφ 0 z, w + 4 φ 0 x, yφ 0 y, zφ 0 z, w, φ 0 x, yφ 0 x, zφ 0 y, z, φ 0 x, yφ 0 z, wφ 0 x, r, φ 0 x, yφ 0 x, zφ 0 x, w φ 0 x, yφ 0 z, wφ 0 r, u, 49
6 6 nd we then hve E Kr, n E Kr, n} 3} = 3 nn E S j }. 50 j= We re thus left with the computtions of ech men ES j, for j 6. We begin with the computtion of multiplictive coefficients, 1 3, nd 6 in following sub-section 3.1, nd will perform the computtions of ech term ES j in sub-section computtion of coefficients, 1 3,... 6 Computing the number of terms ppering in ech sum of Eq. 48, we obtin nn 1 3 = α + j 3 α j 4 α α α 6, 51 where α i = nn 1... n i + 1 is the number of wys to choose n ordered subset of i points mong n. Expnding Eq. 51, we obtin n 6 3n 5 + 3n 4 n 3 = 6 n n Identifying polynomil coefficients in Eq. 5, we hve = 4, 3 j 4 + j 3 n3 3 j 4 n4 3 j 4 3 j 3 + n 3 j 4 + j 3 n, 5 3 j 3 = 3, j 4 = 38, 5 = 1 nd 6 = Then, becuse φ 0 x, y is symmetric function there is 3 3 = 4 wys of writing φ 0x, yφ 0 x, z. Indeed, there is 3 possible positions for φ 0 x, z nd wys to write ech symmetric term. Consequently, 1 3 = 4, nd 3 = j = 8. Similrly, there is 3 3 = 4 wys of writing φ 0x, yφ 0 z, w, but in tht cse, the expression is symmetric in x nd y s well s in z nd w. Consequently, φ 0x, yφ 0 z, w is counted 4 times in x y z w φ 0x, yφ 0 z, w nd 1 4 = 4/4 = 6. Concerning 4, there is 6 3 = 48 wys of writing φ 0 x, yφ 0 x, zφ 0 x, w nd the expression is symmetric in y, w nd z leding to 4 = 48/6 = 8. Finlly, there is 48 wys of writing φ 0 x, yφ 0 y, zφ 0 z, w nd the symmetric role of the couple of points x, y nd z, w leds to 3 4 = 4. We cn check here tht = computtion of E S j } for j computtion of E S } First, ssuming uniform distribution of the points inside Ω nd using α r = nn 1... n r + 1 r µ r, we hve E S } = 4 φ 3 0x, ydα x, y, 54 Ω
7 7 tht we expnd s E S } = 4α φ Ω 3 x, y 3βφ x, y + 3β φx, y β 3 dx, y = 4α φ 3 x, ydx, y 3βI 1 + 3β I 0 β Ω We re thus left with the computtion of I 3 = φ 3 x, ydx, y tht we decompose s in section see Eq. Ω 33 nd 34: I 3 = φ 3 x, ydx, y = urπr + u kh, y + ky, h 3 dydh, 56 Ω 8 0 A h where kh, y nd ky, h re given by Eq. 35. A finite difference scheme with respect to vrible h coupled with Monte-Crlo smpling of y in A h leds to the pproximtion I 3 β ur. 57 Finlly, using Supplementry Tble S1 in Eq. 55, we obtin E S } 4 α β ur 3β ur 3.. computtion of E S 3 } 58 Denoting nd we hve E S3 1 } = α3 φ 0x, yφ 0 x, zdx, y, z 59 Ω 3 E S3 } = α3 φ 0 x, yφ 0 x, zφ 0 y, zdx, y, z, 60 Ω 3 E S 3 } = 4E S 1 3} + 8E S 3 }, 61 Expnding E S3} 1 nd using Monte-Crlo numericl integrtion to ccount for Ω boundries we find tht E S3} 1 0. We then expnd E S 3, s E S3 } Ω3 = α3 φx, yφx, zφy, zdx, y, z 3βI + 3β I 0 3 β 3. 6 nd re left with the computtion of I 4 = φx, yφx, zφy, zdx, y, z. Ω 3 63 First, for x such tht x Ω > r, there is no boundry correction nd we decompose I 4 = I4 in + I4 border where I4 in = 1 x Ω >r} 1 x y <r} 1 x z <r} 1 y z <r} dx, y, z Ω 3 = ur 1 x y <r} 1 x z <r} 1 y z <r} dy, z, 64 Ω nd I border 4 = u 8 r 0 A h kh, y + ky, h A h kh, z + kz, h 1 y z <r} ky, z + kz, y dzdydh. 65
8 8 We then rewrite I in 4 s I in 4 = ur πr Pr y z < r given tht y, z bx, r}. 66 Becuse y nd z re uniformly distributed in bx, r, we hve Pr y z < r given tht y, z bx, r} = 1 πr bx,r by, r bx, r dy 67 where bx, r nd by, r re the bll centered t x nd y with rdius r. Considering locl polr coordintes: y0 r y r, 0 θ y π round x0, 0, we hve r π I4 in = ur Ar y, θ y, rr y dr y, θ y 68 r y=0 θ y=0 where Ar y, θ y, r = bx0, 0, r byr y, θ y, r is equl to [3] Ar y, θ y, r = Ar y, r = r cos 1 r y r y 4r r ry. 69 Finlly, direct integrtion I4 in using Eq. 69 yields I4 in = urπr 4 π 3 3, 70 8 tht is I4 in = ur β ur β 0.587, 71 4π On the other hnd, numericl integrtion of I border 4 gives leding to I 4 = I in 4 + I border I border 4 3 β 1.38 ur, β ur Finlly, reinjecting Eq. 73 in Eq. 6, nd given tht E S 1 3} 0, we obtin E S 3 } 8E S computtion of E S 4 } } = 8 α3 β ur. 73 β We decompose the computtion of E S 4 } s follows E S 4 } = 1 4E S4} E S4} E S4 3 } = 6E S E S 4 + 4E S with nd, E S4 1 } = α4 φ 0x, yφ 0 z, wdx, y, z, w, 76 E S4 } = α4 φ 0 x, yφ 0 x, zφ 0 x, wdx, y, z, w, 77 E S 3 4 } = α4 φ 0 x, yφ 0 y, zφ 0 z, wdx, y, z, w. 78
9 9 First φ Ω 0 z, wdz, w = 0 leds to E S4} 1 = 0. We then expnd E S 4 s E S4 } Ω4 = α4 φx, yφx, zφx, wdx, y, z, w 3βI + 3β I 0 4 β 3, 79 nd decompose φx, yφx, zφx, wdx, y, z, w s φx, yφx, zφx, wdx, y, z, w = ur 3 β 3 Numericl integrtion of Eq. 80 gives + u 8 r φx, yφx, zφx, wdx, y, z, w 4 β 3 nd using Supplementry Tble S1 in Eq. 79, we compute tht Similrly, we expnd E S 3 4} s 0 3 kh, y + ky, h dy dh. 80 A h ur. 81 E S 4} 0. 8 E S4 3 } = α4 φx, yφy, zφz, wdx, y, z, w β I + I β 3 nd we re left with the numericl integrtion of I 5 = φx, yφy, zφz, wdx, y, z, w. For y such tht y Ω > 3r, points x nd z such tht x y < r nd z y < r re t miniml distnce of r from the domin boundry Ω nd there is no boundry correction in I 5. We thus decompose I 5 s I 5 = 3ur πr 3 u 3r + kx, h + kh, x dx 8 0 A h kz, h + kh, z kz, w + kw, z dwdzdh 84 A h A z where A z = bz, r Ω. We then used finite difference lgorithm with respect to the vrible h n h = 3r dh steps of size dh = coupled with Monte-Crlo smpling of x nd z in ech A hj=j.dh for 1 j n h n x = n z = 1000 rndom drws x i, z p, 1 i, p n s well s smpling of w in ech A zp relted to ech rndom drw z p n w = 1000 rndom drws w k, 1 k n w nd pproximte 3r kx, h + kh, x dx 0 A h kz, h + kh, z A h kz, w + kw, z dwdzdh A z leding to n h A hj n z A hj n x n z p=1 n x i kh j, x i + kx i, h j kh j, z p + kz p, h j A z p n w n w k=1 kz p, w k + kw k, z p } 83 dh, 85 I 5 4 β ur. 86 Finlly, reinjecting Monte-Crlo pproximtion of I 5 in Eq. 83, we obtin E S4 3 } α4 β ur, 87 leding to E S 4 } 4E S 3 4 } = 4 α4 β ur,. 88
10 Conclusion Finlly, becuse Ω φ 0 x, ydx, y = 0, we hve nd E S 5 } = 1α 5 φ 0 x, yφ 0 x, rdx, y, r φ 0 z, wdz, w = 0, 89 Ω 3 Ω E S 6 } = α 6 φ 0 x, ydx, y φ 0 z, wdz, w φ 0 r, udr, u = Ω Ω Ω Consequently, using expressions of E S } Eq. 58, E S 3 } Eq. 74 nd E S 4 } Eq. 88 in Eq. 50 we obtin = E Kr, n E Kr, n} 3} = 4 3 n n 1 [ β ur + β 3 α α 4 ur α α which simplifies for n 1 to E Kr, n E Kr, n} 3} = 4 β λ ur n 3 j=4 nn 1 3 E S j } j= + β α 3 α α 3 ur α ], 91 + β ur 4 computtion of E Kr, n E Kr, n} 4} + β n ur. 9 We hve E Kr, n E Kr, n} 4} = 4 nn 1 4 E 4 φ 0 x, y x y, 93 } 4 nd following the method of section 3, we expnd E x y φ 0x, y s E 4 φ 0 x, y x y = 8 j= E Sj 94 where S j is the sum of the terms contining j different points: S = ã φ 4 0x, y, 95 x y S 3 = 1 3 φ 3 0x, yφ 0 x, z + 3 φ 0x, yφ 0x, z x y z x y z x y z φ 0x, yφ 0 x, zφ 0 z, y, 96
11 11 S 4 = ã 1 4 φ 3 0x, yφ 0 z, w + ã 4 φ 0x, yφ 0z, w + ã ã ã 7 4 φ 0x, yφ 0 x, zφ 0 x, w + ã 4 4 φ 0x, yφ 0 x, zφ 0 z, w + ã 6 4 φ 0x, yφ 0 x, zφ 0 y, w φ 0 x, yφ 0 y, zφ 0 z, wφ 0 x, w φ 0 x, yφ 0 x, zφ 0 x, wφ 0 y, z, 97 S 5 = ã 1 5 φ 0x, yφ 0 z, wφ 0 x, r + ã 5 φ 0x, yφ 0 z, wφ 0 z, r + ã ã ã ã 6 5 r r r r r φ 0 x, yφ 0 x, zφ 0 x, wφ 0 x, r φ 0 x, yφ 0 x, zφ 0 x, wφ 0 y, r φ 0 x, yφ 0 x, zφ 0 y, zφ 0 w, r r φ 0 x, yφ 0 x, zφ 0 y, wφ 0 z, r, 98 S 6 = ã 1 6 φ 0x, yφ 0 z, wφ 0 r, u + ã 6 + ã ã 4 6 r u r u r u r u φ 0 x, yφ 0 x, zφ 0 x, wφ 0 r, u φ 0 x, yφ 0 y, zφ 0 z, wφ 0 r, u φ 0 x, yφ 0 x, zφ 0 w, rφ 0 w, u, 99 S 7 = ã 7 φ 0 x, yφ 0 x, zφ 0 w, rφ 0 u, s 100 r u s nd S 8 = ã 8 φ 0 x, yφ 0 z, wφ 0 r, uφ 0 s, r. 101 r u s r 4.1 computtion of coefficients ã, ã 1 3,... ã 8 Computing the number of terms in ech sum of Eq. 94, we get tht nn 1 4 = ã α + 3 ã j 3 α ã j 4 α ã j 5 α ã j 6 α 6 + ã 7 α 7 + ã 8 α 8, 10 where α i = nn 1... n i + 1 is the number of wys to choose n ordered subset of i different points mong n. Expnding Eq. 10 nd identifying polynomil coefficients, we obtin tht ã = 8, ã j 3 = 08, ã j 4 = 65, ã j 5 = 576, ã j 6 = 188, ã 7 = 4 nd ã 8 =
12 1 Using similr counting rguments s in sub-section 3.1, we further obtin tht ã 1 3 = 64, ã 3 = 48 nd ã 3 3 = 96 which verify 3 ã j 3 = 08, 104 ã 1 4 = 16, ã 4 = 1, ã 3 4 = ã 4 4 = 96, ã 5 4 = 19, ã 6 4 = 48 nd ã 7 4 = 19, 7 which verify = 65, 105 ã j 4 ã 1 5 = 96, ã 5 = 48, ã 3 5 = 16, ã 4 5 = 19, ã 5 5 = 3 nd ã 1 5 = 19, 6 which verify = 576, 106 ã j 5 nd ã 1 6 = 1, ã 6 = 3, ã 3 6 = 96 nd ã 4 6 = 48 which verify 4. computtion of E Sj for j computtion of E S 4 ã j 6 = Assuming uniform distribution of the points inside Ω we hve E S = 8 φ 4 0x, ydα x, y, 108 Ω where α = nn 1 µ 4 see Eq. 3, tht we expnd s E S = 8α φ x, y 4βφ 3 x, y + 6β φ x, y 4β 3 φx, y + β 4 dx, y = 8α φ 4 x, ydx, y 4βI 3 + 6β I 1 4β 3 I 0 + β Ω We re thus left with the computtion of φ 4 x, ydx, y tht we numericlly evlute to Ω φ 4 x, ydx, y β ur. 110 Ω Finlly, using Supplementry Tble S1 in Eq. 109, we obtin E S 8 α β ur 4β ur 4.. computtion of E S3 Becuse ã 1 3 = 64, ã 3 = 48 nd ã 3 3 = 96, we cn re-write E S3 s + 6β ur 111 E S3 = 64E S E S E S 3, 11 with E S1 3 } = α 3 Ω 3 φ 3 0x, yφ 0 x, zdx, y, z, 113
13 nd E S 3 = α 3 φ 0x, yφ 0x, zdx, y, z, 114 Ω 3 E S3 3 = α 3 φ 0x, yφ 0 x, zφ 0 z, ydx, y, z. 115 Ω 3 Accounting for Ω boundries t leding order nd using Monte-Crlo numericl pproximtions see sections nd 3, we find tht E S1 3 0, nd E S 3 α 3 β ur β ur. 117 E S3 3 α 3 β ur 4π β ur, 118 leding to 4..3 computtion of E S4 E S3 = α 3 β ur We decompose the computtion of S 4 s follows β ur. 119 E S4 = 7 ã j 4 E Sj 4 10 where coefficients ã j 4 re given by Eq. 105 nd, E S1 4 = α 4 φ 3 0x, yφ 0 z, wdx, y, z, w, 11 E S 4 = α 4 φ 0x, yφ 0z, wdx, y, z, w, 1 E S3 4 = α 4 φ 0x, yφ 0 x, zφ 0 x, wdx, y, z, w, 13 E S4 4 = α 4 φ 0x, yφ 0 x, zφ 0 y, wdx, y, z, w, 14 E S5 4 = α 4 φ 0x, yφ 0 x, zφ 0 z, wdx, y, z, w, 15 nd E S6 4 = α 4 φ 0 x, yφ 0 y, zφ 0 z, wφ 0 x, wdx, y, z, w, 16 E S7 4 } = α 4 φ 0 x, yφ 0 x, zφ 0 x, wφ 0 y, zdx, y, z, w. 17
14 First φ Ω 0 x, ydx, y = 0 leding to E S1 4 = 0. Then, we hve E S 4 = α 4 I1 β leding to see Supplementry Tble S1, E S 4 α 4 β ur Expnsion nd numericl integrtion ner the boundry of E E S4 4 E S Conversely, expnsion of E S6 4 yields E S β. 18 S3 4, E S4 4 nd E S5 4 gives E S3 4 α 4 φx, yφy, zφz, wφx, wdx, y, z, w 4βI 5 + 4β I 4 β 4 } 19 nd we re left with the computtion of I 6 = φx, yφy, zφz, wφx, wdx, y, z, w. 130 First, for x such tht x Ω > 3r, there is no boundry correction nd we decompose I 6 = I6 in + I6 border where I6 in = 1 x Ω >3r} 1 x y <r} 1 x w <r} 1 y z <r} 1 w z <r} dz } dx, y, w = 3ur 1 x y <r} 1 x w <r} 1 y z <r} 1 w z <r} dz } dwdy, 131 Ω 3 nd I border 3r 6 = u kh, y + ky, h kh, w + kw, h 16 0 A h A h 1 y z <r} 1 w z <r} ky, z + kz, y kw, z + kz, w dz } dwdydh. 13 We then rewrite I in 6 s I in 6 = 3ur πr Pr y z < r nd w z < r given tht y, w bx, r}. 133 Denoting dy, w = y w, nd z being uniformly distributed in Ω, we hve Pr y z < r nd w z < r given tht y, w bx, r} = 1 πr by, r bw, r dydw = 1 bx,r πr Ady, w, rdydw 134 bx,r where Ady, w, r is given by Eq. 69 Ady, w, r = r cos 1 dy, w r dy, w 4r dy, w. 135 y nd w re uniformly distributed in bx, r nd we cn thus consider locl polr coordintes: y0 r y r, 0 θ y π nd w0 r w r, 0 θ w π round x0, 0 leding to nd re-write I in 6 s dy, w = dr y, r w, Θ = θ y θ w = r y + r w r y r w cosθ, 136 r r π I6 in = 3urπ A d r y, r w, Θ, t r y r w dr y, r w, Θ. 137 r y=0 r w=0 Θ=0
15 15 Finlly, numericl integrtion of I in 6 with finite differences scheme in r y, r w nd Θ gives Furthermore, we pproximte mumericlly I border 6 s leding to I in 6 3ur0.46β I border 6 3urβ ur, 139 I 6 = I in 6 + I border 6 4 β ur. 140 nd using Supplementry Tble S1 in Eq. 19 we obtin E S6 4 α 4 β ur β Finlly, numericl integrtion of E S7 4 gives E S7 4 0, nd reinjecting Eq. 141 nd Eq. 18 in Eq. 10, we obtin E S4 α 4 1 β ur β + 48β ur 48β 4, 14 tht is E S4 α 4 β ur ur computtion of E S5 We decompose the computtion of E S5 s follows E S5 = 6 ã j 5 E Sj 5 + β ur 36β where coefficients ã j 5 re given by Eq. 106 nd, E S1 5 = α 5 φ 0x, yφ 0 z, wφ 0 x, rdx, y, z, w, r, 145 Ω 5 E S 5 = α 5 φ 0x, yφ 0 z, wφ 0 z, rdx, y, z, w, r, 146 Ω 5 E S3 5 = α 5 φ 0 x, yφ 0 x, zφ 0 x, wφ 0 x, rdx, y, z, w, r, 147 Ω 5 E S4 5 = α 5 φ 0 x, yφ 0 x, zφ 0 x, wφ 0 y, rdx, y, z, w, r, 148 Ω 5 nd E S5 5 = α 5 φ 0 x, yφ 0 x, zφ 0 y, zφ 0 w, rdx, y, z, w, r, 149 Ω 5 E S6 5 } = α 5 Ω 5 φ 0 x, yφ 0 x, zφ 0 y, wφ 0 z, rdx, y, z, w, r. 150
16 First φ Ω 0 x, ydx, y = 0 leds to E S1 5 = E S5 5 = 0. Then, we hve tht is, using Supplementry Tble S1 E S 5 E S 5 = α 5 I1 β I 3 β 151 α 5 β ur ur Finlly, expnsion nd numericl integrtion ner the boundry of E E S4 5 E S leding to tht is E S5 ã 5E S computtion of E S6 We decompose the computtion of E S6 s follows S3 5 }, E S4 5 nd E S6 5 gives E S α 5 β ur ur, 153 E S5 α 5 β ur ur E S6 = 4 ã j 6 E Sj where coefficients ã j 6 re given by Eq. 107 nd, E S1 6 = α 6 φ 0x, yφ 0 z, wφ 0 r, udx, y, z, w, r, u, 156 Ω 6 E S 6 = α 6 φ 0 x, yφ 0 x, zφ 0 x, wφ 0 r, udx, y, z, w, r, u, 157 Ω 6 E S3 6 = α 6 φ 0 x, yφ 0 y, zφ 0 z, wφ 0 r, udx, y, z, w, r, u, 158 Ω 6 nd E S4 6 = α 6 φ 0 x, yφ 0 x, zφ 0 w, rφ 0 w, udx, y, z, w, r, u, 159 Ω 6 Becuse φ Ω 0 x, ydx, y = 0, E S1 6 = E S 6 = E S3 6 = 0, nd we re thus left with the computtion of E S4 6 tht reds E S4 6 = α 6 Ω 6 φ 0 x, yφ 0 x, zφ 0 w, rφ 0 w, udx, y, z, w, r, u nd we hve tht = α 6 I 3 β α6 β ur. 160 ur E S6 = ã 4 6E S4 6 α 6 β
17 Conclusion Becuse Ω φ 0 x, ydx, y = 0, S 7 = S 8 = 0, nd E Kr, n E Kr, n} 4} = 4 nn j= E Sj. 16 Reinfecting expressions of E Sj, for j 6 Eq. 111, 119, 143, 154 nd 161, we hve E Kr, n E Kr, n} 4} = + β α α 4 α α α 4 ur α + β α α 4 α α + α 5 α ur + β 4 36 α α 6 ur α α which simplifies for n 1 to + β n + 4 nn 1 3 β ur α 3 α α 4 α ur α α α 5 ur α α α, E Kr, n E Kr, n} 4} = 1 β λ 4 n ur + β3 n + β n n ur ur n n n n ur n ur 5 Computtion of γ KM r. nd κ KM r + n ur We defined in the min mnuscript the men sttistic K M r = 1 M M K j r, n j 165 where K j r, n j is the modified Ripley s K function tht is evluted on the j th field of view. Thus, } 3 1 M E Kj M r, n j πr γ KM r = 1 M vr K } M j r, n j For i j, Kj is independent of K i nd for ll 1 j M, vr Kj r, n j = 1, thus, 1 vr M M K j r, n j = 1 M M vr Kj r, n j = 1 M, 167
18 18 nd we rewrite Then, we decompose γ KM r = 1 M 3 M Kj r, n j πr M E Kj r, n j πr 3 3 = M Kj r, n j πr 3 Kj r, n j πr Ki r, n j πr j i + 6 Kj r, n j πr Ki r, n j πr Kk r, n j πr, 169 j i k nd becuse E Kj r, n j πr = 0, we hve M E Kj r, n j πr 3 M = E Kj r, n j πr } 3, 170 tht is which leds to Similrly, we hve M E Kj r, n j πr 3 M = γ Kj r, n j, 171 γ KM r = 1 M 3 M γ Kj r, n j. 17 κ KM r = 1 M M E Kj r, n j πr 4, 173 nd we decompose M Kj r, n j πr + 4 j i 4 M = Kj r, n j πr 4 Kj r, n j πr 3 Ki r, n j πr Kj r, n j πr Ki r, n j πr + 3 j i + 6 Kj r, n j πr Ki r, n j πr Kk r, n j πr j i k + 4 Kj r, n j πr Ki r, n j πr Kk r, n j πr Kl r, n j πr 174 j i k l
19 19 which leds to M M E Kj r, n j πr 4 = E Kj r, n j πr } j i E Kj r, n j πr } E Ki r, n i πr }, 175 tht is M E Kj r, n j πr 4 M = κ Kj r, n j + 3M M Reinjecting Eq. 176 in Eq. 173, we obtin κ KM r = 1 M M κ Kj r, n j + 3 M 1 M. 177
20 0 Tble S1: Numericl pproximtions of mjor integrls Denottion Formul Numericl pproximtion I 0 φx, ydx, y Ω β I 1 φx, y Ω dx, y β ur I φx, yφx, zdx, y, z Ω 3 β ur 3 I 3 φ Ω 3 x, ydx, y β ur I 4 φx, yφx, zφy, zdx, y, z 3 β ur Ω 3 4π I 5 φx, yφy, zφz, wdx, y, z, w β ur 4 I 6 φx, yφy, zφz, wφx, wdx, y, z, w β ur 4 References [1] Ripley B 1988 Sttisticl inference for sptil processes. Cmbridge University Press. [] Getis A, Frnklin J 1987 Second-order neighborhood nlysis of mpped point ptterns. Ecology 68: [3] Weisstein E. URL [4] de Chumont F, Dllongeville S, Chenourd N, Hervé N, Pop S, et l. 01 Icy: n open bioimge informtics pltform for extended reproducible reserch. Nt Methods 9: [5] Olivo-Mrin JC 00 Extrction of spots in biologicl imges using multiscle products. Pttern Recognition 35:
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