角度統計配布_final.pptx

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1 01/1/7 1, 1 JST GFP {x 1,x,,,x n } Credit: Elowitz lab {θ 1, θ, θ 3,,, θ n } (+) EB3-GFP π π π θ+π = θ movie Shindo et al., PLoS one, 008

2 (+) beating Shindo et al., PLoS one, 008 Guirao et al., NCB, 010 Landsberg et al., Current Biol, = J. D. Levine et al. Science (00) 1 0h isolated in group ( () ) 18 position Romano et al. Chaos 010,, etc..

3 : / () () von Mises {θ 1, θ, θ 3,,, θ n } 11 1

4 (circular/directional statistics) : N N : 3: θ = θ+180 () 4: (T i, θ i )θ-t / von Mises {θ 1, θ, θ 3,,, θ n } 11 1

5 : {80, 170,175, 00,65, 345 } {1, 359 } 180?? 0 =360 0 ( )/6=06? ( RcosΘ, RsinΘ) = 1 N cosθ, sinθ ( 1/N) R : =191 x 1 N x R ( ) ( R cosθ, RsinΘ) = 1 cosθ, sinθ N = cosθ, sinθ x y x y Im e iθ = cosθ + isinθ xy Re Re iθ = 1 N e iθ = e iθ

6 Im Re R Re iθ = 1 N e iθ = e iθ R circular variance) (standard deviauon) V 1 R (0 V 1) S log(r) R R : θ θ+180 : { 170,175, 160,65, 35 } 0 θ < =180 {1, 179 } 90?? 0 Re iθ = e iθ Re iθ = e iθ { 170,175, 160,65, 35 } : = 6.5 { 340,350, 30,130, 70 } : = 1.9

7 1.1.1 ここまでのまとめ Im ( R cosθ, Rsin Θ ) = Re ReiΘ = 1 cosθ, sin θ N 1 iθ iθ e = e N 複素平面における表記 標準偏差 分散 平均 Θ V 1 R 角度Θ 長さRのベクトル S log(r) (0 V 1) 小 R 大 大 分散 小 ここまでのまとめ Im ( R cosθ, Rsin Θ ) = Re ReiΘ = Guirao et al., NCB, eiθ = eiθ N V 1 R 小 R 大 分散 角度Θ 長さRのベクトル 複素平面における表記 標準偏差 分散 平均 Θ 1 cosθ, sin θ N (0 V 1) S log(r) たくさんの繊毛/1細胞 各繊毛のBasal bodyの角度を測定 細胞ごとに角度Θ 長さRのベクトルを表示 P4 生まれて4日後 とP0で比較 Rのベ 大 小 RP4 < RP0 VP4 > VP0 7

8 von Mises {θ 1, θ, θ 3,,, θ n } 11 1 (µ,κ) R = I 1 (κ)/i 0 (κ) κ 0 κ µ I p (κ)p von Mises ( ( )) exp κ cos θ µ P(θ) = πi 0 κ ( ) exp( κ cos( θ µ )) π - π/ 0 κ = π/ π

9 von Mises {θ 1, θ, θ 3,,, θ n } 11 1 : : : N A, E A, V A N B, E B, V B p (p<0.01)

10 1.1.1 角度データの代表的な検定 Rayleigh test 角度データには偏りがあるか ある角度に偏っているのか Kuiper test 角度データはvon Mises分布に従っているのか Mardia-Watoson-Wheeler test 2群のデータは同じ分布に従っているのか Rayleigh test: 角度データの異方性 A. 角度データに偏り(異方性)があるといえるか iθ ① Rを計算 Re = e iθ 角度Θ 長さRのベクトル ② Rが大きければ一様分布から外れていると言える 一様分布(帰無仮説)のもとでは サンプル数nの時に Z = nr が出る確率は Z Z 4Z 13Z + 76Z 3 9Z 4 Z P = e Z 1 + e 4n 88n (p値) なので Zが大きければ 異方性がある と主張できる (帰無仮説を棄却できる) たくさんの繊毛/1細胞 B. ある角度θ0に偏っているといえるか (角度θ 各繊毛のBasal bodyの角度を測定 0を指定 V- test) 細胞ごとに角度Θ 長さRのベクトルを表示 P4 生まれて4日後 とP0で比較 Rのベ ② R0 が大きければ角度θ0に偏っている度合いが大きいといえる ① R0 =R cos(θ-θ0) を計算 RP4 一様分布(帰無仮説)のもとでのZ = (n)1/ Rが出る確率 (p値) をもとに 帰無仮説を棄却できるか否かを判定する Guirao et al., NCB, 010 < RP0 P 0.05 P<

11 Rayleigh test: A. () R Re iθ = e iθ R R ()nz = nr P = e Z 1+ Z Z 4n 4Z 13Z + 76Z 3 9Z 4 88n e Z (p) Z () B. θ 0 (θ 0 V- test) R 0 =R cos(θ-θ 0 ) R 0 θ 0 ()Z = (n) 1/ R (p) Kuiper test: von Mises Kolmogorov- Smirnov (KS) () KS {x 1,x,x 3,,,x n } ()? x sample F n (x) F(x) D = max F n (x) - F(x) Kuiper {θ 1, θ, θ 3,,, θ n } () von Mises? sample F n (x) F(x) θ V = max (F n (x) - F(x))+max(F (x) - F n (x)) V (p)

12 Mardia-Watson-Wheeler test: U () {x 1,x,x 3,,,x n } {y 1,y,y 3,,,y m } n+m n < m A n = 6 B m = R = 4 (A) U = 7 {x},{y}(p) p () U = nm+n(m+1)/-r(rx) Mardia-Watson-Wheeler test Mann-WhitenyU Mardia-Watson-Wheeler test: MWW {θ 1, θ, θ 3,,, θ n } {ψ 1, ψ, ψ 3,,, ψ m } () n < m n+m0-π πθ 1 n + m, πθ n + m,,, πθ n πψ 1 n + m n + m, πψ n + m,,, πψ m n + m Θ, Ψ 0 n+m-1 A (n=6) B (m=8) R R AR! R

13 von Mises {θ 1, θ, θ 3,,, θ n } 11 1 Circular- circular correlauon : Linear- circular correlauon : ()

14 L ll (a,b), L cl (a,b,c)(a,b,c). - (x,y ) - (S,θ ) y = a + bx S = a + b'cos(θ µ) = a + b cosθ + csinθ L ll (a,b) = y a bx L cl a,b,c ( ) = S a bcosθ c sinθ bx b y y y x x L ll (a,b), L cl (a,b,c)(a,b,c). - (x,y ) - (S,θ ) y = a + bx S = a + b'cos(θ µ) = a + b cosθ + csinθ L ll (a,b) = y a bx L cl a,b,c ( ) = S a bcosθ c sinθ bx b Landsberg et al., Current Biol, 009 x

15 - (x,y )Pearson r xy = Δx Δy Δx Δy - (θ,s )1fidng S = a + b'cos(θ µ) = a + bcosθ + csinθ ()(cosθ,sinθ)fidng () r cs : cosθ-s Pearson ρ θ,s = r cs + r ss r cs r ss r cs 1 r cs r ss : r cs : sinθ -S Pearson cosθ -sinθ Pearson ρ θ,s Bootstrap() (web page) Fisher Sta%s%cal Analysis of Circular Data Mardia & Jupp Direc%onal sta%s%cs Batschelet Circular sta%s%cs in biology 3 100( )

16 MATLAB circular stausucs toolbox by Philippe Berens R circular stausucs package R 0 von Mises {θ 1, θ, θ 3,,, θ n } ρ θ,s = r cs + r ss r cs r ss r cs 1 r cs 11 1 R R

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

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