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1 需要予測のための統計モテルの研究異常値検知のための基本的モテルの考察平成 26 年 3 月東京大学大学院情報理工学系研究科教授博士課程修士課程竹村彰通小川光紀笹井健行特定非営利活動法人ヒューコミュニケーションス副理事長小松秀樹主任研究員池谷貞彦研究員河内敦雄

2 (AR) Change Finder Change Finder Change Finder SDAR Change Finder

3 Change Finder...................... 12 4.4 Change Finder....................... 13 4.5 SDAR....................... 15 5 18 5.1 Change Finder....... 18 5.1.1 1....................... 19 5.1.2 2.

3 Change Finder Change Finder Change Finder A 35 A A A A A B 38 C 43

........... 32 6 33 A 35 A.1 -............. 35 A.1.1....................... 35 A.1.2.................. 35 A.2 -.

4 ARIMA ( ) ARIMA ( ) ARIMA ( ) 1.2 (AR ) AR

5 1 4 Change Finder AR Change Finder

6 5 2 AR [1] T t T x t p (x t ) {p (x t ) t T t T = {1, 2, {x t {x t E (x t ) = µ Cov (x t, x t+1 ) = γ ( h ) {x t V (y t ) = γ (0) ρ (h) = γ (h) γ (0) = γ (t) V (y t ) = ρ ( h) h ρ (h), h = 0, 1, 2,... ρ (h) 1 1

7 S = Cov (x t, x t+h ) = γ (h) h= h= S 2.2 (AR) (AR) {x t t T x t = ϕ 1 x t ϕ p x t p + ε t, {ε t i.i.d.n ( 0, σ 2) (2.1) {x t p AR(p) AR(p) AR(p) ϕ (x) = 1 ϕ 1 x ϕ 2 x 2 ϕ p x p = 0 1 AR(p) (2.1) y t j (j > 0) γ (j) = ϕ 1 γ (j 1) + ϕ 2 γ (j 2) + + ϕ p γ (j p) (2.2) γ (0) p ρ (j) = ϕ 1 ρ (j 1) + ϕ 2 ρ (j 2) + + ϕ p ρ (j p), (j > 0) (2.3) λ 1, λ 2,, λ p c 1, c 2,, c p ρ (j) = p c i i=1 λ j i c 1, c 2,..., c p (2.3) j = 1, 2,..., p 1 ρ (1), ρ (2),..., ρ (p 1)

8 2 7 ρ (1) = ϕ 1 + ϕ 2 ρ (1) + + ϕ p ρ (p 1) ρ (2) = ϕ 1 ρ (1) + ϕ ϕ p ρ (p 2). ρ (p 1) = ϕ 1 ρ (p 2) + ϕ 2 ρ (p 3) + + ϕ p ρ (1). c 1, c 2,.., c p 1 = c 1 + c c p ρ (1) = c 1 + c c p λ 1 λ 2 λ p. ρ (p 1) = c 1 λ p c 2 λ p c p λ p 1 p AR(p) ϕ 1, ϕ 2,..., ϕ p (2.3) j = 1, 2,..., p p. ρ (1) = ϕ 1 + ϕ 2 ρ (1) + + ϕ p ρ (p 1) ρ (2) = ϕ 1 ρ (1) + ϕ ϕ p ρ (p 2). ρ (p) = ϕ 1 ρ (p 1) + ϕ 2 ρ (p 2) + + ϕ p. 1 ρ (1) ρ (p 1) ϕ 1 ρ (1) ρ (1) 1 ρ (p 2) =.... ρ (p 1) ρ (p 2) 1 ρ (p) ϕ p

ρ (p 1) = c 1 λ p 1 1 + c 2 λ p 1 2 + + c p λ p 1 p AR(p) ϕ 1, ϕ 2,..., ϕ p (2.3) j = 1, 2,..., p p.

9 8 3 [2] 3.1 x t, (t = 1, 2,...) µ x τ τ = 7 γ xx (τ) = E [(x t µ x ) (x t+τ µ x )] 3.2 A B A B x t, y t (t = 1, 2,...) µ x, µ y τ x t = 1 y t = 2 τ = 7 1

10 3 9 2 γ xy (τ) = E [(x t µ x ) (y t+τ µ y )] 3.3

11 10 4 Change Finder [3, 4] Change Finder AR AR AR [3, 4] 4.1 x 1 x 2... p x t 1 = x 1 x 2...x t 1 p ( x t x t 1) p p (1), p (2) p ( x t x t 1) = p (1) ( x t x t 1), t < a, p ( x t x t 1) = p (2) ( x t x t 1), t a. t = a -

12 4 Change Finder 11 ( D p (2) p (1)) [ 1 := lim n n E p log p(2) (x n ] ), (2) p (1) (x n ) p (1) = p (2) D = jumping mean p (1), p (2) (µ 1, σ), (µ 2, σ) D ( p (1) p (2)) = (µ 1 µ 2 ) 2 / ( 2σ 2) 2. jumping variance p (1), p (2) (µ, σ 1 ), (µ, σ 2 ) D ( p (1) p (2)) = 1 2 ( σ 2 2 σ log σ2 2 σ 2 1 ) x n 1 = x 1,..., x n t x i, (i = 1,, n) d x t = x 1,..., x t x n t = x t+1,..., x n t x t 1 t k = x t k,..., x t 1 x t p ( [ ] x t x t 1 ) 1 t k = exp (x t w t ) T Σ 1 (x t w t ) (2π) d/2 1/2 Σ 2 Σ AR k α 1, α 2,..., α k, µ w t = k α i (x t i µ) + µ i=1

jumping variance p (1), p (2) (µ, σ 1 ), (µ, σ 2 ) D ( p (1) p (2)) = 1 2 ( σ 2 2 σ 2 1 1 log σ2 2 σ 2 1 ) 2 4.2 x n 1 = x 1,.

13 4 Change Finder 12 AR w t ŵ t I (x n 1 ) = n x t ŵ t 2. t=1 t I (x t 1) Error1 t I ( x n t+1) Error2 t = t δ > 0 I (x n 1 ) ( I ( ) ( x t 1 + I x n t +1)) > δ n t Change Finder 4.3 Change Finder Change Finder Change Finder Step 1 x 1, x 2,... {p t (x) : t = 1, 2,... p t 1 (x) x t 1 = x 1,..., x t 1 t x t Score (x t ), t = 1, 2,... Step 2 W > 0 Step1 {y t : t = 1, 2,... Step {Score (x i ) : i = t W + 1,..., t W - y t

t Change Finder 4.3 Change Finder Change Finder Change Finder Step 1 x 1, x 2,... {p t (x) : t = 1, 2,.

14 4 Change Finder 13 y t = 1 W t t=t W +1 Score (x i ). Step 3 Step1 Step2 y t, t = 1, 2,... {q t (y t ) : t = 1, 2,... t y t Score (y t ), t = 1, 2,... W > 0 t W - Score (t) = 1 t W Score (y i ) i=t T W +1 Change Finder Step2 Step3 Step2 T Change Finder W W W 4.4 Change Finder Change Finder AR Step 1 x t, t = 1, 2,... SDAR {p t (x) : t = 1, 2,... SDAR

15 4 Change Finder 14 t x t Score (x t ) = log p t 1 (x t ) Score (x t ) = d (p t 1, p t ) = ( p (xt 1 ) ) 2 p (x t ) d AR z t = k A i z t i + ε i=1 A i ε 0 Σ {x t : t = 1, 2,... x t = z t + µ x t 1 t k = x t k...x t 1 x t 1 t k x t p ( ( x t x t 1 t k : θ) 1 = exp 1 ) (2π) k/2 Σ 1/2 2 (x t w) T Σ 1 (x t w), k w = A i (x t i µ) + µ i=1 θ = (A 1,..., A k, µ, Σ) x t θ θ t p t = p (, θ t ) θ (SDAR ) Step 2 k (1 r) t i log p ( x i x i 1, θ ). i=1 Step1 {y t : t = 1, 2,... AR SDAR {q t : t = 1, 2,... W > 0 t W -

..x t 1 x t 1 t k x t p ( ( x t x t 1 t k : θ) 1 = exp 1 ) (2π) k/2 Σ 1/2 2 (x t w) T Σ 1 (x t w), k w = A i (x t i µ) + µ i=1

16 4 Change Finder 15 Score (t) = 1 t W ( log q i 1 (y i )) i=t W +1 Step1 Score (t) = 1 t W d (q i 1, q i ) i=t W +1 Score (t) t 4.5 SDAR AR SDAR (Sequencially Discounting AR model learning) n x 1,..., x n x t = z t + µ z 1,..., z n k p (z t θ) t=1 n t=k+1 p ( z t z t 1 t k : θ). k log p (z t θ) + t=1 n t=k+1 log p ( z t z t 1 t k : θ). n k AR ( (n k) log (2π) 1/2 Σ 1/2) 1 2 n t=k+1 ( z t ) T k w i z t i Σ (z 1 t w w i (i = 1,..., k) i=1 k w i C j i = C j (j = 1,..., k). i=1 C j ) k w i z t i. i=1

.., z n k p (z t θ) t=1 n t=k+1 p ( z t z t 1 t k : θ). k log p (z t θ) + t=1 n t=k+1 log p ( z t z t 1 t k : θ).

17 4 Change Finder 16 C j = 1 n k = 1 n k n t=k+1 n t=k+1 z t z T t j (x t µ) (x t j µ) T. s C s = C T s. µ Σ ˆµ = 1 n k n t=k+1 C j µ ˆµ ŵ 1,..., ŵ k x t, ˆΣ = 1 n k = 1 n k n t=k+1 n t=k+1 ( ( z t ) k ŵ i z t i (z t i=1 x t ˆµ ) T k ŵ i z t i i=1 ) ( k ŵ i (x t i µ) x t ˆµ i=1 k ŵ i (x t i µ) SDAR (1 r) : r 0 < r < 1 r SDAR AR SDAR AR SDAR i=1 ) T

18 4 Change Finder 17 SDAR :0 < r < 1 ˆµ, C j, ŵ j (j = 1, 2,...), ˆΣ t (= 1, 2,...) x t ˆµ := (1 r) ˆµ + rx t, C j := (1 r) C j + r (x t ˆµ) (x t j ˆµ) T. k w i C i i = C j (j = 1,..., k). i=1 ŵ 1,..., ŵ k k ˆx t := ŵ i (x t i ˆµ) + ˆµ, i=1 ˆΣ := (1 r) ˆΣ + r (x t ˆx t ) (x t ˆx t ) T

19 18 5 Change Finder 11 Change Finder 5.1 Change Finder SDAR AR Score [x (t)] E [Score [x (t)]] σ [Score [x (t)]] E [Score [x (t)]] + 4σ [Score [x (t)]] Score Score [x (t)] x (t) E [x (t)] 2σ [x (t)] E [x (t)] + 2σ [x (t)] SDAR

02 Score [x (t)] E [Score [x (t)]] σ [Score [x (t)]] E

20 yasai1 sales quantity : 1 yasai1 score :

21 yasai2 sales quantity : 2 yasai2 score : , 17, 25, 52, 29 13, 59

22 yasai3 sales quantity : 3 yasai3 score : , 46, 59, 60 45, 46, 59, 60

23 yasai4 sales quantity : 4 yasai4 score :

24 yasai5 sales quantity : 5 yasai5 score : 5 5

25 niku1 sales quantity : 1 T = 2 AR 3 niku1 score :

26 niku2 sales quantity : 2 niku2 score : 2 2 7, 67

27 niku3 sales quantity : 3 niku3 score : 3 AR 3 3

28 niku4 sales quantity : 4 niku4 score :

29 niku5 sales quantity : 5 niku5 score : 5 5

30 niku6 sales quantity : 6 niku6 score : Change Finder AR

31 Change Finder Change Finder SDAR yasai1 yasai1 sales quantity score : AR AR

32 yasai5 yasai5 sales quantity score : AR AR niku1 niku1 sales quantity score : AR AR

33 Change Finder

34 33 6 Change Finder Change Finder

35 34 [1] 2006 [2] J. P. Peckmann, S. Oliffson Kamphorst, and D. Ruelle, Recurrence Plots of Dynamical Systems, Europhysics Letters, Vol. 4, No. 9, pp , [3] 2009 [4] J. Takeuchi and K. Yamanishi, A Unifying Framework for Detecting Outliers and Change Points from Time Series, IEEE Transaction on Knowledge and Data Engineering, Vol. 18, No. 4, pp , [5] C. M

36 35 A A [5] A.1.1 x h (x) x x p (x) 1 h (x) p (x) 2 x, y h (x, y) = h (x) + h (y) p (x, y) = p (x) p (y) h (x) p (x) h (x) = log p (x) x h (x) 2 A.1.2 H [x] = x p (x) log p (x). N i n i n i n i! N

37 A 36 W = N! i n i! H = 1 N ln W = 1 N ln N! 1 N ln n i! n i /N N i i n i = N H = lim N ln N! N ln N N i ( ni ) ( ni ) ln = N N i p i ln p i p i = lim N (n i /N) i n i /N W H [x] = p (x) ln p (x) dx A p (x) q (x) p (x) q (x) q (x) x x KL (p q) = = p (x) ln q (x) dx { q (x) p (x) ln p (x) dx ( ) p (x) ln p (x) dx p (x) q (x) -

38 A 37 A.3 Change Finder - p (x) θ q (x θ) θ p (x) q (x θ) - p (x) p (x) x n (n = 1, 2,..., N) - KL (p q) 1 N N { ln q (x n θ) + ln p (x n ) n=1 - Change Finder

39 38 B Change Finder R ###.txt ###scorext ###score ##### ###########step1 x <- read.table("yasai1.txt") ### n <- nrow(x) scorext <-matrix(0, nrow=n, ncol=1) pt <-matrix(0, nrow=n, ncol=1) ### Tst1 <- 3 Tst3 <- 5 ###AR kst1 <- 3 kst3 <- 2 ### rst1 < rst3 < xmean <- mean(x[,1]) Cst1 <- matrix(0, nrow=kst1+1, ncol=n) Cst1e <- matrix(0, nrow=kst1, ncol=1) Cmatst1 <- matrix(0, nrow=kst1, ncol=kst1) nmatst1 <- matrix(0, nrow=kst1, ncol=kst1) wst1 <- matrix(0, nrow=kst1, ncol=n) xt <- matrix(0, nrow=n, ncol=1) xtemp <- 0 #must1 <- xmean must1 <- 0 sigmast1 <- matrix(0, nrow=n, ncol=1)

40 B 39 kstartst1 <- kst1+1 for (t in kstartst1 : n ) { must1 <- (1-rst1)*must1 + rst1*x[t,1] for (j in 1 : kstartst1){ Cst1[j,t] <- (1-rst1)*Cst1[j,t-1]+rst1*(x[t,1]-must1)*(x[t-j+1,1]-must1) Cmatst1 <- matrix(0, nrow=kst1, ncol=kst1) for (j in 1 : kst1){ nmatst1 <- matrix(0, nrow=kst1, ncol=kst1) for (i in 1 : kst1){ for (l in 1 : kst1){ if (abs(l-i) == j-1){ nmatst1 <- matrix(0, nrow=kst1, ncol=kst1) nmatst1[l,i] <- 1 Cmatst1 <- Cmatst1 + Cst1[j,t]*nmatst1 for (i in 1:kst1){ Cst1e[i]<-Cst1[i+1,t] wst1[,t] <- solve(cmatst1,cst1e) for (i in 1 : kst1){ xtemp <- xtemp + wst1[i,t]*(x[t-i,1]-must1) xt[t] <- xtemp + must1 xtemp <- 0 if (t == kstartst1){ sigmast1[t] <- (x[t,1]-xt[t])*(x[t,1]-xt[t]) if (t!= kstartst1){ sigmast1[t] <- (1-rst1)*sigmast1[t-1] + rst1*(x[t,1]-xt[t])*(x[t,1]-xt[t]) pt[t] <- exp((-(xt[t]-x[t,1])^2)/(2*sigmast1[t]))*(1/(2*pi*sqrt(sigmast1[t]))) scorext[t] <- -log(pt[t])

41 B 40 #######step2 y <- matrix(0, nrow=n, ncol=1) ytemp <- 0 step2kst1 <- kst1+tst1 for (t in step2kst1 : n ) { kstartst2 <- t-tst1+1 for (i in kstartst2 : t){ ytemp <- ytemp + scorext[i] y[t] = ytemp/tst1 ytemp <- 0 #########step3 scoreyt <-matrix(0, nrow=n, ncol=1) qt <-matrix(0, nrow=n, ncol=1) ytempst3 <- 0 Cst3 <- matrix(0, nrow=kst3+1, ncol=n) Cmatst3 <- matrix(0, nrow=kst3, ncol=kst3) Cst3e <- matrix(0, nrow=kst3, ncol=1) nmatst3 <- matrix(0, nrow=kst3, ncol=kst3) wst3 <- matrix(0, nrow=kst3, ncol=n) yt <- matrix(0, nrow=n, ncol=1) ysum <- sum(y) #must3 <- ysum/(n-tst1-kst1+1) must3 <- 0 sigmast3 <- matrix(0, nrow=n, ncol=1) kstartst3 <- kst1+tst1+1+kst3 kstartst32 <- kst3+1 for (t in kstartst3 : n ) { must3 <- (1-rst3)*must3 + rst3*y[t]

42 B 41 for (j in 1 : kstartst32){ Cst3[j,t] <- (1-rst3)*Cst3[j,t-1] + rst3*(y[t,1]-must3)*(y[t-j+1,1]-must3) Cmatst3 <- matrix(0, nrow=kst3, ncol=kst3) for (j in 1 : kst3){ nmatst3 <- matrix(0, nrow=kst3, ncol=kst3) for (i in 1 : kst3){ for (l in 1 : kst3){ if (abs(l-i) == j-1){ nmatst3 <- matrix(0, nrow=kst3, ncol=kst3) nmatst3[l,i] <- 1 Cmatst3 <- Cmatst3 + Cst3[j,t]*nmatst3 for (i in 1:kst3){ Cst3e[i]<-Cst3[i+1,t] wst3[,t] <- solve(cmatst3,cst3e) ytempst3 <- 0 for (i in 1 : kst3){ ytempst3 <- ytempst3 + wst3[i,t]*(y[t-i]-must3) yt[t] <- ytempst3 + must3 if (t==kstartst3){ sigmast3[t] <- (y[t]-yt[t])*(y[t]-yt[t]) if (t!=kstartst3){ sigmast3[t] <- (1-rst3)*sigmast3[t-1] + rst3*(y[t]-yt[t])*(y[t]-yt[t]) qt[t] <- exp(((yt[t]-y[t])^2)/(2*sigmast3[t])*(-1))*(1/(2*pi*sqrt(sigmast3[t]))) scoreyt[t] <- -log(qt[t]) score <- matrix(0, nrow=n, ncol=1) scoretemp <- 0 winsst3 <- kst1+tst1+tst3 for (t in winsst3 : n ) {

43 B 42 kstartscore <- t - Tst3 +1 for (i in kstartscore : t){ scoretemp <- scoretemp + scoreyt[i] score[t] = scoretemp/tst3 scoretemp <- 0 #plot(x[,1],xlab="",ylab="sales quantity",type="o",main="niku6") #par(new=t) plot(score,xlab="",ylab="score",col="red",main="niku1",type="o")

44 43 C

チュートリアル：ノンパラメトリックベイズ

チュートリアル：ノンパラメトリックベイズ { x,x, L, xn} 2 p( θ, θ, θ, θ, θ, } { 2 3 4 5 θ6 p( p( { x,x, L, N} 2 x { θ, θ2, θ3, θ4, θ5, θ6} K n p( θ θ n N n θ x N + { x,x, L, N} 2 x { θ, θ2, θ3, θ4, θ5, θ6} log p( 6 n logθ F 6 log p( + λ θ F θ