7/7 2 / 34

Size: px

Start display at page:

Download "7/7 2 / 34"

れれまつかた
5 years ago
Views:

1 W707 1 / 34

2 7/7 2 / 34

3 3 / 34

4 1 4 / 34

5 (...) 5 / 34

6 Kernel Density Estimation (gauss) p(x) x 6 / 34

7 O(n) 7 / 34

8 1 8 / 34

9 {X i } n i=1 : ( ) : K ˆp(x) = 1 nh n ( x Xi K h i=1 ). 1 h > 0 2 K : K(x)dx = 1, xk(x)dx = 0, x 2 K(x) > 0. 9 / 34

10 p(x) x 10 / 34

11 . 1 Gaussian: 2 Rectangular: 3 Triangular: 4 Epanechnikov: K(x) = 1 2π exp K(x) = K(x) = K(x) = ( x 2 2 ). { 1 2 ( x 1), 0 (otherwise). { x ( x 1), 0 (otherwise). { 3 4 (1 x 2 ) ( x 1), 0 (otherwise). 11 / 34

12 K(x) kernel functions Rectangular Triangular Gaussian Epanechnikov x 12 / 34

13 Kernel Density Estimation p(x) x 13 / 34

14 Kernel Density Estimation p(x) x 13 / 34

15 Kernel Density Estimation p(x) x 13 / 34

16 n = 100 Kernel Density Estimation (rectangular) Kernel Density Estimation (triangular) Kernel Density Estimation (gauss) p(x) p(x) p(x) x x x rectangular triangular gauss 14 / 34

17 n = Kernel Density Estimation (rectangular) Kernel Density Estimation (triangular) Kernel Density Estimation (gauss) p(x) p(x) p(x) x x x rectangular triangular gauss 14 / 34

18 n = Kernel Density Estimation (gauss) p(x) x gauss 14 / 34

19 1 15 / 34

20 p(x) band width true x 16 / 34

21 LSCV, Least Squares Cross Validation (ˆp h h ) (ˆp h (x) p(x)) 2 dx = ˆp h (x) 2 dx 2 ˆp h (x)p(x)dx } {{ } =:J(h) + p 2 (x)dx. J(h) p(x) ˆp h ˆp h,( i) : i X i. Ĵ(h) = ˆp h (x) 2 dx 2 n n ˆp h,( i) (X i ). i=1 Ĵ(h) h 17 / 34

22 Silverman 1 : ŝ = 1 n n i=1 (X i X ) 2. 2 q(0.25): 0.25 { } ˆq(0.75) ˆq(0.25) ˆσ = min ŝ, 1.34 : ĥ = 1.06ˆσ n. 1/ (1.06 Scott 0.9 Silverman ) [ ĥ = C K n( (p (x)) 2 dx)] 1/5, K(x) ( C K = 2 dx ( ) x 2 K(x)dx) 2 p (x) 2 dx 18 / 34

23 p(x) band width true CV Silverman (1.06) x 19 / 34

24 1 h K(x X i h ) ) X (j) i d j=1 ( 1 K h j x X (j) i h j X i j library(mass) d=kde2d(x,y,c(bandwidth.nrd(x),bandwidth.nrd(y)),n=80) image(d,xlab="latitude",ylab="longitude") 20 / 34

25 1 21 / 34

26 h > 0 ˆp(x) = 1 nh n ( ) x Xi K h ˆp x (Mean Squared Error) : i=1 MSE(ˆp(x), h) := E[(ˆp(x) p(x)) 2 ]. E[ ] {X i } n i=1 x (Integrated MSE): IMSE(ˆp(x), h) := E[(ˆp(x) p(x)) 2 ]dx. Q: h 22 / 34

27 h n h 0. MSE(ˆp(x), h) = E[(ˆp(x) p(x)) 2 ] = E[(ˆp(x) E[ˆp(x)] + E[ˆp(x)] p(x)) 2 ] = E [ (ˆp(x) E[ˆp(x)]) 2 2(p(x) E[ˆp(x)])(ˆp(x) E[ˆp(x)]) + (p(x) E[ˆp(x)]) 2] = E [ (ˆp(x) E[ˆp(x)]) 2 ] + E[(p(x) E[ˆp(x)]) 2 ], }{{}}{{} 23 / 34

28 [ 1 n ( ) ] x Xi E[ˆp(x)] = E K nh h i=1 = 1 n [ ( )] 1 x E n h K Xi h i=1 [ ( )] 1 x X = E h K = K(u)p(x hu)du h 24 / 34

29 p(x hu) p(x hu) = p(x) hu dp(x) dx K(u)p(x hu)du p(x) = h dp(x) dx uk(u)du + h2 2 + h2 u 2 2 d 2 p(x) d 2 x uk(u)du = 0 d 2 p(x) d 2 x + o(h 2 ) u 2 K(u)du + o(h 2 ) ( h 2 = 2 d 2 p(x) d 2 x u 2 K(u)du) 2 + o(h 2 ). 25 / 34

30 ˆp(x) ( Var( ) ) E [ (ˆp(x) E[ˆp(x)]) 2 ] = Var(ˆp(x)) ( n ( ) ) 1 x = Var h K Xi h i=1 = 1 ( ( )) 1 x X n Var h K h { [ ( = 1 ( )) ] 2 1 x X E n h K h = 1 nh = p(x) nh K(u) 2 p(x hu)du 1 n E[ˆp(x)]2 K(u) 2 du + o(1/(nh)). [ ( )] } 2 1 x X E h K h 26 / 34

31 MSE(ˆp(x), h) = p(x) nh K(u) 2 du+ h4 4 ( d 2 p(x) d 2 x u 2 K(u)du) 2 +o(1/(nh)+h 4 ). x IMSE(ˆp(x), h) = 1 nh K(u) 2 du+ h4 4 ( d 2 p(x) d 2 x ) 2 u 2 K(u)du dx+o( 1 nh +h4 ). h ( h = 1 ) 1/5 K(u) 2 du n 1/5 (p (x)) 2 dx ( u 2 K(u)du ) 2 p (x) 27 / 34

32 h IMSE IMSE(ˆp(x), h ) = 5 4n 4/5 = O(n 4/5 ) [ (p (x)) 2 dx ( u 2 K(u)du ) 2 ] 1/5 [ K(u)2 du ] 4/5 + o(1/n 4/5 ) 28 / 34

33 d ( ) 1 IMSE = O n 4/(d+4) d 29 / 34

34 1 30 / 34

35 R 1 density(x,bw,kernel): : bw = nrd0 ( 0.9 ), bw= nrd bw= ucv ( ), bw= bcv : kernel= gaussian epanechnikov, rectangular, triangular, biweight, cosine, optcosine bkde: KernSmooth kde2d: MASS 2 bkde2d: KernSmooth kde: ks 3 MASS bandwidth.nwd bw.nwd 4 31 / 34

36 area e+07 4e+07 6e+07 8e+07 price 32 / 34

37 area e 10 5e e 10 4e 10 3e 10 4e e e e 10 1e 10 0e+00 2e+07 4e+07 6e+07 8e+07 1e+08 price 32 / 34

38 estimated density area price 32 / 34

39 1 CV 33 / 34

40 34 / 34

k- W707 1 / 39

k- W707 1 / 39 k- W707 s-taiji@is.titech.ac.jp 1 / 39 7/7 2 / 39 k- k- 3 / 39 1 k- k- k 2 k- 3 4 / 39 k- 5 / 39 k- k- 6 / 39 1 k- k- k 2 k- 3 7 / 39 k- {X i } n i=1 : (d ) k- (k-nearest neighbor) 1 k 1 2 x k {X (1),