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4 q f (x; α) = α j B j (x). j=1 min α R n+2 n ( d (Y i f (X i ; α)) 2 2 ) 2 f (x; α) + λ dx 2 dx. i=1 f B j 4 / 39

5 : q f (x) = α j B j (x). j=1 : x = [x 1,..., x d ] d f (x) = s k (x k ), k=1 q k s k (x k ) = α k,j B k,j (x k ). j=1 5 / 39

6 : d f (x) = β k x k. k=1 : d f (x) = s k (x k ), k=1 q k s k (x k ) = α k,j B k,j (x k ). s k (x k ) = β k x k j=1 6 / 39

7 d q k f (x) = s k (x k ), (s k (x k ) = α k,j B k,j (x k )). k=1 j=1 s j : min α k,j ( n y i i=1 2 d d s k (x i,k )) + λ k=1 k=1 (s k (t)) 2 dt. 7 / 39

8 f (x, y) = xy. s ab (x a, x b ) (2 ): f (x) = a>b s ab (x a, x b ) + k s k (x k ). s abc (x a, x b, x c ) 8 / 39

9 {(x (i), y i )} n i=1 : f (x) = a>b s ab (x a, x b ) + k s k (x k ). min f n (y i f (x (i) )) 2 i=1 + λ a>b { ( 2 s ab x 2 a ) 2 ( 2 ) 2 s ab x a x b ( 2 ) 2 } s ab dx a dx b x 2 b ( 2 ) 2 s k dx k. + λ k x 2 k 9 / 39

10 s ab (x a, x b ) 1 (tensor product spline) 2 (thin plate spline) B ab,j : q ab s ab (x a, x b ) = α ab,j B ab,j (x a, x b ). j=1 10 / 39

11 b j (x) ( ) B ab,j (x a, x b ) = b ja (x a )b jb (x b ) q a q b s ab (x a, x b ) = α ja,j b b ja (x a )b jb (x b ). j a =1 j b =1 q a q b q c s abc (x a, x b, x c ) = α ja,j b,j c b ja (x a )b jb (x b )b jc (x c ). }{{} j a =1 j b =1 j c =1 11 / 39

12 I m (> I /2) : η m,i (r) := { m+1+i /2 ( 1) 2 2m 1 π I /2 (m 1)!(m I /2)! r 2m I log(r) (I ), Γ(I /2 m) 2 2m π I /2 (m 1)! r 2m I (I ), B a1 a 2...a m,i(x a1,..., x am ) = η m,i ( [x a1,..., x am ] [x (i) a 1,..., x (i) a m ] ). [x (i) a 1,..., x (i) a m ] ( ) m = I = 2 (2 ) r i = [x a1, x a2 ] [x (i) a 1, x (i) a 2 ]. B a1a 2,i(x a1, x a2 ) = r i 2 8π log(r i), 12 / 39

13 0 / -, * ) 0 / -, * ) ) *, - / 0 13 / 39

14 : {(x (i), y i )} n i=1. s(x) = n M α i η m,i ( x x (i) ) + β j ϕ j (x), i=1 M = ( ) m+d 1 d ϕj m α = [α 1,..., α n ] T = (η m,i ( x (i) x (j) )) n i,j=1 α T = 0 j=1 min s n (y i s(x (i) )) 2 +λ i=1 ν 1+ +ν d =m ( m! m ) 2 s ν 1!... ν d! x ν x ν dx d 1... dx d, d 14 / 39

15 : {(x (i), y i )} n i=1. m = 2. n { ( min (y i s(x (i) 1, x (i) 2 ) 2 ( s 2 ) 2 ( s 2 ) 2 } s 2 ))2 +λ + + dx 1 dx 2. x 1 x 2 i=1 x 2 1 x / 39

16 I m + 1 m. s(x 1,..., x I ) : J(s) = : f = j 1 ν 1+ +ν I =m ( m! m ) 2 s ν 1!... ν I! x ν x ν dx I 1... dx I. I s j1 (x j1 ) + j 2,j 3 s j2 j 3 (x j2, x j3 ) + j 4,j 5,j 6 s j4 j 5 j 6 (x j4, x j5, x j6 )... J(s j1 ) + J(s j2 j 3 ) + J(s j4 j 5 j 6 ) +... j 1 j 2,j 3 j 4,j 5,j 6 16 / 39

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18 : ϵ i 0 y i = f (x i ) + ϵ i, = : y i P θ=g(f (xi ))(Y ). 1 P θ (Y ). 2 g 1 ( ) f 18 / 39

19 y y x x (a) (b) 19 / 39

20 : Po θ (Y ) = θy e θ Y! (θ > 0, Y = 0, 1, 2,... ). y i Po(θ = exp(f (x))) g(f (x)) = exp(f (x)), g 1 (θ) = log(θ) = f (x):. f (x) : Bin θ N (Y ) = ( N Y) θ Y (1 θ) N Y (θ (0, 1), Y = 0, 1,..., N). ( ) 1 y i Bin θ = 1 + exp( f (x)) N g(f (x)) = 1 1+exp( f (x)), f (x) = g 1 (θ) = log ( θ 1 θ ) :. 20 / 39

21 : l(y, u) = log(p θ=g(u) (y)). : n min l(y i, f (x i )) + λ J(s j1 ) + J(s j2 j s j1,s j2,j 3,... 3 ) + J(s j4 j 5 j 6 ) +..., j 1 j 2,j 3 j 4,j 5,j 6 i=1 f = j 1 s j1 (x j1 ) + j 2,j 3 s j2j 3 (x j2, x j3 ) +... = ( ) 21 / 39

22 logistic(x) g(u) = 1 1+exp( u) x 22 / 39

23 fit <- gam(y~s(x1,x2),data=artdata,family=binomial(link=logit)) X X1 23 / 39

24 fit <- gam(y~s(x1,x2),data=artdata,family=binomial(link=logit)) X X1 23 / 39

25 fit <- gam(y~s(x1,x2),data=artdata,family=binomial(link=logit)) X X1 23 / 39

26 gam Y~s(X1)+s(X2) glm link glm 24 / 39

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28 chouse <- read.csv("cal_house.csv",header=true) 20640, MedHouseValue : ( ) MedIncome : MedHouseAge : TotalRooms : TotalBedrooms : Population : Households : Latitude : Longitude : 26 / 39

29 > lmfit <- gam(log(medhousevalue)~medincome+...+longitude, > data=chouse) Parametric coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e e < 2e-16 *** MedIncome 1.782e e < 2e-16 *** MedHouseAge 3.261e e < 2e-16 *** TotalRooms e e < 2e-16 *** TotalBedrooms 4.798e e < 2e-16 *** Population e e < 2e-16 *** Households 2.493e e e-11 *** Latitude e e < 2e-16 *** Longitude e e < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 64.3% GCV score = Scale est. = n = / 39

30 Predicted 線形回帰の結果 0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 Actual price あまり当てはまりが良くない 28 / 39

31 addfit <- gam(log(medhousevalue) ~ s(medincome)+ s(medhouseage) + s(totalrooms) + s(totalbedrooms) + s(population) + s(households) + s(latitude) + s(longitude), data=chouse) s( ) 29 / 39

32 > summary(addfit) Approximate significance of smooth terms: edf Ref.df F p-value s(medincome) < 2e-16 *** s(medhouseage) < 2e-16 *** s(totalrooms) e-15 *** s(totalbedrooms) < 2e-16 *** s(population) < 2e-16 *** s(households) e-11 *** s(latitude) < 2e-16 *** s(longitude) < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 74.8% GCV score = Scale est. = n = GCV ( ) 30 / 39

33 4e+05 2e+05 0e+00 Predicted 6e+05 加法モデル回帰の結果 0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 Actual price 31 / 39

34 plot(addfit,scale=0,se=2,shade=true,pages=1) s(medincome,8.71) s(medhouseage,8.81) s(totalrooms,6.55) MedIncome MedHouseAge TotalRooms s(totalbedrooms,8.95) s(population,8.15) s(households,5.69) TotalBedrooms Population Households s(latitude,8.93) s(longitude,8.84) Latitude Longitude 32 / 39

35 : addfit2 <- gam(log(medhousevalue) ~ s(medincome) + s(medhouseage) + s(totalrooms) +s(totalbedrooms) + s(population) + s(households) + s(longitude,latitude), data=chouse) s(longitude,latitude) 33 / 39

36 > summary(addfit2) Approximate significance of smooth terms: edf Ref.df F p-value s(medincome) < 2e-16 *** s(medhouseage) < 2e-16 *** s(totalrooms) ** s(totalbedrooms) < 2e-16 *** s(population) < 2e-16 *** s(households) < 2e-16 *** s(longitude,latitude) < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * R-sq.(adj) = Deviance explained = 77.8% GCV score = Scale est. = n = GCV ( ) 34 / 39

37 4e+05 3e+05 2e+05 1e+05 Predicted 5e+05 6e+05 7e+05 交互作用有り加法モデル: 結果 0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 Actual price 35 / 39

38 Latitude Longitute 36 / 39

39 交互作用なしとありとの違い Latitude Latitude 緯度経度のみから価格を予測 Longitute (c) 交互作用あり (MSE= 0.180) Longitute (d) 交互作用なし (MSE= 0.201) 37 / 39

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() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()