:EM,,. 4 EM. EM Finch, (AIC)., ( ), ( ),. 1. 1990. Web,,.,., [1].,. 2010,,,, 5 [2]., 16,000.,..,,. (,, )..,,. (socio-dynamics) [3, 4]. Weidlich Haag. [5]. 606-8501,, TEL:075-753-5515, FAX:075-753-4919, E-mail:aki@i.kyoto-u.ac.jp 1
. push pull [6]. push, ( ). pull, ( ).,., 1 16,000,, EM, [7].,,,.. 2,. 3,. 4,,. 5 4. 6. 2,., Web Web API, csv. (http://www.jalan.net). 16,000.. (i). (ii) Web. (iii) Web,. (iv)., Web 2 1.. 1.,.,,. 2009 12 24 2010 11 4.. 1 3 1,000., 2010 1 1 8,000, 2010 3 4 9,000, 2010 8 31 9,000,,.. 2. ( ),,.,., 7 8 2
1: ( ) URL.,., 1 2 6.,.,., z m (t) = max Y m (t) + 10 Y m (t)., Y m (t) t m. ( ),,, ( ), ( ), ( ). 3. 3 3.1 t, m z m (t). Pr Zm (
histgram 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 20000 40000 60000 80000 100000 price 2010-01-01 2010-03-04 2010-08-31 1: 2010 1 1, 2010 3 4, 2010 8 31. z m (t) (t = 1,..., T ) m t., (1) L m (a m1,..., a mkm, r m1,..., r mkm ) = (K m (Mr mi ) zm(s) log a mi e ), Mrmi (2) z m (s)! s=1. r mi, a mi (i = 1,..., K m ), K m a mi = 1 {â m1,..., â mk, ˆr m1,..., ˆr mk } = arg max {a mi },{r mi } L m (a m1,..., a mkm, r m1..., r mk ) (3). A, (3)., a (ν+1) mi = 1 T r (ν+1) mi = 1 M F (ν) mi a (ν) mi F (ν) mi (z m(t)) m (z m (t)) T z m(t) F (ν) mi (z m(t)) m (z m (t)) T F (ν) mi (z m(t)) m (z m (t)) (Mr(ν) (x) = x! K m m (x) = mi )x e Mr(ν) mi a (ν) mi F (ν) mi (i = 1,..., K m ), (4) (i = 1,..., K m ). (5), (6) (x), (7) 4
180000 160000 140000 120000 100000 80000 60000 40000 20000 0 Jan-2010 Feb-2010 Mar-2010 Apr-2010 May-2010 Jun-2010 Jul-2010 Aug-2010 Sep-2010 Oct-2010 Nov-2010 # of opportunities 2: 2009 12 24 2010 11 4.. EM [9, 10]. 3.3 Finch EM, EM,.,.,,., Finch [11]. Finch. a (0) mi K m a(0)., a (0) mi mi = 1 s z m (t s ) (s = 1,..., T ) K m a (0)) mi., i r (0) im µ mi, r (0) im = µ mi/m., a (0) m1,..., a(0) mk m [0, 1] 1., r (0) m1,..., r(0) mk m Finch. (18)., EM.. 5
600 500 010502 (Otaru) 072005 (Aizu-Kohgen,Yunogami,Minami-Aizu) 136812 (Shiragane) 171408 (Yuzawa) 400 constant - # of plans 300 200 100 0 Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 date 3: 010502 ( ), 072005,, ( ), 136812 ( ), 171408 ( ).,.. (0) maxobj = 0 counter = 0. (1) [0, 1] b mi, a mi a mi = b mi / Km b mi. (2) r mi Finch. counter > MAXCOUNT (6). (3) L m (a m1,..., a mk m, r m1,..., r mk m ) maxobj, maxobj, r mi := r mi a mi := a mi (4). (4). (1). (4) (a m1..., a mkm, r m1,..., r mkm ), (4) (5). (5), maxobj,. (6) counter = counter + 1. counter < MAXCOUNT, (1)., (7). (7) maxobj. 6
3.4 (AIC) K m., ˆL m = s=1 AIC(K m ) = 4K m 2ˆL m, (8) ˆK m = arg min AIC(K m ). (9) K m (K m (M ˆr mi ) z m(s) ) log â mi e M ˆr mi. (10) z m (s)! 2K m. z m (s), R mi (z) = z log M + log ˆr mi M ˆr mi,. ( ) log z!. (11) î s = arg maxr mi (z m (s)). (12) i 4. z(s) (s = 1,..., T ). { r(t) = ri w.p. a i z(t) Pr(l = Z(t) r(t)) = (Mr(t))l l! e Mr(t). (13), K, a i i (i = 1,..., K; K a i = 1). K = 12, M = 100, 000, 000,. 2( ) T = 200... 4( ) K AIC ˆK = 12. Kolmogorov- Smirnov.. 4( ) KS. ˆK = 12 KS 0.327(<1.36), 5%.. 2( ).,.. 5, 8.0 10 6, ˆr i (t) r i (t) /r i (t) 0.3 %. 7
2: ( ). K = 12. EM ( ). AIC ˆK = 12, AIC = 3803.20. r 1 0.000025 a 1 0.109726 r 1 0.000024 a 1 0.09000 r 2 0.000223 a 2 0.070612 r 2 0.000222 a 2 0.07000 r 3 0.000280 a 3 0.073355 r 3 0.000280 a 3 0.05500 r 4 0.000479 a 4 0.077612 r 4 0.000479 a 4 0.06500 r 5 0.000613 a 5 0.094848 r 5 0.000613 a 5 0.08000 r 6 0.000652 a 6 0.073841 r 6 0.000651 a 6 0.09500 r 7 0.001219 a 7 0.090867 r 7 0.001218 a 7 0.10417 r 8 0.001233 a 8 0.062191 r 8 0.001232 a 8 0.08082 r 9 0.001295 a 9 0.077662 r 9 0.001294 a 9 0.08500 r 10 0.001341 a 10 0.102573 r 10 0.001341 a 10 0.10500 r 11 0.001412 a 11 0.085892 r 11 0.001412 a 11 0.08000 r 12 0.001570 a 12 0.080821 r 12 0.001568 a 12 0.09000 5,. M = 1, 000, 000, 000, 4.. 3 4. AIC,. 3., AIC KS., AIC KS 5%. m t p mk (k = 1,..., n m (t)) P m (t) = 1 n m (t) p mk (t) (14) n m (t). t r m (t) t k=1 10000 9000 2 1.8 5% confidence level AIC 8000 7000 6000 5000 KS statistics 1.6 1.4 1.2 4000 1 3000 10 15 20 25 30 K 0.8 6 8 10 12 14 16 18 20 22 24 K 4: K AIC ( ). AIC K = 12 3803.20. K KS ( ). 5%. 8
0.0016 0.0014 0.0012 0.004 0.0035 0.003 relative error r 0.001 0.0008 0.0006 relative error 0.0025 0.002 0.0015 0.0004 0.001 0.0002 0.0005 0 0 20 40 60 80 100 120 140 160 180 200 t 0 0 20 40 60 80 100 120 140 160 180 200 t 5: r ( ). ( ). 3: 4 AIC, (ll), p, KS K AIC ll p-value KS value ( ) 12 3558.31 1756.15 0.807 0.532,, ( ) 10 3009.10 1485.55 0.187 1.088 ( ) 5 2572.33 1277.17 0.107 1.245 ( ) 11 3695.25 1826.62 0.465 0.850.. 6.,,.,. 6, 2009 12 25 2010 11 4. EM Finch,..,.,, 4,. Kolmogorov-Smirnov 5%.,.,,.,,.,,. 9
45000 40000 35000 30000 25000 20000 15000 10000 5000-7 5.0x10 1.0x10-6 probability r (c) 55000 50000 45000 40000 35000 30000 25000 20000 15000 1.0x10-6 2.0x10-6 3.0x10-6 4.0x10-6 probability r 6: rmi. (a)010502 ( ), (b)072005,, ( ), (c)136812 ( ), (d)171408 ( ).,.,., (#21-5341).., Thomas Lux., Dirk Helbing. A EM Poisson EM. Gm(z) Km Fmi(z) (m = 1,..., Km). Fmi(z) = (Mrmi)z Gm(z) = Km z! (d) e Mrmi, (i = 1,..., Km) (15) amifmi(z). (16) 10 35000 mean of rates per night [JPY] 30000 25000 20000 15000 10000 5000 1.0x10-6 2.0x10-6 3.0x10-6 4.0x10-6 probability r] 28000 mean of rates per night [JPY] 26000 24000 22000 20000 18000 16000 (a) 14000 5.0x10-7 1.0x10-6 1.5x10-6 probability r (b) mean of rates per night [JPY] mean of rates per night [JPY]
a mi,. K m a mi = 1. (17) T {z m (t)} a mi, r mi (i = 1,..., K m ). L m (a m1,..., a mr, r m1,..., r mkm ) =. (15) (17) (18),. (K m 1 log (Mr mi ) z m(t) K m 1 a mi e Mr mi + (1 z m (t)! ( log G m zm (t) ). (18) L m (a m1,..., a mkm, r m1,..., r mkm ) = (19) a mi r mi a mi ) (Mr mk m ) z m(t) e Mr mk z m (t)! ). (19) L m a mi = L m r mi = L m r mkm = F mi (z m (t)) F mk (z m (t)) G m (z m (t)) a mi F mi (z m (t)) ( zm (t) ) M G m (z m (t)) r mi (1 K m a mi)f K (z m (t)) ( zm (t) G m (z m (t)) (i = 1,..., K m 1), (20) r mk (i = 1,..., K m 1), (21) ) M. (22). a mi (20) i., a mi /T (23) F mi (z m (t)) G m (z m (t)) = T (i = 1,..., K m), (23) a mi = 1 T. (21) (22), a mi F mi (z m (t)) G m (z m (t)) (i = 1,..., K m ), (24) r mi = 1 M T z m(t) Fmi(zm(t)) G m (z m (t)) T F mi(z m(t)) G m (z m (t)) (i = 1,..., K m ). (25).,, {a (0) {r (0) mi } {a(ν) mi } and {r(ν) mi } a (ν+1) mi = 1 T r (ν+1) mi = 1 M a (ν) mi F (ν) mi (z m(t)) m (z m (t)) T z m(t) F (ν) mi (z m(t)) m (z m (t)) T F (ν) mi (zm(t)) m (z m(t)) mi }, (i = 1,..., K m ), (26) (i = 1,..., K m ), (27) 11
.,. F (ν) mi mi )z (Mr(ν) (z) = z! K m m (z) = a (ν) mi F (ν) mi e Mr(ν) mi, (28) (z), (29) [1] R. Law, Disintermediation of reservations, International Journal of Contemporary Hospitality Management, 21 (2009) 766-772. [2] : http://www.mlit.go.jp/kankocho/siryou/toukei/index. html. [3] W. Weidlich, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences, Taylor and Francis, London (2002). [4] S. Alfrano and T. Lux, A noise trader model as a generator of apparent financial power laws and long memory, Macroeconomic Dynamics, 11 (2007) 80 101. [5] G. Haag, and W. Weidlich, A stochastic theory of interregional migration, Geographical Analysis, 16 (1984) 331-357. [6] Sukbin Cha, Ken W. McCleary, and Muzaffer Uysal, Travel Motivations of Japanese Overseas Travelers: A Factor-Cluster Segmentation Approach, Journal of Travel Research, 34, (1995) 33 39. [7] A.-H. Sato, Patterns of Regional Travel Behavior: An Analysis of Japanese Hotel Reservation Data, International Review of Financial Analysis, in press. [8] V. Hasselblad, Estimation of Finite Mixtures of Distributions from the Exponential Family, Journal of the American Statistical Association, 64 (1969) 1459 1471. [9] A.P. Dempster, N.M. Laird, and D.B. Rubin, Maximum Likelihood from Incomplete Data via the EM Algorithm, Journal of the Royal Statistical Society, Series B, 39 (1977) 1 38. [10] Z. Liu, J. Almhana, V. Choulakian, R. McGorman, Online EM algorithm for mixture with application to internet traffic modeling, Computational Statistics and Data Analysis, 50 (2006) 1052 1071. [11] S.J. Finch, N.R. Mendell, H.C. Thode, Probabilistic measures of adequacy of a numerical search for a global maximum, Journal of the American Statistical Association, 84 (1989) 1020 1023. 12