R/S.5.72 (LongTerm Strage 1965) NASA (?. 2? (-:2)> 2?.2 NB. -: is half

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1 SHIMURA Masato R/S References 22 C.Reiter 5 1 R/S 1.1 Harold Edwin Hurst England) Leicester, Oxford Sir Henry Lyons ( Hurst Black Simaika [Nail Basin] Jonglei 88 1

2 R/S.5.72 (LongTerm Strage 1965) NASA (?. 2? (-:2)> 2?.2 NB. -: is half _1 ( ((-:2)> 2?.2){1 _1 _1 _ _1 _ _1 1 _1 _1 _1 1 _1 1 _1 7{. <\((-:2)> 2?.2){1 _1 NB _1 _1 _1 _1 _1 1 _1 _1 1 1 _1 _ _1 _ _1 _ _ (box ( ) +/\((-:2)> 2?.2){1 _1 _1 _2 _ rw=: 3 : +/\((-:y)>y?y){1 _1.5 2

3 RW plot rwp 1 rwp (3#.5) rwp 1.5 rwp=:4 : NB. Random walk with many probability NB. Usage: plot rwp 1 NB. or.5 ;("1),. +/\ L:({(x * y)> y? (#x)#y){(l:)_1 1 ) 1.3 dt =.1, dx = ±.1 dt, dx = σ dt P(X n = dx) = µ 2σ dx 2 P(X n = dx) = 1 2 µ 2σ dx 2 X n ( 1 E(X n ) = dx 2 + µ ) ( 1 2σ dx dx µ V(X n ) = E(Xn) 2 (E(X n )) 2 = σ 2 dt µ 2 (dt) 2 X n Y n Y =, Y n = Y 1 + Y Y n ) 2σ dx 2 = dx µ dx = µdt 2σ2 3

4 script N.Thomson rho plot.1 count rno,.5, 1,1 +/\ ] a=. rn.5 1 a,. +/\ a _ _ _ _ _ _ _ _ _.7598 _ _1.886 plot((i.1)%1); +/\ rno.1 1 pd eps /temp/bwn_1.eps 4

5 Brown µ = σ = R/S long memory) R/S R t,n = max 1 k N ( k i=1 (x t+i x t,n )) min 1 k N ( k i=1 (x t+i x t,n )) S t,n R t,n S t,n Q t,n cn H MAX MIN R randsn ] a=. randsn 5 _

6 a,.((- mean)a),. +/\ (- mean) a deviation ccd _ _ _ _ _ _ _ e_17 ccd=:[: +/\ (- mean) r a r=:(>./ - <./)@ccd q a q=:r % sd NB. sd is standard deviation * ) C.Reiter S IDI 18 (n = 58) t (time span) * 1 span NB. span=: NB. n=58 span *1 wavelet 2 n 2 n 6

7 ave ave1 ave2 ave3 ave4 4. span,. span rs_test sp NB. 1/ NB. 1/ NB. 1/ span x ( (ˆ. span rs sp1) %. 1,. ˆ. span _ f = lnx.73 calc hurst x H =.769 calc_hurst sp1 _

8 1.4.2 rs_test=:([: ;._3])" 1 NB. same calc_hurst=: 3 : NB. calc_hurst randsn 1 N=: 2+i.<.-:#y RS=. N rs_test y NB. R/S static (ˆ. RS)%. 1,. ˆ. N ) n N=. 2+i.<.-:#y rs test cut 3 5 * 2 (5,:5)<;.(_3) 1? box(<) (5,:5)q ;.(_3) 1? ,: 5 ;._3]) 1? mean=. +/%#) *2 3 cut 8

9 5 ([: ;._3]) 1? rs_test=: ([: ;._3]) 5 rs_test 1? rs_test"( 1) 1? ( ) calc_hurst randsn 1 _ H sunspots C.Reiter S IDC(Solar Influence Data Analsys Center,Bergium)

10 2.2 Autocorreration ACF(k) = Σn t=k 1 (Y t Ȳ)(Y t k Ȳ) Σ n t=1 (Y t Ȳ) 2 EXAMPLE i.1 ( 9) 1 copy * 3 ({@>1+i.8),. }.}: ( (<\.a),..@(<\) a=. i.1) tmp,. +/ L: tmp=. */ L:,. <"2 > ((<\. a),.. <\ a) - L: mean a _ *3 1

11 _.75 _ _1.25 _2.25 _ _1.75 _3.75 _3.75 _ _ _2.25 _5.25 _6.25 _5.25 _2.25 _ _6.75 _8.75 _8.75 _6.75 _ _11.25 _12.25 _11.25 _ _15.75 _15.75 _ _2.25 _ (+/@:ˆ&2@(- +/%#) ) a=. i ac f acfx a _ _ _ _ _ _ Script acfx=:3 : ((+/@:*/)"2>((<\.y),..<\y) - L: mean y) % +/@:(ˆ&2)@dev y 11

12 (15-18) 15 15{.(>:i. # a),.a=.acf sp _ _ _ _ NB. runner-up NB. max _ H hwalk ( 1 2 2H h ) k+1 C.Reiter 1 2 2H 2 ( ) k h interp=: (}. + }:)@:(2: # -:) osz=: %:@-.@(2&ˆ)@+:@<:@[ sz=: osz * %@+:@<:@#@] ˆ [ 12

13 randunif=: : NB. uniform rand randsn=: * NB. normal rand randadd=: ] + sz * randsn@$@] hwalk=: 4 : x ([ randadd interp@]) ˆ: y,(x osz 1)* randsn 1 ) hwalk,h= Hwalk,H=.5 13

14 hwalk,h= R/S a=. ; }. : sp=. > jread.. plot calc_rs_sub a R/S Y R/S X N= ln2.5 ln2.5 ˆ lnr/s vs ln N.77 calc_hurst a _ ln calc_rs_masked a _

15 3.5 rs_sunspots reg lnr/s masked ln Script mean=:+/%# diff=: }. - }: NB. 2 stage Sabun ccd=: [: +/\ (- mean) NB. same avobe sd=: [: %: # % (+/@: *:@ (- mean)) r=:((>./)-<./ )@ccd NB. max - min q=: r % sd rs=: ([: q ;._3 ]) " 1 NB. R/S static rs_test=:([: ;._3])" 1 NB. same calc_hurst=: 3 : NB. calc_hurst sunspots N=: 2+i.<.-:#y RS=. N rs_test y NB. R/S static (ˆ. RS)%. 1,. ˆ. N ) NB calc_rs_sub=: 3 : X=. diff diff y N=. 2+i. <. -: # X NB. komado RS=. N rs_test X NB. rs (ˆ. N);ˆ. RS 15

16 ) calc_rs_masked=: 4 : X=. diff diff y N=. 2+i. <. -: # X NB. komado RS=. N rs_test X NB. rs MASK=. N <: ˆ x NB (ˆ. MASK # RS)%. 1,. ˆ. MASK # N ) plot_rs=: 4 : NB. find 2.5 by own eye at x axis// N RS =: calc_rs_sub y pd reset pd key rs_sunspots reg pd N; RS,: (1,. N) +/. * f=. x calc_rs_masked y pd show ) 2.5 C.Reiter C.Reiter m histories,k m-histries hen=:3 : y,1+(.3*_2{y)+_1.4**: _1{y plot hen ˆ:(4)] ref 2]\ i

17 henon dist dist %:@(+/)@:*:@:-"1 (a=.1 2 3,: 3 4 2), (-/ a),:ˆ&2 -/ a NB. A NB B _2 _2 1 NB. -/ A,B NB. ˆ& / is 9 NB. sum %: 9 is 3 NB. square 17

18 data=. hen ˆ:(4) ] m=2 k=5 size=: 35 ref=2 ]\ size{. data size ref ] x=. (35+1+i.m) {data {. ref dist x ]j=. (k=.5){. /: ref dist x x,y (m+j),. (m+j){data adress data ] coef=. ((m+j){data) %. 1,. j{ref _ f = x x 2 1,. j{ref

19 1 (>:size+i.m){data coef +/. * 1, (>:size+i.m){data 2 non-update type) param= {. a,. (a=.param,"1 ]>:i.2) fracpred data m k siz T result _ update type (c f ps1 m histry m fps1 data;(>:35+i.m){data Script fps1=:4 : ("1) NB. Usage: fps1 data m k sz =. x y x =. y NB. x=. (sz+1+i.m){y ref=. m ]\ sz {. y j=. k{./: ref dist x coef=. ((m+j){y) %. 1,.j{ref coef +/. * 1,x ) 19

20 f ps2 data y fps2=:3 : param fps1 data;y param= (2step) a=. (}.@, fps2) ˆ:(>:i.2) (>:size+i.m){data plot }.{. : a time henon,m=2 param= time henon,m=3 2

21 2.5.2 sp=. ;{: : sp NB. 58 years sunspots data (jread...) fps1 data=: sp (data ) m (m-histry) AR k m m 2 sz n-g-m G umber of steps in our goal forecast a b m c m a,b,:c Script pred_frac=: 4 : NB. Usage: pred_frac sunspots NB. x is m k sz G NB. or/ y is ; }. : sp M K SZ G =: x data=: y NB. should global definition NB calc a=. (>: SZ+M+i.G){ y b=. ((M,K,SZ),("1 )] >:i.g) fracpred y c=.(<:m)}.{. :(}.@, fps2)ˆ:(>:i.g+<:m)(>:sz+i.m){y a,b,:c ) Example pred_frac ; }. : sp plot_frac ; }. : sp m=8 k=2 sz=46 G=35 ( 21

22 a b c fractal prediction m=8,k=2 3 References C.Reiter [Fracral Visualization and J] Jsoftware 2 SIDC Solar Influence Data Analsys Center(Bergium) 22

1 SHIMURA Masato polynomial irr.xirr EXCEL irr

1 SHIMURA Masato polynomial irr.xirr EXCEL irr 1 SHIMURA Masato 2009 12 8 1 2 1.1................................... 2 1.2 polynomial......................... 4 2 irr.xirr EXCEL 5 2.1 irr............................................. 5 2.2 d f, pv...........................................