FORECASTING USING MARKOV CHAIN

Transcription

1 13: Forecastg Usg Markov Cha FORECASTING USING MARKOV CHAIN Rat Kumar aul Ida Agrcultural Statstcs Research Isttute, New Delh Itroducto I a stochastc process {, =,1,2,. } X that takes o a fte or coutable umber of possble values f =, the the process s sad to be state at tme X. We suppose that wheever the process s state, there s fxed probablty that t wll ext be state. That s, we suppose that { X + 1 = X =, X 1 = 1,., X 1 = 1, X = } = { X + 1 = X = } = for all states,,.,,, ad all. Such a stochastc process s kow as a, 1 1 Markov Cha, ad the codtoal dstrbuto of ay future state X +1 gve the past states X, X 1,. X 1 ad the preset state X, s depedet of the past states ad depeds oly o the preset state. The matrx of oe-step trasto probablty deoted by 1 = s where,, ; = = 1, =,1,. Example 1 (Forecastg the Weather) Suppose that the chace of ra tomorrow depeds o prevous weather codtos oly through whether or ot t s rag today ad ot o past weather codtos. Suppose also that f t ras today, the t wll ra tomorrow wth probablty α ; ad f t does ot ra today, the t wll ra tomorrow wth probablty β. Defe the Markov cha ad oe-step probablty trasto matrx. 141

2 13: Forecastg Usg Markov Cha Example 2 (Trasformg a rocess to a Markov Cha) Suppose that whether or ot t ras today depeds o prevous weather codtos through the last two days. Specfcally, suppose that f t has raed for the past two days, the t wll ra tomorrow wth probablty.7; f t raed today but ot yesterday, the t wll ra tomorrow wth probablty.5; f t raed yesterday but ot today, the t wll ra tomorrow wth probablty.4; f t has ot raed the past two days, the t wll ra tomorrow wth probablty.2. Defe the Markov cha ad oe-step probablty trasto matrx. Example 3 (A Radom Walk Model) A Markov Cha whose state space s gve by the tegers =, ± 1, ± 2,. s sad to be a radom walk f, for some umber < p < 1,, + 1 = p = 1, 1, =, ± 1, ± 2,. Chapma-Kolmogorov Equatos The -step trasto probabltes that s the probablty that a process state wll be state after addtoal trastos, ad s defed by { + } = X = X =,,, k k + m m k = k k =,, m,, ( ) Let deote the matrx of -step trasto probabltes, the ( + ) ( ) ( ) = = m m m 142

3 13: Forecastg Usg Markov Cha Classfcato of States State s sad to be accessble from state f > for some Two states ad that are accessble to each other are sad to commucate ad we wrte The relato of commucato satsfes the followg three propertes: ) State commucates wth state, all. ) If state commucates wth state, the state commucates wth state. ) If state commucates wth state, ad state commucates wth state k, the state commucates wth state k. Two states that commucate are sad to be the same class, ad the Markov cha s sad to be rreducble f there s oly oe class, that s, f all states commucate wth each other. Example Cosder the Markov Cha cosstg of the four states, 1, 2, 3 ad havg trasto probablty matrx. Is ths Markov cha rreducble? = Def. For ay state we let process wll ever reeter state, f deote the probablty that, startg state, the { ( X = ) X } f = U =. State s sad to be = 1 recurret f f = 1 ad traset f f < 1. Remarks If state s recurret, the process wll reeter state aga ad aga, ftely ofte. 143

4 13: Forecastg Usg Markov Cha If state s traset, each tme the process eters state, there wll be a postve probablty, 1 f that t wll ever aga eter that state. Therefore, startg state, the probablty that the process wll be state for exactly tme perods equals f ( 1 f ), 1 1. I other words, f state s traset, startg state, the umber of tme perods that the process wll be state has a geometrc dstrbuto wth fte mea ( 1 ) 1. =1 State s recurret f =, traset f = 1 f <. Remarks State s recurret f ad oly f, startg state, the expected umber of tme perods that the process s state s fte. I a fte-state Markov Cha, all states caot be traset, ad at least oe of the states must be recurret. If state s recurret, ad state commucates wth state, the state s recurret. If state s traset ad commucates wth state, the state must also be traset. For f state were recurret ad commucate wth state, the state would also be recurret ad could ot be traset. Not all states a fte-state Markov cha ca be traset leads to the cocluso that all states of a fte rreducble Markov cha are recurret. Lmtg robabltes ) State s sad to have perod d f = wheever s ot dvsble by d, ad d s the largest teger wth ths property, ad a state wth perod 1 s sad to be aperodc. ) If state s recurret, the t s sad to be postve recurret f, startg state, the expected tme utl the process returs to state s fte, ad t ca be show that a fte state Markov cha all recurret states are postve recurret. ) ostve recurret, aperodc states are called ergodc. 144

5 13: Forecastg Usg Markov Cha Theorem For a rreducble ergodc Markov cha lm exsts ad s depedet of. Furthermore, lettg = lm, the s the uque oegatve soluto of = =, ad = = 1 Remarks Gve that Lettg, = lm exsts ad s depedet of the tal state, { } 1 { 1 } { } { } + = = + = = = = = X X X X X = = It ca be show that = =, the lmtg probablty that the process wll be state at tme, also equals the log-ru proporto of tme that the process wll be state. I the rreducble, postve recurret, perodc case we stll have the,, are the uque oegatve soluto of = =, = 1. must be terpreted as the log-ru proporto of tme that the Markov cha s state. Illustrated steps o Markov cha model developmet ad forecastg For smplcty, the Markov cha forecast model s dscussed by takg the partcular case of sugarcae (a oe-year growth perod crop) yeld forecastg. Let there be sx stages cludg the fal harvest stage, the stages havg bee determed o the bass of caledar dates.e. frst stage s 3 moths after platg, secod stage s 4 moths after platg etc. ad sxth stage at harvest. For developmet of model usg the avalable (say, for oe year) data, frstly costruct states wth stages o the bass of percetles upo the data pots o bometrcal characters, say, average plat heght, grth of cae, umber of plats per plot etc. at each of the frst fve stages ad upo (oly) yeld at fal stage. For brevty, let us cosder, oly oe bometrcal character, say, plat heght (X1). At stage 1, let the percetles, say, quartles for the data pots upo X11 (.e. X1 at stage 1) be a11, a12, a13. Thus four states vz. X11<=a11; 145

6 13: Forecastg Usg Markov Cha a11<=x11<=a12; a12<=x11<=a13; X11>a13 (classes deoted as a1, a2, a3, a4) ca be obtaed wth dstrbutos say, f11, f12, f13, f14. Smlarly wth quartles (a21, a22, a23) upo X12 four states wth stage 2 as X12<=a21; a21<=x12<=a22; a22<=x12<=a23; X12>a23 classes deoted as b1, b2, b3, b4) wth dstrbutos say, f21, f22, f23, f24 ca be obtaed. Lkewse four states of other stages ca be defed otg that the fal stage wll have te states because more fer percetles, vz., decles ca be used for formg states o the bass of varable Y(yeld) as ths s the ma character uder study. Now compute the trasto couts of data pots movg from ay state of oe stage to ay of the states of the succeedg stage. For example, to fd trasto probablty matrx (TM) from stage 1 to stage 2, to beg wth, cosder the f11 data pots state-1 (.e. a1) of stage-1 ad cout the correspodg data pots of X12 that satsfy the state codtos b1, b2, b3, b4 of stage-2. I ths way, the data pots whch are state-1 of stage-1 are redstrbuted to the dfferet states of the ext stage-2. Next cosder the data pots state-2 (.e. a2) of stage-1 ad cout the correspodg data pots of X22 that satsfy the state codtos b1, b2, b3, b4 of stage-2. The same procedure ca be followed for fdg other trasto couts as well to form a trasto frequecy matrx say ((ff )) where, =1,2,3,4 for stage-1 to stage- 2. Here, t s oted that the data pots of ay stage ted to rema the same states o trastg to the ext stage but t ca also be oted that they wll be redstrbuted to other (usually adacet) states of the ext stage as well. However such pheomeo may become less apparet whe more tha oe varable/ trasformed data are used. The TM from stage-1 to stage-2 s fally obtaed by dvdg each row elemet of ths matrx by ts correspodg row sum (these row sums are othg but dstrbutos at varous states of stage-1). Ths TM say A12 from stage-1 to stage-2 wll be a matrx of order 4. Other TMs ca smlarly be obtaed. Smlarly, TMs A23, A34, A45 wll be of order four ad A56 wll be of order (4x1). At stage-6, form a (1x1) vector, say, y m, o the bass of mdpots of class tervals of yeld. Now calculate the predcted yeld dstrbutos (YDs) whch wll be a (4x1) vector at each stage as gve here the table provded alogsde the text. At stage YD 1 A 12 A 23 A 34 A 45 A 56 y m 2 A 23 A 34 A 45 A 56 y m 3 A 34 A 45 A 56 y m 4 A 45 A 56 y m 5 A 56 y m I the year for whch yeld forecast has to be foud out, collect observatos upo a 146

7 13: Forecastg Usg Markov Cha sample of plots, formato upo varable X1 oly for the same set up of stages 1 through 5. Classfy these observatos as per states of stages of the model developed already. Ths wll result 'weghts' that fall varous states of ay partcular stage of the developed model. These values at the varous stages for the dfferet states for the Markov cha model ca be wrtte as states () () () (v) Stage-1 w11 w12 w13 w14 Stage-2 w21 w22 w23 w24 Stage-3 w31 w32 w33 w34 Stage-4 w41 w42 w43 w44 Stage-5 w51 w52 w53 w54 Note that the row sum here should be the umber of data pots at each stage the year for whch forecasts has to be obtaed. The mea yeld forecast at ay partcular stage ca be obtaed as the weghted average of the YDs (four umber) at that stage, weghts (four umber) beg the umber of observatos of the forecast year dfferet states ( the partcular stage) of the developed model. Thus the developed model ca be used practce for crop yeld forecastg. ractcal exercse o Markov cha based forecast modelg The data utlzed the study have bee take from the proect lot study o pre harvest forecast of yeld of sugarcae carred out by IASRI, New Delh Two years data havg uform recordg of bometrcal characters was cosdered. I all, 144 plots data were avalable frst year whereas 156 plots data were avalable secod year. The bometrcal character used s : average plat heght per plot (X1). The varous stages of observatos o X1 are 3 4, 4 5, 5 6, 6 7 ad 7 8 moths after platg. At harvest, the actual yeld.e. weght of caes per plot (Y) were also avalable. Let the stages of the basele data be deoted by s1, s2, s3, s4, s5 ad the fal(harvest) stage be deoted by s6. For the sake of smplcty, a FOMC model s take as a llustrato to dscuss about the varous steps volved developg the model:-the frst year data dfferet stages have bee classfed to dfferet plat codto classes (states) for the bometrcal character X1 (average plat heght) of the Markov cha model take at stage (=1,2,3,4,5). 147

8 13: Forecastg Usg Markov Cha Dataset-5-1 lot No. X11 X12 X13 X14 X15 Y Dataset-5-2 lot No. X11 X12 X13 X14 X Cosder Dataset-5-1. Here X1 (average heght metres).at stages =1 to 5 & Y yeld kg/ha) at harvest stage 6.of sugarcae crop. erform the followg tasks. 1. ercetles of varables:- Calculate quartles for the varables X11, X12, X13, X14 ad X15 ad decles for Y. (ht:- ERCENTILE (<A1:A144>,.75) wll gve thrd quartle) 2. Formato of states wth stages () Isert a ew worksheet ad copy the colum of X11 t. ()Classfy the varable X11 to ether group umber '1' f X11<=Q1 or '2' f Q1<X11<=Q2 or '3' f Q2<X11<=Q3 or '4' f X11>Q3 four subsequet adacet colums ad '' otherwse where Q1, Q2 ad Q3 are the quartles of X11. (ht:- IF(B1<=Q1,1,); IF(B1>Q1*ANDB1<Q2,2,) ad so o) Sum these four colums rowwse to get the states as 1,2,3,4. Let ths colum be amed Z1. () Classfy the varable X12 to four states by followg the same procedure as dscussed 2() above but ths tme usg the quartles of X12. Let the resultat columbe amed Z2. 3. Computato of TMs () Copy the colums Z1 ad Z2 obtaed 2() ad 2() a ew worksheet. () Calculate 4x4 frequecy matrx by formg 16 colums ad fdg colum totals. For f11 - IF( (C1=1)*AND(C2=1),1,) For f12 - IF( (C1=1)*AND(C2=2),1,) For f13 - IF( (C1=1)*AND(C2=3),1,) For f14 - IF( (C1=1)*AND(C2=4),1,) For f21 - IF( (C1=2)*AND(C2=1),1,). For f43 - IF( (C1=4)*AND(C2=3),1,) For f44 - IF( (C1=4)*AND(C2=4),1,) ad by arragg a matrx {f11,f12,f13,f14; f21,f22,f23,f24; f31,f32,f33,f34; 148

9 13: Forecastg Usg Markov Cha f41,f42,f43,f44;} fd row sums of ths matrx as f1, f2, f3,f4. Dvde frst row of ths matrx by f1, secod row, by f2 etc., to get the TM of stage-1 to stage-2.e. A12. () Cosder the frequecy matrces of stage- to stage-(+1) for =2,3,4,5 as gve below ad compute correspodg TMs A23, A34, A45, A56.by dvdg each row by ther row totals. F23={3, 7,, 1; 8, 21, 5, 1;, 7, 23, 6;, 1, 9, 25} F34={3, 7, 1, ; 6, 23, 7, ;, 7, 19, 11;,, 9, 24} F45={29, 6, 1, ; 8, 21, 6, 2; 1, 8, 23, 4;,, 5, 3} F56={1, 7, 1, 5, 2, 1, 1,,, 2; 4, 5, 2, 6, 6, 3, 3, 3, 3, ;, 1, 2, 3, 5, 6, 7, 6, 3, 2; 1, 1,, 1, 1, 4, 4, 5, 8, 11} 4. Meas of redcted Yeld Dstrbutos (YDs) () Fd mmum ad maxmum values of Y ad usg the decles obtaed task 1 form the class-frequecy table, to calculate the mdpots of the class tervals as a 1x1 vetor, say, Ym. () Fd mea of YDs as At stage 5:- ys5=a56 * Ym At stage 4:- ys4=a45*a56*ym At stage 3:- ys3=a34*a45*a56*ym At stage 2:- ys2=a23*a34*a45*a56*ym At stage 1:- ys1=a12*a23*a34*a45*a56*ym 5. Mea yeld forecastg () Cosder a ew set of data (see 'dataset6-2') of 156 observatos upo varables X11, X12, X13, X14, X15 at fve stages 1,2,3,4,5 respectvely. () Classfy the observatos upo varable at stage 1.e. X11 as per the 'states' of the stage-1 of the 'ftted' model.e. accordg to the quartles obtaed for dataset6-1 data obtaed task 1.Cout the umber of observatos fallg state-1 of stage-1, the state-2 of stage-1 ad so o ad deote these as weghts w11, w12, w13, w14.. Lkewse compute the weghts w21, w22, w23, w24 at stage 2 etc. () Forecast at each stage (=1,2,3,4,5), by takg weghted average usg the correspodg four weghts from 5() ad the (4x1) vector ys1 at stage1, ys2 at stage2 etc. from 4() to obta mea yeld forecasts at each stage. 149

10 13: Forecastg Usg Markov Cha Selected readgs Leug ad Shag (1989). Modelg raw producto maagemet system: A Dyamc Markov decso approach, Agrcultural Systems, 29, 5 2. Mats, J.H., Sato, T., Grat, W.F. Iwg, W.C. ad Rtche, J.T. (1985). A Markov cha approach to crop yeld forecastg, Agrcultural Systems, 18, Mats, J.H., Brkett, T. ad Boudreaux, D. (1989). A applcato of the Markov cha Approach to forecastg cotto yeld from surveys, Agrcultural Systems, 29, Ramasubramaa V. ad Ja, R.C. (1999). Use of growth dces Markov cha models for crop yeld forecastg, Bom. J., 41, Sgh, Radhr ad Ibrahm, A.E.I. (1996). Use of spectral data Markov cha model for crop yeld forecastg, Jour. Id. Soc. Remote Sesg, 24,