Expert-Statistical Processing of Data and the Method of Analogs in Solution of Applied Problems in Control Theory

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1 Proceedgs of he 7h World Cogress The Ieraoal Federao of Auomac Corol Exper-Sascal Processg of Daa ad he Mehod of Aalogs Soluo of Appled Problems Corol Theory Alexader S. Madel. Alese G. Belyaov. Dmry A. Semeov Trapeov Isue of Corol Sceces RAS, 65, Profsoyuaya sr., 7997, Moscow Russa (Tel , e-mal: Absrac: The paper provdes a oule of he exper-sascal approach o developg corol ad defcao sysems. A exper-sascal mehod of daa processg desged for forecasg shor me seres s dscussed deal. New adapve algorhms for veory corol are descrbed. The usage of hese algorhms appled exper-sascal sysems o suppor decso mag process s dscussed.. INTRODUCTION I he md 990s he Isue of Corol Sceces RAS (ICS a ool for egrao of heerogeeous formao wh he same corol sysem referred o as expersascal mehods (ESM for daa processg was proposed (Madel, 996, 997. Ma applcaos of he ew ool have bee made o socal-ecoomc sysems ad orgaao corol sysems. I 006 he ESM applcaos also cluded problems of corol ad defcao of echologcal processes ad egeerg objecs (Madel, 006. A bref roduco of he exper-sascal approach o daa processg ad exper-sascal corol sysems s gve. The ESM of aalogs frs descrbed (Belyaov e al, 00a ad s poeal applcaos o he soluo of defcao ad corol problems are dscussed. New adapve veory corol algorhms are suggesed as a basc model wh he appled exper-sascal veory corol sysem... Bacgroud. EXPERT-STATISTICAL APPROACH TO DATA PROCESSING Bloc-dagram of he exper-sascal sysem (ESS s gve Fg. Corol objec Basc model u The ESS cosss of hree ma us: basc model u, u of collecg ad processg of exper formao, ad u for egrao of exper daa o he basc model. For varous ad may examples of applcao of ESS o he soluo of problems of orgaao corol, veory corol ad mareg see (Madel, 996, 997, 006. Of a parcular eres s he u for egrao of exper daa o he basc model. I s he srucure ad he algorhmc coe of hs u ha predeerme a possbly of he successful soluo of he respecve corol problem. I he subseco below we wll brefly dscuss he srucure of hs u... U for egrao of he exper daa o he basc model The u of egrao of exper daa o he basc model coas a se of procedures (echques of ervewg, quesoare surveys, ec. for exracg exper owledge ad geerag hereby a sor of exper sysem (ES. I mos cases, however, coras o he coveoal sysems usg owledge, he ES he ESS are as a rule much more sraghforward. The po s ha ESS are developed for he corol objecs wh regard o whch oe ca be que cera ha a pror ad a poseror formao avalable o he desgers ad users o he ways of her fucog ad characerscs s suffce for developg a early complee model of he sysem descrbed. The crero of objecve daa beg almos suffce ( ca well be llusory s he smalless of he resdual dsperso for a prelmary sechy model of he sysem whch s formed a he sage of he prelmary survey usg a pror daa. Expers Corol sysem Sascs ESS Fg.. Bloc-dagram of he ESS. U for egrao of exper daa o he basc model U for exper daa colleco ad processg Regreably, umerous aemps faled o oba accurae esmaes of he resdual dsperso or smlar characerscs whch would srcly verfy us beg he doma where exper-sascal approach s applcable. Obvously wh small values of he resdual dsperso oe ca mos lely choose bewee coveoal mehods of sascal defcao ad ESM. Moreover, wh he growg resdual dsperso oe has o decde o he borderle bewee he /08/$ IFAC / KR

2 exper-sascal approach ad he purely exper mehods whch would eveually ed up ES. The codoaly of ay src coclusos sems from he fac ha eve wh small values of resdual dsperso (obaed he learg sample! here are o guaraees ha a decso maer (DM would carry ou he recommedaos based o he defed model. Ths falure o obey may mas boh a msae of DM, or lac of cofdece, or hs belef he sample beg o sascally represeave (does o coa formao o all modes of fucog of he objec descrbed. A brgh llusrao of he above s he suao wh compuerao of complex processes such felds as chemsry, perochemsry, peroleum processg ad may ohers (see, for sace, (Doorcev e al, 003; Kasav, 97. I hs case classcal successful aemps of auomao ad he precedg defcao sages bol dow o applcao of wo basc approaches. The former reles o obag a se of complex olear sac models usg he pecewse approxmao echque (Kasav, 97. Ths approach ca be used for a easy modelg of he mulmode objecs o a all reflecg her dyamcs. The laer approach (whch, amog oher hgs, s also used for developg smulaors (Doorcev e al, 003 uses he descrpo of he process va a se of local physcal-chemcal model wh he subseque her egrao o a global model respecg he geomery ad physcal-mechacal properes of rasporao ools coecg local processes. A frs, hs approach compared o he former oe seems o be more adequae o he real process, however, does o hadle he mulmodaly feaure equally well. To mae more adequae he smulao expermes are doe for he process volved wh he subseque preseao of he smulao resuls o he process egeers. Havg suded he smulao resuls he process egeers provde her commes ellg how her exper opo he process should have behaved he respecve suaos. Ther recommedaos are preseed as a sor of he ES whch subsues he creaed se of physcalchemcal ad physcal-mechacal models he crcal suaos defed by exper process egeers. As a alerave (or addo o he above approaches oe ca use he exper-sascal approach. 3.. A example of applcao Cosder he above argumes appled o he laes verso of he exper-sascal veory corol sysem ADAPIN ( ADAPve INveores, he frs verso of he ESS ADAPIN was descrbed Boreo e al., 990. ESS ADAPIN s desged o sudy he paer of chage he demad for commodes, esmae he soc-ou probably, ad forecas requess for repleshg he soc. Order ses geeraed by he ADAPIN sysem mae possble o mee he servce requremes specfed by he user. sochasc behavor ad rsc uceray of he demad sascs s offered by adapve corol schemes. To solve he veory plag (schedulg problem, whch meas, geerae scheduled requess for he whole perod of plag (a year, a quarer, a moh a he level of a major warehouse oe may use he followg adapve algorhm (Loosy e al, 987: ˆ 0 x + = x G( p r, ( ˆ + where x s a plaed value of he carryover soc a he begg of he ex plag perod, x s a real value of he al soc vecor he prevous plag perod, p s a vecor (commody ype-wse of he socou probably he prevous perod, r 0 s a vecor of servce levels, whle G s a marx of coeffces a he -h sep whch mees codos: G =, G <. ( For veory corol he ESS ADAPIN uses so-called myopc, paramerc wo-level (S, s-veory corol sraeges. I hs case adapve algorhms for recalculao of he sraegy parameers ca be wre as (Loosy e al, 99: S s + + where = S [ γ (sg( x ' = s γ ( rs T s sg( x r], Q ( x + B], f y 0, 0 f y > S, y < s, sg( y = 0 f y = 0, Q ( y = f, - f 0; y s y S y < x s he soc a he -h sep, s demad a he -h sep, r s he servce level, B s he parameer whch s a fuco of supply ad sorage coss, whle coeffces {γ } ad {γ } mee he codos g э ' γ = γ =, γ <, ( γ <.. (4 g Le K be a umber of seps he plag perod. The ESS ADAPIN uses wo models o evaluae compoes of p : K + ( ( = K = pˆ, (5 (3 A obvous mechasm for veory corol he coex of K x + where = max( 0, x, = K =, ad 38

3 K + ( ( x pˆ = sg(, (6 K = y > 0, where sg ( y = 0 y 0. I he decso mag process a exper s provded wh he resuls of sascal processg, whereas decsos made by he exper are reaed as feedbac he exper-sascal sysems for veory corol. I ESS ADAPIN he user may ( ( choose bewee esmaes ˆp ad ˆp, he also may updae he order se for repleshg he soc. ˆ + Tha s o say, f x s he soc se recommeded by he exper-sascal sysem for he ex sep, whle he exper-user specfes he order se assumg ha he soc should equal x + = xˆ + + Δx, (7 he, usg daa o he exper adjusme Δ x, he model parameers, specfcally, he coeffces {g }are updaed. The ma adapve algorhm s a oe-dmesoal aalog of algorhm (, where he coeffces γ are descrbed as μ x γ =, (8 ν + where μ ad ν are parameers whch may be a fuco of me. Now f he exper-made adjusme Δ x a he -h sep mees (7, he follows from (, (7 ad (8 ha μ x x ( p r xˆ + x, (9 Δ = o + ν + whch s o be reaed as a equao varables μ ad ν. I s exacly he equao ha s solved he ADAPIN sysem whe recalculag he parameers μ ad ν, ad whe he (0 ( (0 ( followg cosras hold: μ μ μ, ν ν ν, where (0 ( (0 ( (0 μ, μ, ν, ν 0, ad μ 0, chose o mee he covergece requremes (4. 3. METHOD OF ANALOGS IN PREDICTION OF SHORT TIME SERIES 3.. Bacgroud I a suffcely geeral case, he me seres ca be represeed by he sum of hree compoes: sysemac compoe, red; relavely smooh oscllaos abou he red whch occur wh a greaer or lesser regulary ( parcular, he seasoal effec; 3 radom (called also somemes osysemac or rregular oscllaos. Tradoally, he sascal mehods of me seres predco mosly come o decomposg he observao sequece, predcg each s compoe, ad mergg he dvdual predcos (Box e al, 970. Obvously, sascally relable predco of me seres s possble (beg rus oly f he predco base perod, ha s, he umber of he ow values of he me seres, s suffce o draw relable coclusos abou he me profle of each compoe. Sascal aalyss suggess ha order o ae care fully o accou all compoes he predco base perod should coa several hudreds of us. For perods of several es of us, sasfacory predcos ca be cosruced oly for he me seres represeable as he sum of he red, seasoal, ad radom compoes. Wha s more, such models mus have a very lmed umber of parameers. Seres made up by he sum of he red ad he radom compoe somemes may be predced for eve a smaller base perod. Fally, for a predco base perod smaller ha some calculaed value N m, a more or less sasfacory predco o he bass of observaos s mpossble a all, ad addoal daa are requred. The value of N m s defed by he desred predco accuracy, s maxmum horo, red aure (model, ad he radom compoe of he me seres. For he gve requremes o predco, we refer o he me seres as shor f s base (observao perod s smaller ha N m. The shor me seres are represeable as he sum of red ad radom compoe. For he shor me seres, dealed sudy of he properes of he radom compoe maes o sese because for small base perods he sascal coclusos prove o be suffcely relable. However, he radom compoe ca o be compleely dscarded because s value shows he msmach bewee he acual values of he me seres over he predco base perod ad hose calculaed from he model. The msmach ca be used o specfy he predco based o he expers opos. I hs case, he decso maers have a her dsposal a very lmed (frsly ofe hollow measureme sascs ad ca resor o he help of exper or exper eam. If he observao sample s lmed ad s formao s oo sca for relable esmao ad predco, he s advsable o ue all he objecve (sascs, measuremes ad subjecve (exper formao avalable o he DMs or, saed dfferely, o mae use of he exper-sascal approach (Madel, 996, Aalog mehod he problems of predco The aalog mehod proceeds from he assumpo ha some owledge domas he expers ry o predc he me seres o he bass of her coceps of objecs or processes 38

4 whose prehsory hey ow well. I s also assumed ha he umber of hese objecs or processes s suffcely grea ad he arbue space of he objecs mag up he core of expers professoal experece yelds o ea or as s ofe he case fuy classfcao (Bauma e al, 98, 999; Bauma, 988. I formal erms, hs meas ha s possble o cosruc algorhms (recurre, parcular o deerme he exremum of he gve merc fucoal ( M p D = p χ, (0 where χ s a covex fuco, A are he po classes, p are he a pror probables of he classes A, ad M are he frs uormaled momes of he classes A. A ha, membershp a class, f ea, s esablshed by a characersc fuco assumg o he objec belogg o he fxed class or 0, oherwse, or by he membershp fuco h ( x : 0 h ( x wh ( ( x =, f fuy. h Ths meas fac ha he exper experece ca be srucured a sese. Geerally speag, however, here s o obvous eed for such classfcao he objec arbue space because he expers are free o maage her experece ad he scheme of auomac classfcao s jus a formal model of he space a had. Neverheless, as wll be see below, he mehods of auomac classfcao ca prove o be very useful for predco. Srucurg ad aalyss of her ow experece eable he expers o geerae for each of he ewly preseed me seres (called below he predco objec, PO a ls of prevously observed objecs ha from her po of vew are aalogs of he PO. The PO preseed o he exper s a segme of a me seres of legh N: y(, = 0,,,, N, ( a specal case of o daa sample, N ca be ero. I respose o he preseed daa sample (ad/or revealed PO, he exper lss aalog objecs represeed he predco sysem daa base by complee me seres, ha s, seres of leghs cosderably exceedg N. Le Z be he se of he umbers of he aalog objecs dcaed by he exper. The exper has he rgh o he oblgao o defe wo more umercal characerscs for each objec: he smlary coeffce l, Z, (by defaul assumed o be uy ad he scale coeffce s, Z, (by defaul assumed o be uy Procedure of Predco by he Aalog Mehod Ieraco of he exper ad he exper-sascal predco sysem (ESPS provdes he se Z of aalogs of he PO uder cosderao. For hs se, he ESPS daabase coas formao abou complee, ha s, represeed by much loger me seres, realaos of operao of he aalog objecs. Ths formao s represeed by he colleco { x (, Z, = 0,,,..., N} where N >> N. Addoally, he ses of values of he smlary, { l, Z}, ad scale, { s, Z}, coeffces are gve. To predc he values of he PO me seres a he sa, > N, he followg formula ca be used ow: y ˆ( = L l s x( Z where L = α, ( l Z. For N > 0, he values of he coeffces α, Z, ( are esablshed from: m { α, Z } L N = Z α l s x(. ( If N = 0, ha s, f here s o daa sample o PO a all, he all α, Z, are assumed o be equal o Aalog mehod: felds of applcao ad praccal recommedaos I s recommeded o use he exper-sascal predco procedures based o he aalog mehod f: here s o sascal formao abou he PO or predco ca be based oly o he subjecve formao; he exper for some reasos s uwllg or fds dffcul o reveal he erval or po esmaes of he fuure values of me seres; here s exper formao abou he PO ha allows oe o classfy (defy wh oe or aoher smlary class; here exss a represeave se of sascal formao abou a subsaal umber of objecs from he gve owledge doma Aalog mehod: relably esmae Esmao of relably (rus of he aalog mehod (Madel, 000 requres may expermes because he degree of rus of he predcos geeraed o he bass of very shor, somemes lacg, samples depeds fac o he exper s compeece ad o he performace of he decso suppor ESPS. The resuls of some expermes wh such he EXPAM sysem ha was developed a Trapeov Isue of Corol Sceces are descrbed (Belyaov e al, 00a. As was oed hs paper, large-scale expermes wh such sysems are very dffcul because hey dsrac hgh-pad slled expers for a log me. Therefore, he followg wo varas of acos o esmae relably ad effecveess of he decso suppor ESPS are possble: 383

5 careful loggg ad aalyss of he resuls of roducg o pracce ad rug such sysems; desg of smulao sysems for he decso suppor ESPS s where compuers smulae behavor of expers eracg wh he ESPS. Oe of such smulao sysems, EXPRIM, was desged a Trapeov Isue of Corol Sceces. I hs sysem, a sxee-parameer model of exper behavor was realed. I smulaes varous levels of professoalsm ad psychologcal ypes of expers dealg wh predco of demad for ew producs o he bass of he EXPAM decso suppor ESPS. The reader s referred o (Belyaov e al, 00a, b. for a dealed descrpo of he expermes wh he EXPRIM ad EXPAM sysems. For he case of hollow (! daa sample, we jus prese oe revealg graph (see Fg. of he predco accuracy vs. exper professoalsm varyg from he lowermos (abscssa s o he uppermos ( ward exper, abscssa s Fg.. The rms predco error devao for he frs po vs. exper professoalsm. The accuracy of predco proved o be suffcely hgh already a he frs po (! of he fuure seres (we recall ha he sample s empy a all for he frs po. We draw aeo o he fac ha he smulao experme eve he expers who acually have he ero level of professoalsm (professoalsm parameer ad choose aalogs hggledy-pggledy correced choce of aalogs by meas of he EXPAM sysem so ha a he frs sep her predco accuracy was que passable 60 %. 4. ADAPTIVE ALGORITHMS FOR INVENTORY CONTOL AND THEIR APPLICATIONS 4.. Bacgroud Several groups of adapve algorhms have already bee developed for he soluo of varous problems he veory corol heory, see, for sace (Loosy e al, 987, 99; Belyaov e al, 005. Ially hese have bee algorhms eded for sysems o corol supples by he crero of meeg he specfed level of servces provded o cosumers of ype que close o algorhm (: [ ˆ π ρ] xˆ = γ, (3 + x where x ˆ + s a esmae of he recommeded soc a he (+-h sep, x s a real soc a he -h sep, ρ s he specfed servce level, πˆ s he esmae of he probably = of o shorage a he -h sep, whle { γ } s a sequece of o-egave coeffces meeg he ow codos (4. Nex syhesed were adapve algorhms o solve he problem of so cold myopc veory corol (for oe sep plag perod (see algorhm ( New adapve algorhms Whe veory corol s doe a mul-sep process usg he crero of he mmal oal average cos a opmal sraegy of choosg he order se belogs as a rule (Hadley e al, 969 o he class of (R, r-sraeges. I s assumed ha he dsrbuo fuco F(x of he demad ξ for oe sep s a pror uow ad durg he fucog of he supply sysem a sequece of demad values s regsered ξ, ξ,, ξ. For a saoary fucog mode of he veory sysem he approxmae recurre algorhms have bee obaed as (see Madel e al, 008: rˆ + + = + ( c + ( ' γ [( c + h ˆ '' = rˆ + γ [( с ( / + ( h + d η (, rˆ ; ξ d( {( A + ( c + h + d + rˆ ( с dˆ ( A+ ( c + h + d + rˆ ( c d { / + ( h + d η(, rˆ ; ξ d( / }], ( A+ ( c + h cr + ( c h /}], (4 (5 where A s a cosa cos of placg a order, c s he prce of u veory, h s he u cos of veory holdg, d s he u shorage cos, ' '' γ ad γ are coeffces meeg he codos of (5, η( R, r; ξ s he fuco of he ype ( R r / ξ ( R r f ξ r, η( R, r; ξ = ( R ξ / ξ( R r f r < ξ R, (6 0, f R ξ. whle recurre esmaes of he average demad value ad of he secod mome are obaed from he formulas: 384

6 = ( / + ξ /, (7 = ( / + ξ /. ( Adapve exper-sascal sysems Whe mplemeg algorhms (4 (8 wh he expersascal veory corol sysem he expers ca correc esmaes of parameers ad rˆ, as well as he mpac o he coeffces of adapve algorhms (4 ad (5 by specfyg he parameers λ ad μ, =,, he formula whch resembles (8: ' '' γ = λ /( μ + ad γ = λ /( μ +. (9 5. CONCLUSIONS The exper-sascal mehod mples ha expers corbue o he soluo of he forecasg problem based o he defcao by he expers of he aalogs of he process forecased amog he processes ha hey have observed earler. I s assumed ha for he earler observed processes here exss raher represeave sascal formao whch ca be uled alog wh raher lmed sascal maeral drecly wh regard o he process forecased. Oe of he ma mahemacal ools of ESS basc models creag are adapve or robus models. New adapve veory corol algorhms uder crera of he mmal oal average cos a plag perod are he example of such basc model. 6. AKNOWLEGMENT Ths wor s parally suppored by Russa Foudao for Basc Research, projec a. REFERENCES Bauma, E.V. (988. Mehods of Fuy Classfcao (Varaoal Approach. Auomao ad Remoe Corol, v. 49, No.. Bauma, E.V. ad A.A. Dorofeyu (98. Recurre Algorhms of Auomac Classfcao. Auomao ad Remoe Corol, v. 43, No. 3. Bauma, E.V. ad A.A. Dorofeyu (999. Classfcao Daa Aalyss. I: Ibraye Trudy Mehduarodo Koferes po Problemam Upravleya (I. Cof. o Corol Sceces, Seleced Papers, v.. SINTEG, Moscow ( Russa. Belyaov, A.G. ad A.S. Madel (00a. Predco of Tme Seres o he Bass of Aalog Mehod (Elemes of he Theory of Exper-Sascal Sysems. Prepr of Isue of Corol Sceces, Moscow ( Russa. Belyaov, A.G. ad A.S. Madel (00b. Aalyss of Relably of Coclusos Geeraed by Meas of Exper-Sascal Sysems. Prepr of Isue of Corol Sceces, Moscow ( Russa. Belyaov, A.G., A.V. Lap ad A.S. Madel' (005. Iveory Corol for Goods Speculave Demad. Problemy Upravleya, v. 6. ( Russa. Boreo N.I., V.A. Loosy ad A.S. Madel' (990. Exper Sascal Sysems for Demad Predco ad Iveory Corol. I: Evaluao of Adapve Corol Sraeges Idusral Applcaos - IFAC Worshop Seres, No. 7. Pergamo Press, Oxford. Box, G.E.P. ad G.M. Jes (970. Tme Seres Aalyss Forecasg ad Corol. Holde-Day, Sa Fracsco. Doorcev, V.M. ad D.V. Keller (003. Sadard Trag Smulaor for Trag Process Operaors. Avomaasya v Promyshleos, ( Russa. Hadley, G. ad Т.M. Wh (969. Aalyss of Iveory Sysems. Prece Hall, Ic. Eglewood Clffs, New Jersey. Kasav, A.D. (97. Adapve Mehods of Pecewse Approxmao he Problem of Idefcao. Auomao ad Remoe Corol, v. 33, No.. Loosy, V.A. ad A.S. Madel' (987. Adapve Iveory Corol. I: Preprs of X IFAC World Cogress (Much, v. 0, subv. 6. DI/VDE, Dusseldorf. Loosy, V.A. ad A.S. Madel' (99. Models ad Iveory Corol Techques. Naua publshers, Moscow ( Russa. Madel, A.S. (996. Exper-Sascal Sysems Corol ad Iformao Processg. Par I. Prbory&Ssemy Upravleya, No. ( Russa. Madel, A.S. (997. Exper-Sascal Sysems Corol ad Iformao Processg. Par II. Prbory&Ssemy Upravleya, No. ( Russa. Madel, A.S. (000. Mehods for Improvg Relably of Coclusos Exper-Sascal Sysems. I: Trudy Isua Problem Upravleya. Isue of Corol Sceces, v. 0, Moscow, ( Russa. Madel, A.S. (004. Mehod of Aalogs Predco of Shor Tme Seres: A Exper-sascal Approach. Auomao ad Remoe Corol, v. 65, No. 4. Madel, A.S. (006. Exper-sascal Mehods of Daa Processg Iegraed Produco ad Process Corol Sysems. Problemy Upravleya, No. 6 ( Russa. Madel, A.S. ad D.A. Semeov (008. Adapve Algorhms for Esmao of Opmal Iveory Corol Sraegy Parameers uder Resrced Soc-Ou. Auomao ad Remoe Corol, v. 69, No. 5 ( pr. 385