MULTIDIMENSIONAL HETEROGENEOUS VARIABLE PREDICTION BASED ON EXPERTS STATEMENTS. Gennadiy Lbov, Maxim Gerasimov

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1 Iteratoal Boo Seres "Iformato Scece ad Computg" 97 MULTIIMNSIONAL HTROGNOUS VARIABL PRICTION BAS ON PRTS STATMNTS Geady Lbov Maxm Gerasmov Abstract: I the wors [ ] we proposed a approach of formg a cosesus of experts statemets for the case of forecastg of ualtatve ad uattatve varable I ths paper we preset a method of aggregatg sets of dvdual statemets to a collectve oe for the geeral case of forecastg of multdmesoal heterogeeous varable Keywords: multdmesoal varable expert statemets coordato ACM Classfcato Keywords: I6 Artfcal Itellgece - owledge acusto Coferece: The paper s selected from IV th Iteratoal Coferece "Kowledge-alogue-Soluto" KS 008 Vara Bulgara ue-uly 008 Itroducto Let Γ be a populato of elemets or obects uder vestgato By assumpto L experts gve predctos of values of uow m-dmesoal heterogeeous feature for obects a Γ beg already aware of ther descrpto (a) We assume that ( a) = ( ( a) ( a) ( a)) ( a) = ( ( a) ( a) m ( a)) where the sets ad may smultaeously cota ualtatve ad uattatve features = ; or doma of the feature = m ; respectvely Let be the doma of the feature = = m The feature spaces are gve by the product sets: = = be the m = = By assumpto exactly combato of values ( a) ( a) m ( a) s mportat so we have to estmate the whole set smultaeously We shall say that a set s a rectagular set f = = a uattatve feature s a fte subset of feature values f rectagular sets are defed ad = [ α β ] f s s a omal feature I the same way G Fg The wor was supported by the RFBR uder Grat N a

2 98 Algorthmc ad Mathematcal Foudatos of the Artfcal Itellgece I ths paper we cosder statemets ( a) G where S s a rectagular set each statemet S has ts ow weght measure of cofdece = M ; represeted as seteces of type f G s a rectagular set ( a) the (see Fg ) By assumpto w ( 0 < w for dvdual statemets) Such a value s le a Let us remar that the statemet f ( a) the ( a) s eual to the statemet I ow othg about (a) f ( a) Wthout loss of geeralty we may assume that experts themselves have eual weghts Settg of a Problem We beg wth some deftos eote by : = = = ( ) where s the Cartesa o of feature values ad for feature ad s defed as follows Whe s a omal feature s the uo: = Whe s a uattatve feature s a mmal closed terval such that (see Fg ) Fg I the wor [3] we proposed a method to measure the dstaces betwee sets (eg ad ) heterogeeous feature space Cosder some modfcato of ths method By defto put ρ ( ) = = ρ ( ) or ρ ( ) = = ( ρ ( )) where 0 = = Values ρ ( ) are gve by: Δ ρ ( ) = f s a omal feature r + θ Δ α + β α + β ρ ( ) = f s a uattatve feature where r = It ca be proved that the tragle eualty s fulflled f ad oly f 0 θ The proposed measure ρ satsfes the reuremets of dstace there may be Note that we ca use aother measure of dffereces (for example see [4]) I ths paper we assume that dstace betwee rectagular sets s ow Cosder some atural algorthm of formg a cosesus of experts statemets (deote t by A )

3 Iteratoal Boo Seres "Iformato Scece ad Computg" 99 Let for some pot x we have two statemets S ad S wth the weghts w ad w Suppose G ad G are the mages prescrbed by these statemets to the pot x If ρ ( G G ) < ε where ε s a threshold the t may be assumed that the set G G s aturally prescrbed to the pot x Note that f these statemets are gve by dfferet experts the we more cofdece resulted statemet so the weght of ths statemet s hgher tha w ad w (t may be eve more tha ) Otherwse f ρ( G G ) ε the t may be assumed that oly oe statemet wth hgher weght s remaed ad our cofdece t (ad the weght of t) s decreased If for some pot x we have more tha two statemets the algorthm A coordates them the same way M Sce there are M statemets we have up to sets wth dfferet prescrbed mages These sets are the form of or \ ( 3 ) where are rectagular sets Cosder algorthms B of formg a cosesus of experts statemets uder restrctos o amout of resulted statemets The value F( B) ( ρ ( G ( x) G ( x)) dx estmates a ualty of the algorthm B Here = A B G A (x) G B (x) are the mages prescrbed to the pot x by algorthms A ad B respectvely I the * geeral case the best algorthm B = arg m B F( B) s uow Further o the heurstc algorthm of formg a cosesus of experts statemets s cosdered Prelmary Aalyss We frst treat each expert s statemets separately for rough aalyss Let us cosder some specal cases Case ( cocdece ): max max( ρ ( ) ρ ( )) < δ ad ρ ( G G ) < ε where δ ε are thresholds decded by the user { M} I ths case we ute statemets S ad S to resultg oe: f ( a) the ( a) G G Case ( cluso ): m( max( ρ ( )) max( ρ ( ))) < δ ad ρ ( G G ) < ε where { M} I ths case we ute statemets S ad S too: f ( a) the ( a) G G Case 3 ( cotradcto ): max max( ρ ( ) ρ ( )) < δ ad ρ ( G G ) > ε where ε s a threshold decded by the user { M} I ths case we exclude both statemets S ad S from the lst of statemets Coordato of Smlar Statemets ml Cosder the lst of l -th expert s statemets after prelmary aalyss Ω ( l) = { S ( l) S ( )} eote by Ω = L l = Ω( l) M = Ω eterme ow dstace betwee rectagular sets ad G correspods to far sets ad the the feature eterme values l from ths reaso: f far sets G s more valuable tha aother features

4 00 Algorthmc ad Mathematcal Foudatos of the Artfcal Itellgece hece value M M u = v = s hgher We ca use for example these values: u v u v τ = ρ( G G ) ρ ( ) = eote by : ( r = d ) The value d ( F) s defed as follows: d( F) = max (see Fg 3) dam( ) = max ρ( x y) xy ' \ F ' m dam( ) τ = τ = where where ' s ay rectagular set (') 3 (') (') Fg 3 By defto put { }{ } v I = I = {{ r u } δ ad { M v ρ ( G G u ) < ε u v = } where δ ε are thresholds decded by the user = Q ; Q M Let us remar that the reuremet u v uv u v r δ s le a crtero of sgfcace of the set \( ) Notce that someoe ca use aother value d to determe value r for example: m( dam( F ') dam( F) dam( G ') dam( G)) d( F G) = max ' \( F G) dam( ) Further tae ay set = { } of dces such that I ad Δ= Q +Δ I+Δ Now we ca aggregate the statemets S +Δ S = f ( a) the ( a) G where By defto put to the statemet S the weght S to the statemet = G S : = G G c w w = where c = ρ( ) c The procedure of formg a cosesus of sgle expert s statemets cossts aggregatg to statemets for all uder prevous codtos = Q Let us remar that f for example < the the sets ad (see Fg 4) are more sutable to be uted (to be precse the relatve statemets) tha the sets F ad F uder the same aother codtos Note that we ca cosder aother crtero of ufcato (stead of S to the statemet u v S r δ ): aggregate statemets S S oly f w > ε ' where ε ' s a threshold decded by the user

5 Iteratoal Boo Seres "Iformato Scece ad Computg" 0 After coordatg each expert s statemets separately we ca costruct a agreemet of several depedet experts The procedure s as above except the weghts: statemets the more we trust resulted statemet) eote the lst of statemets after coordato by Ω M : = Ω w = c w (the more experts gve smlar F F Fg 4 Coordato of No-smlar Statemets After costructg of a cosesus of smlar statemets we must form decso rule the case of tersected o-smlar statemets The procedure such cases s as follows () ( h) To each h = M cosder statemets S S Ω such that ~ h () ( ) : = h where () are related sets to statemets () S ~ h eote I l) = { S ( l) Ω ( l) ( l) } ( where (l) are related sets to statemets S (l) Cosder related sets G (l) where l = L ; I(l) eote by w (l) the weghts of statemets S (l) ( ) ( ) As above ute sets G ( l ) G ( l ) f ρ ( u v G G ) < ε u v = eote by G G ~ G ~ Λ the sets after procedure of ufcato of the sets G (l) Cosder the statemets S ~ ~ h : f ( a) the ~ ( a) G I order to choose the best statemet we tae to cosderato these reasos: ) smlartes betwee sets (l ; h ~ ad ) ) smlartes betwee sets G ~ ad G (l) ; 3) weghts of statemets S (l) ; 4) we must dstgush cases whe smlar / cotradctory statemets produced by oe or several experts ( ) ~ ( ) ( ) ~ h L ( ρ( G ( l) G ))( ρ( ( l) ) w ( l) I ( l) We ca use for example such values: w = ( ) ~ h ( ρ( ( l) ) eote by * : = arg max w l= h Thus we ca mae decso statemet: S ~ = f * ~ w h : = w max w * eote the lst of such statemets by Ω 3 a ~ I ( l) h ( ) the ~ ~ * ( a) G wth the weght

6 0 Algorthmc ad Mathematcal Foudatos of the Artfcal Itellgece Fal decso rule s formed from statemets Ω ad Ω 3 Cocluso Suggested method of formg of uted decso rule ca be used for coordato of several experts statemets ad dfferet decso rules obtaed from learg samples ad/or tme seres Notce that we ca rage resulted statemets by ther weghts ad the exclude gorable statemets from decso rule or ure for more formato for correspodg sets from experts Bblography [] GLbov MGerasmov Costructg of a Cosesus of Several xperts Statemets I: Proc of II It Cof Kowledge-alogue-Soluto 006 pp [] GLbov MGerasmov Iterval Predcto Based o xperts Statemets I: Proc of III It Cof Kowledge-alogue- Soluto 007 Vol pp [3] GSLbov MKGerasmov etermg of stace Betwee Logcal Statemets Forecastg Problems I: Artfcal Itellgece 004 [ Russa] Isttute of Artfcal Itellgece Urae [4] AVet ev Measure of Refutato ad Metrcs o Statemets of xperts (Logcal Formulas) the Models for Some Theory I: It oural Iformato Theores & Applcatos 007 Vol 4 No pp 9-95 Authors' Iformato Geady Lbov - Isttute of Mathematcs SB RAS Koptyug St bl4 Novosbrs Russa; e-mal: lbov@mathscru Maxm Gerasmov - Isttute of Mathematcs SB RAS Koptyug St bl4 Novosbrs Russa e-mal: max_post@gsru