THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 19, Number 3/2018, pp

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 9 Number 3/ MATHEMATICS ENTROPY AND DIVERGENCE RATES FOR MARKOV CHAINS. III. THE CRESSIE AND READ CASE AND APPLICATIONS Vlad Stefa BARBU Alex KARAGRIGORIOU 2 Vasle PREDA 3 Uversté de Roue LMRS Frace; e-mal: barbu@uv-roue.fr 2 Uversty of the Aegea Deartmet of Mathematcs Greece; e-mal: alex.karagrgorou@aegea.gr 3 Uversty of Bucharest ad ISMMA of the Romaa Academy Romaa; e-mal: vaslereda0@gmal.com Corresodg author: Vlad Stefa BARBU Uversté de Roue Laboratore de Mathématues Rahaël Salem UMR 6085 Aveue de l Uversté BP.2 F7680 Sat-Étee-du-Rouvray Frace e-mal: barbu@uv-roue.fr Abstract. I ths work we cosder geeralzed Alha ad Beta dvergece measure for Markov chas as troduced [2] where the weghted versos have bee vestgated [3]. I cotuato to that work we reset geeralzed Cresse ad Read ower dvergece class of measures obta ther lmtg behavor ad umercally vestgate some roertes of all these geeralzed dvergece measures ad rates. Key words: dvergece measures formato measures Markov chas etroy dvergece rates Cresse ad Read dvergece.. PRELIMINARIES I ths secto we remd some deftos ad results from [2] related to Alha ad Beta dvergece measures ad rovde the basc results o ther rates. Let ( X ) N be a ergodc tme-homogeeous Markov cha wth fte state sace χ = { M}. For ths Markov cha we cosder two dfferet robablty laws. Uder the frst law let = P( X = ) χ deote the tal dstrbuto of the cha ad = P( Xk + = j Xk = ) j χ the assocated trasto robabltes. Let also deote the jot robablty dstrbuto of ( X X 2 X ).e. ( = 2 χ were we deoted by : the -tule ( ) χ. Smlarly we defe uder the secod law ( ad. Uder ths settg of fte state sace Markov chas the Alha-Gamma measure betwee the two models s defed as the Alha-Gamma measure betwee the two jot robablty dstrbutos ad (cf. [2]) ad s wrtte uder the ormalzed form as where DAG ( ) = log ( ( ( ) () : χ = = 2 2 wth ad j χ defed by

2 44 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 2 (2) = = / / ( ( ) ) ( ( ) χ : χ (3) = =. / / ( ( ) ) ( ( ) χ : χ Smlarly the Beta-Gamma measure betwee the two Markov models s defed by (cf. [2]) where wth ad DBG ( ) = log ( : ) ( : ) (4) : χ j χ defed by = = 2 2 (5) = = / ( + ) / ( + ) + + ( ( ) ) ( ( ) χ : χ (6) = =. / ( + ) / ( + ) + + ( ( ) ) ( ( ) χ : χ The followg theorems rovde the corresodg dvergece rates. THEOREM (cf. [2]). Uder the settg of the reset secto we have lm DBG ( ) = log λ( ) where λ( ) := lm λ( ) (assumed to exst) where λ ( ) s the largest ostve egevalue of R ( ) = ( r ( )) j χ where wth r ( ) = = / ( + ) / ( + ) + + ( ( ) ) ( f ( ) ) χ : χ : ad defed Euatos (5) ad (6) resectvely. THEOREM 2 (cf. [2]). Uder the settg of the reset secto we have lm DAG ( ) = log λ( ) ( ) where λ( ) s the largest ostve egevalue of R = ( r ( )) j χ where ( ) = = / / r ( ( ) ) ( ) χ χ ( ) ( )

3 3 Etroy ad dvergece rates for Markov chas. III. The Cresse ad Read case ad alcatos 45 wth ad defed Euatos (2) ad (3) resectvely. 2. CRESSIE AND READ DIVERGENCE RATES FOR MARKOV CHAINS Let ( A Ω µ ) be a measurable sace ad µ ad µ some fte measures (ot ecessarly robablty measures) defed o ths sace wth destes ad wth resect to a certa measure µ. I ths secto we are terested the famly of ower dvergeces troduced deedetly by Cresse ad Read (984) ad Lese ad Vajda (987) whch s gve by CR I ( ) = ( ) ( ) d µ R (7) where for = 0 ad t s defed by cotuty. The same rologato by cotuty wll be used for all the dvergece measures cosdered the rest of the aer. Note that the trasformato aled to Alha ad Beta dvergeces (as doe [2]) ca also be aled to the Cresse ad Read measure gve (7). The resultg measure s gve by D ( ) log ( x) = ( x) d µ (x) (8) ( ) whch s the ormalzed Lese ad Vajda s measure defed by ( x) x ( ) = lace of ( ) ( x) dµ ( x) wth ( / ) R I ( ) = log dµ 0 (9) ( ) x. I fact ote that CR dvergece ca be vewed as a secal case of D A dvergece. Let us ow troduce the D dvergece for Markov chas. Ths measure troduced Euato (8) the..d. settg takes the followg form the Markov cha framework: D ( ) =. ( ( ) ) χ : ( ) log ( ) ( ) : : χ Ths ca be wrtte uder the ormalzed form (0) D ( ) = log ( ( ( ) () : χ where = wth ad 2 j χ defed by (2) = =. / ( ) / ( ) ( ( ) ) ( ( ) χ : χ

4 46 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 4 The followg result cocers the lmtg behavor of D. THEOREM 3. Uder the settg reseted before we have lm D ( ) = log λ( ) ( ) where λ( ) s the largest ostve egevalue of R = ( r ( )) wth ad defed (2). r a ( ) / ( ( ) χ = = 3. NUMERICAL APPLICATIONS I ths secto we wll cosder umercal examles order to llustrate the results obtaed the revous secto. Let ( X ) N be a tme-homogeeous two-state Markov cha. As revously descrbed for ths Markov cha we cosder two dfferet robablty laws the frst oe gve by a Markov trasto matrx = ( j) j = 2 ad a tal dstrbuto = ( 2 ) whle the secod oe s govered by a Markov trasto matrx = ( j) j = 2 ad a tal dstrbuto = ( 2 ). We cosder the trasto matrces gve by = ad = whle for the tal dstrbutos we take the corresodg statoary oes amely ( ) = ( ) ( ) = ( ) 6/7 /7 ad /9 8/ Frst the results of Theorem cocerg the dvergece rate of Beta-Gamma measure are llustrated Table. Table The rate of Beta-Gamma dvergece = 0 = 5 = 20 rate = BG/ = BG/ = BG/ = = 0.5 BG/ =.0845 BG/ =.0707 BG/ = = 0. BG/ =.39 BG/ =.263 BG/ = = 0.0 BG/ =.306 BG/ =.73 BG/ = = 0.00 BG/ =.295 BG/ =.6 BG/ = KL KL/ =.293 KL/ =.60 KL/ = Secod Table 2 we llustrate the covergece of Alha-Gamma measure to the KL measure as goes to (cf. Remark 3). Note that the results Tables ad 2 demostrate both the covergece of the arorate measure to the corresodg rate as well as the covergece to KL for ay value of (cludg the lmt).

5 5 Etroy ad dvergece rates for Markov chas. III. The Cresse ad Read case ad alcatos 47 Dvergece measures lke the oes dscussed ths work are used as dces of smlarty or dssmlarty betwee oulatos. As a result they ca be used as a way to evaluate the dstace (dvergece) betwee ay two oulatos or fuctos. Measures of dvergece ca be used statstcal ferece for estmatg uroses (Toma [9] ad [0]) the costructo of test statstcs for tests of ft (e.g. Zografos et al. [2] Huber-Carol et al. [7] ad Zhag []) or statstcal modelg for the costructo of model selecto crtera lke the Kullback-Lebler measure whch has bee used for the develomet of varous crtera (e.g. Akake [] ad Cavaaugh [4]). Table 2 Covergece of Alha-Gamma measure to the KL measure as = 0 = 5 = 20 rate = AG/ = AG/ = AG/ = = 0.5 AG/ =.098 AG/ = AG/ = = 0. 9 AG/ =.50 AG/ =.352 AG/ = = 0.95 AG/ =.49 AG/ =.279 AG/ = = 0.99 AG/ =.32 AG/ =.87 AG/ = KL KL/ =.293 KL/ =.60 KL/ = Oe of the most oular statstcs s the Cresse ad Read ower dvergece statstcs (CR). The rate of the geeralzed form of ths dvergece for Markov sources was derved Theorem 3. The CR famly of statstcs was orgally roosed for testg the ft of observed freueces to exected freueces. Through ths famly of statstcs Cresse ad Read succeeded rovdg a ufed aroach to goodess-of-ft testg for multomal models. The mortace of the roosed statstcs les o the fact that several goodess-of-ft tests ca be reduced to test a ull hyothess from a multomal oulato ad therefore a statstc that measures how much two dstrbutos dffer s of hgh mortace. Several well-kow test statstcs are members of the Cresse ad Read famly of dvergeces lke the Pearso s ch-suare the lkelhood dsarty (geeratg the log-lkelhood rato statstc) the (twce ad suared) Hellger dstace (Freema ad Tukey [6]) the Kullback-Lebler dvergece ad the Neyma modfed ch-suare whch are dexed by = 2 (by cotuty) /2 0 (by cotuty) ad resectvely. I referece to Theorem 3 some results related to CR famly of measures are reseted below. More recsely for the Cresse ad Read Geeralzed measure we reset the covergece of the measure ad assocated rate to the arorate KL measure ad rate as goes to 0 > 0 ad goes to < (cf. Table 3). Note that lm D( ) = DKL( ) a 0 whle lm D( ) = DKL( ) a whch s clearly cofrmed by the results Table 3. I Table 4 we llustrate the rates of three artcular cases of the Geeralzed Cresse ad Read measure: Pearso's χ ( = 2 ) Freema-Tukey's F ( = 05. ) ad Neyma χ (Euclda log-lkelhood rato statstc) ( = ). We have also cluded the secal case = 2/ 3. Note that based o a comaratve study ths secal value was recommeded by Read ad Cresse [8] (a value betwee the Pearso's ch-suare ad the Neyma's ch-suare statstc) as a comromse caddate amog the dfferet

6 48 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 6 test statstcs although they oted several desrable roertes of the other test statstcs cludg the Pearso's ch-suare (see e.g. Secto 4.5 Secto 6.7 ad Aedx A of Read ad Cresse [8]). Table 3 Covergece of Cresse ad Read Geeralzed measure/rate to the KL measure/rate as a 0 a > 0 ad a a< = 0 = 5 = 20 rate = 0. / = / = / = = 0.0 / = / = / = = 0.00 / = / = / = KL() KL/ = KL/ = KL/ = = 0.9 / =.50 / =.352 / = = 0.99 / =.32 / =.87 / = = / =.296 / =.62 / = KL() KL/ =.293 KL/ =.60 KL/ = Table 4 The rate of the Geeralzed Cresse ad Read measure for some mortat secal cases: Pearso's χ 2 ( a = 2) Freema-Tukey's F 2 ( a = 0.5) Cresse ad Read ( a = 2/3) ad Neyma χ 2 ( a = ) = 0 = 5 = 20 rate = 2 / = / = 0.77 / = = 0.5 / =.098 / = / = = 2/3 / = / = / = = KL/ = KL/ = KL/ = I Fg. the Geeralzed Cresse ad Read dvergece rate s llustrated as a fucto of for the two robablty laws of the Markov cha cosdered at the begg of ths secto. We also rereseted the value of D for several values of. Notce the fast covergece of D to the rate accordg to Theorem 3. Notce further that eve for small values of D gves a good aroxmato of the rate. I referece to the secal value of = 2/ 3 we observe Fg. that ths s ot the value of the dex that dscrmates the most betwee the two Markov chas. For ths artcular examle the value that maxmzes the dvergece rate s * * = Although may be of lmtg sgfcace f the two sources are well searated t wll be of great mortace case the two sources are close to each other. Ideed cosder the followg examle for whch the dvergece rate wll be exected to be close to 0. Let a tmehomogeeous two-state Markov cha ( X ) evolve uder two dfferet robablty laws tha those of the N begg of ths secto the frst oe gve by a Markov trasto matrx = ( j) j = 2 ad a tal = whle the secod oe s govered by a Markov trasto matrx = ( j) j = 2 dstrbuto ( 2 ) ad a tal dstrbuto ( ) =. We cosder the trasto matrces gve by = ad = whle for the tal dstrbutos we take the corresodg statoary oes.

7 7 Etroy ad dvergece rates for Markov chas. III. The Cresse ad Read case ad alcatos 49 Fg. The covergece of D w.r.t.. Fgure 2 resets the Geeralzed Cresse ad Read dvergece as a fucto of for ths examle. * Note that Fgure 2 cofrms the closeess of the sources but at the same tme rovdes the value of the * dex for whch the rate s maxmzed ( =. 57 ). I cocluso for dscrmatory uroses ad coseuetly for statstcal ferece (.e. goodess of ft tests model selecto etc.) we recommed the use * of the dvergece rate wth the dex take to be eual to the value. Note that the same recommedato ales ot oly to the rate but also to the dvergece tself. Let us ow cosder two addtoal examles of two dfferet robablty laws goverg a Markov cha. Frst we are terested a two-state Markov cha ad we set the Markov trasto matrces ad = ad = whle the tal dstrbuto = ( ) ad ( ) 2 = 2 are take to be the assocated statoary oes. I Fg. 3 we reset both the Geeralzed Cresse ad Read dvergece rate comuted for ( ) ad also for ( ) as a fucto of. Fg. 2 The Geeralzed Cresse ad Read dvergece rate for the secod examle.

8 420 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 8 Fg. 3 Reflecto roerty of the GCR dvergece rate for the thrd examle. Secod we cosder aother examle of two laws goverg ow a three-state Markov cha. Let ad be two Markov trasto matrces gve by = ad = = 2 are the assocated statoary oes. As for the revous examle Fg. 4 we reset both the Geeralzed Cresse ad Read dvergece rate comuted for ( ) ad also for ( ) as a fucto of. Note that both Fgs. 3 ad 4 there s a symmetry betwee the two grahs. I fact ths heomeo s due to a reflecto roerty of the GCR dvergece. More recsely let us deote by D; ( ) the GCR dvergece evaluated at. The oe ca easly verfy that D; ( ) = D ; ( ). Obvously whle the tal dstrbuto = ( ) ad ( ) due to Theorem 3 ths roerty holds true also for the dvergece rate. For ths reaso Fgures 3 ad 4 we have a reflecto wrt the le x = 05.. Fg. 4 Reflecto roerty of the GCR dvergece rate for the three-state examle.

9 9 Etroy ad dvergece rates for Markov chas. III. The Cresse ad Read case ad alcatos 42 ACKNOWLEDGEMENTS The authors would lke to thak ther colleague Ncolas Verge from Laboratore de Mathématues Rahaël Salem Uversty of Roue Frace for hs suggestos ad hel o some techcal roblems related to ths aer. The frst two authors would also lke to exress ther arecato to the Uversty of Roue ad Uversty of Aegea for the oortuty to exchage several vsts both sttutos. The research work of Vlad Stefa Barbu was artally suorted by the rojects XTerM Comlex Systems Terrtoral Itellgece ad Moblty ( ) ad MOUSTIC Radom Models ad Statstcal Iformatcs ad Combatorcs Tools ( ) wth the Large Scale Research Networks from the Rego of Normady Frace. REFERENCES. H. AKAIKE Iformato theory ad a exteso of the maxmum lkelhood rcle Proceedg of the Secod Iteratoal Symosum o Iformato Theory B.N. Petrov ad F. Csak (eds.) Akadema Kado Budaest V.S. BARBU A. KARAGRIGORIOU V. PREDA Etroy ad dvergece rates for Markov chas: I. The Alha-Beta ad Alha-Gamma case submtted V.S. BARBU A. KARAGRIGORIOU V. PREDA Etroy ad dvergece rates for Markov chas: II. The weghted case submtted J.E. CAVANAUGH Crtera for lear model selecto based o Kullback s symmetrc dvergece Australa ad New Zealad Joural of Statstcs N. CRESSIE T.R.C. READ Multomal goodess-of-ft tests J. R. Statst. Soc M.F. FREEMAN J.W. TUKEY Trasformatos related to the agular ad the suare-root A. Math. Statst C. HUBER-CAROL N. BALAKRISHNAN M.S. NIKULIN M. MESBAH Goodess-of-ft Tests ad Model Valdty Brkhäuser Bosto T.R.C. READ N. CRESSIE Goodess-of-Ft Statstcs for Dscrete Multvarate Data New York Srger-Verlag A. TOMA Mmum Hellger dstace estmators for multvarate dstrbutos from the Johso system J. Statst. Pla. ad Ifer A. TOMA Otmal robust M-estmators usg dvergeces Statstcs ad Prob. Letters J. ZHANG Powerful goodess-of-ft tests based o lkelhood rato J. R. Stat. Soc. Ser. B K. ZOGRAFOS K. FERENTINOS T. PAPAIOANNOU Φ -dvergece statstcs: Samlg roertes multomal goodess of ft ad dvergece tests Comm. Statst. Theor. Meth Receved March 3 207

ENTROPY AND DIVERGENCE RATES FOR MARKOV CHAINS: II. THE WEIGHTED CASE

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 9 Number /08 3 0 ENTROPY AND DIVERGENCE RATES FOR MARKOV CHAINS II THE WEIGHTED CASE Vlad Stefa BARBU Alex