The Distribution of Minimizing Maximum Entropy: Alternative to Weibull distribution for wind speed

Transcription

1 Proceedgs o the 9th WSEAS Itertol Coerece o Appled Mthetcs, Istul, urey, My 7-9, 6 (pp65-6) he Dstruto o Mzg Mxu Etropy: Altertve to Weull dstruto or wd speed ALADDI SHAMILOV, ILHA USA, YELIZ MER KAAR Deprtet o Sttstcs Adolu Uversty Essehr, 647 URKEY Astrct: I the preset study, we hve costructed MMxEt dstruto ltertve to the Weull dstruto or the wd speed. MMxEt dstruto zes the specl xu etropy uctol the cocrete set o oet uctos, thereore, xzes the out o orto geerted y the gve rdo sple. MMxEt dstruto o wd speed dt esured Essehr s lso oted d copred wth the Weull dstruto y usg vrous sttstcl crters. Cosequetly, t s detered tht MMxEt dstrutos s ore coveet or wd speed dstrutos, copred to the Weull whch s wdely used wd studes. Keywords: MxEt; MxEt dstruto; MMxEt dstruto, Wd speed dt, root e squre error, ch-squre, correlto coecet, Ae orto crter, Byes orto crter. Itroducto Recetly the ethods sed o orto theory to ot the dstruto o rdo vrle re very populr. he xu etropy (MxEt) [] d the u cross etropy (MxEt) re such o these ethods [], [3], [4]. here hs ee lot o study out pplcto o these ethod lterture. I[5], MxEt ethod ws eployed to pproxte the sze dstruto o U.S. ly coe d MxEt dstrutos ws copred wth two covetol coe dstruto. I [6]-[7], t ws used the MxEt prcple - sttstcl erece ethod or the rst te to the eergy eld, d proposed theoretcl pproch to detere dstruto o wd speed dt lytclly. I etoed studes, MxEt ethod operted ccordg to wd s ss, oetu, eergy [6] s proposed tht the ly o dstrutos s developed y troducg pre-expoetl ter to the theoretcl dstruto derved ro MxEt prcple. I [7], MxEt odels re used to detere the dstruto o durl, othly, sesol d yerly wd speed. I ths study, t s gve the deto o MMxEt d coprso o the MMxEt dstruto d Weull dstruto whch ws wdely used wd power studes [6-], y usg sttstcl crters d cocluded tht MMxEt dstruto c e ore coveet to detere dstruto o wd speed dt. he rest o the pper s orgzed s ollows. Secto troduces MxEt ethod. I secto 3, t s deed uctol o oet uctos d propertes o ths uctol re lso gve s theores. It s show tht there s dstruto whch zes the specl xu etropy uctol o oet uctos. hs dstruto s clled s MMxEt. I Secto 4, the estto o the dstruto o wd speed dt esured Esşehr s oted usg deed dstruto. he secto 5 cocludes the pper wth soe suggestos or o urther reserch.. MxEt Method. MxEt ethod or dscrete syste

2 Proceedgs o the 9th WSEAS Itertol Coerece o Appled Mthetcs, Istul, urey, My 7-9, 6 (pp65-6) he MxEt s ethod whch proposes to estte p (x) y xzg the etropy suect to soe oet costrts. he estted p (x) s sd to e MxEt desty ucto. he MxEt desty ucto o dscrete rdo vrle (RV) X s oted y xzg Sho s etropy esure H ( X ) = p l p () = suect to the costrts: = p = E{ g} = = g( x ) p = µ, () () ( ) where g = ( g,..., g ) s oet vector ucto, µ = ( µ,..., µ ) s oet vector vlue. By usg Lgrge ethod, t s esly dcted tht the resultg uctol or o MxEt desty ucto s p = = exp( λ λ g ( x )), (3) where λ, λ,...,λ re Lgrge ultplers whch c e oud y ewto ethod.. MxEt Method or cotuous syste he MxEt desty ucto o cotuous rdo vrle X s oted y xzg Sho s etropy esure = H l dx (4) suect to the costrts: dx = (5) = E { g} g( x) dx = µ (6) () ( ) where g = ( g,..., g ) s oet vector ucto, µ = ( µ,..., µ ) s oet vector vlue. Oted MxEt desty ucto s = = exp( λ λ g ( x)). I ths secto, we gve the deto o MMxEt Dstruto or dscrete d cotuous rdo vrles(rvs) 3. MMxEt Dstruto or Dscrete RV Let x,..., x e rdo sple (RS) wth the correspodg proltes p,..., p. It s requred to ot out o xu orto geerted y gve RS o the ss o the gve oet vector ucto ( ) ( ) g = ( g,..., g ). RS hs etropy = H = p l p (7) H hs xu vlue costrt p = l suect to = (8) Mxu o H suect to costrt (8) d oet costrts (9) represets uctol whch s deoted y S (g) = p g ( x ) = µ, (9) () ( ) ( g,..., g () ( ) ( µ,..., µ ) where g = ) s oet vector ucto, µ = s oet vector vlue. hs uctol possesses soe propertes expressed y heore. I S (g) s the uctol derved y xzg H suect to costrts (8), (9), the S (g) s cotuous o the set o cotuous vector oet uctos C[,]. heore. I S (g) s the uctol derved y xzg H suect to costrts (8), (9), the S (g) s cotuous uctol reches ts lest d gretest vlues the gve copct set K C[,]. () Suppose tht g vector oet ucto relzes u vlue o uctol S (g),.e. () S( g) = S( g ) g K he dstruto p,..., p correspodg to () g s clled s MMxEt dstruto d represeted the ollowg 3 Deto o MMxEt Dstruto

3 Proceedgs o the 9th WSEAS Itertol Coerece o Appled Mthetcs, Istul, urey, My 7-9, 6 (pp65-6) or () = = p = exp( λ λ g ( x )),,...,. Deto o MMxEt dstruto s sed o otg soe oet vector ucto () g ro the gve set K. By vrtue o the etoed oet vector ucto, the xu orto s te ro the gve RS. I ths sese, MMxEt dstruto relzes possle xu orto ro RS. I geerl, otg lest vlue o MxEt uctol S (g) copct set o oet vector uctos c e doe y pplyg ethods o clculus o vrtos (see [3]). 3. MMxEt Dstruto or Cotuous RV I order to xze H deed y (4) suect to costrts (5), (6) t s ppled Lgrge ultplers ethod. hs ethod requres to costruct the ollowg uxlry uctol H or H = = λ ( = l dx λ ( λ ( g = g ( x) dx µ ) l dx ( x) dx µ ), dx ) () where µ =, g ( x) =. Soluto o Euler equto or uctol () hs the or = exp( λ λ g ( x)). () = Susttutg (x) (5), (6) t s possle to ot λ,...,λ. he S = H x = λ + λµ λµ. () heore 3 I S (g) s the uctol derved y xzg H gve y (4) suect to costrts (5), (6), the S (g) s cotuous o the set o cotuous vector oet uctos C [, ]. heore 4. I S (g) s uctol derved y xzg H suect to costrts (5), (6), the S (g) reches ts lest d gretest vlues the gve copct set K C[, ] o cotuous oet vector uctos. ow, we pply the deed MMxEt dstruto to wd speed dt esured Essehr 5. 4 MMxEt Dstruto ucto: etter ltertve to Weull dstruto ucto or wd speed he wd eergy s oe o the ost sgct d rpdly developg reewle eergy sources the world d t provdes cle eergy resource. For ths reso, the prolty dstruto o wd speed s oe o the ost portt wd chrcterstcs or ssesset o wd eergy potetl d or the perorce o wd eergy coverso systes. I ths secto, t s show tht the MMxEt dstruto ot oly gree etter wth vrety o the esured wd speed dt th the covetolly used eprcl Weull dstruto, ut lso c preset the wd power desty uch ore ccurtely. Furtherore, the MMxEt dstruto oted y MxEt theory s ore sutle or the ssesset o the wd eergy potetl. he ethodology preseted ths study s ppled to wd speed dt. I order to clculte MMxEt dstruto, the ollowg steps re relzed.. Detere oet costrs (x, x, lx, l(x+), (lx), l(x +) ).Clculte MxEt dstrutos suect to ech o oet costrs 3.Clculte etropy o MxEt dstruto 4.Clculte orto coted y MxEt dstrutos. 5. Select oet costr whch hs xu orto. 6. Detere the MMxEt dstruto y selected oet costrt It s repeted ove steps or two oet costrts. he correspodg oet costrt to xu orto re lsted le, le usg wd dt esured o Deceer. hs process c e repeted y tes. It s cler tht detered MMxEt 3

4 Proceedgs o the 9th WSEAS Itertol Coerece o Appled Mthetcs, Istul, urey, My 7-9, 6 (pp65-6) suect to the ore costrts shows the etter ttg. le. Etropy o the clculted MxEts suect to oe costrt or esured dt o Deceer Costrs Etropy x x 4.99 lx 4.37 l(+x) 4.9 (lx) l(+x ) 4.93 le. Etropy o the clculted MxEts suect to two costrts or esured dt o Deceer Costrs Etopy Costrs Etropy x & x 4.49 x & l(+x ) x & lx lx & l(x+) 4.53 x & l(x+) 4.87 lx & (lx) x & (lx) 4.75 lx & l(+x ) 4.7 x & l(+x ) l(x+) & (lx) 4.94 x & lx l(x+) & l(+x ) 4.7 x & l(x+) (lx) & l(+x ) 4. x & l(x) le 3. Dstruto clculted ro Weull, MMxEt dstrutos or Deceer V Dec. w x x (/s) (h) Fro le d le, we coclude tht MMxEt uder oe costrt s the clculted MxEt suect to x d MMxEt uder two costrt s the clculted MxEt suect to x &l(x) he wd speed dt te-seres ort s usully rrged the requecy dstruto ort. It s coveet sttstcl lyss. We t the Weull dstruto to wd speed dt, oud the MMxEt dstrutos uder the oe d two oet costrts or ech oths o 5. Arrgeet o the esured hourly teseres dt requecy dstruto ort or Deceer d the requecy dstrutos clculted the Weull d the MMxEt dstrutos re gve le 3. I the le 3, s prolty desty ucto (PDF) o oserved dt, w (x) s the Weull desty ucto, x ( x) s the MMxEt PDF suect to oe costrt, x ( x) s the MMxEt PDF suect to two costrt. ow, order to copre MMxEt dstrutos wth the Weull dstruto, we c e use o sutlty crter to dety the est dstruto ro ogst cosdered dstrutos Root e squre error (RMSE) [8], Chsqure (X ), Correlto coecet (R ) Ae s orto crtero (AIC) [5],[5] Byes s orto crtero (BIC) Kullc- Leler esure (K-L) [4] wll e used sttstclly evlutg the perorce o MMxEt d Weull dstrutos. he orul o etoed sutlty udget crter re RMSE = = ( y x ), (5) ( y x ) = χ =, (6) 4

5 Proceedgs o the 9th WSEAS Itertol Coerece o Appled Mthetcs, Istul, urey, My 7-9, 6 (pp65-6) R ( y x ) = =, (7) ( y z) = eprcl Weull dstruto ters o ll crters. For ths reso the MMxEt dstrutos re uch coveet or wd speed dt th the Weull sce ll crteros vour MMxEt dstutos. AIC = log( L( ( x, θ )) +, (8) BIC = log( L( ( x, θ )) + log, (9) Fgure. Hstogr d oted Weull d MMxEt destes or wd speed dt K L = = y y log () x where y s the th prolty o ctul dt, x s the th predcted prolty, s uer o ll oserved wd speed dt, L s log lelhood ucto, s the uer o preters or the uer o costrs. he est dstruto ucto c e detered ccordg to the lowest vlues RMSE, X, K-L esure, AIC, BIC, the hghest vlues R. It s oted tht coprso o MMxEt d Weull dstrutos or every oths o 5 ut gve oly Octoer, Deceer due to the spce ltto. le 4. Coprso o the ctul prolty dstruto wth MMxEt dstrutos d the Weull dstruto or Octoer Octoer RMSE X R AIC BIC K-L w x x le 5. Coprso o the ctul prolty dstruto wth MMxEt dstrutos d the Weull dstruto or Deceer Deceer RMSE X R AIC BIC K-L w x x e le 4, le 5 shows the excellet greeet o the MMxEt destes wth wd dt, uch etter th the correspodg Fgure reports the hstogr o the wd speed dt d the estted MMxEt destes wth oe d two costrts d the Weull desty. Fro ths Fgure, oted MMxEt destes deostrtes good ttg whe wd speed vlues<5 d t oth tls. 4 Cocluso I the preset study, we hve costructed MMxEt dstruto ltertve to the Weull dstruto or the wd speed. By coprg the MMxEt dstrutos wth the Weull dstruto o wd speed dt, we cocluded tht MMxEt dstrutos s ore coveet or wd speed dstruto. Moreover, t c e show tht estte o wd power c e studed sed o MMxEt dstruto. Reereces: [] Jyes, E.. Ppers o Prolty, Sttstcs, d Sttstcl Physcs. etherlds: D. Redel, Dordrecht,

6 Proceedgs o the 9th WSEAS Itertol Coerece o Appled Mthetcs, Istul, urey, My 7-9, 6 (pp65-6) [] Kullc, S. d Leler, R. A. O Iorto d Sucecy, Als o Mthetcl Sttstcs,, 95, [3] Kullc, S. Iorto heory d Sttstcs. Wley, 959. [4] Kpur, J.. d Kesv, H. K. (99), Etropy Optzto Prcples wth Applctos. Lodo: Acdec Press, Ic. [5] Wu, X. Clculto o xu etropy destes wth pplcto to coe dstruto, Jourl o Ecooetrcs, 3, 5, [6]L M., L X. MEP-type dstruto ucto: etter ltertve to Weull ucto or wd speed dstruto, Reewle eergy, 3, 5, -4. [7] L M., L X. Ivestgto o wd chrcterstcs d ssesset o wd eergy potetl or Wterloo rego, Eergy coverso geet, 46, 5, [8] Cel A.. A sttstcl lyss o wd power desty sed o the Weull d Rylegh odels t the souther rego o urey, Reewle eergy, 9, 3, [9] Apır E.K., Apır S. A ssesset o sesol lyss o wd eergy chrcterstcs d wd ture chrcterstcs, Eergy coverso geet, 46, 5, [] Wesser D. A wd eergy lyss o Gred: estto usg the Weull desty ucto, Reewle Eergy, 8, 83-8, 3. [] Al-ssr W., Alhr S, Al-Ez A., Al-Awdh L.. Potetl wd power geerto the Stte o Kuwt, 3, 5, [] Ulge K, Hepslý A. Deterto o Weull preters or wd eergy lyss o Izr, urey. It J Eergy Res, 6,, [3] Elsgolts, L. Deretl Equtos d the clculus o vrtos, Mr Pulsher, 97 [4] Shlov A., Ktr Mert Y. O Dstruto Mzg Mxu Etropy, Itertol coerece, Ordered Sttstcl Dt, 5. [5] Keeth B. Model selecto d Mult- Model Ierece, Sprger-Verlg,. 6