Bivariate drought analysis using entropy theory

Transcription

1 Purue University Purue e-pus Symposium on Dt-Driven Approhes to Droughts Drought Reserh Inititive Network -3- Bivrite rought nlysis using entropy theory Zengho Ho exs A & M University - College Sttion, hz7@tmu.eu Vijy P. Singh exs A & M University - College Sttion, vsingh@tmu.eu Follow this n itionl works t: Prt of the Bioresoure n Agriulturl Engineering Commons Ho, Zengho n Singh, Vijy P., "Bivrite rought nlysis using entropy theory" (). Symposium on Dt-Driven Approhes to Droughts. Pper his oument hs een me ville through Purue e-pus, servie of the Purue University Lirries. Plese ontt epus@purue.eu for itionl informtion.

2 Bivrite rought nlysis using entropy theory Zengho Ho Deprtment of Biologil & Agriulturl Engineering exs A & M University, 3E Sotes Hll, 7 AMU, College Sttion, exs Emil: hz7@tmu.eu Vijy P. Singh Deprtment of Biologil & Agriulturl Engineering exs A & M University, 3 Sotes Hll, 7 AMU, College Sttion, exs vsingh@tmu.eu Astrt Drought urtion n severity re two properties tht re usully neee for rought nlysis. o hrterize the orreltion etween the two rought properties, ivrite istriution is neee. A new metho se on entropy theory is propose for onstruting the ivrite istriution tht is ple of moeling rought urtion n severity with ifferent mrginl istriutions. Prmeters of the joint istriution re estimte with Newton s metho. Monthly stremflow t from Brzos River t Wo, exs, re employe to illustrte the pplition of the propose metho to moel rought urtion n severity for rought nlysis. Keywors: rought nlysis; joint istriution; entropy theory; priniple of mximum entropy Introution Drought nlysis is importnt for wter resoures plnning n mngement. Yevjevih [967] use the run theory to efine rought s sequene of intervls where wter supply remins elow wter emn. his enles the hrteriztion of rought in simple terms, suh s urtion n severity, using hyrologil vriles (e. g., stremflow). Drought urtion n severity re two min hrteristis tht hve often een use for rought nlysis. Drought urtion n severity n e regre s rnom vriles n thus the proility istriution, either seprte or joint, n e use for hrterizing rought. A tritionl wy to hrterize the rought urtion or severity is se on fitting proility ensity funtion. Severl pprohes hve een propose for univrite rought nlysis. he rought urtion n e moele y geometri istriution [Kenll n Drup, 99; Mthier et l., 99] when it is trete s isrete rnom vrile or y n exponentil istriution when it is trete s ontinuous rnom vrile [Zelenhsi n Slvi, 987]. he gmm istriution is generlly use to esrie rought severity [Shiu n Morres, 9]. However, the orreltion etween rought urtion n severity nnot e hrterize y univrite nlysis n lterntive multivrite pprohes hve therefore een use to moel the orreltion of rought vriles [González n Vlés, 3; Sls et l., ; Kim et l., 6; Shiu, 6; Nrjh, 7; Nrjh, 9]. Some ivrite istriutions hve een use for the joint istriution of rought urtion n severity, suh s the ivrite Preto istriution [Nrjh, 9]. he joint istriution of rought urtion n severity my e moele using ifferent istriutions, in whih se the generl ivrite istriution oes not work. o ress this issue, the opul metho hs een pplie to onstrut the istriution tht is ple of linking two univrite istriutions to form ivrite istriution [Shiu, 6; Shiu et l., 7]. he joint istriution n lso e onstrute from the onitionl istriution n mrginl istriution [Shiu n Shen, ]. Furthermore, nonprmetri methos hve lso een propose for ivrite rought nlysis [Kim et l., 3; Kim et l., 6].

3 his rtile proposes new metho for onstruting the ivrite istriution of rought urtion n severity with ifferent mrginl istriutions se on the priniple of mximum entropy. he propose metho is pplie for rought nlysis se on the monthly stremflow of Brzos River t Wo, exs. Metho. Priniple of mximum entropy For ontinuous rnom vrile X with proility ensity funtion (PDF) f( efine on the intervl [, ], entropy is efine s mesure of unertinty expresse s [Shnnon, 948]: H ln x (.) For ontinuous rnom vriles X n Y with PDF f(x, efine over the spe [, ] [, ], the Shnnon entropy n e efine s: H ln xy (.) he priniple of mximum entropy ws propose y Jynes [97] whih sttes tht the proility ensity funtion shoul e selete mong ll the istriutions tht stisfy the onstrints. Generlly, the onstrints n e speifie s: x () where λ i (i=,,,m) re the Lgrnge multipliers. Differentiting L with respet to f n setting the erivtive to zero, the mximum entropy istriution n e otine s [Kpur, 989]: f ( m m x exp g ( g (... g ( ) () It hs een shown tht mny of the ommonly use istriutions n e erive from entropy theory with ifferent onstrints n the mximum entropy istriution in eqution () inorportes these istriutions s speil ses [Singh, 998]. For exmple, if the first n seon moments re speifie s the onstrints, the mximum entropy istriution is the norml istriution. It n e seen tht the mximum entropy istriution is quite flexile.. Joint istriution o erive the joint ensity funtion f(x, of rought urtion (X) n severity (Y), onstrints for vriles X n Y nee to e speifie seprtely n jointly. Consiering the onstrints use for eriving the ommonly use istriutions n the efinition of the Person orreltion oeffiient tht hs een ommonly use for mesuring the epenene of rnom vriles, the following seprte onstrints of vrile X n Y n e speifie oringly s: x x x (6) g ( x i g i i=,, m (3) y y y (7) where m is the numer of onstrints; g i ( is the funtion of x ng i is the expettion of funtion g i (. he proility ensity funtion for the univrite se n e erive oring to the priniple of mximum entropy y mximizing the entropy efine in eqution (.) sujet to the onstrints in equtions () n (3). he mximiztion n e hieve using the metho of Lgrnge multipliers y introuing the Lgrnge funtion L: L ( log x (4) m ) ( ) i ( gi ( gi ) i ln y y ln y (8) he joint onstrint n e speifie to moel the orreltion through the prout XY s: xy xy XY (9) With these onstrints from equtions (6) n (9), the joint PDF n e otine y mximizing the entropy in eqution (.). Following similr steps in eriving the mximum entropy istriution in the univrite se, the joint PDF in the ivrite se n e otine s:

4 f ( 4 x, exp( x y 3 ln y x ().3 Mrginl istriution he mrginl istriution for rought urtion X n e otine y integrting the joint PDF f(x, given y eqution () over Y s: ( 3 4 f exp( x y ln y x y () Similrly, the mrginl istriution for rought severity Y n e otine s: f exp( x y ln y x x () ( Prmeter estimtion he entropy theory n e pplie for prmeter estimtion n Singh [998] gve n introution of the entropy se metho for estimting the Lgrnge multipliers for the ommonly use istriutions. Lgrnge multipliers n lso e estimte y mximizing the funtion [Me n Ppniolou, 984]: m g i i (3) Newton s metho n e pplie for mximizing the funtion Г y upting λ () with some initil vlue λ () eqution elow: λ ( λ ) () H i through the, i=,,,m (4) where λ=[λ,,λ m ]; the grient is expresse s: g i i gi( x, i=,,,m () n H is the Hessin mtrix whose elements re expresse s: H i, j gi( g j ( x (6) gi( x g j ( x, i, j,..., m. Comments he steps ove emonstrte how to onstrut the ivrite istriution of rought urtion n severity with the onstrints of men n logrithmi men. In rel pplition, ifferent onstrints n e speifie for the urtion n severity seprtely to form ifferent joint istriutions. Aoring to ertin mesures (e.g., the men squre errors of empiril n theoretil proilities), the most suitle onstrints for eh vrile n e use to form the joint ensity funtion of the urtion n severity, whih n then e use for rought nlysis. he hrteristi of the propose metho is tht rought urtion n severity n e moele with ifferent mrginl istriutions (with ifferent onstrints). 3 Results n isussion Monthly stremflow t of Brzos River t Wo, X (USGS 896) for the perio from Jnury 94 to Deemer 9 ws use for rought nlysis. he men stremflow of eh month is use s the truntion level to efine the rought event. Signifint orreltion exists etween rought urtion n severity n the ivrite ensity funtion in eqution () ws use to moel them jointly. Histogrms n the fitte mrginl istriution in eqution () for rought urtion n in eqution () for rought severity re shown in Figure. he fitte PDFs pture the generl pttern of the histogrms. he empiril proility estimte from the Gringorten s plotting position formul n theoretil proility re shown in Figure. It n e seen tht generlly theoretil proilities fitte empiril proilities well. he Kolmogorov- Smirnov (K-S) gooness-of-fit test ws use to further test whether the oserve t n e moele with the propose moel. he ritil vlue t % signifine level ws.7 n the K-S sttistis were. n. for rought urtion n severity, respetively, initing the propose moel n e pplie to moel the rought urtion n severity t. he return perio for rought urtion D greter or equl to ertin vlue n for rought severity S greter or equl to ertin vlue s n e efine s [Shiu, 3; Shiu, 6]: E( L) P ( D ) D (7.) D S E( L) P ( S s) (7.) where E(L) is the expete rought intervl time tht n e estimte from oserve roughts; D, S re the return perios efine for rought urtion n rought severity, respetively; S 3

5 P D (D ) n P S (S s) re exeene proility of rought urtion n rought severity tht n e estimte from equtions () n (), respetively. he univrite return perios of,,,, n yers efine y seprte rought urtion n severity n then e estimte from equtions (7.) n (7.) n re summrize in le. For exmple, the rought urtion for the yer return perio is roun 3. months n the rought severity for the yers return perio is roun fs months. he joint return perio of the rought urtion n severity n e efine y the rought urtion n severity exeeing speifi vlues. Speifilly, the joint return perio DS of rought urtion D n severity S n e efine s [Shiu, 3; Shiu, 6]: DS E( L) P( D, S s) (8) where P(D,S s) is the exeene proility of rought urtion n severity tht n e otine from the joint ensity funtion in eqution (). he joint return perios efine y eqution (8) for ifferent urtion n severity vlues re shown in Figure 3. he onitionl return perios re lso neee to ssess the risk of wter resoures systems. he onitionl return perio D S s for rought urtion given rought severity exeeing ertin vlue n e efine s [Shiu, 3; Shiu, 6]: D S s (9) S P( D, S s) Similrly, the onitionl return perio S D for rought severity given rought urtion exeeing ertin vlue n e efine s [Shiu, 3; Shiu, 6]: S D () D P( D, S s) he onitionl return perios re shown in Figure 4. For exmple, given the rought severity s> 4 fs month, the onitionl return perio of the rought urtion exeeing 6 month is roun yers. 4 Summry n Conlusion A ivrite istriution se on entropy theory is propose for onstruting the joint istriution of rought urtion n severity. he vntge of the propose metho is tht it is flexile to inorporte ifferent forms of mrginl istriutions of the rought urtion n severity. Drought t efine y the monthly stremflow t Brzos River t Wo, exs re use to illustrte the pplition of the propose metho for rought nlysis. A goo greement is oserve etween the empiril n theoretil proilities of the rought urtion n severity. he ivrite istriution is then pplie to moel rought urtion n severity jointly. Return perios of rought urtion seprtely n jointly n the onitionl return perios re then estimte. he results show tht the propose metho is useful tool to erive the joint istriution of rought urtion n severity for rought nlysis. Referene González, J., n J. Vlés (3), Bivrite rought reurrene nlysis using tree ring reonstrutions, J. Hyrol. Eng., 8(), Jynes, E. (97), Informtion heory n Sttistil Mehnis, Physil Review, 6(4), Kpur, J. (989), Mximum-entropy moels in siene n engineering, John Wiley & Sons, New York Kenll, D., n J. Drup (99), On the genertion of rought events using n lternting renewl-rewr moel, Stohsti Hyrology n Hyrulis, 6(), -68. Kim,., J. Vles, n C. Yoo (6), Nonprmetri pproh for ivrite rought hrteriztion using Plmer rought inex, J. Hyrol. Eng., (), Kim,., J. Vlés, n C. Yoo (3), Nonprmetri pproh for estimting return perios of roughts in ri regions, J. Hyrol. Eng., 8(), Mthier, L., L. Perreult, B. Boée, n F. Ashkr (99), he use of geometri n gmm-relte istriutions for frequeny nlysis of wter efiit, Stohsti Hyrology n Hyrulis, 6(4),

6 Proility PDF PDF Me, L., n N. Ppniolou (984), Mximum entropy in the prolem of moments, J. Mth. Phys, (8), Zelenhsi, E., n A. Slvi (987), A metho of stremflow rought nlysis, Wter Resour. Res, 3(), Nrjh, S. (7), A ivrite gmm moel for rought, Wter Resour. Res, 43(8), W8, oi:8.9/6wr64 Nrjh, S. (9), A ivrite preto moel for rought, Stohsti Environmentl Reserh n Risk Assessment, 3(6), 8-8. Sls, J., C. Fu, A. Cnelliere, D. Dustin, D. Boeet l., (), Chrterizing the severity n risk of rought in the Poure River, Coloro, Journl of Wter Resoures Plnning n Mngement, 3(), Shnnon, C. E. (948), A mthemtil theory of ommunitions, Bell Syst. eh. J., 7(7), Shiu, J. (3), Return perio of ivrite istriute extreme hyrologil events, Stohsti Environmentl Reserh n Risk Assessment, 7(), Drought urtion (month) Drought severity ( 4 fs month) Figure Comprison of the histogrms n fitte PDFs Shiu, J. (6), Fitting rought urtion n severity with twoimensionl opuls, Wter resoures mngement, (), Shiu, J., S. Feng, n S. Nrjh (7), Assessment of hyrologil roughts for the Yellow River, Chin, using opuls, Hyrologil Proesses, (6), Shiu, J., n H. Shen (), Reurrene nlysis of hyrologi roughts of iffering severity, Journl of Wter Resoures Plnning n Mngement, 7(), Shiu, J.., n R. Morres (9), Copul se rought severity urtion frequeny nlysis in Irn, Meteorologil Applitions, 6(4), Singh, V. P. (998), Entropy-Bse Prmeter Estimtion in Hyrology, Kluwer Aemi Pulishers, Dorreht, he Netherlns 3 4 Drought urtion (month) 4 6 Figure Comprison of the empiril n theoretil proility Yevjevih, V. (967), Ojetive pproh to efinitions n investigtions of ontinentl hyrologi roughts, Hyrology Pper 3, Coloro Stte U, Fort Collins

7 Return perio (yers) Drought Severity ( 4 fs month) Return perio (yers) Drought Durtion (month) le Return perio efine y rought urtion n severity seprtely Return Perio (Yer) Drought urtion Drought severity (Month) ( 4 fs months) Figure 3 Bivrite rought urtion n severity return perios 7 6 s>.x 4 fs month s>x 4 fs month s>.x 4 fs month 6 > months >4 months > 6 months *he mnusript ws upte on Drought Durtion (month) 3 4 Drought Severity ( 4 fs month) Figure 4 Conitionl return perios 6