BILINEAR TIME SERIES MODEL FOR ESTIMATING A DISEASE DEATH RATE. J. F. Ojo University of Ibadan, Ibadan, Nigeria.

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1 BILINEAR TIME SERIES MODEL FOR ESTIMATING A DISEASE DEATH RATE J. F. Oo Uvy of Iaa, Iaa, Nga E-mal: fmyoo@yahoo.co.k Aac W compa w m- mho fo mag h ah a of a mgg -mgg a o appoach a o O-Dmoal Iga Aogv Bla Tm S Mol Galz Iga Aogv Bla Tm S Mol. Th paam of h popo mol a ma g Nwo-Rapho av mho acal pop of h v ma a vga. A algohm wa popo fo fg h wo mol. To m h o of h mol, Akak Ifomao Co (AIC) wa aop. Ral vaac wa o whch mol pfom. W lla h w cocp wh al lf aa. O-Dmoal Iga Aogv Bla Tm S Mol opfom Galz Iga Aogv Bla Tm S Mol h mao of ah a of a a Kywo: Dah a, mol, al vaac, mgg -mgg a Ioco Th wol coaly h ha of mgg o -mgg a. Dah a mam gv a cao o qaly of halh vc avalal wh a localy a wll a h a of ppa of h halh pov o c h mac of a gv a. Th mo mpoa o halh ca pov o how o ma h ah a of a gv a. A mpoa pmologcal paam of ay a ah a.a mly acca ma of h ah a of a mgg a mpoa. (Oo, 009). (Oo, 0) popo o-moal ga aogv la m mol ha col achv aoay fo all o la m. Alo, Oo & Shagooy (00) popo galz ga aogv la m mol ha col achv aoay fo all o la m. I h lgh of h aov, a amp ma o compa wo la m mol fo mag ah a of a a o o hav a pc of how o ma h ah a. Th ma compa wh h Wol Halh Ogazao (WHO) covcoal fomla fo compao of ah a whch mply h ao of h m of kow ah o h oal m of cofm ca h fomla lkly o ma h ah a ca h ocom of may ca mgh kow o ca a h m h fg a compl. Bla mol log o a cla of pamoo mol ha hav fo o pov a f fo om ologcal, cologcal mcal aa (Mohl, 973) (B, Dpllo & Koch, 974). (Ma, 997) (Goclav, Jaco & M- Lop, 000) hav how ha h cla of mol pov a opmal foca fo val p aha ca pfom ha a la mol. Vao yp of la mol ha col o achv Mol pcfcao aoay fo all o la m hav co y (Pham & Ta, 98), (Ga & Sa L am h followg mpl poalc Rao, 98), (Rao, Sa Rao & Walk, 983), mol. L p h poaly ha a cofm (Shagooy & Oo, 003), (Wag & W, 004), ca p a ah o h h ay af h (Booack, Shol, K. & Shol, E., 005), cofmao of h a. Smlaly, l q (Dokha, Lao, & Oach, 006), (Do, Akk, h & Wk, 007), (Uoo & Omkaa, 008) copog poaly of g cov 9

2 chag fom h hopal o h h ay. Th p 0 p h ah a of h a, q p q h covy a. All ca a 0 am o p of ach oh hav cal ah a poaly o of m o ah (chag). L C h m of cofm ca o h h ay, h copog m of ah y D ha of cov chag ca y R. Each ah o ay m com fom a al cofm ca. Th poaly ha fom ay qal p h a ca o ha ay, o h cooal ma of qal 0 p C C D gv C, = 0,,, 3,... () Fo a f gal la poc, h xpo () w a z l l p C () whl, l 0 a kow low pp o of h m o ah. Th lag l l a cho o ffcly pa h ag of o-zo p '. Th coffc p a xpc o vay moohly wh h lag. I, h copoao of h mooh ampo h mao al ca h mol coa a lag m of lag of C ha ca aly oc mlcollay. Hc, a coa f wll gally yl vy ac ma of h p. Th mooh ampo may ffc y polag om pamoo paamc cla of mol (Tog & Cha, 006). A cla of pamoo mol ha hav fo o pov a f fo ologcal, cologcal mcal aa h la m mol. W hall compa wo mol amly: o-moal ga aogv la mol IARBL(p,,0,,) galz ga aogv la mol GIARBL (p,,0,,) (ca of h fl achvg aoay fo all o la ) o h lf h of qao (); fo al (Oo, 0) (Shagooy & Oo, 00). Mol W w o-moal ga aogv la mol fo mag h ah a a m a: ( B) ( B) wh z p... p k p k z C. (3),... p a h paam of h ga aogv compo;,..., a h paam of h ola compo, p h ah a mol h o m. Mol W w h galz ga aogv la mol fo mag h ah a a m a: ( B) ( B) wh z p... (4) l p kl p k l l Z kl k l 30

3 ,... p a h paam of h ga aogv compo;,..., a h paam of h ola compo, p h ah a mol h o m. wh pc o h paam G,..., ; B,..., B ; p,..., p ). Fo ( p covc, w hall w G, G,..., G R p, wh R = p+++. Th h paal vav of Q( a gv y Mol mao Th mao of h mol a mla, w hall po h mao of galz yp c m,,3..., fo h galz ca cl m h o moal ca. Sppo ha a ga y qao (4), h qc of col m fom h om va lao... p p l To ma h kow paam qao (4), w mak h followg ampo: () Th o a p cally wh ma zo vaac wh f ko. () Th oo of h p( B) p B polyomal l o h ccl, h aoay ofc. () Th val of ' ' ha vly coo q of h la poc af. Fo al (Oo & Shagooy, 00). Th maxmzg h lklhoo fco qval o mmzg h fco Q (, whch a follow: kl kl k Q( G m G,..,R) (5) Q( ( G G m G ( =, l p C W ( ), f =,,, p+ (7) G m ) G G (6) wh h paal vav of afy h cv qao p W ( ) p C,f =,,, (8) W ( ) k m B B (k=,,, ; m =,,...,) (9) ( ) ' W ',, p+) (0) p ' p W ( ) ' p p,) () 0 (, ' =, 0 (, ' =,, m Q(, B W ( ) B k m 0 3

4 (=,,,p+ ; k =,,,; m =,,...,) () p B W ( ) B p k p m h qao followg Kzaowk (998), w hav ˆ G G H ( V(, 0 hy oag a av qao gv y ( ) ( k) ( k) ( k) G G H ( G ) V( G ), (=,,,p+ ; k =,,,; m =,,...,) (3) ' m ( ) ' W ' k B B B B B (k, k' =,,, ; m m ' =,,.) (4) W ( ) B, W ( ) B k W am ha = 0 ( =,,, m-) G,,, m-) 0, G G 0, Fom h ampo h qaly: (, =,,, R; = W ( ) k m (k=,,, ; B B m =,,.) follow ha h co o vav wh pc o ( =,,, p+), p ( =,,, ) zo. Fo a gv of val { }, {B } { p } o ca vala h f co o vav g h cv qao (7), (8), (9) (4). L Q( Q( Q( V (,,..., G G G R l H( [ Q( / G G ] a max of co paal vav. Expg V(, a G G ˆ a Taylo, w oa [ V( ] 0 ( ) ( )( ˆ ˆ V G H G G. Rwg GG k 3 (k ) wh G h of ma oa a h k h ag of ao. Th ma oa y h aov av qao ally covg. Fo ag m h ao, w o hav goo of al ' B val of h paam. Th ca oa a follow: Sppo w wh o f la mol IARBL (p,, 0,, ) GIARBL (p,, 0,, ). W choo h coffc of h ga aogv mol (IAR) pa of h mol qal o h copog IAR mol. Th coffc a a h al val fo ag h ao of h Nwo-Rapho av qao. Algohm fo fg h la m mol Fo h ak of mplcy, w wll ak h algohm ow o h followg p. Sp : F vao o of ga aogv mol of h fom... p p Sp : Choo h mol fo whch Akak Ifomao Co (AIC) mmm amog vao o f p. Th hgh o o f p o h o wh w hav mmm AIC. Sp 3: F pol of cho mol p q g appoach (Haga & Oy, 980). Sp 4: Choo h mol fo whch AIC mmm amog h f mol p 3 o hav h mol h paam of h mol fom h al val. Sp 5: F vao o of ga aogv la mol of h fom... p p...

5 Thfo, fom qao (4), p 0. 04, ao 4%,... p p... R = , AIC =.39960, Ral Vaac = g h mao chq co 3 choo h mol fo whch AIC mmm fo By g h WHO fomla fo ah a o comp h ah a fo h a h o-moal galz mol hopal; w oa 0.0 ao %. pcvly. Th hgh o o f p o Mol h mol wh w hav mmm AIC. ˆ Sp 6: Th paam of h la mol f p 5 fom h al val fo mag ah a g h mao chq a wll. Daa aaly co Th w mhoology hall co o h al lf aa g malaa a ag a; h aa wa collc fom Uvy Tachg Hopal (UCH), Iaa, Nga. Th qaly aa fom 997 o 006 p a o-aoay, h la mol f h pap appl. F la mol a o h algohm co 4 h mol gv a Mol ˆ Th aov mol o ma h ah a. Havg h lco co, h ma mol gv a: ˆ (4) Th aov mol o ma h ah a. Havg h lco co, h ma mol gv a: ˆ C Tal : Dv Sac fo O-Dmoal Galz Mol O-Dmoal Galz Mol Mol Ral Vaac Akak Ifomao Co Coffc of Dmao Dah Ra 0.04 = 4% 0.03 = 3% 3 (5) Thfo, fom qao (5), p 0. 03, ao 3%, R = , AIC =.4077, Ral Vaac = Th aov fomao fo h wo mol a p h al low C Tal aov how ha h al vaac vaac aa a ah a of 4% h, WHO aach o O-Dmoal Bla mol fo fomla lkly o ma h ah a mag ah a mall ha h al ca h ocom of om ca mgh o vaac aach o Galz Bla mol. Th kow a h m of complao qao mplcao ha o-moal la (3) may ack h poc m of ah opfom galz la mol fo lavly wll. mag ah a. Thfo, y g h Wol Halh Ogazao (WHO) fomla fo ah a o Coclo comp h ah a fo h a h hopal; w oa 0.0 ao %. Sc mmm 33

6 Th y foc o wo w la mol fo mag ah a of a gv a wh h am of mg a mol o of h wo wh mag h ah a of a gv a. Ral vaac aach o h mol w o m h mol. I wa fo o ha O-Dmoal Iga Aogv Bla mol opfom h Galz Iga Aogv Bla mol. Th lklhoo of mao ha hav occ g WHO fomla wa mpov po wh h mol wa appl o al lf aa. Th mao of paam of h mol w a q, co covg mao ha ha pv h mol fom xplog, hy makg aoay pol. Rfc Booack, K. Shol, B.K. & Shol E. (005). Mlvaa Bla Tm S: A Sochac Alav Poplao Dyamc. Gophycal Rach Aac. Volm 7, Gocc Uo 005. B, C., Dpllo, G. & Koch, G. (974). Bla Sym: A Appalg Cla of Naly La Sym Thoy Applcao. IEEE Ta. Ao Cool. 9, Dokha, P., Lao, A., & Oach, D., (006). A Smpl g Val Bla Tm S Mol. Avac Appl Poaly. 38, Do, F.C., Akk, R. V. & Wk, B. J. M. (007). No o Ig Val Bla Tm S Mol. Ecoomc Fac gop C. ISSN , Tlg Uvy, h Nhl. Ga, M. M.& Rao, T. S. (98). Th Emao Pco of S Bla Tm S Mol wh Applcao. Joal of Tm S Aaly. (3), Goclav, E., Jaco P. & M-Lop, N. (000). A Dco Poc fo Bla Tm S Ba o h Aympoc Spaao. Sac, Hagga, V. & Oy, O. B. (980). O h Slco of S Aogv Tm S Mol: UMIST (Dp. of Mahmac). Tchcal Rpo. No. 4. Kzaowk, W.J. (998). A Ioco o Sacal Mollg. Aol: h UK. Ma, C. M. (997). A No o h Th-O Mom Sc of a Bla Mol wh No-Ip Shock. Pogala Mahmaca. Volm 56, Mohl, R. R. (973). Bla Cool Poc. Nw Yok: Acamc P. Oo J. F. (009). Th Emao Pco of Aogv Movg Avag Bla Tm S Mol wh Applcao. Gloal Joal of Mahmac Sac. (), -7. Oo, J. F& Shagooy, D. K.(00). Thoy of Galz Iga Aogv Bla Tm Mollg. Joal of Egg Appl Scc. Volm, Oo, J. F (0). O h Thoy of O-Dmoal Iga Aogv Bla Tm S Mol. If Joal of Scc. 3(), Pham, T.D. &Ta, L.T. (98). O h F O Bla Tm S Mol. Jo. Appl. Poaly. 8, Rao, M. B., Rao, T. S. & Walk, A. M. (983). O h Exc of om Bla Tm S Mol. Joal of Tm S Aaly. 4(), Uoo, A. E. & Omkaa, C. O. (008). Low Dagoal Bla Movg Avag Vco Mol. Avac Appl Mahmacal Aaly. 3(), Tog, H & Cha, K (006): Emag h Dah Ra of a Emgg Da y Tm S Aaly: Dpam of Sac & Acaal Scc, Uvy of Iowa, Iowa, USA. Tchcal Rpo. Wag H.B. & W B.C. (004) Spaal Low Tagla Bla Mol. J. Appl Poaly. 4(),