DATA MINING IN CANADIAN LYNX TIME SERIES

Transcription

1 Jounal of Reliabiliy and Saisical Sudies; ISSN (Pin): , (Online): Vol. 5, Issue 1 (01): DATA MINING IN CANADIAN LYNX TIME SERIES R.Kanaboopahy and D.Venkaesan* Depamen of Saisics, Annamalai Univesiy E Mail: *si_san@homail.com (Received Januay 04, 01) Absac This pape sums up he applicaions of Saisical model such as ARIMA family ime seies models in Canadian lynx daa ime seies analysis and inoduces he mehod of daa mining combined wih Saisical knowledge o analysis Canadian lynx daa seies. Key Wods: Daa Mining; Canadian Lynx Daa; Auoegessive Model; FRAR Model; Foecasing. 1. Inoducion The Time Seies Daa Mining (TSDM) famewok is a fundamenal conibuion o he fields of ime seies analysis and daa mining in he ecen pas. Mehods based on he TSDM famewok ae able o successfully chaaceize and pedic complex, non peiodic, iegula, and chaoic ime seies. The TSDM mehods ove come limiaions namely including saionay and lineaiy equiemens of adiional ime seies analysis echniques by adaping daa mining conceps fo analyzing ime seies. A ime seies {X, = 1,.,,,,,N} is a sequence of obseved daa, usually odeed in ime whee is a ime index, and N is he numbe of obsevaions. Reseaches sudy sysems as hey evolve hough ime, hoping o discove hei undelying pinciples and develop models useful fo pedicing o conolling hem. Tadiional ime seies analysis mehods such as he Box-Jenkins o Auoegessive Inegaed Moving Aveage (ARIMA) mehod can be used o model such ime seies. Howeve, he ARIMA mehod is limied by he equiemen of saionaiy of he ime seies, nomaliy and independence of he esiduals. Residuals ae he eos beween he obseved ime seies and he model geneaed by he ARIMA mehod. The esiduals mus be uncoelaed and nomally disibued. Fo eal-wold ime seies such as sock make pices, he condiions of ime seies saionaiy and esidual nomaliy and independence ae no me. A sevee dawback of he ARIMA appoach is is inabiliy o idenify complex chaaceisics. This limiaion occus because of he goal of chaaceizing all ime seies obsevaions, he necessiy of ime seies saionaiy and he equiemen of esidual nomaliy and independence. The TSDM famewok innovae daa mining conceps fo analyzing ime seies daa. This allows he TSDM mehods o pedic non saionay, non peiodic, iegula ime seies. The TSDM mehods ae applicable o ime seies ha appea sochasic, bu occasionally (hough no necessaily peiodically) conain disinc, bu possibly hidden, paens ha ae chaaceisic of he desied evens.

2 Jounal of Reliabiliy and Saisical Sudies, June 01, Vol. 5 (1) The following is he oganizaion of he pape. Secion wo povides eview of ime seies models associaed wih Canadian lynx daa. Secion hee povides idenifying he paen in he Canadian lynx daa hough FRAR ime seies model. Secion fou povides a compaaive sudy and summay of he pape is pesened in Secion five.. Review of Time Seies Model This secion eviews some paameic models ha have been fied o he Canadian lynx daa, which consiss of he annual ecod of numbes of Canadian lynx apped in he Mackenzie Rive disic of Noh-Wes Canada fo he peiod , boh yeas inclusive, giving heefoe a oal of 114 obsevaions. These daa oiginally appeaed in a pape by Elon and Nicholson (194), which gave a deailed discussion of flucuaions of he size of he lynx populaion in vaious egions of Canada ove a peiod of some 00 yeas. I conains a lage amoun of Saisical infomaion elaing o apping ecods bu his was eaed puely descipively; See Moan (1953), Hannan (1960), Kashyap (1973) and Bulme (1974). Moan (1953) was he fis o analysis he lynx daa. Because he cycle in he aw daa {X } is vey asymmeical wih a shap and lage peak and a elaively smooh and small hough, he used a common log-ansfomaion of he daa and esimaed he AR () model. X = X X - + e Wih e ~ WN (0, ); heeafe WN (0, ) sands fo a whie noise pocess wih mean 0 and vaiance. Moan noing ha one-sep-ahead pedicos fo he daa wee no paiculaly good, howeve, he suggesed ha he pocess would be bee epesened by some kind of non-linea model Since, a fied AR () model does no povide a vey good mach o he sample auo-covaiance funcion, he auo-covaiance funcion of he model damping much moe apidly han ha compued fom he daa. In he ealy days of ime seies analysis, owing o he limiaion of compuing faciliies, mos of he model fied was esiced o hose of vey low odes. Howeve, wih he advanage of high-speed compues, hee is no longe any gound fo his esicion and ou e-examinaion of he lynx daa clealy e-affims he necessiy fo models of he ype Full Range Auoegessive models in some cases. An exensive accoun of he Saisical and hisoical aspecs of he modeling of he lynx daa is conained in Campbell and Walke (1977), whee he daa ae epoduced. Akaike s infomaion cieion (AIC) and he idea of subse model sel ecion wee used fis by Tong (1977) o idenify and esimae seveal AR models fo he lynx daa, obained X = 1.13 X X X X X X X X X X X e

3 Daa Mining in Canadian Lynx Time Seies 3 Whee X = (Y ) fo = 1,, 3,, 114 wih e ~ WN (0, 0.04) Lae, using he andom coefficien auo egessive model, Nicholls-Quinn (NQ), obained he model X = ( B 1 ()) X -1 + ( B ()) X -1 + e () B ( ) B () B ( ) B1 Whee E and especively. and ae esimaed as 3. The Full Range Auoegessive Model This secion povides a bief eview of FRAR models ha ae needed hee. We X, define a family of models by a discee-ime Sochasic pocess 0, 1,,..., called he Full Range Auo Regessive (FRAR) model, by he diffeence equaion X a X e 1 whee a k sin cos /, 1,, 3,... paamees, e 1, e, 3, k,, and ae e, ae independen and idenically disibued nomal andom vaiables wih mean zeo and vaiance. The iniial assumpions abou he paamees ae as follows: of I is assumed ha X will influence X n fo all posiive n and he influence X on X will decease, a leas fo lage n, and become insignifican as n n becomes vey lage, because moe impoan fo he ecen obsevaions and less impoan fo an olde obsevaions. Hence a mus end o zeo as n goes o infiniy. This is achieved by assuming ha 1 magniudes of influence on X n n. The feasibiliy of X having vaious, when n is small, is made possible by allowing k o ake any eal value. Because of he peiodiciy of he cicula funcions sine and cosine, 0,. he domain of and ae esiced o he ineval Thus, he iniial assumpions ae 1, k R, and, 0,, k,, S *, whee *, k,, k R, 1,, 0,. i.e., S. Fuhe esicions on he ange of he paamees ae placed by examining he idenifiabiliy of he model and is finally deduced ha he egion of idenifiabiliy of he model is given by S, k,, kr, 1, 0,, 0, / infomaion on hese opics, we ecommend Venkaesan and Gallo (01).. Fo moe

4 4 Jounal of Reliabiliy and Saisical Sudies, June 01, Vol. 5 (1) FRAR is fied fo he Canadian lynx daa fo he fis one hunded obsevaions obained he model ˆ = X = 1 ( ) sin (.5611) Cos (0.559 ) (3.640) x -1 + e whee and values ae in degee and sandad deviaion of e esimaed as I should be poined ou ha he main pupose of ime seies models is o pedic he fuue. So, he suiabiliy of he new soluion o he lynx daa analysis should be examined only by he abiliy of he soluion o coecly pedic he fuue. This aspec is sudied hough he Bayesian appoach. Venkaesan and Gallo (0 1) have obained he Bayesian pedicive disibuion of he FRAR models. Bayesian pedicive disibuion of he (+l) h obsevaion, using he fis obsevaions, is obained. The mean of his disibuion is aken o be he (+l) h pediced value of he Lynx daa. Since he diec evaluaion of he mean of he one-sep ahead pedicive disibuion involves fou dimensional numeical inegaion, insead of he maginal pedicive disibuion of X N+1, he condiional pedicive disibuion of X N+1, given by Venkaesan and Gallo(01) go by fixing he paamees K,, and a hei esimaes, is used and he mean is calculaed. The poseio mean of he pedicive disibuion is compued numeically afe fixing he paamees a hei especive esimaed value. This pedicion is done fo he cases = 11, 1,, 114 and ae given in he Table I. Table I conains boh he ue values and he one-sep ahead pediced values fo he ansfomed daa 4. Compaaive Sudy Nicholls and Quinnon (198) have used he above daa o compae he qualiy of he pediced values obained by seveal mehods, viz., (1) Moan -1 () Tong (3) NQ-1 (4) Moan- and (5) NQ- as pesened above. Moan-1 efes o he linea pedico obained fom he second ode auoegessive model, Tong efes o he linea pedico fom auoegessive model of ode elevenh, NQ-1 denoes he linea pedico obained fom he second ode andom coefficien model while Moan- and NQ- denoes he non-linea pedicos fo he lynx daa. The models and ohe deails can found in he Nicholls and Quinn (198). Nicholls and Quinn (198) have used hese mehods o pedic he las 14 values of he Canadian lynx daa and calculaed he eo sum of squaes. To compae he efficiency of pedicion of he new FRAR model wih hose of he ohes saed above is given in Table II. The eo sum of squaes fo he las 14 pediced values is unde he FRAR model wheeas hey ae 0.531, 0.541, 0.561, and especively unde he ohe mehods. So, a leas in he above conex he supeioiy of he FRAR model is esablished beyond doub.

5 Daa Mining in Canadian Lynx Time Seies 5 5. Conclusion FRAR model povides he bes fi fo he lynx daa and heefoe, he FRAR model ceainly povides a viable alenaive o he exising ime seies mehodology, fom he pedicive powe of he model and fom he poin of view of pue daa analysis, compleely avoiding he poblem of ode deeminaion in he case of Canadian Lynx ime seies daa. S.No Y Ŷ S.No Y Ŷ S.No Y Ŷ Y - Lynx daa (Tansfomed) Ŷ - One-sep-ahead Pediced value Table I : One-Sep-ahead pediced values of he ansfomed Lynx daa

6 6 Jounal of Reliabiliy and Saisical Sudies, June 01, Vol. 5 (1) S.No Yea Lynx daa Moan-I Tong NQ-1 Moan- NQ- FRAR Eo sum of squaes Table II: One-Sep a head pedicos of he ansfomed lynx daa wih ohe models Acknowledgemen We ae gaeful o he Edio and efeees fo hei many helpful commens ha shaped he syle and pesenaion of his pape. Refeences 1. Bulme, M.G. (1974). A saisical analysis of he 10 -yea cycle in Canada, Jounal of Animal. Ecology, 43, p Cambell, M.J and A.M. Walke (1977). A suvey of saisical wok on he Mackenzie Rive seies of Annual Canadian lynx Tappings fo he yeas and a new analysis, Jounal of Royal Saisical Sociey, A, p Elon, C and M. Nicholson (194). The en yea cycle in numbes of lynx in Canada, Jounal of animal Ecology. 11, p Hannan, E.J.(1960). Time Seies Analysis, Mehuen, London. 5. Kashyap, R.L. (1973). Validaion of sochasic diffeence equaion models fo empical ime seies. Poceeding 1973 Confeence on Decision and conol, san Diego, Califonia, USA. 6. Moan, P.A.P (1953). The Saisical anal ysis of he Canadian lynx cycle, Ausalian Jounal of Zoology, 1, p Nicholls,D.F and B.G.Quinn (198). Random Coefficien Auoe gessive Models: An inoducion, Spinge-Velag, New Yok. 8. Tong, H.(1977a). Some Commens on he Canadian lynx daa, Jounal of he Royal Saisical Sociey, A, 140, p Tong, H.(1977b). Discussion on Sochasic modeling of ive flow ime seies by Lawance and Koegoda. Jounal of he Royal Saisical Sociey, A, p. 140.