Naure an Science ;8() Analysis of Non-Saionary Time Series using Wavele Decomposiion Lineesh M C *, C Jessy John Deparmen of Mahemaics, Naional Insiue of Technology Calicu, NIT Campus P O 673 6, Calicu, Kerala, Inia. lineesh@nic.ac.in, essy@nic.ac.in Absrac: The increase compuaional spee an evelopmens in he area of algorihms have creae he possibiliy for efficienly ienying a well-fiing ime series moel for he given nonsaionary-nonlinear ime series an use i for preicion. In his paper a new meho is use for analyzing a given nonsaionary-nonlinear ime series. Base on he Muliresoluion Analysis (MRA) an nonlinear characerisics of he given ime series a meho for analyzing he given ime series using wavele ecomposiion is iscusse in his paper. Afer ecomposing a given nonsaionary-nonlinear ime series in o a ren series an a eail series he ren series an he eail series are separaely moele. Moel T() represening he ren series an he Threshol Auoregressive Moel of orer (TAR()) represening eail series are combine o obain he Tren an Threshol Auoregressive(T-TAR) moel represening he given nonsaionary-nonlinear ime series. The scale epenen hreshols for he T- TAR moel are obaine using he eail series an using he ren series. Also simulaion suies are one an he resuls reveale ha he evelope meho coul increase he forecasing accuracy. [Naure an Science ;8():53-59] ( ISSN: 545-74). Keywors: Non Saionary-nonlinear Time Series; Wavele Decomposiion; Tren Moels; Threshol Auoregressive Moels; Scaling Coefficiens; Wavele Coefficiens.. Inroucion Many sochasic sysems are observe o be nonlinear which governs o nonsaionary nonlinear ime series or signals. The Annual sunspo ime series, he Canaian lynx series (Priesley, 988) Financial ime series (Hynman, 8) are examples of nonsaionary nonlinear ime series. So moeling nonsaionary-nonlinear ime series/signals for preicion is nee of he ay. Curvilinear regression moels, Threshol Auoregressive (TAR) moels, Sae Depenen Moels, ec are use for moeling nonsaionarynonlinear ime series ( Mariais, 99; Priesley,988). Bu accuracy in preicion of nonsaionary-nonlinear ime series/signals was one of he main issues associae wih he exising moels. The increase compuaional efficiency leas o he applicaion of wavele ecomposiion meho as a ool for moeling nonsaionary-nonlinear ime series (Kans, 3; Kuo, 994; Minu, 8; Nason, 999; Papoulis, 99). This meho leas o high accuracy in preicion. This paper iscusses he ecomposiion of a given nonsaionarynonlinear ime series in o a ren series an eail series. The Wol ecomposiion heorem (Hayes, 4) saes ha a given ime series can be splie in o ren series an eail series. I is esablishe (Lineesh, 8) ha he resulan ime series obaine by wavele ecomposiion are he same as he ren an eail series ue o Wol (Hayes, 4). Here insea of using he convenional reconsrucion of he ime series using wavele, he ren series an eail series are moele separaely an he moel represening he given ime series is obaine as a combinaion of boh he moels (Lineesh, 8) which aes care of he ime epenencies of he series an his combine Tren an Threshol Auoregressive moel (T-TAR) is use for preicion.. Review of Lieraure The fiing of moels for nonsaionarynonlinear ime series raises some complex issues lie he eerminaion of he bes fie moel o he given ime series. Srang (998) iscusse how o ecompose a signal in o is wavele coefficiens an reconsruc he signal from he coefficiens. Brocwell an Davis (995) hp://www.sciencepub.ne 53 marslanpress@gmail.com
Naure an Science ;8() iscusse a meho for ienying he orer of he ime series moel which is base on he paerns presen in higher orer cumulans. Sesay an Subba Rao (988) erive ule-waler ype ference equaions for higher orer momens an cumulans for cerain class of Bilinear ime series moels. Subba Rao, M. Euara, e al. (99) exene he iea for he enaive ienicaion of he orer of he Bilinear moel. uan Li an hongie ie (98) engage in he suy of he ienicaion of he hreshols an ime elay of TAR moels by checing feren empirical waveles of he given aa. Oye () inrouce a new approach for moeling nonlinear ime series base on wavele smoohing. Saner Solani (), Hayes (4), Ko an Vannucci (6) are also conribue o he wavele analysis echniques for he analysis of nonsaionary-nonlinear ime series. Nason an Von Sachs (999) give an overview of he wor on wavele applicaions of ime series. Iniially he applicaions of wavele ransform for ime series analysis were focuse on perioogram analysis an cycles evaluaion. 3. Esimaion of T-TAR Moels using Wavele Decomposiion Meho 3. Wavele Decomposiion of Nonsaionarynonlinear Time series To obain a moel for preicion of he given nonsaionary-nonlinear ime series i is require o ecompose he given ime series in o he ren series an he eail series so ha an are orhogonal. A given ime series : =,,,..., N { } can be ecompose as = +, =,,,..., N is he ren series an series given by, = M = is he eail, (), is he h level eail series, using wavele ecomposiion echnique. Lineesh (8) prove ha he componens C M, = M = = an of ime series obaine by wavele ecomposiion saisfy he requiremens of Wol's ecomposiion of he ime series. 3. Esimaion of Threshol Auoregressive Moel Using Wavele Techniques The l hreshol auoregressive moel of orer, i.e. TAR () moel (Priesley, 988) is efine as, = a, + ai i () i= R for =,,..., l, R being a given subse of he real line R. In (), e, =,,..., l an ( he se of inegers) is a sequence of inepenen an ienically isribue ranom variables wih mean, consan variance σ an a, i are consan coefficiens. The TAR () moel is esimae for represening he eail series by applying wavele ecomposiion meho. Deerminaion of he coefficiens an hreshol are he main issues while analyzing a nonsaionary-nonlinear ime series using TAR moel. In his paper he coefficiens an hreshols are esimae as follows. The scale an wavele coefficiens are efine as; φ, =, ψ, =, = φ( ),..., = ψ( ),..., J; = J; =,,...,,,...,,, i.. (3) hp://www.sciencepub.ne 54 marslanpress@gmail.com
Naure an Science ;8() φ ψ an ( + / ) ( + / ) ( + ) ( + / ) ( + / ) ( + ) N,,. = Define = ψ (4) (5) β (6) is he eail series obaine by ecomposing using wavele ecomposiion. Then using (6) J = = =,.ψ, β (7) 3.. Esimaion of he Threshol The hreshol of he TAR moel is esimae as follows. For =,, efine..., J λ = log( #, ), (#, ) enoes he carinaliy of { } λ = Here log #C M,,. Also efine λ enoes he hreshol for he h level eail series an λ enoes he hreshol of he TAR moel. 3.. E simaion of TAR moel The Threshol Auoregressive moel is given by, represening he eail series { } b + b +... + b b + b +... + b e (8) ( i) he coefficiens { b } are efine by, b ( i), an ψ, anψ =,, = ψ,,, =,. ψ = ψ. ψ,,,, 3.3 Moel for Tren Series,,,,. (9) () () The bes fiing ARMA (p, q) moel, linear regression moel an curvilinear regression moel are consiere for he analysis of ren series. The moel hus obaine for ren series is enoe by T (). 3.4 Tren an Threshol Auoregressive Moel (T-TAR) The T-TAR moel represening he given nonsaionary-nonlinear ime series, using wavele ecomposiion is obaine by combining he moel represening he ren series an he eail series which is given by; + b + b T +... + b T... + b + b + b () hp://www.sciencepub.ne 55 marslanpress@gmail.com
Naure an Science ;8() Here T() an TAR () preserves orhogonaliy. 4. Applicaion of T-TAR Moels for Preicion Preicion using ime series originae from a sochasic sysem is he very aim of moeling a ime series. The esimaion of T-TAR moel by applying wavele heory is emonsrae wih feren real worl ime series an he resuls are presene here. 4. Analysis of he Time Series of Annual Sunspo Numbers The ime series of annual sunspo numbers uring years 7 955 (Priesley, 988) is aen for illusraing he esimaion of T-TAR moel explaine in his paper. The plo of he ime series is shown in figure. 4. Analysis of Soc Exchange Time Series To see variey of applicaions he meho is applie for he analysis of soc exchange ime series. The ime series represening monhly weighe-average exchange value of U. S. Dollar saring from Sepember 977 o December 998 is aen for illusraing he meho iscusse in his paper. This is a seconary aa (Hynman, 8). The plo of he aa is given in figure. 6 5 4 3 Figure : Plo of soc exchange ime series 8 6 4 Figure : Plo of ime series of sunspo numbers 9 8 5 5 5 3 8 6 4 5 5 5 3 4.. T-TAR Moel Esimaion of he Time Series of Annual Sunspo Numbers Using he meho explaine in his paper T- TAR moel is esimae for he ime series of sunspo numbers using he wavele meho an i is given in Table. 4.. T-TAR moel Esimaion of he Soc Exchange Time Series Using he meho explaine in his paper T- TAR moel is esimae for soc exchange ime series using he wavele meho an i is given in able 3. 4.. Moel Esimaion of Soc Exchange Time Series using he Exising Meho Using he exising meho he moel represening he ime series is esimae. The analysis resuls of he soc exchange ime series using he exising meho ue o Priesley is inclue in Table 4. 4.. Moel Esimaion of he Time Series of Sunspo Numbers using he Exising Meho The commonly use meho for analyzing nonsaionary-nonlinear ime series is ue o Priesley. Using his meho he moel represening he ime series of annual sunspo numbers is esimae. The analysis resuls using Priesley s meho is inclue in Table. hp://www.sciencepub.ne 56 4.3 Analysis of IBM Soc Price Time Series The ime series of aily closing IBM soc prices (Hynman, 8) is aen for illusraing he esimaion of T-TAR moel explaine in his paper. The plo of he aa is shown in Figure 3. marslanpress@gmail.com
Naure an Science ;8() 65 Figure 3: Plo of IBM soc price ime series 4.3. Moel Esimaion of IBM Soc Price Time Series using he Exising Meho 6 55 5 45 Using he exising meho ue o Priesley he moel represening he IBM soc price ime series is esimae. The analysis resuls using Priesley's meho is inclue in Table 6. 5. Conclusions 4 35 5 5 5 3 4.3. T-TAR Moel Esimaion of he IBM Soc Price Time Series Using he meho explaine in his paper T- TAR moel is esimae for IBM soc price ime series using he wavele meho an he T- TAR moel esimae is given in Table 5. In his paper a new meho for analyzing nonsaionary-nonlinear ime series using wavele ecomposiion is inrouce. Uner his meho he given nonsaionary-nonlinear ime series is ecompose ino ren an eail series. Afer ecomposiion of he given ime series he resulan series are moele separaely an hen he T-TAR moel for he given ime series is obaine by combining he moels represening he ren series an eail series. This meho gives a comprehensive algorihm for analyzing nonsaionary-nonlinear ime series which is an avanage over he exising meho. The evelope meho is verie using feren ime series. The evelope meho is compare wih he exising meho an he error analysis in Table 7 shows he efficiency of he meho in improving he accuracy in preicion. Table : Esimae T-TAR moel for he ime series of annual sunspo numbers Threshol Esimae Moel MAPE MSE 3.33.99.3 5.6.79 4.484 < 3.33 8.89.7 3.99 +.3 4.8 3.33 7.9 + 5.48 3 Table : Analysis of ime series of sunspo numbers using Priesley s meho Threshol Esimae Moel MAPE MSE 35 3.88 6.537.539.96 +.483 3.54.7 +.7 3 +.5 4 +.9 5 +.45 6 < 35 35 hp://www.sciencepub.ne 57 marslanpress@gmail.com
Naure an Science ;8() Table 3: Esimae T-TAR moel for he soc exchange ime series Threshol Esimae Moel MAPE MSE 3.5..48.99.57.79.58.99.57. +.499 < 3.5 3.5 Table 4: Analysis of soc exchange ime series using Priesley s meho Threshol Esimae Moel MAPE MSE 9.964.55.3.44 +.5 3.9 4 +.3 5.3 6 +.3 7.8 8 +.39 9.49 +.58.8 +.4 3.69.6 3 3 < 9 9 Table 5: Esimae T-TAR moel for he IBM soc price ime series Threshol Esimae Moel MAPE MSE 8.744.4 7.696.99 +.74 3..659 8. 3 37.34 4.99.74 +.86 +.76 +.9.36 4 3 < 8.744 8.744 Table 6: Analysis of IBM soc price ime series using Priesley s meho Threshol Esimae Moel MAPE MSE 56.466 84.48.93.93.3.338 +.76 3 +.45 + 4.8 5.6.6 7 +.57 8 6 < 56 56 Table 7: Error Comparison of T-TAR moel an Moel ue o Priesley Sr. No. Time Series T-TAR Moel Moel ue o Priesley MAPE MSE MAPE MSE. Sunspo.79 4.484 3.88 6.537. Soc Exchange..48.964.55 3. IBM Soc Price.4 7.696.466 84.48 hp://www.sciencepub.ne 58 marslanpress@gmail.com
Naure an Science ;8() Corresponence o: Lineesh M C, C Jessy John, Deparmen of Mahemaics, NIT Calicu, Kerala, Inia Telephone: 9-495 8657 Cellular Phone: 994953864 Emails: lineesh@nic.ac.in, essy@nic.ac.in References: [] Brocwell, P.J., Davis, R.A. Time Series: Theory an Mehos. Springer. 995. [] Hayes, M.H. Saisical Digial Signal Processing an Moeling. John Wiley an Sons. 4. [3] Hynman R. J. Time Series Daa Library,www.robhynman.com/TDSL/. 8. [4] Ip, W. C, Wong, H, Li,, ie,. Threshol Variable Selecion by Waveles in Open loop- Threshol Auoregressive Moels. Saisics an Probabiliy Leers. 999. Vol 4:4: pp.375-39(8). [5] Kans, H., Schreiber, T. Nonlinear Time Series Analysis. n Eiion, Cambrige Universiy Press. 3. [6] Ko, K., Vannucci, M. Bayesian wavele analysis of auoregressive fracionally inegrae movingaverage processes. Elsevier, Science Direc. 6: 36: 345-3434. [7] Kopsinis,., McLaughlin, S. Empirical Moe Decomposiion Base Sof Thresholing. Proceeings of he 6 h European Signal Processing Conference, EUSIPCO. 8. [8] Kuo, R. J. Auomae Surface Propery Inspecion using Fuzzy Neural Newors an Time Series Analysis. Worl Congress on Neural Newors, San Diego. Vol. 994. [9] Lineesh, M. C., Jessy John, C. Ienicaion of Threshol Auoregressive Moels using Wavele Basis. Proceeings of he Naional Conference on Recen Developmens an Applicaions of Probabiliy Theory, Ranom Process an Ranom Variables in Compuer Science. 8. [] Mariais, S. Sliing Simulaion: A New Approach o Time Series Forecasing. Managemen Science.99: Vol 36: No. 4. [] Minu, K. K., Jessy John, C. A Mahemaical Discussion on Oulier Eliminaion an Gibbs Error in Wavele Neural Newors. Proceeings of he Naional Conference on Recen Developmens an Applicaions of Probabiliy Theory, Ranom Process an Ranom Variables in Compuer Science. 8. [] Nason, G. P., von Sachs, R. Waveles in Time Series Analysis. Phil. Trans. R. Soc. Lon, A, 999: 357, 5-56. [3] Oye, A. J. Nonlinear Time Series Moeling: Orer Ienicaion an Wavele Filering. Inersa. Journals.. [4] Papoulis, A. Probabiliy, Ranom Variables, an Sochasic Processes, 3 r Eiion. McGraw-Hill. 99. [5] Percival, D. B., Walen, A. T. Wavele Mehos for Time Series Analysis. Cambrige Universiy Press.. [6] Priesley, M. B. Non-linear an Nonsaionary Time Series Analysis. Acaemic Press.988. [7] Rao, R. M., Bopariar, A.S. Wavele Transforms Inroucion o Theory an Applicaions. Pearson Eucaion. 998. [8] Rao, T. S., Euara, S. Ienicaion of Bilinear Time Series Moels. Saisica Sinica.99:: 465-478. [9] Sesay, S. A. O., Rao, T.S. ule-waler Type Dference Equaions for Higher Orer Momens an Cumulans for Bilinear Time Series Moels. Journal of Time Series Analysis.988: 9: 385-4. [] Solani, S. On he use of he wavele ecomposiion for ime series preicion. Elsevier, Neurocompuing.: 48: 67-77. [] Srang. Long-Term Preicion, Chaos an Aricial Neural Newors. Journal of Roayal Saisical Sociey. 998. [] Sysel, P., Misurec, J. Esimaion of Power Specral Densiy using Wavele Thresholing. Proceeings of he 7h Conference on Circuis. Sysems, Elecronics, Conrol an Signal Processing, 8: p.7-. Tenere Canary Islans, Spain. [3] Wei, W.W.S. Time Series Analysis Univariae an Mulivariae Mehos. Aison-Wesley Publishing Company. 99. [4] Malla, S. A wavele our of signal processing. Acaemic Press. 999. [5] Chrisopouloul, E.B., Soras, A. N., Georgailas, A. A. Time Series Analysis of Sunspo Oscillaions Using he Wavele Transform. Digial Signal Processing. :: pp.893-896. [6] www.freelunch.comhp://www.eonomy.com/freelunch. Dae of Submission: 7--9 hp://www.sciencepub.ne 59 marslanpress@gmail.com