Polynomial Representations for a Wavelet Model of Interest Rates

Transcription

1 Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No.5, Deceber 015 Polyoial Represetatios for a Wavelet Model of Iterest Rates Deis G. Lleit Departet of Matheatics, Adaso Uiversity 1000 Sa Marcelio Street, Erita, Maila, Philippies deis_arwid@yahoo.co Date Received: Deceber, 015; Date Revised: Jauary 15, P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research Vol. 3 No.5,6-71 Deceber 015 Part III P-ISSN E-ISSN Abstract - I this paper, we approxiate a o polyoial fuctio which proises to be a essetial tool i iterest rates forecastig i the Philippies. We provide two uerical schees i order to geerate polyoial fuctios that approxiate a ew wavelet which is a odificatio of Morlet ad Mexica Hat wavelets. The firss the Polyoial Least Squares ethod which approxiates the uderlyig wavelet accordig to desired uerical errors. The secod is the Chebyshev Polyoial approxiatio which geerates the required fuctio through a sequece of recursive ad orthogoal polyoial fuctios. We seek to deterie the lowest order polyoial represetatios of this wavelet correspodig to a set of error thresholds. Keywords: Chebyshev polyoials, polyoial least squares, Torre Escaer wavelet INTRODUCTION A waveles a atheatical fuctio thas used to divide a give fuctio or a cotiuous- tie sigal ito differet scale copoets. Sice wavelet trasfors are represetatio of fuctios siilar to those of Fourier trasfors, they are used i data copressio ad sigal processig. Over the years, wavelets have bee eployed i research udertakigs i a broad rage of fields fro electrical egieerig ad coputer sciece to ecooics ad fiace. The utility of wavelets lie i their capacity to ake sese of real world data which are ostly ostatioary, o-periodic, ad oisy. The data are trasfored such that aalyses ad ifereces o the ca be doe easily. Such trasforatio echais is ot iheretly uique to wavelets. Joseph Fourier coceptualized the Fourier trasfor i The ethod uses orthogoal basis fuctios to represet cotiuous ad periodic fuctios. For p R, 1 p < +, let x t L p be the sigal fuctio to be aalyzed. The its Fourier trasfor is defied as f ξ = where ξ is i Hertz. + x t e πiξ dt Fourier trasfor was a success i fuctio trasforatio but has a ajor shortcoig. Is ot copactly supported i tie. Thus, is ot a suitable trasforatio for o-statioary data. The Short Tie Fourier Trasfor (STFT) was coceptualized to correct this drawback. Is give by S τ, f = x t w t τ e πift dt where w(t) is the widowig fuctio, τ is the tie axis, f ad t are frequecy ad traslatio paraeters, respectively, ad is the coplex cojugate operator. The STFT is copactly supported i tie. However, it uses the sae widow for the aalysis of the etire sigal. This brought difficulties i data aalyses. If the sigal to be aalyzed has high frequecy copoets for a short tie spa, arrow widows are used. But such arrow widows ea wider frequecy bads which result to poor frequecy resolutio. O the other had, if the sigal features low frequecy copoets of loger tie spa, the a wider widow eeds to be used to obtai good frequecy resolutio. This liitatio posed by the STFT was the drivig force i fidig aother fuctio trasfor. I 1970, Jea Morlet, a geophysical egieer, ad Alex Grossa, a theoretical physicist costructed aother fuctio trasfor which has versatile widow fuctio. This ew fuctio trasfor was aed wavelet ad it was so successful that several variats

2 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates have bee costructed to suit the eeds of a wide variety of pheoea ad applicatios. I 010, Noei Torre ad Jose Maria Escaer, IV, developed a ew wavelet which we call the Torre Escaer Wavelet. This ew wavelet fits historical data of Philippie 90 day T bill rates fro 1987 up to 008. It was costructed usig the Morlet ad Mexica Hat wavelets ad logically iherits their itrisic properties. The Morlet waveles kow to offer iproved detectio ad localizatio of scale over the Mexica Hat. O the other had, the Mexica Hat wavelet provides better detectio ad localizatio of patch ad gap evets over the Morlet wavelet. The Torre Escaer waveles give by W t = 3 π 1 4(1 1.1t) e (1.1t) e i7.7t (1) where t stads for tie ad W(t) for frequecy.this ew wavelet fits historical data of Philippies 90 day T bill rates ad ca be potetially used to odel Philippie iterest rates accordig to Torre ad Escaer [1]. I the areas of coputig ad uerical aalysis, polyoial fuctios receive preferetial treatet. This is because their structures, beig based o eleetary operatios of additio ad ultiplicatio, are siple to huas ad to coputers. I the cotext of coputatioal, tie ad space coplexities, polyoial fuctios are uch efficiet copared to their o polyoial couterparts. Thas why, if possible, o-polyoial fuctios are trasfored as polyoials via uerical approxiatio ethods. As is the case with such approxiatio techiques, there is always the error trade off. Thas, the error varies iversely to the degree of the approxiatig polyoial. OBJECTIVES OF THE STUDY I this paper, we ited to (a) represet the Torre Escaer wavelet by a polyoial fuctio P (t) via uerical ethods; ad (b) deterie the root-ea-squared errors, E RMS = 1 W t P (t), of these polyoial represetatios. We are iterested i polyoial fuctios as approxiatios for the Torre Escaer wavelet for three reasos. Firstly, polyoial fuctios are geerally siple ad easy to aipulate. Secodly, polyoial fuctios are algorithically easy to ipleet ad coputatioally efficiet. Note that these two are the reasos why we oly wat a polyoial approxiatio of practical degree. Although a plethora of uerical schees ca easily give a approxiatio for (1), they are ostly higher ordered that they becoe ipractical for our goal. Thirdly, we wat to provide a uerical basis or treatet for the wavelet that ca odel Philippie T bill rates. METHODS We eploy techiques ad ethods provided i Nuerical Aalysis such as Polyoial Least Squares (PLS) ad Chebyshev Polyoial approxiatios to costruct the approxiatig or iterpolatig fuctio P (t). Polyoial Least Squares Approxiatio For the Polyoial Least Squares (PLS) approxiatio, we seek to costruct a degree polyoial, P t = a 0 + a 1 t + a t + + a t () fro the Torre Escaer wavelet W(t), such that the root- ea squared error, E RMS is as sall as possible, i.e., Let E RMS = 1 W t P t < ε (3) G a 0, a 1,, a = W P t i = [W a 0 a 1 a a ] a 1 = 0, a 0 = 0, 63 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

3 Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No.5, Deceber 015 To iiize E RMS, we set the followig: ad solve the subsequet equatios which siplifies to where a = 0, = W t a i a 0 a 1 a t i a = 0, 0 = W t a i a 0 a 1 a t i a = 0, 1 = W t a i a 0 a 1 a t i a t i = 0, a 0 + a a 0 + a 1 a 1 a 1 + a t i = W( ), t i + a t +1 i = W( ), t 1 i + a t i = W( ). (4) Accordig to Atkiso ad Ha [], (4) is a syste of oral equatios which is equivalet to X T Xa = X T W (5) X = 1 t 1 t 1 1 t t 1 t t t 1 t t ad W = W(t 1 ) W(t ) W(t ) 64 P-ISSN E-ISSN

4 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates Expressio (5) is also equivalet to 1 W( ) a 0 a 1 a = W( ) W( ), which ca be expressed as La = b (6) where 1 L = a = a 0 a 1 a, ad b = W( ) W( ) W( ). Now, if L is ivertible, the L 1 exists ad atrix a ca be solved. 65 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

5 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates Chebyshev Polyoial Approxiatio Istead of ooial basis, Chebyshev Polyoial approxiatio uses differet bases vectors. Let where 1 t 1. We kow that Thus, we have T t = cos θ = cos (arccos(t)) (7) cos( + 1)θ + cos( 1)θ = cos θ cos θ. T +1 t = tt t T 1 t. (8) Hece, the basis vector is of the for {T 0 t, T 1 t, T t,, T t } which is also a basis vector of polyoials i ters of t. Accordig to a theore by Weierstrass [3], if P terpolates a give fuctio f (t) o the zeros of T +1 (t), the f t P t ax + 1! t 0 t t f +1 t (9) where t 1,1. This theore guaratees that a good fit of f (t) ca be foud. Hece is iperative to deterie the zeros of T +1 t. That ca be doe by siply settig (7) to zero ad solvig for the correspodig θ values. cos θ = 0 θ = k + 1 π, k = 0,1,,, 1, Let θ = k + 1 π, k = 0,1,,, 1. t k = cos θ = cos( k + 1 π ) ; k = 0,1,,, 1 (10) be the kth zero of (7). The the set geerated by (10) is called the set of Chebyshev odes of (7). Next, we ote that sice {T 0 t, T 1 t, T t,, T t } is a basis for the set of polyoials, the the set is liearly idepedet ad T i (t) is orthogoal to T j (t) for i j fro Kola ad Hill[4]. Hece, we ca set P t = c k T k t, (11) k=0 66 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

6 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates a liear cobiatio of these basis vectors. Thus, we eed to deterie each coefficiet c k to coe up with the Chebyshev polyoial approxiatio P (t) for f t. P (t) iterpolates f t at the ( +1) Chebyshev odes so that at every ode t k, Hece, j =0 f t k = P t k. f t j T k t j = c i T i (t j )T k (t j ) i=0 j =0 = c i K i δ ik (orthogoiality of T i ad T k ) i=0 = c k where K i = Thus, the coefficiets ca be obtaied by c k = + 1 j =0 f t j T k t j (1) wheret j = cos j +1 π +1. RESULTS AND DISCUSSION After cosiderig several choices, we coe up with the coo iterval of iterest for (1). The iterval is [-1,1] because this is the oly iterval i which our Matlab codes give quality results. For oe, the graph outside [-1,1] explodes. For copariso purposes, we set a axiu root- ea square error, E RMS, of 0.10 correspodig to 10 percet. This threshold ight be cosidered large but settig a very sall value ight result i very high polyoial degrees. Least Square Approxiatio Results Usig Matlab, the iiu degree of the approxiatig polyoial is = = 0. For values larger tha 0, the Matlab progra returs a warig ote that the solutio to the atrix syste (6) ay be iaccurate. This is because L is a Hilbert atrix [5]. As the atrix becoes large, its deteriat teds to zero. Thus L becoes o- ivertible. For this particular degree, the Matlab progra returs the polyoial represetatio for (1): P 0 t = i t i t i t i t i t i t i t i t i t i t i t i t i t it t it t it 3 7.9t 6.68it (13) 67 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

7 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates Its graph is However, its root- ea- squared error is Figure 1: Graph of P 0 (t) with W(t) i circles E RMS = 1 W t P 0 t = , uch bigger tha the threshold root- ea squared error. Chebyshev Polyoial Approxiatio Results For the Chebyshev Polyoial approxiatio, Matlab was able to fid a polyoial approxiatio that satisfies the threshold error. I fact, it was able to fid ore tha oe polyoial i the iterval [-1,1]. Sice we wat a polyoial of iiu degree, we oly get the three lowest, i ters of degree, polyoial represetatio. The lowest degree polyoial represetatio is P 10 t = 97.36t it t it t it t it t 5.8it (14) 68 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

8 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates Its graph is ad the correspodig root- ea- square error is Figure : Graph of P 10 (t) with W(t) i circles E RMS = 1 W t P 10 t = 0.079, which is less tha the threshold error The ext lowest degree polyoial represetatio is P 11 t = it t it t it t it t it 3.97t + 6.6it (15) Its graph is Figure 3: Graph of P 11 (t) with W(t) i circles 69 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

9 Lleit, Polyoial Represetatios for a Wavelet Model of Iterest Rates while its root- ea- squared error is E RMS = 1 W t P 11 t = , expectedly lower tha P 10 t.+ Its graph is Lastly, the third lowest degree polyoial represetatio is P 1 t = 100.6t it t it t it t it t it t it (16) Its root- ea- squared error is Figure 4: Graph of P 1 (t) with W(t) i circles E RMS = 1 W t P 1 t = , uch lower tha P 11 t ad P 10 t. 70 P-ISSN E-ISSN Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No. 5, Deceber 015

10 Asia Pacific Joural of Multidiscipliary Research, Vol. 3, No.5, Deceber 015 CONCLUSION The purpose of this paper was to fid polyoial approxiatios to the Torre- Escaer wavelet with ideal root- ea- squared errors. We first used the Polyoial Least Squares (PLS) approxiatio schee ad the results showed that the oly resultig represetatio (13) did ot satisfy our threshold E RMS. This eat that PLS failed to provide a good polyoial approxiatio for (1). O the other had, the Chebyshev Polyoial approxiatio schee was able to provide good polyoial approxiatios for (1) at lower polyoial degrees. This is a validatio of the eariiax ature of the Chebyshev Polyoial approxiatio. Uder this schee, the lowest degree polyoial represetatio for the Torre Escaer wavelet that satisfied the threshold E RMS was P 10 t = 97.36t it t it t it t it t 5.8it For future works, we would like to exaie other polyoial approxiatio schees such as the Legedre, Gegebauer, Jacobi ad other orthogoal polyoials whether they ca provide better approxiatios tha the Chebyshev schee. REFERENCES [1] N. Torre ad J.M.L. Escaer IV. A New Wavelet based o Morlet ad Mexica Hat Wavelets. Matiyas Mateatika, 010 [] K. Atkiso ad W. Ha. Eleetary Nuerical Aalysis, Third Editio. Joh Wiley ad Sos, New York, 004. [3] E. W. Cheey. Itroductio to Approxiatio Theory. AMS Chelsea Publishig, Rhode Islad, [4] B. Kola ad D. Hill.Eleetary Liear Algebra. Pearso Educatio, New Jersey, 008. [5] W. Yag, W. Cao, T. Chug, ad J. Morris. Applied Nuerical Methods Usig MATLAB. Joh Wiley ad Sos, New Jersey, 005. Copyrights Copyright of this article is retaied by the author/s, with first publicatio rights grated to APJMR. This is a opeaccess article distributed uder the ters ad coditios of the Creative Coos Attributio licese ( coos.org/liceses/by/4.0/) 71 P-ISSN E-ISSN