Sains Malaysiana 41(12)(2012): 1635–1642

 

Parameter Estimation of Stochastic Differential Equation

(Penganggaran Parameter Persamaan Pembeza Stokastik)

 

 

Haliza Abd. Rahman*, Arifah Bahar & Norhayati Rosli

Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia

81310 Skudai, Johor, Malaysia

 

Madihah Md. Salleh

Department of Biotechnology Industry, Faculty of Biosciences and Bioengineering

Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia

 

Received: 11 January 2011 / Accepted: 21 July 2012

 

ABSTRACT

Non-parametric modeling is a method which relies heavily on data and motivated by the smoothness properties in estimating a function which involves spline and non-spline approaches. Spline approach consists of regression spline and smoothing spline. Regression spline with Bayesian approach is considered in the first step of a two-step method in estimating the structural parameters for stochastic differential equation (SDE). The selection of knot and order of spline can be done heuristically based on the scatter plot. To overcome the subjective and tedious process of selecting the optimal knot and order of spline, an algorithm was proposed. A single optimal knot is selected out of all the points with exception of the first and the last data which gives the least value of Generalized Cross Validation (GCV) for each order of spline. The use is illustrated using observed data of opening share prices of Petronas Gas Bhd. The results showed that the Mean Square Errors (MSE) for stochastic model with parameters estimated using optimal knot for 1,000, 5,000 and 10,000 runs of Brownian motions are smaller than the SDE models with estimated parameters using knot selected heuristically. This verified the viability of the two-step method in the estimation of the drift and diffusion parameters of SDE with an improvement of a single knot selection.

 

Keywords: Bayesian approach; regression spline; stochastic differential equation; truncated power series basis

 

ABSTRAK

Permodelan tak-berparameter adalah satu kaedah yang sangat bergantung kepada data dan dimotivasi oleh ciri kelicinan dalam menganggar fungsi yang melibatkan pendekatan splin dan bukan-splin. Pendekatan spline terdiri daripada splin regresi dan splin pelicinan. Pendekatan pertama dengan kaedah Bayesian digunakan dalam langkah pertama untuk kaedah dua-langkah bagi menganggar parameter struktur persamaan pembeza stokastik (SDE). Pemilihan buku dan tertib splin boleh dilakukan secara heuristik berdasarkan rajah. Untuk mengatasi proses pemilihan bilangan buku dan tertib splin yang subjektif dan memakan masa, satu prosedur penyelesaian dikemukakan. Knot tunggal terbaik dengan nilai pengesahan silang teritlak (GCV) minimum dipilih daripada semua titik kecuali data pertama dan terakhir. Penggunaannya ditunjukkan menggunakan data cerapan saham pembukaan Petronas Gas Bhd. Hasil kajian menunjukkan nilai min ralat kuasadua bagi model stokastik yang menggunakan knot tunggal terbaik sebagai penganggaran parameter bagi larian gerakan Brown 1,000, 5,000 dan 10,000 adalah lebih kecil berbanding model stokastik dengan parameter dianggar menggunakan knot yang dipilih secara heuristik. Ini mengesahkan kebolehjayaan kaedah dua-langkah dalam menganggar parameter hanyut dan jerap SDE dengan menambahbaik kaedah pemilihan buku tunggal.

 

Kata kunci: Basis siri kuasa terpangkas; pendekatan Bayesan; persamaan pembeza stokastik; regresi splin

REFERENCES

Allen, E. 2007. Modelling with Itô Stochastic Differential Equations. New York: Springer-Verlag.

Brunel, N. 2008. Parameter estimation of ODE’s via nonparametric estimators. Electronic Journal of Statistics 21: 1242-1267.

Budiantara, I.N. 2001. Aplikasi spline estimator terbobot. Jurnal Teknik Industri 3(2): 57-62.

Calderon, C.P., Martinez, J.G., Carrol, R.J. & Sorensen, D.C. 2010. P-Splines using derivative information. Multiscale Modelling and Simulation 8(4): 1562-1580.

Eubank, R. L. 1988. Spline Smoothing and Non parametric Regression, New York: Marcel-Dekker.

Hunt, D.L. & Li, C-S. 2006. Aregression spline model for developmental toxicity data. Toxicological Sciences 92(1): 329-334.

Lang, S. & Brezger, A. 2004. Bayesian P-Splines. Journal of Computational and Graphical Statistics 13: 183-212.

Leathwick, J.R., Rowe, D., Richardson, J., Elith, J. & Hastie, T. 2005. Using multivariate adaptive regression splines to predict the distributions of New Zealand’s Freshwater Diadromous fish. Freshwater Biology 50: 2034-2052.

Lee, T.C.M. 2002. On algorithms for ordinary least squares regression spline fitting: Acomparative study. Journal of Statistical Computation and Simulation 72(8): 647-663.

Li, M. & Yu, Y. 2006. Arobust approach to the interest rate: Term structure estimation. Journal of Data Science 4: 169-188.

Molinari, N., Durand, J.F. & Sabatier, R. 2004. Bounded optimal knots for regression splines. Computational Statistics & Data Analysis 45: 159-178.

Poyton, A.A., Varziri, M.S., McAuley, K.B., McLellan, P.J. & Ramsay, J.O. 2006. Parameter estimation in continuous-time dynamic models using principal differential analysis. Computers and Chemical Engineering 30: 698-708.

Ramsay, J.O & Silverman, B.W. 1997. Functional Data Analysis. New York: Springer.

Ramsay, J.O. 1996. Principal differential analysis: Data reduction by differential operators. Journal of the Royal Statistical Society Series B58: 495-508.

Ramsay, J.O., Hooker, G., Campbell, D. & Cao, J. 2007. Parameter estimation for differential equations: Ageneralized smoothing approach (with discussions). The Journal of the Royal Statistical Society. Series B (Statistical Methodology) 69: 741-796.

Taylan, P., Weber, G-W. & Kropat, E. 2008. Approximation of stochastic differential equations by additive models using splines and conic programming. International Journal of Computing Anticipatory Systems 21: 341-352.

Varah, J. M. 1982. Aspline least squares method for numerical parameter estimation in differential equations. SIAM Journal of Scientific Statistical Computing 3: 28-46.

Varziri, M.S., McAuley, K.B. & McLellan, P.J. 2008. Parameter and state estimation in nonlinear stochastic continuous-time dynamic models with unknown disturbance intensity. The Canadian Journal of Chemical Engineering 86(5): 828-837.

Wallstrom, G., Liebner, J.L. & Kass, R.E. 2008. An implementation of Bayesian adaptive regression splines (BARS) in C with S and R wrappers. Journal of Statistical Software 26(1): 1-2.

 

 

*Corresponding author; email: halizarahman@utm.my