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
Diserahkan: 11 Januari 2011 / Diterima: 21 Julai 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
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*Pengarang
untuk surat-menyurat; email: halizarahman@utm.my