Ahmad, Afida
(2018)
Preserving Positivity And
Monotonicity Of Real Data Using
Bézier-Ball Function And Radial Basis
Function.
PhD thesis, Universiti Sains Malaysia.
Abstract
In this thesis, a rational cubic Bézier-Ball function which refers to a rational
cubic Bézier function expressed in terms of Ball control points and weights are used
to preserve positivity and monotonicity of real data sets. Four shape parameters are
proposed to preserve the characteristics of the data. A rational Bi-Cubic Bézier-Ball
function is introduced to preserve the positivity of surface generated from real data set
and from known functions. Eight shape parameters proposed can be modified to
preserve the positivity of the surface. Interpolating 2D and 3D real data using radial
basis function (RBF) is proposed as an alternative method to preserve the positivity of
the data. Two types of RBF which are Multiquadric (MQ) function and Gaussian
function, which contains a shape parameter are used. The boundaries (lower and
upper limit) of the shape parameter which preserves the positivity of real data are
proposed. Comparisons are made using the root-mean-square (RMS) error between
the proposed interpolation methods with existing works in literature. It was found that
MQ function and rational cubic Bézier-Ball is comparable with existing literature in
preserving positivity for both curves and surfaces. For preserving monotonicity, the
rational cubic Bézier-Ball is comparable but the MQ quasi-interpolation introduced
can only linearly interpolate the curve and the RMS values are big. Gaussian function
is able to preserve positivity of curves and surfaces but with unwanted oscillations
which result to unsmooth curves.
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