P.H. Kloppers | Tshwane University of Technology (original) (raw)

Papers by P.H. Kloppers

Transcendental models are often solved by using a different approach, which can be a derivative f... more Transcendental models are often solved by using a different approach, which can be a derivative free, direct optimisation or iterative linearization method. All these approaches require guess values for the unknown parameters to start the iteration procedure. However, if the transcendental model involves
several parameters, some of these methods become very cumbersome and computationally expensive. A new method for computing parameter estimates which are then used as initial values for the unknown model parameters to start the iteration process was proposed. Confidence intervals for the estimated parameters were constructed using the bootstrap method. We generated two randomised datasets that
simulated the decay and growth processes. A three parameterized single exponential model f (x)  exp(x)  was identified using the simulated datasets in each case. The absolute percentage errors were used as a measure of comparison between the proposed method and the
current Levenberg-Marquardt (L-M) method. Tables and figures were used to present results from both methods. The proposed method appeared to produce better results than the current L-M method. The superiority of the proposed method over the current methods is that it does not require initial guess
values and it guarantees convergences. Thus the proposed method could be adopted to solve real life problems.

Estimation of the unknown mean, μ and variance, σ2 of a univariate Gaussian distribution N (μ ,σ ... more Estimation of the unknown mean, μ and variance, σ2 of a univariate Gaussian distribution
N (μ ,σ 2 ) given a single study variable x is considered. We propose an approach that does not require
initialization of the sufficient unknown distribution parameters. The approach is motivated
by linearizing the Gaussian distribution through differential techniques, and estimating, μ and σ2
as regression coefficients using the ordinary least squares method. Two simulated datasets on
hereditary traits and morphometric analysis of housefly strains are used to evaluate the proposed
method (PM), the maximum likelihood estimation (MLE), and the method of moments (MM). The
methods are evaluated by re-estimating the required Gaussian parameters on both large and
small samples. The root mean squared error (RMSE), mean error (ME), and the standard deviation
(SD) are used to assess the accuracy of the PM and MLE; confidence intervals (CIs) are also constructed
for the ME estimate. The PM compares well with both the MLE and MM approaches as
they all produce estimates whose errors have good asymptotic properties, also small CIs are observed
for the ME using the PM and MLE. The PM can be used symbiotically with the MLE to provide
initial approximations at the expectation maximization step.

South African Journal of Science, 2013

ABSTRACT

International Journal of Mathematical Education in Science and Technology, 2001

ABSTRACT The well-known physicist A. A. Michelson started quite an interesting correspondence in ... more ABSTRACT The well-known physicist A. A. Michelson started quite an interesting correspondence in the journal Nature in 1898. He complained about the convergence of continuous Fourier series approximations to a discontinuous function as being 'utterly at variance with the physicist's notions of quantity'. J. W. Gibbs essentially settled matters in 1899 and this situation has become to be called the Gibbs' phenomenon. It is discussed in many texts but appears to be always focused on the discontinuity of a simple step function. The details for an arbitrary jump discontinuity are illuminating and provide an interesting example to discuss the difference between pointwise convergence and uniform convergence. Two proofs are given that the Gibbs' phenomenon only depends on the size of the jump and is a multiple of the integral ƒπ0(sin x/x) dx. The demonstration and calculations are suitable for an advanced calculus class and provide very nice applications of Riemann sums and uniform convergence.

International Journal of Mathematical Education in Science and Technology, 2003

International Journal of Mathematical Education in Science and Technology, 2006

ABSTRACT This note considers the four classes of orthogonal polynomials – Chebyshev, Hermite, Lag... more ABSTRACT This note considers the four classes of orthogonal polynomials – Chebyshev, Hermite, Laguerre, Legendre – and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same as that for Fourier series expansions. Each class of polynomials has features which are interesting numerically. Finally a plausibility argument is included showing that this phenomenon for the Gibbs constants should not have been unexpected. These findings suggest further investigations suitable for undergraduate research projects or small group investigations.

Proceedings of the Annual Congresses of the Grassland Society of Southern Africa, 1974

Irrigated Ariki ryegrass produced 13,7 and 9,2 m t/ha green dry matter during the first and secon... more Irrigated Ariki ryegrass produced 13,7 and 9,2 m t/ha green dry matter during the first and second growing seasons respectively. It was most productive in spring, because the growth rate decreased throughout the growing season. This necessitated increasingly long periods of absence of 23, 39, 39 and 48 days under grazing conditions, to maximize dry matter production. During the same growing periods growth rates were 94, 73, 56 and 43 kg DM/ha/day respectively.A highly significant, negative correlation was obtained between dry matter production and mean minimum temperature. The poorer late season production is probably associated with local temperature regime characteristics. More of this type of research under better controlled conditions is required.

Applied Mathematics and Computation, 2013

and sharing with colleagues.

This paper proposes a modified method (MM) for computing initial guess values (IGVs) of a single ... more This paper proposes a modified method (MM) for computing initial guess values (IGVs) of a single exponential class of transcendental least square problems. The proposed method is an improvement of the already published multiple goal function (MGF) method. Current approaches like the Gauss-Newton, Maximum likelihood, Levenberg-Marquardt etc. methods for computing parameters of transcendental least squares models use iteration routines that require IGVs to start the iteration process. According to reviewed literature, there is no known documented methodological procedure for computing the IGVs. It is more of an art than a science to provide a “good” guess that will guarantee convergence and reduce computation time. To evaluate the accuracy of the MM method against the existing Levenberg-Marquardt (LM) and the MGF methods, numerical studies are examined on the basis of two problems that’s; the growth and decay processes. The mean absolute percentage error (MAPE) is used as the measure of accuracy among the methods. Results show that the modified method achieves higher accuracy than the LM and MGF methods and is computationally attractive. However, the LM method will always converge to the required solution given “good” initial values.
The MM method can be used to compute estimates for IGVs, for a wider range of existing methods of solution that use iterative techniques to converge to the required solutions.

Two new approaches (method I and II) for estimating parameters of a univariate normal probability... more Two new approaches (method I and II) for estimating parameters of a univariate normal probability density function are proposed. We evaluate their performance using two simulated normally distributed univariate datasets and their results compared with those obtained from the maximum likelihood (ML) and the method of moments (MM) approaches on the same samples, small n = 24 and large n = 1200 datasets. The proposed methods, I and II have shown to give significantly good results that are comparable to those from the standard methods in a real practical setting. The proposed methods have performed equally well as the ML method on large samples. The major advantage of the proposed methods over the ML method is that they do not require initial approximations for the unknown parameters. We therefore propose that in the practical setting, the proposed methods be used symbiotically with the standard methods to estimate initial approximations at the appropriate step of their algorithms.