Mathematics Abstracts of Presentations

A Further Study on Chen--Qin's Test for Two-Sample Behrens--Fisher Problems for High-Dimensional Data

Tianming Zhu

A further study on Chen--Qin's test, namely CQ-test, for two-sample Behrens--Fisher problems for high-dimensional data is conducted, resulting in a new normal-reference test where the null distribution of the CQ-test statistic is approximated with that of a chi-square-type mixture, which is obtained from the CQ-test statistic when the null hypothesis holds and when the two samples are normally distributed. The distribution of the chi-square-type mixture can be well approximated by a three-cumulant matched chi-square-approximation with the approximation parameters consistently estimated from the data. The asymptotical power of the new normal-reference test under a local alternative is established. Two simulation studies demonstrate that in terms of size control, the new normal-reference test with the three-cumulant matched chi-square-approximation performs well regardless of whether the data are nearly uncorrelated, moderately correlated, or highly correlated and it performs substantially better than the CQ-test. A real data example illustrates the new normal-reference test.

Discrimination of Olive Oils by Geographical Origin Using a Regularized Logistic Regression Model

Soh Chin Gi

The identification of geographical origin of a given sample of olive oil is a challenging task that has implications in the field of food fraud detection. Spectroscopic techniques are able to capture chemical information that may be useful in identifying the geographical origin of an oil sample, but the resulting data is challenging to analyse due to issues with high-dimension and multicollinearity. Traditional approaches rely on projection-based methods such as principal components analysis combined with discriminant analysis. This talk presents an alternative method for modelling spectroscopic data to solve this classification problem via regularized logistic regression models, along with the relevant optimization algorithm. The regularization penalties enforce sparsity, smoothness and group structure in the model coefficients. Some interesting fitted models will also be presented, and comparisons to the results obtained using traditional approaches will be made.

On Henstock-Kurzweil Integration Theory: Moving Between Stochastic and Non-Stochastic Case

Toh Tin Lam

The classical Riemann integral is well-known. However, the Riemann integral is unable to handle functions which are highly oscillating, as it uses uniform meshes in its definition of the Riemann sums. However, a slight modification of the Riemann integral, by replacing the uniform mesh with one that varies from point to point, results in an integral that is much more general than the Riemann integral, and even the Lebesgue integral. This was independently discovered by Henstock and Kurzweil, hence the integral was termed Henstock-Kurzweil integral. The tag in the interval-pair point can be any point within the interval. In extending this notion of non-uniform mesh to stochastic integral, much restrictions on the tag occurs. Still, the modified Henstock-Kurzweil stochastic integral has been shown to encompass the classical stochastic integral, namely, the Ito integral and the Stratonovich integral. Through the work of the stochastic integral, a re-visit to the non-stochastic integral brings new light on the approach to the integral. This talk discovers the gist in the journey of the work described above.

Henstock Ito’s Approach to Non-Stochastic Integral

Clara Lim Ying Yi

We are all familiar with the Riemann integral. In this presentation, we explore the generalized Riemann approach using non-uniform mesh, which gives rise to integrals that are more general than the Riemann integral. We consider the special interval-point pair in defining the Riemann sum, where the point (or the tag) is the left-hand point of the interval. As a result, the integration-by-substitution and by-parts formulae become easy consequences of the definition.

Probabilistic Powerdomains

Mark Lim

In my Masters research, I aim to understand the domain-theoretic properties of the probabilistic powerdomains and their topological selves, the spaces of valuations. The probability theory we are all familiar with involves finite sets such as coins tosses, or the tossing of a dice and are deterministic. The event space is mapped to the interval [0, 1]. Valuations become important when dealing with non-deterministic events. One of the major results in Valuations is the Jone’s Lemma. In the course of my research I have studied this lemma and will now be the theories leading up to the lemma. While it is unlikely for my presentation to cover the entire lemma deeply, I hope that it will at least spark interest in this beautiful area of Mathematics.