Two different aspects of the problem of selecting measurements for statistical pattern recognition are investigated. First, the evaluation of features for multiclass recognition problems by using measures of probabilistic dependence is examined. Secondly, the problem of evaluation and selection of features for a general tree type classifier is investigated. Measures of probabilistic dependence are derived from pairwise distance measures such as Bhattacharyya distance, divergence, Matusita's distance, and discrimination information. The properties for the dependence measures are developed in the context of feature class dependency. Inequalities relating the measures are derived. Also upper and lower bounds on error probability are derived for the different measures. Comparisons of the bounds are made. Feature ordering experiments are performed to compare the measures to error probability and to each other. A fairly general tree type sequential classifier is examined. An algorithm which uses distance measures for clustering probability distributions and which uses dependence and distance measures for ordering features is derived for constructing the decision tree. The concept of confidence in a decision in conjunction with backtracking is introduced in order to make decisions at any node of the tree tentative and reversible. Also, the idea of re-introducing classes at any stage is discussed. Experiments are performed to determine the storage and processing requirements of the classifier, to determine effects of various parameters on performance, and to determine the usefulness of procedures for backtracking and reintroducing of classes. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

In this thesis we investigate three projects within in the field of KPZ universality class and integrable probability. The first project studies the weak KPZ universality for half-space directed polymers in dimension 1+1. This is the half-space analogue of the full-space polymers studied in [AKQ]. The novelty is the extra random environment introduced at the boundary. The new technical challenges are the estimates for half-space heat kernels which are super-probability measures and accurate estimates on visits to origin (weighted by the boundary randomness) for a simple symmetric walk in dimension 1 and 2 respectively. The second project introduces a framework to prove tightness of a sequence of discrete Gibbsian line ensembles $\mathcal{L}^N = \{\mathcal{L}_k^N(u), k \in \mathbb{N}, u \in \frac{1}{N}\mathbb{Z}\}$, which is a countable collection of random curves. The sequence of discrete line ensembles $\mathcal{L}^N$ we consider enjoys a resampling invariance property, which we call $({\ensuremath{\mathbf{H}}}^N,\textup{H}^{\textup{RW},N})$-Gibbs property. We assume that $\mathcal{L}^N$ satisfies technical assumptions A1-A4 on $({\ensuremath{\mathbf{H}}}^N, {\textup{H}^{\textup{RW},N}})$ and the assumption that the lowest labeled curve with a parabolic shift, $\mathcal{L}_1^N(u) + \frac{u^2}{2}$, converges weakly to a stationary process in the topology of uniform convergence on compact sets. Under these assumptions, we prove our main result Theorem 3.1.13 that $\mathcal{L}^N$ is tight as a sequence of line ensembles and that the $\ensuremath{\mathbf{H}}$-Brownian Gibbs property holds for all subsequential limit line ensembles with $\ensuremath{\mathbf{H}}(x)= e^x$. As an application of Theorem 3.1.13, under the weak noise scaling, we show that the scaled log-gamma line ensemble $\overbar{\mathcal{L}}^N$ is tight, which is a sequence of discrete line ensembles associated with the log-gamma polymer model via the geometric RSK correspondence. The $\ensuremath{\mathbf{H}}$-Brownian Gibbs property (with $\ensuremath{\mathbf{H}}(x) = e^x$) of its subsequential limits also follows. The third project proves an analogue of the classical Komlós-Major-Tusnády (KMT) embedding theorem for random walk bridges and it serves as a key technical input for the second project. The random bridges we consider are constructed through random walks with i.i.d jumps that are conditioned on the locations of their endpoints. We prove that such bridges can be strongly coupled to Brownian bridges of appropriate variance when the jumps are either continuous or integer valued under some mild technical assumptions on the jump distributions. Our arguments follow a similar dyadic scheme to KMT's original proof, but they require more refined estimates and stronger assumptions necessitated by the endpoint conditioning. In particular, our result does not follow from the KMT embedding theorem, which we illustrate via a counterexample.

A research report submitted in partial fulfilment of the requirements for the degree of Master of Science in Education to the Wits School of Education, Faculty of Science, University of Witwatersrand, Johannesburg, 2017 / This research problematised the teaching and learning of the grade 8 topic of simplifying algebraic expressions via the errors and misconceptions learners’ show on that topic. The study conducted at a secondary school in Johannesburg identified the nature of grade 8 learners’ errors and misconceptions on simplifying algebraic expressions. A teaching intervention through using those errors as resource to help learners reduce them was undertaken. There was an implementation of discovery learning as an intervention strategy to help learners to explore algebraic concepts with the minimum involvement of the researcher. The researcher used constructivism, sociocultural learning and variation theories since these theories affect the learners learning of algebra. The researcher used an interpretive paradigm which is concerned about the individuals’ interpretation of the world around them. Purposive and convenience sampling were used in the study. Data was collected using a sample of thirty grade 8 learners. The learners wrote a pre-test as one of the assessment task in the study. The purpose of the pretest was to identify learners’ errors on simplifying algebraic expressions. After the learners’ errors were identified and analysed, the researcher conducted a semi-structured, focus group interview with six learners in the study. The selection of the interviewees depended on the type and frequency of errors they have displayed in their pre-test scripts. The purpose of the interview was to investigate the reasons behind the learners’ errors as identified in the pre-test. An intervention strategy which implemented guided discovery learning was employed to learners with the use of the identified errors as a resource to help learners reduce them. After the intervention, the learners wrote a post-test to check if there was an improvement in learners’ performance after the intervention. Pre- and post-tests results were analysed for errors revealed by learners. The teaching intervention periods were introduced to create learning opportunities for learners. The findings of the study revealed that before intervention learners encountered a lot of difficulties when simplifying algebraic expressions but the learners’ performance improved after the intervention. The recommendations of the study are, teachers should welcome learners’ errors in teaching and learning of mathematics and use them as a resource to help learners reduce them in solving mathematical problems Key words: Learners’ errors, misconceptions, simplification of algebraic expressions. / XL2018

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