South African learners perform poorly in the geometry sections of both national and international assessments. Numerous assessment reports mention multiple errors that keep re-occurring and play a big role in the learners’ poor performance. For this research, the link between the grade 9 learners Van Hiele levels of thought and the typical errors that they made were investigated. In this mixed method study, 194 grade 9 learners in two schools in Port Elizabeth, South Africa were tested using a Van Hiele based test. A test was set up containing multiple-choice and open-ended questions and was used to determine firstly, the predominant level of geometric reasoning of the learners and secondly, to determine their typical errors. Semi-structured interviews were held with six learners to gain more insight into some of the typical errors uncovered in the tests. The quantitative data revealed that the learners’ predominant levels of geometric thought were low. Furthermore, the qualitative data revealed typical error patterns concerning angles and sides, parallel lines, hierarchy of quadrilaterals and incorrect reasons in the proofs. The quantitative and qualitative data was merged to determine if the errors could be linked to the Van Hiele levels. From the findings, it was concluded that most of their typical errors could be linked to the Van Hiele levels of the learners.

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Modern data analysis requires the study of tensors, or multi-way arrays. We consider the case where the dimension d is large and the order p is fixed. For dimension reduction and for interpretability, one considers tensor decompositions, where a tensor T can be decomposed into a sum of rank one tensors. In this thesis, I will describe some recent work that illustrate why and how to use decompositions for orthogonally decomposable tensors. Our developments are motivated by statistical applications where the data dimension is large. The estimation procedures will therefore aim to be computationally tractable while providing error rates that depend optimally on the dimension. A tensor is said to be orthogonally decomposable if it can be decomposed into rank one tensors whose component vectors are orthogonal. A number of data analysis tasks can be recast as the problem of estimating the component vectors from a noisy observation of an orthogonally decomposable tensor. In our first set of results, we study this decompositionproblem and derive perturbation bounds. For any two orthogonally decomposable tensors which are ε-perturbations of one another, we derive sharp upper bounds on the distances between their component vectors. While this is motivated by the extensive literature on bounds for perturbation of singular value decomposition, our work shows fundamental differences and requires new techniques. We show that tensor perturbation bounds have no dependence on eigengap, a quantity which is inevitable for matrices. Moreover, our perturbation bounds depend on the tensor spectral norm of the noise, and we provide examples to show that this leads to optimal error rates in several high dimensional statistical learning problems. Our results imply that matricizing a tensor is sub-optimal in terms of dimension dependence. The tensor perturbation bounds derived so far are universal, in that they depend only on the spectral norm of the perturbation. In subsequent chapters, we show that one can extract further information from how a noise is generated, and thus improve over tensor perturbation bounds both statistically and computationally. We demonstrate this approach for two different problems: first, in estimating a rank one spiked tensor perturbed by independent heavy-tailed noise entries; and secondly, in performing inference from moment tensors in independent component analysis. We find that an estimator benefits immensely— both in terms of statistical accuracy and computational feasibility — from additional information about the structure of the noise. In one chapter, we consider independent noise elements, and in the next, the noise arises as a difference of sample and population fourth moments. In both cases, our estimation procedures are determined so as to avoid accumulating the errors from different sources. In a departure from the tensor perturbation bounds, we also find that the spectral norm of the error tensor does not lead to the sharpest estimation error rates in these cases. The error rates of estimating the component vectors are affected only by the noise projected in certain directions, and due to the orthogonality of the signal tensor, the projected errors do not accumulate, and can be controlled more easily.

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As data-driven modeling becomes more prevalent for representing the uncertain dynamical systems, concerns also arise regarding the reliability of these methods. Recent developments in approximation theory provide a new perspective to studying these problems. This dissertation analyzes the convergence of two kernel-based, data-driven modeling methods, the reproducing kernel Hilbert space (RKHS) embedding method and the empirical-analytical Lagrangian (EAL) model. RKHS embedding is a non-parametric extension of the classical adaptive estimation method that embeds the uncertain function in an RKHS, an infinite-dimensional function space. As a result the original uncertain system of ordinary differential equations are understood as components of a distributed parameter system. Similarly to the classical approach for adaptive estimation, a novel definition of persistent excitation (PE) is introduced, which is proven to guarantee the pointwise convergence of the estimate of function over the PE domain. The finite-dimensional approximation of the RKHS embedding method is based on approximant spaces that consist of kernel basis functions centered at samples in the state space. This dissertation shows that explicit rate of convergence of the RKHS embedding method can be derived by choosing specific types of native spaces. In particular, when the RKHS is continuously embedded in a Sobolev space, the approximation error is proven to decrease at a rate determined by the fill distance of the samples in the PE domain. This dissertation initially studies scalar-valued RKHS, and subsequently the RKHS embedding method is extended for the estimation of vector-valued uncertain functions. Like the scalar-valued case, the formulation of vector-valued RKHS embedding is proven to be well-posed. The notion of partial PE is also generalized, and it is shown that the rate of convergence derived for the scalar-valued approximation still holds true for certain separable operator-valued kernels. The second part of this dissertation studies the EAL modeling method, which is a hybrid mechanical model for Lagrangian systems with uncertain holonomic constraints. For the singular perturbed form of the system, the kernel method is applied to approximate a penalty potential that is introduced to approximately enforce constraints. In this dissertation, the accuracy confidence function is introduced to characterize the constraint violation of an approximate trajectory. We prove that the confidence function can be decomposed into a term representing the bias and another term representing the variation. Numerical simulations are conducted to examine the factors that affect the error, including the spectral filtering, the number of samples, and the accumulation of integration error. / Doctor of Philosophy / As data-driven modeling is becoming more prevalent for representing uncertain dynamical systems, concerns also arise regarding the reliability of these methods. This dissertation employs recent developments in approximation theory to provide rigorous error analysis for two certain kernel-based approaches for modeling dynamical systems. The reproducing kernel Hilbert space (RKHS) embedding method is a non-parametric extension of the classical adaptive estimation for identifying uncertain functions in nonlinear systems. By embedding the uncertain function in a properly selected RKHS, the nonlinear state equation in Euclidean space is transformed into a linear evolution in an infinite-dimensional RKHS, where the function estimation error can be characterized directly and precisely. Pointwise convergence of the function estimate is proven over the domain that is persistently excited (PE). And a finite-dimensional approximation can be constructed within an arbitrarily small error bound. The empirical-analytical Lagrangian (EAL) model is developed to approximate the trajectory of Lagrangian systems with uncertain configuration manifold. Employing the kernel method, a penalty potential is constructed from the observation data to ``push'' the trajectory towards the actual configuration manifold. A probabilistic error bound is derived for the distance of the approximated trajectory away from the actual manifold. The error bound is proven to contain a bias term and a variance term, both of which are determined by the parameters of the kernel method.

An analysis of computer diagnostic systems shows that most systems use answer data (product) for their analyses. This process of determining an error pattern, in addition, does little in the way of telling a teacher what should be done to help the child. This two-fold problem, extant in all computerized arithmetic diagnostic systems to date, prompted this study which sought other data sources in order to bring about more accurate computer analyses. A cognitive orientation suggested that the use of clinical diagnostic techniques should be explored as an alternative to error analysis. Essentially, these two approaches were compared. That is, to what extent does error pattern diagnosis (an essentially product oriented approach) and clinical mathematical diagnosis (a process oriented approach) interrelate? Participants for this study were five, eight year olds from southwest Virginia. These children completed a test that was developed by Van Lehn (1982). This test was analyzed for error patterns and the children were selected on the basis of their error patterns. These children were then tested in a clinical setting using a measure developed for this study in cooperation with a clinical mathematics diagnostician. The analysis was done on the results of these two measures and the protocols collected during the clinical interviews. The results indicated that there was no clear connection between the two types of diagnosis, but the analysis did yield a broader description of each individual participant. That is, error analysis or clinical mathematics alone does not completely describe an individual's knowledge of mathematics. / Master of Arts

Experimental modal analysis (EMA) techniques have become a popular method of studying the dynamic characteristics of structures. A survey of literature available reveals that experimental modal models resulting from EMA may suffer from inaccuracy due to a host of reasons. Every stage of EMA could be a potential source of errors - from suspension of the test structures, transduction to parameter estimation phase. Though time-domain methods are actively being investigated by many researchers and are in use, fast Fourier transform (FFT) methods, due to their speed and ease of implementation, are the most widely used in experimental modal analysis work. This work attempts to quantify errors that result from a typical modal test. Using a simple beam with free-free boundary conditions simulated, three different modal tests are performed. Each test differs from the other chiefly in the excitation method and FRF estimator used. Using finite element models as the reference, correlation between finite element and experimental models are performed. The ability of the EMA process to accurately estimate the modal parameters is established on the basis of level of correlation obtained for natural frequencies and mode shapes. Linear regression models are used to correlate test and analysis natural frequencies. The modal assurance criterion (MAC) is used to establish the accuracy of mode vectors from the modal tests. The errors are further quantified spatially (on a location-by-location basis) for natural frequencies and mode shapes resulting from the EMA process. Finally, conclusions are made regarding the accuracy of modal parameters obtained via FFT-based EMA techniques. / Master of Science

Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model x_ij = μ_i + β_ij + ε_ij, i=1,2,...n; j=1,2,...,r, where x_ij = observation on the i^th treatment of the j^th block, μ_i = true mean of the i^th treatment, β_j = bias of the j^th block, ε_ij = random error, distributed normally with means zero and variance σ²_j, and E(x_ij) = μ_i + β_j. The estimator σ̂²_t, for σ²_t, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Q_t = [r(r-1)∑_i(x_ij-x_i.-x_.t+x_..)²-∑_i∑_j(x_ij-x_i.-x_.j+x_..)²] ÷ [(n-1)(r-1)(r-2)] where x_i., x_.j and x_.. are the means of i^th treatment, j^th block and grand mean respectively. σ̂²_t and Q_t were shown to be identical when σ²_t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Q_j is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Q_t/σ² = [(r-1)²x_(n-1)²-x_(n-1)(r-2)²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Q_t/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑_i∑_j(x_ij-x_i.-x_.j+x_..)². It was noted that this statistic may be used to test H_o: σ²_t = σ²against one of H_a₁: σ²_t > σ²; H_a₂: σ²_t < σ² and H_a₃: σ²_t ≠ σ² assuming σ²_j = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case. / Ph. D.

In this dissertation, we provide a unified framework for analyzing immersed finite element methods in one spatial dimension, and we design a new geometry conforming IFE space in two dimensions with optimal approximation capabilities, alongside with applications to the elliptic interface problem and the hyperbolic interface problem. In the first part, we discuss a general m-th degree IFE space for one dimensional interface problems with many polynomial-like properties, then we develop a general framework for obtaining error estimates for the IFE spaces developed for solving a variety of interface problems, including but not limited to, the elliptic interface problem, the Euler-Bernoulli beam interface problem, the parabolic interface problem, the transport interface problem, and the acoustic interface problem. In the second part, we develop a new m-th degree finite element space based on the differential geometry of the interface to solve interface problems in two spatial dimensions. The proposed IFE space has optimal approximation capabilities, easy to construct, and the IFE functions satisfy the interface conditions exactly. We provide several numerical examples to demonstrate that the IFE space yields optimally converging solutions when applied to the elliptic interface problem and the hyperbolic interface problem with a symmetric interior penalty discontinuous Galerkin formulation. / Doctor of Philosophy / Interface problems appear naturally in many physics and Engineering applications where a physical quantity is considered across materials of different physical properties, such as heat transfer or sound propagation through different materials. Typically, these physical phenomena are modelled by partial differential equations with discontinuous coefficients representing the material properties. The main topics of this dissertation are about the development and analysis of immersed finite element methods for interface problems. The IFE method can use interface independent meshes, and employs approximating functions that capture the features of the solution at the interface. Specifically, we provide a unified framework for analyzing one-dimensional IFE problems, and we design a new framework to construct geometry conforming IFE spaces in two dimensions, with applications to the elliptic interface problem and the hyperbolic interface problem.