Satellites provide humanity with data to infer properties of the earth that were impossible a century ago. Humanity can now easily monitor the amount of ice found on the polar caps, the size of forests and deserts, the earth’s atmosphere, the seasonal variation on land and in the oceans and the surface temperature of the earth. In this thesis, new hypertemporal techniques are proposed for the settlement detection problem in South Africa. The hypertemporal techniques are applied to study areas in the Gauteng and Limpopo provinces of South Africa. To be more specific, new sequential (windowless) and non-sequential hypertemporal techniques are implemented. The time-series employed by the new hypertemporal techniques are obtained from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor, which is on board the earth observations satellites Aqua and Terra. One MODIS dataset is constructed for each province. A Support Vector Machine (SVM) [1] that uses a novel noise-harmonic feature set is implemented to detect existing human settlements. The noise-harmonic feature set is a non-sequential hypertemporal feature set and is constructed by using the Coloured Simple Harmonic Oscillator (CSHO) [2]. The CSHO consists of a Simple Harmonic Oscillator (SHO) [3], which is superimposed on the Ornstein-Uhlenbeck process [4]. The noise-harmonic feature set is an extension of the classic harmonic feature set [5]. The classic harmonic feature set consists of a mean and a seasonal component. For the case studies in this thesis, it is observed that the noise-harmonic feature set not only extends the harmonic feature set, but also improves on its classification capability. The Cumulative Sum (CUSUM) algorithm was developed by Page in 1954 [6]. In its original form it is a sequential (windowless) hypertemporal change detection technique. Windowed versions of the algorithm have been applied in a remote sensing context. In this thesis CUSUM is used in its original form to detect settlement expansion in South Africa and is benchmarked against the classic band differencing change detection approach of Lunetta et al., which was developed in 2006 [7]. In the case of the Gauteng study area, the CUSUM algorithm outperformed the band differencing technique. The exact opposite behaviour was seen in the case of the Limpopo dataset. Sequential hypertemporal techniques are data-intensive and an inductive MODIS simulator was therefore also developed (to augment datasets). The proposed simulator is also based on the CSHO. Two case studies showed that the proposed inductive simulator accurately replicates the temporal dynamics and spectral dependencies found in MODIS data. / Thesis (PhD(Eng))--University of Pretoria, 2012. / Electrical, Electronic and Computer Engineering / unrestricted

After giving a brief background on the subject we introduce in section two the Phase-Integral Method of Fröman & Fröman in terms of the platform function of Yngve and Thidé. In section three we derive a different form of the radial Bohr-Sommerfeld condition in terms of the apsidal angle of the corresponding classical motion. Using the derived expression, we then show how easily one can calculate the exact energy eigenvalues of the hydrogen atom and the isotropic three-dimensional harmonic oscillator, we also derive an expression for higher order quantization condition. In section four we derive an expression for the angular frequencies of a restricted (0≤φ≤β) soap bubble and also give a proposition concerning the parameters l and m of the associated Legendre differential equation. / Vi använder Fröman & Frömans Fas-Integral Metod tillsammans med Yngve & Thidés plattformfunktion för att härleda kvantiseringsvilkoret för högre ordningar. I sektion tre skriver vi Bohr-Sommerfelds kvantiseringsvillkor på ett annorlunda sätt med hjälp av den så kallade apsidvinkeln (definierad i samma sektion) för motsvarande klassiska rörelse, vi visar också hur mycket detta underlättar beräkningar av energiegenvärden för väteatomen och den isotropa tredimensionella harmoniska oscillatorn. I sektion fyra tittar vi på en såpbubbla begränsad till området 0≤φ≤β för vilket vi härleder ett uttryck för dess (vinkel)egenfrekvenser. Här ger vi också en proposition angående parametrarna l och m tillhörande den associerade Legendreekvationen.

In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)