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内容記述 |
In data analysis and engineering applications, one often comes across unknown<br />densities which are complex and multimodal. In such situations, it is natural and<br />intuitive to break up the original density into a mixture of simpler, structurally less<br />complex densities, so as to facilitate analysis and modelling. In this thesis, we demonstrate<br />that it is possible to decompose a multimodal density into simpler densities via<br />the novel concept of M-decomposability. The letter M derives from “multimodal”<br />or “mixture”.<br /> <br /> For clarity of presentation, this thesis is divided into two parts. Part one consists<br />of Chapters 1 to 4, and solely considers densities in one-dimension. In Chapter 2,<br />we introduce the notion of M-decomposability in one-dimension. We say that a density<br />f is M-decomposable if it is possible to rewrite f as a mixture of two densities<br />g and h such that the sum of the standard deviations of g and h is less than the<br />standard deviation of f. If f does not satisfy the above condition, we say that f<br />is M-undecomposable. To clarify matters, we then provide examples to illustrate<br />the concept of M-decomposability. We also derive a theorem that states that “All<br />uniform densities in one-dimension areM-undecomposable” (Theorem 2.1). In Chapter<br />3, we demonstrate that unimodal densities in one-dimension can be approximated<br />to an arbitrary level of accuracy using a specially constructed mixture of uniform<br />densities. In Chapter 4, we make use of Theorem 2.1 and the representation in Chapter<br />3 to derive a theorem which states that “All symmetric unimodal densities in<br />one-dimension are M-undecomposable” (Theorem 4.1).<br /><br /> The second part of the thesis builds up on the results derived in the first and<br />extends to d-dimensions. To avoid confusion of notation, we provide a fresh set of<br />notations in Chapter 5 and a list of theorems and definitions to apply to the second<br />part of the paper. In Chapters 6 and 7, we provide the theoretical aspects<br />of M-decomposability in d-dimensions. In Chapter 6, we define the uniform density<br />in d-dimensions to be the elliptical uniform. To extend the definition of M-undecomposability<br />to apply d-dimensions, the “standard deviation” that appears<br />in the first part is replaced by the “square-root of the determinant of covariance”<br />of the underlying density. This step is crucial to the future development of M-decomposability<br />in d-dimensional. We derive a theorem that says that “All elliptical<br />uniform densities in d-dimension are M-undecomposable”. In Chapter 7, we extend<br />Theorem 4.1 derived in Chapter 4 to d-dimensions, i.e., “All elliptical unimodal<br />densities in d-dimension are M-undecomposable” (Theorem 7.2).<br /> <br /> In Chapter 8, we derive a theorem which links M-decomposability with Kullback-<br />Leibler divergence. This provides justification of using M-decomposability in a number<br />of statistical applications, namely clustering and density estimation. Simulation<br />examples of both clustering and density estimation are provided in the chapter. On<br />top of that, we also demonstrate the application of M-decomposability to real data<br />cluster analysis, using the Iris dataset as test data. The results not only show that <br />M-decomposability can be used to improve cluster analysis and density estimation, but<br />also suggest that M-decomposability is a viable criterion for cluster discrimination.<br />Concluding remarks are given in Chapter 9. |
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