Mathematics http://hdl.handle.net/1903/2261 2022-08-09T23:31:18Z SMALL AREA ESTIMATION OF SUBDISTRICT POVERTY RATES ON JAVA ISLAND, INDONESIA http://hdl.handle.net/1903/29032 SMALL AREA ESTIMATION OF SUBDISTRICT POVERTY RATES ON JAVA ISLAND, INDONESIA Yuniarti, Fnu Poverty representing the deprivation of well-being is a global concern. The provision of poverty statistics for small area levels becomes crucial and mandatory to accelerate the poverty eradication program. This research undertakes poverty rate estimation for subdistrict levels on Java Island using small area estimation techniques. The results would give opportunities to better understand the poverty situation in small areas levels on Java Island. The main data source to calculate the poverty estimates is the Social and Economic Survey of March 2020. The direct estimation method, logistic fixed effects, and the logistics mixed effect models are employed to generate the poverty estimates. The model-based techniques use datasets from Population Census 2010 and Village Potential Enumeration Survey 2014 to derive the covariates. The direct technique does not show plausible estimates due to the sample size insufficiency. The logistic model improves the direct estimates but the variation between subdistricts is quite small which is unreasonable. The logistic mixed-effects model does not work perfectly but it gives good estimates for several subdistricts. 2022-01-01T00:00:00Z Geometric and Topological Reconstruction http://hdl.handle.net/1903/29011 Geometric and Topological Reconstruction Rawson, Michael G. The understanding of mathematical signals is responsible for the information age. Computation, communication, and storage by computers all use signals, either implicitly or explicitly, and use mathematics to manipulate those signals. Reconstruction of a particular signal can be desirable or even necessary depending on how the signal manifests and is measured. We explore how to use mathematical ideas to manipulate and represent signals. Given measurements or samples or data, we analyze how to produce, or \emph{reconstruct}, the desired signal and the fundamental limits in doing so. We focus on reconstruction through a geometric and topological lens so that we can leverage geometric and topological constraints to solve the problems. As inaccuracies and noise are present in every computation, we adopt a statistical outlook and prove results with high probability given noise. We start off with probability and statistics and then use that for active reconstruction where the probability signal needs to be estimated statistically from sampling various sources. We prove optimal ways to doing this even in the most challenging of situations. Then we discuss functional analysis and how to reconstruct sparse rank one decompositions of operators. We prove optimality of certain matrix classes, based on geometry, and compute the worst case via sampling distributions. With the mathematical tools of functional analysis, we introduce the optimal transportation problem. Then we can use the Wasserstein metric and its geometry to provably reconstruct sparse signals with added noise. We devise an algorithm to solve this optimization problem and confirm its ability on both simulated data and real data. Heavily under-sampled data can be ill-posed which is often the case with magnetic resonance imaging data. We leverage the geometry of the motion correction problem to devise an appropriate approximation with a bound. Then we implement and confirm in simulation and on real data. Topology constraints are often present in non-obvious ways but can often be detected with persistent homology. We introduce the barcode algorithm and devise a method to parallelize it to allow analyzing large datasets. We prove the parallelization speedup and use it for natural language processing. We use topology constraints to reconstruct word-sense signals. Persistent homology is dependent on the data manifold, if it exists. And it is dependent on the manifold's reach. We calculate manifold reach and prove the instability of the formulation. We introduce the combinatorial reach to generalize reach and we prove the combinatorial reach is stable. We confirm this in simulation. Unfortunately, reach and persistent homology are not an invariant of hypergraphs. We discuss hypergraphs and how they can partially reconstruct joint distributions. We define a hypergraph and prove its ability to distinguish certain joint distributions. We give an approximation and prove its convergence. Then we confirm our results in simulation and prove its usefulness on a real dataset. 2022-01-01T00:00:00Z STATISTICAL INFERENCE ACROSS MULTIPLE NETWORKS: ADVANCEMENTS IN MULTIPLEX GRAPH MATCHING AND JOINT SPECTRAL NETWORK EMBEDDINGS http://hdl.handle.net/1903/28981 STATISTICAL INFERENCE ACROSS MULTIPLE NETWORKS: ADVANCEMENTS IN MULTIPLEX GRAPH MATCHING AND JOINT SPECTRAL NETWORK EMBEDDINGS Pantazis, Konstantinos Networks are commonly used to model and study complex systems that arise in a variety of scientific domains.One important network data modality is multiplex networks which are comprised of a collection of networks over a common vertex set. Multiplex networks can describe complex relational data where edges within each network can encode different relationship or interaction types. With the rise of network modeling of complex, multi-sample network data, there has been a recent emphasis on multiplex inference methods. In this thesis, we develop novel theory and methodology to study underlying network structures and perform statistical inference on multiple networks. While each chapter of the thesis has its own individual merit, synergistically they constitute a coherent multi-scale spectral network inference framework that accounts for unlabeled and correlated multi-sample network data. Together, these results significantly extend the reach of such procedures in the literature. In the first part of the thesis, we consider the inference task of aligning the vertices across a pair of multiplex networks, a key algorithmic step in routines that assume a priori node-aligned data. This general multiplex matching framework is then adapted to the task of detecting a noisy induced multiplex template network in a larger multiplex background network.Our methodology, which lifts the classical graph matching framework and the matched filters method of Sussman et al. (2018) to the multiple network setting, uses the template network to search for the most" similar subgraph(s) in the background network, where the notion of similarity is measured via a multiplex graph matching distance. We present an algorithm which can efficiently match the template to a (induced or not induced) subgraph in the background that approximately minimizes a suitable graph matching distance, and we demonstrate the effectiveness of our approach both theoretically and empirically in synthetic and real-world data settings. In the second part of the thesis, we present a joint embedding procedure for multi-scale spectral network inference.In our proposed framework, a collection of node-aligned networks are jointly embedded into a common Euclidean space by embedding a specialized omnibus matrix which contains the information across the entire collection of networks. Our approach---which builds upon the work of Levin et al. (2017)---is flexible enough to faithfully embed many different network collection types (e.g., network time-series data; test-retest network data, etc.) and is theoretically tractable. Indeed, our method is among the first joint embedding methods in which statistical consistency and asymptotic normality are established across correlated network collections. Moreover, we are able to identify (and fully analyze in our setting) the phenomenon of induced correlation in the embedded space, which is an artifact of joint embedding methodologies. We examine how both the induced (by the method) and inherent correlation can impact inference for correlated network data, and we provide network analogues of classical questions such as the effective sample size for more generally correlated data. In the final part of the thesis, we consider a constrained optimization problem called corr2OMNI, and we present an algorithm that approximates generalized omnibus matrix structure (for jointly embedding networks) that best preserve (in the embedded space) a given correlation structure across a collection of networks. Moreover, we analyze theoretically an important special case of corr2OMNI where we desire a fixed level of correlation across the networks. 2022-01-01T00:00:00Z Limiting Configurations for the SU(1,2) Hitchin Equation http://hdl.handle.net/1903/28959 Limiting Configurations for the SU(1,2) Hitchin Equation Na, Xuesen This dissertation studies the SU(1,2) Higgs bundle and a limiting behavior of solutions of the SU(1,2) Hitchin's self-duality equation. On a closed Riemann surface $X$ of genus $g\ge 2$, an SU(1,2) Higgs bundle consists of the following data: a rank two holomorphic vector bundle $F$ and the holomorphic maps $\beta: L\otimes K_X^{-1}\to F$, $\gamma: F\to L\otimes K_X$ where $L=\det F^\ast$. The Hitchin map of the moduli space of SU(1,2) Higgs bundles takes $(F,\beta,\gamma)$ to the quadratic differential $q=\gamma\circ\beta$. For an SU(1,2) Higgs bundle $(F,\beta,\gamma)$, the Hitchin equation is a non-linear PDE of hermitian metric $h$ on $F$. The existence of a unique solution follows from the stability condition. For a stable SU(1,2) Higgs bundle $(F,\beta,\gamma)$, we give an explicit description of the behavior of $h_t$, the unique solution of SU(1,2) Hitchin equation for the family $(F,t\beta,t\gamma)$ in the case where $q$ has simple zeros in the limit $t\to\infty$. In Chapter 1, we review the notion of $G$ Higgs bundles and focus on the case $G=$SU(1,2). The simple zeros of $q=\gamma\circ\beta$ are one of the three types: (1) a zero of $\beta$, (2) a zero of $\gamma$, or (3) neither. We present a stability condition in terms of the number of zeros of each type. We also review notions of the filtered bundle and the wild harmonic bundle. In Chapter 2, we give an explicit description of the fiber of the Hitchin map in terms of a fiber bundle over the Jacobian of $X$ with unirational fibers. The fiber is a GIT quotient of a $\mathbb{C}^\times$-action on $(\mathbb{P}^1)^{4g-4}$. The base parametrizes the choice of a line bundle $L$. The fiber gives parameters for a Hecke modification $\iota: F\to V$ which realizes $F$ as a rank-two locally free subsheaf of $V=L^{-2}K_X\oplus LK_X$. We show that the stable locus is a coarse moduli space of the appropriate moduli functor. In Chapters 3 and 4, we study the Hitchin equation for the family $(F,t\beta,t\gamma)$ as $t\to\infty$. In particular, we show that the limiting configuration $h_\infty$ satisfies the decoupled Hitchin equation and is induced from a harmonic metric $h_L$ on $L$ via the Hecke modification $\iota: F\to V$. The metric $h_L$ is adapted to a filtered line bundle $(L,\underline{\lambda_\infty})$ where the weights $\underline{\lambda_\infty}$ are specified by a rule depending on the types of zeros and their count. We prove the convergence of $h_t$ to $h_\infty$ after appropriate normalizing by gluing local model solutions constructed from wild harmonic bundles on $\mathbb{P}^1$ over disks around the zeros to a solution of the decoupled equation on the complement. 2022-01-01T00:00:00Z