On Consistency of Graph-based Semi-supervised Learning

Graph-based semi-supervised learning is one of the most popular methods in machine learning. Some of its theoretical properties such as bounds for the generalization error and the convergence of the graph Laplacian regularizer have been studied in computer science and statistics literatures. However, a fundamental statistical property, the consistency of the estimator from this method has not been proved. In this article, we study the consistency problem under a non-parametric framework. We prove the consistency of graph-based learning in the case that the estimated scores are enforced to be equal to the observed responses for the labeled data. The sample sizes of both labeled and unlabeled data are allowed to grow in this result. When the estimated scores are not required to be equal to the observed responses, a tuning parameter is used to balance the loss function and the graph Laplacian regularizer. We give a counterexample demonstrating that the estimator for this case can be inconsistent. The theoretical findings are supported by numerical studies.Comment: This paper is accepted by 2019 IEEE 39th International Conference on Distributed Computing Systems (ICDCS

The Hydrodynamics and Heat Transfer of Impinging Jet Flow and Circular Hydraulic Jump

The laminar axisymmetric flow and heat transfer of a circular impinging jet and hydraulic jump on a solid surface is analyzed theoretically using boundary-layer and thin-film approaches. Liquid jet impingement features many applications such as jet rinsing, jet cooling, liquid atomization and chemical reactors. The associated hydraulic jump dramatically affects the performance of the heat and mass transfer in such applications. In the current thesis, the effects of inertia, surface tension, surface rotation, gravity and heat transfer are comprehensively explored for impinging jet flow and the formation of hydraulic jump. The boundary-layer heights and film thickness are found to diminish with inertia. The wall shear stress is found to decrease with radial distance for on a stationary impingement surface but can increase for a rotary surface for large rotation speeds. When the surface is in rotation, a maximum liquid thickness occurs, reflecting the competition between inertia and rotation effects. The location of the hydraulic jump is determined for both low- and high-viscosity liquids. For low-viscosity liquid, the location of the jump is determined subject to the thickness near the trailing edge under static condition, reflecting the importance of surface tension. For high-viscosity liquids, the jump coincides with a singularity caused by gravity in the thin-film equation when surface tension is neglected. Downstream of the hydraulic jump, the recent finding of a constant ‘jump Froude number’ is also justified. The heat transfer analysis of impinging jet flow involves a two-way coupling due to the temperature-dependent viscosity and surface tension. To consider this non-linear coupling which is largely missing in the existing theoretical approaches, we develop a simple and iteration-free model, making exploring the influence of heat transfer on the flow field and the hydraulic jump feasible theoretically. Both the hydrodynamic and thermal boundary layers are found to decrease with a higher heat input at the solid surface. Enhanced heating is also found to push the hydraulic jump in the downstream direction. The Marangoni stress causes the hydraulic jump to occur earlier. The hydraulic jump leads to shock-type drops in the Nusselt number, confirming previous findings in the literature

Arvbarhet og biologisk systemdynamikk

The concept of heritability is rooted in the observation that relatives resemble one another more than expected by chance. Narrow-sense heritability is defined as the proportion of phenotypic variance that is attributable to additive genetic variation (i.e. where an allele substitution has the same effect irrespective of the rest of the genotype), while broad-sense heritability denotes the proportion of phenotypic variance caused by genetic variation including non-additive effects. Both concepts have been highly instrumental in evolutionary biology, production biology and biomedical research for several decades. However, this successful instrumental use should not be equated with deep understanding of how underlying biology shapes narrow- and broad-sense heritability. Nor does it guarantee that these statistical definitions and associated methodology are optimally suited to deal with the recent floods of biological data. Seeking a deeper understanding of the relationship between narrow- and broad-sense heritability in terms of biological mechanisms, I simulated genetic variation in dynamic models of biological systems. A striking result was that the ratio between narrow-sense and broad-sense heritability depended strongly on the type of regulatory architecture involved. Applying the same approach to an ensemble of gene regulatory network models, I showed that monotonicity features of genotype-to-phenotype maps reveal deep connections between molecular regulatory architecture and heritability aspects; connections that do not materialize from the classical distinction between additive, dominant and epistatic gene actions. Lastly, I addressed why genome-wide association studies (GWAS) have failed to identify much of the genetic variation underlying highly heritable traits. By linking computational physiology to GWAS, one can do GWAS on lower-level phenotypes that are mathematically related to each other through a dynamic model. This allows much more precise identification of the causal genetic variation, coupled with understanding of its function.Begrepet arvbarhet gjenspeiler det faktum at slektninger jevnt over ligner mer på hverandre enn på andre individer. Arvbarhet i smal forstand defineres som andelen av fenotypisk varians som kan tilskrives additive effekter av genetisk variasjon (altså der en allel-substitusjon har samme effekt uavhengig av resten av genotypen), mens arvbarhet i vid forstand betegner den samlede andelen som skyldes både additive og ikke-additive effekter. Begge begrepene har vist seg nyttige i evolusjonsbiologi, produksjonsbiologi og biomedisinsk forskning over flere tiår. Denne nytten som verktøy er imidlertid ikke ensbetydende med dyp innsikt i hvordan de to typene av arvbarhet formes av underliggende biologi. Det er heller ikke selvsagt at disse statistisk baserte definisjonene og metodene vil være de beste til å møte dagens flom av nye biologiske data. I mitt doktorgradsarbeid har jeg belyst hvordan forholdet mellom arvbarhet i smal og vid forstand henger sammen med biologiske mekanismer, gjennom å simulere genetisk variasjon i dynamiske modeller av fysiologiske systemer. Et slående resultat var at den regulatoriske arkitekturen til systemet har mye å si for forholdstallet mellom arvbarhet i smal og vid forstand. På lignende vis studerte jeg arvbarhet i et knippe modeller av genregulatoriske nettverk med ulike grader av monotonitet i den matematiske sammenhengen mellom genotype og fenotype. Dette avdekket dype bånd mellom arvbarhetsmønstre og molekylær regulatorisk arkitektur; sammenhenger som ikke er åpenbare ut fra det klassiske skillet mellom additive, dominante og epistatiske gen-effekter. Til sist tok jeg for meg svakheter ved dagens statistiske metoder for å forklare hvordan variasjon i sterkt arvbare trekk styres av genetiske forskjeller mellom individer. Såkalte hel-genom-assosiasjons-studier (genome-wide association studies, GWAS) påviser ofte en mengde relevante loci med genetisk variasjon, men disse forklarer likevel bare en liten del av den observerte arvbarheten i overordnede trekk som f.eks. kroppshøyde eller sjukdomsforekomst. En mer lovende tilnærming er å koble matematisk fysiologi til GWAS. Jeg viser at man ved å gjøre GWAS på lavnivå-fenotyper som er matematisk forbundet gjennom en dynamisk modell, kan identifisere den årsaksbestemmende genetiske variasjonen langt mer presist og samtidig øke forståelsen av dennes funksjon

Load Shifting in the Smart Grid: To Participate or Not?

Demand-side management (DSM) has emerged as an important smart grid feature that allows utility companies to maintain desirable grid loads. However, the success of DSM is contingent on active customer participation. Indeed, most existing DSM studies are based on game-theoretic models that assume customers will act rationally and will voluntarily participate in DSM. In contrast, in this paper, the impact of customers' subjective behavior on each other's DSM decisions is explicitly accounted for. In particular, a noncooperative game is formulated between grid customers in which each customer can decide on whether to participate in DSM or not. In this game, customers seek to minimize a cost function that reflects their total payment for electricity. Unlike classical game-theoretic DSM studies which assume that customers are rational in their decision-making, a novel approach is proposed, based on the framework of prospect theory (PT), to explicitly incorporate the impact of customer behavior on DSM decisions. To solve the proposed game under both conventional game theory and PT, a new algorithm based on fictitious player is proposed using which the game will reach an epsilon-mixed Nash equilibrium. Simulation results assess the impact of customer behavior on demand-side management. In particular, the overall participation level and grid load can depend significantly on the rationality level of the players and their risk aversion tendency.Comment: 9 pages, 7 figures, journal, accepte

Integrating Energy Storage into the Smart Grid: A Prospect Theoretic Approach

In this paper, the interactions and energy exchange decisions of a number of geographically distributed storage units are studied under decision-making involving end-users. In particular, a noncooperative game is formulated between customer-owned storage units where each storage unit's owner can decide on whether to charge or discharge energy with a given probability so as to maximize a utility that reflects the tradeoff between the monetary transactions from charging/discharging and the penalty from power regulation. Unlike existing game-theoretic works which assume that players make their decisions rationally and objectively, we use the new framework of prospect theory (PT) to explicitly incorporate the users' subjective perceptions of their expected utilities. For the two-player game, we show the existence of a proper mixed Nash equilibrium for both the standard game-theoretic case and the case with PT considerations. Simulation results show that incorporating user behavior via PT reveals several important insights into load management as well as economics of energy storage usage. For instance, the results show that deviations from conventional game theory, as predicted by PT, can lead to undesirable grid loads and revenues thus requiring the power company to revisit its pricing schemes and the customers to reassess their energy storage usage choices.Comment: 5 pages, 4 figures, conferenc