Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

Computation of (3)J(HH) coupling constants with a combination of density functional theory and semiempirical calculations. Application to complex molecules

This work evaluates the efficiency of economic levels of theory for the prediction of (3)J(HH) spin-spin coupling constants, to be used when robust electronic structure methods are prohibitive. To that purpose, DFT methods like mPW1PW91. B3LYP and PBEPBE were used to obtain coupling constants for a test set whose coupling constants are well known. Satisfactory results were obtained in most of cases, with the mPW1PW91/6-31G(d,p)//B3LYP/6-31G(d,p) leading the set. In a second step. B3LYP was replaced by the semiempirical methods PM6 and RM1 in the geometry optimizations. Coupling constants calculated with these latter structures were at least as good as the ones obtained by pure DFT methods. This is a promising result, because some of the main objectives of computational chemistry - low computational cost and time, allied to high performance and precision - were attained together. (C) 2012 Elsevier B.V. All rights reserved.

Spin-glass behaviour on random lattices

The ground-state phase diagram of an Ising spin-glass model on a random graph with an arbitrary fraction w of ferromagnetic interactions is analysed in the presence of an external field. Using the replica method, and performing an analysis of stability of the replica-symmetric solution, it is shown that w = 1/2, corresponding to an unbiased spin glass, is a singular point in the phase diagram, separating a region with a spin-glass phase (w < 1/2) from a region with spin-glass, ferromagnetic, mixed and paramagnetic phases (w > 1/2).

Segmental dataset and whole body expression data do not support the hypothesis that non-random movement is an intrinsic property of Drosophila retrogenes

Background: Several studies in Drosophila have shown excessive movement of retrogenes from the X chromosome to autosomes, and that these genes are frequently expressed in the testis. This phenomenon has led to several hypotheses invoking natural selection as the process driving male-biased genes to the autosomes. Metta and Schlotterer (BMC Evol Biol 2010, 10:114) analyzed a set of retrogenes where the parental gene has been subsequently lost. They assumed that this class of retrogenes replaced the ancestral functions of the parental gene, and reported that these retrogenes, although mostly originating from movement out of the X chromosome, showed female-biased or unbiased expression. These observations led the authors to suggest that selective forces (such as meiotic sex chromosome inactivation and sexual antagonism) were not responsible for the observed pattern of retrogene movement out of the X chromosome. Results: We reanalyzed the dataset published by Metta and Schlotterer and found several issues that led us to a different conclusion. In particular, Metta and Schlotterer used a dataset combined with expression data in which significant sex-biased expression is not detectable. First, the authors used a segmental dataset where the genes selected for analysis were less testis-biased in expression than those that were excluded from the study. Second, sex-biased expression was defined by comparing male and female whole-body data and not the expression of these genes in gonadal tissues. This approach significantly reduces the probability of detecting sex-biased expressed genes, which explains why the vast majority of the genes analyzed (parental and retrogenes) were equally expressed in both males and females. Third, the female-biased expression observed by Metta and Schltterer is mostly found for parental genes located on the X chromosome, which is known to be enriched with genes with female-biased expression. Fourth, using additional gonad expression data, we found that autosomal genes analyzed by Metta and Schlotterer are less up regulated in ovaries and have higher chance to be expressed in meiotic cells of spermatogenesis when compared to X-linked genes. Conclusions: The criteria used to select retrogenes and the sex-biased expression data based on whole adult flies generated a segmental dataset of female-biased and unbiased expressed genes that was unable to detect the higher propensity of autosomal retrogenes to be expressed in males. Thus, there is no support for the authors' view that the movement of new retrogenes, which originated from X-linked parental genes, was not driven by selection. Therefore, selection-based genetic models remain the most parsimonious explanations for the observed chromosomal distribution of retrogenes.

Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

The Libor Market Model: from theory to calibration

This thesis is focused on the financial model for interest rates called the LIBOR Market Model. In the appendixes, we provide the necessary mathematical theory. In the inner chapters, firstly, we define the main interest rates and financial instruments concerning with the interest rate models, then, we set the LIBOR market model, demonstrate its existence, derive the dynamics of forward LIBOR rates and justify the pricing of caps according to the Black’s formula. Then, we also present the Swap Market Model, which models the forward swap rates instead of the LIBOR ones. Even this model is justified by a theoretical demonstration and the resulting formula to price the swaptions coincides with the Black’s one. However, the two models are not compatible from a theoretical point. Therefore, we derive various analytical approximating formulae to price the swaptions in the LIBOR market model and we explain how to perform a Monte Carlo simulation. Finally, we present the calibration of the LIBOR market model to the markets of both caps and swaptions, together with various examples of application to the historical correlation matrix and the cascade calibration of the forward volatilities to the matrix of implied swaption volatilities provided by the market.

Antipodal random sequences with prescribed second-order statistics: application to Compressive Sensing and UWB system based on DS-CDMA

It is usual to hear a strange short sentence: «Random is better than...». Why is randomness a good solution to a certain engineering problem? There are many possible answers, and all of them are related to the considered topic. In this thesis I will discuss about two crucial topics that take advantage by randomizing some waveforms involved in signals manipulations. In particular, advantages are guaranteed by shaping the second order statistic of antipodal sequences involved in an intermediate signal processing stages. The first topic is in the area of analog-to-digital conversion, and it is named Compressive Sensing (CS). CS is a novel paradigm in signal processing that tries to merge signal acquisition and compression at the same time. Consequently it allows to direct acquire a signal in a compressed form. In this thesis, after an ample description of the CS methodology and its related architectures, I will present a new approach that tries to achieve high compression by design the second order statistics of a set of additional waveforms involved in the signal acquisition/compression stage. The second topic addressed in this thesis is in the area of communication system, in particular I focused the attention on ultra-wideband (UWB) systems. An option to produce and decode UWB signals is direct-sequence spreading with multiple access based on code division (DS-CDMA). Focusing on this methodology, I will address the coexistence of a DS-CDMA system with a narrowband interferer. To do so, I minimize the joint effect of both multiple access (MAI) and narrowband (NBI) interference on a simple matched filter receiver. I will show that, when spreading sequence statistical properties are suitably designed, performance improvements are possible with respect to a system exploiting chaos-based sequences minimizing MAI only.

Coordination and Control of Autonomous Mobile Robots Swarms by using Particle Swarm Optimization Algorithm and Consensus Theory

This thesis presents some different techniques designed to drive a swarm of robots in an a-priori unknown environment in order to move the group from a starting area to a final one avoiding obstacles. The presented techniques are based on two different theories used alone or in combination: Swarm Intelligence (SI) and Graph Theory. Both theories are based on the study of interactions between different entities (also called agents or units) in Multi- Agent Systems (MAS). The first one belongs to the Artificial Intelligence context and the second one to the Distributed Systems context. These theories, each one from its own point of view, exploit the emergent behaviour that comes from the interactive work of the entities, in order to achieve a common goal. The features of flexibility and adaptability of the swarm have been exploited with the aim to overcome and to minimize difficulties and problems that can affect one or more units of the group, having minimal impact to the whole group and to the common main target. Another aim of this work is to show the importance of the information shared between the units of the group, such as the communication topology, because it helps to maintain the environmental information, detected by each single agent, updated among the swarm. Swarm Intelligence has been applied to the presented technique, through the Particle Swarm Optimization algorithm (PSO), taking advantage of its features as a navigation system. The Graph Theory has been applied by exploiting Consensus and the application of the agreement protocol with the aim to maintain the units in a desired and controlled formation. This approach has been followed in order to conserve the power of PSO and to control part of its random behaviour with a distributed control algorithm like Consensus.

Application Platforms for the Internet of Things: Theory, Architecture, Protocols, Data Formats, and Privacy

The Internet of Things (IoT) is the next industrial revolution: we will interact naturally with real and virtual devices as a key part of our daily life. This technology shift is expected to be greater than the Web and Mobile combined. As extremely different technologies are needed to build connected devices, the Internet of Things field is a junction between electronics, telecommunications and software engineering. Internet of Things application development happens in silos, often using proprietary and closed communication protocols. There is the common belief that only if we can solve the interoperability problem we can have a real Internet of Things. After a deep analysis of the IoT protocols, we identified a set of primitives for IoT applications. We argue that each IoT protocol can be expressed in term of those primitives, thus solving the interoperability problem at the application protocol level. Moreover, the primitives are network and transport independent and make no assumption in that regard. This dissertation presents our implementation of an IoT platform: the Ponte project. Privacy issues follows the rise of the Internet of Things: it is clear that the IoT must ensure resilience to attacks, data authentication, access control and client privacy. We argue that it is not possible to solve the privacy issue without solving the interoperability problem: enforcing privacy rules implies the need to limit and filter the data delivery process. However, filtering data require knowledge of how the format and the semantics of the data: after an analysis of the possible data formats and representations for the IoT, we identify JSON-LD and the Semantic Web as the best solution for IoT applications. Then, this dissertation present our approach to increase the throughput of filtering semantic data by a factor of ten.

Essays on the Axiomatic Theory of Matching

This dissertation mimics the Turkish college admission procedure. It started with the purpose to reduce the inefficiencies in Turkish market. For this purpose, we propose a mechanism under a new market structure; as we prefer to call, semi-centralization. In chapter 1, we give a brief summary of Matching Theory. We present the first examples in Matching history with the most general papers and mechanisms. In chapter 2, we propose our mechanism. In real life application, that is in Turkish university placements, the mechanism reduces the inefficiencies of the current system. The success of the mechanism depends on the preference profile. It is easy to show that under complete information the mechanism implements the full set of stable matchings for a given profile. In chapter 3, we refine our basic mechanism. The modification on the mechanism has a crucial effect on the results. The new mechanism is, as we call, a middle mechanism. In one of the subdomain, this mechanism coincides with the original basic mechanism. But, in the other partition, it gives the same results with Gale and Shapley's algorithm. In chapter 4, we apply our basic mechanism to well known Roommate Problem. Since the roommate problem is in one-sided game patern, firstly we propose an auxiliary function to convert the game semi centralized two-sided game, because our basic mechanism is designed for this framework. We show that this process is succesful in finding a stable matching in the existence of stability. We also show that our mechanism easily and simply tells us if a profile lacks of stability by using purified orderings. Finally, we show a method to find all the stable matching in the existence of multi stability. The method is simply to run the mechanism for all of the top agents in the social preference.

Probabilistic Recursion Theory and Implicit Computational Complexity

In this thesis we provide a characterization of probabilistic computation in itself, from a recursion-theoretical perspective, without reducing it to deterministic computation. More specifically, we show that probabilistic computable functions, i.e., those functions which are computed by Probabilistic Turing Machines (PTM), can be characterized by a natural generalization of Kleene's partial recursive functions which includes, among initial functions, one that returns identity or successor with probability 1/2. We then prove the equi-expressivity of the obtained algebra and the class of functions computed by PTMs. In the the second part of the thesis we investigate the relations existing between our recursion-theoretical framework and sub-recursive classes, in the spirit of Implicit Computational Complexity. More precisely, endowing predicative recurrence with a random base function is proved to lead to a characterization of polynomial-time computable probabilistic functions.

Structural patterns for document engineering: from an empirical bottom-up analysis to an ontological theory

This thesis aims at investigating a new approach to document analysis based on the idea of structural patterns in XML vocabularies. My work is founded on the belief that authors do naturally converge to a reasonable use of markup languages and that extreme, yet valid instances are rare and limited. Actual documents, therefore, may be used to derive classes of elements (patterns) persisting across documents and distilling the conceptualization of the documents and their components, and may give ground for automatic tools and services that rely on no background information (such as schemas) at all. The central part of my work consists in introducing from the ground up a formal theory of eight structural patterns (with three sub-patterns) that are able to express the logical organization of any XML document, and verifying their identifiability in a number of different vocabularies. This model is characterized by and validated against three main dimensions: terseness (i.e. the ability to represent the structure of a document with a small number of objects and composition rules), coverage (i.e. the ability to capture any possible situation in any document) and expressiveness (i.e. the ability to make explicit the semantics of structures, relations and dependencies). An algorithm for the automatic recognition of structural patterns is then presented, together with an evaluation of the results of a test performed on a set of more than 1100 documents from eight very different vocabularies. This language-independent analysis confirms the ability of patterns to capture and summarize the guidelines used by the authors in their everyday practice. Finally, I present some systems that work directly on the pattern-based representation of documents. The ability of these tools to cover very different situations and contexts confirms the effectiveness of the model.

Random lattice models

This thesis deals with three different physical models, where each model involves a random component which is linked to a cubic lattice. First, a model is studied, which is used in numerical calculations of Quantum Chromodynamics.In these calculations random gauge-fields are distributed on the bonds of the lattice. The formulation of the model is fitted into the mathematical framework of ergodic operator families. We prove, that for small coupling constants, the ergodicity of the underlying probability measure is indeed ensured and that the integrated density of states of the Wilson-Dirac operator exists. The physical situations treated in the next two chapters are more similar to one another. In both cases the principle idea is to study a fermion system in a cubic crystal with impurities, that are modeled by a random potential located at the lattice sites. In the second model we apply the Hartree-Fock approximation to such a system. For the case of reduced Hartree-Fock theory at positive temperatures and a fixed chemical potential we consider the limit of an infinite system. In that case we show the existence and uniqueness of minimizers of the Hartree-Fock functional. In the third model we formulate the fermion system algebraically via C*-algebras. The question imposed here is to calculate the heat production of the system under the influence of an outer electromagnetic field. We show that the heat production corresponds exactly to what is empirically predicted by Joule's law in the regime of linear response.

Comparison of Model Chemistry and Density Functional Theory Thermochemical Predictions with Experiment for Formation of Ionic Clusters of the Ammonium Cation Complexed with Water and Ammonia; Atmospheric Implications

The G2, G3, CBS-QB3, and CBS-APNO model chemistry methods and the B3LYP, B3P86, mPW1PW, and PBE1PBE density functional theory (DFT) methods have been used to calculate ΔH° and ΔG° values for ionic clusters of the ammonium ion complexed with water and ammonia. Results for the clusters NH4+(NH3)n and NH4+(H2O)n, where n = 1−4, are reported in this paper and compared against experimental values. Agreement with the experimental values for ΔH° and ΔG° for formation of NH4+(NH3)n clusters is excellent. Comparison between experiment and theory for formation of the NH4+(H2O)n clusters is quite good considering the uncertainty in the experimental values. The four DFT methods yield excellent agreement with experiment and the model chemistry methods when the aug-cc-pVTZ basis set is used for energetic calculations and the 6-31G* basis set is used for geometries and frequencies. On the basis of these results, we predict that all ions in the lower troposphere will be saturated with at least one complete first hydration shell of water molecules.