Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observedfor each individual. A particular case of FDA is when the observed functions are densityfunctions, that are also an example of infinite dimensional compositional data. In thiswork we compare several methods for dimensionality reduction for this particular typeof data: functional principal components analysis (PCA) with or without a previousdata transformation and multidimensional scaling (MDS) for diferent inter-densitiesdistances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (householdsincome distributions)

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Refinement criteria for global illumination using convex funcions

In several computer graphics areas, a refinement criterion is often needed to decide whether to goon or to stop sampling a signal. When the sampled values are homogeneous enough, we assume thatthey represent the signal fairly well and we do not need further refinement, otherwise more samples arerequired, possibly with adaptive subdivision of the domain. For this purpose, a criterion which is verysensitive to variability is necessary. In this paper, we present a family of discrimination measures, thef-divergences, meeting this requirement. These convex functions have been well studied and successfullyapplied to image processing and several areas of engineering. Two applications to global illuminationare shown: oracles for hierarchical radiosity and criteria for adaptive refinement in ray-tracing. Weobtain significantly better results than with classic criteria, showing that f-divergences are worth furtherinvestigation in computer graphics. Also a discrimination measure based on entropy of the samples forrefinement in ray-tracing is introduced. The recursive decomposition of entropy provides us with a naturalmethod to deal with the adaptive subdivision of the sampling region

Negative Fukui functions: new insights based on electronegativity equalization

The occurrence of negative values for Fukui functions was studied through the electronegativity equalization method. Using algebraic relations between Fukui functions and different other conceptual DFT quantities on the one hand and the hardness matrix on the other hand, expressions were obtained for Fukui functions for several archetypical small molecules. Based on EEM calculations for large molecular sets, no negative Fukui functions were found

Critical thoughts on computing atom condensed Fukui functions

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory

Linear response functions for a vibrational configuration interaction state

Linear response functions are implemented for a vibrational configuration interaction state allowing accurate analytical calculations of pure vibrational contributions to dynamical polarizabilities. Sample calculations are presented for the pure vibrational contributions to the polarizabilities of water and formaldehyde. We discuss the convergence of the results with respect to various details of the vibrational wave function description as well as the potential and property surfaces. We also analyze the frequency dependence of the linear response function and the effect of accounting phenomenologically for the finite lifetime of the excited vibrational states. Finally, we compare the analytical response approach to a sum-over-states approach

Splitting statistical potentials into meaningful scoring functions: testing the prediction of near-native structures from decoy conformations

Background: Recent advances on high-throughput technologies have produced a vast amount of protein sequences, while the number of high-resolution structures has seen a limited increase. This has impelled the production of many strategies to built protein structures from its sequence, generating a considerable amount of alternative models. The selection of the closest model to the native conformation has thus become crucial for structure prediction. Several methods have been developed to score protein models by energies, knowledge-based potentials and combination of both.Results: Here, we present and demonstrate a theory to split the knowledge-based potentials in scoring terms biologically meaningful and to combine them in new scores to predict near-native structures. Our strategy allows circumventing the problem of defining the reference state. In this approach we give the proof for a simple and linear application that can be further improved by optimizing the combination of Zscores. Using the simplest composite score () we obtained predictions similar to state-of-the-art methods. Besides, our approach has the advantage of identifying the most relevant terms involved in the stability of the protein structure. Finally, we also use the composite Zscores to assess the conformation of models and to detect local errors.Conclusion: We have introduced a method to split knowledge-based potentials and to solve the problem of defining a reference state. The new scores have detected near-native structures as accurately as state-of-art methods and have been successful to identify wrongly modeled regions of many near-native conformations.

Speed of convergence of recursive least squares learning with ARMA perceptions

This paper fills a gap in the existing literature on least squareslearning in linear rational expectations models by studying a setup inwhich agents learn by fitting ARMA models to a subset of the statevariables. This is a natural specification in models with privateinformation because in the presence of hidden state variables, agentshave an incentive to condition forecasts on the infinite past recordsof observables. We study a particular setting in which it sufficesfor agents to fit a first order ARMA process, which preserves thetractability of a finite dimensional parameterization, while permittingconditioning on the infinite past record. We describe how previousresults (Marcet and Sargent [1989a, 1989b] can be adapted to handlethe convergence of estimators of an ARMA process in our self--referentialenvironment. We also study ``rates'' of convergence analytically and viacomputer simulation.

Recursive contracts

We obtain a recursive formulation for a general class of contractingproblems involving incentive constraints. Under these constraints,the corresponding maximization (sup) problems fails to have arecursive solution. Our approach consists of studying the Lagrangian.We show that, under standard assumptions, the solution to theLagrangian is characterized by a recursive saddle point (infsup)functional equation, analogous to Bellman's equation. Our approachapplies to a large class of contractual problems. As examples, westudy the optimal policy in a model with intertemporal participationconstraints (which arise in models of default) and intertemporalcompetitive constraints (which arise in Ramsey equilibria).

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of functions and we relate it to a system for real numberrepresentation which we call (t, t-1) expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.

Optimal taxation without state-contingent debt

To recover a version of Barro's (1979) `random walk'tax smoothing outcome, we modify Lucas and Stokey's (1983) economyto permit only risk--free debt. This imparts near unit root like behaviorto government debt, independently of the government expenditureprocess, a realistic outcome in the spirit of Barro's. We showhow the risk--free--debt--only economy confronts the Ramsey plannerwith additional constraints on equilibrium allocations thattake the form of a sequence of measurability conditions.We solve the Ramsey problem by formulating it in terms of a Lagrangian,and applying a Parameterized Expectations Algorithm tothe associated first--order conditions. The first--order conditions andnumerical impulse response functions partially affirmBarro's random walk outcome. Though the behaviors oftax rates, government surpluses, and government debts differ, allocationsare very close for computed Ramsey policies across incomplete and completemarkets economies.

Location of multiple server common service centers or public facilities for minimizing general congestion and travel cost functions

We propose a model and solution methods, for locating a fixed number ofmultiple-server, congestible common service centers or congestible publicfacilities. Locations are chosen so to minimize consumers congestion (orqueuing) and travel costs, considering that all the demand must be served.Customers choose the facilities to which they travel in order to receiveservice at minimum travel and congestion cost. As a proxy for thiscriterion, total travel and waiting costs are minimized. The travel costis a general function of the origin and destination of the demand, whilethe congestion cost is a general function of the number of customers inqueue at the facilities.

The domain and interpretation of utility functions: An exploration

This paper proposes an exploration of the methodology of utilityfunctions that distinguishes interpretation from representation. Whilerepresentation univocally assigns numbers to the entities of the domainof utility functions, interpretation relates these entities withempirically observable objects of choice. This allows us to makeexplicit the standard interpretation of utility functions which assumesthat two objects have the same utility if and only if the individual isindifferent among them. We explore the underlying assumptions of suchan hypothesis and propose a non-standard interpretation according towhich objects of choice have a well-defined utility although individualsmay vary in the way they treat these objects in a specific context.We provide examples of such a methodological approach that may explainsome reversal of preferences and suggest possible mathematicalformulations for further research.

Generating functions for the numbers of Abelian extensions of a local field

The aim of this paper is to give an explicit formula for the num- bers of abelian extensions of a p-adic number field and to study the generating function of these numbers. More precisely, we give the number of abelian ex- tensions with given degree and ramification index, and the number of abelian extensions with given degree of any local field of characteristic zero. Moreover, we give a concrete expression of a generating function for these last numbers