radial distribution function within the framework of the tsallis statistical mechanics abstract this study is conducted to obtain the radial distribution function rdf within the tsallis statistical mechanics to this end probability distribution functions are applied in the first and fourth versions of the tsallis statistics moreover a closed formula is proposed for rdf the power nature of the probability distribution in the tsallis statistics makes it difficult to separate kinetic energy and configurational potential parts by using the taylor expansion around q 1 of the power distribution it is possible to show the independency of momenta and coordinates through integrating over the phase space variables in addition at low densities numerical calculations have been performed for the rdf our results show that the correlation increases as q values increase

sparse additive models summary we present a new class of methods for high dimensional non parametric regression and classification called sparse additive models our methods combine ideas from sparse linear modelling and additive non parametric regression we derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size sparse additive models are essentially a functional version of the grouped lasso of yuan and lin they are also closely related to the cosso model of lin and zhang but decouple smoothing and sparsity enabling the use of arbitrary non parametric smoothers we give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data showing that they can be effective in fitting sparse non parametric models in high dimensional data

dirichlet laplace priors for optimal shrinkage penalized regression methods such as l1 regularization are routinely used in high dimensional applications and there is a rich literature on optimality properties under sparsity assumptions in the bayesian paradigm sparsity is routinely induced through two component mixture priors having a probability mass at zero but such priors encounter daunting computational problems in high dimensions this has motivated continuous shrinkage priors which can be expressed as global local scale mixtures of gaussians facilitating computation in contrast to the frequentist literature little is known about the properties of such priors and the convergence and concentration of the corresponding posterior distribution in this article we propose a new class of dirichlet laplace priors which possess optimal posterior concentration and lead to efficient posterior computation finite sample performance of dirichlet laplace priors relative to alternatives is assessed in simulated and real data examples

identification of time delays using a polynomial identification method abstract an algorithm for the exact least squares identification of an approximate continuous time time delay system is derived and its operation verified by simulation

universality in the chaotic dynamics associated with saddle centers critical points abstract we describe a new statistical pattern characteristic of saddle center critical points in the dynamics of non integrable hamiltonian systems associated with the partition of the hamiltonian into rotational motion energy and hyperbolic motion energy pieces in a linear neighborhood of the saddle center the distribution of hyperbolic energy of orbits visiting a neighborhood of the saddle center defines a statistical law that for a large range of values of the hyperbolic energy has the form p z z in a regime of high nonintegrability we present numerical evidence to support the conjecture that this pattern is universal for any hamiltonian system with one or several saddle centers the parameter appears as a universal characteristic associated with the chaotic dynamics of saddle centers

identification of partially known systems abstract the concepts of partially known system identification are introduced and the advantages of a method whereby process information can be utilized in a linear transfer function representation of the system are highlighted and discussed in particular an algorithm for identifying a single unknown physical system parameter associated with a linear system is given it is assumed that the system is polynomial as opposed to linear in the physical parameter the bond graph representation is used as the basis for deriving a system dependent algorithm for each partially known system the method is illustrated by simulated examples

spike and slab group lassos for grouped regression and sparse generalized additive models abstract we introduce the spike and slab group lasso ssgl for bayesian estimation and variable selection in linear regression with grouped variables we further extend the ssgl to sparse generali

bayesian regression tree ensembles that adapt to smoothness and sparsity ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions examples of these methods include gradient boosted decision trees random forests and bayesian classification and regression trees two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality we show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic we implement this in the context of the bayesian additive regression trees framework and illustrate its promising performance through testing on benchmark data sets we provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate up to a logarithmic factor for sparse functions and functions with additive structures in the high dimensional regime where the dimensionality of the covariate space is allowed to grow nearly exponentially in the sample size our method also adapts to the unknown smoothness and sparsity levels and can be implemented by making minimal modifications to existing bayesian additive regression tree algorithms

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

bayes and empirical bayes multiplicity adjustment in the variable selection problem this paper studies the multiplicity correction effect of standard bayesian variable selection priors in linear regression our first goal is to clarify when and how multiplicity correction happens automatically in bayesian analysis and to distinguish this correction from the bayesian ockham s razor effect our second goal is to contrast empirical bayes and fully bayesian approaches to variable selection through examples theoretical results and simulations considerable differences between the two approaches are found in particular we prove a theorem that characterizes a surprising aymptotic discrepancy between fully bayes and empirical bayes this discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical bayes estimate indeed even at the extreme when the empirical bayes estimate converges asymptotically to the true variable inclusion probability the potential for a serious difference remains

supremum norm posterior contraction and credible sets for nonparametric multivariate regression in the setting of nonparametric multivariate regression with unknown error variance we study asymptotic properties of a bayesian method for estimating a regression function f and its mixed partial derivatives we use a random series of tensor product of b splines with normal basis coefficients as a prior for f and the error variance is either estimated using the empirical bayes approach or is endowed with a suitable prior in a hierarchical bayes approach we establish pointwise l2 and supremum norm posterior contraction rates for f and its mixed partial derivatives and show that they coincide with the minimax rates our results cover even the anisotropic situation where the true regression function may have different smoothness in different directions using the convergence bounds we show that pointwise l2 and supremum norm credible sets for f and its mixed partial derivatives have guaranteed frequentist coverage with optimal size new results on tensor products of b splines are also obtained in the course

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

the spike and slab lasso abstractdespite the wide adoption of spike and slab methodology for bayesian variable selection its potential for penalized likelihood estimation has largely been overlooked in this paper we bridge this gap by cross fertilizing these two paradigms with the spike and slab lasso procedure for variable selection and parameter estimation in linear regression we introduce a new class of self adaptive penalty functions that arise from a fully bayes spike and slab formulation ultimately moving beyond the separable penalty framework a virtue of these non separable penalties is their ability to borrow strength across coordinates adapt to ensemble sparsity information and exert multiplicity adjustment the spike and slab lasso procedure harvests efficient coordinate wise implementations with a path following scheme for dynamic posterior exploration we show on simulated data that the fully bayes penalty mimics oracle performance providing a viable alternative to cross validation we develop theory for the s

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

optimal finite dimensional recursive identification in a polynomial output mapping class abstract a time varying polynomial output equation expressing an implicit dependence of the observed signal on the parameter vector admits a finite dimensional recursive representation for the optimal on line estimator both continuous and discrete observations result in the differential systems which give the estimate deterministic exponentially weighted least squares is applied exponential stability of the identifier and convergence to the true parameter value are shown two examples are presented a continuous application originates in a ph control process the discrete identification is applied in a dynamic arma type model which is quadratic in the parameter

comment regularization via bayesian penalty mixing ridge regression and the lasso illustrate some of the remarkable successes of penalized likelihood regularization for regression analysis when viewed through the bayesian lens of posterior maximiz

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

study of chaos in hamiltonian systems via convergent normal forms abstract we use moser s normal forms to study chaotic motion in two degree hamiltonian systems near a saddle point besides being convergent they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity both aspects combined allowed a precise computation of the homoclinic interaction of stable and unstable manifolds in the full phase space rather than just the poincare section the formalism was applied to the henon heiles hamiltonian producing strong evidence that the region of convergence of these normal forms extends over that orginally established by moser

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

local asymptotics for regression splines and confidence regions in this paper we study the local behavior of regression splines in particular explicit expressions for the asymptotic pointwise bias and variance of regression splines are obtained in addition asymptotic normality for regression splines is established leading to the construction of approximate confidence intervals and confidence bands for the regression function

fast bayesian factor analysis via automatic rotations to sparsity abstractrotational post hoc transformations have traditionally played a key role in enhancing the interpretability of factor analysis regularization methods also serve to achieve this goal by prioritizing sparse loading matrices in this work we bridge these two paradigms with a unifying bayesian framework our approach deploys intermediate factor rotations throughout the learning process greatly enhancing the effectiveness of sparsity inducing priors these automatic rotations to sparsity are embedded within a pxl em algorithm a bayesian variant of parameter expanded em for posterior mode detection by iterating between soft thresholding of small factor loadings and transformations of the factor basis we obtain a dramatic accelerations b robustness against poor initializations and c better oriented sparse solutions to avoid the prespecification of the factor cardinality we extend the loading matrix to have infinitely many columns with the indian buffet process ibp prior the factor dimen

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators