Search results for "exponential family"
showing 10 items of 14 documents
KFAS : Exponential Family State Space Models in R
2017
State space modelling is an efficient and flexible method for statistical inference of a broad class of time series and other data. This paper describes an R package KFAS for state space modelling with the observations from an exponential family, namely Gaussian, Poisson, binomial, negative binomial and gamma distributions. After introducing the basic theory behind Gaussian and non-Gaussian state space models, an illustrative example of Poisson time series forecasting is provided. Finally, a comparison to alternative R packages suitable for non-Gaussian time series modelling is presented.
Order statistics-based parametric classification for multi-dimensional distributions
2013
Traditionally, in the field of Pattern Recognition (PR), the moments of the class-conditional densities of the respective classes have been used to perform classification. However, the use of phenomena that utilized the properties of the Order Statistics (OS) were not reported. Recently, in [10,8], we proposed a new paradigm named CMOS, Classification by the Moments of Order Statistics, which specifically used these quantifiers. It is fascinating that CMOS is essentially ''anti''-Bayesian in its nature because the classification is performed in a counter-intuitive manner, i.e., by comparing the testing sample to a few samples distant from the mean, as opposed to the Bayesian approach in whi…
The fundamental theory of optimal "Anti-Bayesian" parametric pattern classification using order statistics criteria
2013
Author's version of an article in the journal: Pattern Recognition. Also available from the publisher at: http://dx.doi.org/10.1016/j.patcog.2012.07.004 The gold standard for a classifier is the condition of optimality attained by the Bayesian classifier. Within a Bayesian paradigm, if we are allowed to compare the testing sample with only a single point in the feature space from each class, the optimal Bayesian strategy would be to achieve this based on the (Mahalanobis) distance from the corresponding means. The reader should observe that, in this context, the mean, in one sense, is the most central point in the respective distribution. In this paper, we shall show that we can obtain opti…
A Software Tool for the Exponential Power Distribution: The normalp Package
2005
In this paper we present the normalp package, a package for the statistical environment R that has a set of tools for dealing with the exponential power distribution. In this package there are functions to compute the density function, the distribution function and the quantiles from an exponential power distribution and to generate pseudo-random numbers from the same distribution. Moreover, methods concerning the estimation of the distribution parameters are described and implemented. It is also possible to estimate linear regression models when we assume the random errors distributed according to an exponential power distribution. A set of functions is designed to perform simulation studi…
Maximum probability estimators in the case of exponential distribution
1975
In 1966–1969L. Weiss andJ. Wolfowitz developed the theory of „maximum probability” estimators (m.p.e.'s). M.p.e.'s have the property of minimizing the limiting value of the risk (see (2.10).) In the present paper, therfore, after a short description of the new method, a fundamental loss function is introduced, for which—in the so-called regular case—the optimality property of the maximum probability estimators yields the classical result ofR.A. Fisher on the asymptotic efficiency of the maximum likelihood estimator. Thereby it turns out that the m.p.e.'s possess still another important optimality property for this loss function. For the latter the parameters of the exponential distribution—…
Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models
2017
Cyclic models are a subclass of graphical Markov models with simple, undirected probability graphs that are chordless cycles. In general, all currently known distributions require iterative procedures to obtain maximum likelihood estimates in such cyclic models. For exponential families, the relevant conditional independence constraint for a variable pair is given all remaining variables, and it is captured by vanishing canonical parameters involving this pair. For Gaussian models, the canonical parameter is a concentration, that is, an off-diagonal element in the inverse covariance matrix, while for Ising models, it is a conditional log-linear, two-factor interaction. We give conditions un…
Extended differential geometric LARS for high-dimensional GLMs with general dispersion parameter
2018
A large class of modeling and prediction problems involves outcomes that belong to an exponential family distribution. Generalized linear models (GLMs) are a standard way of dealing with such situations. Even in high-dimensional feature spaces GLMs can be extended to deal with such situations. Penalized inference approaches, such as the $$\ell _1$$ or SCAD, or extensions of least angle regression, such as dgLARS, have been proposed to deal with GLMs with high-dimensional feature spaces. Although the theory underlying these methods is in principle generic, the implementation has remained restricted to dispersion-free models, such as the Poisson and logistic regression models. The aim of this…
Differential geometric least angle regression: a differential geometric approach to sparse generalized linear models
2013
Summary Sparsity is an essential feature of many contemporary data problems. Remote sensing, various forms of automated screening and other high throughput measurement devices collect a large amount of information, typically about few independent statistical subjects or units. In certain cases it is reasonable to assume that the underlying process generating the data is itself sparse, in the sense that only a few of the measured variables are involved in the process. We propose an explicit method of monotonically decreasing sparsity for outcomes that can be modelled by an exponential family. In our approach we generalize the equiangular condition in a generalized linear model. Although the …
Exponential and bayesian conjugate families: Review and extensions
1997
The notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting feat…
Some Remarks on Exponential Families
1987
Abstract The following facts may serve to provide a feeling about how restrictive the assumption of an exponential family is. (a) A one-parameter exponential family in standard form with respect to Lebesgue measure is a location parameter family iff it is normal with fixed variance. (b) It is a scale parameter family iff it is gamma with fixed shape parameter. Both facts are known (see Borges and Pfanzagl 1965; Ferguson 1962; Lindley 1958) but may not have received as much attention as they deserve. Under the assumption of differentiable densities, short and elementary proofs are given.