scholarly journals Geometry of Higher-Order Markov Chains

2012 ◽  
Vol 3 (1) ◽  
Author(s):  
Bernd Sturmfels
Keyword(s):  
Markov Chains ◽  
Prime Ideal ◽  
Linear Space ◽  
Projective Variety ◽  
Likelihood Function ◽  
Distribution Model ◽  
Segre Variety ◽  
Linear Forms ◽  
Transition Distribution ◽  
Binary Case

We determine an explicit Grobner basis, consisting of linear forms and determinantalquadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains.When the states are binary, the corresponding projective variety is a linear space, the model itselfconsists of two simplices in a cross-polytope, and the likelihood function typically has two localmaxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.

scholarly journals Confidence Intervals for the Mixture Transition Distribution (MTD) Model and Other Markovian Models

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 351
Author(s):  
André Berchtold
Keyword(s):  
Markov Chains ◽  
Markov Model ◽  
Em Algorithm ◽  
Hidden Markov Model ◽  
Confidence Intervals ◽  
Hidden Markov ◽  
Simple Calculation ◽  
Standard Methods ◽  
Transition Distribution ◽  
Simulation Results

The Mixture Transition Distribution (MTD) model used for the approximation of high-order Markov chains does not allow a simple calculation of confidence intervals, and computationnally intensive methods based on bootstrap are generally used. We show here how standard methods can be extended to the MTD model as well as other models such as the Hidden Markov Model. Starting from existing methods used for multinomial distributions, we describe how the quantities required for their application can be obtained directly from the data or from one run of the E-step of an EM algorithm. Simulation results indicate that when the MTD model is estimated reliably, the resulting confidence intervals are comparable to those obtained from more demanding methods.

scholarly journals The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 314 ◽  
Author(s):  
Alessandra Bernardi ◽  
Enrico Carlini ◽  
Maria Catalisano ◽  
Alessandro Gimigliano ◽  
Alessandro Oneto
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Tensor Decomposition ◽  
Tensor Rank ◽  
Segre Variety ◽  
Secant Varieties ◽  
Open Problems ◽  
Veronese Variety ◽  
Strong Motivation

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

scholarly journals Linear models for the impact of order flow on prices. II. The Mixture Transition Distribution model

2018 ◽  
Vol 18 (6) ◽  
pp. 917-931 ◽  
Author(s):  
Damian Eduardo Taranto ◽  
Giacomo Bormetti ◽  
Jean-Philippe Bouchaud ◽  
Fabrizio Lillo ◽  
Bence Tóth
Keyword(s):  
Linear Models ◽  
Distribution Model ◽  
Order Flow ◽  
Transition Distribution ◽  
The Impact

scholarly journals The maximum likelihood degree of Fermat hypersurfaces

2015 ◽  
Vol 6 (2) ◽  
Author(s):  
Daniele Agostini ◽  
Davide Alberelli ◽  
Francesco Grande ◽  
Paolo Lella
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Projective Space ◽  
Critical Points ◽  
Projective Variety ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Topological Invariant ◽  
Fermat Curve ◽  
Computational Comparison

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section. 

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