scholarly journals Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Tomack Gilmore
Keyword(s):  
Total Positivity ◽  
Combinatorial Interpretation ◽  
Totally Positive ◽  
Labelled Trees ◽  
Planar Networks

We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and rooted labelled trees on $n+1$ vertices where $k$ children of the root are lower-numbered than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in eight indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.

scholarly journals Covariance Matrix Estimation under Total Positivity for Portfolio Selection*

2020 ◽  
Author(s):  
Raj Agrawal ◽  
Uma Roy ◽  
Caroline Uhler
Keyword(s):  
Covariance Matrix ◽  
Total Positivity ◽  
General Purpose ◽  
Shrinkage Estimators ◽  
Positive Dependence ◽  
Matrix Estimation ◽  
Sample Covariance ◽  
Totally Positive ◽  
Markowitz Portfolio ◽  
Previous State

Abstract Selecting the optimal Markowitz portfolio depends on estimating the covariance matrix of the returns of N assets from T periods of historical data. Problematically, N is typically of the same order as T, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general-purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 (MTP2). This constraint on the covariance matrix not only enforces positive dependence among the assets but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock market data spanning 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.

Optimal couplings are totally positive and more

2004 ◽  
Vol 41 (A) ◽  
pp. 321-332 ◽  
Author(s):  
Paul Glasserman ◽  
David D. Yao
Keyword(s):  
Transportation Problem ◽  
Random Variables ◽  
Total Positivity ◽  
Bivariate Distribution ◽  
Discrete Distributions ◽  
Optimal Solutions ◽  
Totally Positive ◽  
Maximal Correlation ◽  
Optimal Coupling

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

Optimal couplings are totally positive and more

2004 ◽  
Vol 41 (A) ◽  
pp. 321-332
Author(s):  
Paul Glasserman ◽  
David D. Yao
Keyword(s):  
Transportation Problem ◽  
Random Variables ◽  
Total Positivity ◽  
Bivariate Distribution ◽  
Discrete Distributions ◽  
Optimal Solutions ◽  
Totally Positive ◽  
Maximal Correlation ◽  
Optimal Coupling

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

scholarly journals On Total Positivity of Catalan-Stieltjes Matrices

10.37236/6270 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Qiongqiong Pan ◽  
Jiang Zeng
Keyword(s):  
Sufficient Conditions ◽  
Hankel Matrix ◽  
Total Positivity ◽  
Combinatorial Interpretation

Recently Chen-Liang-Wang (Linear Algerbra Appl. 471 (2015) 383—393) proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix $A$ we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of $A$ is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Catalan-Stieltjes matrix $A$.

scholarly journals TOTAL POSITIVITY IN SPRINGER FIBRES

2020 ◽  
Author(s):  
G Lusztig
Keyword(s):  
Reductive Group ◽  
Total Positivity ◽  
Flag Manifold ◽  
Cell Decomposition ◽  
Unipotent Element ◽  
Positive Part ◽  
Totally Positive ◽  
Natural Cell

Abstract Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.

Introduction

2011 ◽  
Author(s):  
Shaun M. Fallat ◽  
Charles R. Johnson
Keyword(s):  
Total Positivity ◽  
Nonnegative Matrices ◽  
Extant Literature ◽  
Totally Positive ◽  
Positive Matrices ◽  
Class Of Matrices ◽  
Totally Positive Matrices

This introductory chapter is an overview of totally positive (or nonnegative) matrices (TP or TN matrices). Positivity has roots in every aspect of pure, applied, and numerical mathematics. The subdiscipline, total positivity, also is entrenched in nearly all facets of mathematics. At first it may appear that the notion of total positivity is artificial; however, this class of matrices arises in a variety of important applications. Historically, the theory of totally positive matrices originated from the pioneering work of Gantmacher and Krein in 1960. The chapter explores the extant literature on total positivity since then, before proceeding to the definitions and notations to be used in the rest of this volume. It also provides a brief overview of the succeeding chapters.

Max-infinite divisibility and multivariate total positivity

1994 ◽  
Vol 31 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Abdulhamid A. Alzaid ◽  
Frank Proschan
Keyword(s):  
Distribution Function ◽  
Total Positivity ◽  
Positive Function ◽  
Infinite Divisibility ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Positive Dependence ◽  
Totally Positive

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

On Density of Generalized Polynomials

1991 ◽  
Vol 34 (2) ◽  
pp. 202-207
Author(s):  
N. Dyn ◽  
D. S. Lubinsky ◽  
Boris Shekhtman
Keyword(s):  
Total Positivity ◽  
Generalized Polynomials ◽  
Totally Positive

AbstractWe consider the density in C[a, b] of generalized polynomials of the form The main point of this note is that total positivity of K(x, t) has little relationship to density: There is a symmetric, analytic, totally positive (in fact ETP (∞)) kernel K for which these generalized polynomials are not dense.

scholarly journals Total Positivity of the Cubic Trigonometric Bézier Basis

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xuli Han ◽  
Yuanpeng Zhu
Keyword(s):  
General Framework ◽  
Total Positivity ◽  
Point Of View ◽  
Shape Parameters ◽  
Positive Basis ◽  
Corner Cutting ◽  
Curve Design ◽  
Totally Positive ◽  
Chebyshev Space ◽  
Cutting Algorithm

Within the general framework of Quasi Extended Chebyshev space, we prove that the cubic trigonometric Bézier basis with two shape parametersλandμgiven in Han et al. (2009) forms an optimal normalized totally positive basis forλ,μ∈(-2,1]. Moreover, we show that forλ=-2orμ=-2the basis is not suited for curve design from the blossom point of view. In order to compute the corresponding cubic trigonometric Bézier curves stably and efficiently, we also develop a new corner cutting algorithm.

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