Column Sums Research Articles

Estimation and Inference in Spatial Models with Dominant Units

In spatial econometrics literature estimation and inference are carried out assuming that the matrix of spatial or network connections has uniformly bounded absolute column sums in the number of units, n, in the network. This paper relaxes this restriction and allows for one or more units to have pervasive effects in the network. The linear-quadratic central limit theorem of Kelejian and Prucha (2001) is generalized to allow for such dominant units, and the asymptotic properties of the GMM estimators are established in this more general setting. A new bias-corrected method of moments (BMM) estimator is also proposed that avoids the problem of weak instruments by self-instrumenting the spatially lagged dependent variable. Both cases of homoscedastic and heteroskedastic errors are considered and the associated estimators are shown to be consistent and asymptotically normal, depending on the rate at which the maximum column sum of the weights matrix rises with n. The small sample properties of GMM and BMM estimators are investigated by Monte Carlo experiments and shown to be satisfactory. An empirical application to sectoral price changes in the US over the pre- and post-2008 financial crisis is also provided. It is shown that the share of capital can be estimated reasonably well from the degree of sectoral interdependence using the input-output tables, despite the evidence of dominant sectors being present in the US economy.

Matrix structure of unified mathematical model of electric AC machines at control

The matrix structure of the equations of a generalized electric alternating current machine is proposed, which, based on the Parke equations, is written in the coordinate axes of the machines rotating with the rotor speed. In the matrix structure, the column matrices of the derivatives of the stator, excitation and rotor windings are equal to the product of diagonal matrices consisting of the machine parameters and the column matrices of the flux links themselves and the sum of the matrix columns of the control parameters which are the matrix columns of the stator voltage, excitation voltage, and rotor voltage. It is shown that the matrix structure of a generalized controlled AC machine is transformed into mathematical models of almost all encountered AC electric machines, namely, into a synchronous machine with two excitation windings - a longitudinal and a transverse one; in a synchronous machine with a longitudinal field winding (classic); in an asynchronous machine with a squirrel-cage rotor; into an asynchronous machine with a phase rotor. It has been shown that the matrix structure includes the controls of these machines both from the stator and from the rotor. On the stator side for synchronous machines, it is a frequency control that regulates both the amplitude and frequency of the applied voltage, and on the rotor side, a constant voltage control is supplied to the longitudinal and transverse windings. For asynchronous machines, the stator and rotor are frequency-controlled. The following are examples of frequency control of an asynchronous machine both from the stator and from the rotor.

Inequalities for permanents and permanental minors of row substochastic matrices

In this paper, some inequalities for permanents and permanental minors of row substochastic matrices are proved. The convexity of the permanent function on the interval between the identity matrix and an arbitrary row substochastic matrix is also proved. In addition, a conjecture about the permanent and permanental minors of square row substochastic matrices with fixed row and column sums is formulated.

Inequalities for permanents and permanental minors of row substochastic matrices

In this paper, some inequalities for permanents and permanental minors of row substochastic matrices are proved. The convexity of the permanent function on the interval between the identity matrix and an arbitrary row substochastic matrix is also proved. In addition, a conjecture about the permanent and permanental minors of square row substochastic matrices with fixed row and column sums is formulated.

A generalization of Heffter arrays

AbstractIn this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let be a positive integer, where divides , and let be the subgroup of of order . A Heffter array over relative to is an partially filled array with elements in such that (a) each row contains filled cells and each column contains filled cells; (b) for every , either or appears in the array; and (c) the elements in every row and column sum to . Here we study the existence of square integer (i.e., with entries chosen in and where the sums are zero in ) relative Heffter arrays for , denoted by . In particular, we prove that for , with , there exists an integer if and only if one of the following holds: (a) is odd and ; (b) and is even; (c) . Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.

Factoring a 2 x 2 contingency table.

We show that a two-component proportional representation provides the necessary framework to account for the properties of a 2 × 2 contingency table. This corresponds to the factorization of the table as a product of proportion and diagonal row or column sum matrices. The row and column sum invariant measures for proportional variation are obtained. Geometrically, these correspond to displacements of two point vectors in the standard one-simplex, which are reduced to a center-of-mass coordinate representation, . Then, effect size measures, such as the odds ratio and relative risk, correspond to different perspective functions for the mapping of (δ, μ) to . Furthermore, variations in δ and μ will be associated with different cost-benefit trade-offs for a given application. Therefore, pure mathematics alone does not provide the specification of a general form for the perspective function. This implies that the question of the merits of the odds ratio versus relative risk cannot be resolved in a general way. Expressions are obtained for the marginal sum dependence and the relations between various effect size measures, including the simple matching coefficient, odds ratio, relative risk, Yule’s Q, ϕ, and Goodman and Kruskal’s τc|r. We also show that Gini information gain (IGG) is equivalent to ϕ2 in the classification and regression tree (CART) algorithm. Then, IGG can yield misleading results due to the dependence on marginal sums. Monte Carlo methods facilitate the detailed specification of stochastic effects in the data acquisition process and provide a practical way to estimate the confidence interval for an effect size.

Matrices in [formula omitted] with minimum t-term ranks

Let R and S be two sequences of nonnegative integers in nonincreasing order which have the same sum, and let A(R,S) be the class of all (0,1)-matrices which have row sums given by R and column sums given by S. For a positive integer t, the t-term rank of a (0,1)-matrix A is defined as the maximum number of 1's in A with at most one 1 in each column and at most t 1's in each row. In this paper, we address conditions for the existence of a matrix in A(R,S) that realizes all the minimum t-term ranks, for t≥1.

Lattice SUSY for the DiSSEP at λ2 = 1 (and λ2 = −3)

We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by (Fendley et al 2003 J. Phys. A 36, 12399424; Yang and Fendley 2004 J. Phys. A 37, 893748; Hagendorf and Fendley 2012 J. Stat. Phys. 146, 112255) and (Hagendorf et al 2013 J. Stat. Phys. 150, 60957; Hagendorf and Liénardy 2017 J. Phys. A 50, 185202; Hagendorf and Liénardy 2018 J. Stat. Mech. 033106) might also exist for the Markov matrices of any one-dimensional exclusion processes, which have the additional constraint of zero column sums by comparison with the spin chains. We find that the DiSSEP (Dissipative Symmetric Simple Exclusion Process), introduced by Crampe et al in (Crampe et al 2016 Phys. Rev. E 94, 032102; Vanicat [arXiv:1708.02440]), provides one such example for suitably chosen parameters. The DiSSEP Markov matrix admits the supersymmetry in these cases because it is conjugate to Ising (λ2 = 1) and Δ = −1/2 XXZ (λ2 = −3) spin chain Hamiltonians which also possess the supersymmetry. The consequences for the spectrum of the DiSSEP Markov matrix are discussed. We also note that the length-changing supersymmetry relation for the DiSSEP Markov matrix ML and the supercharge QL† for L sites, , is reminiscent of a ‘transfer matrix’ symmetry that has been observed in other exclusion processes and discuss the similarity.

Nonnegative sum-symmetric matrices and optimal-score partitions

The main result of the note describes certain optimal-score partitions. This result is based on the fact that any nonnegative square matrix whose column sums are the same as the corresponding row sums can be represented as the sum of circuit matrices.

SPC product codes, graphs with cycles and Kostka numbers

The SPC product code is a very popular error correction code with four as its minimum distance. Over the erasure channel, it is supposed to correct up to three erasures. However, this code can correct a higher number of erasures under certain conditions. A codeword of the SPC product code can be represented either by an erasure pattern or by a bipartite graph, where the erasures are represented by an edge. When the erasure contains erasures that cannot be corrected, the corresponding graph contains cycles. In this work we determine the number of strict uncorrectable erasure patterns (bipartite graphs with cycles) for a given size with a fixed number of erasures (edges). Since a bipartite graph can be unequivocally represented by its biadjacency matrix, it is enough to determine the number of non-zero binary matrices whose row and column sum vectors are different from one. At the same time, the number of matrices with prescribed row and column sum vectors can be evaluated in terms of the Kostka numbers associated with Young tableaux.

Classes of (0,1)-matrices Where the Bruhat Order and the Secondary Bruhat Order Coincide

Given two nonincreasing integral vectors R and S, with the same sum, we denote by $\mathcal {A}(R,S)$ the class of all (0,1)-matrices with row sum vector R, and column sum vector S. The Bruhat order and the Secondary Bruhat order on $\mathcal {A}(R,S)$ are both extensions of the classical Bruhat order on Sn, the symmetric group of degree n. These two partial orders on $\mathcal {A}(R,S)$ are, in general, different. In this paper we prove that if R = (2,2,…,2) or R = (1,1,…,1), then the Bruhat order and the Secondary Bruhat order on $\mathcal {A}(R,S)$ coincide.

CARTAN SUBALGEBRAS IN DIMENSION DROP ALGEBRAS

AbstractWe completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous$\text{C}^{\ast }$-algebras with exactly$n$Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

Transpose of N\xf6rlund matrices on the domain of summability matrices

Let E=(E_{n,k})_{n,kgeq 0} be an invertible summability matrix with bounded absolute row sums and column sums, and let E_{p} denote the domain of E in the sequence space ell _{p}(1le p<infty ). In this paper, we consider the transpose of Nörlund matrix associated with a nonnegative and nonincreasing sequence as an operator mapping ell _{p} into the sequence space E_{p} and establish a general upper estimate for its operator norm, which depends on the ell _{1}-norm of the rows and columns of the matrix E. In particular, we apply our result to domains of some summability matrices such as Fibonacci, Karamata, Euler, and Taylor matrices. Our result is an extension of those given by G. Talebi and M.A. Dehghan (Linear Multilinear Algebra 64(2):196–207, 2016). It also provides some analogues of the results by G. Talebi (Indag. Math. 28(3):629–636, 2017).

The maximum, supremum, and spectrum for critical set sizes in (0,1)‐matrices

AbstractIf is a partially filled‐in (0,1)‐matrix with a unique completion to a (0,1)‐matrix (with prescribed row and column sums), then we say that is a defining set for . A critical set is a minimal defining set (the deletion of any entry results in more than one completion). We give a new equivalent definition of critical sets in (0,1)‐matrices and apply this theory to , the set of (0,1)‐matrices of dimensions with uniform row and column sum . The smallest possible size for a defining set of a matrix in is [N. Cavenagh, J. Combin. Des. 21 (2013), pp. 253–266], and the infimum (the largest‐smallest defining set size for members of ) is known asymptotically [N. Cavenagh and R. Ramadurai, Linear Algebra Appl. 537 (2018), pp. 38–47]. We show that no critical set of size larger than exists in an element of and that there exists a critical set of size in an element of for each such that . We also bound the supremum (the smallest‐largest critical set size for members of ) between and .

COLOR FEATURE WITH SPATIAL INFORMATION EXTRACTION METHODS FOR CBIR: A REVIEW

Inn then last two decades the Content Based Image Retrieval (CBIR) considered as one of the topic of interest for theresearchers. It depending one analysis of the image’s visual content which can be done by extracting the color, texture and shapefeatures. Therefore, feature extraction is one of the important steps in CBIR system for representing the image completely. Color featureis the most widely used and more reliable feature among the image visual features. This paper reviews different methods, namely LocalColor Histogram, Color Correlogram, Row sum and Column sum and Colors Coherences Vectors were used to extract colors featurestaking in consideration the spatial information of the image.

An interval-valued intuitionistic fuzzy DEMATEL method combined with Choquet integral for sustainable solid waste management

The decision making trial and evaluation laboratory (DEMATEL) is a pragmatic method used to construct the structural correlation of criteria in a multi-criteria decision making (MCDM) problem. This paper proposes modifications to DEMATEL. Different from the typical DEMATEL which utilizes crisp numbers, this modification introduces interval-valued intuitionistic fuzzy numbers to enhance judgements in group decision-making environment. We use the interval-valued intuitionistic fuzzy weighted averaging operator instead of the typical averaging operator to aggregate decision-makers’ preferences. This paper proposes a combination of interval-valued intuitionistic fuzzy numbers, DEMATEL and the Choquet integral to allow a strong interrelationship between the sum of rows and the sum of columns of criteria in structural correlation analysis. The feasibility and applicability of the proposed method is illustrated by a numerical example about sustainable solid waste management. The result indicates that ‘collaboration and synergy’ is the most influential criterion for sustainable solid waste management. Comparative results are presented to show the validity of the proposed method.

Condition estimation for regression and feature selection

Regression and feature selection require the minimisation of Xβ−y2 with respect to β, where X∈Rn×p, n<p in feature selection and n≥p in regression. The vector β contains the coefficients of the basis functions in regression, and the weights of the features in feature selection. This paper considers the stability of β, as measured by the ratio of its relative error with respect to the relative error in y, and it is shown that the condition number κ(X) of X is not a good measure of this stability. In particular, a large value of κ(X) may lead to incorrect conclusions about the stability of β and it may be thought regularisation must be applied to the normal equation XTXβ=XTy if κ(X)≫1, but its application may lead to a large error in β. It is shown in this paper that (a) the presence of noise in y or the condition κ(X)≫1 do not imply that regularisation must be applied to the normal equation, and (b) the condition κ(X)≫1 does not imply that a small relative error in y yields a large relative error in β. These disadvantages of κ(X) lead to the effective condition number η(X,y), which provides a better measure of the stability of β due to a perturbation in y, but it is difficult to compute it reliably in some circumstances. Regularisation requires that a constraint be imposed on the solution of the normal equation, and it is shown that a constraint on β1 can be interpreted in terms of the column sums of X, and that a constraint on β2 can be interpreted in terms of the singular value decomposition of X. The paper contains several examples that illustrate the theoretical results.

Efficient numerical algorithms for constructing orthogonal generalized doubly stochastic matrices

A real square matrix is generalized doubly stochastic (g.d.s.) if all of its row sums and column sums equal one. We propose efficient numerical algorithms for generating such matrices having possibly orthogonality property or/and satisfying Yang-Baxter equation (YBE). Additionally, an inverse eigenvalue problem for finding orthogonal generalized doubly stochastic matrices with prescribed eigenvalues is solved here. The tests performed in MATLAB illustrate our proposed algorithms and demonstrate their useful numerical properties.

Premagic and Ideal Flow Matrices

Several interesting properties of a special type of matrix that has a row sum equal to the column sum are shown with the proofs. Premagic matrix can be applied to strongly connected directed network graph due to its nodes conservation flow. Relationships between Markov Chain, ideal flow and random walk on directed graph are also discussed.

The persistent statistical structure of the US input–output coefficient matrices: 1963–2007

ABSTRACTThe paper finds evidence for the existence of a statistical structure in the US input–output coefficient matrices for 1963–2007 and characterizes the identified statistical regularities. For various aspects of matrices, we find smooth and unimodal empirical distributions (EDs) with a remarkable stability in their functional form for most of the samples. The EDs of all entries, row sums, and the entries of the (left- and right-hand) Perron–Frobenius eigenvectors are well described by fat-tailed distributions, while the EDs of column sums and eigenvalues' moduli are explained by the normal and the beta distribution. The paper provides several economic interpretations of these statistical results as well as some implications and potential uses for structural and stochastic input–output analysis.