Comparing Rectangular Data Matrices

1982 ◽  
Vol 14 (8) ◽  
pp. 1087-1095 ◽  
Author(s):  
L J Hubert ◽  
R G Golledge
Keyword(s):  
Comparison Method ◽  
Crime Rates ◽  
Significance Testing ◽  
Testing Strategy ◽  
Metropolitan Statistical Areas ◽  
Numerical Example ◽  
The Matrix ◽  
Comparison Strategy ◽  
Index Crime ◽  
Data Matrices

A procedure is discussed for comparing two rectangular n × m data matrices. The two matrices would typically represent data on the same n objects (for example, cities or subjects) and the same m attributes (for example, crime rates or attitudinal variables). An index that measures the degree to which both matrices are similar is presented along with a significance testing strategy that takes into account the possible dependency among the m attributes. To illustrate the strategy, a numerical example is given that compares the seven index crime rates for a set of twenty standard metropolitan statistical areas for the years 1976 and 1977. In addition to giving several possible generalizations of the basic comparison method, including a natural procedure for comparing three or more data matrices, we show in some detail how the matrix comparison strategy encompasses and extends the work of Tjøtheim on measuring association for spatially related variables.

scholarly journals A New Solution to the Matrix EquationX−AX¯B=C

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Caiqin Song
Keyword(s):  
Unique Solution ◽  
Canonical Form ◽  
Matrix Equation ◽  
Explicit Solution ◽  
Symmetric Operator ◽  
Operator Matrix ◽  
Numerical Example ◽  
Controllability Matrix ◽  
The Matrix

We investigate the matrix equationX−AX¯B=C. For convenience, the matrix equationX−AX¯B=Cis named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix. The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

scholarly journals Another look at matrix correlations

2019 ◽  
Vol 35 (22) ◽  
pp. 4748-4753 ◽  
Author(s):  
Ahmad Borzou ◽  
Razie Yousefi ◽  
Rovshan G Sadygov
Keyword(s):  
Matrix Decomposition ◽  
Data Matrix ◽  
Supplementary Information ◽  
Biological Processes ◽  
Single Experiment ◽  
Sample Number ◽  
The Matrix ◽  
Rv Coefficient ◽  
Systems View ◽  
Data Matrices

Abstract Motivation High throughput technologies are widely employed in modern biomedical research. They yield measurements of a large number of biomolecules in a single experiment. The number of experiments usually is much smaller than the number of measurements in each experiment. The simultaneous measurements of biomolecules provide a basis for a comprehensive, systems view for describing relevant biological processes. Often it is necessary to determine correlations between the data matrices under different conditions or pathways. However, the techniques for analyzing the data with a low number of samples for possible correlations within or between conditions are still in development. Earlier developed correlative measures, such as the RV coefficient, use the trace of the product of data matrices as the most relevant characteristic. However, a recent study has shown that the RV coefficient consistently overestimates the correlations in the case of low sample numbers. To correct for this bias, it was suggested to discard the diagonal elements of the outer products of each data matrix. In this work, a principled approach based on the matrix decomposition generates three trace-independent parts for every matrix. These components are unique, and they are used to determine different aspects of correlations between the original datasets. Results Simulations show that the decomposition results in the removal of high correlation bias and the dependence on the sample number intrinsic to the RV coefficient. We then use the correlations to analyze a real proteomics dataset. Availability and implementation The python code can be downloaded from http://dynamic-proteome.utmb.edu/MatrixCorrelations.aspx. Supplementary information Supplementary data are available at Bioinformatics online.

Development of a Multistage Reliability-Based Design Optimization Method

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Eric J. Paulson ◽  
Ryan P. Starkey
Keyword(s):  
Objective Function ◽  
Design Optimization ◽  
Technology Development ◽  
Computational Cost ◽  
Optimization Method ◽  
Comparison Method ◽  
Reliability Based Design Optimization ◽  
Numerical Example ◽  
Reliability Based Design ◽  
System Acquisition

Complex system acquisition and its associated technology development have a troubled recent history. The modern acquisition timeline consists of conceptual, preliminary, and detailed design followed by system test and production. The evolving nature of the estimates of system performance, cost, and schedule during this extended process may be a significant contribution to recent issues. The recently proposed multistage reliability-based design optimization (MSRBDO) method promises improvements over reliability-based design optimization (RBDO) in achieved objective function value. In addition, its problem formulation more closely resembles the evolutionary nature of epistemic design uncertainties inherent in system design during early system acquisition. Our goal is to establish the modeling basis necessary for applying this new method to the engineering of early conceptual/preliminary design. We present corrections in the derivation and solutions to the single numerical example problem published by the original authors, Nam and Mavris, and examine the error introduced under the reduced-order reliability sampling used in the original publication. MSRBDO improvements over the RBDO solution of 10–36% for the objective function after first-stage optimization are shown for the original second-stage example problem. A larger 26–40% improvement over the RBDO solution is shown when an alternative comparison method is used than in the original. The specific implications of extending the method to arbitrary m-stage problems are presented, together with a solution for a three-stage numerical example. Several approaches are demonstrated to mitigate the computational cost increase of MSRBDO over RBDO, resulting in a net decrease in calculation time of 94% from an initial MSRBDO baseline algorithm.

X.—The Factorial Analysis of Multiple Item Tests

1944 ◽  
Vol 62 (1) ◽  
pp. 74-82 ◽  
Author(s):  
D. N. Lawley
Keyword(s):  
Factorial Analysis ◽  
Numerical Example ◽  
Multiple Item ◽  
The Matrix ◽  
The Individual ◽  
The Cost

Summary6. The expected score of an individual on a test consisting of a large number of items is assumed to be given by a formula involving the ability of the individual and also two quantities constant for the test. An expression is then derived for the covariance between two tests measuring different abilities. It appears that if a factorial analysis is performed on a set of tests of unequal difficulty, using the matrix of variances and covariances, a spurious factor will tend to be introduced depending mainly on the differences in difficulty. The effect of this is removed by transforming the variances and covariances to a new set of coefficients. A numerical example of the process is given.In conclusion I should like to thank the Carnegie Trust for the Universities of Scotland for a grant to cover the cost of the setting and printing of mathematical formulæ in a paper previously published in the Society's Proceedings (LXI, A, 1943, 273–287).

scholarly journals Clustering Objects Described by Juxtaposition of Binary Data Tables

2008 ◽  
Vol 2008 ◽  
pp. 1-13
Author(s):  
Amar Rebbouh
Keyword(s):  
Survey Data ◽  
Binary Data ◽  
Probability Distributions ◽  
Hierarchical Classification ◽  
Classification Method ◽  
Physical Systems ◽  
Numerical Example ◽  
Data Tables ◽  
Ascending Hierarchical Classification ◽  
Data Matrices

This paper seeks to develop an allocation of 0/1 data matrices to physical systems upon a Kullback-Leibler distance between probability distributions. The distributions are estimated from the contents of the data matrices. We discuss an ascending hierarchical classification method, a numerical example and mention an application with survey data concerning the level of development of the departments of a given territory of a country.

A control method for the matrix converter based on virtual AC/DC/AC conversion using carrier comparison method

2005 ◽  
Vol 152 (3) ◽  
pp. 65-73 ◽  
Author(s):  
Jun-Ichi Itoh ◽  
Ikuya Sato ◽  
Hideki Ohguchi ◽  
Kazuhisa Sato ◽  
Akihiro Odaka ◽  
...  
Keyword(s):  
Control Method ◽  
Comparison Method ◽  
Matrix Converter ◽  
The Matrix

Unidimensional Seriation: Implications for Evaluating Criminal Justice Data

1981 ◽  
Vol 13 (3) ◽  
pp. 309-320 ◽  
Author(s):  
L J Hubert ◽  
T Kenny ◽  
R G Golledge ◽  
G D Richardson
Keyword(s):  
Criminal Justice ◽  
Homicide Rate ◽  
Raw Data ◽  
Metropolitan Statistical Areas ◽  
Absolute Value ◽  
Numerical Example ◽  
Single Dimension ◽  
Unidimensional Scale ◽  
Formal Test ◽  
Asymmetric Proximity

The problem of validating a given unidimensional scale (that is, an ordering of a set of objects along a single dimension) is discussed in terms of a few simple properties of the data used to obtain the scale. Based on a set of asymmetric proximity values as raw data, a distinction between analyzing absolute-value information or sign information is presented that leads to a formal test of whether a given scale is being reliably represented. In short, a scale is generated from absolute-value information, but validated through sign information. A numerical example which deals with the perception of homicide rate over fifteen of the largest Standard Metropolitan Statistical Areas is included as an illustration of the general methodological discussion.

scholarly journals EXTENDED CONSISTENCY ANALYSIS FOR PAIRWISE COMPARISON METHOD

2018 ◽  
Vol 10 (1) ◽  
Author(s):  
Sahika Koyun Yılmaz ◽  
Vildan Ozkir
Keyword(s):  
Decision Maker ◽  
Probability Function ◽  
Pairwise Comparison ◽  
Comparison Method ◽  
Analysis Procedure ◽  
Acquisition Phase ◽  
Measurement Unit ◽  
Consistency Analysis ◽  
The Matrix ◽  
Distinct Features

Pairwise comparison (PC) is a widely used scientific technique to compare criteria or alternatives in pairs in order to express the decision maker’s judgments without the need for a unique common measurement unit between criteria. The method constructs a PC matrix by requesting the assessments of the decision maker(s) in the judgment acquisition phase and calculates an inconsistency measure to determine whether the judgments are adequately consistent with each other before subsequent phases. Although the method requires the decision maker to make all judgments in a PC matrix, it does not force him/her to make a judgment for each element of the matrix. If any judgment in a PC matrix is absent, for this reason, the judgment acquisition phase yields an incomplete PC matrix rather than a complete one. Missing judgments are calculated by multiplication of the judgments made by the decision maker. If the judgements of the decision maker are transitive and well-proportioned, missing judgments will not disturb the consistency of the resulting PC matrix. In other words, consistency of a PC matrix relies on the judgments made by the decision maker. Since the current consistency analysis procedure is designed for complete PC matrices, the suitability for evaluating the inconsistency of incomplete PC matrices is questionable. Probability density functions of random PC matrices with altering numbers of missing judgments show distinct features, indicating an incomplete PC matrix and a complete PC matrix do not come from the same probability function, and their mean consistency index (RI) is different. Consequently, we propose an extended consistency analysis procedure to evaluate the consistency of incomplete PC matrices.

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