Terms Of Matrix Algebra Research Articles

Multivalued Classification of Computer Attacks Using Artificial Neural Networks with Multiple Outputs

Modern computer networks (CN), having a complex and often heterogeneous structure, generate large volumes of multi-dimensional multi-label data. Accounting for information about multi-label experimental data (ED) can improve the efficiency of solving a number of information security problems: from CN profiling to detecting and preventing computer attacks on CN. The aim of the work is to develop a multi-label artificial neural network (ANN) architecture for detecting and classifying computer attacks in multi-label ED, and its comparative analysis with known analogues in terms of binary metrics for assessing the quality of classification. A formalization of ANN in terms of matrix algebra is proposed, which allows taking into account the case of multi-label classification and the new architecture of ANN with multiple output using the proposed formalization. The advantage of the proposed formalization is the conciseness of a number of entries associated with the ANN operating mode and learning mode. Proposed architecture allows solving the problems of detecting and classifying multi-label computer attacks, on average, 5% more efficiently than known analogues. The observed gain is due to taking into account multi-label patterns between class labels at the training stage through the use of a common first layer. The advantages of the proposed ANN architecture are scalability to any number of class labels and fast convergence.

INTERPOLATION OF TABULAR FUNCTIONS FROM ONE INDEPENDENT VARIABLE USING THE TAYLOR POLYNOMIAL

A method of local interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree in arbitrarily located interpolation nodes has been developed. This makes it possible to calculate intermediate values of tabular functions between interpolation nodes. The conducted analysis of the latest research and publications in the field of interpolation of tabular functions showed that the main part of the research is a strict theory of interpolation, i.e. clarification of its fundamental mathematical provisions. Some features of the interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree are considered, namely: the solution algorithm and mathematical formulation of the interpolation problem are given; its formalized notation is given, as well as the matrix notation of interpolation procedures for certain values of the argument. A scalar algorithm for solving the problem of interpolation of tabular functions from one independent variable using the Taylor polynomial of the 2nd, 3rd, and 4th degrees has been developed. The simplicity and clarity of this algorithm is one of its advantages, but the algorithm is inconvenient for software implementation. The mathematical formulation of the problem of interpolation of tabular functions in terms of matrix algebra is given. The interpolation task is reduced to performing the following actions: based on the values of nodal points known from the table, it is necessary to calculate the Taylor matrix of the nth degree; based on the function values specified in the table a column vector of interpolation nodes should be formed; solve a linear system of algebraic equations, the root of which is the numerical coefficients of the Taylor polynomial of the nth degree. A method of calculating the coefficients of the interpolant, given by the Taylor polynomial of the nth degree for one independent variable has been developed. The essence of the method reduces to the product of the matrix, inverse of the Taylor matrix, which is determined by the nodal points of the tabular function, by a column vector containing the values of the interpolation nodes. Specific examples demonstrate the peculiarities of calculating the interpolant coefficients of the 2nd, 3rd and 4th degrees for one independent variable, and for each of them the interpolated value of the function at a given point is calculated. Calculations were performed in the Excel environment, which by analogy can be successfully implemented in any other computing environment.

Deconvolution of mass spectra measured with a non-uniform detector array to give accurate ion abundances.

Array detectors are under active development since they offer large improvements in the efficiency of mass spectrum measurements. High quality is always a requirement whether array detectors are used for ions, electrons, UV, x-rays, etc., but in mass spectrometry the very high accuracy frequently needed in ion abundance measurements for isotope ratios is unique. These demands necessitate modelling the measurement process and careful deconvolution of the measured data. A linear model in terms of matrix algebra is presented in which the incident spectrum (unknown) and the measured spectrum are represented by column matrices and the action of the detector array on the incident spectrum is represented by an experimentally measurable square 'A' matrix. Residual noise in this matrix can be minimized following a singular value decomposition procedure and its use in 'forward deconvolution' of measured spectra is discussed. The random error in the incident ion counts is accounted for after the deconvolution since this is not an error from the perspective of the detector. The microchannel plate electron multiplier gain distribution is an important feature of the deconvolution.

Transmission through two-dimensional tight-binding lattices

A methodology of transmission through two-dimensional tight-binding lattices is presented. The theory is formulated in terms of matrix algebra and the relationships between the matrices are examined in detail. The features specific to tight-binding systems are contrasted to those for more general transmission problems. Illustrative examples are given to indicate the wide applicability of the methodology.

Search tree method for the determination of similarity operators between arbitrary lattices

Lattice similarity operators are transformation matrices which describe a linear mapping of two crystal lattices on to each other such that: (i) points of a common derivative lattice of reasonably low index in both lattices form a coincidence site lattice ; (ii) the deformation occurring with the mapping is tolerably small; (iii) the determinant of the transformation matrix is fixed by the ratio of the formula units in the unit cells of both crystals. These conditions are formulated in terms of matrix algebra. To tackle the discrete optimization problem of searching for lattice similarity operators between two given arbitrary lattices, a search tree algorithm is proposed. It minimizes the number of trial matrices to be investigated by systematic use of geometrical criteria which are derived from the above conditions. The search is based on the identification of triplets of lattice vectors of one lattice in the other. The triplets are limited to those uniquely defining all derivative lattices of interest. Applications of the method are envisaged in all situations where geometrical relationships between lattices or crystal structures have to be described, in particular for the real space description of structural transitions if the transformation matrix cannot be deduced clearly from experimental observations. The method is demonstrated using the martensitic transformation in zirconia and the reconstructive transformation between the (Mn, …)SiO 3 polymorphs Rhodonite and Pyroxmangite as examples.

Theoretical mechanisms for synthesis of carcinogen-induced embryonic proteins: XX embryonic gene perturbations expressed in terms of matrix algebra

Simple matrix expressions can be devised for gene repressor associations that lend themselves to manipulations such as linear transformation matrices. Such transformation matrices act in perturbing representations for given repressed genic states and may be analogous to carcinogens. Although the matrix algebraic expressions are developed by using simple repressor theory, it can equally serve to represent modifications of chromatin domains that may be more consistent with mechanisms of derepression of embryonic genes. In general, it is proposed that the potentially exploitable algebras such as abstract, geometric, matrix, vector and tensor be a subset of mathematical biology termed “Bioalgebraic Field Theory”.

A Mathematical Interpretation of Marxian Unproductive Labour

Marx's theory of unproductive labour is incomprehensible without a proper understanding of his dual theory of value. Hence, the mathematical analysis of his model of value and growth is a useful and instructive investigation. Up to now, however, no attempt has been made to grasp the basic distinction between productive and unproductive labour in terms of matrix algebra. We hope this paper will fill this gap. It will be shown that Marx's distinction between productive and unproductive labour follows from his dual theory of labour value and use-value. Once this theory is formulated in terms of matrix algebra, it will be shown that Marx's results are confirmed without any further ideological digression. We will also consider the rate of profit formula developed by Morishima and Seton (1961) in the light of our previous results. In a last section some hints will be given for a less abstract mathematical analysis of unproductive labour today.

TESTS OF ASSOCIATION IN TERMS OF MATRIX ALGEBRA

Initially it is shown how the calculations involved in tests of association on contingency tables can be expressed in terms of matrix algebra. Several advantages which follow from doing so are then described and illustrated. The problem of estimating correlation coefficients between attributes which have truncated distributions is considered, and the paper concludes by comparing the results from two different populations.

Internal Migration and Population Change in Hungary between 1959 and 1965

Population change within the various regions of Hungary is a function of spatial differentials in fertility, mortality and migration. The work of Hungarian demographers has been almost solely in the fields of fertility and mortality, and although extensive internal migration data have been collected since 1955, the analysis of the relationships between migration and population change has been neglected. The analysis presented in this paper uses an ergodic Markovian model to evaluate the potential of current inter-county migration streams in redistributing the population between these areas. A closed system, in the sense that total population within the country as a whole is held constant, is assumed. The model is formalized in terms of matrix algebra. The Markovian model has the advantage that, by examining gross migration streams, the interactions within each annual migration system can be included in the analysis. The implications of the seven annual migration systems for the period 1959 to 1965 are computed. The processes working are such that, within each system, an equilibrium situation is eventually established, from which an equilibrium population distribution for the counties can be calculated. The counties are then classified according to the trend and magnitude of their equilibrium populations for which tentative explanations are advanced. These values are not to be regarded as population forecasts. It is suggested that they are meaningful indices of migration between 1959 and 1965. It is further suggested that the speed of convergence to equilibrium is an index of migration intensity. AT the beginning of 1966 the population of Hungary numbered o. I6 million, making it one of the smaller countries of Europe. The rate of population increase is low, being 2.4 per thousand in 1965 and is entirely attributable to low fertility. Although international migration is negligible, internal mobility is high, and since I957 almost I million internal migrations have been recorded annually. Expressed in another way this means that the average Hungarian can be expected to make just over eight migratory moves between the age of one and death and this figure would be considerably higher if residential mobility were included. The result is a vast redistribution of population, the dominant net movement being from the villages to the towns. The former have consistently lost population since the second world war although, owing to their relatively high rate of natural increase, this was translated into an absolute decline only after I959. Virtually all urban settlements have grown in size through the influx of rural migrants, with the exception of the large Alfold centres which, although having urban status, have most of their gainfully employed population in the agricultural sector of the economy. Trends in the exodus of population from the villages during the post-war period are difficult to determine, because annual data have only been available since 1955, but examination of this material shows that the maximum outflow occurred in 1960, when the villages lost 52,000 persons. In the eleven years from I949 to 1959, 490,000 people abandoned the villages and, since 200,000 left these rural settlements between 1955 and 1959 (a period which includes the abnormally low value of 16,000 for 1955), this would suggest that the annual loss of population between 1949 and 1954 was no higher than that between I956 and I959. Further, even though the average yearly outflow after 1961 was lower than in the preceding five years44,500 compared with 46,400-495,000 persons abandoned the villages in the eleven-year

Multivariate Techniques of Use in Biological Research

SUMMARYMultivariate statistical methods are used increasingly in biological research to investigate the responses of organisms considered as a whole, whereas established statistical methods are usually concerned with measured characteristics considered one at a time. Multivariate techniques are mostly explained in terms of matrix algebra, which is a way of dealing with groups of numbers rather than individual ones. A brief description is given of some elementary results of matrix algebra and a method is presented whereby hypotheses can be generated about interrelations within an organism. Two techniques, principal component analysis and canonical analysis, are described in greater detail. It is emphasized that hypotheses need to be tested even though they have been generated by objective statistical means.

Analysis of the Discriminant Function in Educational and Psychological Research

AN EXTREMELY useful statistical tool that has been largely ignored by educational and psychologi cal researchers is the analysis of the discriminant function. In the December, 1963, issue of the Re view of Education Research, which is devoted en tirely to statistical methodology, discriminant anal ysis is not even mentioned, although there is a chap ter on regression and correlation, to which the dis criminant function is closely alliedc Of books on statistics familiar to educators and psychologists, only that by McNemar (4) and another by Guilford (2) devote some space to the topic. Further evi dence of the neglect of this technique is furnished by the fact that Tatsuoka and Tiedeman, in their chap ter on statistics in the Handbook of Research on Teaching (5) mentioned no more than five studies that used discriminant analysis since 1938. By con trast, the same authors said that factor analysis, which is another multivariate technique, was exten sively used in educational and psychological re search. This phenomenon is puzzling in view of the fact that discriminant analysis is both useful and com paratively easy to comprehend as a statistical con cept. While the usual theoretical discussion of the rationale of the technique is couched in terms of ma trix algebra, an understanding of it is possible to anyone with a background of high school algebra? It is the belief of the author that as researchers be come more familiar with the technique they will make greater use of it than they have hitherto. While discriminant analysis can be used for more than two groups, this paper will deal with only the two group situation. Even with the two group situa tion the potentials of the technique are practically unlimited. In the study of high school dropouts, for instance, discriminant analysis can be used to dif ferentiate successful high school students and poten tial dropouts. Over achievers and underachievers can be studied in a similar fashion. School counse lors, seeking information as to which of two similar occupations or professions a counselee belongs i n, may find himself in a better position to counsel if he knows the discriminant function that best differenti ates the two occupations. In brief, the discriminant function is useful in any situation where subjects can be dichotomized in any meaningful way, such as suc cess or failure, extraversion or introversion, med icine or dentistry, etc. This paper will not go into a detailed discussion of the theoretical basis of discriminant analysis, as excellent articles are available for those with a knowledge of matrix algebra as, for instance, one by Travers (6). Rather, this paper will attempt to de scribe what the discriminant function does and illus trate its use by a concrete example that c an be fol lowed by anyone familiar with high school algebra. The computations, which were made on a Monroe calculator, were checked against a computer pro gram1 which made use of matrices and inverse ma trices in its calculations. The results were identi cal within rounding errors. In a real sense, discriminant analysis is the mul tiple regression technique with a different purpose. Whereas in multiple regression the purpose is to seek a composite score (predicted score) that is a linear function of several independent variables such that it has maximum predictive power for agiven group, in discriminant analysis the purpose is to seek a composite score (the discriminant function) that is a linear function of several independent vari ables such that it can best differentiate two groups. In either case the problem is to find the proper co efficients (weights) for the independent variables such that the composite score is functionally op timal. Algebraically, if we let the composite score be Z, and Xi, X2, X3 . . . Xfc be the independent vari ables, then Z is a linear function o?XlyX2f. . . Xfc, where k is the number of independent variables.

Turbulent and Inertia Flow in Slider Bearings

A computer solution is developed in terms of matrix algebra in which the equations of a slider bearing are analyzed which include the inertia, turbulent, and viscous terms. For small values of turbulence the results indicate that the dominant term after the viscous term is the inertia term, and the inclusion of the Reynolds turbulent stress term actually causes a slight decrease in load capacity. A numerical example is considered which clearly indicates that the inertia effects alone can be considerable and cannot be neglected. The role of turbulence effects is still undecided.

THE THEORY OF THE METHOD OF LEAST SQUARES

AbstractThis paper gives derivations of some of the more important results in the method of least squares in terms of matrix algebra. The following topics are covered: derivation of normal equations and their solution; residuals; estimation of standard errors; dependent unknowns; probability distribution of least‐square estimates of unknowns.

Simultaneous adjustment of triangulation and trilateration: An investigation of Tellurometer lengths

The present development of distance-measuring instruments suitable for geodetic surveys raises the problem of simultaneously adjusting observed lengths and angles. The basic mathematical tools for carrying out such adjustments according to the method of least squares have existed for many years. In recent years excellent material, written in terms of matrix algebra, has been published on the general theory of least squares adjustment by such authors as Marchant, Bjerhammar, Wolf, and Gotthardt. In some of this literature the approach has been to proceed from the general problem to particular cases as opposed to the synthesis of particular cases into a general method.

Application of network theory to the analysis of rotating machinery part I - Synchronous and asynchronous machines

AS the technology of engineering progresses a point is sometimes reached where, by the expansion and generalization of certain basic concepts, two or more seemingly unrelated areas of study can be brought together and shown to have a common origin. The common origins are very important to those in the teaching profession, where the rapid expansion of the engineering profession demands that an ever-increasing amount of technical understanding be developed in the student with little or no increase in student contact time. Without these common origins, the necessary increase in teaching efficiency would be difficult to attain. One of the purposes of this, the companion paper,1 and a conference paper, ?General Network Theory in Terms of Matrix Algebra? by M. B. Reed, is to show that by expanding our general network theory and concepts as we now think of them, and by taking a slightly different and more general viewpoint toward the rotating machinery problem, these two areas of technology can be built on a common and rigorous foundation which results in a clearer and broader understanding of both areas, and at the same time will perhaps provide for a more efficient teaching program.