Conditional Probability Matrices Research Articles

Reliability of multicomponent assemblages

The reliability of multicomponent assemblages is of interest to engineers in many disciplines. It has been shown that the reliability of a multicomponent assembly—a simple frame—may be estimated through the use of a causal network. The causal network approach does not require the modelling of a structure as series or non-series; it only requires that the network of component reliabilities by acyclical. Assumptions about local member influence are implicitly included only through the use of conditional probability matrices throughout the network. These matrices may be generated through the use of a noisy OR gate and/or a rule-based system. It is postulated that the system reliability of any multicomponent assemblage can be estimated in this manner. Testing of the application of the concept to problems of realistic dimensions is currently under way.

A Markovian Approach to the Study of the Canadian Cattle Industry

LIKE any livestock industry, the Canadian beef and dairy cattle industry is characterized by a cyclical pattern in terms of the number of cattle on farm, the number of cattle slaughtered and the number of cattle exported. Traditionally, the analysis and forecast of cattle stocks are based either on econometric models which often include the biological life cycle of cattle or on the pure biological nature of cattle.' In this paper, we investigate the behavior of the cattle industry through the use of a third approach-the Markov chain technique. The Markov chain technique is by itself a very mechanical procedure, but one which can incorporate economic justifications. It can then, in our view, provide a very fruitful view of the industry. Basically, the Markov chain technique allows us to construct flow matrices of beef and dairy cattle according to their biological sequences. For instance, a male calf born during any time period t can, in the same period, be slaughtered, exported, die or remain on farm as a calf. The decision to retain a calf as a steer (for future slaughter) or as a bull (for future reproductive purposes), to export the calf, or to slaughter the calf (for veal), is basically an economic decision. The outcome of such decisions is translated into the elements of the Markovian transition flow matrix. Based upon the biological sequences of the different categories of cattle and the structure of the beef and dairy cattle industry, we can set up transition matrices for Western and Eastern Canada. Table I indicates the structure of such matrices for Western Canada. Cells representing possible flows are identified by numbers while cells representing impossible flows are left blank.2 Transition probability matrices of cattle movement can be constructed by dividing each row element in the matrix by its corresponding row total. These probabilities reflect the probabilities of cattle moving from one category to another. It is also through the use of such probabilities that we will carry out our simulation analysis. This paper is divided into seven sections. In the second section a model of demand, supply and inventory for beef and dairy cattle is presented. Section III discusses some general empirical results based on the transition probability matrices. Section IV discusses the procedures for simulation using the conditional transition probability matrices. Section V presents the results of the historical simulation while section VI presents the results of some sensitivity analysis experiments. The last section is for concluding remarks.

Darwinian selection and “altruism”

Models are proposed for evolution at a single locus affecting altruistic behavior in which genotypic fitnesses are Darwinian and frequency (but not density) dependent. The fitnesses are composed, either in a multiplicative or an additive way, of factors which depend on the receipt and donation of altruistic behavior. The factors are determined from the matrices of conditional probabilities which describe the genotypes of relatives. Since selection occurs, these probabilities are in terms of genotype frequencies. The relationship between the risk to helper and benefit to recipient which allows altruism to evolve is shown to depend on the kinship coefficient between helper and helped, the particular fitness function proposed and the degree of dominance of the altruism. The commonly accepted criteria of W. D. Hamilton [ J. Theor. Biol. 7 (1964), 1–16, 17–52] apply only in the additive case. A second class of models of social cooperation independent of relationship and its evolutionary dynamics are discussed.

Locating Community Boundaries: an Integration of Theory and Empirical Techniques

Utilizing the Kaufman (1959) and Sutton-Kolaja (1960a, 1960b) conceptualization of community, empirical techniques are developed to identify community boundaries. These techniques are unique because they allow simultaneous consideration of boundary and interaction aggregation problems. A sociological conceptualization of community is coordinated with layman respondent's perceptions so empirical data, faithful to both perspectives, may be gathered from community residents. Community boundaries are identified from conditional probability matrices applied to residents identified as business, education, and government leaders in ten rural areas of Illinois. Township boundaries regularly coincide with community boundaries. Techniques developed here appear widely applicable to identifying community and other boundaries.

Creation of conditional dependency matrices based on a stochastic streamflow synthesis technique

The construction of transition probability matrices by the streamflow synthesis technique developed by the Hydrologie Engineering Center of the U.S. Army Corps of Engineers was tried. In certain cases it was found to be necessary to use a transform different from the Wilson‐Hilferty transform used in the stated routine in order to produce reasonable results. Other details of the numerical considerations necessary for the definition of discrete conditional probability matrices between monthly flow volumes are described.

The Derivation of Joint Distribution and Correlation between Relatives by the Use of Stochastic Matrices

The absolute frequencies of the various genotypic parent-offspring and sib-pair combinations with respect to one pair of genes, autosomal or sex-linked, are well known to geneticists, for they are of practical value in certain types of studies in human heredity. These frequencies are, however, obtained by a rather long process, even for the simple case of full sibs. The procedure of obtaining the frequencies of unclenephew or first cousinl combinations is entirely too tedious evenly with the help of matrix notations (Hogben, 1933). The purpose of the present communication is three-fold. The first is to give a simple procedure of finding the frequencies of the various genotype combinations of near relatives by using matrices of conditional probabilities. The second purpose is to express such matrices of conditional probabilities of relatives in the form of a linear function of some basic matrices. The third is to deduce the correlations between relatives from such linear functions of the basic matrices. The meaning of these statements will be made clear in later sections. For many years there existed two apparently very different methods of obtaining the genotypic correlations between relatives. One is a straightforward but long procedure by which the frequencies of the various combinations of the stated relatives in the general population are first found and then the is calculated from such a correlation table. On the other hand, they may be obtained by the method of path coefficients, developed by Wright (1921 and later). The latter method gives the required coefficient almost instantly once the relationship is specified, but does not give us any information about the frequencies of the various combinations of the relatives in the population. Moreover, one has to be familiar with the mathematical theorems concerning the path coefficients before he can use them to derive the required correlations. The method of stochastic process and its final reduction to some basic matrices, as described in the following pages, will give us both the frequencies of relative-pairs and their in a very simple manner. It is hoped that the present method will bridge the gap between the two existing procedures.