Method Of Principal Coordinates Research Articles

Principal component analysis and its generalizations for any type of sequence (PCA-Seq)

In the 1940s, Karhunen and Loève proposed a method for processing a one-dimensional numeric time series by converting it into multidimensional by shifts. In fact, a one-dimensional number series was decomposed into several orthogonal time series. This method has many times been independently developed and applied in practice under various names (EOF, SSA, Caterpillar, etc.). Nowadays, the name ‘SSA’ (Singular Spectral Analysis) is the most often used. It turned out that it is universal, applicable to any time series without requiring stationary assumptions, automatically decomposes time series into a trend, cyclic components and noise. By the beginning of the 1980s, Takens had shown that for a dynamical system such a method makes it possible to obtain an attractor from observing only one of these variables, thereby bringing the method to a powerful theoretical basis. In the same years, the practical benefits of phase portraits became clear. In particular, it was used in the analysis and forecast of animal abundance dynamics. In this paper we propose to extend SSA to a one-dimensional sequence of any type of elements, including numbers, symbols, figures, etc., and, as a special case, to a molecular sequence. Technically, the problem is solved using an algorithm like SSA. The sequence is cut by a sliding window into fragments of a given length. Between all fragments, the matrix of Euclidean distances is calculated. This is always possible. For example, the square root of the Hamming distance between fragments is a Euclidean distance. For the resulting matrix, the principal components are calculated by the principal-coordinate method (PCo). Instead of a distance matrix, one can use a matrix of any similarity/dissimilarity indexes and apply methods of multidimensional scaling (MDS). The result will always be PCs in some Euclidean space. We called this method ‘PCA-Seq’. It is certainly an exploratory method, as is its particular case SSA. For any sequence, in cluding molecular, PCA-Seq without any additional assumptions allows presenting its principal components in a numerical form and visualizing them in the form of phase portraits. A long history of SSA application for numerical data gives all reason to believe that PCA-Seq will be not less useful in the analysis of non-numerical data, especially in hypothesizing. PCA-Seq is implemented in the freely distributed Jacobi 4 package (http://jacobi4.ru/).

Board level drop test: exact solution to the problem of the nonlinear dynamic response of a PCB to the drop impact

An analytical predictive model has been developed for the evaluation of the nonlinear dynamic response of a printed-circuit-board (PCB) to the drop impact during board-level testing. The hypothesis of “heavy-and-flexible” PCB is used in the analysis: the surface-mounted-devices (SMDs) are assumed to be small enough not to affect the PCB’s flexural rigidity, but their masses have been considered and accounted for by “spreading out” the SMD total mass over the PCB surface. The analysis is restricted to the fundamental mode of vibrations, and the method of principal coordinates is used to evaluate the response. The exact solution to the nonlinear differential equation for the principal coordinate has been obtained. Another important finding is that the nonlinear amplitudes were determined even without solving the nonlinear differential equation of motion. The main objective of the analysis is to provide design guidelines for constructing a feasible experimental setup. A simply supported board is suggested as the most appropriate structure for an adequate test vehicle: the experimental data for such a board, as far as the behavior of the solder material in the second level of interconnections is concerned, can be easily and reliably interpreted and extrapolated for the practical use. The developed model enables one to predict the induced bending moments and the in-plane (membrane) forces that could be applied in the subsequent analyses to the PCB areas in the proximity of the package and its solder joint interconnections.

Combining and congruence evaluation of phylogenetic signals from different genes based on geometric approach

A new Euclidean distance based algorithm is used for analysis of congruence and combining molecular genetic data. This approach is called geometric, since Euclidean distance satisfies all metric axioms and the points representing the sequences can be placed in a geometric space without distorting the mutual distances and can be endowed with the coordinates in this space. Geometricness of Euclidean distances allows to apply to molecular data methods of multivariate analysis, which are relevant for intra- and interspecies variability investigating, visualization of possible directions of evolution, combining data and evaluation of the congruence of phylogenetic signals. The algorithm is used for the analysis of more than 1500 nucleotide sequences of two nuclear (apoB, brca1) and two mitochondrial (co1, cytb) genes of 15 Palaearctic and Nearctic shrews species of genus Sorex (Soricidae, Eulipotyphla). All sequences of each gene are represented as a set of points in Euclidean space. Centroids of a set of points belonging to the same species are calculated. The matrix of Euclidean distances between the species centroids is calculated for each gene. Mantel test is applied to estimate pairwise similarity (congruence) of interspecies distances matrices relating to different genes. nDNA genes congruence is equal 0.961, mtDNA – 0.748. All matrices of the interspecies distances are combined into a joint matrix by weighing. Joint genetic space for all species is built by principal coordinate method from the joint matrix. Several variability directions reflecting evolutionary events of different scale are visualized in a joint genetic space. In addition, the joint matrix of interspecies distances is used for building a phylogenetic tree which is consistent with the zoological systematics accepted for today. This confirms the efficiency of our proposed method.

The coordinate function in the problem of the nonlinear dynamic response of an elongated printed circuit board (PCB) to a drop impact applied to its supports

We consider an elongated printed circuit board (PCB) whose long edges are support-free and the short edges are either simply supported or rigidly clamped. The board experiences a significant impact loading applied to the short edges. In such a situation the reactive in-plane (“membrane”) forces (stresses) are large and result in a geometrically nonlinear dynamic response of the board. The analysis is limited to the fundamental mode of vibrations, and the method of principal coordinates is used to evaluate the dynamic response of the board. The objective of the analysis is to find out if it is sufficient to consider the nonlinear behavior of the principal coordinate (because of the in-plane stresses) only, or it is necessary to account also for the effect of these stresses on the coordinate function. Based on the developed simple and physically meaningful analytical (“mathematical”) stress model, we conclude that the effect of the in-plane tension on the coordinate function does not have to be considered for a simply supported board, but has to be accounted for in the case of a clamped board. This could be done, if necessary, using the results of the analysis.

Part-Time Farming in Wales: A Typological Study

This study seeks to identify types of part-time farming systems within Wales. Based on a sample of 211 agricultural holdings and a mixed-mode set of five variables, the investigation structures a Q-matrix of transformed general coefficients of similarity and subjects it to cluster analysis and the method of principal coordinates. Six types of part-time farming are isolated. The motives that underpin the decision to farm on a part-time basis are also considered. Analysts of the agrarian environment have, over the years, drawn attention to the distinctiveness and complexity of part-time farming as a socio-economic system. Numerous studies have shown that farms managed on a part-time basis can vary quite significantly both in their structure and functional organization (Layton, 1978; Mage, 1976), and that part-time farmers themselves can differ in their occupational attitudes, motivations and aspirations (Gasson, 1967b and 1977). It is now clear that part-time farming need not necessarily constitute a short-term career objective-a stepping stone, as it were, into full-time agriculture or out of farming altogether. For many farmers it represents a meaningful adaptation to prevailing employment opportunities, and as such is a positive career commitment. For others, however, part-time farming is an avocation-a hobby, a form of recreation or an exercise in 'rural retreating' (Troughton, 1975; Gasson, 1966; McQuin, 1978). Given this diversity, it is perhaps not surprising that a major focus of research interest has been related to the issue of identifying types or classes within the part-time farming sector. The intention of this paper is to contribute to this line of inquiry, and in particular to effect a multivariate classification of part-time farming systems, using a set of variables which previous studies have suggested are of diagnostic significance. RESEARCH CONTEXT AND DESIGN Definitions of part-time farming are legion, as are reviews of the part-time farming literature (Frauendorfer, 1966; Fuguitt et al., 1977). It is not necessary to extend this debate here; suffice it to say that, despite broad areas of agreement, no consensus has been reached in regard to a typology of part-time farming systems. Specification of the unit of analysis has proved to be a major problem, as has the identification of discriminating variables. Whilst some studies have focused on the farm (holding) as a production unit, others have either referred to selected characteristics of individual part-time farmers or have widened their range of reference to include the part-time farmer and his family. Inevitably, descriptions of part-time farming Trans. Inst. Br. Geogr. N.S. 7, 88-97 (1982) Printed in Great Britain This content downloaded from 157.55.39.235 on Fri, 07 Oct 2016 05:59:11 UTC All use subject to http://about.jstor.org/terms Part-time farming in Wales 89 systems have been based on a wide variety of attributes. However, certain attributes have established themselves as key indicators. Four of these are recognized in this study.

Chemical components of the environment and Late Campanian microfossil frequencies

Abstract Using Late Campanian subsurface data from western Nigeria, the frequencies of species of foraminifers and ostracods were weighed against 14 trace elements of the enclosing sediments (Si, Fe, Mg, Ca, Na, K, Ti, P, Mn, V, Mo, Sr, Pb and Zn) by standard methods of multivariate analysis. The method of canonical correlation requires the interpretation of two canonical variates. The first of these suggests a dipolar relationship between carbonate sediment and clastic components. The method of principal coordinates for Euclidean distances produces meaningful clusters, grouping the organisms into environmentally significant subsets. A largely unidirectional, ecological trend could be identified.