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Theoretical problems with the factor analysis model have resulted in increased interest in component analysis as an alternative. It is therefore of interest to assess empirically some of the asserted differences between the methods. One of the positive features sometimes attributed to factor analysis, as opposed to component analysis, is superior stability of results under sampling from a population of variables. The comparative stability of three methods, Maximum Liklihood Factor Analysis, principal component analysis, and rescaled image analysis, is investigated. Random samples are drawn from a "population" of variables. The sample pattern is then compared with the corresponding population pattern. The results suggest that none of the three methods can be considered superior with respect to stability.

The value of a meteorological parameter at a grid point can be derived from Fourier analysis of data at station observations around the grid point. But the inhomogeneity of data distribution does not warrant such an elaborate and rigorous approach. Cressman's method which is shown as a simplified version of local-point Fourier analysis is found to be more feasible from rms error statistics of wind field analyses by both methods.

Summary Both the analysis of variance and principal component analysis divide sums of squares into orthogonal components. The major difference is that the first uses external criteria, the second internal. The calculations involved in these techniques are demonstrated with an example. The way in which data may be standardized before principal component analysis is discussed. In some cases, the information discarded on standardization may be examined by an analysis of variance, and related to the results of the principal component analysis.

1. A theoretical analysis of the errors in multicomponent spectrophotometric analysis of nucleoside mixtures, by a least-squares procedure, has been made to obtain an expression for the error coefficient, relating the error in calculated concentration to the error in extinction measurements. 2. The error coefficients, which depend only on the library'ofspectraused→fittheexperimentalcurves,havebeencomputedforaνmberoflibrary′ofspectraused→fittheexperimentalcurves,havebeencomputedforaνmberoflibraries' containing the following nucleosides found in s-RNA: adenosine, guanosine, cytidine, uridine, 5-ribosyluracil, 7-methylguanosine, 6-dimethylaminopurine riboside, 6-methylaminopurine riboside and thymine riboside. 3. The error coefficients have been used to determine the best conditions for maximum accuracy in the determination of the compositions of nucleoside mixtures. 4. Experimental determinations of the compositions of nucleoside mixtures have been made and the errors found to be consistent with those predicted by the theoretical analysis. 5. It has been demonstrated that, with certain precautions, the multicomponent spectrophotometric method described is suitable as a basis for automatic nucleotide-composition analysis of oligonucleotides containing nine nucleotides. Used in conjunction with continuous chromatography and flow chemical techniques, this method can be applied to the study of the sequence of s-RNA.

The case of a function of time is considered which is periodic with known period. A classical analysis of variance, based on a pattern involving d observations in each of the r.c cells formed by crossing r rows with c columns is studied. Periodic time functions with a fixed period and a stationary joint distribution are also considered.

An investigation of the numerical variational analysis method is made for a case of “weak constraint” where the subsidiary condition is given in the form of an approximation. The simple example of a system moving with an optimized velocity is used to illustrate the theoretical development. The method is applied to analysis of the National Severe Storms Laboratory mesonetwork data of the severe storm gust that passed over the network on May 31, 1969.

It is clear that a p-variate body of data arranged in a q-way classification will formally look llke a (p+q)-dimensional contingency table, but a distinction can be made between a 'variate' and a ‘way of classification’ in that along the direction of a 'variate' the marginal frequencies are supposed to be stochastic variates while along a 'way of classification' the marginal frequencies are supposed to be fixed. When, along certain directions, the marginals are fixed, an approach based on a conditional probability argument has been used. In the present paper (i) the conditional probability approach is abandoned and we start either from a single multinomial distribution or a product of an appropriate number of different multinomial distributions according as, with multi-way frequency data, all ways are 'variates' or some are 'variates' and some are 'ways of classification'. (ii) Also the hypotheses that are posed are of different kinds altogether according as we have a 'multivariate analysis' situation or an 'analysis of variance' situation. The hypotheses that are meaningful for one situation would not be too meaningful for the other and vice versa. Since the conditional probability approach is altogether abandoned, the mathematical theorems to which appeal is made are the two theorems as stated and proved by Cramer (1946, chapter 30) and a number of other such theorems which have been proved the same way and which, between them, take care of all the hypotheses discussed in this paper. When all ways are 'variates' the hypotheses are analogous to those in the usual multivariate analysis, and when some ways are 'variates' and some are 'ways of classification' the hypotheses are analogous to those in the analysis of variance. The general methods discussed in this paper arose out of an attempt to analyse a large mass of categorical data. The analysis has been carried out, and a few typical cases (together with the numerical analysis) illustrating different parts of the theoretical development will be presented in a subsequent paper. In this paper only the large-sample tests are considered. How 'large' the sample size has to be for the validity of the use of these asymptotic techniques, or in other words, some results on the nature of the approximation involved will also be discussed in a later paper.

A new computer approach to handling large numbers of variables in biological systems is introduced. The method, called “Direct Analysis” or “Finite Element Analysis”, reduces the need for closed form solutions, or deriving solutions based on differential or integral equations. Rather, fundamental laws are allowed to interact via computer program in incremental form and can be merged by combining programs to form, in the ultimate, a “bio” system. An overview of the “Method” and “Concepts” underlying this approach is followed by a listing and description of its applications to physical, chemical and biological properties. Its significance and contribution to evolving computer techniques in the life sciences is evaluated.

Factor analysis has been proposed and used as a method of statistical analysis of several measurements made on one individual repeatedly over time. This paper discusses some difficulties in applying factor analysis to multiple time series and attempts to indicate to what extent such methods can accomplish the goals of time series analysis. Some other methods are suggested.

S-UMMARY Models and methods for analyzing the relations among a set of polytomous variables, when some of the variables are posterior to others, are presented. The techniques proposed here yield 'path diagrams' that are somewhat analogous to those used in path analysis. Earlier path analysis models are not suited to the case where the variables are polytomous, or dichotomous, but the models proposed herein are. For each of our models, methods are presented for (a) testing whether the model fits the data, (b) partitioning the test statistic into components that can be used to test submodels within the overall model, and (c) estimating the parameters in the model. An illustrative application is also included.

SUMMARY This paper describes and discusses graphical techniques, based on the primitive empirical cumulative distribution function and on quantile (Q-Q) plots, percent (P-P) plots and hybrids of these, which are useful in assessing a one-dimensional sample, either from original data or resulting from analysis. Areas of application include: the comparison of samples; the comparison of distributions; the presentation of results on sensitivities of statistical methods; the analysis of collections of contrasts and of collections of sample variances; the assessment of multivariate contrasts;_ and the structuring of analysis of variance mean squares. Many of the objectives and techniques are illustrated by examples. This paper reviews a variety of old and new statistical techniques based on the cumulative distribution function and its ramifications. Included in the coverage are applications, for various situations and purposes, of quantile probability plots (Q-Q plots), percentage probability plots (P-P plots) and extensions and hybrids of these. The general viewpoint is that of analysis of data by statistical methods that are suggestive and constructive rather than formal procedures to be applied in the light of a tightly specified mathematical model. The technological background is taken to be current capacities in data collection and highspeed computing systems, including graphical display facilities. It is very often useful in statistical data analysis to examine and to present a body of data as though it may have originated as a one-dimensional sample, i.e. data which one wishes to treat for purposes of analysis, as an unstructured array. Sometimes this is applicable to ' original' data; even more often such a viewpoint is useful with 'derived' data, e.g. residuals from a model fitted to the data. The empirical cumulative distribution function and probability plotting methods have a key role in the statistical treatment of one-dimensional samples, being of relevance for summarization and palatable description as well as for exposure and inference.

A method of analysis of nonhomogeneous elastic solids involving general three-dimensional states of stress is presented. The displacement equations of equilibrium based on the finite-element variational procedure are derived in the usual manner. The element shape considered is a tetrahedron with linear displacement approximations. p ]The main feature of this paper is the method of solution of the equilibrium equations. This method, called the alternating component iterative method, belongs to the class of block iterative schemes and is particularly suited for problems involving several dependent variables such as the ones encountered in elasticity. The method is presented in a form sufficiently general to permit the utilization of higher-order displacement approximations in the tetrahedral elements. p ]The alternating component iterative method is discussed in detail with respect to the following aspects: the general procedure, convergence criterion, refinement of the solution, and convergence acceleration procedure.

On the basis of some fundamental work due to Barnard (1952) and Cox (1952), Johnson (1953) has derived a procedure for applying sequential tests to the general linear hypothesis. Hoel (1955) obtains a similar test by a rather different method. The general linear hypothesis underlies a number of common analysis of variance situations. It is the purpose of this paper to provide tables which will make it possible to carry out the suggested procedures in the cases of (a) one-way classification by groups, and (b) randomized blocks. In the course of construction of these tables, we have considered a number of approximations to the confluent hypergeometric function and assessed their accuracy. Conjectural approximations to the expected sample sizes of the sequential processes considered are also given.

The stability behavior of a synchronous machine, connected to an infinite bus, is considered. Direct and quadrature axis damper-windings are taken into account and the input torque is said to be zero. The behavior of this machine is characterized by a system of five first-order nonlinear ordinary differential equations containing periodic nonlinear terms. These equations are shown to have two different types of equilibrium points. By means of linearized analysis it is shown that all points of the first type are (locally) asymptotically stable, whereas all points of the second type are (locally) unstable. Stability behavior of the system is analyzed for all initial conditions. This analysis is performed by a combination of various techniques: the second method of Liapunov, the topological methods of Wa[zdot]ewski, extension theory and results of the topological properties of the regions of attraction. These techniques, combined, allow a complete characterization of the stability behavior of the machine. Both global results (the set of all equilibrium points is a global attractor), as well as local results (the trajectory leaving one unstable equilibrium point tends to an asymptotically stable equilibrium point), are proved.

The factor analysis model and Lazarsfeld's latent structure scheme for analyzing dichotomous attributes are derived to show how the latter model avoids three knotty problems in factor analysis: communality estimation, rotation, and curvilinearity. Then the latent structure model is generalized into latent profile analysis for the study of interrelations among quantitative measures. Four latent profile examples are presented and discussed in terms of their limitations and the problems of latent metric and dimensionality thereby raised. The possibility of treating higher order empirical relations in a manner paralleling their various uses in the latent structure model is indicated.

A set of rules is given for finding average variances and covariances from mean squares, and the reverse. These rules serve the same purpose as the recursion formulas provided by Stanley and Wiley (1962), but do not require a knowledge of matrix algebra. A four-factor design is used to illustrate their application.

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.

This program for three-factor analysis of variance with repeated measures on two factors is based on an unweighted means analysis, permitting the use of unequal sample sizes. Each factor may be fixed or random since the program will select an appropriate error term or will calculate a quasi- F ratio. The number of possible levels of each factor is extended over those in other programs. Rounding errors are minimized.

The objective of the present work has been to demonstrate by analyses of experimental data that a postulated quadratic functional polynomial model is a reasonable representation of the added ship resistance produced by waves. The evidence so far produced is fairly strong that this is so, and that the model shows considerable promise as a unifying concept for the analysis and interpretation of added ship resistance in random waves. In the course of the work cross-bispectral analysis methods were developed and applied to the derivation of or "identification" of the "added resistance operator" from data obtained in irregular waves. These methods appear to have further promise as a research tool, but, at their present stage of development, do not appear to be an efficient testing tool.

A method for systematic analysis and modelling of interacting species is developed. An iterative algorithm for solving the inverse problem in ecological modelling (the problem of using empirical population data to evaluate the parameters of a given model) is first presented. It is then shown that certain results from the theory of statistical hypothesis testing provide a method by which the many possible interspecies interactions in an ecosystem can be examined and the dominant interactions can be identified. This analysis leads to the development of mathematical models which are optimum for a given set of empirical data. An example analysis of a four-species webbed ecosystem is discussed.

A case study is presented that demonstrates the usefulness of graphical techniques in the analysis of data. Crossplotting, probability plotting, and graphical multiple comparisons procedures are discussed. It is shown how graphical displays both motivate the use of and help interpret the results of more usual numerical techniques. The entire discussion is czntered around the analysis of a particular set of experimental data involving a comparison of treatments. It is shown step by step how the various graphical techniques, used in conjunction with more classical techniques, extract the information contained in the data.

In many important physical systems involving both diffusion and nonlinearity it often occurs that initially diffusion is the dominant mechanism. The question then arises as to whether or not linearization provides a uniformly valid first approximation for large times. A detailed examination of several model equations, both deterministic and stochastic, reveals that often the nonlinearity has a cumulative effect on a long time scale which must eventually be included in the first approximation. In particular, it is proved that linearization is not uniformly valid for Burgers’ model of turbulence. A major part of the analysis involves constructing asymptotic expansions for an interesting class of multidimensional integrals.

A polynomial shape function of triangular elements for finite element analysis of general deformation of sheet metals with or without orthotropic anisotropy is proposed. And general formulation for large deformation and large strain analysis with non-linearity of material property, say plasticity, is given. Some numerical examples are examined with respect to hydraulic bulging.

It is known that in the random effect model, exact F-tests are not obtainable from the analysis of variance table for testing the main effects in a three-way layout. Similarly exact F-tests are not obtainable for the main effects and certain interactions in higher-way layouts. In this paper level-α: tests obtainable from the analysis of variance table are given for testing these effects. Tables are provided to illustrate the size function and the power function of the tests.

An error analysis is presented for Gaussian elimination when the matrix is arbitrarily sparse. Error nalyses for elimination on band matrices and full matrices follow as special cases.

The size distribution data obtained by screen analysis of a non-uniform substance cannot be used directly in calculating the various average diameters of the material since this method of analysis gives the distribution in terms of weight rather than count. The distribution curve by weight however bears a definite relation to the regular size-frequency curve and suitable transformation equations are presented by means of which one is able to calculate the various average diameters from the parameters of the curve given by the screen analysis. The screens are calibrated in terms of the actual size of the material retained rather than the dimensions of the screen openings or an arbitrary method of calculating the “size of separation.”

In econometric literature a parameter in a theoretical model has been called identifiable if it can be uniquely determined in terms of the joint probability distribution of the observed variables. In this paper the identifiability of parameters in four different factor analysis models is considered. The last of these four models corresponds to Thurstone's factor analysis. In Sections 7 and 11, the possibility of a statistical testing of the models is discussed. Section 10 deals with the problem of actually determining the parameterr (the number of common factors) in terms of the probability distribution of the observed variables.

A numerical method for the dynamic analysis of structural frameworks has been discussed. A new procedure of numerical integration has been presented and its advantage over existing procedures has been shown. The method is convenient for use with a digital computer and can also be used for solving simple problems with a desk machine. The dynamic analysis has also been extended to include elastic non-linear behaviour of frameworks due to the effects of axial forces and change of geometry. The possibility of using the dynamic analysis for obtaining static solutions of structural frameworks by arbitrarily damping the vibrating system has been investigated. A basis has been presented for the choice of suitable coefficients of damping and time interval of numerical integration in order to ensure rapid convergence to the static solutions. Static solutions of a non-linear problem obtained by this method have been verified with the solutions obtained by static analysis.

Summary The simple one‐dimensional example of stress analysis in a beam can be used to illustrate the sophisticated methods involved in the various types of finite element stress analysis— displacement, hybrid displacement, stress and hybrid stress formulations and Reissner's principle. We can also demonstrate the bounds on the energy produced by the displacement and stress formulations and the use of Synge's hypercircle method.

An error analysis of projection methods for solving linear integral equations of the second kind is presented. The relationships between several direct methods for solving integral equations are examined. It is shown that the error analysis given is applicable to other methods, including a modified Nyström method and certain degenerate kernel methods.

The theory of beams supported by elastic foundations has been shown to be particularly well-suited to the analysis of grillage beams [1, 2, 3[.2 This theory leads to a far more general formulation regarding the types of structures amenable to analysis than that offered by the orthotropic plate theory. As compared to the methods of finite element theory, the theory of beams supported by elastic foundations decreases the computer time by an order of magnitude, thus making the analysis particularly valuable as a design tool. This paper shows that the measurement of the grillage beam analysis based upon the elastic foundation concept can be further simplified by certain matrix transformations that uncouple the deflection equations. The problem has, in fact, been simplified to such an extent that many large grillages can be analyzed with the aid of a desk calculator. Whenever comparison with results obtained from finite element methods has been possible, it has been found that, for all practical purposes, complete correlation exists.

This paper offers a new methodology for analyzing individual differences in preference judgments with regard to a set of stimuli prespecified in a multidimensional attribute space. The individual is modelled as possessing an “ideal point” denoting his most preferred stimulus location in this space and a set of weights which reveal the relative saliences of the attributes. He prefers those stimuli which are “closer” to his ideal point (in terms of a weighted Euclidean distance measure). A linear programming model is proposed for “external analysis”i.e., estimation of the coordinates of his ideal point and the weights (involved in the Euclidean distance measure) by analyzing his paired comparison preference judgments on a set of stimuli, prespecified by their coordinate locations in the multidimensional space. A measure of “poorness of fit” is developed and the linear programming model minimizes this measure overall possible solutions. The approach is fully nonmetric, extremely flexible, and uses paired comparison judgments directly. The weights can either be constrained nonnegative or left unconstrained. Generalizations of the model to consider ordinal or interval preference data and to allow an orthogonal transformation of the attribute space are discussed. The methodology is extended to perform “internal analysis,”i.e., to determine the stimuli locations in addition to weights and ideal points by analyzing the preference judgments of all subjects simultaneously. Computational results show that the methodology for external analysis is “unbiased”—i.e., on an average it recovers the “true” ideal point and weights. These studies also indicate that the technique performs satisfactorily even when about 20 percent of the paired comparison judgments are incorrectly specified.

There exist many experimental situations where one has a set of observation vectors from a number of different experimental groups and the investigator is interested in determining what differences may be present among the groups. A univariate statistical analysis on each observed random variable ignores the dependence existing between variables which may well be a source of differentiation. Canonical analysis is one of the statistical techniques available for the analysis of muitivariate data. In this article results from a simulation study of canonical analysis are given in order to display this muitivariate technique and see how it is able to differentiate among muitivariate populations under various distributional assumptions. In addition, an example is presented in which canonical analysis is applied to the investigation of a set of data arising from research in the ecological sciences studying the effect of water temperature ana recovery time on the blood sera of fish.

The spectral analysis of stationary point processes in one dimension is developed in some detail as a statistical method of analysis. The asymptotic sampling theory previously established by the author for a class of doubly stochastic Poisson processes is shown to apply also for a class of clustering processes, the spectra of which are contrasted with those of renewal processes. The analysis is given for two illustrative examples, one an artificial Poisson process, the other of some traffic data. In addition to testing the fit of a clustering model to the latter example, the analysis of these two examples is used where possible to check the validity of the sampling theory.

1. (i) The circuit model proposed in the preceding paper is revised and the application of Boolean algebra to the analysis of reactions in vivo is extended. The behaviour of mutant i s of E. coli is considered. 2. (ii) A relay circuit representing enzyme induction in the heterogenote, i s o + z + Fi + o c z + of E. coli is considered. 3. (iii) The behaviour of a constitutive mutant of E. coli is considered. According to Jacob & Monod (1961) , enzyme formation in mutant o c is partly repressed to a quarter of the normal value when induced. For an explanation of this fact, substance X is invoked. This may be a species of RNA and we call it μRNA. 4. (iv) Analogue or continuous behaviour of the organism which is controlled digitally is considered. For this purpose, a combination of an analogue computer and a relay circuit is tried (Plate I). Using this combined system: 4.1. (a) enzyme formation in o c is simulated; 4.2. (b) an explanation of the experiments of Ono, Iwama & Kameyama (1962) and Ono, Oda & Kameyama (pers. comm.) on the RNA synthesized in a cell-free system is attempted; 4.3. (c) the feedback repression of tryptophan synthetase formation is simulated; and 4.4. (d) the concept of Matsushiro et al. (1962) based on their experiments using strain φ80 Anth del of phage is formulated mathematically. 5. (v) A mathematical theory of such a combined system is given. Let the quantity of matter in a compartment i be m i , and the fluxes be J ji and J ik , then we have the differential equation (5.1). Mathematically speaking J ji or J ik is a function of m i S, and f ji or f ik of (5.2) may be the continuous or quasi-analogue part of this function. The characteristics of this function, which is sometimes strongly non-linear, is represented approximately by R ji or R ik taking the value 1 or 0. If such non-linearity is too strong, then the quasi-analogue analysis should have a limited applicability, even though a digital computer or digital differential analyzer is used, because the methods of numerical analysis for solving differential equations cannot be used.

A distinction is drawn between the method of principal components developed by Hotelling and the common factor analysis discussed in psychological literature both from the point of view of stochastic models involved and problems of statistical inference. The appropriate statistical techniques are briefly reviewed in the first case and detailed in the second. A new method of analysis called the canonical factor analysis, explaining the correlations between rather than the variances of the measurements, is developed. This analysis furnishes one out of a number of possible solutions to the maximum likelihood equations of Lawley. It admits an iterative procedure for estimating the factor loadings and also for constructing the likelihood criterion useful in testing a specified hypothesis on the number of factors and in determining a lower confidence limit to the number of factors.

The experimental design considered in this paper is one in which each of a group of several subjects is observed in the presence of each of the other subjects of the group, the entire set of possible pairings being repeated or replicated on several occasions. Analysis of variance formulas are described for this somewhat unusual design. Both the model of constants and mixed model are considered. Reliability formulas growing out of the analysis of variance calculations are developed.

The effect of inequality of variance on the standard F-tests for one-way and two-way analysis of variance classification has been examined previously. Certain results found by Box for the one-way classification are examined. A Bayesian analysis is shown to throw light on the consequences of Box's work in a given experimental situation.