Rotation in the dynamic factor modeling of multivariate stationary time series

A special rotation procedure is proposed for the exploratory dynamic factor model for stationary multivariate time series. The rotation procedure applies separately to each univariate component series of aq-variate latent factor series and transforms such a component, initially represented as white noise, into a univariate moving-average. This is accomplished by minimizing a so-called state-space criterion that penalizes deviations of the rotated solution from a generalized state-space model with only instantaneous factor loadings. Alternative criteria are discussed in the closing section. The results of an empirical application are presented in some detail.This research was supported by the Institute for Developmental and Health Research Methodology, University of Virginia.     

Dynamic factor analysis of nonstationary multivariate time series

A dynamic factor model is proposed for the analysis of multivariate nonstationary time series in the time domain. The nonstationarity in the series is represented by a linear time dependent mean function. This mild form of nonstationarity is often relevant in analyzing socio-economic time series met in practice. Through the use of an extended version of Molenaar's stationary dynamic factor analysis method, the effect of nonstationarity on the latent factor series is incorporated in the dynamic nonstationary factor model (DNFM). It is shown that the estimation of the unknown parameters in this model can be easily carried out by reformulating the DNFM as a covariance structure model and adopting the ML algorithm proposed by Jöreskog. Furthermore, an empirical example is given to demonstrate the usefulness of the proposed DNFM and the analysis.     

A cognitive multivariate time series and its analysis

The method of analyzing a multivariate time series, by factoring a matrix polynomial of linear transfer function coefficients, is briefly outlined and shown to be potentially applicable to cognitive experiments where the inputs are numerically predictable and the outputs suitably quantified. An experiment on sequences of 90 figures, judged in terms of similarity and of the probability of exceeding or not exceeding some size, was analyzed individually for 12 subjects and their dynamic characteristics expressed in pole zero diagrams. Some dynamic components neglected by classical time-independent psychological methods are identified.     

Classical conditioning as a nonstationary,multivariate time series analysis: A spreadsheet model

The implementation of the Gallistel (1990) model of classical conditioning on a spreadsheet with matrix operations is described. The model estimates the Poisson rate of unconditioned stimulus (US) occurrence in the presence of each conditioned stimulus (CS). The computations embody three implicit principles:additivity (of the rates predicted by each CS),provisional stationarity (the rate predicted by a given CS has been constant over all the intervals when that CS was present), andpredictor minimization (when more than one solution is possible, the model minimizes the number of CSs with a nonzero effect on US rate). The Kolmogorov-Smirnov statistic is used to test for non-stationarity. There are no free parameters in the learning model itself and only two parameters in the formally specified decision process, which translates what has been leamed into conditioned responding. The model predicts a wide range of conditioning phenomena, notably: blocking, overshadowing, overprediction, predictive sufficiency, inhibitory conditioning, latent inhibition, the invariance in the rate of conditioning under scalar transformation of CS-US and US-US intervals, and the effects of partial reinforcement on acquisition and extinction.     

Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences

A class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed. In each of the models, the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix. The component scores series reveal the latent data structure in the course of time. The interpretation of the components is based on the loading matrix. The simultaneous component models model not only intraindividual variability, but interindividual variability as well. The four models can be ordered hierarchically from weakly to severely constrained, thus allowing for big to small interindividual differences in the model. The use of the models is illustrated by an empirical example.This research has been made possible by funding from the Netherlands Organization of Scientific Research (NWO) to the first author. The authors are obliged to Tom A.B. Snijders, Jos M.F. ten Berge and three anonymous reviewers for comments on an earlier version of this paper, and to Kim Shifren for providing us with her data set, which was collected at Syracuse University.     

The use of factor analysis in the statistical analysis of multiple time series

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.Research sponsored by contract AF41(657)-214 between the USAF School of Aerospace Medicine and Teachers College, Columbia University.     

A multivariate model for discrimination methods

We describe a multivariate model for a certain class of discrimination methods in this paper and discuss a multivariate Euclidean model for a particular method, the triangular method. The methods of interest involve the selection or grouping of stimuli drawn from two stimulus sets on the basis of attributes invoked by the subject. These methods are commonly used for estimation and hypothesis testing concerning possible differences between foods, beverages, odorants, tastants and visual stimuli.Mathematical formulation of the bivariate model for the triangular method is provided as well as extensive Monte Carlo results for up to 10-dimensional cases. The effect of correlation structure and variance inequality are discussed. Results from these methods (as probability of a correct response) are not monotonically related to the distance between the means of the stimulus sets from which the stimuli are drawn but depend in a particular way on dimensionality, correlation structure, and the relative orientation of the momentary sensory values in a multidimensional space. The importance of these results to the validity of these methods as currently employed is discussed and the possibility of developing a new approach to multidimensional scaling on the basis of this new theory is considered.     

Detecting correlation changes in multivariate time series: A comparison of four non-parametric change point detection methods

Change point detection in multivariate time series is a complex task since next to the mean, the correlation structure of the monitored variables may also alter when change occurs. DeCon was recently developed to detect such changes in mean and\or correlation by combining a moving windows approach and robust PCA. However, in the literature, several other methods have been proposed that employ other non-parametric tools: E-divisive, Multirank, and KCP. Since these methods use different statistical approaches, two issues need to be tackled. First, applied researchers may find it hard to appraise the differences between the methods. Second, a direct comparison of the relative performance of all these methods for capturing change points signaling correlation changes is still lacking. Therefore, we present the basic principles behind DeCon, E-divisive, Multirank, and KCP and the corresponding algorithms, to make them more accessible to readers. We further compared their performance through extensive simulations using the settings of Bulteel et al. (Biological Psychology, 98 (1), 29-42, 2014) implying changes in mean and in correlation structure and those of Matteson and James (Journal of the American Statistical Association, 109 (505), 334-345, 2014) implying different numbers of (noise) variables. KCP emerged as the best method in almost all settings. However, in case of more than two noise variables, only DeCon performed adequately in detecting correlation changes.     

Generalized multivariate lens model analysis for complex human inference tasks

A generalized multivariate lens model is presented which will permit the analysis of complex human inference tasks. Such tasks, occurring in their natural ecology, may involve judgments or decisions on multiple criteria and/ or where the influence of theoretically interesting partitions or augmentations of cue profiles needs to be systematically delineated. The generalized model incorporates the standard, hierarchical, and fully partialed lens models, the initial elaborations of which were made by N. J. Castellan (1972, Organizational Behavior and Human Performance, 8, 242–261) and T. R. Stewart (1976, Psychometrika, 41, 101–120). Both multivariate and univariate lens model equations are presented within a common notational system. An example which demonstrates some aspects of the generalized model is discussed in detail. Other potential applications for the model are outlined.     

Influence analysis for the factor analysis model with ranking data

Influence analysis is an important component of data analysis, and the local influence approach has been widely applied to many statistical models to identify influential observations and assess minor model perturbations since the pioneering work of Cook (1986) . The approach is often adopted to develop influence analysis procedures for factor analysis models with ranking data. However, as this well‐known approach is based on the observed data likelihood, which involves multidimensional integrals, directly applying it to develop influence analysis procedures for the factor analysis models with ranking data is difficult. To address this difficulty, a Monte Carlo expectation and maximization algorithm (MCEM) is used to obtain the maximum‐likelihood estimate of the model parameters, and measures for influence analysis on the basis of the conditional expectation of the complete data log likelihood at the E‐step of the MCEM algorithm are then obtained. Very little additional computation is needed to compute the influence measures, because it is possible to make use of the by‐products of the estimation procedure. Influence measures that are based on several typical perturbation schemes are discussed in detail, and the proposed method is illustrated with two real examples and an artificial example.     

Fitting the factor analysis model

When the covariance matrix (p×P) does not satisfy the formal factor analysis model for m factors, there will be no factor matrix (p×m) such that =(-) is diagonal. The factor analysis model may then be replaced by a tautology where is regarded as the covariance matrix of a set of residual variates. These residual variates are linear combinations of discarded common factors and unique factors and are correlated. Maximum likelihood, alpha and iterated principal factor analysis are compared in terms of the manner in which is defined, a maximum determinant derivation for alpha factor analysis being given. Weighted least squares solutions using residual variances and common variances as weights are derived for comparison with the maximum likelihood and alpha solutions. It is shown that the covariance matrix defined by maximum likelihood factor analysis is Gramian, provided that all diagonal elements are nonnegative. Other methods can define a which is nonGramian even when all diagonal elements are nonnegative.A modified version of this paper forms part of a Ph.D. thesis submitted to the University of South Africa.Presently at the National Institute for Personnel Research, South Africa.     

A time series analysis of the relation between motor skill acquisition and sleep in infancy

To systematically examine the relation between motor milestone onset and disruption of night sleep in infancy, three families kept microgenetic, prospective, daily checklist diaries of their infants’ motor behavior and sleep (197-313 observation days; 19,000 diary entries). Process control and interrupted time series analyses examined whether deviations from the moving average for night wakings and evening sleep duration were temporally linked to motor skill onset and tested for meaningful differences in individual sleep patterns before and after skill onset. Model assumptions defined skill onset as first day of occurrence or as mastery and moving average windows as 3, 7, or 14 days. Changes in infants’ sleep patterns were associated with changing expertise for motor milestones. The temporal relation varied depending on infant and sleep parameter. Intensive longitudinal data collection may increase our understanding of micro-events in infant development.     

A latent variable model for discrete multivariate psychometric waiting times

A version of the discrete proportional hazards model is developed for psychometrical applications. In such applications, a primary covariate that influences failure times is a latent variable representing a psychological construct. The Metropolis-Hastings algorithm is studied as a method for performing marginal likelihood inference on the item parameters. The model is illustrated with a real data example that relates the age at which teenagers first experience various substances to the latent ability to avoid the onset of such behaviors.We thank Michael Newton and Daode Huang for their helpful comments and suggestions.