 Latent Differential Equations (LDE) Differential equations models of time series data can be estimated from noisy or multivariate data by using a confirmatory factor model with fully constrained loadings and regression coefficients between the estimated factors. When the loading constraints are properly specified, the factors scores are estimates of the derivatives of the observed time series and the covariances between the factors allow estimation of regression coefficients in a differential equations model. This approach was first presented at the 2002 Notre Dame Series on Quantitative Methodology, handouts of which can be downloaded here. A more complete description of the method will be In Recent Developments on Structural Equation Models: Theory and Applications,
K. van Montfort, H. Oud, & A. Satorra (eds), Amsterdam: Kluwer. 151174. A preprint of the chapter is here. Demonstration scripts using Mathematica, R, and Mx that accomplish an LDE analysis with a simulated data set can be found here
 Generalized Local Linear Approximation (GLLA) is a filtering method for estimating derivatives from data that uses time delay embedding and a variant of SavitzkyGolay filtering to accomplish the task. A chapter describing the method can be found in Boker, S. M., Deboeck, P. R., Edler, C., & Keel, P. K. (2009) Generalized Local Linear Approximation of Derivatives from Time Series. In Statistical Methods for Modeling Human Dynamics: An Interdisciplinary Dialogue, S.M. Chow & E. Ferrar (Eds). Boca Raton, FL: Taylor & Francis. Two R functions that accomplish the task are contained in GLLAfunctions.R and an example script that uses them is ExampleGLLA.R
 Differential Structural Equation Modeling of Intraindividual Variability Recent advances in modeling and analysis have enabled statistical tests of dynamical systems based theories for behavioral phenomena. The use of statistical models to test proposed dynamical or selfregulating explanations of intraindividual variability has the potential to lead to new and more tightly focused understandings of cognitive, developmental and social systems. This chapter provides an introduction to the use of differential structural equation models (dSEM) to fit data from repeated observations of individuals: intraindividual variability. Example data from an experiment in the development of postural control are analyzed in order to give a practical demonstration of these methods. Models are fit to individual trials in a postural control experiment and the resulting estimated parameters are analyzed using a random coefficients approach to elucidate development in the dynamics of intraindividual variability. A pdf version of this chapter can be downloaded here.
These same methods can be extended to treat the dynamic parameters as random coefficients so that each individual might have his or her own attractor shape while still preserving the model across the population. An example of the use of this method appeared in the Multilevel Modelling Newsletter Volume 13 Number 1.
Another extension of differential equations modeling involves assuming equal dynamics between individuals. Given this assumption, one may estimate the parameters of a linear oscillator model with as few as three occasions of measurement per individual. A simulation demonstrating this method is presented in a 2002 article in Multivariate Behavioral Research 37:1.
 Linear and Nonlinear Dynamical Systems Data Analytic Techniques and their Application to Physiological and Developmental Data proposes a practical methodology for the analysis of longitudinal data that has been postulated to have an underlying dynamical process. A novel analysis is proposed which builds models based on a mutual information matrix in place of a covariance matrix. Several nonlinear analyses are applied to data from an experiment studying the development of visually guided postural control in infants. A pdf version of this Ph.D. dissertation can be downloaded here.
 Selfregulation of Mental Health in Recent Widows A project is underway in collaboration with Toni Bisconti from the University of New Hampshire and Cindy Bergeman from Notre Dame studying the selfregulation of an omnibus selfreport measure of mental health. The first data from this project have proved to be consistent with a dynamical systems interpretation of selfregulation and a multilevel differential equations model has provided evidence that particular types of social support are related to an estimated coefficient of resiliency in the model. An article is in the Journal of Gerontology and the scripts used in the analysis can be found here.
 Selfregulation of Intimacy in Married Couples. A coupled dynamical systems analysis was performed on six weeks of daily diary data from 95 married couples in collaboration with JeanPhilippe Laurenceau from the University of Miami. Evidence of cyclicity was found as well as evidence that spousal level of intimacy had a significant phase synchronizing effect on intimacy in both wives and husbands. A chapter is in Models for Intensive Longitudinal Data, T. A. Walls & J. L. Schafer (Eds). Oxford University Press.
 Local Time Structure of the Coordination of Posture and Gesture during Dyadic Conversation When people converse, they use head motions, postural adjustments and gestures that convey information. It has been hypothesized that individuals coordinate their movements while they converse in a sort of "conversational dance". Much more about this project, as well as articles and demo movies for download can be found at the Human Dynamics Laboratory website
 Interlimb Synchronization in Unstructured Dance A project is currently underway in Dr. Boker's lab using multiple motion tracking sensors to measure human movement during improvisational dance. Participants are asked to dance to stimuli from the preceding two rhythm perception experiments. The hypothesis is that individuals will segment their limb movements in time with the rhythm such that times of high limb acceleration will correspond to beats that were highly likely to be chosen as segmentation points in the two experiments above. A brief introduction to the issues of symmetry formation and symmetry breaking during dyadic dance is presented in this chapter from the 2003 book Mirror Neurons and the Evolution of Brain and Language, M. Stamenov & V. Gallese (eds).
 Dr. Boker is lead PI on The OpenMx Project. Funded by the NIH Roadmap, OpenMx is a software development project for an open source Structural Equation Modeling (SEM) package that will be free of charge and tied into the R statistical system. The OpenMx project involves collaborators at the University of Virginia, Medical College of Virginia, University of Chicago, University of Houston, McMaster University and University of Edinburgh. The project is scheduled for open beta release in the fall of 2009. Statistical researchers, quantitative psychologists, and software developers who may be thinking about writing extensions to SEM are encouraged to visit the OpenMx Wiki where they can learn about the project and possibly either contribute to the project or find software and open source code that may help them in their own work.
 Mx Software. Mx has been replaced by OpenMx, but is listed here for historical reasons. Mx is a sophisticated statistical modeling package written by Mike Neale at Virginia Commonwealth University. Dr. Boker participated in the design and development of the MXWIN project to create a graphical user interface for Mx running under MS Windows. The MXWIN software allows the user to specify a statistical model by drawing its equivalent path diagram and then estimate and graphically display the parameters and goodness of fit of that model given a particular dataset. MXWIN is free and can be obtained here.
 The RAMPath Algorithm is a method for automatically calculating the components of covariance in a structural model as well as a method for organizing path diagrams for consistent display. This algorithm has been used in the RAMPath software (McArdle & Boker, 1990) as well as in the Mx structural modeling software (Neale et al, 1998). An introduction to the algorithm is published (2002) in Structural Equation Modeling 9:2 and an open source S language implementation of the algorithm can be found here.
 Windowed Cross Correlation and Peak Picking are methods for estimating the strengths and phase lags of time varying linear effects between two physiological data vectors. The article is published in Psychological Methods (2002) 7:1, and so is unavailable for download at this time. However, the source code for the windowed cross correlation and peak picking are in an archive file that can be downloaded here. These source files have been tested under Apple OSX and Linux, but are generic and should be able to be compiled under Windows with a minimum of effort.
 Nonlinear Dependency is a method for calculating mutual information between all pairs of variables in a multivariate time series. C language source code for the nldcalc algorithm can be downloaded here and an OS X binary can be downloaded here. These source files have been tested under Apple OSX and Linux, but are generic and should be able to be compiled under Windows with a minimum of effort.
 Statistical Vector Fields are a way of displaying and summarizing mixed longitudinal and crosssectional data into a graphic display. The method is locally parametric and thus is robust with respect to outliers and mixed distributions and is described in a 1995 article in Experimental Aging Research, 2:1. The Statistical Vector Fields software for DOS and source code that can be compiled under Unix can be downloaded here.
 When I draw path models I use a commercial software package called OmniGraffle. It is very easy to use, but just runs on Mac OS X. I have created a free path modeling stencil for OmniGraffle that may be useful to you. Copy it into 'Library/Application Support/OmniGraffle/Stencils' and relaunch OmniGraffle and away you go.
 AgeBased Comparisons of Nonlinear Dependency in Postural Control
Recent advances in data analytic techniques for the study of nonlinear dynamical systems have allowed the estimation of fractal dimension, a measure analogous to the complexity of a time series, and nonlinear dependency, a measure of predictability similar to covariance in linear systems. A third technique, surrogate data analysis, can be used to estimate the proportion of the nonlinear dependency that can be accounted for by linear statistical methods. This report illustrates the use of these three nonlinear techniques applied to data from the study of agebased changes in postural control. The current work examines nonlinear dependency between head, trunk and hip movement during a postural control task performed by young, middleaged and old adult subjects. Visual and auditory inputs to the postural control system were manipulated and postural sway measured by motion tracking equipment. The results of this preliminary experiment suggest that visual coupling to the environment is much stronger than auditory coupling. The nonlinear component of postural control does not show an agebased difference over the lifespan while the fractal dimension does show an increase. In addition, there appears to be a change in intersegmental sequencing such that the hips lead the head in middleaged subjects while the head leads the hips in older subjects. A postscript version of a report based on the talk at the 1998 Gerontological Association annual meeting can be downloaded here.
 Dynamical Analyses of Postural Development is a talk which was presented to the June 1995 annual meeting of The North American Society for the Psychology of Sport and Physical Activity in Monterey California.
 Nonlinear Analysis of PerceptualMotor Coupling in the Development of Postural Control presents evidence for nonlinearity in Center of Pressure (COP) measurements of sitting infants. This chapter appears in Nonlinear Techniques in Physiological Time Series Analysis, MeyerKress, G., Kantz, H. \& Kurths, J. (Eds.), Berlin: SpringerVerlag. A postscript version of this chapter can be downloaded by clicking here.
 A Psychotelemetry Experiment in Fluid Intelligence describes the results of an experiment in cognitive abilities that was distributed on the internet. Individual volunteers were tested by interacting with a computer program that was sent to participants via an email enclosure. Results were automatically returned via email and compiled.
 Using the Internet and the World Wide Web for Archiving Psychological Data was presented at the Symposium on Data Archiving in Psychology as part of the American Psychological Association Annual Convention in Chicago on August 16, 1997. This report describes a proposed process of archiving data from psychological experiments and large scale surveys and making these data available over the world wide web for secondary analyses by other behavioral scientists.
