Dynamics from Multivariable Longitudinal Data

We introduce a method of analysing longitudinal data in variables and a population of observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space . At each time, information of each observation is coded to a point , where is the physical condition of the observation and is an ordering of variables. Orbit of each observation in is described by a map that dynamically rearranges order of variables at each time step, eventually placing the most stable, least frequently changing variable to the left and the most frequently changing variable to the right. By this operation, we are able to extract dynamics from data and visualise the orbit of each observation. In addition, clustering of data in the stable variables is revealed. All possible paths that any observation can take in are given by a subshift of finite type (SFT). We discuss mathematical properties of the transition matrix associated to this SFT. Dynamics of the population is a nonautonomous multivalued map equivalent to a nonstationary SFT. We illustrate the method using a longitudinal data of a population of households from Agincourt, South Africa. 1. Introduction Analysis of multivariable longitudinal data involves either statistical or nonstatistical methods. Statistical methods include multivariate Markov chain model [1], regression model [2], and mixed models [3], while some nonstatistical methods involve extraction of dynamical system using state space reconstruction technique [4] or visual methods such as motion charts [5, 6] and parallel coordinate plots [7, 8]. A motion chart shows additional dimensions of the data at different time points, where the size and color of the bubble (among others) are used as variables. PCP represents variables as parallel axes, where a sequence of line segments intersects each axis at a point corresponding to the observation’s value at the associated variable. Both methods aim to identify correlation among variables and identification of clusters and patterns among observations in the data. We present a novel nonstatistical method of analysis that is useful particularly for collecting information of change. State space reconstruction method mainly requires use of delay coordinates from data to come up with models for prediction. Our method does not rely on delay coordinates, and our aim is not prediction. Contrasting to motion charts, there is, in principle, no limitation to the number of variables studied in our method. Contrasting to PCP, for large and large number of observations, orbits over our