Dynamics from Multivariable Longitudinal Data

Abstract

We introduce a method of analysing longitudinal data in n≥1 variables and a population of K≥1 observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space Sn. At each time, information of each observation is coded to a point (x,y)∈Sn, where x is the physical condition of the observation and y is an ordering of variables. Orbit of each observation in Sn 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 Sn 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.