581 – Advancement in Theoretical and Applied Aspects of Modeling
Time Series for Boolean Random Sets
Khalil H. Shafie
University of Northern Colorado
Kofi K. Wagya
University of Northern Colorado
M. Khazaei and K. Shafie in 2006 introduced regression models for Boolean random sets, to be able to model the effect of explanatory variables on the distribution of a Boolean random set. However, the measurement of random sets representation of objects are mostly taken over time, which introduces correlation in the observations. One solution to deal with the correlated observations is to fit a time series to the random sets of the Boolean-model type, which is the goal of this paper. Two methods are introduced and maximum likelihood estimation of the parameters are applied to the log link function using the two methods of estimation i.e. n_{t}, the number of points and n^{+}_{t}, the number of lower tangent points in window W_t. A simulation study is used to analyze the behavior of this time series. The distribution of the estimates approaches approximate normality, confirming the asymptotic behavior of maximum likelihood estimators. When the \Beta_{1} and \alpha_{1 }have opposite signs, multivariate normality is achieved faster (at T=1000) than when the signs are the same (at T=2500). The model is then applied to the Mountain Pine Beetle Data for a ten-year period (2001 to 2010). Both methods produced similar results, however, method I has lower standard errors.