Shafie, Khalil

Committee Member

Yu, Han

Committee Member

Merchant, William

Committee Member

Mostowfi, Mehrgan


College of Educational and Behavioral Sciences; Department of Applied Statistics and Research Methods


University of Northern Colorado

Type of Resources


Place of Publication

Greeley, (Colo.)


University of Northern Colorado

Date Created



139 pages

Digital Origin

Born digital


Although Random closed sets (RACS) are rich in modeling complex objects, its model parameter inference is simple. The most important type of the RACS is Boolean random set (BRS). BRS is a parametric model that is formed by placing random closed sets (grains) at points of a Poisson process (germs), and taking union of these sets. The radius of these grains may be fixed and known, fixed but unknown, and random. Furthermore, the intensity parameter l, which is one source of randomness in the BRS model, has been estimated by the method of Intensity, method of Minimum contrast, method of moments, and ordinary and generalized least squares regression for the independent BRS. A time series model was then developed for the intensity estimation of the correlated BRS using maximum-likelihood, and method of moments for the radius estimation. The model used past observations (n+ t or ˆ nt ), and past intensity to estimate current intensity. In addition, twelve parameter schemes were employed to study the properties of these parameter estimates, including biasness, consistency, and asymptotic normality. Simulation results showed that the parameter estimates inherited the properties of maximum likelihood estimators. Thus, the future intensity of phenomena that are correlated random sets (Boolean) in nature can be predicted, using the model built in this study. This model was built for the Rocky Mountain Pine Beetle data from 2001 to 2010, which consisted of seasonal attacks of trees by Pine Beetles in the Rocky Mountain region. This was then used to predict the intensity for the year 2010.

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