Department of Electrical and Computer Engineering. Air Force .... is persistence. For example if a support technician arrived at work between 07:00am and.
Introduction to Temporal Bayesian Networks y and Eugene Santos Jr. Department of Electrical and Computer Engineering Air Force Institute of Technology Wright-Patterson AFB, OH 45433-7765 jdyoung@a t.af.mil, esantos@a t.af.mil Joel D. Young
Presented At The Seventh Midwest AI and Cognitive Science Conference April 26-28, 1996 Abstract
A Bayesian network is a directed acyclic graph in which nodes are random variables and the edges indicate that the source exerts direct causal inuence on the destination. A problem with the Bayesian network is that there is no natural mechanism for representing temporal relations between and within the random variables. This paper introduces a new technique for representing when a random variable holds a particular state, e.g. when an event happens, as well as techniques for applying temporal constraints (precedes, during, etc.) to the edges. Allen's interval structure is used to provide the formal basis. A restricted model is presented along with a corresponding inferencing algorithm based on a linear constraint formulation.
1 Introduction Complex systems consist of collections of interacting processes. These processes change over time in response to both internal and external stimuli as well as to the passage of time itself. There is great variety in the behavior of processes. Some processes are simple events such as going to lunch or ipping a switch. Others are complex. One example being a communication channel, in which errors may occur due to lightning strikes and in which errors are more likely following previous errors. Processes can also be recurrent or periodic, such as the passing of day into night or shifts in a work schedule. Prior temporal modeling techniques have often had diculty modeling uncertainty as to when and if an event occurs. Techniques able to model uncertainty often do not have strong semantics. One such method, the Temporal Abduction Problem (TAP) of 4] uses a cost based approach to model the uncertainty in an events occurrence, however the costs are adhoc and TAP does not model the uncertainty as to when the event happened or for how long. y
This research was supported in part by AFOSR Project #940006.
1
Bayesian networks 2] provide a robust, probabilistic method of reasoning with uncertainty which has become popular with articial intelligence researchers. Bayesian networks, however, do not provide a direct mechanism for representing temporal dependencies. For example, it is dicult to represent a situation such as the variability of when an employee arrives at work and the causal relationships between the time of arrival and later events. This paper presents a new system, the Temporal Bayesian Network (TBN), for representing temporal and atemporal information while maintaining probabilistic semantics. The technique allows representation of time constrained causality, of when and if events occur, and of the periodic and recurrent nature of processes. Bayesian networks lie at the foundation of the system and provide the probabilistic basis. Allen's interval system 1] and his 13 relations provide the temporal basis.
2 Theoretical Structures In probabilistic reasoning, random variables (RVs) are used to represent events and objects. By making various assignments to these RVs, we can model the current state of the world and weight the states according to the joint probabilities. A Bayesian network is a directed acyclic graph. Directed arcs between RVs represent conditional dependencies. When all the parents of a given RV are instantiated, that RV is said to be conditionally independent of the remaining RVs given it's parents. Allen's interval algebra is governed by 13 relations on the intervals. Basically, there is a time interval in which each event occurs denoted by a b] where a is the starting time point and b is the termination point. Temporal relationships between events are expressed as relations between their intervals. The relations between intervals, denoted A, are f= < > m mi d di s si f fi o oig (gure 1). For example, event A = a b] preceding event B = c d] is denoted A < B indicating that a < b < c < d. The set of 13 relations is mutually exclusive and exhaustive. Uncertainty in the exact relationship between intervals is expressed as disjunctions. For example, \interval A precedes or meets interval B " is written as Af
