limited to $0\leq k\leq M-1$ when dealing with the Markov chain. The choice of $M$ affects how learned patterns relate to subsequent predictions; if space usage patterns vary significantly across the weekdays, one might want to prevent occupancy observations on Monday from influencing control actions on Tuesday, in which case a one-week chain would be more appropriate. For this study, the one-day Markov chain is trained using Monday through Friday occupancy data from the MERL data set, ignoring weekends. In practice, one could switch to a different Markov chain or use occupancy-triggered control over weekends.

Training
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In contrast to batch training, which uses a fixed-size history (Figure \[fig:trainingSchematic\]a) , on-line incremental training proceeds without user intervention. It uses observations to update density functions for each of the transition probabilities; the expected values of these density functions in turn populate the Markov chain’s transition matrix (Figure \[fig:trainingSchematic\]b).

![\[fig:trainingSchematic\]Conventional batch training (a) versus the proposed on-line incremental Bayesian training algorithm with forgetting (b).](training)

Boolean occupancy lacks granularity that could otherwise make predictions more accurate. For example, occupancy for the entirety of the previous hour implies different future occupancy compared to just a few minutes. The question we wish to answer is: 