between $\lambda$ and one-hour prediction error using the simulation occupancy data, with $\lambda=0.974$ giving the best prediction accuracy. Of course, there is no guarantee that the best past value of $\lambda$ will work well in the future; nonetheless, the convexity suggests that $\lambda$ could be calibrated on-line with an extremum-seeking algorithm [@ariyur2003real].

![\[fig:Influence-of-forgetting\]Influence of forgetting factor $\lambda$ on one-hour root-mean-square prediction error. These results were obtained by training the model incrementally over the MERL Belady conference room occupancy data from February 12 to April 10, 2007 and simultaneously comparing each observation to the prediction made in the previous hour.](predErrorVsFF)

Performance Comparison
----------------------

Figure \[fig:Simulation-results\] shows simulation results for three identically-tuned MPC implementations:

1.  a purely occupancy-triggered controller,

2.  a scheduled controller supplemented with occupancy triggering, and

3.  an on-line trained occupancy-predicting controller with one week of pre-training ($\lambda=97.4\%$).

The occupancy-triggered controller (green) maintains $\tau=23^{\circ}\textrm{C}$ during occupied hours and $10^{\circ}\textrm{C}$ during vacant hours. The scheduled controller uses the same setpoint from to and any time the space is occupied. To simplify the simulation, all three controllers ignore occupancy and control to $10^{\circ}\textrm{\textrm{C}}$ ($50^{\circ}\textrm{F}$) over weekends.

### Energy and Comfort

The occupancy-triggered controller consumes by far the least energy because it does 