\[sub:Choosing\] will explore the relationship between $\lambda$ and prediction accuracy.

![\[fig:forgettingFactor\]A particular occupancy transition probability distribution when trained with alternating data using no forgetting ($\lambda=1$) (a) and with considerable forgetting ($\lambda=0.85$) (b). The forgetting factor broadens the distribution and allows it to shift laterally even when extensively trained, reflecting greater ability to adjust to changes in space usage. For each case, the initial distribution is uniform (black horizontal trace). Higher levels of training are shown as brighter color.](trainingDist)

### Using Fractional Occupancy

Measuring the percentage of occupancy over each time makes occupancy predictions more precise. To convert the asynchronous pulses from PIR sensors (Figure \[fig:discretization\]a) into a discrete-time sequence of fractional values, we apply a simple two-step heuristic. First, we merge closely-spaced pulses using a minimum dwell time to get a square wave signal $\gamma(t)$ (Figure \[fig:discretization\]b). Then we superimpose a fixed time grid over the signal and average it over each step to obtain the discrete sequence $$\Gamma_{k}=\frac{1}{t_{k}-t_{k-1}}\int_{t_{k-1}}^{t_{k}}\gamma(t)\,\textrm{d}t\equiv{\mathbb{P}\left(\gamma_{k}=1\right)}\in[0,1],\label{eq:bigGamma}$$ essentially treating $\Gamma_{k}$ (Figure \[fig:discretization\]c) as the duty cycle sequence of the pulse width modulated signal $\gamma(t)$. We subsequently pretend that $\gamma(t)$ is sampled probabilistically through $\Gamma_{k}$ with a distribution over the Markov state space $\pi_{k}\in\mathbb{R}^{1\times2M}$. The statements $\Gamma_{k}=60\%$ and ${\mathbb{P}\left(\gamma_{k}=1\right)}=0.6$ 