. 2017 Dec 27;32(3):787–829. doi: 10.1007/s10618-017-0548-4

Table 1.

Formulae to compute the transition probabilities of the Markov chain MD from abstract mobility trajectories

Transition, $s \to s^{'}$	Frequency, ${MD}_{s s^{'}}$
$(h, 1) \to (h + 1, 1)$	$\frac{\sum_{u \in U} \sum_{a \in A_{u}} δ_{h}^{u} (a) δ_{h + 1}^{u} (a)}{\sum_{u \in U} \sum_{a \in A_{u}} δ_{h}^{u} (a)}$
$(h, 1) \to (h + τ, 0)$	$\frac{\sum_{u \in U} \sum_{a \in A_{u}} δ_{h}^{u} (a) [1 - δ_{h + 1}^{u} (a)] \prod_{i = 1}^{τ - 1} {\hat{δ}}_{h + i}^{u} (a) [1 - {\hat{δ}}_{h + τ}^{u} (a)]}{\sum_{u \in U} \sum_{a \in A_{u}} δ_{h}^{u} (a)}$
$(h, 0) \to (h + 1, 1)$	$\frac{\sum_{u \in U} \sum_{a \in A_{u}} [1 - δ_{h}^{u} (a)] δ_{h + 1}^{u} (a)}{\sum_{u \in U} \sum_{a \in A_{u}} [1 - δ_{h}^{u} (a)]}$
$(h, 0) \to (h + τ, 0)$	$\frac{\sum_{d \in D} [1 - δ_{h}^{u} (a)] [1 - δ_{h + 1}^{u} (a)] [1 - {\hat{δ}}_{h}^{u} (a)] \prod_{i = 1}^{τ - 1} {\hat{δ}}_{h + i}^{u} (a) [1 - {\hat{δ}}_{h + τ}^{u} (a)]}{\sum_{u \in U} \sum_{a \in A_{u}} [1 - δ_{h}^{u} (a)]}$