Defining a sequence of stopping times: begin{aligned}& tau_{1}=infbigl{ tgeq0: x t geq theta_{2}bigr},& tau_{2k}=infbigl{ tgeqsigma_{2k-1}: x(t)leq theta_{1} bigr}, qquadtau_{2k+1}=infbigl{ tgeq sigma_{2k}: x(t)geqtheta_{2} bigr},quad k=1,2,ldots.
Let us now define a sequence of stopping times begin{aligned}& tau_{1}=infbigl{ tgeqslant0 : x_{i}(t geqslant2 epsilonbigr},qquad tau_{2k}=infbigl{ tgeqslant tau_{2k-1} : x_{i}(t)leqslant epsilonbigr}, & tau_{2k+1}=infbigl{ tgeqslanttau_{2k} : x_{i}(t geqslant 2epsilonbigr},quad k=1,2,ldots.
In its simplest version, a Pedibus line is a sequence of stops starting from a child home, visiting a sequence of other children's home, and ending at the school.
In the United Kingdom, for example, postwar policy involved a sequence of "stop-go" moves by government for stabilization; unfortunately these often occurred too late and had unintended destabilizing effects.
In order to do so, we convert the raw GPS locations into a sequence of stops at places.
The prediction task can now be re-formulated as follows: given a sequence of stops up to step n, can we predict your next place at step (n+1)?
The process described above converts the raw location history into a sequence of stops at places (Additional file 1 Figures 4 and 5).
This produces a sequence of stops, each one with a centroid calculated as the median of the locations coordinates, and a duration equal to the time between the last location and the first location sample.
where H n is an increasing sequence of stopping times with H0=0, and the mapping (t,x ↦v n ((t 1,x 1),⋯, t n,x n );t,x) is in C 1,2.
implying that the sequence of stopping times τ U is increasing.
Define the sequence of stopping times begin{aligned} &tau_{1}=infbigllbrace tgeqslant 0 : x_{i}(t) geqslant2varepsilonbigrrbrace,qquad tau_{2k}=infbigllbrace t geqslanttau_{2k-1} : x_{i}(t)leqslant varepsilonbigrrbrace, &tau_{2k+1}=infbigllbrace tgeqslanttau_{2k} : x_{i}(t geqslant 2varepsilonbigrrbrace,quad k=1,2,ldots.
