temporalAtRisk(obj, ...)as.integer on both
observation times and time limits t_1 and t_2 (which may be
stored as non-integer values). The functions that create
temporalAtRisk objects therefore return piecewise cconstant
step-functions. that can be evaluated for any real t in
[t_1,t_2], but with the restriction that mu(t_i) = mu(t_j)
whenever as.integer(t_i)==as.integer(t_j).A temporalAtRisk object may be (1) 'assumed known', or (2) scaled to a particular dataset. In the latter case, in the routines available (temporalAtRisk.numeric and temporalAtRisk.function), the stppp dataset of interest should be referenced, in which case the scaling of mu(t) will be done automatically. Otherwise, for example for simulation purposes, no scaling of mu(t) occurs, and it is assumed that the mu(t) corresponds to the expected number of cases during the unit time interval containnig t. For reference purposes, the following is a mathematical description of a log-Gaussian Cox Process, it is best viewed in the pdf version of the manual.
Let $\mathcal Y(s,t)$ be a spatiotemporal Gaussian process, $W\subset R^2$ be an observation window in space and $T\subset R_{\geq 0}$ be an interval of time of interest. Cases occur at spatio-temporal positions $(x,t) \in W \times T$ according to an inhomogeneous spatio-temporal Cox process, i.e. a Poisson process with a stochastic intensity $R(x,t)$, The number of cases, $X_{S,[t_1,t_2]}$, arising in any $S \subseteq W$ during the interval $[t_1,t_2]\subseteq T$ is then Poisson distributed conditional on $R(\cdot)$, $$X_{S,[t_1,t_2]} \sim \mbox{Poisson}\left{\int_S\int_{t_1}^{t_2} R(s,t)d sd t\right}$$ Following Brix and Diggle (2001) and Diggle et al (2005), the intensity is decomposed multiplicatively as $$R(s,t) = \lambda(s)\mu(t)\exp{\mathcal Y(s,t)}.$$ In the above, the fixed spatial component, $\lambda:R^2\mapsto R_{\geq 0}$, is a known function, proportional to the population at risk at each point in space and scaled so that $$\int_W\lambda(s)d s=1,$$ whilst the fixed temporal component, $\mu:R_{\geq 0}\mapsto R_{\geq 0}$, is also a known function with $$\mu(t) \delta t = E[X_{W,\delta t}],$$ for $t$ in a small interval of time, $\delta t$, over which the rate of the process over $W$ can be considered constant.