There are two types of forecast models: Model without regression and Regression models. For each type of models, we both use time and space discretization. The time discretization is given by time windows of 30 mins while the user can choose among 3 different discretizations denoted by Rect 10 (regular space discretization into 10x10=100 rectangles), Uber 7 (provided by Uber discretization library with scale parameter 7), and District (the districts of Rio de Janeiro). There are four sections: Line plot for the number of emergency calls, heatmap of the mean number of emergency calls, and empirical distributions of the number of calls (histograms).

Figure 1: Space discretization of a rectangle containing the city of Rio de Janeiro in $10\small{\times}10=100$ rectangles, 76 of which have intersection with the city and are represented in the figure.

Figure 3: Space discretization of a rectangle containing the city of Rio de Janeiro in Hexagones using Uber library for space discretization in hexagones with scale parameter equal to 7.

Figure 5: Space discretization of the city of Rio de Janeiro into its 144 administrative districts.

Methodology for forecast

MODEL WITHOUT REGRESSION

The region under study is partitioned into a set $\mathcal{I}$ of zones. Different zones may have different shapes, sizes, and other attributes. For the basic model described here, it is assumed that arrival rates vary over time according to the time of the week. Time during the week is partitioned into a set $\mathcal{T}$ of time intervals. The model may impose constraints on the differences between arrival rates in different time intervals. For example, it may be required that the arrival rate during [8:00,8:30) on all weekdays be the same, or be close to each other. To facilitate such constraints, partition $\mathcal{T}$ into a collection $\mathcal{G}$ of subsets. For example, $G \in \mathcal{G}$ may be $G =\big\{$Monday [8:00,8:30); Tuesday [8:00,8:30); Wednesday [8:00,8:30); Thursday [8:00,8:30); Friday [8:00,8:30).$\big\}$ There are a variety of classification systems for emergency calls, including ICD-10, ICD-11, MPDS, and APCO. Let $\mathcal{C}$ denote the set of arrival classes.

For each $c \in \mathcal{C}$, $i \in \mathcal{I}$, and $t \in \mathcal{T}$, let $N_{c,i,t}$ denote the number of observations for arrival class $c$ in zone $i$ during time interval $t$, and let these observations be indexed $n = 1,\ldots,N_{c,i,t}$.

For each $c \in \mathcal{C}$, $i \in \mathcal{I}$, $t \in \mathcal{T}$, and $n \in \{1,\ldots,N_{c,i,t}\}$, let $M_{c,i,t,n}$ denote the number of arrivals for observation $n$ of arrival class $c$, zone $i$, and time interval $t$, and let $M_{c,i,t}:=\sum_{n=1}^{N_{c,i,t}} M_{c,i,t,n}$ denote the total number of arrivals over all observations for arrival class $c$, zone $i$, and time interval $t$.

Assume that$\big\{M_{c,i,t,n},c \in \mathcal{C},i \in \mathcal{I}, t \in \mathcal{T}, n=1, \ldots,N_{c,i,t}\big\}$are independent Poisson distributed random variables. Denoting by $\lambda_{c,i,t}$ the mean number of calls per time unit (say an hour)for time interval $t$ arrival class $c$ and zone $i$ and by $\mathcal{D}_t$ the duration (in time units) of time interval $t$, $M_{c,i,t,n}$ is Poisson with parameter (mean) $\lambda_{c,i,t} \mathcal{D}_t$. We calibrate the intensities $(\lambda_{c,i,t})$ solving $\displaystyle \min_{\lambda \geq 0}\;\ell(\lambda)$ with objective $\ell(\lambda)$ given by $\ell(\lambda) \quad=$ $\quad \displaystyle\sum_{c \in \mathcal{C}} \sum_{i \in \mathcal{I}} \sum_{G\in\mathcal{G}} \sum_{t \in G}\left[N_{c,i,t} \lambda_{c,i,t} \mathcal{D}_t - M_{c,i,t} \log\left(\lambda_{c,i,t} \mathcal{D}_t \right) + \frac{W_{G}}{2} \sum_{t' \in G} \left( \lambda_{c,i,t} - \lambda_{c,i,t'} \right)^2 \right]$$\qquad \displaystyle + \sum_{c \in \mathcal{C}} \sum_{i \in \mathcal{I}} \sum_{G \in \mathcal{G}} \sum_{t \in G} \sum_{j \in \mathcal{I}} \frac{w_{i,j}}{2} \left(\lambda_{c,i,t} - \lambda_{c,j,t}\right)^2$ for some weights $W_G, w_{i,j} \geq 0$.

MODEL WITH REGRESSION

Each arrival class $c$, zone $i$, and time interval $t$ may have covariates, and $\lambda_{c,i,t} \mathcal{D}_t$ can be modeled as a function of these covariates. For each $c \in \mathcal{C}$,$i \in \mathcal{I}$, and $t \in \mathcal{T}$, let $x_{c,i,t} := (x_{c,i,t,1},\ldots,x_{c,i,t,K})$ denote the covariate values of class $c$, zone $i$, and time interval $t$. Then consider the model $\lambda(x_{c,i,t}) \mathcal{D}_t = \beta^{\top} x_{c,i,t}$where $\beta = (\beta_{1},\ldots,\beta_{K})$ are the model parameters. To facilitate such a model, let $N$ denote the total number of observations, and let these observations be indexed $n = 1,\ldots,N$. For each observation $n \in \{1,\ldots,N\}$, let $M_{n}$ denote the number of arrivals for observation $n$ and let $x^{n}:=(x^{n}_{1},\ldots,x^{n}_{K})$ denote the covariate values of observation $n$. Typically $x^{n}$ will indicate the class of emergency, the zone, the time interval, and other covariates used to explain the number of arrivals $M_{n}$. Then the negative log-likelihood function is given by $\mathscr{L}(\beta) = \sum_{n=1}^{N} \left[\beta^{\top} x^{n} - M_{n} \log\left(\beta^{\top} x^{n}\right)\right]$.

We calibrate $\beta$ maximizing the likelihood. We use as regressors the areas of the different land types (add a filter to choose the SAMU and show the corresponding heatmap of land types), the population, and holidays.