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\title%
    [The Random Spread Model]% short title (maximum of 50 characters)
    {The Random Spread Model}% full title

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\begin{Authors}
    \Author
        {Jean Serra}
        \Affil[ESIEE]
          {Laboratoire L2ASI, ESIEE, France \\ \email{serraj@esiee.fr}}
\end{Authors}

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\begin{Abstract}
The paper proposes the new stochastic model of a Random Spread for
describing the spatial propagation of sequential events such as
forest fires. A Random Spread is double Markov chain, whose each
step is a (random) set operator $\beta $ combining a Cox process
with a Boolean random closed set. Under iteration, operator $\beta $
provides the time evolution of the Random Spread, which turns out to
be a birth-and-death process. Average sizes, and the probabilities
of extinction are derived. The random spread model was applied to
the analysis of the fires that occurred in the State of Selangor
(Malaysia) from 2000 to 2004. It was able to predict all places
where burnt scars actually occurred, which is a strongly significant
verification.
\end{Abstract}

\begin{Keywords}
    \Index{Random Spread},
    \Index{Boolean RACS},
    \Index{Markov chain},
    \Index{birth-and-death},
    \Index{random closed sets},
    \Index{forest fires},
    \Index{simulations},
    \Index{mathematical morphology}.
\end{Keywords}


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\section{Introduction}

In a large number of wild forests, such as typically in South East
Asia, forest fires propagate less under the action of the wind, as
in Mediterrenan countries, than under almost isotropic causes. For
instance, the fires that occurred in the state of Selangor
(Malaysia, see Fig.\ref{init}) from 2000 to 2004 seem to evolve
randomly: the initial seats vanish sometimes, but may also give
birth to new seats at some distance. The foresters use to describe
this Malaysian context by means of two key maps \cite{FOR92}
\cite{HAI05}, namely the daily \textit{spread rate\ }of the fire,
and the\textit{\ fuel amount} of the vegetation, as depicted in
Fig.\ref{init}. Clearly, a straightforward use of such key maps does
lead to a pertinent description. By starting from \textit{any} \
point seat, one always arrives to burn the whole space in a finite
time. Indeed, one must use the rate information in some restrictive
way, to be able to reach actual
events. It is exactly the purpose of the present paper\footnote{%
A more complete study of the random spread has been submitted to the
Journal of Applied Probability, and some simulations were presented
at the DMAI congress on disaster monitoring \cite{HAI05} in November
2005.}. Two maps of the state of Selangor a)\textit{\ }map of the
spread rate, i.e. of the radius $r$\ of the daily circular
propagation of the fire; b) map $f_{w}$ of the fuel consumption.

\begin{figure}% [!tb]
 \begin{center}
  \includegraphics[width=\linewidth]{\fullpaperpdirectory/images/inittriple.pdf}
 \end{center}
 \caption{a) State of Selangor in Malaysia; b)\textit{\ }map of the spread rate, i.e. of the radius $r$\ of the daily circular propagation of the fire; c) map $f_{w}$ of the fuel consumption.}
 \label{init}
\end{figure}



\section{Stochastic models for growth}

Since C.\ J.\ Preston's pioneer work on\textit{\ spatial
birth-and-death} \cite{PRE77}, one finds in the literature several
point stochastic processes for describing joint evolutions in space
and time. The characteristic functionals $Q(K)$ are generally
inaccessible, but they yield instructive simulations \cite{LAN02}.
The second class of growth RACS is the concern of "thick"
structures, i.e. which do not reduce to points. A particular case
can be found in \cite{Serra82:Book}, p.562, under the name of
hierarchical
RACS, with several variants such as the following: \textit{the RACS at time }%
$t$\textit{\ is }$X_{t}$\textit{, and }$X_{t+dt}$\textit{\ is
generated by
adding to }$X_{t}$\textit{\ any boolean grain which occurs during }$[t,t+dt]$%
\textit{\ and whose center hits }$X_{t}$. In spite of its outward
simplicity, this variant is not tractable, i.e. one cannot express
the functional $Q_{t}$ of $X_{t}$\ by means of the initial
conditions, as proved by D.Jeulin in \cite{JEU91}, Ch.6.


However, a number of phenomena, including tumors and forest fires,
follows the same type of behavior. Each time that in the mineral,
vegetal or animal words, seeds move and then develop a new colony,\
they involve some random sequential growth. But how to model it by
tractable RACS? The trouble with the hierarchical RACS comes from
their evolution from step $n$ to $n+1$ refers to the \textit{whole
past}, from $0$ to $n$. Another difficulty is due to the increment
between $X_{n}$ and $X_{n+1}$, which is viewed globally, whereas it
turns out to be the Booleanization of a collection of primary grains
which could also be viewed \textit{individually}. If we change these
two axioms, can we reach sequential growth RACS which become
calculable? In addition, we must take into account that the space
parameters which govern the evolution laws (e.g. the fuel amount for
forest fires)
usually vary from place to place, so that the future model should not be%
\textit{\ a priori} translation invariant, but accept imposed laws
of heterogeneity, for both intensity and primary grain. This
constraint sets a non-trivial problem because the set which appears
in functional $Q$ of a Boolean RACS is not the primary grain, but
its\textit{\ }reciprocal version in the sense of Equivalence
(\ref{Randomdil1}). When translation invariance is assumed, this
reciprocal version reduces to a symmetry, but what about non
stationary cases ?

That are the questions we consider in this paper, by means of the \textit{%
Random Spread }RACS.

\section{Reminders}

\paragraph{Set dilation}

A \emph{structuring function, }i.e. an arbitrary mapping
$x\longmapsto \delta (x)$ from $E$ to $\mathcal{P}(E)$ generates a
set dilation $\delta $
from $\mathcal{P}(E)$ into itself \cite{Serra88:Book}, \cite%
{Heijmans90:Ronse} by the relation
\begin{equation}
\delta (X)=\cup \left\{ \delta (x)\,\,,\,\,x\in X\right\} \qquad
\qquad X\in \mathcal{P}(E). \label{Randomdil0}
\end{equation}
In addition, we introduce the following compacity
assumption~\cite{SER07}

\begin{definition}
\label{regulardil} A structuring function $\delta
:\mathbb{R}^{d}\longmapsto \mathcal{P}(\mathbb{R}^{d})$ is said to
be \emph{compact} when

i) it is u.s.c. from $\mathbb{R}^{d}$ into $\mathcal{K}$,

ii) the union $\cup \{\delta _{-x}(x),x\in \mathbb{R}^{d}\}$ has a
compact
closure $\delta $%
\begin{equation}
\overline{\delta _{0}}=\overline{\cup \{\delta _{-x}(x),x\in \mathbb{R}^{d}\}%
}  \label{Randomdil37}
\end{equation}
\end{definition}

A duality between dilations plays an important role below. For each
structuring function $\delta $, associate the \textit{reciprocal
structuring function} $\zeta $ by writing
\begin{equation}
y\in \zeta (x)\qquad \mathrm{if\,\ and}\text{ }\mathrm{only\,\
if}\qquad x\in \delta (y)\qquad \qquad x,y\in E.  \label{Randomdil1}
\end{equation}

The algorithm that expresses $\zeta $ in function of $\delta $ is
therefore
as follows%
\begin{equation}
\zeta (x)=\cup \{y:x\in \delta (y)\}.  \label{Randomdil38}
\end{equation}

\paragraph{ Random Closed Sets (Euclidean case)}

In the Euclidean space $\mathbb{R}^{d}$ of dimension $d$, denote by the $%
\mathcal{F}=\mathcal{F}(\mathbb{R}^{d})$ (resp. $\mathcal{K}$) the
family of all closed (resp. compact) sets of $\mathbb{R}^{d}$. Given
an element $K\in \mathcal{K}$, consider the class
\begin{equation*}
\mathcal{F}^{K}=\left\{ F:F\in \mathcal{F}\ ,\ F\cap K=\emptyset
\right\}
\end{equation*}%
of all closed sets that miss $K$.\ As $K$ spans the family
$\mathcal{K}$, the classes $\left\{ \mathcal{F}^{K}\,,\,K\in
\mathcal{K}\right\} $ are sufficient to generate the $\sigma
$-algebra.\ Moreover, as set $\mathcal{F}$ is compact for the
hit-or-miss topology, there exist probabilities, $Pr$ say, on
$\sigma _{f}$, and each triplet $(\mathcal{F},\sigma _{f},P)$
defines a RACS~\cite{Matheron75:RS}~\cite{MOD04}~\cite{STO87}.
Matheron made this rather abstract definition of a RACS more
tractable by proving the following fundamental result
(\cite{Matheron75:RS} p.30)

\begin{theorem}
\label{Thranddil2}Every RACS $X$ is \emph{characterized} by the
datum of the probabilities
\begin{equation}
Q(K)=\Pr \left\{ K\subseteq X^{c}\right\}  \label{Randomdil5}
\end{equation}%
as $K$ spans the class $\mathcal{K}$ of the compact sets of $\mathbb{R}^{d}$%
.\ Conversely, a family $\left\{ Q(K),\,K\in \mathcal{K}\right\} $
defines a (necessarily unique) RACS if and only if $1-Q(K)=T(K)$ is
an alternating
Choquet capacity of infinite order such that $0\leq T\leq 1$ and $%
T(\emptyset )=0$.\

\ The mapping $Q:\mathcal{K\rightarrow }[0,1]$ is called the \emph{%
characteristic functional }of the RACS.
\end{theorem}

The characteristic functional\emph{\ }$Q$ plays w.r.t. a random closed set$%
\emph{\ X}$ the same role as the distribution function for a random
variable $x$. It is sometimes called capacity functional, or Choquet
functional.

Set mappings may be used in the framework of the RACS as soon as
they are semi-continuous (\cite{Matheron75:RS} p.\ 28).\ If $\left\{
X_{i},i\in
I\right\} $ stands for all possible realizations of the RACS $X$, and if $%
\psi :\mathcal{F}\longmapsto \mathcal{F}$ is semi-continuous, then
the family $\left\{ \psi (X_{i}),\,i\in I\right\} $ characterizes in
turn all realizations of a RACS, that we denote by $\psi (X)$. This
basic result allows us to play with RACS just as with deterministic
sets, to intersect them, to dilate them, etc... instructive
introductions to RACS\ may be found in \cite{MOD04},\ or
\cite{STO05}.

\paragraph{Boolean RACS}
\label{booleanmodel}

The Boolean RACS, introduced by G.\ Matheron \cite{Matheron75:RS},
is very popular and led to many presentations and variants (e.g.
\cite{SER80} \cite{Serra82:Book} \cite{JEU00}, \cite{AYA89}
\cite{STO05}). Consider the two primitives of

i/ a Poisson process $J(\theta )$, of intensity $\theta $,

ii/ a "primary grain", namely a compact structuring function
$\delta$.

The Boolean RACS $X$ is constructed in two steps. First, take a realization $%
J$ of Poisson points, which provides the set of points $x_{j},$ $x_{j}\in J.$%
Second, take the union $X$ of all primary grains whose centers
belong to the Poisson realization
\begin{equation*}
X=\cup \{\delta _{x_{j}}\,,x_{j}\in J\}
\end{equation*}

This union generates a realization of the Boolean RACS $X$. The
characteristic functional $Q$ of RACS $X$ derives easily form the
above definition \cite{Matheron75:RS}, and equals
\begin{equation}
Q(K)=\exp -\int \theta (dz)1_{\delta (z)\cap K}=\exp -\int \theta
(dz)1_{z\cap \zeta (K)}=\exp -\theta \lbrack \zeta (K)].
\label{Lantu2}
\end{equation}

If we restrict the previous Poisson points to those which occur in a
given compact set $X_{0}$, then this comes back to change the
intensity $\theta (dx)$ into $\theta ^{\ast }(dx)=$ $\theta
(dx).1_{X_{0}}(x) $, the Boolean structure being preserved, and we have that%

\begin{equation}
\Pr \{K\subseteq X_{1}^{c}\}=Q_{1}(K)=e^{-\theta ^{\ast
}[\zeta(K)]}=e^{-\theta \lbrack \zeta (K)\cap X_{0}]}.
\label{Lantu8}
\end{equation}

\section{Random Spreads}
\label{Randomspread}

\subsection{Definition}

The Random Spread model generalizes Matheron's Boolean RACS\ by
introducing
a genetic dimension, namely the successive steps, according to which the $%
(n+1)^{th}$ Boolean RACS depends on the realization of the $n^{th}$
one.

Consider an initial random seat $I_{0}$ made of an a.s. locally
finite
number of initial point seats in $\mathbb{R}^{d}$. The fire evolution from $%
I_{0}$ is the concern, on the one hand, of the fire the initial
seats provoke, or\textit{\ fire spread} $X_{1}=\delta (I_{0})$, and
on the other hand of the generation of subsequent \textit{seats
spread} $I_{1}=\beta (I_{0})$. These secondary seats will develop
new fires in turn. Both aspects refer to some compact dilation
$\delta $. We propose to model the seats spread $\beta (I_{0})$ by
picking out, randomly, a few points in each dilate $\delta (x_{i})$,
for all points $x_{i}\in I_{0}$. The double\ spread process is then
written

\begin{align}
\text{for the fire spread }X_{1}(I_{0}) &=\delta (I_{0})={\cup }%
\{\delta (x_{i}),x_{i}\in I_{0}\}  \label{Lantu13} \\
\text{for the seats spread }I_{1}(I_{0}) &=\beta (I_{0})={\cup }%
\{\left( \delta (x_{i})\cap J_{i}\right) ,x_{i}\in I_{0},J_{i}\in
\mathcal{J}(\theta )\}  \label{Lantu11}
\end{align}%
where each point $x_{i}$ of the set $I_{0}$ induces a bunch of seats
$\delta (x_{i})\cap J_{i}$ independent of the others, since a
different realization\ $J_{i}$ is associated with each point
$x_{i}$. These two equations mean that though the fire from a seat
$x$ does burn the zone $\delta (x)$ around $x$, only a few points of
the scar $\delta (x)$ remain active seats for the next step.

Under iteration, Rel.(\ref{Lantu13}) and (\ref{Lantu11}) become%
\begin{eqnarray}
X_{2}(I_{0}) &=&\delta (I_{1})=\cup \{\delta (y_{k}),y_{k}\in I_{1}\}=\cup \{\delta (y_{k}),y_{k}\in \delta (x_{i})\cap J_{i}%
\text{ };\text{ \ }x_{i}\in I_{0}\}  \nonumber \\
I_{2}(I_{0}) &=&\beta (I_{1})=\beta \lbrack \beta (I_{0})]={\cup }_{i}%
{ \{}{\cup}_{k}[\delta \left( \delta (x_{i})\cap J_{i}\right) ]\cap
J_{k}{ \}},x_{i}\in I_{0}\}.  \label{Lantu25}
\end{eqnarray}

Fig.\ref{propagations4} depicts the first three steps of a random
spread, for which:

\begin{figure}% [!tb]
 \begin{center}
  \includegraphics[width=3.8597in]{\fullpaperpdirectory/images/propagations4.pdf}
 \end{center}
 \caption{Three generations of fires stemming from point $x_{0}=$ $I_{0}$.}
 \label{propagations4}
\end{figure}

\begin{itemize}
\item the initial seat $I_{0}$ is the point $x_{0}$,\ and the first spread,
or front, the dark grey disk $X_{1}(I_{0})=\delta (x_{0})$;

\item then two Poisson points, namely\ $y_{1}$ and $y_{2}$, fall in $\delta
(x_{0})$. They generate%
\begin{eqnarray*}
\text{ }X_{2}(x_{0}) &=&\delta (y_{1})\cup \delta (y_{2})=\delta (I_{1})%
\text{, in medium grey,} \\
\text{and }I_{1}(x_{0}) &=&\{y_{1}\}\cup \{y_{2}\};
\end{eqnarray*}

\item then again a new Poisson realization generates one point, $z_{1}$ in $%
\delta (y_{1})$, and another Poisson realization the three points $%
z_{2,1},z_{2,2},$ and $z_{2,3}$ in $\delta (y_{2})$, hence%
\begin{eqnarray*}
X_{3}(x_{0}) &=&\delta (z_{1}){ \cup }[\delta (z_{2,1})\cup \delta
(z_{2,2})\cup \delta (z_{2,3})]=\delta (I_{2})\text{, in light grey.} \\
\text{and }I_{2}(x_{0}) &=&\{z_{1}\}{ \cup }[\{z_{2,1}\}\cup
\{z_{2,2}\}\cup \{z_{2,3}\}].
\end{eqnarray*}
\end{itemize}

The doublet spread $(X_{n},I_{n})$ of order $n$, is the $n^{th}$
element of the chain depicted in Fig.\ref{chain}, which is Markov:
as soon as $I_{n}$
is known, the previous links do not serve in the creation of link $%
(X_{n+1},I_{n+1})$. This Markov assumption means that the fire of
tomorrow can only be caused by points seats stemming from the zone
which burns today. What burnt yesterday, before yesterday, etc., has
no longer importance. In this space-time process, the successive
sets $X_{1},X_{2}...X_{n}$ are at least as descriptive as the seats
$I_{1},I_{2}...I_{n}$ themselves, because
they show the extension of the fire front through the time steps $i$. The $%
(n+1)$ step is obtained by the two induction relations%
\begin{eqnarray}
X_{n+1}(I_{0})=\delta \lbrack I_{n}(I_{0})]  \label{Randomdil25} \\
I_{n+1}(I_{0})=\beta \lbrack I_{n}(I_{0})]={ \cup }\{\left( \delta
(x_{i,n})\cap J_{i,n}\right), \text{ } x_{i,n}\in I_{n},\text{ }%
J_{i,n}\in \mathcal{J}(\theta )\}  \label{Randomdil26}
\end{eqnarray}

We observe that, in Rel.(\ref{Lantu11}), and the ulterior ones, each
small zone $dz$ intervenes as many times as $dz$ belongs to
different $\delta
(x_{i})$.\ Therefore, the random measure%
\begin{equation}
\tau _{n+1}(dz)=\theta (dz)\sum \{1_{\delta (x_{i,n})}(z),\text{ \ }%
x_{i,n}\in I_{n}\}  \label{Randomdil24}
\end{equation}%
turns out to be a realization of the Cox process of intensity $\tau
_{n+1}$, if $I_{n+1}$ is finite. Now, are we sure that all these
handlings and derivations really lead to a RACS, in
Matheron-Kendall's sense, i.e. to something we can characterize by a
functional $Q(K)$?\ Are we sure, for example, that the intensity
$\tau _{n+1}$ is always a.s. finite? The answer
is positive, as soon as the structuring function $\delta $ is \textit{compact%
}, in the sense of Definition \ref{regulardil}.Then one can easily
prove
that the reciprocal function $\zeta $ of $\delta $ is also compact \cite%
{SER07}, so that we can state the following

\begin{figure}% [!tb]
 \begin{center}
  \includegraphics[width=4.4529in]{\fullpaperpdirectory/images/chain.pdf}
 \end{center}
 \caption{The Markov chain of the spreads.}
 \label{chain}
\end{figure}


\begin{proposition}
Let $I_{0}$ be a a.s. locally finite number of points, let $\theta $
be a
Poisson intensity, and $\delta :\mathbb{R}^{d}\rightarrow \mathcal{K}(%
\mathbb{R}^{d})$ be a structuring function. If function $\delta $ is
compact, then both families $\{X_{n}(I_{0}),n>0\}$ and
$\{I_{n}(I_{0}),n>0\}$ of fire spreads and seats spreads are
composed of RACS.
\end{proposition}

\begin{proof}
Consider the sum $\tau _{1}(z)$. Its value at point $z$ is
proportional to
the number of points of set $I_{1}(I_{0})$ that fall in the reciprocal set $%
\zeta (z)$. As the structuring function $\delta $ is compact, $\zeta
(z)$ is compact, and as set $I_{1}(I_{0})$ involves only finite
union of Poisson points (Rel.(\ref{Lantu11})), the number of points
of $I_{1}(I_{0})$ that
are taken into account in $\tau _{1}(z)$\ is a.s. finite. According to Rel.(%
\ref{Randomdil24}), this number of points is weighted by
non-negative and
finite value $\theta (z)$, which results in a a.s. finite sum $\tau _{1}(z)$%
. Therefore function $\tau _{1}$ is a convenient intensity for making $%
I_{1}(I_{0})$ a Cox process. The same proof applies, by induction,
to the passage from $\tau _{n}$ to $\tau _{n+1}$.

The set $X_{n+1}(I_{0})=\delta \lbrack I_{n}(I_{0})]$, which is the
dilate of a RACS by the compact structuring function $\delta $, i.e.
a semi-continuous mapping, is in turn a RACS.
\end{proof}

Since the two spreads $I_{n}$ and $X_{n}$ are RACS, we shall
characterize them by their functionals $Q_{n}(K)$. In the
"Boolean-Cox RACS" $X_{n}$, the
primary grain \textit{only }is independent of step $n$, since at each time $%
n $, the intensity $\tau _{n}$ is a new one. This circumstance
simplifies the theoretical study of the time evolution. Now, the
time progression being
a Markov chain, we are suggested to find induction relations between $%
Q_{n}(K)$ and $Q_{n+1}(K)$ that reflect the two definitions by
induction of Rel.(\ref{Randomdil25}) and (\ref{Randomdil26}).

\subsection{Characteristic functional}

The additivity property of the random spread allows us to take for
$I_{0}$ a point initial seat, $x_{0}$ say, of dilate $X_{1}=\delta
(x_{0})$, and whose intersection of the dilate with Poisson points
$\theta $ provides the first random set $I_{1}=$ $\delta (x_{0})\cap
J$. The functional $Q_{n}(K\mid x_{0})$\ of the random fire spread
$X_{n}(x_{0})$, i.e. the probability that set $K$ misses the
$n^{th}$ spread $X_{n}(x_{0})$ of seat $x_{0}$, satisfies an
induction relation between the steps $n$ and $n+1$. Fix the volume
element $dy$ of $\delta (x_{0})$, with $y\in \delta (x_{0})$. To say
that $K$ lies in the pores of the spread of step $n$ stemming from
$y$ is equivalent to saying that $K$ lies in the pores of the spread
of order $n+1$ stemming from $x_{0}$, but conditionally to $y$. By
de-conditioning, we find that
\begin{equation}
dQ_{n+1}(K\mid x_{0})=1-\theta (dy)+\theta (dy)Q_{n}(K\mid y).
\label{Randomdil12}
\end{equation}%
and by summing up in $\delta (x_{0})$%
\begin{equation}
Q_{n+1}(K\mid x_{0})=\exp -\int_{\delta (x_{0})}\theta
(dy)[1-Q_{n}(K\mid y)].  \label{Randomdil3}
\end{equation}

Each step involves an exponentiation more than the previous one. We
find for example for the first steps that

\begin{align}
Q_{2}(K) &=\exp -\theta \lbrack \zeta (K)\cap \delta (x_{0})],
\label{Lantu19} \\
Q_{3}(K) &=\exp -\int_{\delta (x_{0})}\theta (dy){ [}1{ -}%
e^{-\theta \lbrack \zeta (K)\cap \delta (y)]}{ ]}.  \label{Lantu20} \\
Q_{4}(K) &=\exp -\int_{\delta (x_{0})}\theta (dy){ [}1{ -}\exp
\{-\int_{\delta (y)}\theta (dz)[1-e^{-\int_{\delta (z)}\theta
(dw)1_{\zeta
(K)}(w)}]\}{ ].}  \nonumber%\label{Lantu14}
\end{align}%
where $Q_{2}$, but neither $Q_{3}$ nor $Q_{3}$, is equivalent to the
Boolean RACS functional of Eq.(\ref{Lantu8}). Note in passing that a
non compact
function $\delta $ would\ possibly lead to unbounded $\zeta $, hence to $%
Q_{2}(K)\equiv 0$ in Rel.(\ref{Lantu19}), and to
$Q_{3}(K)...Q_{n}(K)$ independent of $K$!

A more concise representation of the probabilities $Q_{n}(K)$ can be
given \cite{SER07}.The seat spread $I_{n+1}$ satisfies the same
induction relation (\ref{Randomdil3}) as the fire spread $X_{n+1}$.
The only change holds on
the first term, for which it suffices to replace $\zeta (K)$ by $K$ in Rel.(%
\ref{Lantu19}).We can summarize the main results on the
characteristic functional by stating:

\begin{theorem}
Let

- $\beta $ be the random spread of parameters $(\theta ,\delta )$,

- $I_{1}=\beta (x_{0})$ be the random seat spread stemming from
point $x_{0}$ of dilate $X_{1}=\delta (x_{0}),$

- $I_{2}=\beta (I_{1})$ and $X_{2}=\delta (I_{1})$ be the iterated
seat spread and its fire spread,

- $I_{n+1}=\beta (I_{n})$ be $n^{th}$ iteration of $\beta $, and $%
X_{n+1}=\delta (I_{n})$ the associated fire spread,

then the characteristic functionals of both RACS $I_{n+1}$ and
$X_{n+1}$
satisfy the induction relation%
\begin{equation*}
Q_{n+1}(K\mid x_{0})=\exp -\int_{\delta (x_{0})}\theta
(dy)[1-Q_{n}(K\mid y)],
\end{equation*}

with initial terms
\begin{eqnarray*}
Q_{1}(K) &=&\exp -\theta \lbrack K\cap \delta (x_{0})]\text{ \ for
the seat
spread }I_{1}\text{ and } \\
Q_{2}(K) &=&\exp -\theta \lbrack \zeta (K)\cap \delta (x_{0})]\text{
\ for the fire spread }X_{2}.
\end{eqnarray*}
\end{theorem}

When both parameters $\theta $ and $\delta $ reduce, we can suspect
a worse propagation of the fire, i.e. a functional $Q_{n}(K\mid
x_{0})$ which increases, and tends toward $1$ as $n$ increases. In
order to make this idea more precise, introduce the sum $u(z)$ of\
the intensity $\theta $ over the set $\delta (z)$, i.e.

\begin{equation*}
0\leq u(z)=\int_{\delta (z)}\theta (dx)\leq \int_{(\overline{\delta _{0})}%
_{z}}\theta (dx)<\infty .
\end{equation*}

We draw from Definition \ref{regulardil} and Rel.(\ref{Randomdil37})
that
\begin{equation}
\overline{u}={ \bigvee }_{z}\{u(z),z\in \mathbb{R}^{d}\}\leq Mes%
\overline{\delta }_{0}.\theta _{\max }<\infty ,  \label{Randomdil41}
\end{equation}%
as $\delta $ is compact and $\theta $ is upper bounded. This supremum $%
\overline{u}$ is the \emph{weight} of dilation $\delta $ w.r.t. measure $%
\theta $. When $\bar{u}<1$ then we draw from the equations
(\ref{Lantu19}) and (\ref{Lantu20}) of the two functionals $Q_{2}$
and $Q_{3}$ that
\begin{equation*}
1-Q_{2}(K\mid x_{0})\leq \bar{u}<1\text{ \ \ \ \ \ ;\ \ \ \ \ \ }1-\
Q_{3}(K)\leq \bar{u}^{2}<1;\text{ \ }etc
\end{equation*}%
so that $1-Q_{n}(K\mid x_{0})$ $\rightarrow 0$ as $n$ increases
indefinitely. At the infinity, the induction relation
(\ref{Randomdil3})
becomes%

\begin{equation*}
Q_{\infty }(K\mid x_{0})=\exp \{-\int_{\delta (x_{0})}\theta
(dy)[1-Q_{\infty }(K\mid y)]\},
\end{equation*}%
which implies, again, that $Q_{\infty }(K\mid x_{0})=1$.

\subsection{Spontaneous extinction}
\label{extinct}

The fire which stems from the point seat $x_{0}$ may go out,
spontaneously, after one, two, or more steps. The description of
this phenomenon is the concerns of the whole space, and involves no
particular compact set $K$. Denote by $h(n\mid x_{0})$ the
probability that the fire extinguishes at most at step $n$. This
event occurs before the second step, when no Poisson
point falls inside set $\delta (x_{0})$, hence when%
\begin{equation*}
\ h(1\mid x_{0})=\exp -\theta \lbrack \delta (x_{0})].
\end{equation*}

The proof by induction that allowed us to link $Q_{n+1}$ with
$Q_{n}$ in Rel.(\ref{Randomdil3}) applies again, and gives, for a
spontaneous extinction before step $n$, the\ probability
\begin{equation}
h(n+1\mid x_{0})=\exp { \{}\int_{\delta (x_{0})}\theta
(dy)[1-h(n\mid y)]{ \}}. \label{Lantu7}
\end{equation}

Solving the induction leads to the above operator $\tau $, but which
now acts on the constant function $c_{1}=1$. For example, the
probability of an extinction before the fourth step is given by the
expression:

\begin{equation*}
h(3\mid x_{0})=\exp -\int_{\delta (x_{0})}\theta (dy){ [}1{ -}%
\exp { \{}-\int_{\delta (y)}\theta (dz)[1-\exp \{-\int_{\delta
(z)}\theta (dw)\}{ ]}{ \}].}
\end{equation*}

Upper bounds are derived just in the same way as for the spreads, with now%
\begin{equation*}
h(n+1\mid x_{0})\leq \exp -\int_{\delta (x_{0})}\theta (dy)\
\bar{u}^{n}.
\end{equation*}%
When $\bar{u}<1$, then the fire extinguishes spontaneously, since
the probability it vanishes before step $n+1$ decreases more rapidly
than the geometrical sequence\ $\bar{u}^{n}$. For having an idea of
what happens in a zone where $\bar{u}\geq 1$, suppose for the moment
that $\theta $ is constant and $\delta $ translation invariant, so
that Rel.(\ref{Randomdil41}) becomes $\int_{\delta (z)}\theta
(dx)=u(z)$ $=\bar{u}$. The extinction probabilities no longer depend
on\ $x_{0}$. Denote by $h=h(\infty )$ the probability that the fire
goes on indefinitely. The infinite iteration of operator $\tau $
leads to the equality
\begin{equation*}
h=e^{\bar{u}(h-1)}\text{ \ \ \ ( with }0\leq h\leq 1\text{).}
\end{equation*}

This equation admits always the solution $h=1$, but when
$\bar{u}\geq 1$ then the straight line $y=\bar{u}(h-1)$ intersects
the curve $y=\log h$ at a second point of abscissa $h^{\prime }<1$.
Now
\begin{equation*}
h(1)=e^{-\bar{u}}\leq ...\leq h(n)\leq ...\leq h(\infty )=h^{\prime }=e^{%
\bar{u}(h^{\prime }-1)}.
\end{equation*}

There is a non zero probability of an infinite propagation. In
practice, both functions $\theta $ and $\delta $ vary through the
space, and $\delta (x_{0})$ is a connected set that contains
$x_{0}$.\ Therefore, if $Z$ is the set of all points where $u(z)\geq
1$, then there is every chance that the
fire invades the connected component of $\delta (Z)$ that contains point $%
x_{0}$.

\ELIMINE{
\section{Boolean upper bounds}
\label{upper bounds}

In Rel.(\ref{Lantu19}), the functional $Q_{1}(K)$ of the first order
spread
is that of a Boolean RACS of primitives $(\delta ,\theta _{1})$, with $%
\theta _{1}(dy)=\theta (dy).1_{\delta (x_{0})}(y)$. This is no
longer the case for the expression $Q_{3}(K)$ of
Rel.(\ref{Lantu20}); however we can
write:%
\begin{equation}
Q_{3}(K)=\exp -\int_{\delta (x_{0})}\theta (dy){ [}1{ -}%
e^{-\theta \lbrack \zeta (K)\cap \delta (y)]}{ ]\geq }\exp
-\int_{\delta (x_{0})}\theta (dy)\int_{\mathbb{R}^{d}}\theta
(dz)1_{\zeta (K)}(z)1_{\delta (y)}(z). \label{Lantu 21}
\end{equation}

As $1_{\delta (y)}(z)=1$ if and only if $1_{\zeta (z)}(y)=1$, the
right
member becomes%
\begin{equation*}
\exp -\int_{\delta (x_{0})}\theta (dy)\int_{\mathbb{R}^{d}}\theta
(dz)1_{\zeta (K)}(z)1_{\zeta (z)}(y)=\exp -\int_{\zeta (K)}\theta
(dz)\int_{\zeta (z)}\theta (dy)1_{\delta (x_{0})}(y),
\end{equation*}

which is nothing but the probability $Q_{3}^{\ast }(K)$ that $K$
lies in the
pores of the Boolean RACS of primitives $(\delta ,\theta _{2})$, with%
\begin{equation*}
\theta _{2}(dz)=\theta (dz)\int_{\zeta (z)}\theta _{1}(dy).
\end{equation*}

Inequality (\ref{Lantu21})means that $K$ lies more easily in the
pores of the random spread $X_{3}$ than in those of the Boolean RACS
$(\delta ,\theta _{2})$, hence that the latter upper bounds the
former, whose particles are more clustered. By using similar upper
bounds in the induction relation (\ref{Randomdil3}),
 one easily derives that the $n^{th}$ spread is a.s. smaller
than the Boolean RACS of primitives $\delta $ and
\begin{equation*}
\theta _{n+1}(dz)=\theta (dz)\int_{\zeta (z)}\theta _{n}(dy).
\end{equation*}

We find again two possible behaviors at infinity, but the switch
from contained to invasion mode is now given by the reciprocal
dilation $\zeta $. If $Z$ is a zone where the weight $\bar{v}$ of
$\zeta $
\begin{equation*}
\bar{v}=sup\{\int_{\zeta (z)}\theta (dy),\ \ z\in \text{$Z$}\}
\end{equation*}

is smaller than $1$, then%
\begin{equation}
\theta _{2}(y)\leq \theta (y)\bar{v}\text{\ \ ; \ }\theta
_{3}(y)\leq \theta
(y)\bar{v}^{2}\text{ \ ; .....\ }\theta _{n+1}(y)\leq \theta (y)\bar{v}^{n}%
\text{ \ } \label{Randomdil11}
\end{equation}

and the successive Boolean upper bounds $(\delta ,$ $\theta
_{1}),(\delta ,\theta _{2}),..(\delta ,\theta _{n})$ tend toward
zero as $n$ increases indefinitely.

In inequalities (\ref{Randomdil11}) the supremum $\bar{v}$ involves
the reciprocal dilation $\zeta $ which is not directly known, unlike
$\delta $. However, when $\delta $ is symmetrical, then the weight
$\bar{v}\leq \bar{u}$ \cite{SER07}. Therefore, when $\bar{u}<1$ in
some zone $Z$, then the Boolean upper bounds of the iterated random
spreads inside $Z$ tend toward zero. If the infimum$\
$\underline{$v$} of the $v(z)=\int_{\zeta (z)}\theta (dy)$ over $Z$
is larger than $1$, the upper bounds are not informative, but we
know from the previous section that there is a non zero probability
that the fire does not extinguish spontaneously: $Z$ is a risky
zone. }

\section{Scar prediction}
\label{scars}

We will now match the data of the actual scars with the model. We
know from the conclusions of the section \ref{extinct} that
the significant parameter, here, is the \textit{weight }$u\left( x\right) $%
.\ In each region $Z$ where all $u\left( x\right) $, $x\in Z$, are
noticeably $\geq 1$, any initial seat invades progressively the
whole region, whereas in the regions with $u\left( x\right) <1$, the
spread stops by itself, all the sooner since $u\left( x\right) $ is
small. In Selangor's case, the expression of $u$ from the two maps
of Fig.\ref{init} is as follows
\begin{equation*}
u\left( x\right) =\int \theta \left( z\right) 1_{\zeta \left(
x\right) }\left( z\right) dz=k\int f_{w}\left( z\right) 1_{\zeta
\left( x\right) }\left( z\right) dz\simeq \pi kf_{w}\left( x\right)
r^{2}\left( x\right) .
\end{equation*}

This expression suggests to introduce the \textit{scar function} $%
s(x)=f_{w}\left( x\right) r^{2}\left( x\right) $, as represented in
Fig.\ref{scarpredict}. Scar function $s$ is accessible from the
experimental data, since functions $f_{w}$ and $r$ are given (see
Fig.\ref{init}c and a). Note that map $s$ is not obtained by
simulations, but comes from a combination of the input parameters of
the random spread model. By putting a threshold on image $s$ at
level $1/$ $\pi k$, one splits the plane into the two regions where,
either fires spontaneously extinguish (when $s(x)<1/$ $\pi k$), or
invade the connected components that contain their initial seats (when $%
s(x)\geq 1/$ $\pi k$).

If we take for $k$ the value $1.61\times 10^{-3}$, which derives
from the hot spots measurements \cite{HAI05}, we get $1/$ $\pi
k=193$. The two sets above thresholds $190$ and $200$ are reported
in Fig.\ref{scarpredict}b,
side by side with the burnt areas (Fig.\ref{scarpredict}c).\ In Fig.\ref%
{scarpredict}b, the fire locations \textbf{A} to \textbf{E}
predicted by the model point out regions of actual burnt scares.
Such a remarkable result could not be obtained from the maps $f_{w}$
and $r$ taken separately: the scare function $s=f_{w}r^{2}$ means
something more, which corroborates the random spread assumption.
Region \textbf{F} is the only one which seems to invalidate the
model. As a matter of fact this zone is occupied by peat swamp
forest, or rather, \textit{was occupied}. It is today the subject of
a fast urbanization, linking the international airport of Kuala
Lumpur to the administrative city of Putra Jaya. Finally, on the
whole, the random spread model turns out to be realistic.

\begin{figure}[h]
 \begin{center}
  \includegraphics[width=\linewidth]{\fullpaperpdirectory/images/scarpredict.pdf}
 \end{center}
 \caption{a) Scar function $s=f_{w}\times r^{2}=$ $u/\protect\pi k$ whose thresholds estimate the burnt scar zones. b) Two thresholds of $\ $function $s$ for $1/\protect\pi k=190$, in dark grey, and $1/\protect\pi k=200$, in black (the simulations suggest value $193$); c) map of the actual burnt areas. Note the similarity of the sets, and of their locations.}
 \label{scarpredict}
\end{figure}


\section{Conclusion}

This paper proposes a new RACS model, the \textit{random spread},
which combines the three theoretical lines of Boolean random sets,
Markov chains, and birth-and-death processes. Its characteristic
functional was established. More than classical spatial
birth-and-death processes, spread RACS depends strongly of the
heterogeneity of the space, which appears via two functions of
intensity $(\theta )$ and extensivity $(\delta )$. As a result, the
process no longer describes a global birth-and-death, but regional
expansions and shrinkages of the sets under study, namely the front,
the seats and the scar of the spread RACS.

The time evolution was introduced in a discrete manner, by the Makov
assumption that the fire front of tomorrow can only be caused by
points seats stemming from the zone which burns today.

We draw from the model and its upper bounds a precise predictor of
scars that actually occurred in the State of Selangor during the
period 2000-2004.


%
% BIBLIOGRAPHY
% for details, see ftp://ftp.ams.org/pub/tex/amsrefs/amsrdoc.pdf
%

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