Stéphane Loisel
email: stephane.loisel@univ-lyon1.fr

DIFFERENTIATION OF SOME FUNCTIONALS OF RISK PRO-CESSES. APPLICATIONS TO RUIN THEORY AND TO DETER-MINATION OF OPTIMAL RESERVE ALLOCATION FOR MUL-TIDIMENSIONAL RISK PROCESSES

Keywords: Ruin theory, Sample path properties, Optimal reserve allocation, Multidimensional risk process, Risk measures AMS 2000 Subject Classication: Primary 60G17 Secondary 60G55, 91B30, 91B32, 62P05

For general risk processes, the expected time-integrated negative part of the process on a xed time interval is introduced and studied. Dierentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below zero studied by Dos Reis [1], and the probability of ruin. Dierentiation of other functionals of unidimensional and multidimensional risk processes with respect to the initial reserve level are carried out. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve which minimizes one of these risk measures, illustrate the variety of application elds and the benets deriving from an ecient and eective use of such tools.

Introduction

For unidimensional risk processes R t = u + X t (representing the surplus of an insurance company at time t, with initial reserve u and with X t = ct -S t , where c > 0 1 {u+Xt<0} dt using Gerber's work [START_REF] Gerber | Mathematical fun with ruin theory[END_REF].

All these random variables are drawn from the innite time ruin theory, or involve the behavior of the risk process between ruin times and recovery times. It seems interesting to consider risk measures based on some xed time interval [0, T ] (T may be innite).

One of the simplest penalty functions may be the expected value of the time-aggregated negative part of the risk process:

E(I T ) = E   T 0 1 {Rt<0} |R t |dt   .
Note that the probability P (I T = 0) is the probability of non ruin within nite time T. I T may be seen as the penalty the company will have to pay due to its insolvency until the time horizon T. From an economical point of view, it seems more consistent to consider

I g,h (u) =   T 0 1 {u+Xt≥0} g(|u + X t |) -1 {u+Xt≤0} h(|u + X t |) dt  
with 0 ≤ g ≤ h, where g corresponds to a reward function for positive reserves, and h is a penalty function in case of insolvency. As for utility functions, g and h should be both increasing and convex in the classical case. g ≤ h because usually the cost of ruin is higher than the reward of the opposite wealth level.

These risk measures may be dierentiated with respect to the initial reserve u, which makes it possible to compute them quite easily as integrals of other functions of u such as the probability of ruin or the total time in the red. Moreover, they have the advantage that the integral over t and the mathematical expectation may be permuted thanks to Fubini's theorem.

Dierentiation of some functionals of risk processes 3 Statement and proofs of dierentiation theorems can be found in sections 1 and 2.

Section 3 presents examples of applications to unidimensional risk measures, in particular a closed-form formula is derived for E(I ∞ (u)) in the Poisson-exponential case.

One can also use these concepts to construct risk measures for multidimensional risk processes, modelling dierent lines of business of an insurance company (car insurance, health insurance, ...). In this framework, determining the needed global initial reserve for the global expected penalty to be small enough requires to nd the optimal allocation of this reserve. Dierentiation of unidimensional risk measures are useful to nd this optimal reserve allocation. All this is illustrated in section 4.

Dierentiation theorems

Theorem 1. Let (X t ) t∈[0,T ) be a stochastic process with almost surely time-integrable sample paths. For u ∈ R, denote by τ (u) the random variable corresponding to the time spent under zero by the process u + X t between the xed times 0 and T:

τ (u) = T 0 1 {u+Xt<0} dt,
Let τ 0 (u) correspond to the time spent in zero by the process u + X t :

τ 0 (u) = T 0 1 {u+Xt=0} dt.
Let I T (u) represent the time-integrated negative part of the process u + X t between 0 and T: (u) represents the time spent by the process u + X t in the interval ] -ε, ε[ between dates 0 and T.

I T (u) =   T 0 1 {u+Xt<0} |u + X t |dt   and f (u) = E(I T (u)). For u ∈ R, if Eτ 0 (u) = 0, then f is dierentiable at u, and f (u) = -Eτ (u). I T (u) is illustrated by gure 1. Proof. Fix u ∈ R. For ε ≥ 0, set τ ε (u) = T 0 1 {|u+Xt|<ε} dt. 4 Stéphane Loisel τ ε
For each sample path (considered as a function of time t), t → 1 {|u+Xt|<ε} pointwise converges, decreasingly to

t → 1 {u+Xt=0} .
Besides, each of the integrals of the indicator functions is bounded by T. From the monotone convergence theorem, τ ε is decreasing with respect to ε and surely converges to τ 0 .

From the monotone convergence theorem (for mathematical expectation this time),

Eτ ε ↓ Eτ 0 as ε ↓ 0, because for all ε ≥ 0, Eτ ε ≤ T . Lemma 1.1. For ε ∈ R, |I T (u + ε) -I T (u) + ετ (u)| ≤ |ε|τ ε (u) Proof of the lemma. For ε > 0, {u + ε + X t < 0} ⊂ {u + X t < 0}, whence I T (u + ε) -I T (u) = T 0 (|u + ε + X t | -|u + X t |) 1 {u+Xt<0} dt - T 0 |u + ε + X t |1 {-ε<u+Xt<0} dt I T (u + ε) -I T (u) = -ε T 0 1 {u+Xt<0} dt - T 0 |u + ε + X t |1 {-ε<u+Xt<0} dt (1)
On the right side of (1), the left term corresponds to -ετ (u). The absolute value under the integral of the second term is less than ε on the support of the indicator function.

Hence |I T (u + ε) -I T (u) + ετ (u)| < T 0 ε1 {-ε<u+ct+St<0} dt,
which proves the lemma for ε > 0. A symmetrical procedure solves the case ε ≤ 0, which ends the proof of the lemma.

From lemma 1.1,

|EI T (u + ε) -EI T (u) + εEτ (u)| ≤ |ε|Eτ ε (u)
and

EI T (u + ε) = EI T (u) -εEτ (u) + εv(u, ε)
where

|v(u, ε)| ≤ Eτ ε (u) → Eτ 0 (u) = 0
as ε → 0, which proves that f is dierentiable with respect to u and that for u ∈ R,

f (u) = -Eτ (u).
Corollary 1. Using notations of theorem 1, let X t = ct -S t , where S t is a jump process such that, almost surely, S t has a nite number of nonnegative jumps in every nite interval, and that X t has a positive drift (X t → +∞ a.s.). Then f dened by

f (u) = E(I T (u)) for u ∈ R is dierentiable on R, and for u ∈ R, f (u) = -Eτ (u). Proof. Only Eτ 0 (u) = T 0 1 {u+ct-St=0} dt = 0
has to be shown. R t = u + ct -S t is a process whose sample paths are almost surely increasing between two consecutive jump instants. The number of jumps is almost surely nite on the time interval [0, T ]. Between two times when the process is in 0, there must be at least one jump instant. This implies that the number of visits of 0 is almost surely nite (it is less than N T + 1 where N T is the number of jumps between 0 and T).

So Eτ 0 = 0 and the result comes from theorem 1.

Proposition 1. More generally, all processes for which the distribution of R t is diuse for all t ∈ R + -N satisfy the condition Eτ 0 = 0, if N is a null subset of R + for the Lebesgue measure. Theorem 1 is also veried for this wide class of processes.

Proof. For T ∈ R, from Fubini's theorem, integrable with respect to t. Let I g be the function from R + into the space of nonnegative random variables, and dened by

Eτ 0 (T ) ≤ E +∞ 0 1 {Rt=0} dt = +∞ 0 P (R t = 0)dt which provides the expected result. Theorem 2. Let g ∈ C 1 (R + , R + ) be a convex function, such that g(0) = 0. Let X t be a stochastic process such that, for u ∈ R, t → g(u + X t )1 {u+Xt<0} is almost surely
I g (u) =   T 0 1 {u+Xt<0} g(-(u + X t ))dt   for u ∈ R and let f (.) = EI g (.). For u ∈ R, if f (u) < +∞, EI g (u) < +∞ and Eτ 0 (u) = 0, then f is dierentiable at point u, and f (u) = -E   T 0 1 {u+Xt<0} g (|u + X t |)dt   Proof. Fix u ∈ R. For ε > 0, {u + ε + X t < 0} ⊂ {u + X t < 0}, whence I g (u + ε) -I g (u) ε = T 0 g(|u + ε + X t |) -g(|u + X t |) ε 1 {u+Xt<0} dt - T 0 g(|u + ε + X t |) ε 1 {-ε<u+Xt<0} dt For t ∈ [0, T ], g(-(u + ε + X t )) -g(-(u + X t )) -ε 1 {u+Xt<0} ↑ g (-(u + X t ))1 {u+Xt<0}
almost surely as ε ↓ 0, from the increase of the rates of increase of convex functions.

From the monotone convergence theorem, for t ∈ [0, T ],

E g(-(u + ε + X t )) -g(-(u + X t )) ε 1 {u+Xt<0} → -g (-(u + X t ))1 {u+Xt<0}
From Fubini's theorem,

E T 0 g(u + ε + X t ) -g(u + X t ) ε 1 {u+Xt<0} dt → -EI g (u)
as ε ↓ 0, where

I g (u) = T 0 g (-(u + X t ))1 {u+Xt<0} dt Hence |f (u + ε) -f (u) + εEI g (u) + εw(u, ε)| ≤ E T 0 g(-(u + ε + X t ))1 {-ε<u+Xt<0} dt
with w(u, ε) → 0 as ε ↓ 0, and

|f (u + ε) -f (u) + εEI g (u) + εw(u, ε)| ≤ εEτ ε (u)E sup t∈[0,ε] g (t) EI g (u + ε) = EI g (u) -εEτ (u) + ε(v(u, ε) -w(u, ε))
where

|v(u, ε)| ≤ KEτ ε (u) → KEτ 0 (u) = 0
as ε ↓ 0, which proves that f is right-dierentiable at point u and that

f r (u) = E T 0 g (-(u + X t ))1 {u+Xt<0} dt .
With similar reasoning, f is left-dierentiable and f l = f r , which ends the proof.

Dierentiation of the average time in the red and other generalizations

Recall that the time in the red is the time spent by the wealth process below 0, between time 0 and some xed time horizon T:

τ (u) = T 0 1 {u+Xt<0} dt.
T is rst supposed to be nite.

h (u) = - 1 c EN 0 (u),
where

N 0 (u) = Card ({t ∈ [0, T ], u + ct -S t = 0}).
Proof. Almost surely in ω, the number of jumps N (T ), and so N 0 (u), is nite.

Consider a sample path (X t (ω)) 0≤t≤T . Let R t = u + X t and denote by T i the i th jump instant. Dene

ε 0 (ω) = inf n≤N (T ),R Tn >0 R Tn
If N 0 (u) = 0, then dene

ε + = inf {u + X t , 0 ≤ t ≤ T } ∩ R + and ε -= -sup {u + X t , 0 ≤ t ≤ T } ∩ R -. ε -and ε + are almost surely positive. If |ε| < inf(ε + , ε -), then τ (u -ε) -τ (u) = 0,
and the following reasoning remains valid.

Otherwise, for 1 ≤ i ≤ N 0 (u), denote by t i the instant of the i th visit of R t in 0, and by t i the instant of the rst jump of R t after t i . The sample paths of the process R t are almost surely right-continuous, and the probability that R T = 0 is zero. So one may consider

ε 1 (ω) = min min 1≤i≤N 0 (u) c(t i -t i ), c(T -t N 0 (u) ) .
Then, for 0 < ε < min (ε 0 (ω), ε 1 (ω)),

{0 < u + ct -S t < ε} = N 0 (u) i=1 {]t i , t i + ε/c[} and so τ (u -ε) -τ (u) = T 0 1 {u-ε+ct-St<0} -1 {u-ε+ct-St<0} dt
Dierentiation of some functionals of risk processes

9 = T 0 1 {0≤u+ct-St<ε} dt = N 0 (u) k=1 ε c
Hence

τ (u -ε) -τ (u) ε → 1 c N 0 (u)
almost surely as ε → 0. Moreover, between two consecutive jumps of R t , the dierence between the two integrals is less than ε c in absolute value, whence

Ti+1 Ti 1 {0≤u+ct-St<ε} dt ≤ ε c .
So for ε > 0 small enough, with notations T N (T )+1 = T and T 0 = 0,

τ (u -ε) -τ (u) ε =   N (T ) i=0 1 ε Ti+1 Ti 1 {0≤u+ct-St<ε} dt   τ (u -ε) -τ (u) ε ≤   N (T ) i=0 1 ε ε c   ≤ 1 c (EN (T ) + 1)
Hence, from the dominated convergence theorem,

E τ (u -ε) -τ (u) ε → 1 c EN 0 (u)
as ε → 0. This proves that h is left-dierentiable on R + * , and that for u > 0,

h l (u) = - 1 c EN 0 (u).
With similar reasoning, h is right-dierentiable on R + * , and h l = h r . Hence h is dierentiable on R + * , and for u > 0, h (u) = -1 c EN 0 (u).

Remark 1. This provides the second-order dierentiate of EI T (.), which appears to be positive. EI T (.) is thus well strictly convex, which will be very important for the minimization in section 4.

Remark 2. This second-order dierentiate corresponds in the general case to the expectation of the local time L T (0) in 0 of the process u + X t up to time T:

L T (0) = lim ε↓0 1 2ε T 0 P (|u + X t | < ε)dt
For all ω and for ε > 0, the function

ω) → τ (u + ε, T ) -τ (u, T ) ε (ω)
is increasing with respect to T, and its limit expectation is equal to -1 c EN 0 (u, T ) as ε ↓ 0. From the monotone convergence theorem,

E lim ε↓0 τ (u + ε, ∞) -τ (u, ∞) ε = - 1 c EN 0 (u, ∞)
Remark 4. In innite time, the probability of ruin may be regarded as the expectation of the local time in 0 of the process (up to multiplication by a constant number).

Applications to the unidimensional case

Theorem 7. In the Poisson-Exponential(1/µ) case,

ψ(u) = (1 + µR)e -Ru , with R = 1 µ 1 -λµ c . Hence, for T = +∞, Eτ (u) = 1 + µR cR µRe -Ru and EI ∞ (u) = 1 + µR cR 2 µRe -Ru
Proof. This comes from a mere integration of the well-known formula for ψ(u), as the considered functions tend to 0 as u → +∞. Besides, as µ is the average claim

amount, R = µ -λ c and ρ = c-λµ c .
This method provides a way to get back the average total time in the red from the integration of the probability of ruin. Dos Reis [START_REF] Reis | How long is the surplus below zero? Insurance Math[END_REF] derived this result for Eτ (u, ∞)

by considering the number of ruins, and using the distributions of the length of the rst period in the red (until recovery), and of those of the following periods in the red, which had been derived by Gerber [START_REF] Gerber | Mathematical fun with ruin theory[END_REF].

Remark 5. Of course, it is possible to derive EI ∞ (u) for Gamma-distributed or phasetype-distributed claim amounts, as we know the probability of ruin in these cases. It is not developed here to keep it concise.

The with the Brownian case is also interesting. The local time of a standard Brownian motion W t in x is dened by

L t (x) = lim ε↓0 1 4ε t 0 1 {|Ws-x|<ε} ds
This provides a density for the occupation time Γ t (B) of a Borelian set B between 0 and t:

Γ t (B) = B 2L t (x)
2L t (0) = lim ε↓0 εD t (ε)
where D t (ε) is the number of downcrossings of the interval [0, ε] by the process W s between 0 and t .

This well-known theorem might be viewed as a limit case of theorem 3.

Multidimensional risk measures and optimal allocation

For a unidimensional risk process, one classical goal is to determine the minimal initial reserve u ε needed for the probability of ruin to be less than ε.

In a multidimensional framework, modelling the evolution of the dierent lines of business of an insurance company by a multirisk process

(u 1 +X 1 t , . . . , u n +X n t ) (u i +X i t
corresponds to the wealth of the i th line of business at time t), one could look for the global initial reserve u which ensures that the probability of ruin ψ satises

ψ(u 1 , . . . , u n ) ≤ ε
for the optimal allocation (u 1 , . . . , u n ) such that

ψ(u 1 , . . . , u n ) = inf v1+•••+vn=u ψ(v 1 , . . . , v n ) with ψ(u 1 , . . . , u n ) = P (∃i ∈ [1, n], ∃t > 0, u + X i t < 0).
Instead of the probability of crossing some barriers, it may more interesting to minimize the sum of the expected cost of the ruin for each line of business until time T, which may be represented by the expectation of the sum of integrals over time of the negative part of the process. In both cases, nding the global reserve needed requires to determine the optimal allocation. It has just been shown in the previous sections how to compute E(I T ) for one line of business, and the linearity of the expectation makes it possible to compute the sum for n dependent lines of business just as in the independent case. The structure of dependence between lines of business has no impact on this risk measure. This may be considered as a problem of optimal allocation of resource under budget constraints as in economics, the goal being to maximize the utility function given by the opposite of the sum of the E(I i T ).

Minimizing the penalty function

Recall that what has to be minimized is

A(u 1 , . . . , u n ) = n i=1 EI i T
where

EI i T = E T 0 |R i t |1 {R i t <0} dt with R i t = u i + X i t under the constraint u 1 + • • • + u n = u.
This does not depend on the dependence structure between the lines of business because of the linearity of the expectation. Denote v i (u i ) the dierentiate of EI i T with respect to u i . Using the Lagrange multipliers implies that if

(u 1 , . . . , u n ) minimizes A, then v k (u k ) = v 1 (u 1 ) for all 1 ≤ k ≤ n. Compute v i (u i ): v i (u i ) = E T 0 |R i t |1 {R i t <0} dt = -Eτ i = - T 0 P {R i t < 0} dt
where τ i represents the time spent in the red between 0 and T for line of business i.

The dierentiation theorem of the previous section justies the previous derivation.

The sum of the average times spent under 0 is a decreasing function of the u i . So A is strictly convex. On the compact space

S = {(u 1 , . . . , u n ) ∈ (R + ) n , u 1 + • • • + u n = u},
A a unique minimum. The optimal allocation is thus the following: there is a subset J ⊂ [1, n] such that for i / ∈ J, u i = 0, and for i, j ∈ J, Eτ i = Eτ j . The interpretation is quite intuitive: the safest lines of business do not require any reserve, and the other ones share the global reserve in order to get equal average times in the red for those lines of business.

Relaxing nonnegativity, on {u 1 + • • • + u n = u}, if (u 1 , . . . , u n ) is an extremum point for A, then for the n lines of business, the average times spent under 0 are equal to one another. If it is a minimum for the sum of the times spent below 0 for each line of business, then the average number of visits are proportional to the c i , and in innite time the ruin probabilities are in xed proportions. However the existence of a minimum is not guaranteed, because (u 1 , . . . , u n ) is no longer compact. It would be more tractable with the average time in the red or with minimization on the c i , because some factors penalize very negative u i in these problems.

Example

In the Poisson-Exponential( 1 µ ) case, recall that

EI u = 1 + µR cR 2 µRe -Ru .
Consider a two-line-of-business model, with the following parameters: A mere modication of the parameter µ 1 makes the optimal allocation strongly vary.

µ 2 = 1, c 1 = c 2 = 1, R 1 = 0.2, R 2 = 0.
When µ 1 = 1 (Figure 2), the optimal allocation is about (u 1 = 6.409748867, u 2 = 3.590251133). When µ 1 = 10 (Figure 3), the optimal allocation is (u 1 = 0, u 2 = 10).

In the second case, line of business 2 is much more risky than line of business 1, which justies the transfer of the whole global initial reserve to line of business 2. Here, the Lundberg exponents are the same in both cases, and heavier claims (with a smaller frequency) are more risky, and the line of business requires a higher initial reserve. For more properties or examples about optimal reserve allocation, the interested reader may consult [START_REF] Loisel | Finite-time ruin probabilities in the Markov-Modulated Multivariate Compound Poisson model with common shocks, and impact of dependence[END_REF]. where

Eτ i (u) = E T 0 1 {R i t <0} 1 { n j=1 R j t >0} dt .
B takes dependence into account, and the following proposition prescribes to do the same kind of reasoning:

Proposition 2. Let X t = ct -S t , where S t satises hypothesis (H1) of theorem 3.

Dene B by B(u 1 , . . . , u n ) = n j=1 E(τ j (u)) for u ∈ R n . B is dierentiable on (R * + ) n , and for u 1 , . . . , u n > 0,

∂B ∂u i = - 1 c i EN 0 i (u),
where

N 0 i (u) = Card {t ∈ [0, T ], R i t = 0 ∩ n j=1 R j t > 0 } .
It is also possible to dierentiate with respect to c instead of u.

Theorem 9. With notations of theorem 1, consider the case X t = ct -S t , where S t satises hypothesis (H1) of theorem 3, and dene f (c) = E(I(c)).

If all c, Eτ 0 (c) = 0, then f is dierentiable on R and for c ∈ R,

f (u) = - T 0 tP (R t < 0)dt.
It is interesting to look for the optimal allocation of the global premium c = c 1 +• • •+c n because if c i is small enough to make the safety loading negative, the process R i t tends to -∞. Quite often, optimizing with the c i will be easier than with the u i for this reason. These examples illustrate how these dierentiation results may be used.

The dierentiation developed here is quite general and may be useful to solve many problems involving multirisk models. For a discussion about multidimensional risk measures, optimal allocation procedures, and impact of dependence between lines of business, the interested reader may consult Loisel [START_REF] Loisel | Finite-time ruin probabilities in the Markov-Modulated Multivariate Compound Poisson model with common shocks, and impact of dependence[END_REF].

Figure 1 : 0 1

 10 Figure 1: The area in red represents IT (u) = T 0

  dx Paul Lévy's Brownian local time representation Theorem with downcrossings states that Theorem 8. (Paul Lévy)

4

 4 and u := 10.

Figure 2 :

 2 Figure 2: µ1 = 1: The two lines of business part of the global reserve

Figure 3 :

 3 Figure 3: µ1 = 10: Line of business 2 receives the whole global initial reserve

  Let X t = ct -S t , where S t is a jump process satisfying hypothesis (H1): almost surely, S t has a nite number of nonnegative jumps in every nite interval, and for each t, the distribution of S t is absolutely continuous. For example, S t might be a compound Poisson process with a continuous jump size distribution. Consider T < +∞ and dene h by h(u) = E(τ (u)) for u ∈ R. h is dierentiable on R + * , and for u > 0,

	8	Stéphane Loisel
	Theorem 3.

Stéphane Loisel

Theorem 4. Let g, h be two convex functions in 1 (R + , R + ), such that for x ≥ 0, g(x) ≥ g(0) and h(x) ≥ h(0). Let X t be a stochastic process such that t → g(u + X t ) and t → h(u + X t ) are almost surely integrable on [0, T ]. Let I + g be the function from R into the space of nonnegative random variables, and dened by

for u ≥ 0 and let f (.) = EI + g (.) -EI h (.).

If, for u ∈ R,

and if Eτ 0 (u) = 0, then f is dierentiable on R + * , and for u > 0,

Corollary 2. With the hypotheses of theorem 4, if besides X t = ct -S t , where S t satises hypothesis (H1) of theorem 3, then the dierentiate may be rewritten for u > 0 as:

Proof of corollary 2. Immediate from theorem 4, after replacing the last term in [START_REF] Gerber | Mathematical fun with ruin theory[END_REF] following the proof of theorem 3.

Proof of theorem 4. Decompose

where Ĩg is obtained from I g by changing X t into -X t . From linearity of expectation and of dierentiation, applying theorem 2 to g -g(0) with -X t and to h -h(0) with X t , and using theorem 3 end the proof of theorem 4.

Theorem 5. If besides the process X t converges almost surely to +∞ as t → +∞, and if for u ≥ 0, EI ∞ < +∞ and Eτ (u, ∞) < +∞, then theorem 1 remains valid with

Dierentiation of some functionals of risk processes 11 Proof. kind of reasoning as previously.

Remark 3. These conditions of integrability are fullled if the time spent below 0 for a single ruin is integrable.

Denote by ψ(u) the probability of ruin in innite time with initial reserve u.

Theorem 6. Theorem 3 remains valid with T = +∞ if besides X t has a positive drift and if τ (u) is integrable for all u > 0. Besides, in the compound Poisson case, for

(N 0 (u, n)) n≥0 is a nondecreasing sequence of random variables which surely converges to N 0 (u, +∞), possibly innite.

Let us show that EN 0 (u, +∞) < +∞.

Almost surely, u + X t → +∞ as t → +∞. Hence, almost surely, N 0 (u, +∞) < +∞ and is equal to the number of ruins:

Indeed, after each ruin, there is a recovery because X t converges almost surely to +∞ as t goes to +∞, and the number of jumps which lead exactly to the value 0 is nite almost surely. Besides, in the compound Poisson case, the number of ruins has the following distribution:

for n ≥ 1 and P (N 0 (u, ∞) = 0) = 1 -ψ(u). So N 0 (u, ∞) follows a zero-modied geometric distribution : P (N 0 (u, ∞) = 0) = 1 -ψ(u) and for n > 0, P (N 0 (u, ∞) = n|N 0 (u, ∞) > 0) = ψ(0) n-1 (1 -ψ(0))

Hence N 0 (u, ∞) is integrable and


Julie Leroy

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Icr

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The objective of this research is to understand the existing relations between the young person, the rap movement and the sociocultural environment of which it is a part.

The (well-known) ethnographic approach

Among the range of qualitative tools a researcher can use in Consumer Research, the ethnography embodies recognized and legitimate methodological perspective to embrace the cultures of contemporary consumption. If, in its premises, the ethnography was mainly practiced to seize the meaning of "primitive" cultures, as in the works of some of its founders: Malinowsky (1922), [START_REF] Mauss | Manuel d'ethnographie[END_REF] then [START_REF] Geertz | Interpretation of Cultures[END_REF], it spread out of its original frame to integrate contexts of research connected to organizations [START_REF] Van Maanen | The Fact of Fiction in Organizational Ethnography[END_REF] and marketing [START_REF] Arnould | Market-Oriented Ethnography: Interpretation Building and Marketing Strategy Formulation[END_REF]. The purpose of ethnography by ethnographers lies in the writing and the publication of a work where the researcher tells his adventure chronologically and mixes it with his own interpretation. It lets the reader the freedom to affix a new layer of interpretations. The purpose of ethnography by the researchers in marketing is quite different. It is to become integrated into a structure of wider report, where the frame of interpretation is somewhat closed. So, in this context, we notice that the dynamic aspect of the ethnography is diluted in a structure built around pillars such as: the theoretical frame, the methodology, the results and the interpretation. In such a structure of publication (whatever it is), the part dedicated to the observations and the results does not allow to bend over the dynamics of the studied phenomenon. Nevertheless, [START_REF] Miles | Le changement peut-il être géré ?[END_REF] underline that there is no guide, no format to arrange what they call the "qualitative report". Every researcher owes "to realize with a sense of art the structure of their report which adapts itself to the local and intellectual context of a particular study" (Miles and Huberman, 2003, p. 546). No matter what the research field is: Ethnology [START_REF] Geertz | Interpretation of Cultures[END_REF], Sociology [START_REF] Latour | Changer de société -Refaire de la sociologie[END_REF], Management Sciences in the broad sense [START_REF] Wacheux | Méthodes qualitatives de recherches en gestion[END_REF][START_REF] Hlady-Rispal | La méthode des cas : application à la recherche en gestion[END_REF] and Marketing [START_REF] Brown | Postmodern marketing 2: Telling tales[END_REF][START_REF] Borghini | Representing BtoB reality in case study research: challenges and new opportunities[END_REF], the authors agree to say that the work of writing an ethnography has not for objective to restore a photography of the reality, but rather to reconstitute a possible explanation of the reality [START_REF] Wacheux | Méthodes qualitatives de recherches en gestion[END_REF], putting into perspective its temporal dimension. And, analyzing a phenomenon from its chronology and from its history allows redrawing the path it took from its birth to the moment of the study or in some cases its death. The objective is here to consider breaking points, falls, which offer elements of understanding to the evolutionary dimension of the studied phenomenon. In this sense, we focus on previous works that can comprise a chronological analysis of data and even emphasize the need for it. However, these techniques focus more on how to comprehend the field rather than how to analyze data dealing with time. We then focus on the global approach here rather than its various components in order to keep the same level of discussion on each approach that deal with time we were able to identify.

Approaches dealing explicitly with time

The approaches we focus on in this work are: the contextualist approach [START_REF] Pettigrew | Contextualist Research: A natural way to link Theory and Practice[END_REF][START_REF] Baumont | Certification Qualité et gestion des ressources humaines : méthodologie[END_REF]Pichault, 2005) and the historical approach [START_REF] Savitt | Historical research in marketing[END_REF][START_REF] Firat | Historiography, scientif method, and exceptional historical events[END_REF][START_REF] Fullerton | HIstoricism: What it is, and what it means for consumer research[END_REF][START_REF] Jones | Historiographic paradigms in marketing[END_REF][START_REF] Smith | Historical Method in Consumer Research: Developing Causal Explanations of Change[END_REF][START_REF] Miles | Le changement peut-il être géré ?[END_REF]. The contextualist approach is meant to study an organizational process in its relation to past, present and future [START_REF] Baumont | Certification Qualité et gestion des ressources humaines : méthodologie[END_REF]. This approach is considered by its supporters as global frame for analyzing organizational phenomena in which different longitudinal study types can fit (Pichault, 2005). It is based on three pillars: contents compose the central phenomenon to understand, contexts designate the factors that may influence the contents, and processes that deal with the initiatives of the actors that will make the phenomenon evolve. For the context, [START_REF] Pettigrew | Contextualist Research: A natural way to link Theory and Practice[END_REF] distinguishes factors dealing with the organization itself, i.e. internal context, or factors dealing with the organization's environment, i.e. external context. In order to comprehend the phenomenon, [START_REF] Pettigrew | Contextualist Research: A natural way to link Theory and Practice[END_REF] recommends both a vertical and horizontal analysis:

• The vertical analysis will include: internal context (structure, culture, technologies, actors, management types), external context (environmental, economic, social and political factors).

Both possess an objective and subjective dimension. The latter being mainly driven by actors' perceptions, interactions and interpretations. • The horizontal analysis will include: the actions, interactions, representations and the different contexts over time. This level comprises the interconnected sequence of past, present and future phenomena. This contextualist approach presents the advantage of taking into account the impact of time on a phenomenon. The main setback of this approach is that it leaves the researcher with no other guidelines than the ones listed above. In this sense, the horizontal analysis needs to be further developed. [START_REF] Smith | Historical Method in Consumer Research: Developing Causal Explanations of Change[END_REF] define the historical and diachronic analysis as a method that "seeks to document and explain changes across time (p.597). The objective is here to reveal facts and events in order to observe their continuity or discontinuity. Our proposal is in line with the work of [START_REF] Smith | Historical Method in Consumer Research: Developing Causal Explanations of Change[END_REF], in particular in the understanding of time as being a social construct and, so, in the central role of the actors. It nevertheless differs slightly in the approach since we focus on different levels of analysis. Beyond observing the changes and the invariants of a phenomenon, our approach is also interested in the evolutions of the interactions between the various levels of observation. As for [START_REF] Miles | Le changement peut-il être géré ?[END_REF], they also explain that the chronological approach is an intrinsic part of the qualitative researcher and that there is a total symbiosis between the construction of the narrative tale and the construction of a relevant and accurate timeline. To achieve this objective, they propose a chronological protocol based on four successive operations: isolate the different event categories; preserve the sequential aspect; underline the importance of some event compared to those following; present the resulting timeline in a clear and faithful manner. Our proposal is in line with Miles and Huberman's view on qualitative work by putting into light the necessity of the timeline to understand a consumption phenomenon. Nevertheless, even though [START_REF] Miles | Le changement peut-il être géré ?[END_REF] point out the role of past events on present or future ones, they don't mention the possible links between the different levels of context. Furthermore, their main proposal is based on a census of events or a chronological matrix that remains merely descriptive.

On a paradigmatic level of understanding the historiographic approach, [START_REF] Jones | Historiographic paradigms in marketing[END_REF] distinguishes two historiographical paradigms: a scientific one and a traditional one (read a positivist one and an interpretive one [START_REF] Firat | Historiography, scientif method, and exceptional historical events[END_REF][START_REF] Fullerton | HIstoricism: What it is, and what it means for consumer research[END_REF]). The data collected by the first one are: demographic data, archives from governments, organizations and companies, census reports, lists of jobs, accounting books, schedules, advertisements. The data collected by the second are: personal and family archives, artifacts and photos, correspondences, personal diaries, non-published manuscripts. In this distinction, a border is marked between what is scientific and what is not. Our position goes beyond this arbitrary order by the use of the observation scale [START_REF] Desjeux | Scale of Observation : A Micro-Sociological Epistemology of Social Science Practice[END_REF]. We state that to understand marketing or consumption phenomena, it is necessary, when multiple data is collected, to classify all dates and durations, not by source of data, but by levels of observation on the phenomenon. So, to complete Jones' classification, we propose that the "scientific data" are observed at more macro and meso levels of observation, while "traditional data" are observed at more micro-social and individual levels. These two types of data then complete each other in a richer understanding of the phenomenon.

Getting dirty: Practicalities on time data from collection to analysis

In this section we address the practicalities of building a timeline. It starts from the data collection and ends with the data analysis that combines both a vertical and horizontal analysis where dates are considered as important as emerging themes.

From a vigilant data collection

The distinctive feature of the ethnographic method lies in the four principles enacted by [START_REF] Arnould | Market-Oriented Ethnography: Interpretation Building and Marketing Strategy Formulation[END_REF], among which the plurality of data collection techniques. The dynamic and chronological approach we develop here builds upon on this principle. In fact, in the two studies presented above, we used a dozen of different techniques to observe our respective phenomena (figure 1.).

Figure 1. Tools of collected data

Event though our diachronic analytical process is close to any ethnographic study, we put the emphasis on temporal data during the data collection phase. In the many occasions when the researcher meets with the field, we made sure to systematically pinpoint the dates, periods or eras embedded in observations, artifacts collection and most importantly narratives and interviews. The objective is related to the ability of tracing the story and the evolution of the phenomenon under study at different observation levels. These temporal data form a body of rich and diversified data that need a double analysis: thematic and diachronic.

To a thorough time data analysis

Our proposal, as it is now, for a chronological approach within the frame of ethnography is formalized in the analytical part of it. The objective is to suggest a tool allowing the researcher building the narrative of a phenomenon while better understanding its different layers of meanings. The analysis is articulated around to successive and complementary steps. They, however, differ slightly from the contextualist approach.

• A deconstructing vertical analysis: From the collected data, the aim is to distribute the units of meaning into categories [START_REF] Bardin | L'analyse de contenu[END_REF]. • A reconstructing horizontal analysis: The categories are the pieces of a new puzzle of meaning that are used to build a comprehensive narrative.

Deconstructing vertical analysis

The dates gathered during the data collection are distributed among four levels of observation [START_REF] Desjeux | Scale of Observation : A Micro-Sociological Epistemology of Social Science Practice[END_REF]:

• Macro-social level: e.g. the national reform of Chambers of Commerce taking place from X to X (several years) • Meso-social level: e.g. collaboration between three regional Chambers of commerce taking place from X to X (several years) • Micro-social level: e.g. transversality attempts between departments of each regional Chamber of commerce (meeting dates during several months) • Micro-individual level: e.g. effective start date, creation date of a document... Among others, the goal here is to identify the types of actors that are more significant on a specific level, where they make more sense. They can be distributed among tangible and intangible actors. The intangible actors are more easily distributed since they are usually gathered on macro or meso levels of observation. The physical actors represent a more delicate matter since a person can move from a level to another depending on the situation. Here again, the time matters! Beyond reconstructing the story and evolution of a phenomenon with the facts and influencing events that compose it [START_REF] Smith | Historical Method in Consumer Research: Developing Causal Explanations of Change[END_REF], our objective is to give actors' narratives and perceptions upon the evolution a more important role. The chronological reconstruction is carried out from the emic collected data. Then, on each level the researcher will distinguish:

• Facts/events : e.g. creation of Chamber of Commerce, product launch • Activities/actions : e.g. meetings, battles between dancing clans • Representations : e.g. the website to be launched can be interpreted as an informative site, a collaborative platform or a network of networks • Normes/strategies : e.g. a decision is made on publishing local business information vs. customer participation vs. adapting to consumer experiences • Affects/attitudes/beliefs : e.g. the website is an "empty shell", "i am proud of my achievement"

Level of observation

Reconstructing horizontal analysis

Two visions of diachronic approach could be presented. Engler (1989, p.200) argued that « the synchrony knows only one perspective, that of the speaking subjects, and all its method consists in collecting their testimony; to know to what extent a thing is a reality, it will be necessary and will be enough search to what extent it exists for the consciousness of the subjects. The diachronic linguistics, on the contrary, has to distinguish two perspectives, the one prospective, which follows the course of time, the other retrospective, which goes back up it. ». And Bibeau (1983, p.95) added that « the diachronic analysis consists in studying the evolution in time of a language on relatively long durations so that we can perceive the changes which occur, either by accommodations to the social changes, or by the influence of the other languages ».

Since we focus on the social semantics embedded in a consumption phenomenon, we build the timeline on situations or episodes faced by individuals, groups and at a larger scale a society. It constitutes the skeleton of the chronology. The structure of the timeline is built according to four types of dates and periods.

First the dates and periods supplied by informants are used to redraw the story of the experience they live(d), from their point of view. This deals with a more individual level of understanding of the phenomenon. These dates and periods compose the spine of the tale. They are mostly based on the actors' representation of the phenomenon and its evolution. The key events they identify are based on their representation of the faced situation (Pichault, 2005), in that sense temporality is subjective and the goal is to reach an intersubjective temporality by multiplying the voices of the actors and by using other types of dates as described hereafter.Then, the dates of group interactions (as in observation, participant or not, sessions) are used, within the story, to describe how representations and practices mix. These specific moments help also better understand the links between one's speech and the according behaviors. They are the organs of the timeline. Finally, two other types of dates complete the body of the chronology. Any tangible creation (documents, homemade objects, etc.) has a date of creation or distribution to the public; they can be used as proofs or outcome of an experience. Last but not least, dates and periods that help better understand an era will also be introduced into the timeline. These can be found in newspapers, magazines, essays, contemporary novels, shows and TV shows, that can be collected during field presence and connected to the phenomenon. They help going upper in the observation scale in order to grasp the macro-level of observation.

The timeline is then edited in periods that contain the dates listed before. They are entitled so as to transcribe the meaning of the period within the process. Each period is constituted of situations. These situations are described in a dense way thanks to the various sources of data collected. The periods, which punctuate the chronology, are not representative of the number of events listed within the totality of the data, but they are representative of their influence in the outcome of the intrigue as seen by its protagonists. The timeline can be used as an intermediary tool to build the narrative and a reading tool crystallizing the narrative. In both cases, the final product is the thematic and chronological ethnographic narrative that encompasses the interpretation of the researcher along the numerous interpretations of the actors of the consumption phenomenon.

Conclusion

This communication suggests a dynamic method of analysis, allowing the researchers realizing an ethnography, going beyond a static reading to comprehend the evolution of a phenomenon. From both presented study cases, we can see that this tool maybe used in different contexts, but also with different objectives. Indeed, the diachronic analysis allows, either to focus on a level of analysis to decipher details, or to have an overview of a phenomenon and its evolution. So, the diachronic analysis joins in the recent evolutions to Consumer Culture Theory by taking into consideration the context in which the practices, the representations and the interactions are born and evolve. However, this proposal is not without raising any limits. First of all, the diachronic analysis of ethnographical data is, by definition, the result of the interpretation of the researcher. So, it is possible that some dates or periods did not draw enough attention or even are forgotten. To face this bias, the plurality of the data collection tools as well as an analysis inter-researchers is desirable. Furthermore, in spite of the triangulation, the interpretation, it is possible that a previous period, or a level of analysis may be under-investigated, being this the risk of any qualitative research. A last limit can be raised on the originality of the approach. This proposal can be seen as the formalization of practices already realized intuitively. However, it is to recall that if some works deal with dynamic approaches (chronologies, historic methods, etc.), they propose only methods of collection, while the diachronic approach appears as a method of analysis of data. To strengthen this analytical approach, we wish to continue on this diachronic track, by returning to the essence of the word, namely its linguistic coloring. To do so, it could be interesting to analyze the meaning of dates to bring an even finer understanding of the phenomena and the practices studied.


Alexandre Nicolas

Fluctuations in pedestrian evacuation times: Going one step beyond the exit capacity paradigm for bottlenecks

For safety reasons, it is important that the design of buildings and public facilities comply with the guidelines compiled in building codes. The latter are often premised on the concept of exit capacity, i.e., the mean pedestrian flow rate through a bottleneck (at congestion). Here, we argue that one should duly take into account the evacuation time fluctuations when devising these guidelines. This is particularly true when the narrowing is abrupt and the crowd may behave competitively. We suggest a simple way to assess the extent of (part of) these fluctuations on the basis of the statistics of time gaps between successive escapes through the considered bottleneck, which in practice could be garnered by analysing recordings of future real evacuations or, perhaps, realistic drills (in the limits of what is ethically possible). We briefly present a test of the proposed strategy using a cellular automaton model and confirm its validity under some conditions, but also disclose some of its limitations. In particular, it may severely underestimate fluctuations in the presence of strong correlations in the pedestrians' behaviours (while still performing better than only the mean capacity).

Introduction

In emergency situations (whether it be a fire outbreak, an earthquake, a terrorist attack, etc.), buildings and public facilities usually need to be evacuated quickly and safely. Building architectures should be designed accordingly. In particular, corridors and doorways on the egress pathway must be wide enough to allow the occupants to egress without excessive delay and to reduce the risk of clogging at the LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France e-mail: alexandre.nicolas@polytechnique.edu bottlenecks. How can this plain condition translate into a practical guideline? Building codes answer this question either by prescribing minimal requirements in terms of bottleneck widths or by setting the minimal standards (performances) that should be met with respect to the evacuation time.

To take a few examples, building codes in the United States (for instance in Florida [START_REF]Florida building code[END_REF]) prescribe a minimal clear width w = max {813, 5.1 N} mm for doorways, where N is the occupant load. Note that w is lowered to max {813, 3.8 N} mm in suitably equipped low-risk buildings. In France, emergency exit doorways and corridors in public buildings must typically be larger than 1 + N 100 × 0.6 m if 200 < N 500, where x ∈ N is the ceiling of x, or N 100 × 0.6 m if N > 500 [START_REF]portant approbation des dispositions générales du règlement de sécurité contre les risques d'incendie et de panique dans les établissements recevant du public (erp) -section 9, co 36[END_REF]. French railway stations must be designed in such a way that they can be evacuated in less than 10 minutes. To evaluate the evacuation time, a walking velocity v = 1.4 m • s -1 is assumed, along with a specific capacity J s = 1.67 m -1 • s -1 , i.e., a maximal flow rate of 1.67 people per second per meter of corridor width. These figures are reduced to v = 1.0 m • s -1 and J s = 1.0 m -1 • s -1 if it is a regional or national station [START_REF]portant approbation des dispositions générales du règlement de sécurité contre les risques d'incendie et de panique dans les établissements recevant du public (erp) -section 9, co 36[END_REF]. Finally, in the United Kingdom, the maximal evacuation time for sports stadiums is set to a value between 2.5 and 8 minutes, depending on the risks, and a specific capacity J s = 1.37 m -1 • s -1 is assumed for level walkways [START_REF]Football League Authority: Guide to Safety at Sports Grounds[END_REF].

These guidelines are thus premised on the concept of exit capacity, or in other words mean flow rate in saturated conditions. Not surprisingly, the values indicated in the above building codes are slightly lower, but comparable to the values measured in controlled experiments where participants were asked to walk through a bottleneck in normal conditions (for instance, J s = 1.85 m -1 • s -1 was reported in [START_REF] Kretz | Experimental study of pedestrian flow through a bottleneck[END_REF]. On the other hand, they are considerably lower than the values measured in controlled competitive conditions through narrow doors: In [START_REF] Pastor | Experimental proof of faster-is-slower in systems of frictional particles flowing through constrictions[END_REF], Zuriguel et al. measured J s ≈ 3.7 m -1 • s -1 . Varying the participants' (prescribed) eagerness to escape, my colleagues and I observed the whole range of values 1.4 J s 3.3 (in m -1 • s -1 ) [START_REF] Nicolas | Pedestrian flows through a narrow doorway: Effect of individual behaviours on the global flow and microscopic dynamics[END_REF]. The underestimates provided in building codes can be interpreted as safety margins, intended to absorb unforeseen delay. But is this enough to ensure that the evacuation time will always be within the chosen bounds?

Here, we show that the presence of significant fluctuations, of diverse origins, undermines this reasoning based on mean values (Section 2). We then propose a simple method to assess these fluctuations in Section 3 and test it in Section 4.

2 Beyond the mean exit capacity: Fluctuations

Importance of fluctuations

Since the dynamics of evacuations depend on many details that are out of control, significant fluctuations should be foreseen. They undermine any reasoning based exclusively on mean values. Indeed, in the presence of strong fluctuations, knowing that evacuations are quick enough on average does not tell you how often they will be excessively lengthy. Even if we focus only on the delays expected at bottlenecks (doorways, relatively narrow corridors, etc.), experiments involving mice that were forced to flee through a narrow orifice showed that realisations conducted in virtually identical conditions exhibit a significant dispersion, with standard deviations (std) that typically amount to 6% to 14% of the total evacuation times of about 90 mice, for a variety of settings [START_REF] Lin | An experimental study of the 'faster-is-slower' effect using mice under panic[END_REF][START_REF] Lin | An experimental study of the impact of an obstacle on the escape efficiency by using mice under high competition[END_REF] (the figure is closer to the upper bound when there are many realisations). Similar figures are reported for the entrance of 85 sheep into a barn through a narrow gate, with an std-over-mean ratio of around 15% for the total time [START_REF] Garcimartín | Flow and clogging of a sheep herd passing through a bottleneck[END_REF], and the importance of the dispersion of the evacuation times was underscored by the authors. Considering a crowd of about 90 participants walking through a narrow (69-cm-wide) door in a controlled experiment, the ratio is found to be around 7%, with moderately competitive participants as well as with highly competitive ones [START_REF] Garcimartín | Flow of pedestrians through narrow doors with different competitiveness[END_REF]. (Note that the given std-over-mean ratios are the results of my own calculations using the data of the cited papers.) Importantly, even though intermittence in the dynamics is favoured by the presence of abrupt narrowings, the fluctuations do not vanish when the constriction gets wider and they may be very considerable if the crowd behaves frantically. A striking example is the huge clog that occurred in the running of the bulls during the 2013 San Fermín in Pamplona, when the crowd running in front of the bulls pushed so hard on the human clog formed at the entrance of the arena that, despite its being a few meters wide, not even a handful of people could enter per second. This very large deviation from the mean flow rate is a sign of anomalously broad statistics.

As a consequence, it is of paramount interest to have an idea of the distribution of evacuation times, beyond its mean value. This distribution corresponds to an ensemble of realisations, for a fixed number of occupants N and a fixed geometry. Unfortunately, it is unrealistic to expect the collection of such statistics for any N and any geometry. In the following, we will propose a strategy to bypass this need. To this end, we will first disentangle the origins of the fluctuations.

Fig. 1 (Left Some guidelines or measurements for the specific capacity of doors and corridors (i.e., the flow rate per metre of cross section). (Right) Time line of a controlled evacuation [START_REF] Nicolas | Pedestrian flows through a narrow doorway: Effect of individual behaviours on the global flow and microscopic dynamics[END_REF] obtained by stitching pixel lines corresponding to the doorway in successive frames; the definition of the time gap ∆t is illustrated.

Origins of the fluctuations

Statistical fluctuations. As any complex system, pedestrian crowds feature an amount of disorder: The pedestrian's morphologies and their preferential velocities are different, their initial positions at the beginning of the evacuation are more or less random, etc. For these reasons, successive realisations (even conducted with identical crowds and in seemingly identical conditions) will not display exactly the same dynamics. The observed differences are statistical fluctuations due to the stochasticity of uncontrolled parameters and are also present in the flow of purely mechanical systems, such as grains discharging from a silo [START_REF] Lozano | Flow and clogging in a silo with an obstacle above the orifice[END_REF].

Extrinsic variations. Besides these statistical fluctuations, extrinsic variations are also expected in real evacuations. In particular, the composition of the crowd that might need to evacuate the building is not the same from day to day, nor will their eagerness or competitiveness to egress be the same. Turning up the last parameter, for instance, was shown to produce more intermittent (bursty) evacuation dynamics [START_REF] Pastor | Experimental proof of faster-is-slower in systems of frictional particles flowing through constrictions[END_REF][START_REF] Nicolas | Pedestrian flows through a narrow doorway: Effect of individual behaviours on the global flow and microscopic dynamics[END_REF].

A practical method to assess statistical fluctuations

Having disentangled the origins of the fluctuations, we now propose a practical way to predict the statistical fluctuations of the global evacuation time.

Distribution of time gaps between successive egresses

Consider a given escape zone. Let t i be the time of the i-th escape (out of N) in a realisation of the evacuation and let ∆t i be the time gap t it i-1 , where we have set t 0 = 0, as illustrated in Fig. 1. The premise of our approach is that it is possible to collect enough statistics about these time gaps to get a decent approximation of their distribution p(∆t) for a typical crowd composition. This requires the observation of a reasonable number of real evacuations, which may be achieved in the near future, owing to the increased monitoring of public facilities. Alternatively, one may choose to perform evacuation drills in conditions as realistic as is ethically possible.

Micro-macro relation

To proceed, we remark that the total evacuation time of the N occupants (i.e., here, the delay at the bottleneck) reads

T esc (N) = N ∑ i=1 ∆t i .
It can thus be regarded as a sum of N random variables drawn from the distribution p(∆t). If we overlook possible correlations between successive time gaps, the distribution of T esc (N) is given by the following micro-macro relation, based on a convolution product ( * ),

P N (T esc ) = p * N (T esc ) . (1)
In particular, provided that the 'microscopic' distribution p(∆t) has a finite mean ∆t and a finite standard deviation σ , the central limit theorem implies that, in the limit of large attendance N 1, P N (T esc ) follows a normal law of mean N∆t and of variance Nσ 2 . Thus, we have managed to assess the global distribution P N (T esc ), whose direct assessment would require a hopelessly large amount of data because of its dependence on N, on the basis of the more accessible p(∆t) [START_REF] Nicolas | Statistical fluctuations in pedestrian evacuation times and the effect of social contagion[END_REF].

Caveats and possible issues

Already at this stage, we ought to mention some caveats with respect to the micromacro relation. First, one may think that p(∆t) actually depends on N. But we believe that, granted that N is large enough (typically N > 100 for a narrow door [START_REF] Garcimartín | Flow of pedestrians through narrow doors with different competitiveness[END_REF]), this dependency can be overlooked (unless huge pressure builds up at the door, in which case pressure itself will cause a tragedy, regardless of the delay at the door). Indeed, in the controlled experiments we performed (see [START_REF] Nicolas | Pedestrian flows through a narrow doorway: Effect of individual behaviours on the global flow and microscopic dynamics[END_REF] for details), the flow rate was not found to vary substantially when there were fewer people left in the room. Secondly, contrary to the assumption of uncorrelated time gaps, we recently demonstrated that an alternation between short time gaps and longer ones is generally expected in bottleneck flows [START_REF] Nicolas | Origin of the correlations between exit times in pedestrian flows through a bottleneck[END_REF]. Nevertheless, these are short-time (anti)correlations and they should not very substantially affect the results (where long clogs play an important role). In any case, the problem can be remedied easily, by reasoning on the distribution of time lapses for a handful of successive escapes (e.g., ∑ 3 p=0 ∆t i+p ), instead of considering the individual time gap ∆t i .

Validity and limits of the micro-macro relation

In this section, we outline a validation test for the micro-macro relation relying on model pedestrian dynamics. More precisely, the model under consideration is a cellular automaton that we designed to reproduce the dynamics of competitive escape through a narrow door observed experimentally by Zuriguel et al. [START_REF] Pastor | Experimental proof of faster-is-slower in systems of frictional particles flowing through constrictions[END_REF], in particular the heavy-tailed, power-law-like distribution of time gaps, viz., p(∆t) ∼ ∆t -α for large ∆t, with α > 0.

Succinct description of the model

In the model, space is discretised into a regular rectangular grid, in which each cell hosts at most one agent. At each time step, all pedestrians start by selecting the site that they target among the adjacent cells (denoted k and including the present one).

A move from site i to site j has probability p i→ j ≡ e A j -A i x

/ ∑ i,k e A k -A i x
, where A k is the attractiveness of the site (how close it is to the exit) and the noise intensity x = 1 has been introduced to avoid the artefacts caused by strictly deterministic moves. If the target site is occupied or if other people have selected it as target site and are therefore competing for it, the agent simply waits. Otherwise, (s)he moves to the target site. Following this round of motion, some sites have been vacated. This allows other agents to move to their target cell. The round is iterated until all possibilities of motion have been exhausted. The limit of strong friction considered in the model is noteworthy: As soon as two or more agents are competing for a site, the conflict is sterile and nobody can move. Further details can be found in [START_REF] Nicolas | Statistical fluctuations in pedestrian evacuation times and the effect of social contagion[END_REF].

Simulations showed that this model is able to capture the exponential distributions of burst sizes (i.e., egresses in rapid succession). But adding one last ingredient was crucial in order to replicate the heavy tails in p(∆t). Indeed, some amount of disorder was required. This was achieved by introducing behaviours: Each agent (i) is endowed with a propensity to cooperate Π i ∈]0, 1[. At each time step, this propensity determines whether agent i cooperates (which occurs with probability Π i ) or not. Nothing changes if the agent behaves cooperatively. In the opposite case, the agent is impatient to move to another site, so (s)he undervalues the attractive-ness of the current site, viz., A(x i , y i ) -→ A(x i , y i ) + 1 2 ln Π i . With this additional ingredient, the model succeeded in describing a 'faster-is-slower' effect [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF] when the crowd becomes more impatient, but also in yielding power-law tails in p(∆t) for impatient crowds and narrow doors [START_REF] Nicolas | Statistical fluctuations in pedestrian evacuation times and the effect of social contagion[END_REF].

Validation of the micro-macro relation

For each distribution of behaviours Π i and each door width, a large number of evacuation realisations were simulated for an arbitrary number of occupants N. The resulting histogram of total evacuation times is plotted as cyan bars in the right box of Fig. 2. In parallel, the distribution of time gaps p(∆t) was computed from the simulation of an evacuation involving a very large crowd. On this basis, the global distribution of evacuation times predicted using the micro-macro relation is shown in the same figure. Clearly, there is excellent agreement between the prediction line and the histogram, which gives credence to the validity of the micro-macro relation.

Limits to the validity

However, the proposed relation was found to fail in some conditions, and sometimes severely so. This notably happened when a process of social contagion was incorporated into the model, whereby the neighbours of an impatient agent tended to behave more selfishly. At high contagion strengths, system-spanning correlations emerged and the crowd sometimes ended up being extremely impatient (thus evacuating more slowly in the model), whereas it remained in its initial, patient state during other realisations, thus evacuating faster. Such scenario splitting completely undermined the micro-macro relation.

More generally, we also expect extrinsic fluctuations, as defined in Section 2.2, to lead to a broader distribution of global evacuation times than the micro-macro prediction. Despite these limitations, by accounting for statistical fluctuations, the latter prediction is more informative regarding the probability of dangerously long evacuations than just the exit capacity.

Conclusion

In this contribution we have argued that estimates for the time spent at doors, gates or in corridors during an evacuation should not be based exclusively on the mean flow rate at the bottleneck under study, i.e., its capacity. Indeed, these times may fluctuate considerably, especially when the relative narrowness of the bottleneck and the competitiveness of the crowd favour highly intermittent dynamics. We have suggested a simple way to assess the extent of these fluctuations (more precisely, their statistical component) on the basis of the 'microscopic' statistics of time gaps between successive escapes, which could be collected in practice. Putting the proposed relation to the test with a cellular automaton model [START_REF] Nicolas | Statistical fluctuations in pedestrian evacuation times and the effect of social contagion[END_REF] has confirmed its validity under some conditions, but also pointed to its limitations: Fluctuations are underestimated when there are strong correlations in the pedestrians' behaviours.

Fig. 2

 2 Fig. 2 Illustration of the micro-macro relation, which takes as input the 'microscopic' statistics of time gaps ∆t at the door and predicts the global distribution of evacuation times T esc for the N occupants.

Acknowledgments -I would like to thank Sebastián Bouzat and Marcelo Kuperman, who co-authored the paper [11] on which this talk was largely based.


Didier Bernard
email: didier.bernard@univ-ag.fr

Adaté Tossa
email: otossa@yahoo.fr

Richard Emilion
email: richard.emilion@univ-orleans.fr

Statistical analysis of the wind speed at the top of La Soufrière volcano

Keywords: atmospheric parameters, La Soufrière, multiplicative cascades, turbulence, wind speed

We first analyze some atmospheric parameters measured in 2004 at the top of La Soufrière volcano: wind speed, pressure and temperature. Next, we study the stochastic process of the wind speed modulus which is of interest for prediction of pollution induced by the plume. We fit to the data various models: FARIMA, periodic trend plus multifractional Brownian noise, multifractal continuous cascades. The estimations show that the parameters are quite constant during some time intervals but that they evoluate during some transition phases in response to environment changes. Thus, we finally suggest, for future works, some settings that should be more appropriate: time series with regime changes and stochastic processes in random environment.

Introduction

Since several years, a strong fumarolic plume is emitted from the south crater of La soufrière volcano, in Guadeloupe. In 2004, in order to characterize gases, OP-FTIR measurements in that volcanic plume were performed, revealing presence of HCl, SO2, SiF4, CH4. Accurate estimations of gas traces were difficult to obtain due to spatial and temporal variability of the emissions, acidity of gases, volcano morphology, and weather conditions. For example, humidity largely contaminated the OP-FTIR spectra while high variations of wind intensity and wind direction disturbed the measurements in the plume. These results and observations can be found in [START_REF] Bernard | Remote and in situ measurements of aid gas release from La Soufrière volcano, Guadeloupe[END_REF]]. As temporal evolution and forecasting of gas traces is a matter of fundamental importance for the assessment of the volcanic risk and for public safety in the volcanic areas, it is of great interest to elaborate models that can predict atmospheric gase concentrations, taking in account the wind process. As a first step to this purpose, we report in the present work a statisticial analysis of measurements on various meteorological parameters, performed near the summit of the volcano. The paper is organized as follows. In Section 2, we shortly present the measurements context. Section 3 is concerned with elementary but useful statistics such as daily or monthly profiles. In Section 4 we perform a fast Fourier transform analysis and we represent the autocorellation function of the wind modulus process. In Section 5 we put in evidence a periodic trend and a high local variability of the noise process by fitting a multifractional Brownian motion (MFBM). Due to the difficulties of identifying a model for the Hurst function in the MFBM, we turn to, in Section 6, a model which has a deep interpretability in physics: the continuous cascades model with fractional stochastic integral drift and standard noise [Schmitt (2001)], [Schmitt (2003)], [Schmitt (2005)]. We also show why the estimations that we obtain suggest an extension of such a model. Conclusion and perspectives on future works are presented in the last section.

Material and methods

Instruments

Since 2000, a meteorological station has been installed by the "Observatoire Volcanologique et Sismologique de Guadeloupe" at the top of the volcano La Soufrière. The station is situated at Piton Sanner, 16 • 02.70'N, 61 • 39.76'W, h=1442 m on the the mount "Sanner". Horizontal wind speed and direction were measured by a wind monitor (Campbell scientific 05103-03) Accuracy : 0.3 m s -1 , 1 to 60 m s -1 ; ±1.0 m s -1 60 to 100 m s -1 ) at z = 2m. This type of anemometer was selected because its design offers improved resistance to corrosion from atmospheric pollutants (volcanic plume) and gusts. Temperature (HMP45C), relative humidity (HMP45C Campbell), barometric pressure (PTB 101B Campbell), radiation (pyranometer SP1110 Campbell) and pluviometry were also measured. Signals from the station were recorded by dataloggers (CR10X Campbell scientific). All the equipments are Campbell. A detailed physical description of these systems is given in "http://www.ovsg.univag.fr/meteo/public/SANNER".

Sampling

The dataset used in this analysis concerns three meteorological parameters, wind speed, pressure, wind direction, supplied by the 'Observatoire volcanologique et Sismologique de Guadeloupe and denoted S, P and D, respectively. The data were discretely sampled every minute but only averaged intervals of ten minutes are recorded. The experiment has provided a time series of 12 month length (from january 1st to december 31th 2004). The mean ten-minute wind speed (in m s -1 ) corresponds to the meteorological mean wind so that variations due to turbulence are filtered. Therefore, measurements made here are well adapted to synoptic and average scale phenomenon representation.

Elementary statistics

Summary statistics

The following table is concerned with some elementary yearly statistics of the three variables S, P and D. As it must be expected, the atmospheric pressure seems to be a stationary process with small variation, at the opposite of the wind speed and wind direction processes. The figure below represents the time series of the wind speed, the atmospheric pressure and the wind direction, respectively. It can be observed that wind speed is more chaotic than atmospheric pressure while wind direction has some random extreme jumps.

Distributions and occupation time

It seems that the connection between the distribution of the sampled values of a continous stochastic process and its local time is not very popular. First recall that the bar histograms as those plotted in Fig. 2 below are computed as follows. The min-max interval of the data values is splitted into regular bins and the surface of each bar is the frequency of the values belonging to the bin. As the measurements were sampled at regular time intervals, we see that this frequency is also a time frequency. Therefore, if the wind speed and pressure are considered as continuous stochastic processes observed during a time interval (t, T ), these bar histograms estimate a probability distribution function (pdf) L(t, T ), the occupation time density of these processes, also called local time in the probabilistic literature.

(S,P,D) occupation time densities

For 2004, the wind intensity varies from 0 to 28 m s -1 with a mean speed of 10.89 m s -1 and a standard deviation of 5.55 m s -1 . Its graphical representation is a bimodal asymetric histogram (Fig. 2). This corresponds to the well known property of the wind speed density to generally be a mixture of two Weibull distributions.

Recall that a peak of the density (for example at 4m s -1 and at 12m s -1 in april 2004) means that the speed is more frequent at the corresponding value during the monitoring interval. The atmospheric pressure varies from 849 to 862.5 mm Hg. Its representation seems to be quite a normal distribution. It is more difficult to characterize the wind direction which seems to be a gamma distribution.

Monthly profiles

Monthly averages of the wind speed, atmospheric pressure and wind direction are represented in Fig. 3. Monthly profiles wind speed vary between 14.70 m s -1 in June to 5.87 m s -1 in October. These profiles are very similar to those recorded and calculated for the synoptic flux in a climatic station of Météo-France in Guadeloupe [START_REF] Brevignon | Brevignon[END_REF]]. Only the order of magnitude is different: the intensity is multiplied by 1.6 at the summit. Monthly profiles could be separated in four parts:

• January to february: we observed a constant flux for wind speed and atmospheric pressure • March-June: the Azores anticyclone grows stronger and stronger while the ITCZ goes up to the north with an increase of the atmospheric pressure accompanied with a strengthening of the wind flux. • July to October: flux intensity decreases to reach a minima in october. A similar trend is also observed for the atmospheric pressure. • October-December: a steady increase of fluxes is observed.

The monthly evolution of the average pressure follows the same profile as the intensity of the wind. Concerning wind direction, it is about 110-115 south until July. From August a progressive rotation towards the south is observed. It reaches its maximum, about 140, in October and then decreases towards the values measured at the beginning of the year.

The following figures show the distributions of S, P and D during two extreme behaviours: June and October 2004. Observe that the probability that the speed is less than 10 m s -1 is small while it is large in October. Concerning the pressure, there is a sliding of the high pressures towards the weak ones with a variation of magnitude around 4 mb. The wind direction is mainly south-east in June and south to south-west in October.

Average day profile (V,P)

Since the sampling is done every 10 min, 6 data are collected per hour and 144 per day. To get a day profile, we have computed for each time t = 1, ...144, the mean and the standard deviation through the 366 days of year 2004. The following figures represent the evolution of the mean of the data computed at each time. Clearly, S and P are highly correlated. Similarities between these evolution curves yield some empirical rules such as the following ones which can be of interest for future measurement experiments. Between 0 and 10 a.m., wind speed varies around 11 m s -1 . From 10 a.m. to 3:00 p.m. it decreases to reach a minima of 9.5 m s -1 between 3:00 and 4:00 p.m.. Then it regularly increases to reach a maxima at 10 p.m.. Concerning the pressure we observe a well shaped "pressure wave" with an amplitude of 2 hPa and a 12 hour phase. We observe two maxima at 10 a.m. and 10 p.m. local time alternated by two minima at 4 a.m. and 4 p.m.. These results agree with published meteorological data [Dhonneur (1978)], [START_REF] Brevignon | Brevignon[END_REF]]. Concerning wind direction, it is constant from 0:00 to 9:00 a.m.. A rotation of about 10 towards the south is observed from 9:00 a.m. to 2:00 p.m.,then wind direction comes back to its initial direction at 6:00 p.m.. Fig. 6 represents standard deviations calculated at each time of the day-profile for the three variables S, P and D. The measure of the distibution dispersion is evaluated by the coefficient of variation. For S, it is 50%, for P it is less than 1% and for D it is about 25 %. Thus S is the most chaotic variable and P is the most stable one.

Profiles at a fixed hour

It is of interest to know how the data vary when they are observed at a fixed hour every day, say e.g. at 8 a.m. The following figures yield some empirical rules. On the 5th april 2004, at 8 a.m., the wind speed modulus has a first peak, then it decreases from 5th to 10th, and from 10th it increases up to reach a second peak on 12th. Next, the speed at 8 a.m. decreases from 12th to 15th and then increases up to the largest peak on 23th. It decreases down on 24th to slightly increase again. This profile is quite the same at any fixed hour during this month.

Fourier transform

Spectral analysis is the most popular tool for observing cyclical phenomena and highliting lead-lag relation within a time series. It is based on Fourier decomposition wich is a way of separating time series into different frequency components that contribute to or influence the time series. This separating procedure is important since dynamic processes operate on different frequencies. In our case, the fast Fourier transform is performed in order to get a periodogram. In this wind speed periodogram, we observed a first peak corresponding to a 12-hours cycle (2.309 e-5 Hz) and a second one corresponding to a daily cycle (1.155 e-5 Hz) . These two periodic peaks are observed at 0.5-day and 1-day. Concerning the atmospheric pressure, we observed two additional peaks corresponding to a 8-hours cycle (3.4688 e-5 Hz) and a 6-hours cycle (4.623 e-5 Hz).

(P,V) autocorrelation

The normalized auto-correlation function (NACF) is defined as

R(τ ) = X(t i )X(t i + τ ) X(t i ) 2 ,
where the time series X(t i ) is centered, τ denoting the delay time parameter. The decay of the NACF against the delay time τ from 0 is an indicator of the relaxation time scale of the system. For the wind speed, it is observed that we have to wait for a long time before flow renewal. The extinction time for the wind flow is about 3,000 minutes, that is 2 days and 2 hours. This specific extinction time and the mean value of the wind speed intensity yield an idea of the whirl size. For the atmospheric pressure, the extinction is softer. The observed time period is 720 minutes, that is a half-day. 5 Wind speed stochastic process

FARIMA model

The slow decay of the autocorrelation function plotted in section 4 shows a long-term memory behaviour of the wind intensity, confirming a well-known property of meteorological time series. Since [START_REF] Raftery | Space -time modeling with long memory dependence: assessing irland's wind power resource[END_REF]] and [Motanari et al (1996)], FARIMA and GARMA models are usually fit to such data. A process (X t ) t>0 is FARIMA(p, d, q) [Hosking (1981)] if:

φ(B)∆ d X t = θ(B)ǫ t ,
where d is a real number (-1/2 < d < 1/2), φ(B) and θ(B) are AR and MA polynomials respectively, and ǫ t is a white noise.

Its autocorrelation function ρ(h) verifies

ρ(h) ≈ c d h 2d-1 as h → ∞. If log(|ρ(h)|) ≡ α + βlog(h)
is a linear regression model estimation, then d can be estimated by d = β + 1 /2.

For our wind speed time series, we obtain α = 0.9069, and β = -0.2794, therefore d = 0.360.

Periodic trend plus noise model

We now consider the following model:

X t = T (t) + N t ,
where N is the noise process and T is a deterministic (one year period) periodic trend which can be expressed as a linear combination of sine and cosine functions:

T (t) = α 0 + m j=1 α j Cos 2πjt K + β j Sin 2πjt K , (1)
where α j and β j for j = 1, ..., m are real constants given by the least squares estimation method and defined as

α 0 = 1 K K n=1 X(n), α j = 2 K K n=1 X(n)Cos 2πjn K , β j = 2 K K n=1 X(n)Sin 2πjn K ,
the integer K denoting the number of data recorded during one year, here K = 52706.

The main problem is the estimation of m, the number of harmonics to be considered. When m is too large we get an overfit model so that we penalize large m by using the Akaike Information Criterion AIC [Akaike (1974)] or the Bayesian Information Criterion BIC [START_REF] Schwarz | Estimating the dimension of a model[END_REF]] defined respectively by

AIC(p) = Ln(σ 2 p ) + 2p K and BIC(p) = Ln(σ 2 p ) + p.Ln(K) K ,
where σ2 p = 1 K K t=1 e 2 tp , e 2 tp = (X(t) -T p (t)) 2 , and T p is the tendancy on p harmonics. The value of m is then the value p that minimizes the choosen criterion. Unfortunately, the above formulas show that the larger the number K is the higher p will be. In our case, this leads to very large value of m so we turned to the mean deviation criterion. We decided that a polynomial of type ( 1) is a good approximation of the wind speed process when its mean deviation is less than 10. The best solution in this way is obtained for m = 72. The noise process (N t ) t≥0 is defined as N t = X t -T (t). In terms of infinitesimal increments we have dX t = T (t)dt + dN t . Usually the noise process is a standard white noise that is the derivative in the sense of Ito calculus of a standard Brownian motion: dN t = dB t . However, while Brownian motion is memoryless (the increments are independent), the measurements show that the process of wind speed modulus has a long term memory behaviour. So, in a first approach we have estimated the noise by using a fractional Brownian motion. A fractional Brownian motion (fBm) (B H t ) t≥0 with (Hurst) parameter H, 0 ≤ H ≤ 1 is a zero-mean Gaussian process such that

Cov[B H t , B H u ] = |t| 2H + |u| 2H -|t -u| 2H 2 .
The parameter H can be estimated by

H K = 1 2 log 2 V K/2 V K + 1 ,
where V K is the generalized quadratic variation defined by

V K = K-2 p=0 B H p+2 K -2B H p+1 K + B H p K 2 .
Unfortunately the above estimations applied to the wind speed data show that this parameter is not constant as it depends on the time parameter t. Hence we have considered a multifractional Brownian motion (mfBm) B

H(t) t
, a richer model for which the Hurst parameter is a function of t. The estimation of the function H is achieved by using the formula

H ǫ,K (t) = 1 2 log 2 V ǫ,K/2 (t) V ǫ,K (t) + 1
for 0 ≤ t ≤ 1, noticing that the process is sampled at discrete time i = 1, 2, ....., K.

The process V ǫ,K (t) is the local generalized quadratic variation defined on a neighborhood

ν ǫ,K (t) = p ∈ Z, p K -t ≤ ǫ of t by V ǫ,K (t) = p∈ν ǫ,K (t) B H(t) p+2 K -2B H(t) p+1 K + B H(t) p K 2 .
In our study we have taken ǫ = 300/K.

The results reveal that the maximal value of H(t) does not exceed 0.5. This confirms the high variability of the noise series, since H(t) is an indicator of the smothness of the sample paths of a mBm. The more H(t) is closed to 1 the smoother the sample paths are. Futher H(t) is not a regular function since it presents several variations in short intervals. A close look at the results (figure 13) sugest that H(t) might be well fited by a stochastic process.

Continuous cascades and stochastic equation model

The results in the preceding section has put in evidence the local high variability of the wind process but due to the difficulties of identifying a model for the Hurst function H(t), we turned to a model which has a deep interpretability in physics: the continuous cascades model with fractional stochastic integral drift and standard noise [Schmitt (2001)] [ Schmitt (2003)] [ Schmitt (2005)].

Our motivation has come from the similarity between our measurements time series representation (Fig. 1) and the simulations in the papers of F.G. Smith [Schmitt (2003)] . The estimation of the parameters are derived from some nice computations of F.G. Schmitt [Schmitt (2003)] whose notations are kept to make comparisons easier. We will also show how our numerical results suggest an extension of this continuous cascade model. Discrete multiplicative cascades model of Yaglom [Yaglom (1966)] appeared after the celebrated pioneering works of Richardson, Kolmogorov and Obukhov on turbulence modelling. Yaglom model can be seen as a weighted random tree depending on an i.i.d. family of random variables (r.v.) W (p,i) : the root node is associated with W 1,1 and with a scale number L = l 0 λ n 1 , each node having λ 1 > 1 children nodes and all the branches having length n. Each of the λ p 1 edges i in layer p is associated with W p,i and with a scale number L/λ p 1 = l 0 λ n-p

1

. Hence each node is associated with the product of the W p,i 's which are in the path from the root to this node, and also to a scale number. In particular, each leaf node x is associated with a r.v.

ε(x) = n p=1 W p,x .
Similar constructions are done for random Polya trees [START_REF] Paddock | A randomised Polya tree models for nonparametric Bayesian inference[END_REF]] or in Kraft construction of a continuous random distribution [Kraft (1964)]. Continuous cascades are obtained by letting the depth n → +∞ while the total scale ratio λ = L/l 0 = λ n 1 is kept fixed (but large) so that λ 1 = λ 1/n → 1 + . When taking all the W 's log-normal, that is

W = exp( µ ln 2g o - µ 2 ln 2),
where g 0 is a standard Gaussian, we obtain a dissipation stochastic process:

ε λ (t) = λ -µ 2 exp(µ 1 2 t t+1-λ (t + 1 -u) -1/2 dB(u)),
and its generator,called singularity process:

γ λ (t) = ln ε λ (t) = - µ 2 ln λ + µ 1/2 t t+1-λ (t + 1 -u) -1/2 dB(u)
has a stochastic drift:

dγ λ (t) = - µ 1/2 2 t t+1-λ (t + 1 -u) -3/2 dB(u) dt+µ 1/2 dB(t)-λ -1/2 dB(t+1-λ).
For large scale ratios λ ≫ 1 we have

dγ λ (t) ≈ - µ 1/2 2 t t+1-λ (t + 1 -u) -3/2 dB(u) dt + µ 1/2 dB(t).
These processes are both stationary with long-range correlations so that under some standard hypothesis, the ergodic theorem ensures the existence of

< ε λ (.) >= lim T →+∞ 1 T T 0 ε(s)ds.
F.G. Schmitt [Schmitt (2003)] (pages 89-90) has computed the moments of the dissipation process

< ε q λ (.) >= λ -qµ/2 λ µq 2 /2 < ε λ (.) >= 1,
while the autocorrelation function of the singularity process satifies < γ λ (.)γ λ (. + τ ) >≈ A λ -µ ln τ under the assumption 1 ≪ τ ≪ λ.

Estimation of the continuous cascades model

Our one-year 52706 measurements of the wind speed modulus during 2004 will be represented by a vector denoted S[1..52706]. Although the wind process is not stationnary, we will assume that we can fit the above process γ λ (.) during a time interval of our study period. However as we need < ε λ (.) >= 1, the vector S has to be translated from a constant c determined by the equation:

1 =< exp(γ λ (.)) >=< exp(S + c)(.) > which gives c = -ln( lim n→+∞ 1 n n 1 exp(S[i])).
In our series, the limit is reached for n ≈ 15000. The estimation of the intermittency parameter µ can be obtained as follows:

< γ λ (.)γ λ (. + τ ) >≈ A λ -µ ln τ and < γ λ (.)γ λ (. + 3τ ) >≈ A λ -µ ln τ -µ ln(3) imply that µ ≈ (< γ λ (.)γ λ (. + τ ) > -< γ λ (.)γ λ (. + 3τ ) >) / ln(3), µ ≈ lim n→+∞ 1 n -1 n-1 1 γ[i]γ[i + 1] - 1 n -3 n-3 1 γ[i]γ[i + 3] / ln(3).
In our case we have taken 1 ≪ τ = 10 ′ = 600" since the logarithmic decrease is observed during 3τ = 30 ′ = 1800". The estimation of the parameter λ is achieved by observing that

< ε λ (t) 2 >=< exp(2γ λ (t)) > implies λ µ = lim n→+∞ 1 n n 1 exp(2γ[i]).
As µ is already estimated, we get an estimate of λ and that of the depth d = ln(λ)/ ln(2) of the continuous cascade model.

Numerical results on continuous cascades model

Using S[1..52706], we first plot the autocorellation of the γ process with a logarithmic scale on the x-axis for the parameter τ . As expected we observe a logarithmic decrease. The above estimations yield µ ≈ 0.3378 and λ ≈ 20, 000, 000 It is important to observe that during some periods the parameters µ and λ are quite constant but that they evoluate in response to environment change, putting in evidence some regime changes. Similar behaviour have been observed when taking windows of size 30000, 35000, 40000 and 45000.

Continuous mixtures and hierarchical models

The above estimations suggest that deeper models should be considered: time series with regime changes or, if the parameters are themselves considered random, hierarchical mixture models. It should be pointed out that the estimations of such models need a more accute sampling, we think of 10 or 20 measurements per second.

We make precise our idea on mixture models. Let M (µ, λ) denote the above cascades model. In order to take in account the environment changes, we assume that the parameters µ and λ are themselves stochastic processes µ = (µ t ) t≥0 , λ = (λ t ) t≥0 . This is connected to the refined notion of stochastic process in random environment. We think of the following hierarchical model (as usual ∼ indicates the distribution and | conditionality):

W indSpeed(t) |µ,λ ∼ M (µ t , λ t ) (µ, λ) ∼ D 0 ,
where the mixing distribution D 0 is the distribution of a pair of stochastic processes, that is, in our case, a distribution on a product of two sets of continuous functions. This model can even be complexified by using a nonparametric Bayesian approach via a continuous-time Ferguson -Dirichlet prior:

W indSpeed(t) |µ,λ ∼ M (µ t , λ t ) (µ, λ) |P ∼ P P ∼ Dirichlet(cD 0 ), where P denotes a random distribution drawn from a Dirichlet prior and c is a positive constant representing the confidence in that the mean distribution of P is D 0 . This model is a generalization to continuous time of some hierarchical models used in classification (see e.g. [Ishwaran (2002)]).

Conclusion

We have analyzed the behaviour of some atmospheric parameters at the top of La soufrière volcano giving some useful profiles and interpretations which can be of interest for future experiments. We have specially put in evidence the complexity of the wind speed modelling by fitting various models: FARIMA, periodic trend plus multifractal noise, and a continuous cascades model with fractional stochastic integral drift, this last one being fit on some specific periods. The estimations suggest the use of more complex models such as time series with regime changes or stochastic processes in random environment, for example continuous time hierarchical mixture models with Dirichlet prior. This is a challenge for future works since a nice model for wind speed is a preliminary step for getting nice models for pollution prediction.

Fig. 1 .

 1 Fig. 1. Time series of (S,P,D), La Soufrière Volcano

Fig. 2 .

 2 Fig. 2. Densities of (S,P,D)

Fig. 3 .

 3 Fig. 3. Monthly average of (S,P,D)

Fig. 5 .

 5 Fig. 4. Densities of (S,P,D) in June 2004

Fig. 6 .Fig. 7 .

 67 Fig. 6. Average Day profile of (S,P,D) for 2004.

Fig. 8 .

 8 Fig. 8. Evolution of the wind speed at 8 a.m in April 2004

Fig. 9 .

 9 Fig. 9. Power spectrum density of (S,P,D)

Fig. 10 .

 10 Fig. 10. Autocorrelation function of (S, P, D)

Figure 11 Fig. 12 .

 1112 Fig. 11. Trend on 72 harmonics 5.3 Multifractional noise

Fig. 14 .

 14 Fig. 14. Autocorrelation function and logarithmic decrease that is d = 23 levels of cascades. Further, as the wind speed is not stationnary we have also considered sliding windows of size 25000 by considering S[1..25000], S[2..25001] . . . S[27706..52706]It is important to observe that during some periods the parameters µ and λ are quite constant but that they evoluate in response to environment change,

Fig. 15 .Fig. 16 .Fig. 17 .

 151617 Fig. 15. Evolution of µ[1 + n..25000 + n] for n = 0, 1, 2, ...

Table 1 :

 1

		Min Max Mean Median Std. dev.
	Wind velocity	0	27.81 10.89	11.39	5.55
	Pressure (mm.Hg) 849 862.2 857.5	857.5	1.676
	Direction (degrees) 21.6 342.4 118.2	107.6	36.23

Summary statistics of (S,P,D) 3.2 Time series (S, P, D)

Acknowledgments

This research was supported by the European Regional Development Funds (ERDF/FEDER), French government and the Guadeloupe Regional council. The authors would like to thank Dr. Beauducel from The observatoire Volcanologique et Sismique de Guadeloupe for the data set.