Isabelle Albert
email: isabelle.albert@paris.inra.fr

Jean-Baptiste Denis
email: baptiste.denis@jouy.inra.fr

Sophie Ancelet

Natalie Commeau

Clémence Rigaux

Dirichlet and multinomial distributions: properties and uses in Jags

Keywords:

Some properties of the Dirichlet and multinomial distributions are provided with a focus towards their use in Bayesian statistical approaches, particularly with the jags software.

Notations

p, q, r refers to probabilities. u, v, w refers to random counts. m, n, N, M refers to sample sizes. i, j refers to generic supscripts and h, k to their upper value. Di () , Be () , Ga () , M u () , Bi () , P o () will respectively denote the Dirichlet, Beta, Gamma, Multinomial, Binomial and Poisson probability distributions.

1 The Dirichlet distribution 1.1 Denition (p 1 , ..., p k ) is said to follow a Dirichlet distribution with parameters (a 1 , ..., a k ) (p 1 , ..., p k ) ∼ Di (a 1 , ..., a k ) if the following two conditions are satised:

1. p 1 + p 2 + ...p k = 1 and all p i are non negative.

[p

1 , p 2 , ..., p k-1 ] = Γ( k i=1 a i) k i=1 Γ(a i ) k-1 i=1 p a i -1 i 1 -k-1 i=1 p i a k -1
It is important to underline some points:

Due to the functional relationship between the k variables (summation to one), their joint probability distribution is degenerated. This is why the density is proposed on the rst k -1 variables, the last one being given as p k = 1 -k-1

1 p i .
But, there is still a complete symmetry 1 between the k couples (p i , a i ) and the density could have been dened on any of the k subsets of size k -1.

All a i are strictly positive values2 .

When k = 2, it is easy to see that p 1 ∼ Be (a 1 , a 2 )3 , this is why Dirichlet distributions are one of the possible generalisations of the family of Beta distributions. We will see later on that there are other links between these two distributions.

A helpful interpretation of this distribution is to see the p i variables as the length of successive segments of a rope of size unity, randomly cutted at k -1 points with dierent weights (a i ) for each. The most straightforward case being when all a i are equal to a: then all random segments have the same marginal distribution and greater is a less random they are.

When all a i = 1, the Dirichlet distribution reduces to the uniform distribution onto the simplex dened by i p i = 1;

p i > 0 into R k .
Possibly a super-parameterization of this distribution could be convenient for a better proximity with the multinomial

(a 1 , ..., a k ) ⇐⇒ (A, π 1 , ..., π k )
where A = i a i and π i = a i A .

Moments

Let denote A = k i=1 a i .

E (p i ) = a i A V (p i ) = a i (A -a i ) A 2 (1 + A) Cov (p i , p j ) = -a i a j A 2 (1 + A) ∀i = j
as a consequence

Cor (p i , p j ) = - a i a j (A -a i ) (A -a j ) ∀i = j and V (p) = 1 A (1 + A) diag (a) - 1 A aa (1)
as a consequence, one can check that V ( i p i ) = 0 as expected. Also when all a i are greater than one, the vector a i -1 A-k is the mode of the distribution. When r i are non negative integers, a general formula for the moment is given by:

E k i=1 (p i ) r i = k i=1 a [r i ] i A [ k i=1 r i]
where the bracketted power is a special notation for

p [n] = n-1 j=0 (p + j) = p (p + 1) (p + 2) ... (p + n -1)

Marginal distributions (and distribution of sums)

When h < k, if (p 1 , ..., p k ) ∼ Di (a 1 , ..., a k ) then (p 1 , ..., p h , (p h+1 + ... + p k )) ∼ Di (a 1 , ..., a h , (a h+1 + ... + a k ))
As a consequence:

Generally ((p 1 + ... + p h ) , (p h+1 + ... + p k )) ∼ Di ((a 1 + ... + a h ) , (a h+1 + ... + a k ))
(2) a Dirichlet of dimension two, and S h = p 1 + ... + p h follows a Beta distribution with the same parameters.

As a particular case: p i ∼ Be (a i , A -a i ).

Conditional distributions When

h < k, if (p 1 , ..., p k ) ∼ Di (a 1 , ..., a k ) then 1 h i=1 p i (p 1 , ..., p h ) | p h+1 , ..., p k ∼ Di (a 1 , ..., a h )
An important consequence of this formulae is that 1 h i=1 p i (p 1 , ..., p h ) is independent of (p h+1 , ..., p k ) since the distribution doesn't depend on its value. So we can also state that

1 h i=1 p i (p 1 , ..., p h ) ∼ Di (a 1 , ..., a h )
with the special case of the marginal distribution of the proportion of the sum of g < h components among the h:

g i=1 p i h i=1 p i ∼ Be ((a 1 + ... + a g ) , (a g+1 + ... + a h )) (3)
A specic property of this kind is sometimes expressed as the complete neutral property of a Dirichlet random vector. It states that whatever is j (p 1 , p 2 , ..., p j ) and

1 1 -(p 1 + ... + p j ) (p j+1 , ..., p k ) are independent.
Then p 1 , p 2 1-p 1 , p 3 1-p 1 -p 2 , ...,

p k-1
1-p 1 -p 2 ...-p k-2 are mutually independent random variables and

p j 1 -j-1 u=1 p u ∼ Be a j , j-1 u=1 a u .
Of course, any order of the k categories can be used for the construction of this sequence of independent variables.

1.5 Construction

1.5.1 With independent Gamma distributions Let λ i , i = 1, ..., k a series of independently distributed Ga (a i , b) variables 4 . Then (p 1 , ..., p k ) = 1 k i=1 λ i (λ 1 , ..., λ k ) ∼ Di (a 1 , ..., a k )
4 Here we use the parameterization of bugs, the second parameter is the rate, not the scale so that

E (λ i ) = ai b and V (λ i ) = ai b 2 .
Remind that a Ga (a, 1) distributed variable is no more than a 1 2 χ 2 2a . Note that the second parameter is a scaling parameter, so it can be set to any common values to all λs, in particular it can be imposed to be unity (b = 1) without any loss of generality.

1.5.2

With independent Beta distributions Let q i , i = 1, ..., k -1 a series of independently distributed Be (α i , β i ) distributions. Let us dene recursively from them the series of random variables

p 1 = q 1 p i = q i i-1 j=1 (1 -q j ) i = 2, ..., k -1 p k = k-1 j=1 (1 -q j ) then (p 1 , ..., p k ) ∼ Di (a 1 , ..., a k )
when for i = 1, ..., k -1 :

α i = a i β i = k u=i+1 a u .
That implies that the following equalities must be true:

β i = β i-1 + α i ∀i = 2, ..., k -1.

The multinomial distribution

In a Bayesian statistical framework, the Dirichlet distribution is often associated to multinomial data sets for the prior distribution 5 of the probability parameters, this is the reason why we will describe it in this section, in a very similar way.

Denition

(u 1 , ..., u k ) is said to follow a multinomial distribution with parameters (N, p 1 , ..., p k ) (u 1 , ..., u k ) ∼ M u (N, p 1 , ..., p k )
if the following conditions are satised:

1. p 1 + p 2 + ...p k = 1 and all p i are non negative.

2. u 1 + u 2 + ...u k = N and all u i are non negative integers.

3.

[u 1 , u 2 , ..., u k-1 ] = Γ(N +1) k i=1 Γ(u i +1) k-1 i=1 p u i i (p k ) N -k-1 1 u i
It is important to underline some points: 5 for conjugation properties, cf. 3.1

Due to the functional relationship between the k variables (summation to N , a known value), their joint probability distribution is degenerated. This is why the density is proposed on the rst k -1 variables, the last one being given as

u k = N -k-1 1 u i .
There is a complete symmetry between the k couples (u i , p i ) .

When P = 2, it is easy to see that u 1 ∼ Bi (N, p 1 ), this is why the multinomial distribution is a generalisation of the binomial distributions. We will see later on that there are other links between these two distributions.

Moments

The expectations are given by

E (u i ) = N p i
The variances are given by

V (u i ) = N p i (1 -p i )
The covariances are given by Cov (u i , u j ) = -N p i p j So the variance matrix reads

V (u) = N (diag (p) -pp )
and V ( i u i ) = 0 as it must be since i u i = N , a constant. Finally, the correlation is

Cor (u i , u j ) = - √ p i p j (1 -p i ) (1 -p j )
for i = j

Marginal distributions (and distributions of sums)

When h < k, if (u 1 , ..., u k ) ∼ M u (N, p 1 , ..., p k ) then (u 1 , ..., u h , (u h+1 + ... + u k )) ∼ M u (N, p 1 , ..., p h , (p h+1 + ... + p k ))
As a consequence:

Generally

((u 1 + ... + u h ) , (u h+1 + ... + u k )) ∼ M u (N, (p 1 + ... + p h ) , (p h+1 + ... + p k )) (4)
a multinomial of size two, and S h = u 1 + ... + u h follows a binomial distribution with the parameters (N, (p 1 + ... + p h )).

As a particular case: u i ∼ Bi (N, p i ).

Conditional distributions

When

h < k, if (u 1 , ..., u k ) ∼ M u (N, p 1 , ..., p k ) then ((u 1 , ..., u h ) | u h+1 , ..., u k ) ∼ M u N - k i=h+1 u i , p 1 h i=1 p i , ..., p h h i=1 p i (5)
An important consequence of this formulae is that (u 1 , ..., u h ) depends on (u h+1 , ..., u k ) only through their sum, (u h+1 + ... + u k ). From this, it is not dicult to see that when g < h:

g i=1 u i | h i=1 u i ∼ Bi h i=1 u i , g i=1 p i h i=1 p i (6)
It can also be shown that

u 1 , (u 2 | u 1 ) , (u 3 | u 1 + u 2 ) , ..., (u k-1 | u 1 + ... + u k-2
) are independent and follow binomial distributions:

u j | j-1 u=1 u u ∼ Bi N - j-1 u=1 u u , p j 1 -j-1 u=1 p u . ( 7
)
2.5 Construction

With independent Poisson distributions

Let w i , i = 1, ..., k a series of independently distributed P o (λ i ) variables, then

w 1 , ..., w k | k i=1 w i = N ∼ M u N, λ 1 k i=1 λ i , ..., λ k k i=1 λ i .
So the (u i ) can be seen as a set of Poisson variables constrained to sum to N and having parameters proportional to the probabilities p i .

Notice that this kind of condition (the sum be equal to a given value) is not aordable with bugs softwares.

With conditional Binomial distributions

From (7), let u i , i = 1, ..., k -1 a series of successive conditional random variables distributed as

u 1 ∼ Bi (N, p 1 ) u 2 | u 1 ∼ Bi N -u 1 , p 2 1 -p 1 ... u i | u 1 , u 2 , ..., u i-1 ∼ Bi N -u 1 -... -u i-1 , p i 1 -p 1 -... -p i-1 then (u 1 , ..., u k ) ∼ M u (N, p 1 , ..., p k ) .
Notice that, with a positive probability, N can be attained with less than k positive u i , so the last binomial can be of size zero. This limit case is perfectly handled in jags.

Table 1: Probability distributions associated to the dag of Figure 1.

(1.a)

(a) → (p) (p) ∼ Di (a) (1.a) ((p) , N ) → (u) (u) ∼ M u (N, p) (1.b) (a) → (λ) λ i ∼ Ga (a i , 1) (1.b) (λ) (p) p i = λ i λu (1.b) ((p) , N ) → (u) (u) ∼ M u (N, p) (1.c) (a) → (λ) λ i ∼ Ga (a i ) (1.c) (λ) (p) p i = λ i λu (1.c) ((λ) , N ) → (u) (u) ∼ M u N, λ i λu (1.d) (a) → (λ) λ i ∼ Ga (a i 1) (1.d) (λ) (p) p i = λ i λu (1.d) (λ) → (w) w i ∼ P o (λ i ) (1.d) (w) (u) (w) = (u) (1.d) (w) N N = w u 3 A global scheme 3.1 Standard results of conjugation Beta-Binomial p ∼ Be (a 1 , a 2 ) and (u | N, p) ∼ Bi (N, p) imply that (p | N, u) ∼ Be (a 1 + u, a 2 + N -u) .
Dirichlet-Multinomial (p 1 , ..., p k ) ∼ Di (a 1 , .., a k ) and (u 1 , ..., u k | N, p 1 , ..., p k ) ∼ M u (N, p 1 , ..., p k ) imply that (p 1 , ..., p k | N, u 1 , ..., u k ) ∼ Di (a 1 + u 1 , ..., a k + u k ) .

Gamma-Poisson λ ∼ Ga (a, b) and (w | λ) ∼ P o (λ) imply that

(λ | w) ∼ Ga (a + w, b + 1) .
These results emphasize the relationships between the probability distributions we have previously considered.

Global presentation

Indeed, we can embed all the previously described distributions in the same framework allowing a better view of the relationships; four ways to attain the joint distribution of ((p) , (u)) are given in Figure 1 and detailed in Table 1.

The construction 1.a is the direct construction while 1.d is associated to the Gamma-Poisson modelling conditionned with the total number (N) giving rise with simple derivations to the desired random variables.

More intricated situations

It can occur that the multinomial vector is partially observed. Let us consider the case of k = 3, and suppose that we only know D = (u 1 , u 2 + u 3 ) that is we are not able to distinguish between the last two categories. The point is to derive the posterior. The model is :

prior (p 1 , p 2 , p 3 ) ∼ Di (a 1 , a 2 , a 3 ) likelihood ((u 1 , u 2 + u 3 ) | (p 1 , p 2 , p 3 )) ∼ M u (N, p 1 , p 2 + p 3 )
Standard posterior computation provides

[(p 1 , p 2 , p 3 ) | (u 1 , N )] ∝ 3 i=1 p a i -1 i p u 1 1 (p 2 + p 3 ) N -u 1 = p u 1 +a 1 -1 1 N -u 1 i=0 (N -u 1 )! i! (N -u 1 -i)! p i+a 2 -1 2 p N -u 1 -i+a 3 -1 3 = N -u 1 i=0 (N -u 1 )! i! (N -u 1 -i)! p u 1 +a 1 -1 1 p i+a 2 -1 2 p N -u 1 -i+a 3 -1 3
The posterior is a mixture of Dirichlet distributions,

Di (u 1 + a 1 -1, i + a 2 -1, N -u 1 -i + a 3 -1), for i = 0, ..., N -u 1 with binomial coecients (N -u 1 )! i!(N -u 1 -i)! .
This could have been seen directly since it is a mixture of the completely observed cases associated to the partial observation, each being weighted with the probability of its occurrence:

[(p 1 , p 2 , p 3 ) | (u 1 , N )] = N -u 1 i=0 [(u 2 = i, u 3 = N -u 1 -i) |u 1 ] [(p 1 , p 2 , p 3 ) | (u 1 , u 2 = i, u 3 = N -u 1 -i)]
and

[(u 2 = i, u 3 = N -u 1 -i) |u 1 ] = (N -u 1 )! i! (N -u 1 -i)! .
4 Implementation with jags Here, we will consider how the above considerations could be used with jags to perform some statistical computations. The main R script used to obtain the presented results is provided in A. There are two series of calculations : simulations and inferences. For the sake of the number of les, inferences have been performed in the same run which constrains to duplicate the data to get independent inferences.

Simulating a multinomial vector

In bugs the multinomial distribution is available with dmulti and it can be also obtained with nested binomial (see 2.5.2) as examplied in the following model code.

Density of p2[1]

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0 4 8 N = 100000 Bandwidth = 0.001479

Density of p2[2]

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0 4 8 N = 100000 Bandwidth = 0.001734 Notice that no information is born by empty data set and that a natural solution is to exclude

Density of p2[3]

  model { # (1) direct simulation mult.dir ~dmulti(p[],N);

Figure 2 :

 2 Figure 2: Simulating a multinomial vector in two dierent ways. First column with Bi () and second column with M u ().

Figure 3 :Figure 4 :

 34 Figure3: Simulating a Dirichlet vector in three dierent ways. First column: with Be (); second column with Di () and third column with Ga ().

Figure 5 :

 5 Figure 5: Inferring the probability vector of multinomial data in two dierent ways.

  in 1:nbds) { YY2[ii,1] ~dbin(pp[1],nb2[ii]); for (jj in 2:(k-1)) { YY2[ii,jj] ~dbin(pp[jj],nb2[ii]-sum(YY2[ii,1:(jj-1)])); } } }Results are proposed in Figure5. Both ways produce the same inference.

Figure 6 :

 6 Figure 6: Inferring the probability vector of multinomial data when one of them is of size zero.

  the parameters p2 ~ddirch(a[]); lambda2 ~dunif(lin,lax); # likelihood of the data for (ii in 1:nbds) { nb2[ii] ~dpois(lambda2); } pp[1] <-p2[1]; for (i in 2:k) { pp[i] <-p2[i]/(1-sum(p2[1:(i-1)]));} # for (ii in 1:nbds) { YY2[ii,1] ~dbin(pp[1],nb2[ii]); for (jj in 2:(k-1)) { YY2[ii,jj] ~dbin(pp[jj],nb2[ii]-sum(YY2[ii,1:(jj-1)])); } } # }

  Figure 1: Dierent constructions of the joint density of ((p) , (u)).. Fixed values are in squared boxes, random variables in circles; parentheses indicate vectors; dashed arcs indicate deterministic links, solid arcs random ones. Relationships are provided in

	Table 1
	(a)

Indeed notice that 1 -k-1 i=1 p i is p k .

If not the Γ function are not dened.

and that p 2 ∼ Be (a 2 , a 1 ) plus p 1 + p 2 = 1.

If the sum of the gamma variables was known, we could multiply the p i by it, but it is not the case.

The binomial way requires some care. The obtained results are proposed in Figure 2: the two ways of simulation are not distinguishable.

Simulating a Dirichlet vector

In bugs the Dirichlet distribution is available with ddirch and it can be also obtained with percentages of independent Gamma (see 1.5.1) or nested beta (see 1.5.2) as examplied in the following model code.

Density of diri.bet[1]

The beta way is quite tricky to implement. The obtained results are proposed in Figure 3: the three ways gave very similar results.

Inferring from multinomial vectors with xed sizes

Using the dierent ways of simulating multinomial distributions, ones can try some inference about the probability parameter when some multinomial data have been observed. Here is the proposed bugs model when the size of the multinomial is considered as xed.

Results are graphically presented in Figure 4. The posterior distributions looks identical.

Inferring from multinomial vectors with random sizes

The same exercice can be made when the size of the multinomial data is supposed random (not been monitored by the experimenter). To do so, to comply bugs requirements, the sizes are duplicated to give direct likelihood to the multinomial vectors and to the sizes; this has no consequence on the statistical analysis as far as no stochastic dependence is introduced between the parameter of the two parts. We will distinguish two cases because the results are dierent.

4.4.1

Every multinomial has got a non null size

Here is the bugs model code.

them from the statistical analysis, making the dmult solution always available.

Inferring from a Dirichlet vector

Even if not a so frequent situation, we can check what occurs when the observations are at the level of the Dirichlet variables. This implies an additional level in the model to focus the inference on the a parameter. An example is given in the following bugs model code considering the three identied ways : (i) the direct ddirch use, (iii) the dbeta construction theoritically exact and (iii) a dgamma approximation. The dbeta construction implies the computation of q i , direct transformations of the observed p i (see 1.5.2) and the code in the Appendix for the details. The dgamma construction is only an approximation. There is no way for a transformation of the observations as done for the dbeta construction 6 . Here the parameterization of the gammas variables is dened such that their expectations and variances be equal to those of a Dirichlet vector, forgetting the existing covariance of (1). More precisely, with the notations of 1.1 and the denition of gamma distributions in Footnote 4 we use

for the ith probability. Higher will be the dimension k of the Dirichlet, better we can imagine will be the approximation.

The obtained results are displayed in Figure 7 , it can be seen that (i) as expected ddirch and dbeta ways are very similar and (ii) the dgamma approximation is quite good even if the dimension of the Dirichlet (k = 4) is not very high.

Density of a3[3]

0 100 200 300 400 500 0.000 0.003 N = 10000000 Bandwidth = 2.592

Density of a3[4]

A Main R script Here is the complete R script used for all calculations related to 4. The called les to describe the model were given in the text above: dirimul.mulsim.jam in 4.1, dirimul.dirsim.jam in 4.2, dirimul.mulinf1.jam in 4.3, dirimul.mulinf2.jam in 4.4.1, dirimul.mulinf3.jam in 4.4.2 and dirimul.dirinf1.jam in 4.5, # 12_06_06 12_06_07 12_06_20 12_06_21 12_07_18 library(rjags) # initialization nbsimu <-100000; nbburnin <-10000; #nbsimu <-1000; nbburnin <-100; files <-TRUE; fitext <-"dirimul.res.txt"; if (files) { sink(fitext); sink();} fifi <-function(nom) { res <-rep(NA,2); res[1] <-paste("dirimul",nom,"jam",sep="."); res[2] <-paste("dirimul",nom,"pdf",sep="."); res; } # Simulating a multinomial vector ".RNG.state" <-c(19900, 14957, 25769); set.seed(1234); nom <-"mulsim"; ff <-fifi(nom); p <-1:4/sum(1:4); k <-length(p); N <-12; tomonitor <-c("mult.dir","mult.bin"); constants <-list(p=p,k=k,N=N); modele <-jags.model(file=ff[1],data=constants); update(modele,10); resultats <-coda.samples(modele,tomonitor,n.iter=nbsimu); if (files) { sink(fitext,append=TRUE); cat("\n\n\n",nom,"\n\n"); } print(summary(resultats)); if (files) { sink();} if (files) { pdf(ff[2]);} par(mfcol=c(4,2)); densplot(resultats,show.obs=FALSE); if (files) { dev.off();} cat(nom,": <Enter> to continue\n"); if (!files) { scan();} # Simulating a Dirichlet vector ".RNG.state" <-c(19900, 14957, 25769); set.seed(1234); nom <-"dirsim"; ff <-fifi(nom); a <-1:4; k <-length(a); tomonitor <-c("diri.dir","diri.gam","diri.bet"); constants <-list(a=a,k=k); modele <-jags.model(file=ff[1],data=constants); update(modele,10); resultats <-coda.samples(modele,tomonitor,n.iter=nbsimu); if (files) { sink(fitext,append=TRUE); cat("\n\n\n",nom,"\n\n"); } print(summary(resultats)); if (files) { sink();} if (files) { pdf(ff[2]);} par(mfcol=c(4,3)); densplot(resultats,show.obs=FALSE); if (files) { dev.off();} cat(nom,": <Enter> to continue\n"); if (!files) { scan();} # Inferring multinomial vectors of fixed sizes ".RNG.state" <-c(19900, 14957, 25769); set.seed(1234); nom <-"mulinf1"; ff <-fifi(nom); a <-rep(1,4); k <-length(a); tomonitor <-c("p1","p2"); YY1 <-matrix(c(1,0,2,7, 70,2,1,27),byrow=TRUE,2,k); nbds <-nrow(YY1); nb <-apply(YY1,1,sum); YY2 <-YY1; if (files) { sink(fitext,append=TRUE); cat("\n\n\n",nom,"\n\n"); } print(YY1); if (files) { sink();} constants <-list(a=a,k=k,nbds=nbds,nb=nb, -c(19900, 14957, 25769); set.seed(1234); nom <-"mulinf2"; ff <-fifi(nom); a <-rep(1,4); k <-length(a); lin <-1; lax <-100; tomonitor <-c("p1","lambda1","p2","lambda2"); YY1 <-matrix(c(1,0,2,7, 0,1,0,0, 5,2,1,3),byrow=TRUE,3,k); nbds <-nrow(YY1); nb1 <-apply(YY1,1,sum); YY2 <-YY1; nb2 -c(19900, 14957, 25769); set.seed(1234); nom <-"mulinf3"; ff <-fifi(nom); a <-rep(1,4); k <-length(a); lin <-1; lax <-100; tomonitor <-c("p2","lambda2"); YY1 <-matrix(c(1,0,2,7, 0,0,0,0, 5,2,1,3),byrow=TRUE,3,k); nbds <-nrow(YY1); nb1 <-apply(YY1,1,sum); YY2 <-YY1; nb2 -c(19900, 14957, 25769); set.seed(1234); nom <-"dirinf1"; ff <-fifi(nom); p <-c(5,1,1,2); k <-length(p); lin <-1; lax <-1000; tomonitor <-c("a1","a2","a3"); p <-p/sum(p); p1 <-p2 <-p3 <-p; q2 <-p2[-4]; q2[2] <-q2[2]/(1-q2[1]); q2[3] <-q2[3]/(1-q2[1])/(1-q2[2]); constants <-list(k=k,lin=lin,lax=lax,p1=p1,q2=q2,p3=p3)


domain: sdv sde

text:
Frédéric Rees

Françoise Watteau

Jean-Louis Morel

Localization of heavy metals immobilized on specific organic and mineral parts of a wood-derived biochar

come

ABSTRACT FOR EGU 2013 (300-500 words)

Session SSS8.6/CL2.9 : Biochar for soil remediation and global warming mitigation Convener: Gabriel Gascó Co-Conveners: Ana Maria Tarquis , Paulo Pereira , Agustin Merino, Ana María Méndez Lázaro , Nick J. Ostle , Pan Genxing , Bruno Glaser , Jorge Paz-Ferreiro , Dimitrios Kalderis Biochar has attracted a great deal of attention in the last years and it has been proposed as a geoengineering approach to promote soil carbon sequestration and thus help to mitigate global warming. Biochar addition to tropical soils resulted in the build up of soil carbon pools and has created a type of anthropic soils (Terras Pretas) which higher carbon stocks and fertility levels than the surrounding ferralsols. However, there are still many uncertainties that prevent biochar widespread use in soils, as biochars prepared from different sources and in different conditions (temperature, etc.) will behave in a different manner when added to soils with contrasting properties. The main aim of this session is to bring together scientist that are trying to develop a better understanding on the benefits and limitations of using biochar as soil amendments. In this session we welcome studies both in Terra Pretas and in other types of temperate or tropical soils as well as well as studies including different scenarios in climate change. Emphasis will be put in the effects of biochar on pollutant fate, carbon sequestration, soil nutrient transformation and leaching and soil microorganisms. Also, studies about the effect of different pyrolysis conditions on biochar are welcomed.

OR Session SSS2.4 : Novel sorbent materials for environmental remediation

Convener: Ioannis D. Manariotis Co-Conveners: David Werner, Hrissi K. Karapanagioti Sorbent materials have various environmental applications, i.e. gas storage, separation, and purification. Rapid progress in nanotechnology and a new focus on biomass-based instead of non-renewable starting materials have produced a wide range of novel engineered sorbents. The development and evaluation of novel sorbents requires a multidisciplinary approach encompassing environmental, nanotechnology, physical, analytical, and surface chemistry. The necessary evaluations encompass not only the efficiency of these materials to remove contaminants from surface waters and groundwater, industrial wastewater, polluted soils and sediments, etc., but also the potential side-effects of their environmental applications. Contributions examining the use of novel sorbents for environmental remediation are welcome. More specifically the contributions may be focused on:

• biosorbents: characterization; evaluation

Localization of heavy metals immobilized on specific organic and mineral parts of a wood-derived biochar

Author: Frédéric REES Co-authors: Françoise WATTEAU, Jean-Louis MOREL Biochar has been intensively investigated over recent years, not only as a promising carbon sequestration or fertilizing agent in soils but also as a possible new sorbent to remediate contaminated soils. A few studies have revealed its high potential for heavy metals immobilization depending on the nature of biochar and trace elements. The mechanisms behind this immobilization remain however unclear: some authors have hypothesized a high sorption capacity due to biochar large surface area while others have suggested that this immobilization is mainly due to soil pH increase. In particular, the distinction between heavy metals specific sorption in biochar pores and heavy metals precipitation in or outside biochar particles is often impossible to make while it is of primary importance to evaluate biochar ability to retain these pollutants on a long-time scale.

In order to evaluate the main heavy metal immobilization effects on a standard biochar and to identify the most successful biochar parts of the sample, we examined biochar particles after heavy metals immobilization in batch experiments designed to mimic real chemical processes in soils. A biochar derived from hard and soft wood and pyrolyzed at about 450°C was put in contact with relatively low concentrations of heavy metals (Pb, Cu, Cd, Zn, Ni) in an initially acidic Ca(NO 3 ) 2 solution. Following a one-week adsorption and a one-week desorption step, we recovered the biochar particles and observed them using scanning electron microscopy coupled to energy dispersive x-ray spectroscopy, focusing especially on the changes in mineral phases and the location of each of the retained heavy metals on biochar particles. We were able to distinguish different structures in the biochar samples which were linked to the degree of pyrolysis and the exact nature of the raw wood biomass. We detected the presence of concentrated metals zones (e.g. lead) in specific locations of the organic particles depending on the original plant tissues, and enlightened metal associations with newly-formed mineral phases such as calcite present on biochar surface. These observations provide new insights in the understanding of metal immobilization mechanisms on biochar such as precipitation and co-precipitation. Our findings also underline the need to consider the heterogeneity of biochar constitution for optimizing the remediation potential of biochar on contaminated sites.


come

Modelling phloem fluxes between sources and sinks


MHC

Relative protein synthesis and breakdown rates HAVE CHANGED

SINGLE body protein

The amino acid composition of body tissues • Different tissues respond differently to a deficient supply:

CONCLUSIONS

• Different tissues respond differently to a deficient supply:

• Liver, distal small intestine and skin respond little

• Longissimus dorsi and rhomboideus and proximal small intestine respond much more 6 th International Met Seminar 2012 Bahia (Brazil)

• The longissimus dorsi muscle has a great plasticity:

• Weight and relative weight

The amino acid composition of different proteins (Pearson & Young, 1989)

	Context Objective Experimental design Results	Conclusions
				Actin	Collagen
		% Amino acid
		Lys	10.7	7.0	2.8
		Met	1.8	4.1	0.8
		Cys	0.9	1.4	-
		Thr	5.5	6.9	2.0	Change in the AA
		Trp Val	0.5 4.6	1.2 5.4	-2.4	content in tissues
	There is no a SINGLE body	Ile Leu Phe Phe Tyr	4.4 11.3 3.1 3.1 2.2	-7.6 3.1 3.1 4.0	1.1 2.4 1.1 1.1 0.2	Contribution of ≠ tissue
		His	2.0	2.0	0.5	proteins HAS CHANGED
		Arg	5.6	4.8	5.1
		Ser	5.6	5.9	4.3
		Gly	3.3	7.5	33.1
		Ala	8.0	8.0	10.6
		Asp	5.3	9.0	4.6
		Glu	13.3	11.3	7.1
		Pro	1.5	4.9	12.1
		Asn	3.8	-	-
		Gln	6.4	-	-
		Hyp	-	-	9.3
		Hyl	-	-	0.6

6 th International Met Seminar

2012 Bahia (Brazil)

6 th International Met Seminar 2012 Bahia (Brazil)

	Context Objective Experimental Design Results Conclusions
		Composition of the experimental diets
	Cereals Evaluate if (and how) a long-term and Week 1 Week 17 Diet -Diet + Diet -Diet +	20% and 16%
	moderate sulfur AA deficiency affects : NE (MJ/kg) 9.50 9.50 9.50 9.50	BELOW
	Peas	N x 6.25 (%) SID AA (%) Lys	19.2 19.3 10.7 10.7 1.10 1.10 0.50 0.50	requirements for Met and Met+Cys
		Met Met	0.24 0.38 0.11 0.15 0.24 0.38 0.11 0.15
	Meat quality and carcass traits TSAA 0.52 0.66 0.26 0.30

Soybean hulls Soybean meal 6 th International Met Seminar 2012 Bahia (Brazil)

	1.1	wk 1	wk 3
			wk x
	0.5

dLys % Time Context Objective Experimental Design Results Conclusions

	6 weeks of age
	(~14 kg)

DIET - •12 Piétrain × (LW×LD) barrows 90 % ad libitum DIET +

	Context Objective Experimental Design Results Conclusions
	Animal performance
			Diet + Diet -P -value
	Body weight (kg)
	Initial		12.3	12.1	0.66
	Final		90.7	79.1	0.01	-13%
	(~14 kg) Daily gain (g/d) Daily gain (g/d)		659 659	563 563	0.01 0.01	-15% -15%
	Daily feed intake (kg/d)	1.82	1.74 0.15
	17 weeks Gain to feed ratio (g/g) 0.36	0.32 0.00	-11%
	Longissimus dorsi	Rhomboideus	Liver
	Skin section
	Ileum	Distal jejunum	Proximal jejunum
	6

th International Met Seminar 2012 Bahia (Brazil)

	Context Objective Experimental Design Results Conclusions
	Relative weight of tissues
		Diet + Diet -P -value
	Final BW (kg)	90.7	79.1	0.01	-13%
	Tissue (g/kg BW)
	Longissimus dorsi muscle	23.7	20.5	0.06	-13%
	Rhomboideus muscle	0.34	0.29 0.09	-14%
	Liver Liver	17.5 17.5	19.7 19.7	0.23 0.23
	Abdominal fat	10.5	9.5	0.41
	Proximal jejunum	6.39	7.71	0.09	+21%
	Distal jejunum	7.58	7.60	0.99
	Ileum	7.78	8.46	0.38
	Skin (section)	0.77	0.78 0.68
	6

th International Met Seminar 2012 Bahia (Brazil)

	Context Objective Experimental Design Results Conclusions
	Protein composition of tissues
		Diet + Diet -P -value
	Protein (g/kg tissue)
	Longissimus dorsi muscle	215	202	0.03	-6%
	Rhomboideus muscle	180	171	0.00	-5%
	Liver	214	209	0.52
	Proximal jejunum Proximal jejunum	132 132	133 133	0.81 0.81
	Distal jejunum	124	125	0.86
	Ileum	115	115	0.99
	Skin (section)	261	262	0.84
	20% of change for longissimus dorsi muscle
	at 36% TSAA deficiency 19 days (Conde-Aguilera et al., 2010)

versus 6 th International Met Seminar 2012 Bahia (Brazil)

	Context Objective Experimental Design Results Conclusions
			Methionine content in protein of body tissues
								Diet +	Diet -
		2.50 3.00	NS	NS	P < 0.05	NS	NS	NS
	Met g/16 g N	2.00 0.50 1.00 1.50						Remethylation and
		0.00
			LM	RH	Liver	Jejunum	Jejunum	Ileum	Skin
						proximal	distal	section

th International Met Seminar 2012 Bahia (Brazil)

	Context Objective Experimental Design Results Conclusions Context Objective Experimental Design Results Conclusions
			Essential amino acid composition of longissimus Essential amino acid composition of proximal
	8.00		dorsi (LM) and rhomboideus (RM) muscle protein jejunum (I1), distal jejunum (I2) and ileum (I3)
	6.00 7.00	8.00 9.00 10.00	*	*			Diet + LM Diet + RH	Diet -RH Diet -LM	*	*	*	Diet + I3 Diet + I2 Diet + I1	Diet -I3 Diet -I2 Diet -I1
	5.00 0.00 1.00 2.00 3.00 I1 I2 I3 6 4.00 6	5.00 6.00 7.00 0.00 1.00 2.00 3.00 4.00 LM RM Lys	Lys Lys Lys Met	Met Cys * Cys	*	Cys Cys * * Thr Thr Thr	Thr * *	Transulfuration could be limited in intestines for protein synthesis * * * Val Ile Leu Phe Tyr His Val * Ile Leu Phe Tyr Leu * * * * * * * * Val Ile Leu Phe Tyr His * * * Val Leu Ile Phe Tyr His Val Phe Tyr

th International Met Seminar 2012 Bahia (Brazil) Diet + Diet -P -value Carcass traits Hot carcass weight (kg)

	Context Objective Experimental Design Results Conclusions Context Objective Experimental Design Results Conclusions Context Objective Experimental Design Results Conclusions
	Carcass traits and composition of carcass Meat quality traits of longissimus dorsi muscle Comparison between TSAA deficient supply studies
	Conde-Aguilera et al. 2010		Diet + Diet -P -value Present study
	Longissimus dorsi muscle (kg) 2.15 1.62 0.01	-25%
	Dressing (%) Backfat thickness (mm) pH u Drip loss (%) 19 days pH 1 TSAA deficient supply of 36%	V	69.8 76.9 12.8 5.63 5.36 0.88 59.9 75.7 0.34 11.5 0.23 5.51 5.60 0.23 119 days (17 weeks) 0.01 -14% 6.21 6.13 0.54 TSAA deficient supply of 16%
	Muscle depth (mm) Glycolytic potential (µmol/g) 163 44.5 E	43.3 195	0.74 0.02	+20%
	Lean meat content (%) Lean meat content (%) Carcass composition (%) Glycogen (µmol/g) Glycogen (µmol/g) Glucose+G6P (µmol/g)	R	61.4 61.4 40.3 40.5 60.4 60.4 40.3 40.5 6.48 11.6	0.14 0.14 0.97 0.97 <0.01	+80%
	Ham Lactate (µmol/g)		24.5 69.8 90.4 24.9	0.36 0.05	+30%
	Loin Colour CIE Shoulder Belly L* Backfat a* b*		27.5 25.0 14.1 56.0 56.9 26.6 25.3 13.4 5.46 8.46 9.44 0.41 0.27 0.30 0.55 0.27 5.99 0.29 6.78 7.37 0.42
	c*		10.8 12.0	0.40
	H°38.8 38.3	0.79

Final body weight 7% Final body weight 13% Relative weight of intestines 46% 6 th International Met Seminar 2012 Bahia (Brazil)

	Daily gain 20%	R	Daily gain 15%
		S
		U
		S

Relative weight of longissimus dorsi muscle 34% Relative weight of longissimus dorsi muscle 13% Relative weight of intestine 21% 119 days (17 weeks) 19 days Context Objective Experimental Design Results Conclusions Comparison between TSAA deficient supply studies Conde-Aguilera et al. 2010 Present study

	TSAA deficient supply of 36%	TSAA deficient supply of 16%
	V
	E
	R

Protein content of longissimus dorsi muscle 20% Protein content of longissimus dorsi muscle 6% 6 th International Met Seminar 2012 Bahia (Brazil)

	R
	S
	U
	S

Met content of longissimus dorsi 12% Met content of liver 6% Cys content of rhomboid muscle, distal jejunum and ileum 6% Cys content of tissues = longissimus dorsi muscle 20% longissimus dorsi muscle 6% Amino acid profile of tissues ≠ Amino acid profile of tissues

≠

Context Objective Experimental Design Results

Conclusions

• The

amino acid content in body protein is not constant • The Met content is preserved during a long-term and moderate Met deficiency

acid composition of proteins • Few meat quality and carcass traits are affected by a long-term and moderate Met deficiency 6 th International Met Seminar 2012 Bahia (Brazil)

• Protein content • Amino acid composition of proteins CONCLUSIONS Context Objective Experimental Design Results Conclusions • Amino

th International Met Seminar 2012 Bahia (Brazil)

th International Met Seminar 2012 Bahia (Brazil) Rhodimet Research Grant 2010-2012

Acknowledgments

Carlos COBO-ORTEGA, Yolande JAGUELIN, Anne PASQUIER, Nadine MEZIERE, Georges GUILLEMOIS, Jean-François ROUAUD, Patrick TOUANEL, Fabien GUERIN, Bernard CARRISSANT, Hervé DEMAY, Jaap van MILGEN


Abdeslam Kadrani

Mohamed Mahmoud

Ould Sidi

Bénédicte Quilot-Turion

Michel M Génard

Francoise F Lescourret

M M Ould-Sidi

Comparison of evolutionary and swarm intelligence-based approaches in the improvement of peach fruit quality

Keywords: Multi-objective optimization, genetic algorithm, particle swarm optimization, decision-making

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Modeling and simulation are increasingly used to tackle the complexity of agricultural production systems . Researchers have to propose innovative perspectives of evolution towards systems respectful of the environment and producing safe food while ensuring the economic viability of farms. They need to deal with strong genotype x environment x practices interactions to design best combinations of genetic resources and cultural practices adapted to specific agro-environmental conditions optimizing the trade-off between strongly antagonistic criteria. In this work, we deal with the case of brown rot which is one of the main diseases causing large economic losses for peach growers. The challenge is to build-up a tool to conceive innovative management strategies that optimize genotype x environment x practices interactions to limit fruit contamination by brown rot while keeping or improving fruit quality (sweetness and fruit fresh mass). Hence design of peach ideotypes for a given set of cultural practices and a specific environment is governed by a set of objective functions and constraints. Several methods have been proposed to deal with multi-objective optimization problems. Two types of approaches can be distinguished: the aggregative approaches and the Pareto dominance based approaches. In this paper, we investigated two major families of the Pareto dominance algorithms: the Multi-Objective Evolutionary Algorithms (MOEAs) and the Multi-Objective Particle Swarm Optimization (MOPSO). These two families are biology inspired and are population-based as use a set of solutions which evolve within the search space. In recent decades, a wide range of MOEAs has been developed. One of the most cited MOEAs is the Non-dominated Sorting Genetic Algorithm II (NSGA-II). This success story of NSGA-II could be attributed, as announced by Deb, to its low computational requirements, its elitist approach, and its simple constraint-handling method. For these reasons, NSGA-II has become the reference for researchers in the field of MOEAs. MOPSO are more recent algorithms which are based on biological metaphor. These algorithms are inspired and imitate the movement of birds, fishes, or others organisms in their research of food. Large number of MOPSO algorithms was proposed in the literature. The diversity maintenance mechanism of MOPSO while seeking global Pareto-optimal solutions is rather poor compared with other MOEAs such as NSGA II. This motivated researchers to introduce into MOPSO algorithms some procedures from the MOEAs such as elitism, diversity operators, mutation operators, constraint handling and crowding distance. Raquel and Naval have proposed an algorithm that called MOPSO-CD which incorporates the crowding distance computation and the constraints handling of NSGA-II. In this study, we choose to compare the performance of the NSGA-II with those of MOPSO-CD in the case of the improvement of peach fruit quality. Both algorithms NSGA-II and MOPSO-CD were interfaced with the 'Virtual Fruit' process-based model which describes quality traits of peach fruit during the growth stage. The performance of each algorithm is analyzed qualitatively by comparing the non-dominated Pareto front and the performance measures. The remainder of this paper is organized as follows. The modeling of the peach fruit quality with the problem formulation is discussed in Section 2. Section 3 presents a brief description of NSGA-II and MOPSO-CD. Section 4 provides the experimental results obtained by the two multi-objective algorithms and discusses their performance measures with a fuzzy logic based strategy for selection of a compromised solution.

Case study description and mathematical formulation

The brown rot is one of the main diseases causing large economic losses for peach growers. Today, no other alternative than chemical treatment is available so fungicide applications are generalized and occur till pre-harvest. Those practices are acceptable neither for environment nor for consumer health. We focus on the model-based design of sustainable peach ideotypes: combination of genetic material (genotypes) and cultural practices optimized for specific environment. We used the 'Virtual Fruit' a process-based model which simulates peach growth, to perform virtual experiments. In past work, six genetic parameters of the 'Virtual Fruit' were identified as the most important and genotype depending. These parameters are considered as the decision variables for our optimization problem and they are described as follows: The objective of our approach is to create virtual genotypes (combination of the six parameters) leading to fruits of good quality (large and sweet) with low skin density of cracks in order to limit sensitivity to brown rot. Therefore, three criteria simulated by the model were chosen: Fruit mass, fruit sweetness, and the density of cracks: i. Fruit fresh mass remains the only criterion of payment for the peach producers. This criterion should be maximized with the range of the required mass in the market:

ii. Fruit taste represents an increasing economic challenge in order to answer a global social demand regarding fresh product quality. The fruit sweetness is then the second criterion to be maximized under reasonable values:

iii. Finally, we also considered the density of cracks of the fruit skin. This criterion may largely influence fruit sensitivity to brown rot which is one of the main polycyclic diseases in stone fruit and causes large economical losses. has to be minimized respecting some thresholds:

The bounds of variation of the three criteria were set to 50 (g) and 300 (g) for the fruit mass , 4% and 20% for sweetness and 0 and 20% for skin crack density . The normalization should be performed in order to take into account the constraints on criteria. Thus, the transformed objective functions will be defined in [0, 1] as follows:

where . The genotype design problem can be stated as follows:

where and are the bounds of the optimization variables (see Table 1).

Multi-objective optimization approaches

3.1. Multi-objective Evolutionary Algorithm: NSGA II NSGA-II is one of the MOEAs, proposed by Deb et al. as a significant improvement to the original NSGA by using a more efficient ranking scheme and improved selection to capture the Pareto front. In NSGA-II, the selection process at various stages of the algorithm toward a uniformly spread-out Pareto optimal front is guided by assigning fitness to chromosomes based on domination and diversity. Domination is determined by ranking all chromosomes in the population, where chromosomes with higher rank are considered to have better fitness. Chromosomes with the same rank are compared based on their diversity which is defined based on a crowding measure for each chromosome. Chromosomes with larger values of crowding distance are preferred more to be selected for next generations. Interested readers may refer to Deb et al. for a detailed description of the algorithm. A brief step-by-step description of the NSGA-II pseudo-code is as follows:

1. Initialization: randomly initialize parameters of the population of size N (N individuals). This population is called parent population and noted P. the rank and the crowding distance. This population will be used as parent population P in the next generation. 9. Go to the step 3 and repeat until stopping criterion is satisfied (number of generations in our case)

3.2. Multi-objective Swarm Intelligence: MOPSO-CD MOPSO-CD is a variant of MOPSO that extends the PSO algorithm to handle MOPs by incorporating the mechanism of crowding distance computation into the algorithm of PSO specifically on global best selection and in the deletion method of an external archive of nondominated solutions. The diversity of non-dominated solutions in the external archive is maintained by using the mechanism of crowding distance together with a mutation operator which was proposed in MOPSO. Crowding distance computation, Global best selection and Mutation operation were incorporated to MOPSO-CD. The main algorithm consists of initialization of the population, evaluation and reiterating the search on swarm by combining PSO operators with Pareto-dominance criteria. The non-dominated solutions found are stored in an external repository (ERP), and are used to guide the search particles. Whenever this archive (external repository) is full, it will be truncated using crowding distance mechanism. At the end of the execution, all the particles stored in the external archive give us an approximation of the true Pareto-optimal front. The pseudo code of MOPSO-CD is given below. More details of this algorithm can be found in .

1

Results and discussion

The 'Virtual Fruit' process-based model is linked to multi-objective optimization approaches to design genotypes adapted to cultural practices. The model runs on daily basis and it is driven by its parameters, which are constant over time and independent of the environment. The simulations were performed from 87 days after bloom (DAB) to 150 DAB in the climate of Avignon in 2009, and the cultural scenarios chosen was the low crop (LC) load combined with well-irrigated (WI) regime. The low crop load contains 4 fruits per fruit-bearing stem and is defined in the initial state of the model. The NSGA II and MOPSO-CD are applied to the multi-objective problem presented above to generate the best combinations of the six parameters of the 'Virtual Fruit' model that maximize fruit mass and sweetness and minimize skin crack density. The optimization is performed at fruit maturity. The initial population chosen for the both algorithms is 400 and the number of the generations is set to 200. Each candidate solution consisting of six parameters of the 'Virtual Fruit' is represented by a particle in MOPSO-CD and a chromosome in NSGA-II. The population is initialized randomly using C function and set as the initial population for both algorithms. After several experiments with different simulation parameters of each algorithm, the parameters listed in Table 2 are found to provide the best performance.

MOPSO-CD NSGA-II

Cognitive and social acceleration=1 Crossover probability=0.9 Inertia weight=0.9 Distribution parameter (for crossover)=20 Mutation rate=0.5 Mutation probability=0.1

Archive size=400

Distribution parameter (for mutation)=20 Table 2

. Parameters of NSGA-II and MOPSO-CD algorithms

The two optimal Pareto fronts obtained in one of the 24 experiments using MOPSO-CD and NSGA-II algorithms are shown in Fig. 3 for qualitative comparison purpose. Both algorithms have generated large numbers of solutions and it can be seen that the Pareto optimal front is showing a nonlinear relationship between the three objectives. The solutions provided to the decision maker are reasonable in terms of the three fitness functions. To check the performance of the multi-objective algorithms, various performance measures for evaluating a set of non-dominated solutions have been proposed in the literature. For the purpose of this paper, we use two metrics, namely set coverage metric (SC) and spacing metric (SP) , to measure performance of NSGA-II and MOPSO-CD. The set coverage metric gives the relative convergence and domination of solutions between two non-dominated sets and. The metric calculates the proportion of solutions in , which are weakly dominated by solutions of. means that all solutions in are weakly dominated by, while represents the situation when none of the solutions in are weakly dominated by. It may be noted that is not necessarily equal to . If this means that the set has better solutions than the set. Table 3 shows the statistical summary of different performance metrics for both MOPSO-CD and NSGA-II algorithms based on 24 independent runs. This table indicates that NSGA-II provides the best results with respect to the set coverage metric, since the average value of is higher than the average value of ( here is NSGA-II and is MOPSO-CD). Thus, in this case, NSGA-II is performing better than MOPSO-CD. NSGA-II also provides the best results of the SP metric in comparison with the MOPSO-CD. It can be observed that the mean value of the spacing metric for NSGA-II is lower than for MOPSO-CD. This indicates that the distribution of Pareto optimal solutions is closer equidistantly distributed in NSGA-II than in MOPSO-CD. Thus NSGA-II results in better performance.

Performance metrics

Set coverage metric

Spacing metric

Statistic

Upon having the Pareto optimal set of non-dominated solutions, it is preferred to get the best compromise solution for implementation. There are many ways to select the compromise solution. In this study for final decision-making a fuzzy set-based approach is adopted. The fuzziness is defined by membership functions which represent the degree of fuzziness in some fuzzy set using values in . means that the membership is entirely not in the set, means completely in the set, and a number in between and means partially in the set. Membership value for th objective of a solution in the final Pareto front is represented by a membership function defined by suggests how far the non-dominated solution is able to satisfy the objective. The accomplishment of each non-dominated solution with respect to all the non-dominated solutions can be obtained as follows:

The solution with maximum of value of is the compromised solution that can be chosen as the best solution or the one having the highest cardinal priority ranking.

Using the fuzzy set-based approach, the best compromised solution is found at points and on the Pareto fronts respectively generated by NSGA-II and MOPSO-CD (see Figure 1). Figure 2 shows the three fruit traits corresponding to the compromised solutions obtained using NSGA-II and MOPSO-CD.

Conclusion

In this paper, a multi-objective evolutionary algorithm (NSGA-II) and a multi-objective swarm intelligence approach are presented to design optimal ideotypes for sustainable fruit production systems. Both algorithms are successfully applied for generating efficient Pareto optimal solutions in the peach fruit quality improvement problem. The performance of NSGA-II is compared with that of MOPSO-CD and it is found that NSGA-II provides a wide spread of Pareto optimal with better convergence than MOPSO-CD. In addition to this, a fuzzy based mechanism is used to find out a compromised solution on the Pareto optimal front.

  2. Evaluation : calculate the values of criteria and constraints for each individual 3. Classification: rank the population using non-domination relation (best individuals in the first front). 4. Preserving the diversity within the population: calculate the crowding distance. The crowding distance computes the relative closeness of a given solution to other solutions on the same front. The maximal the crowding distance the better the solution. 5. Child population: use genetic operators i.e. selection, crossover, mutation to create a child population from the parent population (N children). Child population is noted Q. 6. Evaluation of child population 7. Recombined population: combine the parent and child population and calculate the crowding distance. The recombined population noted R is of size 2N. 8. New parent population: select N best individuals based on

Figure 1 .

 1 Figure 1. Optimal Pareto front obtained in one of 24 experiments using NSGA-II and MOPSO-CD. The spacing metric was proposed as a way of measuring the homogeneity of the front description by computing the mean distance between each element of the Pareto estimation. This indicates how far the generated non-dominated solutions are closer and equidistantly spaced. A null value of SP means that the non-dominated solutions found are equidistantly spaced.Table3shows the statistical summary of different performance metrics for both MOPSO-CD and NSGA-II algorithms based on 24 independent runs.

Figure 2 .

 2 Figure 2. Simulation of the three fruit traits with the compromised solutions obtained using NSGA-II and MOPSO-CD algorithms.

Table 1 .

 1 Lower and upper bounds of the optimization variables

	Variabl	Unit	Definition	Min	Max
	e			value	value
		dimensionles	Leaf structural mass/ leafy shoots structural	0.672 0.842
		s	mass
		degree day -1	Initial relative fruit growth rate	0.001 0.01
		dimensionles	Proportion of carbon as sucrose in phloemic	0.23	0.54
		s	sap
		g	Empirical coefficient relating stone dry mass	6.1	7.46
			and total fruit dry mass
		dimensionles	Parameter of the equation of the relative	1.134 1.386
		s	expansion rate of the cuticle surface area
		gcm -2 bar -1 h -1	Hydraulic conductance per unit of fruit surface 0.076	0.667
				9

Table 3 .

 3 Summary statistics of performance metrics for the results of 24 runs. Bold numbers indicate the best performing algorithm.

	Best	0.700625	0.00354167	0.02107373	0.03020891
	Worst	0.656875	0.000625	0.02692485	0.03020891
	Mean	0.6773	0.00175347	0.0239	0.02974282
		4858		7124
	Variance	0.0001769	1.0768E-06	2.39718E-06	3.67896E-06
	SD	0.01330038	0.00103771	0.00154828	0.00191806


Virginie Hugouvieux

Monique A V Axelos

Max Kolb

Micelle formation, gelation and adsorption of amphiphilic multiblock copolymers: A Monte Carlo study

come

Low hydrophobicity leads to spherical micelles and a weak, hydrophilically connected gel (top), high hydrophobicity to tubular micelles and a strong hydrophobically linked gel (bottom).

Amphiphilic multiblock copolymers are versatile macromolecules able to form a large variety of structures, both intra-or intermolecular. Applications range from stabilizing, emulsifying or foaming agents to rheology modifiers and drug vectors. In this study we focus on the structures and properties of copolymers with a large number of hydrophobic and hydrophilic blocks, both in solution or close to a surface. For this purpose, we use a lattice Monte Carlo simulation approach.

In solution, multiblock copolymers self-assemble because of the hydrophobic effect, which leads to an effective attraction between the hydrophobic blocks. At low concentration they form intramolecular structures ranging from pearl-necklaces of micelles to lamellar structures depending on the quality of the solvent and the ratio of hydrophobic to hydrophilic monomers (1) . At higher concentrations (semi-dilute solutions) intermolecular micelles and percolating networks are observed. The topologies of the networks depend on the hydrophobic content of the copolymers (2)

(see Figure).
In the presence of a solid hydrophobic surface, the hydrophobic blocks tend to adsorb and form isolated surface micelles with a well-defined size whenever bulk micelles are present. With increasing surface hydrophobicity the surface micelles arrange in a regular manner. Surface percolation is favoured by the multiblock nature of the copolymers. Long multiblock copolymers also lead to the presence of a secondary bulk micellar layer attached to the layer of surface micelles.