Mathieu Serrurier
email: mathieu.serrurier@gmail.com

Henri Prade

An informational distance for estimating the faithfulness of a possibility distribution, viewed as a family of probability distributions, with respect to data

Keywords:

An acknowledged interpretation of possibility distributions in quantitative possibility theory is in terms of families of probabilities that are upper and lower bounded by the associated possibility and necessity measures. This paper proposes an informational distance function for possibility distributions that agrees with the above-mentioned view of possibility theory in the continuous and in the discrete cases. Especially, we show that, given a set of data following a probability distribution, the optimal possibility distribution with respect to our informational distance is the distribution obtained as the result of the probability-possibility transformation that agrees with the maximal specificity principle. It is also shown that when the optimal distribution is not available due to representation bias, maximizing this possibilistic informational distance provides more faithful results than approximating the probability distribution and then applying the probability-possibility transformation. We show that maximizing the possibilistic informational distance is equivalent to minimizing the squared distance to the unknown optimal possibility distribution. Two advantages of the proposed informational distance function is that (i) it does not require the knowledge of the shape of the probability distribution that underlies the data, and (ii) it amounts to sum up the elementary terms corresponding to the informational distance between the considered possibility distribution and each piece of data. We detail the particular case of triangular and trapezoidal possibility distributions and we show that any unimodal unknown probability distribution can be faithfully upper approximated by a triangular distribution obtained by optimizing the possibilistic informational distance.

Introduction

Possibility theory, based on max-decomposable set-functions, associated with possibility distributions, may have either a qualitative or a quantitative understanding [START_REF] Dubois | Possibility theory: qualitative and quantitative aspects[END_REF], depending on the nature of the scale used for possibility degrees. Quantitative possibility theory corresponds to the case where the interval [0, 1] is used as a ratio scale. In qualitative possibility theory, only the ordering of the possibility values makes sense. In this paper, we deal with quantitative possibility theory. Quantitative possibility measures can be viewed as upper bound of probabilities. Then, a possibility distribution represents a family of probability distributions [START_REF] Dubois | When upper probabilities are possibility measures[END_REF]. The quantitative view was first suggested by Zadeh [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF] when he expressed the idea that what is probable must be possible. Following this intuition a probability-possibility transformation has been proposed [START_REF] Dubois | On possibility/probability transformations[END_REF]. This transformation associates a probability distribution with the maximally specific possibility distribution which is such that the possibility of any event is indeed an upper bound of the corresponding probability.

We call informational distance any function that evaluates the adequateness between a distribution of probability (or possibility) and a set of data. Thus, the likelihood is a well-known informational distance used for building a probability distribution from a set of data. Given a set of parametrized probability distributions and a set of data, the likelihood function is used for optimizing the choice of the parameters in order to determine the best suited distribution with respect to the data. However, this approach supposes that the shape of the distribution is a priori known. Otherwise, generic probability distributions, such as Gaussian mixtures, can be used. However, it requires to have a large amount of data at our disposal. Moreover, due to the constraints on the probability distributions, induced by their additive nature, there are no simple distribution that allows for a faithful description of any set of data. An informational distance function is also useful for comparing the relative adequateness of two distributions with respect to the same set of data.

There exist different kinds of methods for eliciting possibility distributions from data. For instance, some approaches build directly the possibility distribution on the basis of a proximity relation defined on the universe of the data [START_REF] Dubois | On data summarization with fuzzy sets[END_REF]. This is to be related to the idea of building fuzzy histograms based on fuzzy partitions, which has been fully investigated in [START_REF] Strauss | Rough histograms for robust statistics[END_REF][START_REF] Loquin | Fuzzy histograms and density estimation[END_REF]. Mauris proposes a method for constructing a possibility distribution when only very few data are available (even only one or two) based on probability inequalities [START_REF] Mauris | Inferring a possibility distribution from very few measurements, Soft Methods for Handling Variability and Imprecision[END_REF][START_REF] Mauris | Possibility distributions: A unified representation of usual direct-probability-based parameter estimation methods[END_REF]. This latter method is justified in the probabilistic view of possibility theory. In the same setting, Aregui and Denoeux studies the building of possibility distributions that bound a family of probability distributions on the basis of the confidence intervals of the parameters of these latter distributions [START_REF] Aregui | Constructing consonant belief functions from sample data using confidence sets of pignistic probabilities[END_REF]. This kind of approach has also been studied in the discrete case by using simultaneous confidence intervals [START_REF] Masson | Inferring a possibility distribution from empirical data[END_REF]. These methods, how different they are, have in common to build the distributions directly. In this paper, we investigate another road based on the optimization of an appropriate informational distance function. The proposed informational distance function is in agreement with the view underlying the probability-possibility transformation, as we shall see. This paper is a fully revised and expanded version of [START_REF] Serrurier | Maximum-likelihood principle for possibility distributions viewed as families of probabilities[END_REF], which is also the basis of the regression method at work in [START_REF] Serrurier | Imprecise regression based on possibilistic likelihood[END_REF]. More precisely, the present paper proposes an interpretation of the possibilitstic informational distance function and establishes different properties that formally justify it and supports its practical use.

The paper is organized as follows. Section 2 provides the necessary background about possibility distributions and their interpretations in terms of families of probabilities. Section 3 presents informational distance functions for probability distributions, and in particular a new non-logarithmic one, which can be used for approximating unbounded distributions by bounded ones. Sections 4 and 5 focus on the definition of the possibilistic counterpart of the new informational distance function, in the discrete and continuous settings respectively. The specific cases of triangular or trapezoidal distributions are also discussed in this section. We propose some examples of the construction of a possibility distribution from data using the possibilistic informational distance function in Section 6. Lastly, we emphasize the usability of this function in the conclusion.

Possibility theory: basic settings

In this background section, we first recall the basic definitions of possibility theory. We then present the view of a possibility measure as representing a family of probabilities, and the probability-possibility transformation.

Basic notions

Possibility theory, introduced by Zadeh [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF], was initially created in order to deal with imprecision and uncertainty due to incomplete information as the one provided by linguistic statements. This kind of epistemic uncertainty cannot be handled by a single probability distribution, especially when a priori knowledge about the nature of the probability distribution is lacking. A possibility distribution π is a mapping from to [0, 1] ( may be a discrete universe, i.e., = {C 1 , . . . , C q }, or a continuous one, i.e., = R). The value π(x) is called the possibility degree of the value x in . For any subset of , the possibility measure is defined as follows:

∀E ⊆ , (E) = sup x∈E π(x).
If it exists at least a value x ∈ for which we have π(x) = 1, the distribution is normalized. We can distinguish two extreme cases of information situations:

• complete knowledge: ∃x ∈ such as π(x) = 1 and ∀y ∈ , y = x, π(y) = 0 • total ignorance: ∀x ∈ , π(x) = 1.
The necessity is the dual measure of the possibility measure. We have:

∀E ⊆ , N(E) = 1 -(E).
Let us introduce the α-cuts of π . They are subsets of such that:

A α = {x ∈ , π(x) ≥ α}.
Then, it can be checked that, if = R and the distribution is continuous and normalized, we have ∀α ∈ [0, 1], (A α ) = 1 and N(A α ) = 1α.

Lastly, a possibility distribution π 1 is said to be more specific than a possibility distribution π 2 (π 1 π 2 ) if and only if ∀x ∈ , π 1 (x) ≤ π 2 (x)

A possibility distribution as a family of probability distributions

One view of possibility theory is to consider a possibility distribution as a family of probability distributions (see [START_REF] Dubois | Possibility theory and statistical reasoning[END_REF] for an overview). Thus, a possibility distribution π will represent the family of the probability distributions for which the measure of each subset of will be respectively lower and upper bounded by its necessity and its possibility measures. More formally, if P is the set of all probability distributions defined on , the family of probability distributions P π associated with π is defined as follows:

P π = {p ∈ P, ∀E ⊆ , N(E) ≤ P(E) ≤ (E)}. ( 1
)
where P is the probability measure associated with p. In this scope, the situation of total ignorance corresponds to the case where all probability distributions are possible. This type of ignorance cannot be described by a single probability distribution.

When

= R, this family of probability distributions can also be described in terms of confidence intervals. Given a probability distribution p, a confidence interval I α is a subset of such as P(I α ) = α. We define I * α as the smallest confidence interval with probability measure equal to α. In the following, we will only use the expression 'confidence interval' for referring to I * α . It can be observed that:

∀α, ∃β,

I * α = {x|p(x) ≥ β}.
Moreover, if the distribution has a finite number of modes, I * α is a finite union of intervals. When π is continuous, we have:

∀p ∈ P π , ∀I * α ∈ , I * α ⊆ A 1-α (2)
where A 1-α is the (1α)-cut of π . Indeed we have ∀E ⊆ , N(E) ≤ α ⇒ E ⊆ A 1-α and N I * α ≤ α (Eq. ( 1)). Thus, the α confidence interval of a probability distribution in P π is bounded by the (1α)-cut of π . In this scope, a possibility distribution can be viewed as an upper bound of the confidence interval of a family of probability distributions.

Probability-possibility transformation

According to this probabilistic interpretation, a method for transforming probability distributions into possibility distributions has been proposed in [START_REF] Dubois | On possibility/probability transformations[END_REF]. The idea behind this is to choose the most informative possibility measure that upper bounds the considered probability measure. This possibility measure corresponds to the most specific possibility distribution which bounds the distribution. We denote π sp the probability-possibility transformation of p. This distribution is defined in the following way: ∀π, p ∈ P π ⇒ π sp π.

(3) and is obtained with this equation:

∀x ∈ , π sp (x) = max α,x∈I α (1 -α). (4)
Let us first consider the discrete case ( = {C 1 , . . . , C q }) and a set of data X = {x 1 , . . . , x n } that are realizations of a random variable on . Let α 1 , . . . , α q be the frequency of the elements of X that belong respectively to {C 1 , . . . , C q }. Let us also assume that the frequencies of examples in class C i are put in decreasing order, i.e., α 1 ≥ • • • ≥ α q . In the following, given a possibility distribution π , we note π i the value π(x ∈ C i ) (or π(C i )). It has been shown in [3] that the transformation of p (which is derived from the frequencies) into a possibility distribution π sp (see Eq. ( 4)), is: ∀i ∈ {1, . . . , q}, π sp i = q j=i α j .

(5) This possibility distribution is one of the cumulated functions of p. It is worth noticing that it is the tightest one.

Example 1. For instance, we consider X that leads to the frequency α 1 = 0.5, α 2 = 0.3, α 3 = 0.2. We obtain π sp

1 = 0.5 + 0.3 + 0.2 = 1, π sp 2 = 0.3 + 0.2 = 0.5 and π sp 3 = 0.2.
In the case where is continuous ( = R) given p and its transformed distribution π sp we have:

A * 1-α = I * α
where A * 1-α is the (1α)-cut of π sp . Thus, if p has a finite number of modes, π sp is the possibility distribution for which each (1α)-cuts correspond to the confidence interval of p. When p is unimodal, the unique value x such that π sp (x) = 1 is the mode of p. Fig. 1 illustrates this transformation for a Gaussian distribution. The density function is in green, its possibility transformation in blue. The α-cuts of the possibility distribution corresponds to the (1α) confidence interval of the probability distribution.

Probabilistic informational distance functions

Informational distance function is commonly used for evaluating the adequateness of a probability distribution with respect to a set of data. In the following, the likelihood function is first recalled as a noticeable informational distance function. Then, we shall look for an informational distance of the form I(f , X) where f is a distribution (probabilistic or possibilistic) and X = {x 1 , . . . , x n } is a set of data, which is decomposable, as the likelihood function, i.e.,

I(f , X) = n i=1 I(f , x i ) (6)
Let us consider a set of realizations X = {x 1 , . . . , x n } of a random variable over a discrete universe = {C 1 , . . . , C q }. We note α 1 , . . . , α q the frequency of the elements of X that belong respectively to {C 1 , . . . , C q }. Given a probability distribution p on the discrete space = {C 1 , . . . , C q }, we define p 1 , . . . , p q the probability of belonging to each element of , i.e., p(x ∈ C i ) = p i . The values p i , . . . , p q entirely define p, and are then the parameters of p. The maximization of the informational distances is used for estimating the parameters of a probability distribution with respect to the data. In the continuous case, the shape of the distribution has to be known, and the parameters are obtained through an optimization procedure with respect to the informational distances function. In the discrete case, the parameters are p i , . . . , p q , and obey the constraint

q i p i = 1.
The likelihood is the most used informational distance function for probability distribution. Formally the likelihood coincides to a probability value. The logarithmic-based likelihood is defined as follows (under the strict constraint q i p i = 1):

I log (p|X) = - n i=1 log(p(x i ))
or, when considering frequency directly

I log (p|X) = - q i=1 α i log(p i ).
It is equivalent to compute the joint probability of the elements of x with respect to p. As an informational distance, the likelihood has a strong limitation, since it gives a very high weight to the error when probability is very low. This is especially true when is continuous. Since I log is not defined when p(x i ) = 0, an unbounded density cannot be approximated by a bounded one by optimization of I log . In order to overcome these limitations, we propose another informational distance function, named I surf , that is based on the distance between the probability distribution considered and the optimal one. We have:

I surf (p|X) =   n i=1 p(x i )   - 1 2 * q i=1 p 2 i
or, when considering frequency directly

I surf (p|X) =   q i=1 α i * p i   - 1 2 * q i=1 p 2 i .
Roughly speaking, I surf favors the probability distributions that share the maximum surface with the optimal one. Thus, when, is continuous, it allows the approximation of unbounded densities by bounded ones.

Proposition 1. Given a set of realization X = {x 1 , . . . , x n } of a random variable over a discrete universe = {C 1 , . . . , C q }, finding the distribution p that maximizes I surf is equivalent to find the distribution that minimizes the square of the distance to the optimal one p * .

Proof. Given X = {x 1 , . . . , x n }, the optimal p * is p * i = α i d(p * , p) 2 = d(α, p) 2 = q i=1 (α i -p i ) 2 = q i=1 α 2 i -2 *   q i=1 α i * p i - 1 2 q i=1 p 2 i   = C p -2 * I surf (α, p)
where

C p = q i=1 α 2 i is independent of p. Proposition 2. Given a set of data X = {x 1 , . . . , x n } belonging to a discrete universe = {C 1 , . . . , C q }, we have argmax p∈P (I log (p|X)) = argmax p∈P (I surf (p|X)).
Proof. Let p log = argmax p∈P (I log (p|X)) be the optimal probability distribution given X. This distribution is such as the probability of an event C i is equal to the frequency of element of X in C i , i.e., p log (x

∈ C i ) = p i = α i .
We now look for the probability distribution p surf that maximizes I surf . We have:

∀i = 1 . . . q, δI surf (p|X) δp i = α i -p i thus ∀i = 1 . . . q, δI surf (p|X) δp i = 0 ⇔ p i = α i .
Since the derivative of I surf (p|X) with respect to p i (the parameters of p) does not depend on the other parameters p j , j = i, we obtain p surf (x

∈ C i ) = p i = α i . Thus p surf = p log .
This proposition proves that, given X, the probability distribution that maximizes I log is the same as the one that maximizes I surf . I surf can be extended to the case of = R in the following way:

I surf (p|X) =   n i=1 p(x i )   - 1 2 * R p(t) 2 dt.
Proposition 2 remains true when = R. Fig. 2 shows the triangular density that maximizes I surf given a set of 3000 values that are generated by a Gaussian distribution with mean equal to 0 and standard derivation equal to 1. This triangular probability distribution is parametrized by m, l and r which are respectively the mode, the left and the right spread of the probability distribution. We have used an optimization procedure in order to find the parameters that maximizes I surf . What we obtain is the triangular probability distribution that shares the maximum of surface with the histogram computed from the sample set. If we use I log , the support of the triangle would grow to infinity as the number of values increases.

Possibilistic informational distance: discrete case

In this section we show how to use I surf in order to define an informational distance function for possibility distributions that supports the interpretation of a possibility distribution in terms of a family of probability distributions. Roughly speaking, the idea is to have an informational distance function that is maximal for the result of the probability-possibility transformation of the distribution and that favors the approximations of the distribution that share the maximal surface with the optimal one (as for I surf in the probabilistic case). We first consider the case of a discrete universe, i.e., = {C 1 , . . . , C q }. In this case, it is expected that the possibility distribution π * that maximizes the informational distance is exactly π sp defined in Eq. ( 4). This possibility distribution is one of the cumulated functions of p * . What we want to obtain is a function that is maximized for a possibility distribution π having following properties:

• π is a cumulated function of p • ∀i, j, π i ≥ π j ⇔ p i ≥ p j .
In a first step, we will use I surf for estimating the possibility distribution π as the cumulated function that respects the ordering of the possibility degrees in π . Then, in the following we consider that the ordering of subsets C i and the frequencies α i reflect the possibility ordering, i.e., π 1 ≥ • • • ≥ π q . By considering π as a cumulated function, we suppose that, for all i, the pair (π i , 1π i ) is a binomial probability distribution for the sets q j=i C j and i-1 j=1 C j . By estimating this probability function by I surf we obtain:

I surf ((π i , 1 -π i )|X) =   q j=i α j   * π i +   i-1 j=1 α j   * (1 -π i ) - π 2 i + (1 -π i ) 2 2 = π i *   q j=i α j -   1 - q j=i α j     + i-1 j=1 α j - 2π 2 i + 1 -2π i 2 = 2 * π i *   q j=i α j   -π 2 i + 1 2 + i-1 j=1 α j = 2 * π i *   q j=i α j   -π 2 i + 3 2 - q j=i α j
We obtain an evaluation π as a cumulated function by summing this calculus for all the π i :

I surf (cumul(π)|X) = i I surf ((π i , 1 -π i )|X) = 2 * q i=1   π i *   q j=i α j     - q i=1 π 2 i + q i=1 3 2 - q i=1   q j=i α j   = 2 * q i=1   π i *   q j=i α j     - q i=1 π 2 i + 3 * q 2 - q i=1 α i * i (7)
Proposition 3. Given a set of realizations X = {x 1 , . . . , x n }, I surf (cumul(π)|X) is equal to the same value for any π that is a cumulated function of p * Proof. (sketch) By replacing π by a cumulated function of p * we obtain an expression that only depends on α i 's and that is independent of the ordering of the α i 's.

Proposition 4. Given a set of realizations X = {x 1 , . . . , x n } of a random variable over a discrete universe = {C 1 , . . . , C q } and given a fixed order for the π i 's finding the possibility distribution π that maximizes I surf (cumul(π)|X) is equivalent to find the distribution that minimizes the square of the distance to the cumulated functions of p * which have the α i 's and the π i 's in the same order.

Proof. Given X = {x 1 , . . . , x n }, the cumulated functions of p * which have the α i 's and the π i 's in the same order is

π ′ i = q j=i α j d(π ′ , π ) 2 = q i=1   q j=i α j -π i   2 = q i=1      q j=i α j   2 + π 2 i -2 *   q j=i α j   * π i    = q i=1      1 - i-1 j=1 α j   2 + π 2 i -2 *   q j=i α j   * π i    = q i=1   1 +   i-1 j=1 α j   2 - i-1 j=1 α j -2 *   q j=i α j   * π i    + π 2 i - i-1 j=1 α j = C c -I surf (cumul(π)|X)
where

C c = q i=1 1 + 3 2 + i-1 j=1 α j 2 -i-1
j=1 α j is independent of π . This establishes the proposition.

Then I surf (cumul(π)|X) is a good candidate for the possibilistic information distance since it is maximized by possibility distributions that are cumulated functions of p * , and it can be still considered as the squared distance between the considered distribution and a cumulated function of p * . However, Proposition 3 states that I surf (cumul(π)|X) has the same value for any cumulated function. Thus, maximizing I surf (cumul(π)|X) doesn't not guarantee that the α i 's and the π i 's are in the same order. In order to overcome this issue, we propose the following function:

I pos (π |X) = q i=1 π i *   q j=i α j   - 1 2 q i=1 π 2 i - q i=1 α i * i (8)
Indeed, we have:

I pos (π |X) = 1 2 * I surf (cumul(π)|X) - 1 2 q i=1 α i * i - 3 * q 4
I pos is nothing but I surf of the cumulated function with an additional term that favors the possibility distributions that respect the frequency ordering. This is established in the following proposition.

Proposition 5. Given a set of data X = {x 1 , . . . , x n } belonging to a discrete universe = {C 1 , . . . , C q }, the possibility distribution π * that maximizes the function I pos is the probability-possibility transformation of the optimal probability distribution p * (i.e., ∀i ∈ {1, . . . , q}, p * i = α i ), according to Eq. ( 5) (i.e., π * = π sp ).

Proof. We look for the probability distribution π * that maximizes I pos .

We have:

∀i = 1 . . . q, δI pos (π |X) δπ i = q j=i α j -π i thus ∀i = 1 . . . q, δI pos (π |X) δπ i = 0 ⇔ p i = q j=i α j .
Since the derivative of I pos (π |X) with respect to π i (the parameters of π ) does not depend on the other operator π j , j = i, we obtain π * i = p i = q j=i α j which corresponds to a cumulated distribution of the α i 's. Since the part

q i=1 α i * i is maximized when α 1 ≥ • • • ≥ α q , the
distribution π * corresponds exactly to Eq. ( 5) and we have π * = π sp . This proposition shows that I pos is an acceptable informational distance function for possibility distributions viewed as families of probabilities. This proposition and Proposition 4 provide an interpretation of the possibility informational distance: if the maximization algorithm used does not prevent to have a possibility distribution that respect the frequency ordering, the algorithm will favor the distribution that shares the maximum of area with the optimal one. As for I surf the informational distance depends on the surface shared between the considered possibility distribution and the optimal one.

If we only consider one piece of data x such that x ∈ C j we obtain :

I pos (π |x) = 1 n   j i=1 π i - 1 2 q i=1 π 2 i -j   (9)
It is worth noticing that, when optimal distributions can only be approximated, finding the best approximation with respect to I pos is not equivalent to finding the best probability approximation with respect to a probabilistic informational distance and then turning it into a possibility distribution.

Example 2. For instance, we consider X that leads to the frequency α 1 = 0.5, α 2 = 0.3, α 3 = 0.2. We require that p 3 = 0 and π 3 = 0. In this context, the optimal p with respect to I surf (I log is not applicable here) is p 1 = 0.6, p 2 = 0.4, p 3 = 0. The optimal π with respect to

I pos is π 1 = 1, π 2 = 0.5, π 3 = 0. The transformation π ′ of p is π ′ 1 = 1, π ′ 2 = 0.4, π ′ 3 = 0.
We ob- serve that π ′ is different than π and that π is a better approximation of the optimal possibility distribution given in Example 1. This result is fundamental since it illustrates that using a probabilistic informational distance and then the probabilitypossibility transformation is not an effective approach for constructing a possibility distribution from data. The maximization of I pos is more adapted in this scope. However, if the optimization constraints or the representation bias prevent us to have a possibility distribution that respect the frequency ordering, we may obtain the possibility distribution that shares the maximum of area with the cumulated distribution that corresponds to the ordering of the possibility degrees.

Example 3. We again consider X that leads to the frequency α 1 = 0.5, α 2 = 0.3, α 3 = 0.2. We now require that p 2 = 0 and π 2 = 0. In this context, the optimal p with respect to I surf (I log is not applicable here) is p 1 = 0.65, p 2 = 0, p 3 = 0.35. The optimal π with respect to I pos is π 1 = 1, π 2 = 0, π 3 = 0.5 which corresponds to the approximation of another distribution.

In this paper, the optimization of I pos is performed with a simulated annealing algorithm [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF]. It is interesting to remark that we have not restricted the state space to normalized distributions since the maximization of I pos converges naturally to a normalized possibility distribution. However, the function I pos also applies to non-normalized distributions, but since it evaluates the possibility distribution as a cumulated distribution function, normalized ones are always preferred. Then, if the optimization process returns a non-normalized distribution, normalizing it by fixing the greatest value to 1 will automatically increase the value of I pos .

Possibilistic informational distance: continuous case

We now extend the definition of the possibilistic informational distance to the continuous case where = R.

Definitions

In the continuous case, the consideration of the values of π in an increasing order is naturally replaced by the consideration of α-cuts. Then the part j i=1 (π i )j of Eq. ( 9) becomes:

A π(x) π(t)dt -|A π(x) |
where A π(x) is the π(x)-cut of π and |A π(x) | its size. Naturally, the part -1 2 q i=1 π 2 i becomes:

- 1 2 R π(t) 2 dt
Then, for one piece of data we obtain:

I pos (π |x) = A π(x) π(t)dt -|A π(x) | - 1 2 R π(t) 2 dt (10)
If we consider more than one piece of data, we obtain:

I pos (π |X) = 1 n n i=1 A π(x i ) π(t)dt -|A π(x i ) | - R π(t) 2 2 dt ( 11
)
Proposition 6. Assuming an infinite set of data that follows a probability distribution p, finding the possibility distribution π that maximizes I pos is equivalent to find the distribution that minimizes the square of the distance to the optimal one π * if π respects the ordering of p (i.e., ∀x, y ∈ R if p(x) ≤ p(y) then π(x) ≤ π(y)).

Proof. It is basically the extension of Proposition 4 where the frequency becomes density values of p. Then, we will have:

d(π * , π ) 2 = C -2 * I pos (π |p) + 2 * R p(x) * |A π(x) |dx where C = R (π * (x) 2 dx is independent of π . 2 * R p(x) * |A π(x)
|dx is minimized when values x with greater density are in the smallest α-cuts, i.e., when values x with greater density have greater possibility values. When π respects the ordering of p we have A π(x) = {y ∈ R|p(y) ≥ p(x)} which corresponds to the case when 2 * R p(x) * |A π(x) |dx is minimal.

Proposition 7. The possibility distribution π * that maximizes the function I pos for data that follow a probability distribution p is the probability-possibility transformation of the optimal probability distribution p * according to Eq. ( 4).

Proof. It is the extension of Proposition 5 where the frequency becomes density values of p. We have:

I pos (π |p) = R p(x) A π(x) π(t)dt -|A π(x) | dx - R π(t) 2 2 dt (12) the part R p(x) A π(x) π(t)dt dx -R π(t) 2
2 dt is maximized for possibility distribution that are cumulated distribution of p (i.e., A 1-α = I α ). The part -R p(x) * |A π(x) |dx is maximized when values x with greater density are in the smallest α-cuts, i.e., when values x with greater density have greater density value, thus when π respects the ordering of p. Finally, I pos is maximized when A 1-α = I * α , i.e., when π * = π sp .

Thus, the possibilistic informational distance in the continuous case has the same understanding than in the discrete case. It favors the possibility distribution that minimizes the distance with the probability-possibility transformation of the distribution. However, the main advantage of this measure if that, of course, the optimal probability-possibility transformation has not to be known. Note that, having the informational distance function defined for triangular (resp. trapezoidal) distribution, we can obtain the optimal triangular (resp. trapezoidal) by finding the parameters that maximizes I pos . Since this problem cannot be solved analytically, we have to use a meta heuristics such as simulated annealing [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF] or particle swarm optimization [START_REF] Kennedy | Particle swarm optimization[END_REF]. We will consider the calculus of I pos for triangular and trapezoidal possibility distributions.

Triangular distribution

We define a triangular possibility distribution as the triple π tri = (m, l, r) where is the mode of the triangle and l and r the left and the right spread respectively. We consider a piece of data x ∈ X. We note µ = π tri (x) the possibility degree of x and [a, b] the µ-cut of π tri . The second part of the Eq. ( 10) is computed such as:

- R (π tri (t)) 2 2 dt = - m+r m-l (π tri (t)) 2 2 = - l + r 6 .
There are two cases for the term that depends on π tri (x) in [START_REF] Loquin | Fuzzy histograms and density estimation[END_REF]. We consider the case of x ∈]ml, m + r[. We have:

A µ π tri (t)dt -|A µ | = b a π tri (t)dt -(1 -µ) * (l + r) = m a (π tri (t))dt + b m (π tri (t))dt -(1 -µ) * (l + r) = m a 1 - m -t l dt + b m 1 - t -m r dt -(1 -µ) * (l + r) = -(1 -µ) 2 * l + r 2 .
The problem is when x ∈ ]ml, m + r[. In this case, the 0-cut is infinite. In the settings of the proposition 7, the interval that has to be considered is {y ∈ R|p(y) ≥ p(x)}. If we assume that the probability decreases linearly when the distance to the bound increases, we obtain:

A µ π tri (t)dt -|A µ | = - l + r 2 -C 1 * min(d(x, m -l), d(x, m + r).
where C 1 is a constant and d denotes the Euclidean distance. Finally, we obtain:

I pos (π tri |x) =          -(1 -µ) 2 * l+r 2 -l+r 6 if x ∈ x ∈ ]m -l, m + r[ -l+r 2 -C 1 * d(x, m -l) -l+r 6 if x ≤ (m -l) -l+r 2 -C 1 * d(x, m + r) -l+r 6 if x ≥ (m + r) ( 13
)
Proposition 8. Given an infinite set of data that follows an unimodal symmetric probability distribution p, the triangular possibility distribution that maximizes I pos with C 1 = 2 is the one that minimizes the square of the distance to the optimal one π sp .

Proof. If the distribution is unimodal and symmetric, any triangular possibility distribution which has a mode that corresponds to the mode of the probability distribution will respect the ordering of p on its support. Outside the support,

the value -l+r 2 -2 * min(d(x, m -l), d(x, m + r) is equal to l+r 2 -(l + r + 2 * min(d(x, m -l), d(x, m + r)). l + r + 2 * min(d(x, m -l), d(x, m + r))
is the size of the interval {y|p(y) ≥ p(x)} due to the symmetry of p. Then, the property 6 applies.

This proposition shows that, if the distribution is unimodal and symmetric, the triangular possibility distribution that maximizes I pos is the one that shares the maximal surface with the probability-possibility transformation of the considered distribution. Moreover, it has been shown in [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities[END_REF] that, given an unimodal and symmetric probability distribution, for any value of α, it exists a triangular possibility distribution for which for all β ≤ α, I * β ⊆ A 1-β . This triangular distribution can also be obtained by maximizing I pos with the adequate value of C 1 . In fact, when C 1 increases, the weight of the value outside the support increases and then the size of the support of the optimal distribution will increase too. Finally, the maximization of I pos for a triangular possibility distributions can be used for upper estimating the confidence intervals of an unimodal and symmetric probability distribution, until a confidence level threshold. This threshold increases when C 1 increases. This result does not hold when considering asymmetric unimodal probability distribution. This is essentially due to the fact that the confidence intervals, and then the α-cuts are not centered on the mean of the distribution (as it is considered to be for values outside the support in Eq. ( 13)). We propose to adapt Eq. ( 13) in order to take into account this asymmetry in the following way:

I pos (π tri |x) =          -(1 -µ) 2 * l+r 2 -l+r 6 if x ∈ x ∈]m -l, m + r[ -l+r 2 -1 2 * C 1 * 1 + r l * d(x, m -l) -l+r 6 if x ≤ (m -l) -l+r 2 -1 2 * C 1 * 1 + l r * d(x, m + r) -l+r 6 if x ≥ (m + r) (14)
Even if this version of I pos does not guarantee that the results obtained for the symmetric unimodal probability distribution remain true for asymmetric unimodal ones, we observe that it provides good approximations in most of the cases. Indeed, the optimal triangular possibility distribution with respect to I pos is the approximation of the cumulated distribution that respect as much as possible the ordering of the density function (p(x) ≥ p(x ′ ) ↔ π(x) ≥ π(x ′ )). Triangular distribution can respect this ordering in the symmetric case, it is not always the case in the non symmetric case. If the probability values are proportional to the distance of the mode (i.e., p(x + m) = c * p(xm) where m is the mode and c a constant) the mode of optimal triangular possibility distribution will correspond to the mode of the probability distribution. The less the probability values are proportional to the distance of the mode, the higher the distance between the mode of optimal triangular possibility distribution and the mode of the probability distribution.

Trapezoidal distribution

We define a trapezoidal possibility distribution as the quadruple π trap = (a, b, c, d) where [a, d] and [b, c] are respectively the support and the core of the distribution. We consider a piece of data x ∈ X. We note µ = π trap (x) the possibility degree of x and [a, b] the µ-cut of π trap . We also note lt = (ba), the size of the left part of the support, rt = (dc) the size of the right part of the support and mt = (cb) the size of the core. In the spirit of the triangular case we obtain:

I pos (π trap |x) =          -(1 -µ) 2 * lt+rt 2 -lt+rt 6 -C 2 * mt if x ∈ x ∈]b, c[ -lt+rt 2 -1 2 * C 1 * 1 + rt lt * d(x, a) -lt+rt 6 -C 2 * mt if x ≤ a -lt+rt 2 -1 2 * C 1 * 1 + lt rt * d(x, d) -lt+rt 6 -C 2 * mt if x ≥ d ( 15
)
where C 2 is a constant. Formally, C 2 is equal to 1. However, in such a case, the optimal distribution with respect to I pos will always be a triangular one (i.e., b = c). This is due to the fact that increasing the core of the distribution will never decrease the distance with the optimal distribution. By allowing to have C 2 less than 1, we allow the decreasing of the weight of the core of the distribution and then to have a genuine trapezoidal possibility distribution that maximizes I pos . Notice that it is not in contradiction with the spirit of the possibilistic informational distance since the optimal trapezoidal distribution obtained by decreasing C 2 will always contain the optimal triangular one (for which C 2 = 1). C 1 has the same use and meaning than in the triangular case. Finally, the maximization of I pos for a trapezoidal possibility distribution with adequate values of C 1 and C 2 , can be used for upper estimating the confidence intervals of any multi modal (with a finite number of mode) probability distribution, until a confidence level threshold. As in the triangular case, this threshold increases when C 1 increases. Unlike the discrete case, the use of triangular distribution guarantee by construction that the distribution obtained by optimization of I pos is normalized even the definition of I pos is still relevant if the distribution is not normalized.

Illustrations

In this section, we use the possibilistic informational distance function in order to a build possibility distribution from a set of data. In each case, we will consider a different probability distribution. Thus, we construct a set of data that corresponds as much as possible to the probability distribution chosen. In order to do that, we discretize the density function on a fixed interval. After a normalization step, we obtain a discrete probability distribution. We then maximize the sum of the I pos for each of these points weighted by their probability degree. We choose this approach rather than a sampling because, with a sufficient high discretization range, the result obtained is very close to the optimization of I pos for an infinite set of data that follows the chosen probability distribution. Of course, the method is still applicable to any sample set. As pointed out previously, we have to use a meta heuristic (simulated annealing here) in order to find the maximal triangular or trapezoidal distribution that maximize I pos . For each probability distribution, we discretize the density function with 500 divisions of the interval [-6, 6]. On each figure, the probability distribution is in green, its probability-possibility transformation is in blue In the following, for illustration purpose we suppose that we know the shape of the probability distribution. Obviously, in this case, the squared distance with the probability-possibility transformation can be computed directly. However, I pos does not require the knowledge of the shape of the distribution. Moreover, the result is obtained by summing up the local evaluations of I pos on each piece of data, while the method based on the square of the surface relies on a global evaluation. Contrarily to the method based on the square of the surface, the optimization of I pos still applies when having a finite (even small) amount of data, without knowing the shape of the distribution.

Unimodal symmetric distributions

First, we consider a unimodal symmetric distribution. Fig. 3 corresponds to the building of triangular possibility distribution that maximize I pos with C 1 = 2, C 1 = 4 and C 1 = 10 (from light red to dark red) 1 for data that follow a Gaussian distribution with mean equal to 0 and standard deviation equal to 1. We can first notice that the mode of the triangular distribution corresponds to the mode of the Gaussian distribution. The triangular distribution with C 1 = 2 is very close to the optimal one. As expected, when C 1 increases, the threshold α for which ∀β ≤ α, I * β ⊆ A 1-β increases too. In the following, we will name domination level the value α. Of course we have π

C 1 =2 π C 1 =4 π C 1 =10 . For π C 1 =2 , π C 1 =4 and π C 1 =10
, the domination levels are respectively 0.76, 0.93 and 0.97.

Fig. 4 presents the same approach with data that follow a triangular probability distribution. The results are similar to the ones observed with a Gaussian distribution. The domination levels are very close (respectively 0.76, 0.93 and 0.97). As for the discrete case, this figure illustrates the fact that maximizing possibilistic informational distance provides results that are different from the ones obtained by approximating the probability distribution and then applying the probability-possibility transformation. This is obvious here since the probability-possibility transformation of a triangular probability distribution is not a triangular possibility distribution.

What is interesting to remark here is that the maximum possibilistic informational distance principle allows us to elicitate triangular possibility distributions that upper bound the confidence intervals of any unknown unimodal symmetric probability distribution.

Unimodal non symmetric distributions

We perform the same experience than the previous one with a skewed normal distribution which is unimodal and asymmetric (see Fig. 5). Although the distribution is not symmetric, we can observe similar results to the ones observed with the Gaussian distribution, except that the mode is not identified exactly. Thus, π C 1 =2 upper bounds the confidence intervals I * α for 0.2 ≤ α ≤ 0.76, π C 1 =4 for 0.1 ≤ α ≤ 0.94 and π C 1 =10 for 0.07 ≤ α ≤ 0.98. We can observe that, even if the results are not as precise as the previous ones, maximum possibilistic informational distance principle is still able to upper bound unknown unimodal distributions that are not symmetric. We can also observe that, even the choice of C 1 is empirical at this time, the dominance level remains relatively stable for a given value of C 1 , regardless to the type of the distribution.

Multimodal symmetric distributions

In Figs. 678, the distribution is multi modal. In Fig. 6 we have C 2 = 1, and from light red to dark red C 1 = 2, C 1 = 4 and C 1 = 10. As expected, we obtain, we obtain triangular distributions. We can observe that the mode of the triangular distribution is close the highest mode of the probability distribution. When C 2 = 0.8 (Fig. 7), we obtain genuine trapezoidal distributions, but they still do not upperbound the confidence intervals. A possibility distribution that upper bounds the confidence intervals I * α for α ≤ 0.98 is obtained with C 1 = 10 and C 2 = 0.67 (Fig. 8). This illustrates the fact that, for adequate values of C 1 and C 2 , the maximum possibilistic informational distance principle allows us to elicitate trapezoidal possibility distributions that upper bound the confidence intervals of any unknown probability distribution that has a finite number of modes. As for C 1 , the effect of the value of C 2 does not highly depend on the type of the distribution and then the values C 1 = 10 and C 2 = 0.67 perform well in most of the cases.

Conclusion

In this paper we have proposed a definition of possibilistic informational distance that agrees with the view of possibility distributions as families of probability distributions and with the probability-possibility transformation based on the maximum-specificity principle. We have defined this possibilistic informational distance function both for discrete and for continuous universes. The calculus is described for the cases of triangular and trapezoidal possibility distributions. We have shown that the possibilistic informational distance is related to the distance between the considered possibility distribution and the probability-possibility transformation of the unknown probability distribution that has generated the data. More precisely, by maximizing I pos (under some constraints, such as triangular shaped possibility distributions) we obtain the possibility distribution that respects as much as possible the ordering of the probabilities, and that shares a maximal surface with the cumulated distribution that respects the possibilistic ordering. This type of function is interesting in many respects. It can be used for comparing the faithfulness of two possibility distributions with respect to a set of data. Moreover, the good properties of triangular possibility distributions for bounding the confidence intervals of a unimodal probability distribution makes the building of possibility distributions by optimization of I pos a good approach when no a priori information on the type of distribution is available. In the same way, we can use maximum possibilistic informational distance principle for upper bounding the confidence intervals of any unknown probability distribution that has a finite number of modes. Possibilistic informational distance is particularly promising in the area of machine learning. Indeed, it is common in machine learning to have to estimate probability distributions with few data and without a priori knowledge about the shape of the distribution. It may happen for instance in Bayesian learning or in k-nearest neighbor approaches. In this case, possibilistic informational distance may be a cautious and valuable tool for upper bounding the confidence intervals of these unknown distributions. It this scope, maximizing the possibilistic informational distance is the core of the imprecise regression method [START_REF] Serrurier | Imprecise regression based on possibilistic likelihood[END_REF] that allows us to predict a possibility distribution of the output value from a crisp vector of input values. A similar approach to possibilistic classification is worth investigating too.

In the future, it would be useful to look for alternatives to the meta heuristics that are used in the optimization process since these algorithms are heavy to tune and require high amounts of computation. This may rely on analytical solutions of the optimization problems, on the use of lighter algorithms that provide good approximations. Moreover, another practical issue is the tuning of the parameters C 1 and C 2 and the study of their properties. Lastly, it may desirable to take into account the quantity of data available. Indeed, if only one value is available, the best possibility distribution is a Dirac, and there will be no difference with the case where a thousand of identical values are available. Less data should lead to less specific distributions. This should be taken into account in the computation of the possibilistic informational distance.

Fig. 1 .

 1 Fig. 1. Probability to possibility transformation of a Gaussian distribution.

Fig. 2 .

 2 Fig. 2. Approximation of a Gaussian density with a triangular density by maximization of I surf .

Fig. 3 .

 3 Fig. 3. Triangular possibility distributions that maximize I pos with respect to a set of data that follows a Gaussian distribution.

Fig. 4 .

 4 Fig. 4. Triangular possibility distributions that maximize I pos with respect to a set of data that follows a triangular probability distribution. and optimal possibility distributions are in red. A figure may contain multiple possibility distributions which correspond to different values of the constants C 1 and C 2 .

Fig. 5 .

 5 Fig. 5. Triangular possibility distributions that maximize I pos with respect to a set of data that follows a skewed normal probability distribution.

Fig. 6 .

 6 Fig. 6. Trapezoidal possibility distributions that maximize I pos (C 2 = 1) with respect to a set of data that follows a Gaussian probability distribution.

Fig. 7 .

 7 Fig. 7. Trapezoidal possibility distributions that maximize I pos (C 2 = 0.8) with respect to a set of data that follows a Gaussian probability distribution.

Fig. 8 .

 8 Fig. 8. Trapezoidal possibility distribution that maximizes I pos (C 1 = 10, C 2 = 0.67) with respect to a set of data that follows a Gaussian probability distribution.

For interpretation of colour in figure artwork, the reader is referred to the web version of this article.


Wan-Yu Tsai

Rongying Lin

Shanthi Murali

Li Li

Zhang

John K Mcdonough

Rodney S Ruoff

Pierre-Louis Taberna

Yury Gogotsi

Patrice Simon

Outstanding performance of activated graphene based supercapacitors in ionic liquid electrolyte from À50 to 80 1C

Keywords: Electrochemical capacitors, Activated graphene, Ionic liquid mixture, Wide temperature range

High specific surface area (SSA $2000 m 2 /g) porous KOH-activated microwave exfoliated graphite oxide ('a-MEGO') electrodes have been tested in a eutectic mixture of ionic liquids (1:1 by weight or molar ratio N-methyl-N-propylpiperidinium bis(fluorosulfonyl)imide (PIP 13 -FSI) and N-butyl-Nmethylpyrrolidinium bis(fluorosulfonyl)imide (PYR 14 -FSI)) as electrolyte for supercapacitor applications. By optimizing the carbon/electrolyte system, outstanding capacitive performance has been achieved with high capacitance (up to 180 F/g) and wide electrochemical window (up to 3.5 V) over a wide temperature range from À50 1C to 80 1C. This is the first demonstration of a carbon-ionic liquid system capable of delivering capacitance in excess of 100 F/g below room temperature. The excellent electrochemical response of the proposed couple shows that optimization of the carbon/electrolyte interface is of great importance for improving capacitive energy storage.

Introduction

Electrochemical double layer capacitors (EDLCs), also known as supercapacitors, store electrical charges between electrode materials and electrolyte ions by electrostatic http://dx.doi.org/10.1016/j.nanoen.2012.11.006 attractions. Supercapacitors had their debut mainly in CMOS memory back-up and are now widely used in power electronics for peak power shaving and back-up supplies (as support for power disruption). Today, one of the most promising applications is their use in transportation (in hybrid-electric vehicles (HEVs) in particular) for harvesting braking energy and for faster start-up of the engine [1][2][3]. Their increasing use for energy storage is due to a number of factors including (i) their high power densities (410 kW/kg), (ii) moderate energy density ($5-6 Wh/kg), (iii) that they can be fully charged/discharged in a matter of seconds [4,5], (iv) the cost competitiveness of the materials that they are comprised of, (v) high charge/discharge efficiency, and (vi) their high cycle life, which is generally longer than the life of the device being powered.

Generally, EDLCs are made up of two symmetrical activated carbon electrodes that display symmetric charge/ discharge power capability over the complete range of potential operation (i.e., from the maximum rated voltage to near zero). Carbon/carbon supercapacitors can be used over the complete voltage range without damage for at least 1 million cycles [START_REF] Everett | Ultracapacitors: a mature energy-storage alternative in a dramatically expanding application set, Maxwell Technologies[END_REF]. Carbon-based electrodes are of great interest because they are light, of moderate cost, abundant, easy to process, and possess high electronic conductivity and high specific surface area (yielding high charge storage). A variety of different carbon nanostructures can be produced that have very high surface areas and a wide variety of physical properties, which allows targeting of different types of applications. Templated carbons [START_REF] Kajdos | [END_REF], carbide-derived carbons (CDCs) [8][9][10], 'onion-like' carbons (OLCs) [11], carbon fabrics (cloth, fibers) [12], nanotubes [13], activated graphene [14] and nanohorns [15] have been used as electrodes for EDLC applications [16].

Batteries have much higher energy densities (up to 250 Wh/kg for Li-ion) than current commercial supercapacitors (5 Wh/kg) [17]. One approach to increase energy density of supercapacitors is with ionic liquid electrolytes that offer better electrochemical stability than that of the aqueous (1.23 V) and organic (2.7 V) electrolytes currently used. Ionic liquids (IL) -salts (mainly, organic) that are liquid at room temperature -are being explored as electrolytes for Electrical Energy Storage (EES) systems owing to their high thermal (up to 350 1C) and electrochemical (up to 6 V) stability [18]. Their high heat capacity, a large cohesive energy, and negligible vapor pressure at elevated temperatures make them safe and non-flammable electrolytes. This is especially important in energy storage applications when devices need to perform under extreme temperature conditions that can raise/cause safety issues. The raison d'être for exploring eutectic mixtures of ionic liquids is their large stable liquidus temperature range (the span of temperatures between the freezing point and boiling point of a liquid, such as from À80 1C to Z250 1C). The size and symmetry of cations strongly influence the melting points of ionic liquids. Usually, one of the two pairing ions (usually the cation) is particularly large and possesses a low degree of symmetry, leading to inefficient packing of the large irregular cations with small charge-delocalized anions, which is why organic salts have a reduced crystal lattice energy and hence a lower melting point compared to inorganic salts [START_REF] Angell | Electrochemical Aspects of Ionic Liquids[END_REF]. Binary and ternary eutectic mixtures have been studied for EES systems. Binary eutectic systems first appeared with the discovery by Wilkes et al. in 1982 of a MeEtImCl/AlCl 3 molten salt mixture that was liquid over a wide range of compositions and to temperatures as low as À98 1C for a composition of 67% AlCl 3 [START_REF] Wilkes | [END_REF]. Other binary systems have been investigated by Hou et al. [21] using imidazolium-based ILs with organic salts. Every et al. [22] reported a eutectic mixture of pure ILs (mixing ethylmethyl-immidazolium tri-fluorosulfonyl-imide (EMI-TFSI) and EMI-CF 3 SO 3 À (triflate)) with no evidence of a melting point, indeed only a glass transition was observed. Pyrrolidinium-based mixtures with TFSI-FSI-based ILs were also found by differential scanning calorimetry to form eutectic mixtures [23]. Deeper eutectics (À95 1C) were however observed to be mainly in mixtures of a common cation with two different anions compared to eutectics (À25 1C) in mixtures of a common anion with two different cations. Other binary systems include IL-LiX mixtures as electrolytes for battery applications [24][25][26]. Henderson et al. [24] reported decreasing the eutectic temperatures of pyrrolidinium-based ILs by using different alkyl chain lengths and Li salt concentrations.

As reported before, we have designed IL mixtures with exactly the same anion (FSI) but two different cations: piperidinium (PIP) and pyrrolidinium (PYR) [27]. These cations have the same molecular weight and the same number of atoms of the same nature, with the only difference being their cation molecular structure, namely a five-member (piperidinium) or six-member (pyrrolidinium) heterocycle. Methyl +propyl substituent groups (total of 4 carbons on side chains) are used for synthesizing the PIPbased cation and methyl+ butyl substituent groups (total of 5 carbons on side chains) for synthesizing the PYR-based cation. The concept is to maintain similar intermolecular interactions due to the similar constituent atoms of the two cations, while also increasing disorder and asymmetry among the cations to hinder lattice formation to lower the melting point, but still maintaining good miscibility. Amongst common anions such as hexafluorophosphate (PF 6 À ), tetrafluoroborate (BF 4 À ), bis(trifluorosulfonyl)imide (TFSI À ) and bis(fluorosulfonyl)imide (FSI À ) anions, the imides (TFSI À and FSI À ) have emerged as popular alternatives to the PF 6 À and BF 4

À anions due to the susceptibility of the PF 6 À and BF 4 À anions to hydrolysis, leading to the production of hazardous hydrogen fluoride [28][29][30][31][32]. This is why hydrophobic TFSI À and FSI À anions are increasingly being studied as electrolytes for energy storage [33][34][35]. The FSI À ion has been selected to be the anion for the electrolyte used in this study due to its generally higher conductivity and lower viscosity as compared to the TFSI À ion when paired with the same cation [36], because of the smaller size and hence higher mobility of the FSI À ion. Cations were chosen by comparing their stability. The electrochemical stability of commonly used cations of ILs decreases in the order of AmmoniumEPiperidiniumZPyrrolidi-nium4Imidazolium [START_REF] Wasserscheid | Ionic Liquids, in Synthesis[END_REF]. Hence, cations with functional groups of pyrrolidinium (PYR) and piperidinium (PIP) with relatively higher electrochemical stability were explored.

A eutectic mixture composed of (1:1 by weight or molar ratio) N-methyl-N-propylpiperidinium bis(fluorosulfonyl)imide (PIP 13 -FSI) and N-butyl-N-methylpyrrolidinium bis(fluorosulfonyl)imide (PYR 14 -FSI) was prepared and used as electrolyte. The measured melting points of constituents of this IL mixture were 6 1C (PIP 13 -FSI) and À18 1C (PYR 14 -FSI), in good agreement with published data [23]. The liquid state of this IL eutectic mixture can be maintained at temperatures several tens of degrees lower compared to the individual ILs. Mixing PIP 13 -FSI with PYR 14 -FSI resulted in an IL with two different cationic species and only one anion (namely, (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 ), which expands the liquid range to À80 1C as reported previously [27]. Most importantly, the wide electrochemical stability window is preserved in the mixtures, which contributes to the goal of increasing energy density, as E=½CV 2 , where C is capacitance and V is voltage. This electrolyte mixture was used in our previous work with nanostructured carbon electrodes of either onion-like carbon (OLC) or carbon nanotubes (CNTs) [27]. Excellent capacitive behavior has been observed in an extended operating temperature range of À50 1C to 100 1C. However, the specific capacitance obtained in these exohedral carbons was low ($30 F/g; 4 mF/cm 2 ), because of their moderate SSA and highly graphitic surface. This suggests increasing SSA and improving the carbon/electrolyte interface to increase the magnitude and quality of charge storage, thereby further improving the energy density and in an extended operating temperature range. However, nanoporous activated or carbidederived carbons that possess a high SSA and show excellent performance in organic electrolytes, show poor capacitive performance with IL electrolytes at room temperature or below [10,27,34] due to low mobility of ions in porous networks. To the best of our knowledge, no carbon electrode material has ever shown capacitance of 100 F/g or higher in any IL electrolyte below room temperature.

Ruoff's group recently reported on the activation of microwave exfoliated graphite oxide ('MEGO') using potassium hydroxide [START_REF] Zhu | [END_REF]. This carbon ('a-MEGO') had unique structure and properties, such as high SSA, approaching and even exceeding that of single-layer graphene, as measured by the Brunauer-Emmett-Teller (BET) method [39], along with a continuous, three dimensional pore structure consisting of atom-thick walls and both micro-and mesopores. It showed high capacitance in IL electrolyte at room temperature, but has not been tested at low temperatures. Since a-MEGO possesses high electrical conductivity, low O and H content, and is composed mostly of sp 2 bonded carbon, we reasoned that it might yield a dramatic increase in specific capacitance, compared to OLC or CNTs, with the (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 ionic liquid electrolyte.

Experimental

The synthesis procedure for a-MEGO, as well as its structure, has been described in Ref. [START_REF] Zhu | [END_REF]. MEGO was prepared from graphite oxide (GO) by microwave irradiation [40], then dispersed and soaked in aqueous KOH solution for 20 h. The solid cake obtained after vacuum filtration of the excess KOH was then dried. The dry MEGO/KOH mixture was heated at 800 1C for 1 h in a horizontal tube furnace with an argon flow of 150 sccm at atmospheric pressure. The sample was then washed with de-ionized water, and dried at 65 1C in ambient air for 2 h. Annealing of activated graphene was done in a custom made vacuum furnace (Solar Atmospheres, USA) at 1100 1C, with the heating ramp 10 1C/min, and a vacuum $10 À6 Torr for 10 min.

Nitrogen gas sorption was performed at 77 K using a Quadrasorb Pore Size and Surface Area Analyzer (Quantachrome Instruments, USA). Prior to analysis, samples were outgassed at 200 1C in vacuum at about 10 À2 Torr for 24 h. Surface area was calculated using the BET method in the range of 0.05-0.3P/P 0 . Pore size distributions (PSD) were determined using the quenched solid density functional theory (QSDFT) model, assuming a slit pore geometry, which was part of the QuadraWin software package (Quantachrome Instruments, USA).

Raman spectra were recorded with an Invia confocal Raman spectrometer (Renishaw, UK) using a laser with an excitation wavelength of 514.5 nm and having a spot size of 2 mm.

ILs were prepared at Solvionic SA (France). N-methyl-Nbutylpyrrolidinium bromide, PYR 14 -Br, is first produced from the quarternization of methyl-pyrrolidine by 1-bromobutane to yield its halide salt, PYR 14 -Br. The desired PYR 14 -FSI IL was then obtained through the anionic exchange of the Br À with potassium bis(fluorosulfonyl)imide (KFSI). N-methyl-N-propylpiperidinium bromide, PIP 13 -Br, was obtained from the quarternization of N-methyl-N-propylpiperidine by 1-bromopropane to yield its halide salt, PIP 13 -Br and the subsequent metathesis (with KFSI) to obtain the product PIP 13 -FSI. The eutectic IL mixture (1:1 by weight) was then obtained by mixing N-methyl-N-propylpiperidinium bis(fluorosulfonyl)imide (PIP 13 FSI) and N-butyl-N-methylpyrrolidinium bis(fluorosulfonyl)imide (PYR 14 FSI) [27].

Microcavity electrode measurements that have been described in our previous work [11,41] were done on the a-MEGO powders under glovebox conditions of o0.1 ppm of water and oxygen content. A two-electrode Swagelok s nylon cell with stainless steel pistons was used for quantitative electrochemical characterizations of as-activated MEGO (i.e., a-MEGO) powders, and a three-electrode Swagelok s nylon cell was used to characterize vacuum annealed a-MEGO powders. Each cell was assembled in the eutectic mixture of ILs under glove-box conditions. Pt disks were placed onto stainless steel current collectors to decrease the contact resistance at the as-activated a-MEGO/piston interface. A silver wire was used as a quasi-reference electrode. The as-activated a-MEGO, electrode films (disks) of 1.1 cm 2 , 1.65 mg, and 55 mm thick were made of 95% of the carbon powder and 5% of PTFE (Dupont de Nemours) binder as described in Zhu et al. [START_REF] Zhu | [END_REF], whereas for the annealed a-MEGO, the powder was tested directly without film processing. Electrodes were wetted with the electrolyte and covered by two layers of 25 mm-thick porous Al 2 O 3 separator (Separion s , average pore size of 240 nm, 440% porosity, Evonik Industries, Germany); these separators were wetted by the electrolyte prior to being placed between the electrodes. Swagelok s cells were kept under light pressure during the measurement to buffer thermal expansion/contraction of the stainless steel pistons. Temperature control was achieved using a Votsch climatic chamber (Germany) with the temperature being additionally checked by a thermocouple. After heating to a certain temperature, the system was kept isothermal for a period of no less than 5 h prior to electrochemical measurements to ensure system equilibration. For low temperature experiments, this dwell time was increased to an overnight period (46 h).

Capacitance was calculated from cyclic voltammetry (CV) by measuring the slope of a charge Q versus voltage (V) plot. Charge was calculated from the integration of the area under the voltammogram obtained in discharge.

The current in the CVs in Figs. 3a,4a,c, and 5b was normalized by dividing the current values by the potential scan rate to obtain capacitance (C).

Results and discussion

Single-layer graphene has a theoretical specific surface area of 2630 m 2 /g however in practice the BET SSA is generally much lower because of multi-layer sheets and agglomeration. We approach the theoretical value of a single sheet of graphene, with as-activated a-MEGO having a high surface area of 1901 m 2 /g and all pores less than 5 nm (Figure 1a). The a-MEGO was vacuum annealed at 1100 1C in an attempt to remove the oxygen containing surface functional groups. At this temperature (and for the 'right' exposure time), one might expect that oxygen containing groups would be removed while the structure of the a-MEGO might be preserved. From Figure 1b, after annealing, the BET SSA is 1586 m 2 /g, and the peak in the PSD around 3.25 nm is smaller than that of the as-activated a-MEGO. The microporous region (o2 nm) was unchanged after annealing, indicating the initial pore structure in a-MEGO has been largely preserved. The vacuum annealed a-MEGO has a desirable PSD for supercapacitors, as it has both micropores that give it a high SSA, in the range of values typical for activated carbons, and mesopores that facilitate ion mobility during charging and discharging.

Comparing the pore structure of vacuum annealed a-MEGO to the as-activated a-MEGO material that was recently reported and tested in supercapacitors at room temperature [START_REF] Zhu | [END_REF], the pore size distribution of the vacuum annealed a-MEGO made here tends to a smaller average pore size and a smaller specific surface area, with the previously reported a-MEGO having a 'high end' value of BET SSA of $3100 m 2 /g [START_REF] Zhu | [END_REF]. Both materials have all pores less than 10 nm. Other materials for supercapacitors, such as CDC and activated carbon, have similar surface areas to vacuum annealed a-MEGO, but tend to have an intricate network of pores that limits ion transport rates and prevents fast charging and discharging when viscous IL electrolytes are used [27,34].

The disorder and graphitic nature of the structure before and after annealing was analyzed using Raman spectroscopy (Figure 2). There is very little difference in the two spectra, implying the annealing process did not change the graphene layer structure. The main difference lies in the full width at half maximum (FWHM) of the disorder-induced (D) band, which narrows after annealing from 160 to 140 cm À1 , implying healing of some defects. The main goal of annealing the a-MEGO was to remove oxygen groups on the surface without affecting a-MEGO structure, which seems to be successful from the Raman spectra as the difference before and after annealing is subtle.

Figure 3a shows the CV scanned at 20 mV/s at room temperature of the as-activated a-MEGO electrodes in the IL mixture presented in terms of capacitance versus voltage of the cell. This again confirms the feasibility of using the eutectic IL mixture with as-activated a-MEGO and annealed a-MEGO, in addition to the previous OLCs and CNTs [27] that have been tested. A maximum operating voltage of 3.5 V (same as for the OLCs) [27] was obtained at room temperature with a capacitance of $160 F/g, a 5-fold increase compared to OLCs and CNTs ($30 F/g) [27]. This huge increase can be firstly attributed to the SSA of the as-activated a-MEGO electrodes, which is at least 4 times larger than that of OLC ($400 m 2 /g).

Moreover, the open inter-layer porosity (up to 4 nm) of the asactivated a-MEGO electrodes allows electrolyte ions access to the subnanometer pores in a-MEGO sheets without much limitation in ion transfer, as may occur when using conventional activated carbons with a tortuous pore network and narrow bottlenecks between pores [27].

Figure 3b shows the EIS plot for the a-MEGO electrodes assembled in the ionic liquid mixture. The vertical increase of the imaginary part of the impedance provides evidence of the capacitive storage achieved with a-MEGO. A fairly low ESR of 3.4 O cm 2 (intercept of the X-axis at high frequency) was measured at room temperature (10 O cm 2 ESR was measured for CNT electrodes [27]), mainly due to the high electrical conductivity of the a-MEGO electrode (500 S/cm) compared to commercial activated carbons (0.1-1 S/cm) [42]. Since both sides of the atom-thick pore walls of a-MEGO (graphene-like layers) can be used in charge storage, once the electrolyte gains access to these surfaces, a tremendous amount of charge could be stored as observed in the 5-fold increase in capacitance using a-MEGO instead of OLCs. This capacitance value is in agreement with the value reported by Zhu et al. [START_REF] Zhu | [END_REF] using neat EMI-TFSI with as-activated a-MEGO electrodes.

Capacitance at Elevated Temperature (20rTr80 1C): Figure 4a shows CVs recorded at 100 mV/s to their respective maximum voltages at different temperatures. There is a gradual increase of capacitance (area under the CV curve) with increasing temperature that can be explained by the higher ionic conductivity (ion mobility) of the electrolyte at elevated temperatures, which is also reflected in the more rectangular shape of the CVs with increasing temperature. Electrochemical impedance measurements further confirm the decrease in ionic resistance as shown in Figure 4b and the ESR value decreases from 3.4 to 1.3 O cm 2 upon increasing the temperature from 20 1C to 80 1C. The capacitance increases from 160 to 180 F/g as temperature increases from RT to 80 1C. The operating voltage is limited to 2.9 V at 80 1C, which is linked with the enhanced electrolyte electrochemical activity at platinum current collectors at elevated temperature (see Supplementary Figure S1).

Capacitance at sub-zero temperatures (À40 and À50 1C): Figure 4c shows the CVs recorded at sub-zero temperatures of À40 and À50 1C. About 70% of the capacitance is still retained at 1 mV/s (which is a reasonable scan rate taking into account the reduced ion mobility for such low temperatures) and the CVs show good capacitive behavior at À40 1C with high capacitance values of 100 to 120 F/g obtained when charged to 3 V. Such retention of capacitance over this wide operating temperature range (from À50 1C to 80 1C) with capacitance always above 100 F/g, to our knowledge, has never been reported before in the literature [43][44][45][46][47][48]. EIS measurements still show a capacitive behavior, despite the presence of an expected extra semicircle at high frequency, attributed to the onset of gelation that is related to decreased mobility of the electrolyte ions within the pores of a-MEGO at such low temperatures [49][50][51]. The capacitive behavior of a-MEGO decreases at À50 1C which is evident by a less rectangular CV than that seen in the CNTs.

While the a-MEGO electrodes show outstanding capacitance of $180 F/g (see Figure 3a), the presence of bumps in the CV at about 1.5 V could be due to electrochemically active oxygen-containing functional groups, such as carboxylic acid or hydroxyl groups on their surface. Such bumps were also observed when a-MEGO electrodes were tested in EMI-TFSI electrolyte [START_REF] Zhu | [END_REF]. Although they are suitable for improving the capacitance, such surface groups are known to be unstable during cycling, thus limiting the cycle life of the cells [16]. This motivated annealing a-MEGO powder at 1100 1C under vacuum to try to eliminate functional groups that may be present on its surface. Annealed and asactivated a-MEGO powders were tested in the eutectic IL mixture at room temperature using a microcavity electrode to obtain information on any difference in electrochemical response as a result of vacuum annealing. The CVs collected from both samples in a 3-electrode cell are shown in Figure 5a, scanned within a potential window of À1.7 to 1.3 V/Ref. Bumps observed at $À0.3 V/Ref in the positive scan and at $À0.8 V/Ref in the negative scan are pronounced for the as-activated a-MEGO powder and clearly decreased after annealing.

Annealed a-MEGO powders were then characterized in Swagelok s cells, using a 3-electrode configuration. Figure 5b shows the CV of annealed a-MEGO powders at 1 mV/s at room temperature. Note that in these experiments, raw powders were tested without any binder so that the ohmic drop was not optimized in the cell which is why a low scan rate was used. Annealed a-MEGO powders reached a capacitance of 130 F/g, which is $20% less than the capacitance obtained for the as-activated samples, as expected from a lower SSA of the annealed material. The other point to be taken into account is the processing of the electrode, which could have some effect on capacitance values measured (Figure 5b). In conclusion, the high capacitance of the a-MEGO materials observed in the ionic liquid mixture can be primarily associated with double layer charging as shown in Table 1.

Conclusions

The electrochemical characterization of as-activated and vacuum annealed a-MEGO has been carried out in a eutectic mixture of ionic liquids, (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 . Annealing the a-MEGO yields better electrochemical stability, but the specific capacitance somewhat decreases due to decreased mesoporosity and associated accessible SSA after annealing. For ionic liquid electrolytes, an exceptionally high capacitance of about 150 F/g, with a maximum voltage of 3.5 V was attained over an unprecedented 130 1C (from À50 to 80 1C) temperature window. This is 5 times higher capacitance compared to exohedral nanostructured carbon (nanotubes and onions) electrodes in the same eutectic mixture of ionic liquids. These findings have largely overcast the temperature range offered by conventional organic electrolytes of 110 1C (from À30 1C to 80 1C) and would greatly contribute to the advances in electrical storage based on ILs, pushing the limits of energy storage to greater heights. These studies have not only affirmed our previous findings on the need to design carbon structure for optimizing carbon/electrolyte interface, but also showed that ionic liquids are no longer confined to the category of ''above room temperature'' applications. Even more, they have proved their extended threshold for extreme climatic conditions, which clearly proves the robustness of ILs as electrolytes for electrical storage.

Herein, the research pathway has been paved towards optimizing the carbon accessible surface area (e.g., by using activated graphene, i.e., a-MEGO [START_REF] Zhu | [END_REF], de-bundled singlewalled nanotubes [52], or mesoporous templated carbons [START_REF] Kyotani | Carbon Materials for Electrochemical Energy Storage Systems[END_REF]) together with designing the IL-based electrolyte. Control of the electrolyte composition and its molecular/ ionic structure is required for achieving better compatibilities with various carbon structures and surpassing the current capacitance and energy density limits of EES.

Figure 1

 1 Figure 1 Nitrogen sorption isotherms (a) and corresponding pore size distributions (b) for as-activated (BET SSA 1900 m 2 /g) and vacuum annealed a-MEGO (BET SSA 1590 m 2 /g). After annealing, there is a decrease in the volume of small mesopores around 3 nm, while the micropores within a-MEGO are preserved. Closed symbols in (a) represent the adsorption branch while open symbols represent the desorption branch.

Figure 2

 2 Figure2Raman spectra of as-a-MEGO in the carbon spectral range showing the bands corresponding to disordered structures (D) around 1350 cm À1 and in-plane vibrations of graphite (G) around 1600 cm À1 . Note that there is no significant change of the structure as a result of annealing.

Figure 4

 4 Figure 4 Electrochemical characterization of a-MEGO electrodes in (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 electrolyte: CVs collected at 100 mV/s (a), corresponding EIS Nyquist plots (b) at room and elevated temperatures (RTrTr80 1C), and CVs collected at 1 mV/s (TrÀ40 1C) (c) and the corresponding EIS Nyquist plot at T=À40 1C (d).

Figure 3

 3 Figure 3 Room temperature electrochemical characterization of a-MEGO electrodes in (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 electrolyte: CVs collected at 20 mV/s with the electrochemical window of 3.5 V and showing capacitance of 150 F/g (a) and the corresponding EIS Nyquist plot (b). Inset in (b) shows data at the high-frequency range.

Figure 5

 5 Figure5Cyclic voltammograms of annealed and as-activated a-MEGO electrodes in (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 electrolyte: CVs collected at 500 mV/s at room temperature using CME (a); CV collected at 1 mV/s using a Swagelok s cell (b).

Table 1

 1 Capacitance of annealed a-MEGO electrodes in (PIP 13 -FSI) 0.5 (PYR 14 -FSI) 0.5 electrolyte at room temperature.

	Scan rate	Charging/discharging capacitance
	(mV/s)	(F/g)
	20	100
	5	110
	1	130

Acknowledgments

P. Simon would like to thank the European Research Council for supporting the research (Advanced Grant no. ERC-2011-AdG, Project no. 291543 -IONACES); W.-Y. Tsai was fully supported by the ERC Grant. Y. Gogotsi and J. McDonough were supported as part of the Fluid Interface Reactions, Structures and Transport (FIRST) Center, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences. Work at UT Austin was supported by the US Department of Energy (DOE) under Award no. DE-SC001951.

Appendix A. Supporting information

Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/ j.nanoen.2012.11.006.