Marc Fleurbaey

Stéphane Zuber

Robert E Kuenne

Climate policies deserve a negative discount rate

Keywords: discounting, climate policy, intergenerational equity

We defend a methodology of discounting, for the evaluation of the long-term effects of climate policies, which relies on a social welfare objective, against the view that the market rate of return should be used for that purpose. We also show that in the long run, the discount rate for such policies should focus on the worst-case scenario for the most disadvantaged populations. As a consequence, it is likely that the appropriate discount rate for climate policies should be negative, implying a high priority for the future.

Les Working Papers et les Position Papers de la Fondation Maison des sciences de l'homme ont pour objectif la diffusion ouverte des travaux en train de se faire dans le cadre des diverses activités scientifiques de la Fondation : Le Collège d' études mondiales, Bourses Fernand Braudel-IFER, Programmes scientifiques, hébergement à la Maison Suger, Séminaires et Centres associés, Directeurs d'études associés... Les opinions exprimées dans cet article n' engagent que leur auteur et ne reflètent pas nécessairement les positions institutionnelles de la Fondation MSH.

The Working Papers and Position Papers of the FMSH are produced in the course of the scientific activities of the FMSH: the chairs of the Institute for Global Studies, Fernand Braudel-IFER grants, the Foundation's scientific programmes, or the scholars hosted at the Maison Suger or as associate research directors. Working Papers may also be produced in partnership with affiliated institutions.

The views expressed in this paper are the author's own and do not necessarily reflect institutional positions from the Foundation MSH.

Un taux d'escompte négatif pour les politiques climatiques

Résumé

Nous défendons une méthodologie de l'actualisation, pour l' évaluation des effets à long terme des politiques climatiques, qui repose sur un objectif de bien-être social, contre l'idée que le taux de rendement du marché doit être utilisé à cette fin. Nous montrons aussi que dans le long terme, le taux d'actualisation pour de telles politiques devrait se concentrer sur le scénario le plus défavorable pour les populations les plus défavorisées. En conséquence, il est probable que le taux d'actualisation approprié pour les politiques climatiques doive être négatif, ce qui implique une grande priorité pour le futur.

Mots-clés taux d'escompte, politique climatique, équité intergénérationnelle

Sommaire

The descriptive-prescriptive debate 4

The methodology of discounting 7

Discounting under risk 8

Priority for the poor in the long run 10

Beyond utilitarianism 12 Conclusion 14

References 14

Appendix

Fondation Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19 S ince the Stern Review [START_REF] Stern | The Economics of Climate Change[END_REF]) and the debate it has sparked off, the discount rate has been at the center of heated discussions about climate policies. In the very long run, the discount rate makes a huge difference in the evaluation of policies. The following table (Table 1) shows the minimum return that a $1 investment for the future should have to be considered better than consuming it now, depending on the discount rate that is adopted and depending on the horizon. The 1.4% discount rate is advocated by the Stern Review, but later Stern suggested that 2.7% might be a better figure . The table shows that this hesitation is not innocuous. Obviously, adopting a much higher discount rate as recommended by [START_REF] Nordhaus | The Stern Review on the Economics of Climate Change[END_REF] -around 5.5% -has even more extreme consequences. Legend: With a 1.40% discount rate, a $1 investment today must yield at least $4.02 in 100 years; with a 2.70% discount rate, the number jumps to $14.36, which is 3.57 greater.

The thesis defended in this paper is that climate policies may justify the use of a negative discount rate for their evaluation. There are two important steps in the argument, each of which is an interesting separate thesis: 1) Different policies should be evaluated with different discount rates depending on what populations are impacted; 2) In the long run only the worst scenario for the worst-off fraction of the population counts.

Our thesis is at odds with the conclusion of Chapter 7 in Posner and Weisbach (2010 -hereafter PW), even though we share the same premises: impartiality between generations, compatibility with ethical principles, and taking opportunity costs into account. They advocate using the market interest rate as the discount rate for the selection of particular projects. It is an interesting question to understand how our similar premises can deliver very different practical conclusions.

The main difference is that we disagree on how to make use of the discount rate. For us, it is a tool to assess and compare different consumption paths or money flows in terms of net present value; for PW, it serves to take into account the opportunity cost of the investment.

Our thesis is somewhat closer to Weitzman's and Gollier's arguments in favor of using a small, possibly a negative discount rate [START_REF] Weitzman | Why the far distant future should be discounted at its lowest possible rate[END_REF][START_REF] Gollier | Discounting an uncertain future[END_REF]), but involves different reasons. The paper is structured as follows. In the next section we briefly review the arguments of the advocates of the descriptive and the prescriptive approaches, in the debate about discounting (the "ethicists" and the "positivists", as called by PW), and discuss agreements and disagreements with PW. Then in the next two sections we explain the methodology of computation of the discount and propose a reformulation of the Weitzman-Gollier set of arguments to the context of risk. The following section explains our core arguments. For the sake of an easy presentation, the bulk of the paper is formulated in the context of utilitarian reasoning, but we explain in the penultimate section why the utilitarian approach must be replaced with a more promising approach and how this can affect the discount rate. The final section concludes.

The descriptiveprescriptive debate

The opposition between a descriptive approach and the prescriptive approach is hard to understand when it is labeled in this way, as suggested by [START_REF] Arrow | Intertemporal equity, discounting, and economic efficiency[END_REF]. It is equally puzzling when the "ethicists"-"positivists" labels are used.

As PW write, "in the end, of course, the positivists' approach is worth nothing unless it can be defended on ethical grounds." (p. 150) So, the debate is not between ethics and something else, it is a debate within ethics.

The descriptive approach invokes two ethical arguments. The first is that market rates reflect the preferences of the population, so that it is undemocratic to propose using different rates [START_REF] Nordhaus | The Stern Review on the Economics of Climate Change[END_REF]. The climate economists who propose using lower rates for climate policies are imposing their views on a population that appears to care less about the future than they do. (Gollier, in There are several mistakes in the argument. The first is to seek to impose the population preferences on every evaluator. Obviously, there are many views in the population. If an evaluator wants to examine a development path with a great concern for the future, there must be some people in the population who share this concern.

Even if nobody shared this concern, the evaluator might be right against everyone else. Just as there is freedom of thought, just as different political parties can have their own platforms, there should be a space for economic evaluation that embodies various views about social welfare and the principles of intergenerational equity.

The reply to this objection will certainly be: Any evaluation is admissible, but the government, in its decisions, cannot impose idiosyncratic views on the whole population. This is a powerful argument, even though history contains praiseworthy examples of governments imposing policies against the majority opinion (e.g., the abolition of death penalty in France in 1981). But this powerful democratic argument does not imply that the market interest rate should serve as the discount rate. It only requires a democratic debate to take place. This debate will have to ponder the various arguments underlying the computation of the discount rate. One cannot pretend to know the conclusions of this debate in order to prevent some propositions from reaching the debate. Democratic principles cannot be used to bar some (minimally sound) ethical principles from the forum.

One could still try to argue that the market does tell us something about the population preferences over intertemporal trade-offs. The market interest rates are determined by the joint effect of technical possibilities (the productivity of capital) and the willingness of investors to transfer wealth into the future, with a benefit. Just like the relative market price of oranges and pears implies that all buyers active on both markets are willing to trade oranges for pears at this relative price, the interest rate indicates the investors' and savers' common marginal rate of substitution across time. This is true but investors and savers make decisions to transfer wealth for themselves by a few years. If they were asked to transfer wealth to other people living all over the world in many decades, they might express very different preferences. The financial markets don't ask them this outlandish question, and therefore we cannot pretend to use their answer to a different question for this purpose. Observe, moreover, that the market interest rate also depends on the distribution of wealth in the population, which has no reason to be particularly democratic.

Even if there were markets in which people could express such preferences (private donations to environmental NGOs focused on the climate might be the relevant source of information), it is doubtful that such preferences would be more respectable than the outcome of an outright democratic debate involving the relevant expertise and considering the best ethical arguments.

In conclusion, if experts like Stern propose a series of reasonable arguments leading to the conclusion that climate policies should be evaluated with a discount rate that is much lower than the market interest rate, they cannot be dismissed as undemocratic and off track. They should be admitted to the democratic debate and their arguments should be carefully listened to (without guarantee that they will be adopted). (computed with whatever discount rate seems appropriate) of all options, including ordinary market investments. Clearly, with this methodology, if one option costs less today or pays more tomorrow (or both) than another option, it will be deemed preferable, whatever the discount rate!

There is therefore no danger that adopting a lower discount rate than the market rate may induce inefficient (i.e., dominated at each period) choices. It will only imply making different choices among the efficient (i.e., undominated) options. With a lower discount rate, one will choose to invest more for the future, but one will never be tempted to invest at a low rate of return when a high rate of return is possible. If a business-as-usual investment policy that puts all savings in the financial market brings more benefits to the future generations than a mitigation policy aimed at curbing GHG emissions, even the most devoted disciple of the Stern Review will approve it.

To illustrate this point, Table 2 presents an example of four policies, with their undiscounted benefits and their NPV according to two different values of the discount rate (1.4% is from the Stern Review, 5.5% has been advocated by Nordhaus). Policies A and C are market investments at 5.5%, policy B is a climate policy with impacts equivalent to a 10% monetary return, policy D is a climate policy with impacts equivalent to a 6% return. Policies A and B yield returns in 100 years, while policies C and D pay in 500 years. The 5.5% discount rate enables us to check that climate policies B and D are not dominated by market investments, but is it helpful to choose between B and D? It suggests that B, which has impacts of greater value in the shorter term, is preferable to D. But a lower discount rate at 1.4% suggests otherwise, while still revealing that these climate policies are not dominated by the market. It is therefore important to have a good discount rate, not to check if the market dominates a climate policy -because any discount rate will do for that limited exercise, but to be able to choose between undominated policies.

In conclusion to this point, when PW write: "even if the ethicists' arguments are entirely correct, we must still carefully consider the opportunity cost of projects and pick those with the highest returns," we fully agree, and every reasonable "ethicist" should agree, too. But this does not imply that the market rate of return should be used for the evaluation of projects.

Note that the use of discount rates would be superfluous if the problem were to choose between policies with similar time profiles like policies A and B in Table 2, because their own rates of return can be directly compared. It is only when there are time trade-offs that computing the NPV becomes useful, as in the comparison between policies B and D. (Actually, as we will show later, projects with the same time profile may affect different populations, thus deserving different discount rates, which is equivalent to incorporating social benefits in the computation of their rate of return.)

There is another related methodological issue on which we disagree with PW. They propose to use the discount rate in a limited way: "Discoun-ting… should be seen only as a method of choosing projects, not as a method of determining our obligations to the future." (p. 168) This separation is a direct consequence of the tension produced by their idea that one should use the market discount rate for the choice of projects, but nevertheless follow the ethicists to decide how much to save for the benefit of the future generations. In other words, two discount rates would be used in the methodology proposed by PW, although they do not make it fully explicit (and would perhaps allow for other considerations than standard welfarism to determine how much should be saved).

Horizon A B C D 0 -1 -1 -1 -1 100 211 13781 0 0 500 0 0 4,23E+11 4,50E+12
The low discount rate of the ethicists would serve to check if more should be invested, whereas the choice of particular projects would use the market rate in order to make sure to pick efficient options.

There is no need for such a dichotomous methodology. The "ethically right" discount rate can be used both for the selection of projects and for deciding when to stop saving for the future, which constitute in fact one and the same set of decisions -selecting the projects includes choosing the amount that is invested. One will start with the highest-NPV investment plans (which are those with the highest rate of return for their particular time profile) and go on as long as the NPV of the remaining projects is positive. Note that the discount rate itself goes up in the process, because as more is invested for the future, the future generations grow better off, which tends to raise the discount rate (see below for an explanation of this phenomenon). Therefore a low discount rate advocated now, on the background of a business-as-usual scenario in which the future generations are in jeopardy, need not be the indication of the market rate that will prevail after the recommended investment has been done. The convergence value of the market rate will be somewhere between the initial market rate and the initial discount rate.

Finally, let us briefly consider another objection raised by PW against the ethicists. They claim that choosing projects as the ethicists propose may be futile when private decisions to dissave may partly undo the public investments. Again, the ethicists can only agree and proclaim their innocence. Their criteria are meant to bear on final consequences, not on mistaken estimates of the consequences. If a certain Ricardian equivalence implies that the government cannot influence the macroeconomic savings rate, there is no point for the government to try to change it and no point in applying any discount rate to this kind of decision. If, in a less extreme case, public savings partly crowd out private savings, this must be taken into account, too.

In this section we have defended the ethicists and their prescriptive approach against the attacks of the positivists with their descriptive approach, but we haven't said a word, almost, about what the prescriptive approach says and how it computes its discount rate. To this we now turn.

The methodology of discounting

As announced in the introduction, for simplicity we will provisionally adopt the utilitarian way of defining social welfare, and more precisely assume that social welfare can be computed as the sum of , where is the consumption level of individual . What is important about this approach is that the function is assumed to be the same for all individuals, which means in particular that there is no preference for earlier generations against future generations, and we consider that this function embodies the preferences of the evaluator about inequalities in consumption -which means that, in terms of consumption, the approach is prioritarian rather than utilitarian. We will even adopt an assumption that is common in the economic literature, and gives a special form to the utility function:

, where can be interpreted as a coefficient of aversion for consumption inequality, and is the minimum level of consumption that is required to make utility positive. With this function, the marginal utility is equal to , which makes things quite simple.

For the sake of simplicity we ignore the risk that future generations will not exist. This issue will be introduced later in the paper. and by comparing this expression with the discounted sum , one obtains the discount rate by the formula The discount rate is a direct expression of the relative priority of the two individuals (or generations), modulated by the time distance between the two individuals. If the future individual is better off than the present individual, the expression is greater than one, i.e., the discount rate is positive.

This methodology gives the discount rate that can serve to evaluate small projects. Any project that yields a rate of return greater than the discount rate is beneficial to social welfare. For big projects, the marginal utilities are no longer acceptable in the computation and one has to make a direct evaluation of the change in social welfare.

When the marginal utility is equal to the formula for the discount rate simplifies into where is the annual growth rate of consumption between and . When is small, this formula can be approximated by the famous Ramsey formula .

A reasonable value for inequality aversion is (the Stern Review took ) while a standard estimate for the growth rate is 1.3 (as in the Stern Review), which implies a discount rate of approximately 2.6% (Stern adds a 0.1% term due to the risk of extinction of humanity, we ignore this term for the moment). This discount rate may be much lower than the market rate of return, but, as explained in the previous section, this is not particularly threatening for this methodology and there is no danger of choosing dominated investment plans. [START_REF] Weitzman | Why the far distant future should be discounted at its lowest possible rate[END_REF] has proposed an interesting argument in favor of adopting even lower discount rates for investments that pay in the very long run, when there is a background risk on the growth rate. We first present a variant of it which enables us to connect the argument to Ramsey's formula, in a similar fashion as in [START_REF] Gollier | Discounting an uncertain future[END_REF]. Suppose that there is uncertainty about future consumption, and that our criterion is the expected value of social welfare (which is also, in the case of utilitarianism, the sum of expected utilities). Let us again consider two small changes and . Unlike the level of consumption, these changes are certain. The change in social welfare is now equal to where denotes the expected value, and the formula for the discount rate becomes:

Discounting under risk

( now denotes the discount rate, and we keep the notation for the discount rate in a particular state of nature.) What Weitzman noticed is that this kind of formula involves neither the expected value of nor the expected value of but the expected value of More precisely, recalling from the previous section that in every particular state of the world, , one has

Now, what is remarkable about this expression is that it has the form of a well-known quasi-arithmetic mean of the form which is known to converge to the minimum value of when tends to infinity. Therefore, in the very long run, the discount rate under risk converges to the lowest possible value of the risk-free discount rate. This is a remarkable result. For long term evaluations, one can focus on the worst scenarios in which future consumption is the lowest and the corresponding discount rate is the lowest.

PW propose an intuitive explanation of this reasoning directly based on Weitzman's original formulation. For every possible value of there is a corresponding discount rate (which Weitzman and PW assimilate to the market rate, rather than the Ramsey rate, in the corresponding scenario). The expected present value of the investment, when all possible discount rates are considered, is then equal to and by comparing this expression to , one directly obtains the desired formula. This explanation remains, however, a little mysterious because it is not obvious why one should compute the expected present value of a project rather than some other formula. In fact, in general the expected present value is not correct. What should be computed, in the Ramsey approach, is the ratio of the expected values of the marginal utilities, but when there is no risk about the present consumption the expected value of the marginal social value of is just the sure value and therefore the ratio of expected values is, in this specific case, the expected value of the ratio.

To illustrate the result, consider the situation in which with 80% chance, the growth rate will be 1.3% on average in the future, but there is 20% chance that it will be zero. Let us retain so that the risk-free discount rate is either 2.6% or 0%. This example shows that the convergence to the lowest value may be rather slow, but also that the discount rate is very quickly well below the average discount rate 0.8x2.6 = 2.08%. Admittedly, risk typically affects not just the background growth rate of consumption but also the future yield of the investment. Taking account of this risk further complicates the discount rate, and it is easy to show that the discount rate applied to expected benefits should be lower for projects with greater returns in states in which the marginal social utility of the beneficiaries is greater.

For our discussion of climate policy, it is especially relevant to see if the Weitzman convergence to the lowest discount rate remains valid in this context. If risk takes the form of an uncertain rate of return on the investment, then, keeping the possible rates of return in the various states of nature fixed and varying the horizon, one obtains an interesting generalization of Weitzman's result:

In the long run the discount rate that should be applied to the expected return of the investment tends to

where and are, respectively, the discount rate and the rate of return in the various possible states of the world. When is not a random term, this expression singles out the lowest discount rate, as was explained previously. But when is random, other possibilities appear.

Let us first explain how to derive this result. If one invests $1 in a project that has a random rate of return in periods, the marginal effect on expected social welfare is equal to

In present value, this reads Now, suppose one wants to equate this to a NPV with a discount rate applied to expected returns:

From this equation one immediately derives At the numerator as well as at the denominator, the greatest term dominates when goes to infinity, which implies the generalized formula.

This generalized formula is particularly interesting when it is used to compare different kinds of investments. An investment that is most profitable in good times (such as, typically, a market investment) may have a low ratio in relatively good times in which is high, or in bad times in which is low. In both cases, the appropriate discount rate is greater than the lowest . In contrast, an investment that is most profitable

Horizon

Discount rate 50 1,74% 100 1,35% 200 0,80% 500 0,32% 1000 0,16% in bad times (such as a climate policy that averts dangerous climate change) will definitely have to be evaluated with the lowest because its greatest rate of return will happen in the states with the lowest discount rate. In conclusion, Weitzman's argument may have to be watered down when applied to ordinary investments with returns that are correlated with growth, but seems to retain its full force for projects that are aimed at protecting us against climate hazards.

Weitzman ( 2009) proposed an even more striking ("dismal") thesis by arguing that the discount rate can be made arbitrarily close to -100% even in the short run. But this argument is much less convincing and a discussion of it can be found in [START_REF] Fleurbaey | Intergenerational equity, risk, and discounting[END_REF].

Priority for the poor in the long run

We do agree with Weitzman, however, that a negative discount rate may be justified for climate policies. There is another line of argument that, combined with the phenomenon, described in the previous section, of convergence toward to the lowest rate in the long run, reinforces the presumption that negative values are relevant.

The debate about "the" discount rate is somewhat misleading because there is not a single discount rate but as many discount rates as there are distributions of costs and benefits among different populations. We have already seen this phenomenon when the discount rate to be used depends on the time lag between generations, as in Table 3.

More generally, the formula that determines the discount rate is about changes in the consumption of two individuals The value of the discount rate depends on the consumption levels of these two individuals. Imagine now that two individuals from a future generation, not just one individual, will benefit from a change in their consumption, so that we have to deal with the formula As before, one can compute the present value by dividing by the marginal utility of , and compare this with a formula involving person-to-person discount rates: Now this formula is structurally similar to the formula obtained in the case of risk. Imagine that and share the benefit of the investment in fixed proportions: and The above expression can then be written as for The same argument as in the previous section implies that in the very long run, i.e., when tends to infinity, will converge to the smallest value of person-to-person discount rates. The smallest value is obtained for the individuals who are the worst off in the future generation. Therefore, in the very long run, only the worst off of the future generations matter. More precisely, one must focus on the worst off among those who benefit from the investment. Those whose share is null play no role in the formula.

A complication is that the investment cost is generally paid by several members of the present generation. If one thinks of a public policy such as a mitigation effort to reduce GHG emissions, many individuals may be involved. Let us therefore consider the problem when several people from the present generation contribute in fixed shares:

and But to simplify the presentation, let us come back to a situation in which only one individual from a future generation stands to benefit from the investment. The formula is now the following:

The present value can therefore be written as This formula has the opposite behavior to the previous one. When tends to infinity, tends to the greatest value of the person-to-person discount rates. What is remarkable is that the greatest value is obtained for the worst off of the present generation, among those who share in the cost.

When many individuals share the cost now and many individuals share the benefit in the future, these two results remain jointly valid, even though the formula is more complicated: In the very long run, the discount rate converges to the worst-off-toworst-off discount rate, among the individuals who are affected by the change in consumption to be evaluated (for a formal statement and a proof, see [START_REF] Fleurbaey | Intergenerational equity, risk, and discounting[END_REF]. This holds whatever the shares, although, of course, the speed of convergence is influenced by the shares.

Table 4 illustrates this phenomenon with four individuals, two from each generation. Shares in cost and benefit are supposed to be equal (halfhalf ) in every generation. In the present generation the poor has a consumption of 1 unit, the rich has a consumption of 5 units. The dynasty of the poor has a consumption growing at 1.3% per year, the dynasty of the rich enjoys a growth rate of 1.5%. We keep .

The table shows that the poor-to-poor discount rate is a good indication of the social discount rate in this context, even at a moderate horizon (but this depends on the shares and is not true in general). Observe that the rich-to-poor and the poor-to-rich discount rates change with the horizon because their relative consumption does not evolve according to a constant growth rate.

In the beginning the poor dynasty remains poorer than the rich of the first generation, which justifies a negative rich-to-poor discount rate, whereas the rich dynasty is much richer than the poor of the first generation, which justifies a high poor-to-rich discount rate. In the long run, the relative consumptions tend to follow the growth rate of each dynasty, which explains the convergence toward a discount rate that is specific to the beneficiary rather than specific to the donor. (In this particular setting with separate dynasties with specific growth rates, the discount rate is also a weighted average of the two dynasties' discount rates, and converges to the lowest discount rate, i.e., the discount rate of the dynasty with the lowest growth rate, which ultimately becomes the poorer dynasty even if it starts richer.)

We now come to the main thesis of this paper.

Why should climate policies be evaluated with a negative discount rate? The person-to-person discount rate is negative when the present donor is richer than the future beneficiary. Given the above result, if we consider long-run policies, the discount rate should be negative when the poorest contributors to the policy are richer than the poorest beneficiaries. It is plausible that many climate policies satisfy this condition. Mitigation efforts, when they are well conceived, should put the burden on the high emitters who are typically among the affluent members of the present generation, but they will benefit many members of future generations. Moreover, it is often said that the most vulnerable to climate change are the poorest, so that many beneficiaries in the future will be among the poorest of their generation. Can we hope that the poorest of future generations will be better off than the middle class of the present generation? This appears, sadly, unlikely. Therefore climate policies that avoid imposing a burden on the poor members of the present generation deserve to be evaluated with a negative discount rate. Another element reinforces the thesis. Weitzman's result of a convergence toward the lowest discount rate in the case of risk combines with the result presented in this section. In the very long run, the discount rate converges to the worst-offto-worst-off discount rate of the worst-case scenario. Therefore, even if there are favorable scenarios in which the destitute populations catch up and reach good standards of living, it is enough to assign a positive probability to dark scenarios in which the standards of living of the poorest stagnate in order to validate our conclusion about the negative discount rate for climate, especially mitigation, policies.

Horizon

Of course, this does not mean that such policies should have greater priority than other policies such as redistribution toward the poor members of the present generation [START_REF] Schelling | Intergenerational discounting[END_REF]. The choice of the best policies, as we have seen in the second section, involves a comparison of present values, not just checking that the chosen policy improves on the status quo. At least, however, we want to argue strongly against the popular thesis that the market rate should be applied indiscriminately to the evaluation of all policies, independently of the affected populations.

Beyond utilitarianism

So far we have adopted the utilitarian approach, which indeed dominates in the debate about discounting for the long run. The utilitarian approach is quite acceptable in the absence of risk, because the utility function can then be chosen, as suggested above, to embody the aversion to inequalities in consumption that the evaluator endorses. Formally, the utilitarian social welfare can then also be adopted by prioritarians and egalitarians who accept the property of subgroup separability that underlies the additive form of the criterion. (Subgroup separability means that the evaluation of a change affecting a subgroup of the population can ignore the consumption level of the unconcerned individuals and focus on the affected subgroup only. By an important theorem due to Gorman and Debreu, subgroup separability implies that the evaluation criterion can be represented by an additive function.)

In the presence of risk, things are less easy. The coefficient of inequality aversion also becomes a coefficient of risk aversion if the utilitarian criterion is then applied as the sum of expected utilities (or equivalently, the expected sum of utilities).

There is therefore a dilemma. Either one respects the risk aversion of the population (assuming away a potential heterogeneity of risk preferences across individuals), which severely constrains the degree of inequality aversion, or one adopts a coefficient of inequality aversion on the basis of ethical principles and then potentially imposes on the population a degree of risk aversion that appears paternalist. This is a classical problem in social ethics [START_REF] Harsanyi | Cardinal welfare, individualistic ethics and interpersonal comparisons of utility[END_REF] viewed it as a key justification of utilitarianism), and it has been recently mentioned in the context of discounting by [START_REF] Kaplow | Discount rates, social judgments, individual risk preferences, and uncertainty[END_REF].

Fleurbaey ( 2010) proposed a compromise. The idea is that respecting preferences is much less compelling under uncertainty than in a risk-free context, because in the context of risk, by definition, individuals are not perfectly informed about the consequences of their decisions. In particular, respecting preferences under risk may even appear to betray informed preferences when the evaluator has information about the final distribution. Suppose for instance that individuals are willing to take a risk but that it is known in advance that the only consequence of this risk is a widening of inequalities, without any overall gain. At the individual level the risk may appear attractive, but at the social level it is already known that many will be unlucky and that they actually act against their true interests when they are willing to take the risk. When making a decision under risk, each individual focuses on his own payoffs and ignores the correlation with other individuals. A social evaluator can take account of this correlation and forecast how many individuals will turn out to have acted against their ultimate interests.

This observation leads to the conclusion that respecting risk preferences is not always necessary, but it also suggests that respecting risk preferences remains an attractive idea when there is perfect correlation between individuals, because in such a situation an evaluator cannot forecast if some of them are acting against their interests. Fleurbaey (2010) shows that when the requirement to respect risk preferences is limited to the case of perfect correlation, other criteria than utilitarianism become acceptable, permitting a greater degree of inequality aversion. There is a theorem stipulating that, under minimal conditions of rationality under uncertainty, all such criteria must take the form of the expected value of the equally distributed equivalent (EDE) utility, which is the level of utility that would yield the same social welfare if it were equally distributed across all individuals.

Let us illustrate this with a particular functional form. Suppose that represents the risk preferences of the individuals, assuming away any heterogeneity across individuals in order to keep things simple and in line with the literature on discounting. Suppose that in absence of risk one would like to use the prioritarian criterion Then the expected EDE criterion takes the form where denotes the inverse function and is the number of individuals.

In [START_REF] Fleurbaey | Intergenerational equity, risk, and discounting[END_REF], we study how this kind of criterion can be used in the computation of the discount rate. What is important is that the discount rate can then be approximated by the usual discount rate obtained for the additive social welfare function , to which one has to add a term that depends (positively) on the correlation between the well-being of the beneficiaries of the investment and social welfare at the global level. (There is an additional term reflecting the attitude of the criterion to population size -this issue will be explained later.) It is not easy to figure out whether this result pushes in the direction of raising or lowering the discount rate for climate policies. A first issue is whether climate risks generate common risks for most populations or induce negative correlations. In the case of common risks, the correlation term is positive and tends to raise the discount rate. The case of negative correlations is possible if a change in the climate would actually be beneficial in the high latitudes where the most affluent populations are now settled, whereas it would be dramatic for the subtropical areas in which the most vulnerable populations live.

But even if negative correlations occur, it is still possible for the correlation term to be positive. Indeed, recall that in the long run, the poor members of the future generations are those who matter for the discount rate. If the degree of inequality aversion (i.e., the concavity of function ) is strong, social welfare as measured by the EDE is then close to the lowest utility in society, and therefore directly correlated with the well-being of the worst-off.

Not much is known about the size of the correlation term and simulations are not easy to perform because they require considering scenarios that describe the situation of the whole human species, from beginning to end. An example of simulations is provided in the appendix, showing that the difference between criteria may be far from negligible. This compels us to mention another issue that cannot be ignored when the risk of extinction is considered. In the Stern Review there is a 0.1% additional term that comes from the estimated exogenous 1/1000 risk per annum of extinction of the species due to cosmic phenomena (meteors, eruptions) or unforeseen disruptions of life systems (pandemics). The underlying utilitarian reasoning is that the expected value of total utility is equal to

The computation of the discount rate then involves an additional term in the formula of the marginal change in social welfare: implying which is approximately equivalent to adding 0.001 to the initial value (when the latter is close to 1).

There is, however, an issue that such an approach raises. Different values for the longevity of the human species imply different sizes for the total human population, which requires taking a stance on the question of the optimal size of the population. In the utilitarian galaxy, there are three popular approaches. Total utilitarianism, implicitly adopted above, adds utilities considering that a new member with a positive utility always improves social welfare. Critical-level utilitarianism adds utilities but deducts a fixed amount for every new member. In other words, it computes the sum of , which means that adding a new member to society is considered beneficial only if his utility is above . The introduction of the critical level, however, does not affect the marginal utility of consumption for existing members and therefore does not affect the discount rate. The third approach is average utilitarianism, which divides total utility by the size of the population and considers that adding new members is desirable only when their utility is above average. The computation of the discount rate for this third approach is substantially different and, to the best of our knowledge, has not been explored.

With the EDE criterion introduced in this section, one has various options for the critical level, but unlike utilitarianism there is only one constant critical level that can be taken, and this is the lowest possible utility. Another salient option is to take a critical level that is equal to the EDE itself. For a significant degree of inequality aversion, the EDE is close to the lowest utility in the population, which may not be an unreasonable option for the critical level. Depending on which of these two options is retained, the discount rate contains an additional term that is negatively or positively related to the correlation between the well-being of the beneficiaries of the investment and the population size (see Fleurbaey and Zuber's paper for details).

Conclusion

Let us briefly wrap up the argument of this paper.

The discount rate only serves to measure the relative social priority of different individuals belonging to different generations. Therefore there is no need to worry about comparing the discount rate to the market interest rate, as a rational evaluation in terms of present value at the chosen discount rate will never fail to avoid dominated investments, and never fail to choose those with the greatest rate of return.

The key message of this paper is that discount rates are really to be computed between individuals (person-to-person), which gives a great role to inequalities within and between generations.

In the very long run, Weitzman's observation that the worst-case scenario drives the discount rate has to be supplemented by the fact the situation of the worst off at both ends of the investment will also drive the value of the discount rate.

Therefore, if climate policies such as mitigation efforts are paid by the affluent populations of the present generations and greatly benefit the worst off of the distant future generations in the most catastrophic scenarios, it is very likely that the correct discount rates for the evaluation of such policies should be negative, which means that a dollar of benefit in the distant future is worth more than a dollar of effort today.

In conclusion, we would like to recall that a rigorous evaluation of climate policies is particularly challenging because it requires rethinking the welfare economics of risk, time, and population.

In such endeavor the utilitarian criterion, which remains prominent in the debates about discounting, should be questioned and, perhaps, replaced with other criteria that better combine a certain respect for the risk preferences of the population and a substantial degree of aversion to inequality.

Finally, we should recall a point that has been made already by [START_REF] Stern | The Economics of Climate Change[END_REF]. The discount rate is useful to evaluate small transfers of consumptions across individuals living at different times.

It is not the all-purpose tool that can serve for all evaluations. It is not adapted to large scale changes, and it is also not adapted to evaluating policies that change the size of the population or the probabilities of different scenarios. For such policies one has to go back to the underlying social welfare criteria. This is an additional reason to pay attention to the selection of such criteria on sound ethical principles.

Fondation Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

The main lesson of this table is that the difference between the criteria is far from negligible. The difference is smaller between the EDE and average utilitarianism than between these two criteria and total utilitarianism, but other simulations done by the authors for different consumption paths show that other patterns are possible. The fact that the discount rates are not very different between the utilitarian criteria and the EDE (which introduces additional inequality aversion via function ) is a direct consequence of adopting a utility function that varies very little between the consumptions of one and five units, so that the social priority between the two dynasties is mostly determined by marginal utility, as in utilitarianism.

The discount rates tend to zero in the table for the largest horizons, because the difference in consumption between the donors (5 units) and the worst-off in the worst scenario (1 unit) becomes very small in terms of annual growth rate. Observe that -3% per annum implies a strong priority for the future: it means that it worth sacrificing $4.6 now to transfer $1 fifty years into the future.

Fondation

  Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

  Suppose that is reduced by a small amount and , which occurs periods later, is increased by a small amount . Is this good for social welfare? If the changes are infinitesimal, one can use the marginal utilities to evaluate the changes, and the variation in social welfare is then equal to This can be expressed in present value by dividing every term by the marginal utility of , Fondation Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

Fondation

  Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

for

  Fondation Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

Fondation

  Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

Table 1 :

 1 The implications of different discount rates

	Horizon	1.4%	2.7%	Ratio
	50	2.00	3.79	1.89
	100	4.02	14.36	3.57
	200	16.13	206.11	12.78
	500	1,044	609,848	584
	1000	1,091,327	371,914,916,666	340,791

Table 2 :

 2 How to choose policies with various discount rates

Legend: Policy A costs $1 today and pays $211 in 100 years; at a 1.4% discount rate, the present value is $51.66; at a 5.5% discount rate, it is $0.

Table 3 :

 3 Discount rate for different horizons when the growth rate is either 1.3% or 0%

Table 4 :

 4 The discount rate at different horizons, for society as a whole (column 2) and the person-to-person discount rates (columns 3-6)

	Climate policies deserve a negative discount rate	12/17

Fondation Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.msh-paris.fr -FMSH-WP-2012-19

//www.msh-paris.fr -://www.msh-paris.fr -FMSH

Appendix

In this appendix, we provide a simple illustration of the discount rate obtained by the various utilitarian criteria and by the EDE in a simple twostate scenario. We assume that in the favorable state the human species spans 2 million years, comprising 80000 generations (a generation is 25 years), and we assume that 4000 generations (100,000 years) have already lived up to now. The world population is assumed stable from now on, with 3 billion members per generation (three generations overlap at any given moment in time). The past population since the origins is assumed to have grown from an initial number of two individuals at the growth rate of 0.3% per generation until 10,000 years ago, when the growth rate rose to 2.62% per generation.

There are two dynasties, one consuming 1 unit today and the other consuming 5 units today. The evolution of consumption over time can hardly be assumed to be exponential over such a long horizon. Indeed, assuming that the first generation consumed 0.04 units (per capita), 1 the growth rate of per capita consumption up to now would have been 0.081% on average per generation, which seems very small, but would imply that the last generation in 1.9 million years should consume about 10 26 as much as now, a big number that is probably much greater than the number of planets in the whole universe (the number of stars is estimated below 10 24 ).

We will instead assume that consumption is constant, except for the period 1760-2260, in which the growth rate per year is approximately 1.3% and consumption grows by a factor 625. Inequality remains constant between the two dynasties. After the growth transition, future generations consume 25 times our current consumption level.

In order to introduce risk we also assume that with a 20% probability, consumption will stagnate forever from now on, and that only 40,000 generations live (one million years). This is the unfavorable state.

We retain the utility function and take , with and (This is half the first generation's consumption.) We consider a policy that is paid only by the rich dynasty of the 4000 th generation (i.e., the present generation) and benefits equally to every dynasty in future generations, i.e., we ask how much $1 equally shared between a rich and a poor in the future is worth compared to $1 paid by a rich today. The horizons in Table 5 are expressed in years, and discount rates are also per annum, as in the previous tables.


Kidiyo Kpalma

Cong Bai

Miloud Chikr

E L Mezouar

Kamel Belloulata

Nasreddine Taleb

Lakhdar Belhallouche

Djamal Boukerroui

A New Histogram-based Descriptor for Images Retrieval from databases

Keywords: pattern recognition, image description, image retrieval, texture image, differential-turning angle scale space, turning angle scale space, bull's eye performance

In this paper, we propose a new approach for designing histogrambased descriptors. For demonstration purpose, we generate a descriptor based on the histogram of differential-turning angle scale space (d-TASS) function and its derived data. We then compare the proposed histogram-based descriptor with the traditional histogram descriptors in terms of retrieval performance from image databases. Experiments on three shapes databases demonstrate the efficiency and the effectiveness of the new technique: the proposed technique of histogram-based descriptor outperforms the traditional one. These experiments showed also that the proposed histogram-based descriptor using d-TASS function and the derived features performs well compared with the state-of-theart. When applied to texture images retrieval, the proposed approach yields higher performance than the traditional histogram-based descriptors. From these results, we believe that the proposed histogram-based descriptor should perform efficiently for medical images retrieval so we will focus on this aspect in the future work.

Introduction

With the extraordinary growth of image databases due to the development of digital systems, automatic image retrieval within large databases becomes necessary. To make them more efficient and more easy to use, retrieval systems must be based preferably on images content. In the present study, we use the d-TASS (differential-Turning Angle Scale Space) function and two related measurements to generate their corresponding histograms from which the shape descriptor is built. The d-TASS function introduced in [START_REF] Kpalma | Planar shapes descriptors based on the turning angle scalogram[END_REF] and [START_REF] Kpalma | Turning Angle Based Representation for Planar Objects[END_REF] is known to yield characteristic properties and to be invariant under translation, rotation and scale change. The rest of the paper is organized as follows: after a brief description of the d-TASS construction and the definition of the related features, in section 2, the proposed descriptor is presented in section 3. Section 4 presents some experimental results and compares the proposed and the traditional histogram-based descriptor generation. We also present some comparison of the proposed approach and some existing techniques of the state of the art. Finally, section 5 gives some discussions and concludes the paper.

Description of the features

As presented in [START_REF] Kpalma | Turning Angle Based Representation for Planar Objects[END_REF], the input contour is defined by a set of N points ordered counter clockwise in the plane. Given a starting point P 0 , the points are numbered from P 0 to P N-1 . The vector n V originating at P n and oriented towards P n+1 makes an angle n θ with the x-axis. This angle is called the turning angle. Figure 1, illustrates the principle. From this angle, we define the differential turning angle (d-TA) function n ϕ :

1

- - = n n n θ θ ϕ (1)
P n P n-2

P n-1

P n+1 P n+2 n V r 1 - n V r 2 - n V r ϕ n 1 - k V r k V r P k P k-1 ϕ k θ n θ n-1

Fig.1: Illustration of the differential-turning angle

The d-TA function yields an expression of the curvature of the contour thus its zerocrossing corresponds to that obtained from curvature analysis [START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF] and [START_REF] Mokhtarian | A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves[END_REF].

Generation of the d-TASS function

By definition, the d-TASS function is generated, by progressively smoothening the contour with a Gaussian filter [START_REF] Kpalma | Planar shapes descriptors based on the turning angle scalogram[END_REF], [START_REF] Kpalma | Turning Angle Based Representation for Planar Objects[END_REF] and [START_REF] Kpalma | Multiscale contour description for pattern recognition[END_REF]. Given a Gaussian kernel with a standard deviation σ 0 , a progressive filtering is performed by iterating the operation so that the filtering scale From the d-TA function, three essential points are derived to give the d-TASS map [START_REF] Kpalma | Turning Angle Based Representation for Planar Objects[END_REF]. These essential points are:

• α-points corresponding to local minimums of the d-TA function,

• β-points corresponding to local maximums of the d-TA function,

• γ-points corresponding to zero-crossings of the d-TA function.

Derived features

Generating the d-TASS function and the corresponding map is done according to the following procedure. The input contour is resampled to generate 360 equidistant points [START_REF] Kpalma | Planar shapes descriptors based on the turning angle scalogram[END_REF] and [START_REF] Kpalma | Turning Angle Based Representation for Planar Objects[END_REF] ordered counter clockwise. From this input sequence, the d-TA function is computed to generate corresponding essential points. In this study, we focus on the γpoints only because they will be used to set the stop condition of the iteration process. Like in the case of curvature scale space (CSS) [START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF], the stop condition is that there is no more γ-point at the current scale.

At each scale σ s (iteration s), γ-points are detected and the following desired features are computed:

• an area function: area of the triangle defined by the current point P n , the following one P n+1 and the mid-point P M between two consecutive γ-points (see Fig. These measures are characteristic of the contour segment. By iterating the process for a range of scales (until the stop condition), we obtain the area scale space function (ASSF) and the distance scale space function (DSSF), respectively, for the area function and the distance function. Figure 3 illustrates the three corresponding functions for the contour shown in Fig. 2.a. The superimposed red curves represent the map of γ-points. On this figure, the x-axis represents the curvilinear abscissa and the y-axis the scale (or the iteration number).

Definition of the proposed descriptor

As said before, the proposed shape descriptor is based on the histograms of the d-TASS function and the related ASSF and DSSF functions. The idea of the proposed technique of histogram generation is based on the approach used in [START_REF] Bai | A new descriptor based on 2D DCT for image retrieval[END_REF] and in [START_REF] Zhong | DCT histogram optimization for image database retrieval[END_REF]. The histogram of the whole database is generated to produce a reference histogram from which we calculate decision levels that enable us to generate descriptors. Commonly and particularly in [START_REF] Zhong | DCT histogram optimization for image database retrieval[END_REF], histogram descriptors are designed by simply uniformly quantizing the feature of interest (say the image grey scale) and then counting the number of elements having the same value and then taking the number of desired bins. This approach has proved to provide efficient results. The proposed technique is borrowed from image coding or image restoration where histogram equalization is applied. In image processing, it is shown that this adaptive quantification can help to reduce the bit rate and to enhance signal-to-noise ratio. Starting from this observation, we propose to design the histogram-based descriptor by defining the histogram bins like in the case of adaptive quantization. First, the reference histogram of the database is generated and normalize to sum to the unity. Then the adaptive quantization is designed such a way that the "probability" to get a value between any two consecutive decision levels is equal. Knowing the desired number K of bins, this probability equals PK=1/K. This enables to generate decision levels that are then used to define the descriptor for the query. Finally, for each individual histogram, use the generated decision levels and sum up the bins between two consecutive levels. This leads to the descriptor for the contour under study. Figure 4, illustrates the principle and indicates the decision levels; it also shows an example of generated histogram bins.

The proposed descriptor is thus defined by the concatenation of individual descriptors, corresponding to the three features, as follows:

V=[H dTASS H ASSF H DSSF ] (2)
where H dTASS , H ASSF are H DSSF are the respective generated histograms descriptors of d-TASS, ASSF and DSSF functions. Thus the length of the generated descriptor V is D=3K

Evaluation of the proposed descriptor

To evaluate the ability of the proposed descriptor to discriminate objects, we have conducted experiments of image retrieval from database of planar objects. In order to compare our results with those from other methods, the experiments are done on three well-known databases that are the kimia99 dataset [START_REF] Klein | Recognition of shapes by editing their shock graphs[END_REF], the multiview curve dataset (MCD) [START_REF] Zuliani | Affine-invariant curve matching[END_REF], [START_REF] Zuliani | Drums and Curve Descriptors, British Machine Vision Conference (BMVC)[END_REF] and that extracted from the SQUID marine animals [START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF]. The performance of the descriptors is evaluated using the Precision-Recall curve averaged over the whole database and particularly we present the bull's eye performance (BEP).

The similarity measure is based on Manhattan distance. As proposed in [START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF], some global parameters like eccentricity, aspect ratio and circularity are taken into account.

Based on this, a two-steps hierarchical procedure is performed: given a global parameter g, a threshold th g is set so that all models giving 1 th g

) g , g max( g g M Q M Q < ≤ - are
discarded from the retrieval procedure, where g Q and g M are the parameter corresponding, respectively, to query Q and model M. Then we define the similarity measure between the query and a model, represented by their respective descriptor vectors V Q and V M , as follows:

∑ ∑ = = - + - = D 1 i M Q G 1 i M Q ) i ( V ) i ( V ) i ( g ) i ( g ) M , Q ( Sim β α (3)
where G is the number of global parameters (7 in this case), α and β are the weights applied, respectively, to global parameters distance and to that of the histogram descriptor.

Performance assessment

Precision and Recall are two parameters commonly used to assess the performance of a retrieval system. They are given by relations ( 4) and ( 5)

Retrieved # Relevant) (Retrieved # Precision = (4) Relevant # Relevant) (Retrieved # Recall = (5)
The BEP indicates the percentage of the retrieved relevant contours after we have retrieved twice the number of relevant contours in the database: this corresponds to the Recall for the twice the number of relevant contours in the database. It provides an objective measurement of the efficiency of a retrieval system: the higher it is, the better is the system. Results presented in the next sections correspond to the average values all over the database by using each contour of the database as a query.

Test sets

The evaluation is done on three databases. The first one is proposed by Klein and Kimia [START_REF] Klein | Recognition of shapes by editing their shock graphs[END_REF]. It consists of shapes gathered from different sources to form 9 different categories. Each category contains 11 shapes leading to variations in form, occlusion, articulation, missing parts... This gives a total of 99 shapes. Figure 5 illustrates this dataset. As our method utilizes contour-based analysis, an edge detector is applied to each of these elements to extract the corresponding contours. The MCD dataset presents realistic perspective transformations that one can encounter while creating images of real objects. Indeed, in the construction process, the authors [START_REF] Zuliani | Affine-invariant curve matching[END_REF] have printed the 40 original contours on white paper and for each one, they took images from 7 different view angles using a digital camera and then contours are extracted. By adding random rotations and reflections to these samples, the number of samples in each class is doubled to 14, leading to a total of 560 shapes in the database. The third dataset is the SQUID subset used by F. Mokhtarian and M. Bober in [START_REF] Zhang | Review of shape representation and description techniques[END_REF] and by F. Mokhtarian et al. in [START_REF] Mokhtarian | Curvature Scale Space Representation: Theory, Applications and MPEG-7 Standardization[END_REF]. It consists of 17 classes containing from 6 to 8 shapes selected from the SQUID marine animals database [15] represented on Fig. 7, containing 131 shapes. The evaluation on this database is done by using the same procedure as proposed in [START_REF] Zhang | Review of shape representation and description techniques[END_REF]. Similar to the concept of Recall, the procedure is as follows: 1. for a given class, use each of the contours as the query and determine the first 15 most similar outputs. Count the number of retrieved relevant contours and compute the ratio of this number to the number of relevant contours in the class.

The mean value of this ratio all over the class is the performance measure for the concerned class (see individual columns in table III). 2. repeat 1. for all the classes in the dataset. The overall performance is then obtained by averaging the performance provided by the 17 classes. Column 'Overall' in table III, gives that overall performance for the four involved methods.

Experimental results

In the experiments we have evaluated the effect of the length of the descriptor. This led us to observe that, for both histogram generation techniques, the performance varies a lot at the beginning and then establishes from a limit length above which there is no gain in performance. I summarizes the retrieval results from the three databases. We present the mean, the standard deviation and the maximal values of BEP over the number of bins used. On the right of the table, we give the differences of those respective values between the two techniques: the value obtained from the new technique minus that of the traditional technique. From this table, it appears clearly that the proposed technique gives the best performance with more than 3.58% mean BEP improvement. Notice also that the proposed technique gives more stable descriptor since the BEP values vary the least: we obtain negative difference values indicating standard deviation reduction (of more that 0.30).

Comparison of the two histograms techniques

Comparison of the proposed approach with other approaches

In this subsection, we evaluate the proposed approach and compare it with some existing methods by applying it to appropriate databases.

A -Kimia99 dataset: in this experiment, we retrieve the top 15 most similar for each query. Figure 9 shows an example of this retrieval. The left column represents the query contour. As each category (or class) is composed of 11 contours, there are 10 relevant contours for each query: hence columns 1~10 represent the first top 10. This figure shows that, in most cases, most of the contours from the query category are among the first 10 retrieved contours. In general, the whole class is retrieved after 15 retrievals except for the 8 th category (row 8) that misses one shape. The BEP performance for this database is of 97.43% for 11 bins (leading to a 33-length descriptor). In a recent survey, Chandan Singh and Pooja [START_REF] Singh | Improving image retrieval using combined features of Hough transform and Zernike moments[END_REF] compared their methods with others and showed that their proposal outperforms the best on this dataset. Their proposed descriptor (ZM+HLTC) is hybrid and consists of two kinds of features: 1) local features extracted from the contour by using Hough transform and 2) global (region-based) features extracted from the region of the shape by using Zernike moments (ZM). The BEP obtained is 99.55%: this performance is very challenging. The comparison reported in [START_REF] Singh | Improving image retrieval using combined features of Hough transform and Zernike moments[END_REF] is done with methods using local, global or hybrid descriptors. Table II shows the performance of involved methods: as can be observed, the proposed method ranks second just after the challenging ZM+HLTC method. As our descriptor uses only local features, we can reasonably believe that we could improve our performance to meet the latter if we integrate global features. C -MCD dataset: Figure 10 shows the Precision vs. Recall graphs obtained from the MCD database. Compared to the results reported in recent studies [START_REF] Ekombo | Application of affine invariant Fourier descriptor to shape-based image retrieval[END_REF] and [START_REF] Zuliani | Drums and Curve Descriptors, British Machine Vision Conference (BMVC)[END_REF], the proposed approach performs very well. The Recall-Precision graph indicates more than 91.86% Recall for more than 92% Precision for the proposed approach. This outperforms the reported performance that are 86% Recall for 85% Precision for Helmholtz curve descriptor (HCD) [START_REF] Zuliani | Drums and Curve Descriptors, British Machine Vision Conference (BMVC)[END_REF] and 90% Recall for 90% Precision obtained by Ekombo et al. with the invariant Fourier descriptor [START_REF] Ekombo | Application of affine invariant Fourier descriptor to shape-based image retrieval[END_REF].

Conclusion

The subject of this study is the description of planar objects based on the analysis of their contours. For this purpose, we introduced two contributions: 1) first, we proposed a new descriptor based on the histograms of three features derived from the turning angle scale space analysis of the contours and 2) we adopted another technique for histogram descriptor generation as opposed to traditional histogram-based descriptors.

The proposed technique of histogram generation jointly with the new features provides an efficient and effective descriptor that outperforms the traditional histogram descriptor. Applied to three databases, the proposed technique gave more than 3.58% improvement of the bull's eye performance regarding the traditional histogram descriptors.

Initially designed for d-TASS-based descriptor for shape retrieval, the novel technique of histogram-based descriptor construction is successfully applied for texture retrieval.

These encouraging results suggest a potential application (using an appropriate set of features: DCT, ...) of the novel proposed histogram technique on medical images retrieval. Finally, as demonstrated by experimental results, the proposed descriptor allows reaching state-of-the-art performance. Since this descriptor is based only on contour information, we believe that the descriptor can be improved by combining it with a region-based feature.

From the previous observations, the future developments of this study will be of two main types: the first will be to generate a hybrid descriptor by combining contourbased and region-based features in order to improve the shape descriptor and the second will be to extend the proposed histogram generation technique to medical images description for their retrieval from databases.

  s iterations. In this study, a Gaussian kernel of size 3 (g=[0.25, 0.50, 0.25]) corresponding to used. After each iteration, the d-TA function is computed. Thus, given a range of scales (a range of iterations), one obtains the d-TASS function.

6 P σ, 1 P σ, 2 P σ, 3 P σ, 4 PFig. 2 :

 612342 Fig.2: a: original contour and the illustration of b) the area function and c) the distance function.

Fig. 3 :

 3 Fig.3: Illustration of a) d-TASS function, b) ASSF function and c) DSSF function.

Fig. 4 :

 4 Fig.4: a) Reference histogram of the whole database, b) histogram of a contour ("bonefishes"), and c) its 7-bins' proposed histogram.

Fig. 5 :

 5 Fig.5: Content of the kimia99 dataset The second dataset is the multiview curve dataset (MCD) [14]. It consists of 40 classes drawn from the MPEG-7 Core Experiment Shape-1. Each class contains 14 contours corresponding to different perspective distortions of the original one.Figure 6 shows a sample set taken from this database.

Figure 6

 6 shows a sample set taken from this database.

Fig. 6 :

 6 Fig.6: A sample set of MCD database contours

Fig. 7 :

 7 Fig.7: Dataset-III extracted from the SQUID database

Figure 8 Fig. 8 :

 88 Fig.8: BEP vs. number of bins for a) kimia99 b) SQUID and c) MCD databases TableIsummarizes the retrieval results from the three databases. We present the mean, the standard deviation and the maximal values of BEP over the number of bins used. On the right of the table, we give the differences of those respective values between the two techniques: the value obtained from the new technique minus that of the traditional technique. From this table, it appears clearly that the proposed technique gives the best performance with more than 3.58% mean BEP improvement. Notice also that the proposed technique gives more stable descriptor since the BEP values vary the least: we obtain negative difference values indicating standard deviation reduction (of more that 0.30).

Fig. 9 :

 9 Fig.9: Results of retrieval from kimia99 database

Fig. 10 :

 10 Fig.10: Example of Precision vs. Recall graphs.

Table I :

 I Comparison of the techniques.

		Traditional histogram generation (THi)	Proposed histogram generation (PHi)	Difference (PHi-THi)
		kimia99 SQUID MCD kimia99 SQUID MCD kimia99 SQUID MCD
	Mean	92.82	85.41	93.68	96.70	88.99	97.56	3.88	3.58	3.88
	σ	0.66	0.97	0.45	0.18	0.29	0.15	-0.48	-0.68	-0.30
	Max	94.95	87.63	94.82	97.34	90.35	97.87	2.39	2.72	3.05

Table II :

 II Comparison of average BEP with other methods applied to kimia99. the following table shows the retrieval results from the SQUID database. For comparison purpose, we use results from the well-known curvature scale space (CSS) descriptor, the Fourier descriptor (FD) and the moment invariants (MI) reported in[START_REF] Zhang | Review of shape representation and description techniques[END_REF]. Columns 1~17 indicate the mean success retrieval rate for each class and the column "Overall" indicates the mean success retrieval rate over the whole database. As can be seen on this table, the proposed approach outperforms the FD and MI descriptors and ranks first (ex aequo with CSS method).

	FD	WLD	CPDH	MI	GFD	ZMD	TCS	QLS	ZM+HLTC	Proposed
	local	local	local	global	global	global	hybrid	hybrid	hybrid	local
	84.29	74.39	90.05	8.74	87.93	92.68	84.29	82.81	99.55	97.43
	B -SQUID dataset:

Table III :

 III Comparison of retrieval rate with other methods applied to SQUID dataset

		Class N°	1	2	3	4	5	6	7	8	9 10 11 12 13 14 15 16 17 Overall
	Descriptors	CSS Proposed 88 91 96 100 100 97 93 98 75 97 74 96 92 85 91 99 68 89 86 81 95 100 100 91 98 75 92 81 100 94 89 100 95 69 FD 100 84 100 75 78 42 80 100 36 98 13 70 78 98 73 97 48 MI 34 78 91 100 88 41 72 86 47 53 23 91 53 45 72 73 58	91 91 75 65

Corresponding author: kidiyo.kpalma@insa-rennes.fr

Acknowledgements: This work was partially supported by the Franco-Algerian cooperation program PHC TASSILI grants. Entitled "Système Conjoint de Compression et d'Indexation Basé-Objet pour la Vidéo (SCCIBOV)", the project gathers members of Institut National des Sciences Appliquées (INSA) de Rennes, Université de Technologie de Compiègne (UTC), the University of Sidi Bel Abbès and the University of Mascara.

bell08 bell05 bell09 bell02 bell12 bell03 bell07 bell10 bell14 bell04 bell06 bell11 bell13 apple01 apple08 apple09 apple02 apple11 apple04 apple12 apple05 apple10 apple03 apple13 apple07 apple14 apple06 butterfly01 butterfly08 butterfly12 butterfly05 butterfly02 butterfly09 butterfly03 butterfly10 butterfly04 butterfly11 butterfly13 butterfly06 butterfly07 butterfly14 insect01 insect08 insect10 insect03 insect05 insect12 insect13 insect09 insect02 insect06 insect14 insect07 insect04 insect11 bat01 bat08 bat02 bat09 bat10 bat03 bat04 bat14 bat11 bat07 bat12 bat05 bat13 bat06 bone01 bone08 bone09 bone02 bone14 bone07 bone10 bone03 bone05 bone12 bone11 bone04 bone06 bone13 brick01 brick08 brick04 brick11 brick09 brick02 brick07 brick06 brick14 brick13 brick10 brick03 brick05 brick12 camel01 camel08 camel14 camel07 camel13 camel06 camel10 camel03 camel02 camel09 camel11 camel04 camel12 camel05


Da-Yan Y Liu
email: dayan.liu@kaust.edu.sa

Olivier Gibaru
email: olivier.gibaru@ensam.eu

Wilfrid Perruquetti
email: wilfrid.perruquetti@inria.fr

Taous-Meriem M Laleg-Kirati
email: taousmeriem.laleg@kaust.edu.sa

Fractional order differentiation by integration with Jacobi polynomials

The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discretetime models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises.

I. INTRODUCTION

Fractional models arise in many practical situations ( [START_REF] Cugnet | On Lead-acid battery resistance and cranking capability estimation[END_REF], [START_REF] Pommier | CRONE control of a nonlinear hydraulic actuator[END_REF] for example). Such fractional order systems may also be used for control purposes: CRONE control is known to have good robustness properties (see [START_REF] Oustaloup | The CRONE control of resonant plants: application to a flexible transmission[END_REF], [START_REF] Oustaloup | From fractal robustness to the CRONE approach[END_REF], [START_REF] Oustaloup | From fractal robustness to the CRONE control[END_REF], [START_REF] Pommier | CRONE control of a nonlinear hydraulic actuator[END_REF]). In order to implement such controller one needs to have a good digital fractional order differentiator from noisy signals, which is the scope of this paper.

The fractional derivative has a long history and are now very useful in science, engineering and finance (see, e.g., [START_REF] Oldham | The Fractional Calculus[END_REF], [START_REF] Fliess | Sur les systèmes linéaires à dérivation non entière[END_REF], [START_REF] Hotzel | On linear systems with a fractional derivation: Introductory theory with examples[END_REF]). The fractional order differentiator is concerned with estimating the fractional order derivatives of an unknown signal from its noisy observed data. Because of its importance, various methods have been developed during the last years. They are divided into two kinds: continuous-time model (see, e.g., [START_REF] Podlubny | Analogue realizations of fractional-order controllers[END_REF], [START_REF] Charef | Fractal system as represented by singularity function[END_REF], [START_REF] Oustaloup | Frequency-band complex noninteger differentiator: Characterization and synthesis[END_REF]) and discrete-time model (see, e.g., [START_REF] Tseng | Design of fractional order digital differentiator using radial basis function[END_REF], [START_REF] Samadi | Exact fractionalorder differentiators for polynomial signals[END_REF], [START_REF] Tseng | Improved design of digital fractional-order differentiators using fractional sample delay[END_REF]). Nevertheless, the real case of signals with noises was somewhat overlooked. In order to resolve this problem, an optimization formulation using genetic algorithms was proposed in [START_REF] Machado | Calculation of fractional derivatives of noisy data with genetic algorithms[END_REF] to reduce the noise effect in the estimations of the fractional order derivatives. But, the complex mathematical deduction restricts its application. A novel Digital Fractional Order Savitzky-Golay Differentiator (DFOSGD) was introduced in [START_REF] Chen | Digital Fractional Order Savitzky-Golay Differentiator[END_REF], which was deduced from the Riemann-Liouville fractional order derivative definition [START_REF] Podlubny | Fractional Differential Equations[END_REF] (p. 63) and the Savitzky-Golay filter [START_REF] Savitzky | Smoothing and differentiation of data by simplified least squares procedures[END_REF], [START_REF] Schafer | What is a Savitzky-Golay filter[END_REF]. Using the Savitzky-Golay filter guarantees its accuracy and simplicity for estimating the fractional order derivatives of noisy signals. However, let us recall that the Legendre polynomial filter [START_REF] Persson | Smoothing by Savitzky-Golay and Legendre filters[END_REF] is more recommended than the Savitzky-Golay filter for reasons of simplicity and speed. In particular, it has advantages of being suitable for irregularly spaced or missing data.

The method of differentiation by integration uses an integral of an unknown noisy signal to estimate the integer order derivatives of this signal. For free of noise signals, this method was firstly studied by Cioranescu [START_REF] Cioranescu | La généralisation de la première formule de la moyenne[END_REF] (1938) and became well known for the Lanczos generalized derivative [START_REF] Lanczos | Applied Analysis[END_REF] (p. 324) (1956). The Lanczos generalized derivative proposed an integral of a noisy signal and the Legendre orthogonal polynomial to estimate the first order derivative of the signal. Recently, in the noisy case, Mboup, Join and Fliess [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] applied an algebraic setting to estimate high order derivatives by integration, where Jacobi orthogonal polynomials were introduced in the integral. Hence, we call the obtained differentiator Jacobi differentiator. Moreover, it was shown in [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF] that the Jacobi differentiator greatly improved the convergence rate of the Lanczos generalized derivative. Very recently, it was shown in [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF] that the Jacobi differentiator could also be obtained by taking the derivative of the Jacobi orthogonal polynomial filter considered as the extension of the Legendre polynomial filter [START_REF] Persson | Smoothing by Savitzky-Golay and Legendre filters[END_REF], [START_REF] Meer | Smoothed diferentiation filters for images[END_REF].

Let us recall that the algebraic differentiation method used in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] was introduced in [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF] and also analyzed in [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF], [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF], [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF], [START_REF] Riachy | Multivariate numerical differentiation[END_REF], where the used algebraic manipulations were inspired by the algebraic parametric estimation methods [START_REF] Fliess | Questioning some paradigms of signal processing via concrete examples[END_REF], [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF], [START_REF] Liu | An error analysis in the algebraic estimation of a noisy sinusoidal signal[END_REF], [START_REF] Liu | Parameters estimation of a noisy sinusoidal signal with time-varying amplitude[END_REF]. Additional theoretical foundations can be found in [START_REF] Fliess | Analyse non standard du bruit[END_REF], [START_REF] Fliess | An algebraic framework for linear identification[END_REF]. In particular, by using non standard analysis techniques, Fliess [START_REF] Fliess | Analyse non standard du bruit[END_REF], [START_REF] Fliess | Critique du rapport signal à bruit en communications numériques -Questioning the signal to noise ratio in digital communications[END_REF] showed that these methods exhibit good robustness properties with respect to corrupting noises without the need of knowing their statistical properties. However, these methods have not been used to estimate fractional order derivatives.

The aim of this paper is to generalize the differentiation by integration method from the integer order to the fractional order for estimating fractional order derivatives. In Section II, we recall the method of integer order differentiation by integration with Jacobi polynomials. Then, we deduce a fractional order differentiator from the Riemann-Liouville fractional order derivative definition and the Jacobi orthogonal polynomial filter. This differentiator is exactly given by an integral of Jacobi polynomials. Hence, we call it fractional Jacobi differentiator. In Section III, we compare the fractional Jacobi differentiator to the DFOSGD in some numerical simulations. Finally, we give some conclusions and perspectives for our future work in Section IV.

II. METHODOLOGY

Let y δ = y + ̟ be a noisy signal observed in an interval

I = [a, b] ⊂ R of length h = b -a,
where y ∈ C(I) and the noise1 ̟ is bounded and integrable. We are going to estimate the α th (α ∈ R + ) order derivative of y by using its observation y δ .

A. Differentiation by integration with Jacobi polynomials

In this subsection, we recall the method of integer order differentiation by integration with Jacobi polynomials. This method was introduced in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] and studied in [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF], [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF], [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF], [START_REF] Riachy | Multivariate numerical differentiation[END_REF].

The n th (n ∈ N) order Jacobi orthogonal polynomial defined on [0, 1] is given as follows (see [START_REF] Abramowitz | editeurs. Handbook of mathematical functions[END_REF] p. 775)

P (µ,κ) n (t) = n j=0 n + µ j n + κ n -j (t -1) n-j t j , (1)
where µ, κ ∈] -1, +∞[. Let f and g be two functions belonging to C([0, 1]), then we define the scalar product •, • µ,κ of these functions by (see [START_REF] Abramowitz | editeurs. Handbook of mathematical functions[END_REF] p. 774)

f (•), g(•) µ,κ = 1 0 w µ,κ (t) f (t) g(t) dt, (2)
where w µ,κ (t) = (1t) µ t κ is the associated weighted function. Thus, by denoting its associated norm by • µ,κ , we obtain

P (µ,κ) n 2 µ,κ = Γ(µ + n + 1) Γ(κ + n + 1) Γ(µ + κ + n + 1) Γ(n + 1) (2n + µ + κ + 1) . (3)
where Γ(•) is the classical Gamma function (see [START_REF] Abramowitz | editeurs. Handbook of mathematical functions[END_REF] p. 255).

Let us ignore the noise for a moment. Then, we define an N th (N ∈ N) order approximation polynomial of y by taking its truncated Jacobi orthogonal series expansion

∀ t ∈ [0, 1], D (0) h,µ,κ,N y(a + ht) := N i=0 P (µ,κ) i (•), y(a + h•) µ,κ P (µ,κ) i 2 µ,κ P (µ,κ) i (t). (4)
If we take κ = µ = 0, then the Jacobi orthogonal polynomials become the Legendre orthogonal polynomials. Hence, this Jacobi polynomial filter [START_REF] Meer | Smoothed diferentiation filters for images[END_REF] can be considered as the extension of the Legendre polynomial filter [START_REF] Persson | Smoothing by Savitzky-Golay and Legendre filters[END_REF].

If y ∈ C n (I) with n ∈ N, then we take the n th order derivative of the polynomial D (0) h,µ,κ,N y(a + h•) as an estimate of the n th order derivative of y [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF]

: ∀ t ∈ [0, 1], D (n) h,µ,κ,N y(a + ht) := d n d(ht) n D (0) h,µ,κ,N y(a + ht) = 1 h n d n dt n D (0)
h,µ,κ,N y(a + ht) .

(

For any t ∈ [0, 1], this differentiator can be expressed as follows [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF]

D (n) h,µ,κ,N y(a + ht) = 1 h n 1 0 Q µ,κ,n,N (τ, t) y(a + hτ ) dτ, (6)
with

C µ,κ,n,i = (µ+κ+2n+2i+1) Γ(µ+κ+2n+i+1) Γ(n+i+1) Γ(κ+n+i+1) Γ(µ+n+i+1) , Q µ,κ,n,N (τ, t) = w µ,κ (τ ) N -n i=0 C µ,κ,n,i P (µ+n,κ+n) i (t)P (µ,κ) n+i (τ ).
Finally, we replace y in ( 6) by y δ . Consequently, the n th order derivative of y can be estimated by an integral of Jacobi polynomials. The noise effect on obtained estimations was analyzed in [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF], [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF]. In the next subsection, we are going to generalize this differentiation by integration method from the integer order to the fractional order.

B. Fractional order differentiation by integration

Similarly to [START_REF] Chen | Digital Fractional Order Savitzky-Golay Differentiator[END_REF], we are going to take the Riemann-Liouville fractional order derivative of our approximation polynomial so as to get a fractional order differentiator. The Riemann-Liouville fractional order derivative (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 62) is defined as follows:

∀ x ∈ R * + , 0 D α x f (x) := 1 Γ(l -α) d l dx l x 0 (x -τ ) l-α-1 f (τ ) dτ, (7)
where 0 ≤ l -1 ≤ α < l with l ∈ N * . From now on, we denote the α th (α ∈ R + ) order derivative of f by

∀ x ∈ R * + , 0 D α x f (x) = d α dx α {f (x)} = f (α) (x). (8)
Hence, if we take f (x) = x n with n ∈ N and x ∈ R * + , then we obtain (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 72)

d α dx α {x n } = Γ(n + 1) Γ(n + 1 -α) x n-α . (9)
By using [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF], we obtain the following lemma. Lemma 1: The α th (α ∈ R + ) order derivative of the n th order Jacobi orthogonal polynomial P (µ,κ) n defined in (1) is given as follows: ∀ t ∈]0, 1],

d α dt α P (µ,κ) n (t) = n j=0 n-j l=0 c µ,κ,n,j,l Γ(n -l + 1) Γ(n -l + 1 -α) t n-l-α , (10)
where c µ,κ,n,j,l = (-1) l n+µ j n+κ n-j n-j l .

Proof. By applying the binomial theorem to (1), we get

P (µ,κ) n (t) = n j=0 n-j l=0 c µ,κ,n,j,l t n-l ,
where c µ,κ,n,j,l = (-1) l n+µ j n+κ n-j n-j l

. Hence, this proof can be completed by using [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF] and the linearity of the fractional order differentiation (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 91).

Then, we give the following proposition. Proposition 1: Let y δ = y + ̟ be a noisy observation of y in an interval I = [a, b] ⊂ R of length h = ba, where y ∈ C(I) and the noise ̟ is bounded and integrable. If the α th (α ∈ R + ) order derivative of y exists, then a fractional order differentiator, called fractional Jacobi differentiator, is defined as follows:

∀ t ∈]0, 1] D (α) h,µ,κ,N y δ (a + ht) := 1 h α 1 0 Q µ,κ,α,N (τ, t) y δ (a + hτ ) dτ, (11)
where

h ∈ R * + , N ∈ N, µ, κ ∈] -1, +∞[, Q µ,κ,α,N (τ, t) = N i=0 w µ,κ (τ ) P (µ,κ) i (τ )
P (µ,κ) i 2 µ,κ q µ,κ,i (t), (12)
q µ,κ,i (t) = i j=0 i-j l=0 c µ,κ,i,j,l Γ(i -l + 1) Γ(i -l + 1 -α) t i-l-α , (13)
with c µ,κ,i,j,l = (-1) l i+µ j i+κ i-j i-j l and P (µ,κ) i 2 µ,κ is given in (3).

Proof. Let us take the α th order derivative of the polynomial D (0) h,µ,κ,N y(a + h•) defined in [START_REF] Cioranescu | La généralisation de la première formule de la moyenne[END_REF]. By using the scale change property of the fractional order differentiation (see [START_REF] Oldham | The Fractional Calculus[END_REF] p. 76) we obtain:

∀ t ∈]0, 1], D (α) h,µ,κ,N y(a + ht) := d α d(ht) α D (0) h,µ,κ,N y(a + ht) = 1 h α d α dt α D (0) h,µ,κ,N y(a + ht) . ( 14
)
The linearity of the fractional order differentiation gives us that

D (α) h,µ,κ,N y(a + ht) = 1 h α N i=0 P (µ,κ) i (•), y(a + h•) µ,κ P (µ,κ) i 2 µ,κ d α dt α P (µ,κ) i (t) . (15)
Finally, this proof can be completed by using Lemma 1 and substituting y by y δ in [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF]. Consequently, by calculating the integral of the noisy signal y δ and a sum of the Jacobi polynomials we can estimate the value of y (α) at each point a + ht in the interval ]a, b] for each t ∈]0, 1]. This integral corresponds to a convolution in the discrete case.

If we fix the value of t to 1, then the fractional Jacobi differentiator D (α) h,µ,κ,N y δ (a + ht) only estimates the value of y (α) at point b. Thus, if we increase the length of the interval I = [a, b] ⊂ R, then we can estimate the other values of y (α) . Consequently, the fractional Jacobi differentiator can also be considered as a pointwise causal differentiator which is useful in many on-line applications. Moreover, we can fix the parameters κ, µ and N such that the sum of the Jacobi polynomials can be explicitly calculated by off-line work. Indeed, the fractional Jacobi differentiator only needs a simple convolution for on-line applications. The computation time is significatively improved.

III. SIMULATIONS RESULTS

The comparisons between the DFOSGD and some existing fractional order differentiators were shown in [START_REF] Chen | Digital Fractional Order Savitzky-Golay Differentiator[END_REF], where the DFOSGD was better than the others. In this section, by taking one numerical example considered in [START_REF] Chen | Digital Fractional Order Savitzky-Golay Differentiator[END_REF] we compare the fractional Jacobi differentiator to the DFOSGD so as to show its accuracy and robustness with respect to corrupting noises.

A. Numerical case

From now on, we assume that y δ (x i ) = y(x i ) + c̟(x i ) is a sequence of uniformly sampled noisy data of y with x i = a + T s i and

T s = h M for i = 0, • • • , M . The noise c̟(x i
) is assumed to a zero-mean white Gaussian iid sequence, and c ∈ R + . In this discrete case, we apply the trapezoidal numerical integration method in the fractional Jacobi differentiator to approximate the integral. Then, let us recall that the values of y (α) (x i ) for i = 0, • • • , M are estimated by the DFOSGD as follows

y (α) (x i ) = 1 (T s ) α b i,N X T θ,N X θ,N -1 X T θ,N Y δ θ , (16)
where

N ∈ N, θ ∈ N, b i,N =         1 Γ(1-α) (i + 1) -α 1 Γ(2-α) (i + 1) 1-α Γ(3) Γ(3-α) (i + 1) 2-α . . . Γ(N +1) Γ(N +1-α) (i + 1) N -α         T , Y δ θ =        y δ (x 0 ) y δ (x θ ) y δ (x 2θ ) . . . y δ (x M )        , (17)
and

X θ,N =          1 1 1 • • • 1 N 1 (1 + θ) 1 • • • (1 + θ) N 1 (1 + 2θ) 1 • • • (1 + 2θ) N . . . . . . . . . . . . 1 (M + 1) 1 • • • (M + 1) N          . ( 18
)
Hence, the idea is to obtain an N th order approximation polynomial by using Savitzky-Golay algorithm [START_REF] Savitzky | Smoothing and differentiation of data by simplified least squares procedures[END_REF], [START_REF] Schafer | What is a Savitzky-Golay filter[END_REF] from a subsequence Y δ θ , and then to calculate the α th order derivative of this polynomial at each point x i .

B. Example

In this example, we assume that y δ (x i ) = sin(5x i )+c̟(x i ) where x i ∈ I = [0, 4] for i = 0, • • • , 1000 = M , and c = 0.25 is adjusted in such a way that the signal-to-noise ratio SN R = 10 log 10

|y δ (xi)| 2 |c̟(xi)| 2
is equal to SN R = 10dB. The noisy signal is shown in Figure 1.

Firstly, we estimate the half order derivative of y. For the DFOSGD, we take N = 14 and θ = 5 which are the same values as the ones used in [START_REF] Chen | Digital Fractional Order Savitzky-Golay Differentiator[END_REF]. For the fractional Jacobi differentiator, we take N = 14 and κ = µ = 0. The associated absolute estimations errors in the noise-free case and in the noisy case are given in Figure 2 and Figure 3 respectively, where the black dash-dotted lines represent the errors obtained by the fractional Jacobi differentiator and the red dotted lines represent the ones obtained by the DFOSGD. Moreover, we can see in Figure 4 the associated absolute noise error contributions. Secondly, we estimate the derivative y (α) with α = 1.5 by using the DFOSGD and the fractional Jacobi differentiator with N = 14, θ = 5 and κ = µ = 0. The obtained estimation errors are shown in Figure 5, Figure 6 and Figure 7. Consequently, we can observe that the fractional Jacobi differentiator is more accurate and more robust with noises than the DFOSGD in the noise-free case and in the noisy case.

IV. CONCLUSION

In this paper, we propose a fractional Jacobi differentiator which is a fractional order differentiator exactly given by an integral formula. This differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. It can accurately and easily estimate the fractional order derivatives of noisy signals.

There are some parameters h, µ, κ and N on which the fractional Jacobi differentiator depends. We do not give any analysis on the influence of these parameters on the estimation errors. However, similarly to the integer order differentiation by integration case (see [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator[END_REF], [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF], [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF]), this study can be easily carried out in a further work. Moreover, we do not use the causal propriety of the fractional Jacobi differentiator in the numerical simulations. It will be used in some interesting applications where the sampling data may be irregularly spaced.

Fig. 1 .

 1 Fig. 1. Signal y and noisy signal y δ .

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. Absolut estimation errors in noise-free case.

Fig. 5 .

 5 Fig. 5. Absolut estimation errors in noise-free case.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Absolut estimation errors in noisy case.

More generally, the noise is a stochastic process, which is bounded with certain probability and integrable in the sense of convergence in mean square (see[START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF]).