Minh Ha-Duong

Venance Journé

Manuscript to appear in Environment, Systems and Decisions CALCULATING NUCLEAR ACCIDENT PROBABILITIES FROM EMPIRICAL FREQUENCIES

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The major accident that occurred in Fukushima on 11 March 2011 led to a renewed scientific interest in assessing nuclear risks. In the press, there has been a number of declarations such as Laponche et al. (2011) who claimed that the probability of a serious nuclear accident in the next thirty years was a "statistical certainty" for the European Union. Moreover countries such as Austria, China, Finland, France, Germany, India and Italy have responded with the adoption of very diverse nuclear safety and energy policies. Considering the policy importance of this issue, we recompute these estimates constructing our own data set and using a different method.

Risk has two dimensions: the probability of an accident and the magnitude of its consequences. This paper deals with the first dimension of risk, the probability. The causes that might result in a serious nuclear accident are manifold and they include but are not limited to: design errors, construction problems, material or hardware deficiencies, internal events such as pipe break or fire, external events such as a plane crash or natural hazards (earthquake, floods), and mostly human causes, such as errors in manipulation, insufficient safety information, bad knowledge of instructions. In many cases, the accidents are due to a combination of several factors.

Despite the complexity of causes, nuclear engineers and scientists have tried to estimate the probability of a nuclear accident. One commonly used method is the probabilistic risk assessment (PRA) method that computes, in a bottom-up combination, the probability of an accident as the product of the probabilities of all known different possible failures. This method allows to assess risks and identify potentials to improve safety. However, However, the PRA analysis are bound to be subject to uncertainties since they cannot be totally exhaustive and there are uncertainties in knowledge. This usually results in a small probability.

Table 1 shows how the predictions for the PRA methodology compare to the historical record. In the first column, are indicated given nuclear accident probabilities per reactor per year, denoted ε. The numerical objective defined for core damage frequency, as evaluated from PRA studies, range from 10 -4 to 10 -6 for regulators in different countries (NEA-OECD, 2012). The second column shows at the expected number of accidents in the whole history of the nuclear industry for each value of ε. As of end of 2012, 437 reactors were in operation worldwide, and the cumulative nuclear reactor-years of operation worldwide and including all technologies adds up to 15,080 (European Nuclear Society, 2013). These values should correspond to core damage accidents of 0,15 to 1,5 for 15,080 reactor-years (15,080 times ε). This number is small compared to the real number of accidents. The probability of an accident per reactor per year (first column) remains the same along each line. The probability of seeing an accident depends on how many reactors are observed during how many years.

Accident probability per reactor per year

Expected number of accidents since 1950

Probability of at least one accident in 5 years

Probability of at least one accident in 30 years

ε 15080 * ε 1 -(1-ε) ^ (437*5) 1 -(1-ε) ^ (437*
The third and fourth columns indicate the probabilities of having an accident in five and 30 years for 437 reactors, assuming that the reactor-year accident probability is ε. The number of 437 reactors corresponds approximately to the number of nuclear reactors operating in the world today.

Thus given the complexity of the overall system, and the discrepancy between the values given by PRA analyzes and the number of accidents, a holistic approach such as the use of historical frequencies is used in this study. We assess the frequency of a nuclear accident and of a major accident from past experience.

In section 2, the basis on which nuclear accidents have been selected to construct the data set is defined and the way in which the discrete INES scale does not take into account the complexities of a reality which is "fuzzy" is explained. Section 3 contains a review of some literature on nuclear accident risks with emphasis on probability estimates and nuclear accident inventories. In section 4, the construction of the data set is described. The method and results are presented in section 5 and 6. Section 7 concludes.

INES scale and nuclear accidents

The International Nuclear and Radiological Event Scale (INES) is a worldwide tool for communicating to the public the safety significance of nuclear and radiological events (IAEA, 2009). It was developed by the International Atomic Energy Agency (IAEA) : the IAEA statutes (article II -Objectives) states that "The Agency shall seek to accelerate and enlarge the contribution of atomic energy to peace, health and prosperity throughout the world. It shall ensure, so far as it is able, that assistance provided by it or at its request or under its supervision or control is not used in such a way as to further any military purpose" (IAEA, 2012) -and the OECD Nuclear Energy Agency -whose mission is "To assist its member countries in maintaining and further developing, through international co-operation, the scientific, technological and legal bases required for a safe, environmentally friendly and economical use of nuclear energy for peaceful purposes. To provide authoritative assessments and to forge common understandings on key issues as input to government decisions on nuclear energy policy and to broader OECD policy analyzes in areas such as energy and sustainable development." (NEA, 2013) Events are classified on the scale with 7levels. For communicating to the public, distinct designations have been attributed to each level, from anomaly (level 1) to major accident (level 7): « the severity of an event could increase by about an order of magnitude for each increase in level on the scale » (IAEA, 2009).

According to the IAEA's vocabulary, any event whose severity is rated at INES level ≥ 4 on the INES scale is an accident. Below level 4, the event is called incident. As mentioned above and illustrated on Table 2, there are many types of accidents. The potential consequences of an accident range from those limited to the core, to those contained inside the pressure vessel, to the release of radioactivity in the environment and those affecting the population. Because of this diversity, there are several criteria to assess the severity of an accident and the INES defines a level 4 event by any of the following characteristics:

1. At least one death from radiation.

2. Minor environmental release of radioactive material unlikely to result in implementation of planned countermeasure other than local food controls.

3. Fuel melt or damage to fuel resulting in more than 0.1 % release of core inventory.

4. Release of significant quantities of radioactive material within an installation with a high probability of significant public exposure.

The IAEA criteria provides the only agreed definition of a nuclear accident. However, the construction of an historical nuclear accident data set on the basis of the INES rating is an uncertain process for the following reasons:

• Back-rating: The INES scale was developed only in 1990. Many previous events are not officially rated on the scale. Unofficial ratings may be found in the literature that may not be adequately justified.

• Fuzziness: The INES scale is a communication instrument 2 , not a safety evaluation tool. The INES manual provides guidance for reporting the events using only a single number, but the proper definition of an accident is complex. The fuzziness of the categorization using the INES level is apparent from the above four criteria : the terms "minor", "likely", "significant" are used. It is also the case for higher levels : one of the characteristics of rating is the quantity of radioactivity released and the boundaries between levels from 4 to 7 are set approximately to 500, 5000 and 50 000 tera-becquerels (IAEA, 2009, p. 17). Moreover, the quantitative variables used to rate accidents, such as the released radioactivity, cannot be always precisely estimated either.

• Heterogeneity: The difficulty to characterize precisely an accident and its consequences, and therefore the ensuing fuzziness of the INES scale, makes the historical and geographical uniformity of assessment difficult to ensure, unlike the Richter earthquake magnitude scale for example. This difficulty could be partially alleviated if there were a world entity in charge of the consistent interpretation and rating of an accident. But rating is the responsibility of the power plant owner, or of the regulation entity in certain countries such as France, which is therefore the unique source of primary information. This can lead to important differences in reporting.

• Secrecy: To be notable, an event must have caused well-attested and substantial radioactivity release and contamination. For example, an accident that occurred in September 1957 in Soviet Union has been acknowledged only decades after :the waste storage explosion at Mayak, a plutonium production reactor for nuclear weapons and nuclear fuel reprocessing plant, near Kyshtym. This accident would have been rated at level 6 on the INES scale, 15,000 square kilometers were contaminated. It was made public to the outside world by Soviet dissident Zhores Medvedev in 1976 [START_REF] Medvedev | Chernobyl precursor accidents kept secret[END_REF]. The fact that serious events are not made public may also occur in the West: on 17 December 1987, eighteen months after the Chernobyl major accident, a serious incident occurred in the Biblis power station in West Germany. It was known internationally nine months later in an expert meeting in Vienna and made public one year later in an article published by Nucleonics Week, and rated at level 1 on the INES scale (Hibbs, et al, 1988). Therefore this serious incident in a purely civilian nuclear power plant was not known to the IAEA when it occurred. These examples underline the fact that some events, which could involve important radiation contamination or potential for much more serious consequences, can be kept secret from the public.

• Dual and multiple uses: Some accidents cannot be attributed only to power production. This is the case for those occurring in nuclear reactors producing jointly isotopes and electric power. Also, research and commercial uses cannot be distinguished in the case of large-scale reactor prototypes.

• Simultaneity: Power plants usually have two or more power reactor units.

In our study, several simultaneous accidents at one plant are counted as a single one, such as However, in the case of the Fukushima Daichi major accident, results will be provided counting it as one accident as well as three accidents.

Previous studies on nuclear accident frequencies

There have been many studies on various aspects of nuclear accident frequencies and their scope is restricted to specific types or to a limited number of accidents. Assessments range from the assertion that « nuclear power plants are very robust » (World Nuclear Association, 2012) to the conclusion that « nuclear power comes with the inevitability of catastrophic accidents » [START_REF] Ramana | Beyond our imagination: Fukushima and the problem of assessing risk[END_REF].

As stated above, Laponche and Dessus (2011) argued that nuclear accidents were statistically certain. They wrote that: "On the basis of the major accidents having occurred these last thirty years, the occurrence probability of a major accident would therefore be about 50% for France and more than 100% for European Union". Giving more details on the calculations, Ghys (2011) estimated this later probability to be about 72 %, based the same data (4 catastrophic core failures -1 in Chernobyl, 3 in Fukushima -for 14 000 reactor-years). Stratton (1967) published the first seminal review of criticality accidents. This work is about a subset of nuclear accidents, and does not look at commercial nuclear power plants. It was first revised in 1989 and updated again by [START_REF] Mclaughlin | A Review of Criticality Accidents[END_REF]. The 2000 edition contains 60 accidents, 19 of which are new to it. [START_REF] Lochbaum | Nuclear Plants Risk Studies: Failing the grade, Union of concerned scientists report[END_REF], for the Union of Concerned Scientists, studied the risk assessment reports published by the US Nuclear Regulatory Commission and concluded that the probability assessments were biased towards lower values by the use of many unrealistic assumptions regarding plant design, construction and operation, a very limited number of human errors. [START_REF] Kastchiev | Residual risk, An Account of Events in Nuclear Power Plants Since the Chernobyl Accident in[END_REF] have studied the safety records of nuclear power plants to assess risks and possible consequences of nuclear accidents. Based on the analysis of sixteen events, they concluded that "many nuclear safety related events [...] go either entirely unnoticed by the broader public or remain significantly under-evaluated when it comes to their potential risk".

T. Cochran (2011) gave a testimony where he stated that the historical frequency of core melt accident is about 1 in 1400 reactor-years which is rather far from the probability risk assessment estimate of 2 × 10-5 to 1 × 10-4 events/reactor-year for accidents (INES level ≥ 4) published in 1986 by the US Nuclear Regulatory Commission. Cochran's paper lists 12 nuclear power reactors (including Sodium Reactor Experiment) that have experienced fuel-damage or partial core-melt accidents.

Smythe (2011) proposed a new scale for the magnitude of nuclear accidents. He describes a list of 33 events in which he counts 4 events as catastrophic, leading him to conclude that "A magnitude 7 or greater accident is to be expected every 12-15 years."

Escobar Ranger and Lévêque ( 2013) have used a statistical -Poisson -model approach and performed a methodological study for the same problem, based on continuous time models and reliability theory. They found that the Fukushima major accident leads to a re-evaluation of previous risks assessments.

Construction of the dataset

The Aircraft Crashes Record Office compiles all aircraft accidents in the civilian aviation sector and makes complete and detailed statistics available to the public (ACRO, 2014). On the contrary, int he civilian nuclear energy sector, there is no world data set of nuclear accidents, and there is no way to have harmonized information on such events. The IAEA hosts a website NEWS "The Information Channel on Nuclear and radiological Events" which displays short reports of accidents as they are deposited on the website, but there is no website or directory of accidents including incident and accident statistics with detailed information on specific events. Moreover, in its disclaimer, it declares : "The IAEA makes no warranties, either express or implied, concerning the accuracy, completeness, reliability, or suitability of the information." (IAEA, 2014) Moreover, the IAEA is not legally bound to reply to all queries for information (Kastchier 2007, note page 5).

Many online sources, including the IAEA, have been consulted for this study, but with no assurance that any of them is complete. For instance, the IAEA NEWS (2012) web page states that « event reports are filed and updated by participating countries which remain responsible for all related content » and does not contain an archive which is publicly available. The available information include: event description, INES rating, related technical information and press releases. We also consulted accident lists published online by Wikipedia ( 2012) and other sources, other databases provided by non governmental organizations, or independent researchers, some of them containing a lot of information such as Johnston's (2011) archives andClimatesceptics.org (2012).

In building this sample, the goal was to find and record all events rated at INES level ≥ 3 which occurred at a licensed and commercially operational power plant since 1951, the start of nuclear energy for civilian purposes. Given the lack of standardized and complete world data set, the reconstructed data set suffers from the deficiencies discussed in Section 2 above. There is uncertainty on the number of events and in their rating, since the INES levels are not precise enough, including in terms on released radioactivity, and thus the frontier between accident and incidents is fuzzy. Several accident types are listed in Table 2. Only events of the categories described in the left column pertaining to the production of nuclear electricity are retained in the dataset. The right column shows that many other types of radiological accidents are not taken into account in the present study : military activities as well as those such as medical activities or industrial activities which cannot lead to major accidents were ignored.

For events having occurred before 1991, when the INES scale became operative, accident descriptions were used to assess a level equivalent to the INES level.

We assembled a data set of all events with an INES level ≥ 3 in the power sector. It contains 68 records summarized in Table 3 (see annex). Excluding level 3 events, Table 4a retains all events with an INES level ≥ 4 in all types of nuclear facilities. Excluding events in test facilities or fuel facilities, Table 4b lists the 13 accidents which occurred in nuclear power plants. Those are the only ones kept for this study.

Figure 1 shows the time distribution of events by INES level and shows a decrease in the number of accidents, but the two4 major accidents occur in the later decades. Figure 2 shows the number of nuclear reactors in operation as a function of time : the number of reactors is increasing steadily until the mid 80s, and then remains at a roughly constant level until 2012. The comparison of those two figures shows that the bulk of accidents occurred before 1986 and that few accidents occurred when the number of reactors was at its maximum. The absolute number of accidents decreases, but the damage on environment does not decrease.

Method

An accident being defined as an event with an INES level ≥ 4, the empirical frequency of the number of accidents over T years is computed using the method described below. Over the 61 years from 1951 to 2011, there was 49 years with no accident, 11 with one accident and one year with two accidents. The number of accidents in one year is modeled as a random variable denoted X. This variable takes the value 0, 1, 2 with a probability respectively of 49/61, 11/61 and 1/61.

To consider the number of accidents in T years, each year is independently drawn with re-sampling. This number is described by the random variable Y defined as the sum of T realizations of X.

The probability of Y can be computed from the probability law of X. This is a classical problem of probability. The combinatorial counting required here is mathematically the same as the counting involved in the expansion of polynomial. This leads to use the so-called moment-generating function of a discrete variable X, defined by gX (z) = ∑ k p(X=k) z k .

gX (z) = 49/61 + 11/61 z + 1/61 z²

The moment-generating function of Y can be obtained from the momentgenerating function of X by polynomial expansion:

gY (z) = gX(z) T
The coefficients of gY(z) directly give the probability distribution of Y.

The results are shown in the Table 5a for T = 5, and in Table 5b for T = 30.

Regarding the number of major accidents, where there has been 59 years with none, and 2 years with one, the same method has been used with the generating function 59/61 + 2/61 z. The results are shown on the second lines on Table 5a and table 5b .

Numerical results were verified with a different statistical method: the bootstrap method. This method relies on iterated computer simulations. It involves drawing five years randomly in {1951, ..., 2011} and counting the accidents in these years. This is then repeated for 100,000 times to look at the statistical distribution of results. Since the repetition number is large enough, the statistical distribution of results is a satisfying approximation of the law of Y. The sample is represented in Figure 1 since it contains the number of accidents per year.

The results are shown in the Table 6a for T = 5, and in Table 6b for T = 30.

6 Results

Calculations were performed for a five-year time-span, which correspond to a typical decision-making time horizon, and for the same 30-year time-span, as used by Laponche, Dessus and Ghys.

The results presented in Table 5a show that for a sample of five random years, assuming that probability distribution of accidents is constant over the considered time span, the probability of having one year with one nuclear major accident (INES level = 7) is 14.3 %, and the probability of having two major accidents is 1.0 %. The probability of having at least one nuclear accident (INES level ≥4) in the same five random year sample is 66.6 %.

The results in Table 5b show that for a sample of thirty random years the probability of drawing at least one year with a major accident is 63.2 %, and the probability for two or more major accidents is 25.8 %. These results are in agreement with the previously published studies.

Table 6a and Table 6b show the analogous results using the bootstrap method: the absolute differences with the numbers shown Tables 5 and6 are always less than 0,2%. This validates the numerical computations of this study since the generating functions method is exact, while the bootstrap method is approximate.

Systematic errors due to the inherent ambiguities and uncertainties in the construction of the data set as discussed in Section 2 have been estimated. For example, since the study is limited to operational nuclear power plants, it does not take into account the Kychtym accident nor the Tokai-Mura criticality accident. It could be argued that one should count all accidents related to the nuclear power industry, even those that occurred in reprocessing installations. As a sensitivity analysis study, probabilities were computed including the Kyshtym event to the 13 accidents dataset. The frequency of five-year samples with no accidents drops from 33.4 % to 30.2 %.

The main assumption underlying the computation of accident frequencies from past data is that the accident probability distribution in past years is constant.

Since our sample has low statistics, our discussion on auto-correlation and trends could be only qualitative (see Hofert 2011, section 3.1 for a quantitative analysis of these questions). Auto-correlation could occur for several reasons. The occurrence of an accident in a given year may tend to decrease the risk in the following year, because of increasing awareness of plant managers and regulators worldwide. On the other hand, an accident in a reactor may increase the risk of accident in another reactor of the same plant as was the case in Fukushima.

As a sensitivity test, the same calculation was done considering what happened in Fukushima as three major accidents (INES level = 7). The results are presented on table 7.

Conclusion

Accidents at operational power plants have been considered for this study, an accident being defined as an event rated at INES level ≥ 4, and a major accident, as an INES level 7 event. Drawing samples of five random years from the past and assuming that they are all equi-probable, the probability of having a year with a nuclear major accident in a five-year is 14.3%. The probability of having two major accident in a five-year time span is 1.0%. The probability of having at least one nuclear accident is 66.6% in a five-year sample (Table 5a). For a 30-year sample, we found that the probability of having at least one major accident was 63.2%, and a 25.8% probability of having at least two (Table 5b). Our results are in agreement with previously published studies.

Improving safety in the nuclear industry requires transparency. Since the dataset completeness is a most important parameter in risk studies, we strongly support the establishment of a complete and public international historical report of incidents and accidents worldwide with the obligation for plant owners to report.

We consider that the IAEA should be the most suited organization to host such a data set since this task would very well fit in its objectives recalled above.

Figure 1 (

 1 Figure 1 (top) : Chronology of accidents in this study's dataset of civilian nuclear accidents with INES level ≥ 4 for all facilities

Table 1 :

 1 Four viewpoints on accidents probability.

	30)

Examples of accidents related to nuclear power production Examples of nuclear accidents not related to electricity production, not considered in this paper Uranium mine.

  The present study is only based on accidents with an INES level ≥ 4. 3 Radioactive waste spill. Church Rock,USA, 1979 (no immediate direct fatality)

		Weapons. Criticality accident with Pu metal assembly.
		Los Alamos, USA, 1946 (1 fatality)
	Fuel facility. Criticality accident with Uranium solution.	Naval propulsion. K27 submarine reactor accident.
	Tokaimura, Japan, 1999 (2 fatalities)	Russia, 1968 (9 fatalities)
	Reprocessing facility. Chemical explosion following	Attack. Five RPG-7 rockets fired on nuclear plant
	tank cooling system failure. Kychtym, Russia, 1957	construction site. Creys-Malville, France, 1982 (minor
	(300 × 50km area contaminated)	damage)
	Power reactor. Partial core melt. Saint Laurent des	Industrial. Lost gamma radiography source.
	Eaux, France, 1969. (no immediate direct fatality)	Conceptión, Chile, 2005 (3 workers injured)
		Medical. Radiotherapy overexposure. Zaragoza,
		Spain, 1990 (11 fatalities)

Table 2 :

 2 Types of nuclear accidents. This paper only considers those involving electricity production (left column).

The first lines of the INES reference User's Manual (2008 edition ) state "The need for easily communicating the significance of any event ..." (Foreword, page v.) and "The International Nuclear and Radiological Event Scale is used for promptly and consistently communicating to the public the safety significance of events associated with sources of radiation." (section 1.1) http://www-pub.iaea.org/MTCD/publications/PDF/INES2009_web.pdf)

Since the limits between INES level are fuzzy, accidents rated at level = 3 were also looked for in the research to build the dataset. This will be used, in a later study, to allow more specific tests regarding the influence on the results of this study stemming from a fuzzy frontier between different INES levels.

Or four major accidents if one considers that 3 major accidents occurred in Fukushima Daichi.

Acknowledgments

We acknowledge preliminary research by Arnaud de Myttenaere, ENSAE student intern at CIRED. We thank warmly M. V. Ramana for a careful reading of the manuscript and his valuable comments. All remaining errors are the responsibility of the authors.


A Cousin

A Janon

V Maume-Deschamps

I Niang

ON THE CONSISTENCY OF SOBOL INDICES WITH RESPECT TO STOCHASTIC ORDERING OF MODEL PARAMETERS

In the past decade, Sobol's variance decomposition have been used as a tool -among others -in risk management ([3, 14]). We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol's indices in uncertainty quantification, as one indicator among others.

Introduction

Many models encountered in applied sciences involve input parameters which are often not precisely known. Sobol indices are used so as to assess the sensibility of a model output as a function of its input parameters. In other words, they quantify the impact of inputs' uncertainty on an output. Sobol indices are widely used for example in hydrology (see [START_REF] Varella | Global sensitivity analysis measures the quality of parameter estimation: The case of soil parameters and a crop model[END_REF][START_REF] Pappenberger | Multi-method global sensitivity analysis of flood inundation models[END_REF]). Recently ( [START_REF] Borgonovo | On the quantification and decomposition of uncertainty[END_REF][START_REF] Owen | Sobol' indices and shapley value[END_REF][START_REF] Saltelli | Global Sensitivity Analysis: The Primer[END_REF]), this global sensitivity analysis has been used as a risk management tool, amongst other indicators ( [START_REF] Borgonovo | A new uncertainty importance measure[END_REF]). In this work, our goal is to prove that Sobol indices behave coherently with respect to stochastic ordering theory. This question arises naturally: under some reasonable conditions on the output, uncertainty quantifiers should increase if the uncertainty on the input increases in some way. Thus, our problematic is to find for which kind of stochastic orders and under which sufficient conditions on the output function, the Sobol indices behave coherently. Roughly speaking, given two random variables X and Y , the X-Sobol indeces on Y is given by

S X = Var(E(Y | X)) Var(Y ) .
It is a statistical indicator of the relative impact of X on the variability of Y . If we study the impact of several independent variables X 1 , . . . , X k on Y , then the Sobol indices may be used to provide a hierarchization of the X i 's with respect to their impact on Y . It is an interesting alternative to regression coefficient, which may be hardly interpreted if the relationship between Y and the X i 's is far from linear.

In our context, X = (X 1 , . . . , X k ) is a random vector of R k , with the X i 's being independent. We are interested in the variance and then the Sobol indices of an output function f . We shall assume properties such as convexity and/or monotonicity of the function f . One of our main result is that Sobol indices have a behavior which is compatible with respect to the excess wealth order / or the dispersive order (it depends on the convexity properties of f , see Theorems 2.1 and 2.3). The fact that Sobol index are in accordance with the excess wealth or with the dispersive order confirms that it could be used to quantify some uncertainty, even if, depending on the purpose, moment-independent approaches should be prefered to variance decomposition (see [START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF]). Neverthelss, as we shall see in the examples, the ordering of the Sobol indices heavily depends on the law of the parameters, so that one has to be careful on the conclusions.

To simplify notations, if i ∈ {1, . . . k}, we shall write X -i for the random vector (X 1 , . . . , X i-1 , X i+1 , . . . , X k ), and if α ⊂ {1, . . . k}, we shall write X α for the random vector (X i , i ∈ α) and X -α for the random vector (X i , i ∈ α).

The paper is organized as follows. In Section 1, we study the impact of stochastic orders on the variance. In Section 2, we state our may results concerning the accordance of Sobol indices with respect to the dispersive order. Finally, in Section 3 we provide some examples of illustrations. In Section 4, we give some concluding remarks.

Impact of the stochastic orders on the variance

Let us recall some particular notions of ordering on random variables / vectors. We refer to [START_REF] Müller | Comparison Methods for Stochastic Models and Risks[END_REF][START_REF] Shaked | Stochastic orders[END_REF] or [START_REF] Denuit | Actuarial Theory for Dependent Risks: Measures, Orders and Models[END_REF] for a detailed review on stochastic orders, their relationships and properties.

1.1. Stochastic orders. We shall be mainly interested in the stochastic order, the convex order, the dispersive order, the excess wealth order, the * order and the Lorenz order. For a random variable X, F X denotes its distribution function, and F -1 X the generalized inverse of F X (or the quantile function). The survival function is

F X = 1 -F X .
Definition 1.1. Let X and Y be two random variables, we say that [START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF] X is smaller than Y for the standard stochastic order (X ≤ st Y ) if and only if, for any bounded and non decreasing function f ,

E(f (X)) ≤ E(f (Y )).
(2) X is smaller than Y for the convex order (X ≤ cx Y ) if and only if, for any bounded convex function f ,

E(f (X)) ≤ E(f (Y )).
(3) If X and Y have finite means, then X is smaller than Y for the dilatation order (X ≤ dil Y ) if and only if

(X -E(X)) ≤ cx (Y -E(Y )).
(4) X is smaller than Y for the dispersive order (X ≤ disp Y ) if and

only if F -1 Y -F -
F -1 Y F -1
X is non decreasing. [START_REF] Denuit | Actuarial Theory for Dependent Risks: Measures, Orders and Models[END_REF] If X and Y are non negative with finite mean, X is smaller than Y for the Lorenz (X ≤ Lorenz Y ) if and only if

X E(X) ≤ cx Y E(Y )
.

Remark 1.2. The st and cx orders may be defined in the same way for random vectors.

1.2. Some relationships between variance and stochastic orders. It is well known that the stochastic order and the convex order are not locationfree and may not be compatible with the variance. The dispersive and excess wealth orders are location-free and in accordance with the variance. Below, we give some conditions implying some accordance of the stochastic order or the convex order with respect to the variance.

Proposition 1.3. Let i ∈ {1, . . . , k}. If X * i is a random variable, we shall write X i * for the random vector of R k : X i * = (X 1 , . . . , X i-1 , X * i , X i+1 , . . . , X k ). Let X *
i be a random variable, independent of X. The following holds:

(1) If X * i ≤ st X i then X i * ≤ st X. In particular, if f is non decreasing with respect to its i-th component and E(f

(X i * )) = E(f (X)) then Var(f (X i * )) ≤ Var(f (X)). (2) If X * i ≤ cx X i then X i * ≤ cx X. In particular, if f is convex with respect to its i-th component and E(f (X i * )) = E(f (X)) then Var(f (X i * )) ≤ Var(f (X)).
Proof. The result follows from the definitions and the fact that for any func-

tion f : R k -→ R, E(f (X)) = E E(f (X 1 , . . . , X k )|X -i ) .
In what follows, we will consider the excess wealth order, for which we can prove the ordering of Sobol's indices. Let us remark that the dispersive order implies the excess wealth order. Natural examples of random variables ordered with respect to the dispersive order will be recalled in Section 3. The following results proved in [START_REF] Shaked | Stochastic orders[END_REF], show that the excess wealth order (and thus the dispersive order) is in accordance with the variance. Proposition 1.4. [START_REF] Shaked | Stochastic orders[END_REF] Let X and Y be two random variables with finite means.

( Remark 1.5. Result ( 5) above has been incorrectly stated in [START_REF] Kochar | The total time on test transform and the excess wealth stochastic orders of distributions[END_REF] and [START_REF] Shaked | Stochastic orders[END_REF], where the hypothesis on the left-end points of the supports was missing. This hypothesis is indeed required, as shown by the following example. Consider X which follows a uniform law on [1, 1.9] and Y which follows a uniform law on [0, 1]. Then X ≤ ew Y . Let f = exp which is a convex and increasing function. We have that Var(f (X)) ∼ 1.32 and Var(f (Y )) ∼ 0.24 so that f (X) cannot be less than f (Y ) for the ew order (see [START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF] of Proposition 1.4). The correct statement and proof of ( 5) above may be found in [START_REF] Shaked | A class of locationindependent variability orders, with applications[END_REF], as well as an example showing that the left-end points of the support have also to be finite.

1) If X ≤ disp Y then X ≤ ew Y then X ≤ dil Y
Below, we give two simple counter-examples that show that the hypothesis on f above are necessary to get the inequality on the variance.

Example 1.6. Let X have uniform law on [0, 1] and Y have uniform law on [0, 10]. We consider the function f such that f (t) = t for t ∈ [0, 1] and

f (t) = 1 for t ≥ 1. f is a non decreasing function and X ≤ st Y . But, Varf (X) > Varf (Y ).
Example 1.7. Let X be such that P(X = 0) = 19 20 and P(X = 1) = 1 20 and Y be such that P(Y = 0) = 1 2 and P(Y = 10) = 1 2 . Consider, any function f such that f (0) = 0, f (1) = 10 and f (10) = 1. Then, we have

E(f (X)) = E(f (Y )), X ≤ cx Y but Var(f (X)) > Var(f (Y )).

Impact on Sobol indices

Sobol indices can be used as a tool to quantify the impact of input parameters on the output. They are more accurate than the variance in order to identify the input variables that have the most important impact on the output. Our goal is to explore how an increase of riskness (in the sense of stochastic orders) of the input parameters may have an impact on the output. We begin by recalling definitions on Sobol indices. We refer to [START_REF] Buzzard | Variance-based global sensitivity analysis via sparse-grid interpolation and cubature[END_REF][START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[END_REF] or [START_REF] Janon | Analyse de sensibilité et réduction de dimension[END_REF] for more details on this subject.

2.1. Some facts on Sobol indices. As before, we consider one output Y = f (X 1 , . . . , X k ) with X 1 , . . . , X k independent random variables. In what follows, if α ⊂ {1, . . . , k}, X α is the random vector X α = (X i , i ∈ α). We shall denote µ Xα the law of the random vector X α . For α ⊂ {1, . . . , k}, |α| denotes the length of α, i.e. its number of elements. The function f can be decomposed into

(2.1) f (X 1 , . . . , X k ) = α⊂{1,...,k} f α (X α ), with (1)
f ∅ = E(f (X)), (2)
f α dµ X i = 0 if i ∈ α, (3)
f α • f β dµ X = 0 if α = β.
The functions f α are defined inductively:

f ∅ = E(f (X)), for i ∈ {1, . . . , k} (2.2)
f i (X i ) = E(f (X) | X i ) -f ∅ = f dµ X -i -f ∅ .
If the f β have been defined for |β| < n, let α ⊂ {1, . . . , k} with |α| = n then,

f α (X α ) = f dµ X -α - β α f β (X β ).
With these notations, we have that:

Var(Y ) = Var(f (X)) = α⊂{1,...,k} Var(f α (X α )) = α⊂{1,...,k} E(f α (X α ) 2 ).
This decomposition of variance is often called Hoeffding decomposition ( [START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]).

The impact of X i on Y = f (X) may be measured by the Sobol index:

(2.3)

S i = Var(E(f (X) | X i )) Var(Y ) = E(f i (X i ) 2 ) Var(Y ) .
There are also interactions between the variables X 1 , . . . , X k , they are identified by the f α , with |α| ≥ 2. The total Sobol indices take into account the impact of the interactions:

(2.4)

S T i = i∈α⊂{1,...,k} Var(f α (X α )) Var(Y ) = i∈α⊂{1,...,k} E(f α (X α ) 2 )
Var(Y ) .

2.2.

Relationship with stochastic orders when there is no interactions. In this section, we assume that there is no interactions between the X i 's, that is, f (X) can be expressed in the following additive form:

(2.5) f (X 1 , . . . , X k ) = k j=1 g j (X j ) + K
where g 1 , . . . , g k are real-valued functions and K ∈ R. It is straitghforward to prove that, in that case, decomposition (2.1) reduces to

(2.6) f (X) = k i=1 f i (X i ) + f ∅ ,
so that, for any i = 1, . . . , k, the "individual" Sobol index defined by (2.3) coincides with the total Sobol index defined by (2.4).

As in the previous section, X * i denotes another variable that will be compared to X i . We shall assume X * i ≤ ew X i and study the impact of replacing X i by X * i on Sobol indices. We assume that X * i is independent of X -i and we denote by X * = (X 1 , . . . , X i-1 , X * i , X i+1 , . . . , X k ) the vector X where the i-th component has been replaced by X * i and by

S * i = Var(E(f (X * ) | X * i ))) Var(f (X * ))
the i-th Sobol index associated with f (X * ). Because we shall use the excess wealth order, we assume that X i and X * i have finite means, this hypothesis may be relaxed by considering random variables ordered with respect to the dispersive order. Theorem 2.1. We assume that there is no interactions, i.e. (2.5) is satisfied. Let X * i be a random variable independent of X -i and assume that X * i ≤ ew X i and -∞ < ℓ * ≤ ℓ, where ℓ * and ℓ are the left-end points of the support of X * i and X i . If g i is a non decreasing convex function, then S * i ≤ S i and S * j ≥ S j for j = i.

The proof of Theorem 2.1 makes use of Proposition 1.4.

Proof. As g i is a non decreasing convex function, Proposition 1.4 implies

g i (X * i ) ≤ ew g i (X i ) and Var(g i (X * i )) ≤ Var(g i (X i ))
. Now, the Hoeffding's decomposition of f (X) can be expressed as (2.5) where

f ∅ = E[f (X)] and f j (X j ) = g j (X j ) -E[g j (X j )] and the Hoeffding's decomposition of f (X * ) writes f (X * ) = j =i f j (X j ) + f * i (X * i ) + f * ∅ where f * ∅ = E[f (X * )] and f * i (X * i ) = g i (X * i ) -E[g i (X * i )]. Then, from (2.3), the i-th Sobol indices of f (X) and f (X * ) are such that (2.7) S i =      1 + j =i Var(g j (X j )) Var(g i (X i ))     
and

S * i = Var(g i (X * i )) j =i Var(g j (X j )) + Var(g i (X * i )) (2.8) =      1 + j =i Var(g j (X j )) Var(g i (X * i ))      -1 . (2.9)
We have already noticed that Var(g i (X * i )) ≤ Var(g i (X i )) and thus we conclude that S * i ≤ S i . The result for j = i follows from the fact that Var(g i (X * i )) ≤ Var(g i (X i )) and

S j = Var(g j (X j )) j =i
Var(g j (X j )) + Var(g i (X i )) and S * j = Var(g j (X j ))

j =i Var(g j (X j )) + Var(g i (X * i ))
.

Remark 2.2. Note that, from Proposition 1.4, the previous result also holds for any non decreasing convex or non increasing concave function g i as soon as X * i ≤ disp X i and X * i ≤ st X i . In addition, it is shown in [START_REF] Shaked | Stochastic orders[END_REF] that if X * i ≤ disp X i with common and finite left end points of their support (i.e., ℓ * = ℓ) then X * i ≤ st X i . 2.3. Relationship with stochastic orders when there are interactions. In the case where there are interactions, we have to consider the total Sobol indices as defined by (2.4). We will first show that the i-th total Sobol indices are ordered if X * i ≤ disp X i and X * i ≤ st X i , provided that the function f is a product of functions of one variable whose log is non decreasing and convex. Then we consider some extensions of that case. Theorem 2.3. We assume that f writes:

(2.10) f (X 1 , . . . , X k ) = g 1 (X 1 ) × • • • × g k (X k ) + K
where K ∈ R and g j , j = 1, . . . , k are real-valued functions. Let X * i be a random variable independent of X -i and assume that X

* i ≤ disp X i and X * i ≤ st X i . If log g i is a non decreasing convex or a non increasing concave function, then S * T i ≤ S T i and S * T j ≥ S T j , for j = i.
Proof. Without loss of generality, we may assume that K = 0. With the hypothesis of Theorem 2.3, the decomposition (2.1) satisfies: for all j = 1, . . . , k,

f j (X j ) = (g j (X j ) -E(g j (X j )) ℓ =j E(g ℓ (X ℓ )),
and following, e.g. [START_REF] Kuo | On decompositions of multivariate functions[END_REF], for α ⊂ {1, . . . , k}, we have

(2.11) f α (X α ) = β⊂α (-1) |α|-|β| E(f (X)|X β ).
The form of f then gives:

f α (X α ) = β⊂α (-1) |α|-|β| j∈β g j (X j ) j ∈β E(g j (X j )) = j ∈α E(g j (X j )) j∈α (g j (X j ) -E(g j (X j ))) .
We write

f T i = α∋i f α
Then,

f T i = α∋i j ∈α E(g j (X j )) j∈α (g j (X j ) -E(g j (X j ))) = (g i (X i ) -E(g i (X i )) γ⊂{1,...,k}\{i} j ∈γ E(g j (X j )) j∈γ (g j (X j ) -E(g j (X j ))) . Now, γ⊂{1,...,k}\{i} j ∈γ E(g j (X j )) j∈γ (g j (X j ) -E(g j (X j ))) = j∈{1,...,k}\{i} (g j (X j ) -E(g j (X j )) + E(g j (X j ))) = j∈{1,...,k}\{i} g j (X j ).
So that, finally, (2.12)

f T i (X) = (g i (X i ) -E(g i (X i )) j =i g j (X j ).
We denote by f * α the functions involved in the Sobol decomposition of f (X * ) and by f * T i the sum of the f * α 's over the α for which i ∈ α. Then,

f * T i (X * ) = (g i (X * i ) -E(g i (X * i )) j =i g j (X j ),
and

S T i = Var(f T i (X)) Var(f (X)) and S * T i = Var(f * T i (X * )) Var(f (X * ))
.

As in the proof of Theorem 2.1, this may be rewritten as

S T i =      1 + α ∋i Var(f α (X α ) Var(f T i (X))      -1
and (2.13)

S * T i =      1 + α ∋i Var(f * α (X α ) Var(f * T i (X * ))      -1
.

We have Var(f T i (X)) = Var(g i (X i ))

j =i E(g j (X j ) 2 )
and Var(f

* T i (X * )) = Var(g i (X * i )) j =i E(g j (X j ) 2 ). Also, if i ∈ α, Varf α (X α ) = E(g i (X i )) 2 Var     j =i j ∈α E(g j (X j )) j∈α (g j (X j ) -E(g j (X j )))     and Varf * α (X α ) = E(g i (X * i )) 2 Var     j =i j ∈α E(g j (X j )) j∈α (g j (X j ) -E(g j (X j )))     .
So, the result follows from (2.13) if

Var(g i (X * i )) E(g i (X * i )) 2 ≤ Var(g i (X i )) E(g i (X i )) 2 .
We have (see Proposition 1.4)

log g i (X * i ) ≤ disp log g i (X i ) ⇐⇒ g i (X * i ) ≤ * g i (X i ) so that g i (X * i ) ≤ Lorenz g i (X i ) =⇒ Var(g i (X * i )) E(g i (X * i )) 2 ≤ Var(g i (X i )) E(g i (X i )) 2
and thus S * T i ≤ S T i . When j = i, we have:

Var(f T j (X)) = E(g i (X i ) 2 )Var(g j (X j )) p ∈{i,j} E(g p (X p ) 2 )
and

Var(f (X)) = α⊂{1,...,k},α =∅ Var(f α (X α )) = E(g 2 i (X i )) α⊂{1,...,k}\{i},α =∅ p ∈α E(g p (X p )) 2 p∈α Var(g α (X α )) +Var(g i (X i )) p =i E(g p (X p )) 2 . So that S * T j ≥ S T j if E(g i (X * i ) 2 ) E(g i (X * i )) 2 ≤ E(g i (X i ) 2 ) E(g i (X i )) 2 which holds as above because g i (X * i ) ≤ Lorenz g i (X i ).
The conditions on stochastic ordering between X i and X * i , and on the log-convexity are necessary, as can be seen with the two counter-examples below.

Example 2.4. To see the necessarity of the stochastic ordering, one can consider:

f (X 1 , X 2 , X 3 ) = exp(exp(X 1 )) exp(X 2 ) exp(X 3 ), with X 1 , X 2 and X 3 uniform on [0, 1]. If X * 1 is uniform on [1, 1.9], then X * 1 ≤ disp X 1 but X 1 ≤ st X *
1 and it can be easily checked that S * T 1 > S T 1 (S * T 1 ≈ 0.90 and S T 1 ≈ 0.65). Example 2.5. The log-convexity of g i is also necessary. Indeed, take:

f (X 1 , X 2 , X 3 ) = g 1 (X 1 )X 2 X 3
where

g 1 (x) =    0 if x < 0.45, x/10 if 0.45 ≤ x ≤ 0.5, x else. and X 2 , X 3 uniform on [2, 3], X 1 uniform on [0, 1], X * 1 uniform on [0, 0.5] so that X * 1 ≤ disp X 1 , X * 1 ≤ st X 1 but g 1 is not log-convex.
In that case, we have S * T 1 > S T 1 (S * T 1 ≈ 0.99 and S T 1 ≈ 0.97). Now, we turn to the case where f writes as a sum of product of convex and non decreasing functions of one variable, that is, there are: a finite set A and convex and non decreasing functions g a i , i ∈ {1, . . . , k}, a ∈ A, such that

(2.14) f (X) = a∈A g a 1 (X 1 ) × • • • × g a k (X k ).
Proposition 2.6. Assume that f satisfies (2.14). Then, for any i ∈ {1, . . . , k},

f i (X i ) = a∈A   (g a i (X i ) -E(g a i (X i )) j =i E(g a j (X j ))   (2.15) f T i (X) = a∈A   (g a i (X i ) -E(g a i (X i ))) j =i g a j (X j )   (2.16) Var(f T i ) = a,b∈A Cov(g a i (X i ), g b i (X i )) j =i E(g a j (X j )g b j (X j )). (2.17)
Proof. The proof uses in a straightforward way the computations done in the proof of Theorem 2.3.

We deduce the two following extensions of Theorem 2.3. Proposition 2.7. Let {I a } a∈A be a partition of {1, . . . , k} and assume that

f (X) = a∈A j∈Ia g j (X j )
where the g j 's are real-valued functions. Let X * i be a random variable independent of X and assume that X * i ≤ disp X i and X * i ≤ st X i . If log g i is a non decreasing convex function or a non increasing concave function, then S * T i ≤ S T i and S * T j ≥ S T j , for j = i.

Proof. The proof follows that of Theorem 2.3 in a straighforward way because the I a are disjoints.

Proposition 2.8. Let f (X) = ϕ 1 (X i ) j =i
g j (X j ) + ϕ 2 (X i ) with log ϕ 1 and log ϕ 2 non decreasing and convex. If

• X * i is independent of X, X * i ≤ disp X i and X * i ≤ st X i . • Var(ϕ 2 (X * i )) E(ϕ 1 (X * i )) 2 ≤ Var(ϕ 2 (X i )) E(ϕ 1 (X i )) 2 and Cov(ϕ 1 (X * i ), ϕ 2 (X * i )) E(ϕ 1 (X * i )) 2 ≤ Cov(ϕ 1 (X i ), ϕ 2 (X i )) E(ϕ 1 (X i )) 2 . Then S * T i ≤ S T i . Proof. Proposition 2.6 gives Var(f T i (X)) = Var(ϕ 1 (X i )) j =i E(g j (X j )) 2 + Var(ϕ 2 (X i )) +Cov(ϕ 1 (X i ), ϕ 2 (X i )) j =i E(g j (X j )) and if i ∈ α, f α (X α ) = E(ϕ 1 (X i )) j ∈α E(g j (X j )) × j∈α (g j (X j ) -E(g j (X j ))), Var(f α (X α )) = E(ϕ 1 (X i )) 2 j ∈α E(g j (X j )) 2 ×Var   j∈α (g j (X j ) -E(g j (X j )))   .
We use once more (2.13) to get that S * T i ≤ S T i if and only if

Var(ϕ 1 (X * i ))

j =i E(g j (X j )) 2 + Var(ϕ 2 (X * i )) + Cov(ϕ 1 (X * i ), ϕ 2 (X * i )) j =i E(g j (X j )) E(ϕ 1 (X * i )) 2 α ∋i j ∈α E(g j (X j )) 2 × Var   j∈α (g j (X j ) -E(g j (X j )))   ≤ Var(ϕ 1 (X i )) j =i E(g j (X j )) 2 + Var(ϕ 2 (X i )) + Cov(ϕ 1 (X i ), ϕ 2 (X i )) j =i E(g j (X j )) E(ϕ 1 (X * i )) 2 α ∋i j ∈α E(g j (X j )) 2 × Var   j∈α (g j (X j ) -E(g j (X j )))   .
With our hypothesis, we have that Var

(ϕ 1 (X * i )) E(ϕ 1 (X * i )) 2 ≤ Var(ϕ 1 (X i )) E(ϕ 1 (X i )) 2 , Var(ϕ 2 (X * i )) E(ϕ 1 (X * i )) 2 ≤ Var(ϕ 2 (X i )) E(ϕ 1 (X i )) 2 and Cov(ϕ 1 (X * i ), ϕ 2 (X * i )) E(ϕ 1 (X * i )) 2 ≤ Cov(ϕ 1 (X i ), ϕ 2 (X i )) E(ϕ 1 (X i )) 2 .
This leads to the announced result.

The second condition in Proposition 2.8 is very technical and unsatisfactory. Nevertheless, very simple counter-examples exist as it can be seen below.

Example 2.9.

Let X i ∼ U ([1.5, 3.5]), X * i ∼ U ([0, 1.8]), ϕ 1 (x) = exp(x 2 ) and ϕ 2 (x) = exp(x), g j (x) = 1, j = i. Then one can show that X * i ≤ disp X i and X * i ≤ st X i . However, Var(ϕ 2 (X * i )) E(ϕ 1 (X * i )) 2 ∼ 0.08 Var(ϕ 2 (X i )) E(ϕ 1 (X i )) 2 ∼ 10 -7 Cov(ϕ 1 (X * i ), ϕ 2 (X * i )) E(ϕ 1 (X * i )) 2 ∼ 0.3 Cov(ϕ 1 (X i ), ϕ 2 (X i )) E(ϕ 1 (X i )) 2 ∼ 10 -3 .
and, from the proof of Proposition 2.8, S * T i > S T i . The following result derived in [START_REF] Fagiuoli | A characterization of the dilation order and its applications[END_REF] is mentioned here as a related result on excess wealth orders, even if it is not sufficient to obtain a more general version of Proposition 2.8. Proposition 2.10. [Corollary 3.2 in [START_REF] Fagiuoli | A characterization of the dilation order and its applications[END_REF]] Let X and Y be two finite means random variables with supports bounded from below by ℓ X and ℓ Y respectively. If X ≤ ew Y and ℓ X ≤ ℓ Y then for all non decreasing and convex functions h 1 , h 2 for which h i (X) and h i (Y ) i = 1, 2 have order two moments,

(2.18) Cov(h 1 (X), h 2 (X)) ≤ Cov(h 1 (Y ), h 2 (Y )).

Examples

In this section, we illustrate the previous results on some classical financial risk models. All considered models are associated with a set of parameters. In a context where these parameters are not known with certainty (due to estimation error for instance), the global sensibility analysis is useful to assess which (uncertain) input parameters mostly contribute to the uncertainty of model output and in turns, which parameters have to be estimated with caution. In most of our examples, we will consider truncated distribution functions (ordered with respect to the dispersive and stochastic orders). The use of truncated distribution is motivated by the fact that the distribution of financial parameters have generally bounded support. Let us first recall the conditions under which some particular distribution functions are ordered with respect to the dispersive order. We refer to [START_REF] Müller | Comparison Methods for Stochastic Models and Risks[END_REF] for other classes of distribution functions. Proposition 3.1. Let X and Y be two random variables.

(

1) If X ∼ U [a, b] and Y ∼ U [c, d], then X is smaller than Y for the dispersive order (X ≤ disp Y ) if and only if b -a ≤ d -c. (2) If X ∼ E(µ) and Y ∼ E(λ), then X is smaller than Y for the dispersive order (X ≤ disp Y ) if and only if λ ≤ µ. (3) If X ∼ N(m 1 , σ 2 ) and Y ∼ N(m 2 , ν 2 ), then X is smaller than Y for the dispersive order (X ≤ disp Y ) if and only if σ ≤ ν.
As mentioned above, most of the numerical illustrations will be based on model parameter with truncated distribution functions. We present some properties of such distributions. Definition 3.2. Let X be a random variable with density function f and (a, b) ∈ R 2 . If F denotes the cumulative distribution of X then the truncated distribution of X on the interval [a, b] is the conditional distribution of X given that a < X ≤ b. The truncated density function of X is then given by

(3.1) f (x|a < X ≤ b) = g(x) F (b) -F (a)
where g(x) = f (x) for all x such that a < x ≤ b and g(x) = 0 else.

In what follows, we denote by N T and E T the truncated normal and the truncated exponential laws respectively. Proposition 3.3. Let X and Y be two random variables [START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF] if X ∼ N T (m, σ 2 ) where X is truncated on [a, b] then the quantile function of X is given by

F -1 X (x) = φ -1 (φ(α) + x(φ(β) -φ(α)))σ + m where α = a-m σ , β = b-m σ
and where φ is the standard normal cumulative distribution function.

(2) if Y ∼ E T (λ) is truncated on [a, b] then the quantile function of Y is given by

F -1 Y (x) = - 1 λ log(e -λa + x(e -λb -e -λa ))
where λ denotes the parameter of the exponential distribution.

In the following lemma, we give some conditions that ensure the ordering of two truncated random variables with respect to the dispersive order. Lemma 3.4. Let X and Y be two random variables.

(

1) If X ∼ U [a, b] and Y ∼ N T (m, σ 2 ) where Y is truncated on [c, d], then X is smaller than Y for the dispersive order X ≤ disp Y if and only if b -a ≤ σ √ 2π(φ(β) -φ(α))
where φ represents the cumulative distribution of a standard gaussian law and where α and β are given by α = c-m σ and

β = d-m σ . (2) If X ∼ E T (µ) and Y ∼ E T (λ) are truncated on the same interval then X ≤ disp Y if and only if λ ≤ µ.
Proof. The previous conditions can be easily derived by differentiating the difference of the quantile functions F -1 Y -F -1 X and by using the fact that this derivative should be positive. Also, we recall (see Remark 2.2) that if two random variables ordered for the dispersive order have the same finite left point of their support, then they are ordered for the stochastic order. In the example that we will consider, N T (resp. E T ) denotes the truncated gaussian (resp. exponential) distribution on [0, 2].

3.1. Value at Risk sensitivity analysis. In risk management, the Valueat-risk (VaR) is a widely used risk measure of the risk of losses associated with portfolio of financial assets (such as stock, bond, etc). From a mathematical point of view, if L denotes the loss associated with a portfolio of assets, then V aR α (L) is defined as the α-quantile level of this loss, i.e.,

V aR α (L) := inf {x ∈ R : F L (x) ≥ α}
where F L denotes the cumulative distribution function of L. Let us consider a portfolio loss of the form L = S T -K where K is positive and where S T stands for the aggregate value at time T of a basket of financial assets. This corresponds to the loss at time T of a short position on this portfolio when the latter has been sold at time 0 for the price K. We assume that S follows a geometric brownian motion so that its value at time T can be expressed as

S T = S 0 exp (µT + σW T )
where W T is the value at time T of a standard Brownian motion, µ (resp. σ) is a positive drift (resp. volatility) parameter. Therefore, the α-Valueat-Risk associated with loss L is given by

(3.2) V aR α (L) = S 0 exp µT + σ √ T φ -1 (α) -K.
where S 0 ∈ R * + and where φ -1 is the normal inverse cumulative distribution function of a standard gaussian random variable. Note that, as soon as α ≥ 0.5, the VaR expression (3.2) can be seen as a product of log-convex non-decreasing functions with respect to µ and σ. Our interest is to quantify the sensitivity of the uncertain parameters µ and σ on the Value-at-Risk (V aR α (L)) by evaluating the total Sobol indices S Tµ and S Tσ as defined by (2.4). We then analyze how an increase of uncertainty in the input parameters impacts S Tµ and S Tσ . In the numerical illustrations, we consider a VaR associated with a risk level α = 0.9 and for a portfolio loss with the following characteristics:

T = 1, S 0 = 100, K = 100.
Table 1 illustrates the consistency of total Sobol indices when the distributions of input parameters are ordered with respect to the dispersive order.

Each line of Table 1 corresponds to a scenario where one of the parameter has been increased with respect to both the dispersive and the stochastic order. As can be seen, changing the laws of model parameters µ and σ have a significant impact on the values of total Sobol indices S Tµ and S Tσ . Note that the ordering among Sobol indices is fully consistent with the one predicted by Theorem 2.3. 1. Total Sobol indices of VaR (3.2) when α = 0.9. All digits are significant with a 95% probability.

µ * µ σ * σ S * Tµ S Tµ S * Tσ S Tσ U [0, 1] U [0, 1] U [0, 1] U [0,
3.2. Vasicek model. In risk management, present values of financial or insurance products are computed by discounting future cash-flows. In market practice, discounting is done by using the current yield curve, which gives the offered interest rate as a function of the maturity (time to expiration) for a given type of debt contract. In the Vasicek model, the yield curve is given as an output of an instantaneous spot rate model with the following risk-neutral dynamics

(3.3) dr t = a(b -r t )dt + σdW t
where a, b and σ are positive constants and where W is a standard brownian motion. Parameter σ is the volatility of the short rate process, b corresponds to the long-term mean-reversion level whereas a is the speed of convergence of the short rate process r towards level b. The price at time t of a zero coupon bond with maturity T in such a model is given by (see, e.g., [START_REF] Brigo | Interest Rate Models -Theory and Practice[END_REF]):

(3.4) P (t, T ) = A(t, T )e -rtB(t,T )

where

A(t, T ) = exp (b - σ 2 2a 2 )(B(t, T ) -(T -t)) - σ 2 4a B 2 (t, T )
and

B(t, T ) = 1 -e -a(T -t) a .
The yield-curve can be obtained as a deterministic transformation of zerocoupon bond prices at different maturities.

In what follows, we quantify the relative importance of the input parameters {a, b, σ} affecting the uncertainty in the bond price at time t = 0. In the following numerical experiments, the maturity T and the initial spot rate r 0 are chosen such that T = 1 and r 0 = 10%. Table 2 and 3 reports the total Sobol indices of the parameter a, b, σ under two different risk perturbations in the probability laws of these parameters. Table 2 illustrates the effect of an increase of the mean-reverting level with respect to the dispersive order and the stochastic dominance order, i.e., b * ≤ disp b and b ≤ st b * . We observe from Table 2 that the relative importance of the mean-reverting level b increases from 0.52 to 0.57. Although the total Sobol index of σ decreases, the total index of a increases. Note that the assumptions of Theorem 2.3 are not satisfied here : one can show that the output function (3.4) 2. Total Sobol indices as a result of a risk perturbation of b. All digits are significant with a 95% probability. Table 3 displays the total Sobol indices when σ * ≤ disp σ and σ * ≤ st σ. The law of σ is taken as a truncated gaussian random variable on [0, 2] and has a variance of 0.3. We observe an increase in the total Sobol index of σ and a decrease in total index of a and b.

parameter law total index parameter law total index a U [0, 1] 0.41 a U [0, 1] 0.25 b U [0, 1] 0.52 b U [0, 1] 0.13 σ * U [0, 1] 0.18 σ N T (0.5, 2)
0.70 Table 3. Total Sobol indices as a result of a risk perturbation of σ. All digits are significant with a 95% probability.

3.3. Heston model. In finance, the Heston model is a mathematical model which assumes that the stock price S t has a stochastic volatility σ t that follows a CIR process. The model is represented by the following bivariate system of stochastic differential equations (SDEs) (see, e.g., [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF])

dS t = (r -q)S t dt + √ σ t S t dB t (3.5) dσ t = κ(θ -σ t )dt + σ √ σ t dW t (3.6)
where d B, W t = ρdt. The model parameters are

• r: the risk-free rate,

• q: the dividend rate, • κ > 0: the mean reversion speed of the volatility, • θ > 0: the mean reversion level of the volatility, • σ > 0: the volatility of the volatility, • σ 0 > 0: the initial level of volatility, • ρ ∈ [-1, 1]: the correlation between the two Brownian motions B and W .

The numerical computation of European option prices under this model can be done by using the fast Fourier transform approach developed in [START_REF] Carr | Option valuation using the fast fourier transform[END_REF] which is applicable when the characteristic function of the logarithm of S t is known in a closed form. In this framework, the price at time t of a European call option with strike K and time to maturity T is given by (3.7) C(t, K, T ) = S t e -qτ P 1 -Ke -rτ P 2

where for j = 1, 2

P j = 1 2 + 1 π ∞ 0 ℜ e -iφ log K f j (φ; x t , σ t ) iφ dφ f j (φ;
x t , σ t ) = exp (C j (φ; τ ) + D j (φ; τ )σ t + iφx t )

C j = (rq)iφτ + κθ σ 2 (b jρσiφ + d j )τ -2 log 1g j e d j τ 1g j

D j = b j -ρσiφ + d j σ 2
1e d j τ 1g j e d j τ g j = b jρσiφ + d j b jρσiφd j d j = (ρσiφb j ) 2σ 2 (2u j iφφ 2 )

u 1 = 1 2 , u 2 = - 1 2 , b 1 = κ -ρσ, b 2 = κ, x t = log S t , τ = T -t.
Given that the input parameter are not known with certainty, which one mostly affect the uncertainty of the output pricing function (3.7)? Table 4 displays the total Sobol indices of each parameter under two assumptions on the distribution of input parameter. In the first case (3 first columns), all parameters are assumed to be uniformly distributed. In the second case (3 last columns), we only change the distribution of the interest rate parameter r is such as way that r * ≤ disp r and r * ≤ st r. The option characteristics are taken as follows: T = 0. As can be observed in Table 4, the influence of the input factors κ, θ, σ and ρ is negligible under the two considered assumptions. Note that most of the uncertainty in the option price is due to the dividend yield q and the interest rate r. Similar conclusions are outlined in [START_REF] Saltelli | Global Sensitivity Analysis: The Primer[END_REF] but the difference here is that we analyze how the total Sobol indices are affected by a change in the law of some input parameters. Interestingly, when the law of the interest rate parameter r * is changed to r, this parameter becomes more important than the dividend yield in terms of output uncertainty.

Concluding remarks

We have enlightened the fact that Sobol indices are compatible with the stochastic orders theory, and more precisely with the excess wealth order, or the dispersive order, provided that the output function satisfies some monotonicity and convexity properties. The Vasicek and Heston model examples suggest that the hypothesis on the form of the output function f might be relaxed (hypothesis of Theorem 2.3 are not fulfilled, especially in the Heston case). In other words, the compatibility between Sobol indices and stochastic orders should hold for more general functions than those considered in the present paper. On an other hand, as shown in the examples, the choice of the input laws is crucial: the most influential parameters (in the sense that it corresponds to the greater Sobol index) may change as some input distributions are perturbed. A way to overcome this difficulty could be to use our result on the consistency of Sobol index with stochastic orders to get universal bounds on Sobol indices for a given class of input laws.

  and thus if X and Y admit an order 2 moment, Var(X) ≤ Var(Y ).

	(2) If X and Y are non negative and X ≤ * Y then X ≤ Lorenz Y and then Var(X) E(X) 2 ≤ Var(Y ) E(Y ) 2 .
	(3) If X and Y are non negative then X ≤ * Y if and only if log X ≤ disp log Y .
	(4) If X ≤ disp Y and X ≤ st Y then for all non decreasing convex or non increasing concave function f , f (X) ≤ disp f (Y ). (5) If X and Y are continuous random variables with supports bounded
	from below by (resp.) ℓ * and ℓ, X ≤ ew Y and -∞ < ℓ * ≤ ℓ, then for all non decreasing and convex function f , for which f (X) and
	f (Y ) have finite means, we have f (X) ≤ ew f (Y ). [Theorem 4.2 in [18]]

  is log non decreasing and log convex in b but the multiplicative form (2.10) does not hold.

	parameter	law	total index parameter	law	total index
	a b * σ Table	U [0, 1] U [0, 1] U [0, 1]	0.41 0.52 0.18	a b σ	U [0, 1] U [0, 2] U [0, 1]	0.48 0.57 0.06

Table 4 .

 4 [START_REF] Buzzard | Variance-based global sensitivity analysis via sparse-grid interpolation and cubature[END_REF], S 0 = 100, K = 100. Total Sobol indices for the price of a call option in the Heston model. All digits are significant with a 95% probability.

	parameter	law	total index parameter	law	total index
	r * q κ θ σ σ 0 ρ	U [0, 1] U [0, 1] U [0, 1] U [0, 1] U [0, 1] U [0, 1] U [0, 1]	0.32 0.39 0.0036 0.0082 0.0012 0.30 0.0011	r q κ θ σ σ 0 ρ	U [0, 2] U [0, 1] U [0, 1] U [0, 1] U [0, 1] U [0, 1] U [0, 1]	0.73 0.20 0.0009 0.0020 0.0004 0.08 0.0004


Yongxin Liao
email: ayongxin.liao@pucpr.br

Mario Lezoche
email: bmario.lezoche@univ-lorraine.fr

Eduardo Rocha Loures
email: ceduardo.loures@pucpr.br

Hervé Panetto
email: dherve.panetto@univ-lorraine.fr

Nacer Boudjlida
email: enacer.boudjlida@loria.fr

A Semantic Annotation Framework to Assist the Knowledge Interoperability along a Product Life Cycle

Keywords: Knowledge Engineering, Semantic Annotation, Semantic Interoperability, Product Life Cycle, Formalization

The interoperability among a variety of systems, in or across manufacturing enterprises, has been widely accepted as one of the important factors that affects the efficiency of production. The aim of this study is to deal with the semantic interoperability issues in a product lifecycle management environment. Through the investigation of related works, the need for the formalization of semantic annotation was discovered. This paper addresses this drawback and introduces a framework that uses formalized semantic annotations to assist the knowledge interoperability along a product life cycle.

Introduction

In recent years, more and more manufacturing enterprises have discovered and considered the Product Lifecycle Management (PLM) approach as one of essential solutions to create competitive products. A major current focus in PLM researches is how to ensure a seamless interoperability among different enterprise systems at each stage of a Product Life Cycle (PLC) [START_REF] Ameri | Product lifecycle management: closing the knowledge loops[END_REF]. Interoperability is an ability of two or more systems or components to exchange information and to use the information that has been exchanged [START_REF] Geraci | IEEE standard computer dictionary: Compilation of IEEE standard computer glossaries[END_REF]. However, because stakeholders who operate those systems along a PLC have different backgrounds, heterogeneous expertise, unique knowledge, particular needs and specific practices, which over increase the difficulty to achieve interoperability in semantic level [3].

Ontology is a kind of common agreement on the conceptualization of terms in a specific domain of interest [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF]. One of the widely recognized methods to deal with semantic gaps is to semantically enrich the exchanged information through an ontology. There exist many studies that highlighted and proved the advantages of applying semantic annotations on various kinds of models, which represent a product from different perspectives. For example, research [START_REF] Song | An ontology-driven framework towards building enterprise semantic information layer[END_REF] and [START_REF] Bergamaschi | The Open Source release of the MOMIS Data Integration System[END_REF] focused on annotating data models to support data integration. Research [START_REF] Attene | Characterization of 3D shape parts for semantic annotation[END_REF] and [START_REF] Li | Ontology-Driven Semantic Annotations for Multiple Engineering Viewpoints in Computer Aided Design[END_REF] developed annotation tools to demonstrate the possibility of sharing and reusing the knowledge that is represented in a computer-aided design model. Research [START_REF] Francescomarino | Semantic annotation of business process models[END_REF] and [START_REF] Lin | Semantic annotation for process models: Facilitating process knowledge management via semantic interoperability[END_REF] suggested to annotate business process models with semantic contents for assisting designers and analysts in the management of their models. And more generally, research [START_REF] Boudjlida | DTG4. 2: Experimental Semantic Enrichment of Enterprise Models for Interoperability and its Practical Impact[END_REF] and [START_REF] W3c | Semantic Annotations for WSDL and XML Schema[END_REF] proposed high level semantic annotation methods for the semantic enrichment of all kinds of enterprise models. Even though lots of efforts were made for addressing the issues related to semantic interoperability, few researchers have paid attention to the formalization perspective of semantic annotations.

The objective of this study is to deal with the interoperability problems, mainly the issue of semantic interoperability, through introducing a formal semantic annotation framework. This framework is then being adapted in a new application scenario, to firstly realize the semantic enrichment of models, to secondly assist the knowledge interoperation, and to lastly show its adaptability.

Methodology

In this paper, we focus on identifying the essential elements of a semantic annotation through its meta-model and presenting a semantic annotation framework to support the mutual understanding of the semantics that's embedded in the exchanged models along the life cycle of a product.

Knowledge Representation. Knowledge is an awareness of things that brings to its owner the capability of grasping the meanings (semantics) from received information [START_REF] Ackoff | From data to wisdom[END_REF]. In our work, it is considered as a kind of intangible thing, which has to be made perceptible and afterward to be expressed under certain representations. Knowledge representation is the result of embodying the knowledge from its owner's mind into explicit forms. It gives a possibility for external entities to perform some specific operations and achieve their particular needs. In this study, all relevant models that are generated by stakeholders through different kinds of enterprise systems are all being considered as knowledge representations.

Semantic Interoperability. It signifies an unambiguous understanding of exchanged knowledge representation considering the point of view of both the senders and the receivers [START_REF] Pokraev | Semantic service modeling: Enabling system interoperability[END_REF]. A model "is often presented as a combination of drawings and text" [START_REF] Miller | MDA Guide Version 1.0. 1[END_REF]. The mutual understanding of models requires the comprehension of the semantics of both the "combination of drawing" and the "text". Which is why two aspects of semantics are needed to be made explicit by a semantic annotation: (1) Structure semantics, which describes the interrelations between an annotated element and the other elements that it related to; (2) Domain semantics, which describes the context and the meaning of an annotated element in a selected domain.

Meta-model of the Semantic Annotation. Several important concepts that are used throughout this paper need to be cited. In this study, the targets of semantic enrichment are models, which are named as Target Knowledge Representations (TKRs). Ontologies represent real-world semantics that enable a human or a machine to use meaningful terminologies to perform semantic enrichment on TKRs. They are named as Ontology-based Knowledge Representations (OKRs). And more precisely, two types of ontologies are categorized: (1) PLC-related ontologies, which represent the knowledge that related to a product during one or more stages its life cycle; (2) Meta-model ontologies, which represent the knowledge of how models are constructed based on different specifications. A Semantic Annotation is acting as a bridge to describe the semantic relationships between TKRs and OKRs. The meta-model of the semantic annotation is given in Fig. 1, which describes its main components (the essential elements) and their relations:

 A "Target Knowledge Representation" is the composition of one or more "Element of a TKR".  The "Ontology-based Knowledge Representation" is the generalization of the "Meta-model Ontology" and the "PLC-related Ontology".  A "Meta-model Ontology" is the composition of one or more "Element of a Meta-model Ontology".  A "PLC-related Ontology" is the composition of one or more "Element of a PLC-related Ontology".  An "Element of a TKR" can be annotated by zero or more "Semantic Annotation".  A "Semantic Annotation" contains one "Structure Semantics".  A "Semantic Annotation" contains zero or more "Domain Semantics".  A "Structure Semantics" is the aggregation of one "Element of a Meta-model Ontology"  A "Domain Semantics" is the aggregation of one or more "Element of a PLC-related Ontology". Semantic Annotation Framework. In the interest of applying formal semantic annotations on multiple TKRs in a PLM environment, as it is shown in Fig. 2, a semantic annotation framework is introduced.

On the left side, there is a series of processes that describes a linear product life cycle. It represents the TKR Creation and Management module, which is composed of a number of enterprise systems. Stakeholders in or across enterprises along the product lifecycle use those systems to create and manage TKRs. Those systems need to provide sufficient APIs to enable the communications between themselves and the Semantic Annotation and Processing Agent module.

Figure 2 The Proposed Semantic Annotation Framework in a PLM Environment.

On the right side, the OKR Creation and Management module is in charge of capturing, formalizing and managing knowledge related to the life cycle of a product into a knowledge base, namely, Knowledge Cloud. The OKRs are supposed to be stored in a platform-independent form, which ensures they can be imported, mapped, merged and interrelated with each other. The Knowledge Cloud module acts as a knowledge repository. As shown in Fig. 3, three main kinds of knowledge are stored in it:

(1) The OKRs, as it is shown in the Fig. 3 The Reasoning Engine module is an external call pattern-matching search engine. It performs inference on the semantic annotations, the OKRs and the reasoning rules according to different stakeholders' requests.

The Semantic Annotation and Processing Agent (SAPA) model is in charge of the definition of semantic relationships. It also acts as a mediator to support the communications (requests and feedbacks) between enterprise systems in different processes of a PLC (TKR Creation and Management module) and the other three modules (Knowledge Cloud module, OKR Creation and Management module and Reasoning Engine module) in the framework.

This framework makes use of formal semantic annotations to describe the semantic relationships between TKRs and OKRs. It makes explicit the implicit semantics of the TKRs, gives possibilities to detect inconsistencies between semantic annotations and to identify possible conflicts among annotated elements, and makes explicit the relations among all disperse TKRs.

Application Scenario

In order to show the adaptability of this framework, beside the preceding application scenario in our previous research [START_REF] Liao | Semantic Enrichment of Models to Assist Knowledge Management in a PLM environment[END_REF], we further prototyped it in a new environment. The lifecycle of a nursery toy As it is shown in Fig. 4, three enterprise systems are selected: The SolidWorks system 2 , which responds of designing the product and generating its technical information; The Bizagi BPM Suite 3 , which in charge of defining and modelling the process models; The Infor Visual Manufacturing system 4 , which is responsible for the production planning and control. They represent the TKR Creation and Management module. For the rest of the framework, the Protégé ontology editor is used as the OKR Creation and Management module, the Microsoft windows folder system is used as the Knowledge Cloud module, the SAP-KM (Semantic Annotation Plugin for Knowledge Management) is considered as the Semantic Annotation and Processing Agency module, and the Jena Reasoner is employed as the Reasoning Engine module. Through this prototyped framework, the formalized semantic annotations are contributing in: (1) acquiring the initial semantics that the stakeholders, who manipulate the SolidWorks system, wanted to express; (2) verifying, semi-automatically, the semantic consistency between the contents in the received models and in the developing models in the Bizagi BPM Suite; and (3) guaranteeing the semantics correctness of the under development models for the stakeholders who use the Infor Visual Manufacturing system.

Conclusion and Perspectives

In summary, we have introduced a semantic annotation meta-model that unambiguously describes its major components and their interrelations, given a semantic annotation framework to apply semantic annotations in a PLM environment, and prototyped this framework in a new application scenario to assist knowledge interoperation. Despite some limitations (such as, the time-consuming ontology creation process, the difficulties in the measurement of semantic relationships, the not perfect annotation process), this paper shows the semantic annotation framework is able to be used in different application scenarios to support the knowledge interoperation along the lifecycle of a product. Furthermore, in future, under the context of the PLM, the potentials of this approach will be used in the following directions: the formalization of requirements from different dimensions; to 1 Laboratory for Manufacturing Automation: http://www.pucpr.br/graduacao/engproducao/curitiba/las01.php 2 SolidWorks: https://www.solidworks.com/ 3 Bizagi BPM Suite: http://www.bizagi.com/en/products/bizagi-bpm-suite/overview-bpm-suite 4 enable the traceability of requirements along a PLC; and the verification of consistencies among changed and existed requirements.

Figure 1

 1 Figure 1 The Meta-model of the Semantic Annotation

  (a), which are structured into a traditional three-levels: a. Top level ontology. It contains the most common terminology that can be used in different domains. b. Domain level ontology. It is classified into two aspects: the PLC-related ontologies and the Meta-model ontologies. c. Application level ontology. Concerning the above-mentioned two aspects of domain level ontologies, the corresponding application level ontologies contain certain specific terms in different enterprises and in different modelling tools respectively. (2) The semantic annotations, as it can be seen in Fig. 3 (b), which are created by stakeholders along the product lifecycle via using the concepts and relationships in the OKRs (3) The reasoning rules, as it is shown in the Fig. 3 (c), which are generated based on the OKRs, the implemented semantic annotation schema and corresponding reasoning mechanisms.

Figure 3

 3 Figure 3 The Knowledge Repository: Knowledge Cloud.

  is manufactured in the Laboratory for Manufacturing Automation 1 is being chosen.

Figure 4

 4 Figure 4 The Application Scenario of the Case Study

  Infor Visual Manufacturing: http://go.infor.com/visual/

	SAP-KM	Protégé
		Ontology Editor	Legend
	SAs		Ontologies (OKRs)
			SAs	Semantic Annotations
		Knowledge Cloud
			Reasoning Rules
	SAs		Interrelations
		SAs	SAs
		SAs SAs SAs SAs	The Increment of
	s		Semantic Annotations
			Interfac
		Jena Reasoner	Request and feedback
	SAs		API
	s

Acknowledgements

The authors would like to thank the financial support provided by the Charles Hermite Research Federation (CNRS FR 3198) and the "Région Lorraine" local government in France, and by the Coordination of Improvement of Higher Education Personal (CAPES) in Brasil, under the project of Science without Borders Program.