Alexandre Gohin
email: alexandre.gohin@rennes.inra.fr

Keywords: path independent, second order approximation Superlative, Path Independent, Second Order Approximation: 18 -51_ 09 ' Welfare Decomposition, Computable General Equilibrium, Equivalent Variation, Second Best Theory JEL Classifications: D58, Fl 1, F14

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Introduction: Problem Statement and Literature Review

Welfare is very likely the favorite notion of economists ; accordingly it has always received considerable attention, both theoretically and empirically. A significant part of the economic literature deals with the issue of defining the good measure of economic welfare. Most frequently used indicators are real wages, real GDP, real income, consumer surplus as well as the Hicksian Compensating Variation (CV) and Equivalent Variation (EV) measures. Ali these indicators have both pros and cons ; then one must start choosing a particular one. This choice obviously depends on the problem at hand. In this paper, we adopt the last one (EV) as our main objective is to explain the welfare effects which are simulated using Computable General Equilibrium (CGE) models. Given the on-going creativity on this research area, our opinion is that it may be useful to defend this choice. In addition to be the most widely reported welfare indicator by CGE modelers, we choose EV because ail other mentioned welfare measures either suffer from some pitfalls, either are not best-suited in the context of this paper.

In particular, changes in real wages are often used to measure welfare changes but, as demonstrated by [START_REF] Thierfelder | A note on Taxes, Prices, Wages and Welfare in General Equilibrium Models[END_REF] a problem arises when analyzing the impact of changes in taxes in the presence of many distortions/taxes. For instance, a policy reform with increases of indirect taxes and decreases of direct taxes may reduce real wages while being revenue neutral and without any effects on aggregate absorption and hence on agent well-being. Because policy reforms are always contemplated in a second-best world, thus we do not retain real wages.

In a similar vein, changes of real GDP are often reported in country comparisons but [START_REF] Kohli | Real GDP, real domestic income, and terms-of-trade changes[END_REF] demonstrates that this welfare indicator may be in fact misleading as it underestimale the increase of real domestic income and welfare when one country experiences some terms of trade improvements. Real GDP focuses only on production possibilities and thus is unable to capture the beneficial effect for an economy of an improvement in its terms of trade (say a decrease of import prices). Because policy reforms may translate into terms of trade changes, we also discard the real GDP indicator.

Real income index number and consumer surplus are distinct but quite close concepts [START_REF] Hicks | Consumer's Surplus and Index Numbers[END_REF] ; both will be disregard for the same reason in our paper. We focus the discussion on the latter one (for a very recent paper on real income index, see [START_REF] Neary | Rationalising the Penn World Table: True Multilateral Indices for International Comparisons of Real Income[END_REF]. Originally proposed by [START_REF] Bennet | The Theory of Measurement of Changes in Cost of Living[END_REF], the consumer surplus has been for a long time the most widely used measure, partly thanks to the strong support offered by [START_REF] Harberger | Three Basic Postulates for Applied Welfare Economies: An Interpretive Essay[END_REF] in his open letter to the economic profession. Its main advantages corne from its simplicity and relative sparse data requirements, namely only prices and quantities consumed in the two periods of comparison (be they observed or simulated). But it has been criticized by Chipman and Moore (1976) on the ground that it is a val id/exact measure of consumer well-being only for homothetic preferences. This is clearly a serious problem since now almost 150 years of empirical evidence demonstrates that demand patterns are inconsistent with homotheticity. [START_REF] Diewert | Harberger's Welfare Indicator and Revealed Preference Theory[END_REF] also greatly reduces the relevance of the consumer surplus concept by demonstrating that it can be made positive or negative simply by scaling price in either period. Finally [START_REF] Mckenzie | Exact Measures of Welfare and the Cost of Living[END_REF] argue that consumer surplus is empirically unsatisfactory because it is only a second-order approximate measure of welfare. Subsequent works try to resurrect consumer surplus with slightly modified version of the original version but this proves again controversial. For instance, [START_REF] Willig | Consumer's Surplus Without Apology[END_REF] shows that, in the case of a single price change, observable estimates of consumer's surplus can be used to provide a good approximation to the theoretically superior EV and CV measures but [START_REF] Hausman | Exact Consumer's Surplus and Deadweight Loss[END_REF] did find that Willig's approach is inaccurate in case of large income effects. [START_REF] Weitzman | Consumer's Surplus as an Exact Approximation when Prices are appropriately Deflated[END_REF] also tries to bring back to life the consumer surplus approach by offering a device (a "Paasche normalization" of price in period one) to the normalization issue raised by [START_REF] Diewert | Harberger's Welfare Indicator and Revealed Preference Theory[END_REF] but it finally appears that this device requires the assumption of homothetic preferences [START_REF] Diewert | Exact and Superlative Welfare Change Indicators[END_REF]. More recently, [START_REF] Chambers | Consumer's Surplus as an Exact and Superlative Cardinal Welfare Indicator[END_REF] shows that another norrnalized version of the Bennet consumer surplus measure is an exact and superlative cardinal welfare indicator if preferences are of translation-homothetic generalized quadratic form (the associated Engel curves are linear in expenditures without necessarily departing from the origin as with homothetic preferences). Without doubts, this represent a new support for the consumer surplus but the present proliferation of econometric estimations of rank three demand systems may attenuate this contribution.

To summarize, the main reason why the consumer surplus is criticized in the literature lies in the fact that it is only exact and superlative for some restricted structures of preferences. This is not our reason for rejecting this measure because many CGE evaluations are still performed with these restricted structures. Our concern is different and may be explained as follows. By and large, previous efforts on the consumer surplus try to define an accurate measure of welfare changes that only depends on the observed price and quantity vector for two periods. In other words, they want to avoid the arbitrary specification of household preferences/ demand fonctions / substitutability patterns between goods which are required for Hicksian welfare measures. However CGE models are built on these arbitrary specifications and accordingly the computation of Hicksian welfare measures can be made without additional costs (if we have only implicit representation of preferences, they can still be evaluated using Vartia's approach). Thus it is no longer useful to limit oneself to these indicators when theoretically superior ones are already available.

Finally, we are left with the different versions of Hicksian welfare measures [START_REF]Measuring Welfare Changes with Distortions[END_REF] and must choose between i) the compensated versus uncompensated notion of surplus (also labeled direct or money metric) and ii) CV and EV. As we want our approach to be applicable to multi-country CGE models, we prefer to adopt the most common money metric versions. Finally, because we usually want to compare different policy reforrns to a same benchmark, we concentrate on the EV measure. That is, we compare the impact of policy reforms using the same (initial) price vector.

Once one particular welfare measure has been adopted, the critical challenges are to analyze the welfare impacts of policy changes as well as to define optimal policies. These challenges may be much simplified if it would be possible to identify and quantify the sources of welfare and their respective contributions with respect to a given policy scenario. This is basically the purpose of a growing Iiterature on the decomposition of welfare effects.1 To date, several studies have proposed ways of attributing changes in welfare to sources corresponding to the alleviation, or exacerbation, of existing market imperfections and 'distortions'. In general, they extend the pioneer decomposition proposed by [START_REF] Harberger | Three Basic Postulates for Applied Welfare Economies: An Interpretive Essay[END_REF] in a highly simplified economy. Recent contributions include: [START_REF] Coady | Evaluating Transfer Programmes within a General Equilibrium Framework[END_REF] who evaluate transfer programs in Mexico and decomposes welfare effects into three components (redistribution, reallocative and distortionary) ; [START_REF] Diewert | The Gains from Trade and Policy Reform Revisited[END_REF] who show in particular how the introduction of new goods into the economy generate welfare gains ; Kohler (2004) who assesses impacts of Eastern enlargement of the EU and decomposes welfare effects into three components too (static gains from trade, dynamic effects following capital accumulation with growth externalities and employments effects in labor markets characterized by unemployment due to costly search) and Kreickmeier (2005) who derives the welfare effects of tariffs and import quotas in the presence of involuntary unemployment and highlights the crucial production elasticities.

As far as we are aware of, ail these decompositions are based on first-order approximations of underlying CGE specifications. Without doubts, these decompositions are extremely useful in order to understand some economic mechanisms at work. However, these locally-based decompositions may have a rather poor explanatory power from a empirical standpoint in the case of " multiple and large shock" experiments, which is the traditional purpose of CGE models. In other words, linearized representation of multiple non-linear relations may be prone to erroneous conclusions [START_REF] Hertel | Mending the family tree. A reconciliation of the Iinearization and levels schools of AGE modeling[END_REF]. In order to cope with this potential empirical issue, current practice is to split the original experiment into many smaller ones, so as to update welfare elasticities in the decomposition equation. While this seems a priori crafty, this solution leads to another problem: the welfare decomposition becomes path dependent [START_REF] Fane | Alternative ways of measuring and decomposing equivalent variation[END_REF]. Appendix I presents this issue in a very simplified setting using a graphical analysis. The consequence is that the decomposition is not unique and thus may be of little value to explain the welfare impacts of any given policy experiment.

Our main objective in this paper is to propose a new approach for decomposing welfare effects of CGE models. The main idea is to develop Taylor series approximations to CGE specifications rather than relying on first order marginal conditions only. This idea was already present in [START_REF] Harberger | Three Basic Postulates for Applied Welfare Economies: An Interpretive Essay[END_REF] but, as far as we are aware of, have never been exploited in empirical analysis. This way, we want to avoid the arbitrary sharing out of policy experiments (hence the qualifier path independent). As will be apparent below, the attractive feature of our approach is that it encompasses previous decompositions and still retains the suggestion of Harberger to express welfare in terms of distortions. It may also be applied to any representation of globally regular preferences (hence the qualifiers exact and superlative). From a practical point of view, it can be implemented in both level and linearized models. Finally comparisons of welfare across individuals are made easier if one makes use of multiple household CGE mode!.

On the other hand, we must acknowledge that our decomposition includes a residual term which captures Taylor series approximation terms of order greater than 2 (hence the qualifier second order). Depending on the degree of non-linearity, second-order Taylor series approximations may still be insufficient to provide an accurate decomposing of a particular function. For instance, McKenzie et Pearce show with a simple quadratic utility function that even a fifth order approximation to one given welfare measure is still an approximation. However they also show that the error, expressed in percentage of expenditure, significantly declines from the first (1.54%) to the second order approximation (0.08%). Consequently our approach remains an approximation that must be empirically appreciated in real situations. In this paper, we offer some numerical examples using a widely used CGE mode!.

The paper is organized as follows. The first section presents the basics of our Taylor based approach in a highly simplified CGE mode! with few distortions. We also contrast it with those currently used. In the second section, we derive a new welfare decomposition equation for a widely used CGE mode!. Finally we perform various policy reforms in order to illustrate the empirical benefits of the proposed decomposition.

Analytical framework.

CGE models are now extensively employed m order to evaluate the market and welfare impacts of policy reforms. Many theoretical improvements have been introduced in such models, including dynamic behavior, imperfect competition, risk aversion, ... lt is beyond the scope of this paper to provide a welfare decomposition that incorporates ail these possible features. In this section, we prefer to adopt a quite standard CGE mode! in order to ease the comparison between our decomposition and previous ones. In fact, we adopt the synthetic mode! described in [START_REF] Fane | Alternative ways of measuring and decomposing equivalent variation[END_REF] where they compare different welfare indicators. We first briefly present the notations, then provide the usual first-order decomposition of the EV and finally detail our Taylor based approach.

Notations

We consider a static open economy mode! with only one representative consumer, 1 goods (indexed by i) and mono-product activities and F primary factors of production (indexed by f ). Extension to multiple household CGE models is described in appendix 2. Producers are assumed to maximize their profit subject to constant return to scale production technologies. Likewise, the representative consumer maximizes utility subject to budget constraint. His income is given by the primary factor returns and the net product of specific taxes/subsidies on production, primary factor use, consumption and trade. Primary factors of production are perfectly mobile between activities and are in fixed supply. Perfect competition prevails in ail markets. Finally we assume an uncompensated setting where transfer from/to abroad is fixed, that domestic and foreign goods are perfect substitutes and that our economy is potentially large in world markets. This is clearly a simplified CGE mode!, where we abstract from savings, investment and intermediate use in production technologies. Nevertheless it is a useful setting to derive our welfare decomposition and contrasts it to current ones. Mathematically, such a mode! is represented by the following eleven equations:

x i,/ = x f.i (Y, ,W +rf) F P; .Y, = L (w1 + t/;,1 }xi,/ /=1 ci= c;(Q,R) R = E(Q,U) PW, = PW,(M, z) P; = PW, +tm; -ty; Q; = PW, + tm; +te; Ix,,J = x i C; = .Y; + M ; F F / I R = Iw1 .X1 + LL f/;,1 ,X;J + Lty,.r; + tc,.C, +tm,.M; + b /=I /=1 i=I i=I I b = IPW,.M, i=I (1)
(2)

(3) (4) (5) (6) (7) (8) (9) (10) (11)
with the following notations for the endogenous variables (always written with upper case letters):

X ;,1 the use of primary factor f by activity i , W 1 the market price of primary factor f, P;

the producer price of good i , Y, the domestic production of good i, C; the domestic consumption of good i , Q; the consumer price of good i, R the total income or expenditure, U the consumer utility, PW, the world price of good i , M , the trade volume of good i, and the following notations for the exogenous variables (always written with lower case letters): tf,1 the specific tax on primary factor f used by activity i, ty; the specific tax on production i , te, the specific tax on consumption i , tm; the specific tax on trade i , b the balance of tracte deficit and x I the fixed endowment of factor.

Equations 1 and 2 form the production block and include the primary factor derived demands and the zero profit condition. Equations 3 and 4 form the consumption block and include the final demands as well as the expenditure fonction, which implicitly define utility. Note that we still do not restrain the structure of preferences. The only restriction is to be globally regular. Equations 5 to 7 form the price block where ail taxes on good are introduced. Equations 8 and 9 are the market equilibrium conditions. Finally equations 10 and 11 are macro-economic conditions expressing the economy budget constraint and the balance of payments. As expected, one equation in this mode) is redundant and must be skipped when solving the mode 1. This has no incidence of results and is indeed an ex post powerful way to check the consistency of the CGE model. It may be finally noted that in general modelers also remove equation 4 when solving the mode) as it only serves to determine the ordinal utility.

"First-order" decomposition of welfare

First-order decomposition of welfare usually start from total differentiation of the income equation ( 10) around the initial point (for instance, [START_REF] Huff | Decomposing Welfare Changes in the GTAP Mode![END_REF]:

F F l dR = L dW/"x f + wf .dx f + L L dif.f .xi.f + if.j"dXi,f /=l /=1
+ I_dW 1 .x 1 + L,L,dif.r X ;,1 + I_dty,.Y, +(dtc, -dQ,).C, +dtm;.M; /= 1 /=I i=I 1=1
The left hand side of this equation simply gives the change of real income (measured with initial consumption quantities). The first two elements of the right hand side represent exogenous flows of income to the economy (from the rest of the world or from an increase of primary factor endowment). The fourth following terms measure the alleviation, or exacerbation, of existing distortions. Other terms are much more difficult to interpret but can be much more simplified by proceeding in the following way. Total differentiation of the zero profit condition (equation 2) gives:

F P, .dY, + dP, .Y, = L (w 1 + if,1 )dX;,1 + (dw 1 + dtf ,1 )x,,1 (14) f =I
Because production technologies are constant return to scale, we have from the Euler Theorem:

F dP, .Y, = L (dw 1 + dif ,1 )xi,/ (15)
/ =1

If we sum this last identity over ail production sectors and use the market clearing equation ( 8), then we get:

I F I F LdP,.Y, = LdWr x 1 + LLdifJ .X;,1 (16) i=I / = I i = I /=1
We are now in a good position to simplify the terms on the right hand side of equation ( 13). If we substitute the right hand side of ( 16) for the left hand side in equation ( 13), next use first order differentiation of price equations ( 6) and ( 7) and make use of product market equilibrium equation ( 9), the equation 13 becomes: The last new term reduces to classical terms of trade effect which obviously have some impacts on real income. If the prices of import increase, then real income decreases as expected. This last term can be further decomposed into two terms using equation ( 5) in order to reflect changes due to the evolution of trade flows and changes due to the evolution of foreign market conditions (for instance, see [START_REF] Fane | Alternative ways of measuring and decomposing equivalent variation[END_REF].

I F F I / dR -
Finally, EV is related to the real income just decomposed in the following manner:

EV= E(P 0 ,U 1 ) -E(P 0 ,U 0 ) ( 18
)
where superscript O refers to the initial situation and superscript 1 to the final situation. At this stage, let' s recall that the current practice is to split the whole experiment into smaller ones. Let's assume that there are K sub-experiments (indexed by k ). For each sub-experiment, we have:

dEV* = dE(P 0 uk )= aE(Po ,Uk Lu* ' au (19)
Let's now totally differentiate equation 4 at the same sub-experiment:

(20)

The second equality is satisfied only locally because the derivative of the expenditure fonction with respect to price gives the hicksian demand fonction and not the marshallian demand function. Combining equations 19 and 20 and summing over ail sub-experiments, we finally obtain:

aE(P 0 ,Uk) K EV= L (~ k ).(dR k -Ck .dPk) k=I aE P ,u au ( O k) F F ! aE p ,u db k + ""w k .d.:/ + """" ,,rk .dXk K -~-~ L, f / L,L,'1 1,f 1,/ = I au 1=, 1=1 i=I k=I aE(Pk, uk )' ~ k k k k k k k k au + L,ty;
.df; + te; .dC; + tm; .dM; -dPW .M, i=I

As [START_REF] Fane | Alternative ways of measuring and decomposing equivalent variation[END_REF] indicate, the issue with the uncompensated version of EV is that the terms of the real income decomposition must be weighted by a term involving the inverse of marginal utility of income. Accordingly, uncompensated EV can not be decomposed simply in terms of changes in the levels of distorted activities, each multiplied by the excess of the marginal social benefit of the activity over its marginal cost. Our main issue here is that this decomposition is based on local approximation and thus may be empirically poor for experiments including large changes, unless many sub-experiments are contemplated. But the decomposition may not be unique, thus not really facilitating the interpretation of welfare effects.

Taylor based decomposition ofwelfare

Like [START_REF] Harberger | Three Basic Postulates for Applied Welfare Economies: An Interpretive Essay[END_REF] and [START_REF] Weitzman | Consumer's Surplus as an Exact Approximation when Prices are appropriately Deflated[END_REF], we start our procedure by developing Taylor series approximations to the EV and to the direct utility fonction:

(

) 22
where O(L\U) stands for ail polynomial terms of third order or higher.

Even if preferences are such that it is impossible to express the direct utility function (like the implicit CDE demand system), we always have:

(23)

From utility maximization, we have the following first order condition for ail goods:

au= J.Q ac ' 1 ( 24
)
with À the marginal utility of income.

au (25) (26)
Harberger (1971) suggests to use only the first two terms for measuring welfare (which is an second order approximation to the consumer surplus) and that the others may be neglected. But equation ( 26) makes clear that even with homothetic preferences (such as the last two terms of the last line vanish), this does not strictly correspond to EV. It remains terms involving changes in the marginal utility of income. These terms generalize to N commodities the difference between EV and the consumer surplus derived by Boadway and Bruce (1984, p. 218) in a single commodity context.

We now concentrate on the first two terms. Using total differentiation of equations 6, 7 and 9, we can express them as follows: On can already remark that the first three lines of the right hand side represent changes in one flow multiplied by the corresponding average tax. The purpose of subsequent derivations is to show that the terms of the last two lines can also be expressed in this form. In that respect, we first totally differentiate equation l l , rearrange terms to get:

I I I I L,Q 1 .LiC 1 + 0.5.
I ! / 1
L,PW.!1M; + 0.5.I,MW.!1M; = Lib -I,MW.M, -0.5.I,MW.!1M;

(28)

r=I i=I i=I i=I
Finally, in order to simplify the terms involving changes in producer prices and domestic productions (penultimate line of equation 27), we again use a Taylor series approximation to the primai production technologies, in the same manner as we did for preferences:

(29)

The first-order conditions of profit maximization are:

(30)

Total differentiating these first order conditions gives:

(31) lntroducing 31 into 29 and making appropriate arrangements, we obtain:

(32)

The left hand side of this last equation can also be expressed using another Taylor approximation:

M.Af>.~Y P,.~Y,.

M =P,.~Y, . + 0.5.M,.~Y,. + 0.25. ' ' ' +0(Af>) 1-0.5.-i P,

(33)

P,

Combining (32) and ( 33) and summing over ail production activities, we get:

I F IP,

+ o(~u.~c,M>,M)

This expression must be compared to expression 21 which gives the EV decomposition in the usual first-order approximation. The first thing to note is that expression 35 was obtained using ail CGE equations with the exception of the income equation ( equation 10) while expression 21 makes use of ail CGE equations, with the exception of the balance of payments condition (equation 11). As stated previously, this has no incidence of the resolution of the CGE model and hence the computation of welfare. The second thing to note is that the proposed decomposition is path independent, so that the explanation of welfare impacts of a given policy reform package is unique. The third thing to note is that the evolution of the marginal utility of income enters as an additional term rather than a multiplicative term. This make things more understandable: we can <livide the decomposition of EV between a Marshallian part (corresponding to CS and observables) and a Hicksian one (not directly observable, only with modeling). The fourth thing to note is that the "Marshallian" parts are very similar with ail distortions evaluated at the average rather than the initial level.

Graphically, this can be seen as computing triangles rather than rectangles. Obviously triangles may not be very good approximations if supply and demand fonctions are highly non-linear. That' s the reason why third order terms appear at the bottom of expression (35). [START_REF] Harberger | Three Basic Postulates for Applied Welfare Economies: An Interpretive Essay[END_REF] simply neglects ail terms of order higher than three (that is, the last three terms in 35). This is an empirical issue to which we tum now in some "real" policy experiments.

Empirical illustrations

Empiricalframework

ln order to assess the benefits of our decomposition, we perform some illustrative simulations using one simple version of the well known GT AP CGE mode! (Hertel, 1997). This version is described in [START_REF] Rutherford | GTAPinGAMS and GTAP-EG: Global Datasets for Economie Research and Illustrative Models[END_REF] and is static, perfectly competitive and with constant return to scale production technologies. This CGE mode! differs from the stylized mode! described in the preceding section on the following aspects (we simultaneously present the new notations).

This is a multi-country mode] (regions indexed by r = 1, ... , R) where bilateral trade is modeled with a nested Armington structure using CES fonctions. lt thus distinguishes imports and exports (denoted by E ;,r,s for export of good from region r to s ). We assume that domestic production and export are perfect substitute. At the import side, we assume that domestic consumers first combine, according to a CES fonction, imports from the different zones into an aggregate bundle (denoted by MT, ,r with price PM;,r ). The latter is then combined, according to a new CES fonction, to domestic production to determine domestic demand.

In addition to the final demand derived from the maximization of a Cobb Douglas utility fonction , the domestic demand includes intermed iate use by activities (denoted by IDi,J,r for the demand of good i by the activity j in region r ), public demand (denoted by GDi,r) and investment demand ( denoted by l i,r ). Investment by commodity is assumed to be fixed in this static version. Public demands are derived from a Cobb Douglas public utility fonction and the macro-economic closure are such that public utility is fixed. Intermediate demands are proportional to the level of activity and a Cobb-Douglas production fonction relates activity level and primary factor inputs.

In this mode!, ail distortions are represented by ad valorem taxes. They include ail taxes described above and taxes on intermediate demands (denoted by ti;,J,r ), on exports (denoted by tei,r,J and on public consumption (denoted by tg;,r ). Finally the mode! includes an international transport sector responsible for trade flows between regions (denoted by T,,r,, ).

International transport services are assumed to be proportional to trade and are a Cobb Douglas aggregate of national transport services. The price of imports is thus given by the net price from the exporting country plus the associate transport cost to the importing country.

In this mode!, first order decomposition of EV for a given region is given by: Few comments of this expression are in order. Terms in the three first lines are very similar to those previously identified. We only take into account for ad valorem taxes rather specific taxes. Subsequent terms represent the new distortions on intermediate demands, exports and public demands. Terms of trade effects must be now decomposed between price of exports and price of imports. Finally the last term is related to the selling of transport services to the international trade transport.

F F / db; + I,
The Taylor decomposition of the same EV is given by:

F EV, = /lb,+ Lwf,r '!),.xf,
• • • + ~ • • • + ~ • • • i=I P,,, i=I Qi,r 1=! PMi,r + O(/lU, /lC, M , /lY, M , MM,MfT) (37)
Again this expression is very similar to 35. We only remark that in the penultimate line we have now three terms which correspond to three distinct fonctions and behavior: profit maximization, cost minimization of total consumption and finally cost minimization of total imports. These do not correspond to distortions. Like Harberger, we neglect them in the following numerical examples ; we also neglect them as well as the polynomial fonctions of order strictly higher than two (the last line of expression 37).

Simulations

In order to implement the CGE mode!, we use the GT AP 4.0 database which captures economic flow of the year 1995. Like [START_REF] Huff | Decomposing Welfare Changes in the GTAP Mode![END_REF], we use a crude sectoral and regional aggregation. It features three produced sectors: food, manufactures and services, and three regions: the United States (USA), the European Union (EU) and the Rest of the World (ROW). We perform three experiments which focuses on trade instruments. In the first one, we decrease trade taxes (both import and export) by only 0.1 % in ail regions. This first simulation intends to show similarities between the two decompositions. The second experiment is more " policy-minded" and assume a 50% decrease of these taxes. Finally the last experiment is for testing purpose (the quality of approximations) where we assume full removal of ail trade taxes. In the last two non marginal experiments, we test different paths when implementing the first order decomposition.

Welfare results of the first experiment are reported in Table 1. As expected, there are few differences between the two approaches This illustrates that our proposed approach encompasses current ones. On can still remark small deviations in the case of first order decomposition. For instance, EU welfare is 29.070 million US dollars and the first order decomposition gives a value of 29.079 white the Taylor based decomposition fits perfectly.

Results of the second experiment are reported in ! Without any surprise, the contributions of each distortion to the global welfare effects are overstated with the first order decomposition. For instance, the alleviation of tariffs distortions in the EU amounts to 4128 millions US dollar according to this decomposition white it amounts to 3102 according to the Taylor decomposition (difference of 33%).

Let's turn now to the case where this policy reform is implemented in two steps. There are obviously many ways to define it and we explore here three possibilities. The first path assumes that in the first step export taxes are reduced by 50% white tariffs are unchanged.

The second path is symmetric to the first one by assuming only tariff reductions in the first step. Finally the third path assumes a 25% reduction of trade taxes in the first step. For these three paths, the second step is rationally defined in order to complete the simulation. As expected, decompositions are sensitive to the path choice. For instance, the alleviation of export taxes distortions in the EU takes the value of 2195, 2880 and 1896 millions US dollars.

There is as much as 50% variation between the higher and the lower. Aggregate figures reported in Table 2 tends to suggest that the third linear path performs better than the two others (providing that the Taylor decomposition is a good candidate for comparison). However, when detailing further the decompositions, it appears that this is often but not always the case. For instance, we report in Table 3 the contributions of tariff distortions for each commodity. In two of nine cases (Manufactures in the USA and Services in the ROW), the third linear path is outperformed.

Results of the third large experiment are reported in table 4. They mainly confirm previous results and hence are not repeated here. We just underline that our decomposition performs still reasonably well for the EU and the USA. On the other hand, the approximation is poorer for the ROW. In order to discover the main reasons, we implement our decomposition step by step and find that the main issue lies in the second order approximation of the Armington-type import demand fonctions (derived from CES utility specifications). This is not surprising once we give a look to market effects and observe dramatic changes in trade flows (as much as 160% increase of imports of food from the USA to the RoW). As stated previously, our

Taylor based decomposition still remains an approximation and it may not fit perfectly in some rather extreme cases. When such cases occur, it is still attractive to reveal the places where more details are needed.

Concluding remarks

CGE models are widely used in order to measure the welfare effects of policy scenarios and thus to identify the winners and losers in each case. Recently CGE modelers quite systematically report some decompositions of welfare effects in order to understand why such or such effect appears. To date, these welfare decompositions are based on first order approximations of the CGE specifications. These locally-based decompositions may have a poor explanatory power from an empirical standpoint when one contemplates large policy shocks. The current device to circumvent this issue is to split the original shock into many smaller ones, where first order approximations are more adequate. Unfortunately this strategy rai ses another problem, mainly the path dependency of the welfare decomposition.

In that context, this paper proposes a new decomposition of welfare effects simulated by CGE models. Our approach overcomes previous issues and is based on Taylor series approximations of CGE specifications. Then it is a generalization of current ones which still allows to attribute changes in welfare to sources corresponding to the alleviation, or exacerbation, of existing market imperfections and distortions. Our decomposition approach is also attractive because that i) it can be applied to any globally regular representation of preferences, ii) it can be implemented in both level and linearized CGE models and iii) it eases the comparisons of welfare effects across individuals.

We implement our approach to the widely used GTAP CGE model. We simulate many experiments and find that our approach represents a substantial empirical improvement compared to standard practices. Moreover it performs well in most cases in the sense that the decompositions are quite close to the true welfare effects.

As usual, many works remain to do before providing definitive statements on the proposed approach. We suggest here two main directions. The first one, quite obvious, is to test the proposed approach on many policy scenarios simulated with "standard" CGE model. The second, much challenging, is to extend it to CGE models with more complex sources of market imperfections (for instance with imperfect competition, dynamic considerations, ... ) and more complex specifications (for instance with more flexible preference structures, production technologies, . ... ).

•••••••••••••••••••••••••••••" p d' •••••••••••••••••••••••••••••••••••••••••1e' d'' 1 •••••••••• ••••••••••••••••••••••••••••• •••••••••••••✓ -- o, x, X2 Xo X3
02

  Total differentiation of the above first order condition gives:

	+ 0.5. a 2 E(P	0 ;U

au Li ac = Lil.Q; + J.LiQ; + Lil.LiQ; 1 Substituting (24) and (25) into (23) and next the resulting expression into (22), we get: I I EV= LQ,.LiC, + 0.5.LLiQ,.LiC; i;J i;J +0.5.Lil{ tQ;.LiC;.+ tLiQ;.LiC;) 0 ).(LiU)2 + O(LiU,LiC)

  L,LiQ;.LiC; = L,lC;.LiC 1 + 0.5.I,Mc 1 .LiC;

	i=I	1=1	i=I	i=I
		I	!
		+ L,0'1 .LiY, + 0.5.I,Liryl .LiY,
		r=I	r=I
		I	!
		+ I,tm,.!1M; + 0.5.I,Mm;.!1M 1	(27)
		1=1	i=I
		I		I
		+ I,P;.LiY; + 0.5.I,M,.LiY;
		i=I	r=I
		I	!
		+ L,PW;.11M; + 0.5.I,MW,.!1M;
		r=I	i=I

  r +0.5./lWf,r '!),.xf, r

		f=I
	F	I
	+ LLif,1,, •wJ,r'MiJ,r + 0.5./l(Wf,r .tf 1J,r )Mi,f,r
	f=I 1=1
	I + Io1;,, .P,_ , ./lY,., + o.5./l(ty;., .P,., }/lr;_,
	l=I
	I
	+ Itci,r'Qi,r '/lCl,r + 0.5./l(tc,,,.Q;,rJ/lCi,r
	i=I
	I
	+ Lfg 1 ,,.Q 1 ,, ./lGDi,r + 0.5./l(tgi,r 'Qi,r }/lGDi,r
	i=I
	I + L L tml,r,s .PW1,r,s ./lA(,,,s + 0.5./l(tmi,r,s .Pwi,r,s }iliWi,r,s R
	i=I s=I
	/	R
	i=I s=I
	I	R
	-LLMi,r,s •MW,,r,s + 0.5.fuWi,r,s'/lPW,,r,s
	i=I s=I
	/	R
	1= l s=I
	( ' + 0.5./lÂ, . LQt,r '/lCi,r ' + L/lQi,r '/lCi,r + 0.5. ' J a 2 i=I i = I	E(P r; r ,(/lU,)2 0 u 0 ) au

+ L L fei,r,s .P,,, .

Mi,r,s + 0.5 .. /l(tei,r,s .P,,, }Mi,r,s

+ LL Ei,r,s ,/l((I + tei,r,s }P,,, )+ 0.5./lEi,r,s ./l((I + te 1 ,,,s }P,,,) i=I s=I I R + LL'.l;,,,s •/lP,,, + 0.5./lT;,r,s'/lP,,r ( ~ /lP, ,./lP, , ./l}'; r ~ /lQ; , ./lQ; ,./lC; r ".' MM; , .MM,r./lAfI; r J -0.25. ~

Table 2 .

 2 The differences between the two approaches are much more pronounced. Let's focus initially on the decomposition when the policy reform is fully implemented in one time. The first order approach does not accurately measure the welfare impacts. Above ail, the first order decomposition overstates the welfare impacts by 2053 millions US dollars in case of the EU, which is equivalent to a 16.2% error. Corresponding figures for the USA and ROW are respectively 523 (9.5%) and 8822 (59.1 %). Our Taylor based decomposition performs much better in this case. It first provides the correct measure of welfare (which is eased by the Cobb Douglas specification). Moreover the decomposition is quite close. The errors are now limited to 32 millions US dollars (0.2%) in case of the EU, to 118 millions US dollars (2.1 %) in case of the USA and finally to 367 millions US dollars (2.4%) in case of the ROW. For the EU, the error is divided by a factor 64

Table 1 . Welfare effects and decompositions of a 0.1 % reduction of tariffs and export taxes (millions US dollars)

 1

	EU	USA	ROW

Table 2 . Welfare effects and decompositions of a 50% reduction of tariffs and export taxes (millions US dollars)

 2

	E U	USA	ROW

Table 3 . Decomposition of the contribution of tariff distortions by commodities for a 50% reduction of tariffs and export taxes (millions US dollars)

 3

	EU	USA	ROW

Table 4 . Welfare effects and decompositions of a full removal of tariffs and export taxes (millions US dollars

 4

	EU	USA	ROW

g 1 ........... .J..~1 • • --

lt is worthy to note in parait el the growing literature on the decomposition of index numbers ( see for instanceHallerbach,

2004).

Appendix 1. The path dependency of welfare decomposition: A graphical presentation

This appendix shows in a highly simplified context the path dependency issue of welfare decompositions that are based on multi-step simulations. We assume one market (primary factor or good) whose supply is fixed and two demands/consumers (say 1 and 2). Let's consider that initially there are no distortions on this market. We would like to understand the welfare effects of a policy reform which introduces differentiated taxes on the two demands. Market and welfare impacts are fully reported in the table A. l. In this context, it is obvious that the final allocation may be obtained with only one tax (given by the absolute difference between the two taxes) but we maintain the introduction of both in order to illustrate the path dependency issue. ln order to explain the welfare Joss from these taxes, one can examine first the impacts of the tax on demand 1 and next the additional impacts of the tax on demand 2. In the table below, we label it path 1. One can also examine the welfare impacts of the reform by looking first at the impacts of the tax on demand 2 and next the additional impacts of the tax on demand 1 (path 2). In our graphical illustration, it appears that tax 2 appears to be welfare improving with path 1 (by area thjb') and is welfare decreasing with path 2 (by area obb"). Tax one is always welfare decreasing but the magnitude varies with the path. Thus the two "extreme" path decompositions lead to very different explanations and thus do not help much to understand the welfare impacts of the policy reform.

The decomposition we propose in this paper is as follows. Tax 1 is welfare decreasing by half of the area b" hbs while tax 2 is welfare increasing by half the area b"jbt. Total welfare is still decreasing by the area hbj.

Appendix 2: Welfare decomposition in a multiple-household CGE model

The analytical framework developed in the main text considers only one representative consumer for the economy. However applications of multiple household CGE models are mounting ; accordingly we briefly describe in this appendix the welfare decompositions in this context.

Incorporating many households in a CGE mode) is a great challenge, mainly in terms of data collection (both economic flows and elasticities). On the other hand, behavioral specifications may remain quite simple. In this appendix, we assume that there are H households (indexed by h) who consume goods and who own some primary factors. For simplicity, we assume that they ail face the same price of goods and factors. In fact households differ with respect to factor endowments and preferences. In our stylized CGE mode), they are introduced by modifying equations ( 3), ( 4), ( 8), ( 9) and ( 10) as follows:

where 0,, is the share of the product of net taxes that the household h receives (pays) and we The welfare decomposition for a given household is still negatively dependent of the welfare for other household, this time in a more direct way. Finally it is interesting to remark that both decompositions can not attribute the distortionary effects to any particular household. This is unsurprising because these effects depend on market equilibrium which in turn depend on the behavior of ail agents.


Mohamed El Koujok

Rafael Gouriveau

Noureddine Zerhouni

Mohamed El-Koujok

From monitoring data to remaining useful life: an evolving approach including uncertainty

Keywords: Prognostic, Degradation, Reliability Modeling, Evoloving Neuro-Fuzzy System, Uncertainty

HAL is

Introduction

The growth of reliability, availability and safety of a system is a determining factor in regard with the effectiveness of industrial performance. As a consequence, the high costs in maintaining complex equipments make necessary to enhance maintenance support systems and traditional concepts like preventive and corrective strategies are progressively completed by new ones like predictive and proactive maintenance. Thereby, prognostic is considered as a key feature in maintenance strategies as the estimation of the provisional reliability of an equipment as well as its remaining useful life allows avoiding inopportune spending. From the research point of view, many developments exist to support the prognostic activity [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance[END_REF][START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF][START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF]. However, in practice, choosing an efficient technique depends on classical constraints that limit the applicability of the tools: available data-knowledgeexperiences, dynamic and complexity of the system, implementation requirements (precision, computation time, etc.), available monitoring devices... Moreover, implementing an adequate tool can be a non trivial task as it can be difficult to provide effective models of dynamic systems including the inherent uncertainty of prognostic. That said, developments of this paper are founded on the following two complementary assumptions. 1) On one hand, real systems increase in complexity and their behavior is often non-linear, which makes harder a modeling step, even impossible. Intelligent Maintenance Systems must however take it into account. 2) On the other hand, in many cases, it is not too costly to equip dynamic systems with sensors, which allows gathering real data online. Furthermore, monitoring systems evolve in this way. According to all this, neuro-fuzzy (NF) systems appear to be very promising prognostic tools: NFs learn from examples and attempt to capture the subtle relationship among the data. Thereby, NFs are well suited for practical problems, where it is easier to gather data than to formalize the behavior of the system being studied. Actual developments confirm the interest of using NFs in forecasting applications [START_REF] Wang | Prognosis of machine health condition using neurofuzzy systems[END_REF][START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF][START_REF] Zhang | Forecasting with artificial neural networks: the state of the art[END_REF]. In this context, the paper deals with the definition of a prognostic system for which any assumption on its structure is necessary: it starts from monitoring data and goes through provisional reliability and remaining useful life by characterizing the uncertainty following from the degradation process. More precisely, the paper emphasizes on the development of an evolving neuro-fuzzy predictor that, not only "gives" an approximation of the degradation of an equipment but also associates to it a confidence measure. The model is well adapted to perform a priori reliability analysis and thereby optimize maintenance policies. The paper is organized in three main parts. In the first part, the concept of "prognostic" is clarified and replaced within the maintenance strategies. The relationship between prognostic, prediction and online reliability is also explained: the efficiency of a prognostic system is highly dependent on its ability to perform "good" predictions as reliability indicators follow from it. This is a central point of the work. Following that, the use of Takagi-Sugeno neurofuzzy systems in prognostic applications is justified and the ways of building such models are briefly discussed in the second part. An evolving neuro-fuzzy model for prognostic is thereby proposed and presented. Unfortunately, neuro-fuzzy predictors do not provide uncertainty indicators. Thus, in the third part, statistical estimation techniques are adapted to the evolving neuro-fuzzy predictor in order to provide a confidence measure on prediction and thereby enable reliability analysis. The whole is illustrated with an example extracted from literature.

Prognostic and reliability

From maintenance to prognostic

Maintenance activity combines different methods, tools and techniques to reduce maintenance costs while increasing reliability, availability and security of equipments. Thus, one usually speaks about fault detection, failures diagnosis, and response development (choice and scheduling of preventive and/or corrective actions). Briefly, these steps correspond to the need, firstly, of "perceiving'' phenomena, secondly, of "understanding"' them, and finally, of "acting'' consequently. However, rather than understanding a phenomenon which has just appeared like a failure, it seems convenient to "anticipate'' its manifestation in order to take adequate actions as soon as possible. This is what could be defined as the "prognostic process''. The relative positioning of detection, diagnosis, prognostic and decision / scheduling can be schematized as proposed in Figure 1.

From prognostic to prediction

Although there are some divergences in literature, prognostic can be defined as proposed by the International Organization for Standardization: "prognostic is the estimation of time to failure and risk for one or more existing and future failure modes'' [START_REF]Condition monitoring and diagnostics of machinesprognostics -Part1: General guidelines[END_REF]. Prognostic is also called the "prediction of a system's lifetime'' as it is a process whose objective is to predict the remaining useful life ( RUL ) before a failure occurs given the current machine condition and past operation profile [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. Thereby, two salient characteristics of prognostic appear: (1) prognostic is mostly assimilated to a prediction process (a future situation must be caught), [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF] prognostic is based on the failure notion, which implies that it is associated with a degree of acceptability. A central problem can be pointed out from this: the accuracy of a prognostic system is related to its ability to approximate and predict the degradation of an equipment: starting from a "current situation'', a prognostic tool must be able to forecast the "future possible situations'' and the prediction phase is thereby a critical one.

From prediction to reliability

As mentioned earlier, an important task of prognostic is to predict the degradation of an equipment. Following that, prognostic can also be seen as a process that allows the a priori reliability modeling. Reliability ( R t ( ) ) is classically defined as the probability that a failure does not occur before time t . If the random variable ϑ denotes the time to failure and F t Pr t ( ) ( )

ϑ ϑ = ≤ its cumulative distribution function, then: R t 1 F t ( ) ( ) ϑ = - (1)
Let assume now that the failure is not characterized by a random variable but by the fact that a degradation signal ( y ) overpass a degradation limit ( lim y ), and that this degradation signal can be predicted ( y ˆ) with a degree of uncertainty (Figure 2). At any time t , the failure probability can be predicted as follows:

[ ] lim F t Pr y t y ( ) ˆ( ) = ≥ (2)
Let note g y t ( ˆ/ ) the probability distribution function that denotes the prediction at time t . Thereby, by analogy with reliability theory, the reliability modeling is:

[ ] lim y lim R t 1 Pr y t y 1 g y t dy ( ) ˆ( ) ( ˆ/ ). ∞ = - ≥ = -∫ (3)
The remaining useful life ( RUL ) of the system can finally be expressed as the remaining time between the time in which is made the prediction ( p t ) and the time to underpass a reliability limit ( lim R ) fixed by the practitioner. This can be generalized with a multi-dimensional degradation signal. See [START_REF] Chinnam | A neurofuzzy approach for estimating mean residual life in condtition-based maintenace systems[END_REF] or [START_REF] Wang | Reliability prediction based on degradation modeling for systems with multiple degradation measures[END_REF] for more details. Finally, in order to perform a priori reliability analysis, an effective prognostic tool should provide the probability distribution function of prediction for time t . Moreover, this would enable to build confidence intervals on predictions, which can help practitioners in judging from the degradation state of the system and thereby, in taking adequate decisions. This point is again discussed in part 4.

Fuzzy models for prediction

Takagi-Sugeno system: a fitted prediction tool

According to some authors, the methods presented in this section are sometimes labeled as "prognostic techniques''. However, most of them refer to what, in this paper, is called "prediction''. Note that the aim of this part is not to dress an exhaustive overview of prediction techniques but to explain the orientations of works that are taken. Various prognostic approaches have been developed ranging in fidelity from simple historical failure rate models to high-fidelity physics-based models [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance[END_REF][START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF]. Briefly, similarly to diagnosis, prognostic methods can be associated with one of the following two approaches, namely model-based and data-driven.

Model-based methods assume that an accurate mathematical model for the analyzed system can be constructed. The main advantage of these approaches is their ability to incorporate physical understanding of the system. However, this closed relation with a mathematical model may also be a strong weakness: it can be difficult, even impossible to catch the system's behavior.

Data-driven approaches use real data (like online gathered with sensors or operator measures) to approximate and track features revealing the degradation of components and to forecast the global behavior of a system. The strength of data-driven techniques is their ability to transform high-dimensional noisy data into lower dimensional information for prognostic decisions: in many applications, measured input/output data is the major source for a deeper understanding of the system degradation.

Real systems are complex and their behavior is often non linear, non stationary. These considerations make harder a modeling step, even impossible. Yet, a prediction computational tool must deal with it. Moreover, monitoring systems have evolved and it is now quite easy to online gather data. According to all this, data-driven approaches have been increasingly applied to machine prognostic. More precisely, works have been led to develop systems that can perform nonlinear modeling without a priori knowledge, and that are able to learn complex relationships among "inputs and outputs'' (universal approximators). Indeed, artificial neural networks (ANNs) have been used to support the prediction process [START_REF] Zhang | Forecasting with artificial neural networks: the state of the art[END_REF], and research works emphasize on the interest of using it. Nevertheless, some authors remain skeptical as ANNs are "black-boxes'' which imply that there is no explicit form to explain and analyze the relationships between inputs and outputs. According to these considerations, recent works focus on the interest of hybrid systems: many investigations aim at overcoming the major ANNs drawback (lack of knowledge explanation) while preserving their learning capability. In this way, neuro-fuzzy systems are well adapted. More precisely, first order Tagaki-Sugeno (TS) fuzzy models have shown improved performances over ANNs and conventional approaches [START_REF] Wang | Prognosis of machine health condition using neurofuzzy systems[END_REF]. Thereby, they can perform the degradation modeling step of prognostic.

Takagi-Sugeno models: principles

A first order TS model provides an efficient and computationally attractive solution to approximate a nonlinear input-output transfer function. TS is based on the fuzzy decomposition of the input space. For each part of the state space, a fuzzy rule can be constructed to make a linear approximation of the input. The global output is a combination of the whole rules. In others words, a TS model can be seen as a multimodel structure consisting of linear models that are not necessarily independent [START_REF] Angelov | An approach to online identification of takagisugeno fuzzy models[END_REF].

Consider Figure 3 to explain the first order TS model. In this illustration, two inputs variables are considered, two fuzzy membership functions (antecedent fuzzy sets) are assigned to each one of them, and the TS model is finally composed of two fuzzy rules. (Note that a TS model can be generalized to the case of n inputs and N rules).

Π Π Ν Ν Σ y x 1 x 2 R 1 R 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A Π Π Ν Ν Σ y x 1 x 2 R 1 R 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A Π Π Π Π Ν Ν Ν Ν Σ y x 1 x 2 R 1 R 2 R 1 R 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 1 1 A 1 2 A 2 2 A 2 1 A Figure 3. First order TS model.
The rules perform a linear approximation of inputs as follows: [ ]

1 n i 1 i n i i i0 i1 1 in n R if x is
q 1 n , =
. Let assume Gaussien antecedent fuzzy sets (this choice is justified by its generalization capabilities) to define the regions of fuzzy rules in which the local linear sub-models are valid:

i i 2 4 x x j i j j exp * ( ) ( ( ) ) σ µ - - = (5)
where i j σ is the spread of the membership function, and i x * is the focal point (center) of the th i rule antecedent. The firing level ( i τ ) and the normalized firing level ( i λ ) of each rule are obtained as follows:

i i1 1 in n x x ( ) ... ( )
τ µ µ = × × , N i i j j 1 λ τ τ = = ∑ (6)
The model output is the weighted averaging of individual rules' contributions:

T N N i i i e i i 1 i 1 y y x λ λ π = = = = ∑ ∑ (7)
where

[ ] i i0 i1 in a a a , , , π = K
is the vector parameter of the th i sub-model, and

T T e x 1 X [ ] =
is the expanded data vector.

A TS model has two types of parameters. The non-linear parameters are those of the membership functions (a Gaussian membership like in eq. ( 5) has two parameters: its center x * and its spread deviation σ ). These kinds of parameter are referred to as premise or antecedent parameters. The second types of parameters are the linear ones that form the consequent part of each rule ( iq a in eq. 4).

Choosing a TS system: exTS for prognostic application

Assuming that a TS model can approximate an input-output function (previous section), in practice, this kind of model must be tuned to fit to the studied problem. This implies two task to be performed: (1) the design of the structure (number and type of membership functions, number of rules), (2) the optimization of the model's parameters. For that purpose, different approaches can be used to identify a TS model (mosaic scheme: by an expert [START_REF] Espinosa | Fuzzy Logic, Identification and Predictive Control[END_REF], gradient descent [START_REF] Jang | Neuro-fuzzy modeling and control[END_REF], genetic algorithms [START_REF] Angelov | Fuzzy systems design: direct and indirect approaches[END_REF],

clustering methods [START_REF] Bezdek | A convergence theorem for the fuzzy isodata clustering algorithms[END_REF], evolving algorithms [START_REF] Angelov | On-line design of takagi-sugeno models[END_REF][START_REF] Kasabov | Denfis: Dynamic evolving neural-fuzzy inference system and its application for time-series prediction[END_REF]). All approaches can not be detailed in the paper, so that the following paragraphs only briefly justify the choice of a prediction technique.

According to the degradation modeling problem, a prediction technique for prognostic purpose should not be tuned by an expert as it can be too difficult to catch the behavior of the monitored equipment. Thereby, the first approach for identification (mosaic scheme) should be leaved aside. Descent gradient and genetic algorithms approaches allow updating parameters by a learning process but are based on a fixed structure of the model, which supposes that an expert is able to indicate the adequate architecture to be chosen. Unfortunately, the accuracy of predictions is fully dependent on this. In opposition, clustering approaches require less a priori structure information as they automatically determine the number of membership functions and of rules. However, in practical applications, the learning process is effective only if sufficient data are available. In addition to it, when trained, such a TS model is fixed and if the behavior of the monitored system changes (like in a degradation phase), predictions can suffer from the lack of representative learning data.

Considering the applicative restrictions that supposes the implementation of a prognostic tool, evolving TS models appear to be the more promising for prognostic applications. Firstly, they are able to update the parameters without the intervention of an expert. Secondly, they can be trained in online mode as they have a flexible structure that evolves with the data gathered from the system: data are collected continuously which enables to form new rules or to modify an existing one. This second characteristics is very useful to take into account the non-stationary aspect of degradation. According to all this, an accurate TS prediction technique for online reliability modeling is the evolving one. A particular model is this one proposed by [START_REF] Angelov | Evolving fuzzy systems from data streams in real-time[END_REF]: the "evolving eXtended Takagi-Sugeno'' system (exTS).

Learning procedure of exTS

The learning procedure of exTS is composed of two phases: (1) an unsupervised data clustering technique is used to adjust the antecedent parameters, (2) the supervised recursive least squares learning method is used to update the consequent parameters.

Clustering phase: partitioning data space.

In opposition to other approaches, the exTS clustering phase processes on the global input-output data space:

T T T z x y [ , ] = , n m z R + ∈
, where n m + defines the dimensionality of the input/output data space. Each one of the sub-model of exTS operates in a sub-area of z . This clustering algorithm is based on the calculus of a "potential'' which is the capability of a data to form a cluster (antecedent of a rule). The procedure starts from scratch and, as more data are available, the model evolves by replacement or upgrade of rules [START_REF] Angelov | Evolving fuzzy systems from data streams in real-time[END_REF]. This enables the adjustment of the antecedent parameters (the non-linear ones): the center x * and the spread deviation σ of the membership functions (eq. 5).

RLS phase: update of the consequent parameters.

The exTS model is used for online prediction: at prediction time k , eq. 7 can be expressed as follows:

T T N N k 1 i i i e i k k i 1 i 1 y y x λ λ π ψ θ + = = = = = ∑ ∑ (8)
T T T T k 1 2 N k ˆ[ ˆ, ˆ,..., ˆ] θ π π π =
are parameters of the sub-models. The following RLS procedure is applied:

T k k 1 k k k 1 k k 1 C y k 2 3 ˆˆ( ) ; , ,... θ θ ψ ψ θ - + - = + - = (9)
T T k k 1 k 1 k k k 1 k k 1 k C C C C 1 C ψ ψ ψ ψ - - - - = - + (10)
with initial conditions

T T T T 1 1 2 N k 1 0 C I [ , ,..., ]
, andk θ is an estimation of the parameters based on k data samples.

θ π π π = = = Ω, where Ω is a large positive number, 1 C is a R n 1 R n 1 ( ) ( ) + × + co-variance matrix,

Including uncertainty to exTS prediction model 4.1 Principles

Unfortunately, neuro-fuzzy predictors do not provide uncertainty indicators. Thus, in this part, a way to adapt statistical estimation techniques to the evolving neuro-fuzzy predictor (exTS) is proposed in order to supply a confidence measure on prediction and thereby enable reliability analysis.

As explained before, an exTS model can predict the degradation state of an equipment: in this way, exTS approximate an input-output function by optimizing non-linear and linear parameters. Yet, a modeling error must be taken into account:

k 1 k k k 1 y f X * * ( , , ) λ θ ε +
+ = + (11)
where, k 1 y + is the real degradation situation, k X is the input vector at prediction time k , * λ is the optimal vector of the normalized firing level of the rules that depends on the non-linear parameters, k * θ is the optimal vector of the linear parameters, and k 1 ε + traduces the modeling error of the exTS system. k 1 ε + is assumed to be independently and identically distributed following a Gaussian distribution 2 N 0 s ( , ) with mean zero and variance 2 s [START_REF] Yu | Feature signature prediction of a boring process using neural network modeling with confidence bounds[END_REF]. The prediction error ( k 1 ε + ) can be expressed as the difference between the real degradation ( k 1 y + ) and the output of the exTS predictor ( k 1 y ˆ+ ):

k 1 k 1 k 1 k k k 1 k 1 y y f X y * * ˆˆ( , , ) ε λθ ε + + + + + = - = + - (12)
k 1 ε + is assumed to be distributed following a normal distribution

2 k 1 N 0 ( , ) ε σ + .
Thereby, a confidence interval of the prediction error can be expressed as:

2 k 1 2 k 1 k 1 ˆ/ / . . α ε α ε ϕ σ ε ϕ σ + + + - ≤ ≤ + (13)
where, 2 / α ϕ is the inverse of the normal cumulative distribution function for the confidence bounds α . Following that, the real degradation can be bounded:

k 1 2 k 1 k 1 2 k 1 k 1 y y y ˆ/ / ˆ. ˆ. α ε α ε ϕ σ ϕ σ + + + + + - ≤ ≤ + (14)
k 1 ε σ + can be calculated at each prediction step:

2 2 k k k 1 k 1 k 1 f X y * * ˆ[ ( , , ) ˆ] ε σ σ λ θ ε + + + = + - (15)
Assuming that * λ is tuned by the clustering phase of the learning process, the inputouput function f is linear with regards to the consequent parameters k * θ . According to eq. 8, the degradation state is approximated as follows:

T k 1 k k k 1 k k k 1 y f * * ( , ) ψ θ ε ψ θ ε
+ + + = + = + (16)
where, k ψ is the vector of the inputs weighted by normalized firing proposed. Thus,

2 2 T 2 T T k k k 1 k 1 k k k 1 k k k 1 y * * ˆ[ ˆ] [ ] ε σ σ ψ θ ε σ ψ θ ε ψ θ + + + + = + - = + - ( 17
)
2 2 T 2 2 T 2 T 2 k k k k 1 k k k k k k 1 s C s * ˆˆ[ ( )] [ ] [ ] ε σ σ ψ θ θ σ ε σ ψ θ ψ ψ + + = - + = + = + (18)
k C and k θ are obtained with eq. 9 and 10. Further information can be found in [START_REF] Angelov | Evolving fuzzy systems from data streams in real-time[END_REF].

As a synthesis, it is possible to provide the exTS output ( k 1 y ˆ+ ) (eq. 8). In addition, the normal distribution of the prediction error ( k 1 ε + ) can also recursively be obtained (eq. 18). The whole permit: (1) to dispose from a distribution function of the prediction ( k 1 k 1 y ε

+ + +
) and thereby perform the reliability analysis (like proposed in section 2.3), (2) to build confidence intervals on the real degradation estimation and thereby take more reliable decisions (eq. 14). This proposition is illustrated in next sections.

Experimentations

A real experimental data set has been used to show the performance of exTS when used as a prediction system, and to illustrate the integration of the uncertainty to build the confidence interval. The aim of the predictions is to approximate a physical phenomenon by learning data gathered from the system. The data set is issued from an hair dryer. It has been contributed by W. Favoreel from the KULeuven University (ftp://ftp.esat.kuleuven.ac.be/sista/data/mechanical). This data set contains 1000 samples. The air temperature of the dryer is linked to the voltage of the heating device. For simulations, exTS has been used with five inputs variables. Predictions concern the air temperature and were made at different horizons h : at ).

A first way of assessing the prediction performance is to use the root mean square error criterion (RMSE) which is the most popular prediction error measure, or the mean absolute scaled error (MASE) (table I). In order, to judge from the predictions, the confidence intervals were also dressed up by using the proposed methodology.

The results are shown in Figure 4. For clarity, this figure is reduced to an extract of all the prediction times: from sample 450 to sample 600. According to table I, exTS provides very good predictions, which illustrates the accuracy of this evolving neuro-fuzzy system. However, both RMSE and MASE increase with the horizon of prediction. Thus, these aggregated indicators are not sufficient to judge from the adequacy of the prediction system and thereby, from the opportunity of using it in prognostic applications. Yet, Figure 4 illustrates that the real output is almost in the confidence interval of predictions. Finally, the exTS system improved by the methodology developed to build the confidence interval enable to make accurate predictions which is of good omen with regard to the online reliability estimation (as stated in part 2).

Conclusion

In maintenance field, prognostic is recognized as a key feature as the estimation of the remaining useful life of an equipment allows avoiding inopportune maintenance spending. However, it can be difficult to define and implement an adequate and efficient prognostic tool that includes the inherent uncertainty of the prognostic process. Indeed, an important task of prognostic is that of prediction since prognostic can also be seen as a process that allows the reliability modeling.

In this context, the paper deals with the definition of a prognostic system for which any assumption on its structure is necessary: it starts from monitoring data and goes through provisional reliability and remaining useful life by characterizing the uncertainty following from the degradation process. More precisely, the paper emphasizes on the use of the evolving neuro-fuzzy predictor exTS. A method to associate a confidence measure to the prediction is proposed and illustrated. This procedure is based on the adaptation of statistical techniques. The model is thereby well adapted to perform a priori reliability analysis since it provides the distribution functions of prediction. Developments are at present extended in order to ensure a confidence level by modifying the learning algorithms. The underlying idea is that a compromise between generalization and approximation should be pursued: practitioners surly prefer a "well known constant" error than a "sometimes catastrophic ones". This work is led with the objective of being integrated to an e-maintenance platform at a French industrial partner (em@systec).

Figure 1 .

 1 Figure 1. Prognostic within maintenance activity.

Figure 2 .

 2 Figure 2. Prediction and reliability modeling.

Figure 4 .

 4 Figure 4. Accuracy of predictions and prediction intervals (95% confidence).

Table I :

 I Simulation results -RMSE and MASE.

	Prediction measure / horizon	t 1 +	t 5 +	t 10 +
	RMSE	0,01560 0,12816 0,22997
	MASE	0,47768 1,97647 3,66373


J Rochat

R Goebel

Elisabeth Tabone

J M Bègue

E Fernandez

R Tibère

J C Gauvin

Bernard Vercambre

Integrated control of the sugarcane spotted stalk borer Chilo sacchariphagus (Lep : Pyralidae) in Reunion Island

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The spotted stalk borer, Chilo sacchariphagus Bojer, is a major pest of sugarcane in southern Asia and on islands in the southwest of the Indian Ocean. In Reunion, Mauritius and Madagascar, the borer has been a serious pest since it was introduced with sugarcane during the 19 th century.

The Entomology Laboratory of CIRAD, which has been working for nearly 30 years on insect pests, including stem borers, has focused its research programme on the control of the spotted stalk borer since 1994. About 17 species of natural enemies of C. sacchariphagus were introduced into Reunion from various countries in the 1970s, but only one third of them established themselves and none have successfully controlled populations of the borer. Therefore, the bio-ecology of the borer is being re-examined, from population dynamics in the field to laboratory studies, with biological control of the borer as the main goal.

Life tables studies indicate that the borer is more likely to develop large populations in the lowlands of the island, where the temperature exceeds 20°C nearly all year round (optimum temperature for development of C. sacchariphagus is 26°C).

The population dynamics of the borer were precisely determined, on the two main varieties of sugarcane (R570 and R579) grown in Reunion, at two sites on the island representing distinct climatic zones [START_REF] Goebel | Caractéristiques biotiques du foreur de la canne à sucre Chilo sacchariphagus (Bojer, 1856) (Lepidoptera: Pyralidae) à l'île de la Réunion. Facteurs de régulation de ses populations et conséquences pour la lutte contre ce ravageur[END_REF]. On a regional scale, the population dynamics of C. sacchariphagus have been studied in relation to damage caused to sugarcane. Such damage is expressed as the number of infested sugarcane internodes in sampled stalks [START_REF] Goebel | Caractéristiques biotiques du foreur de la canne à sucre Chilo sacchariphagus (Bojer, 1856) (Lepidoptera: Pyralidae) à l'île de la Réunion. Facteurs de régulation de ses populations et conséquences pour la lutte contre ce ravageur[END_REF]. These factors have been examined in relation to variety, cultural practices, and climate for four consecutive years at 50 sites distributed over the island. Similarities in infestation level observed between neighbouring sites were frequent, and revealed a regional pattern, probably related to the local climate.

Crop loss studies showed that only R579 was affected in terms of cane yield, expressing a reduction in growth rate, height and diameter of the stalks at harvest. In heavy infestations, yield loss can reach 30% of the mass of commercial cane, compared with losses in resistant variety R570.

Repeated studies over several crop cycles and field trials also confirmed that the variety R579 was more susceptible to C. sacchariphagus than the more common variety R570. The latter is resistant to the borer and various aspects of this resistance (antibiosis, antixenosis, and penetrometry of stalks) have been investigated (see for example [START_REF] Vercambre | Méthodes applicables à la recherche de la résistance variétale de la canne à sucre aux chenilles mineuses des tiges -cas de la relation entre la R 570 et Chilo sacchariphagus (Lep: Pyralidae) à la Réunion[END_REF][START_REF] Morin | Lepidoptera : Pyralidae). Etude de la résistance de la variété R 570 par rapport à la variété R 579, variétés cultivées à la Réunion[END_REF]. Resistance appears to be due, at least in part, to the mechanical characteristics of cane stalks arising from the structure of plant tissues and components of cell walls (tannin, lignin or silicate). This is being studied using artificial diets with sugarcane powder to rear C. sacchariphagus larvae.

The impact of predators, predominantly ants of the genus Pheidole, on borer egg masses was quantified: 80-90% of the eggs are eaten by predators [START_REF] Goebel | Caractéristiques biotiques du foreur de la canne à sucre Chilo sacchariphagus (Bojer, 1856) (Lepidoptera: Pyralidae) à l'île de la Réunion. Facteurs de régulation de ses populations et conséquences pour la lutte contre ce ravageur[END_REF][START_REF] Goebel | Prédation par Pheidole megacephala (Fabricius) (Hym: Formicidae) des oeufs de Chilo sacchariphagus (Bojer) (Lep: Pyralidae), foreur de la canne à l'ile de la Réunion[END_REF][START_REF] Morin | Lepidoptera : Pyralidae). Etude de la résistance de la variété R 570 par rapport à la variété R 579, variétés cultivées à la Réunion[END_REF]. It is believed that the practice of burning the crop, before or after harvesting, has negative effects on natural enemies; moreover borer larvae and pupae, living inside the stalks, are less affected by the burning practice. Observations have shown that when burning was banned, natural enemy populations increased and damage due to borers was reduced [START_REF] Goebel | Caractéristiques biotiques du foreur de la canne à sucre Chilo sacchariphagus (Bojer, 1856) (Lepidoptera: Pyralidae) à l'île de la Réunion. Facteurs de régulation de ses populations et conséquences pour la lutte contre ce ravageur[END_REF].

C. sacchariphagus and Galleria mellonella eggs are used to trap egg-parasitoids in the field in order to identify parasitoids and to examine their population dynamics. Natural parasitism of borer eggs is about 30-40% [START_REF] Goebel | Caractéristiques biotiques du foreur de la canne à sucre Chilo sacchariphagus (Bojer, 1856) (Lepidoptera: Pyralidae) à l'île de la Réunion. Facteurs de régulation de ses populations et conséquences pour la lutte contre ce ravageur[END_REF], with maximum parasitism observed from November to February. To date, a single species, Trichogramma chilonis Ishii, has been recovered in sugarcane fields in Reunion. Bionomics of the species and/or strains of Trichogramma collected in the fields are being examined in the laboratory to select appropriate strains for mass production and augmentative release programmes.

Field experiments with inundative releases of Trichogramma are planned to test for their efficiency. Predation of borer eggs by ants is lower during early growth stages of sugarcane (December -February) [START_REF] Goebel | Caractéristiques biotiques du foreur de la canne à sucre Chilo sacchariphagus (Bojer, 1856) (Lepidoptera: Pyralidae) à l'île de la Réunion. Facteurs de régulation de ses populations et conséquences pour la lutte contre ce ravageur[END_REF], which corresponds to the highest attack rate of the borer on the plant. Hence, releases of Trichogramma should take place during this period of reduced vulnerability to predators.

Acknowledgements

The authors wish to thank MC Gauvin and A Ruttee (CIRAD, Reunion) for technical help in lab and field, N Lezcano (INRA, Antibes, France) for Trichogramma cultures and insect preparations, and B Pintureau (INRA/INSA, Lyon, France) for identification of Trichogramma specimens.


Laurent Marche

Lionel Renault

Christophe Tastet

Didier Mugniery

Quarantine root-knot nematodes : specific determination and study of proteins implied in virulence/avirulence

By pI and molecular weight estimation, a first investigation in protein databases (NCBI, Swissprot and TrEMBL) was done. The results showed that most part of candidate proteins could match with i.e. cuticle or secreted proteins.

Possible family and function of candidate proteins putatively implied in virulence/ avirulence process.

Proteins family Fonction The sensibility and the specificity of the immuno reaction were tested. For the detection, a minimum of 8 ng of protein is necessary. Dot blots upon individual females give very good results especially at high dilution level.

T he identification of quarantine root-knot nematodes that is a preliminary aspect of studies, management and phytosanitary controls is now possible and reliable with a serological test. The results of proteomics studies on virulence/avirulence of quarantine root-knot nematodes allow identification of virulent/avirulent population and enhancing knowledge upon the virulence/avirulence process to improve resistance crop management.

The final study's step is the obtention of mass fingerprint for the candidate proteins by mass spectrometry, in order to match proteins in databases or de novo sequencing if necessary.

Two major proteins, Mcf-A67 and Mcf-B66, have been observed as differentials between quarantine nematodes and others. They have been isolated and sequenced by HPLC after endoprotease digestion.

Virulence (+) and avirulence (-) of the 4 lines (V#) of M. chitwoodi on S. bulbocastanum clones.

  des organismes et des populations appliquée à la protection des plantes, 35653 Le Rheu, France Quarantine root-knot nematodes in Europe awarness, resistance, management and phytosanitary policy Workshop. 9-10 october 2003, Wageningen, The Netherlands B reeders and official phytosanitary services need to identify root-knot nematode as Meloidogyne chitwoodi and M. fallax with reliable methods. A test to discriminate quarantine nematodes could be now effective and the study of virulence/avirulence of those nematodes allows to identify proteins implied in such process. A part of Mcf-A67 sequence, composed of 20 amino acids, shows homology with fatty acid biding proteins (FABPs) of Caenorhabditis elegans. Mcf-A67 was used for serological diagnostic. Serological identification of M. chitwoodi and M. fallax Studied species : M. incognita, M. javanica, M. arenaria, M. mayaguensis, M. exigua, M. arabica, M. naasi, M. hapla race B Including quarantine species : M. chitwoodi, M. fallax Bidimensional electrophoresis (E2D) have been realised upon 10 species of Meloidogyne. Spot specifically present in V1 line in a 3-5 pH range E2D gel. Proteome differential analysis of virulent/avirulent lines of M. chitwoodi Proteome of 4 lines of M. chitwoodi were obtained from the same population virulent related to RMc1 resistance gene of Solanum bulbocastanum.

  were analysed with Kepler software. From 3 pH ranges, 3-5, 5-8, 8-10, 36 proteins were localised as specifically present or absent in one line.