Keywords: Data envelopment analysis, quasi-fixed factors, subvector technical efficiency

When quasi-fixed factors exist, only subvector measures of technical efficiency are consistent with short-run economic efficiency. They are shown to be less than or equal to radial measures and to increase as the number of quasi-fixed factors is increased. Nonparametric estimates for French cereal farms evaluate robustness of technical efficiency scores with respect to quasi-fixed factors. While the distribution of scores changed substantially, their mean and rank ordering did not change significantly. Nonetheless, both theoretical and empirical results highlight the importance for the DEA approach to technical effrciency estimation of accurate specification of the salient characteristics of the technology and economic environment faced by firms.

Introduction

An important objective of nonparametric estimation of technical ineffrciency is the development of recommendations for adjustment of factors of production. Implicitly, these recommendations must be limited to adjustments which are economically efficient. Where technologies are homothetic, expansion paths are rays from the origin. V/ithin this context, recommendations for radial adjustments for technical efficiency are consistent with achieving economic efficiency if adjustment is proposed along the expansion path relevant under the current price structure. In this sense, measures of radial technical efficiency (RTE) would be of interest to managers because they provide recommendations that are consistent with economic efficiency. This correspondence between radial technical adjustment and economic effrciency dissolves when technology is nonhomothetic. A similar situation occurs when the set of technical possibilities is constrained by quasi-fixity of factor services. In this case, even for homothetic technologies, the expansion path will deviate from a radial path. Such fixity might occur as a result of temporal, regulatory, or other constraints (e.g. exogoneity of flows) on adjustment. Within decision horizons where such factor services can not be controlled, recommendations for their radial adjustment would be of little interest to managers. Where the marginal productivity of such services is nonnegative at their constrained level, radial adjustment through disposal of such services would be economically inefficient.

This problem was recognizedby Fanell (1957) though he maintained a focus on radial technical adjustment.t Hull and Winsten (1959) reconsidered the issue by distinguishing between controllable and noncontrolable variables. More generally, the issue is a case of measurement of subvector technical efficiency (STE) as studied by [START_REF] Kopp | Measuring the Technical Efficiency of Production: A Comment[END_REF]; Frire, Lovell, and Zieschang (1983); and applied by Banker andMorey (1986), Kamakura (1988), Adolphonson, et al. (1990). For the case of a single quasi-fixed factor, [START_REF] Banker | Efficiency Analysis for Exogenously Fixed Inputs and Outputs[END_REF] demonstrated that the resulting estimate of subvector technical efficiency will be less than or equal to a corresponding estimate of unrestricted or radial technical efficiency.

In this paper, a formal proof is provided of the relationship between RTE and STE for the case of an arbitrary vector of quasi-fixed factors. Next, the paper establishes that, in general, a type of magnification effect occurs as the vector of quasi-fixed factors is expanded.

That is, for a particular firm, as the number of quasi-fixed factors is increased (decreased) the STE estimate decreases (increases). In analogy to the Le Chatelier -Samuelson principle, the proposition states that as the opportunity for adjustment of variable factors is reduced, so the extent of technical efficiency is reduced. Going further, we evaluate the impact of fixity of factors on the efficiency rank of firms.

The presence of efficient markets and opportunities for instantaneous adjustment of factor services in many industries limits the applicability of subvector technical efficiency. However, where ideal economic conditions fail to exist, or where economic conditions are distorted by regulatory constraints, subvector measures of technical efficiency are warranted.

To evaluate the empirical implications of use of RTE measures in the presence of quasi-fixed factors, we present estimates of RTE and STE for a sample of French cereal farms drawn from the FADN (Farm Agricultural Data Network) farm record system for 1990. The bias of the RTE measure is evaluated. Given the conditionality of STE estimates on the composition of the subvector of quasi-fixed factors, the robustness of firm efficiency rank is evaluated as the number of quasi-fixed factors is varied. t

The issue was also considered in a wider context by [START_REF] Fâre | Measuring the Technical Efficiency of Production[END_REF] who introduced input specific measures of input technical inefficiency to generalize the Fanell measure. These < Russell > input efficiency measures are interpretable as nonradial and were shown to provide identification of efficient firms from the primal. In contrast, the Farell input and output measures provide different rankings of firm efficiency. While Fâre and Lovell's Russell measure achieved its objective of providing a unique means of identiffing technical efficiency from the primal, Kopp noted that the measure lacks a useful dual interpretaton. J Technical Efficiency with Quasi-Fixed Factors Input based and output based measures of technical efficiency popularized by Farrell indicate an identical proportion by which either all inputs or all outputs may be reduced to place a firm on a particular technical frontier of reference. The usefulness of such aggregative indicators has been criticized because they fail to provide input specific recommendations for adjustment (Fernandez-Cornejo,1994). While the Russell measure of technical efficiency introduced by FÈire and Lovell provides input specific estimates, [START_REF] Kopp | Measuring the Technical Efficiency of Production: A Comment[END_REF] noted it lacks a corresponding dual measure. More generally, it may be of interest to access the technical efficiency of subvectors of inputs or outputs. The presence of quasi-fixity of subvectors is an important example. The notion of subvector technical efficiency was considered by Fiire, Grosskopf, and Zeischang (1983) for the a single factor case and by Fiire, Grosskopf, and Lovell (1994) for the general case. The input based measure of subvector efficiency can be defined following these studies for a set of .Ifirms indexedj= 1,....J, each with access to the same technology that transfonns a vector of inputs x, e R! into a vector of outputsy, e Ry . More generally, for the set of firms define a (J x N/ input matrix X and a (J x M) output matrix I . Suppose the input matrix is partitioned so X = (Xu ,Xi) where v c. {1,2,...N} with v oi = A and y u v = {1,2,...,N}. For convenience, the partition is identical for all firms.

The elements of Xu are assumed variable and those of Xi are assumed quasi-fixed. Each row of X can be equivalently written x, = (xj,xl). Suppose the technology satisfies the augmented regularity conditions adopted by [START_REF] Banker | Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis[END_REF]2. The production possibilities set P can be written:

'

That is, the set of production possibilies is defined by: P: (x,)):) ) 0 may be produced from r > 0) satisffing: lPtl: (x,,y,) e P Yi: l, ...., J

[P2]: P is a regular set, i.e. nonempty, closed, and satisSing the origin condition lhat y j =0 if X j = 0

[AE],Pisconsistentwithfreedisposalof inputsandoutputs:V(x,y) e P if I2x and y (y,then (î,y) e P .

lPal: P isconvex: V(x,y) ePand(1,y) eP,Vcr e[0,1] ; aQc,y)+(1-c]x,y; ef IP 5l : P is the intersection of all sets satis$ing P ] to P4.

More formally, P2, P3 and P4 each define linear constraints that define closed subspaces. The intersection of these subspaces satisfies P I and, defines a convex envelop that may be written: p: {x' ,x' ,y):x' >}"Xu,xn > },Xt ,y <Xf ,fX,= 1,1, .Rj } tll j=t where X: (7"1 ,7"r,... ,1", ) is the intensity vector. Each ),, denotes the intensity at which activity 7 is undertaken. By equation (l), the jth firm'sproduction plart (xl,xrY,yr) b.longs to the production possibilities set, if, and only if, (*1,*1,!)eP. Input based radial technical efficiency (RTE) is defined following [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF], [START_REF] Charnes | Measuring the Efficiency of Decision Making Units[END_REF], and Fâre, Grosskopf, and Lovell (1994):

R,(x,,!)-min{Rr:R,x, el(y),Ri.&} l2l where /(yr) is the input requirements set derived from (1). The interpretation of .R, as a radial measure follows from its scalar dimension which implies each factor will be reduced by the same nonnegative proportion. As an alternative, Fâre, Lovell and Zeischang defined asymmetric technical ineffrciency where the scalar reduction \ is applied to only one element ofxr. More generally, Fiire, Grosskopf, and Lovell define subvector technical efficiency (STE) as follows:

si@l,xi ,11, = rnitr{si :sJx' el"(xo, ,y,),si . & } t3l where I'(x',,1)is the input requirements set conditional upon the levels of the quasi-fixed factors. The distinction between R, *d ry rests squarely on the application of a common proportion of reduction (thus, subaggregative) to a subvector of inputs which by maintained hypothesis are considered variable and controllable by the manager. Efficient adjustment is defined as conditional upon the levels of quasi-fixed inputs. A direct comparison of these two measures can be made within the context of linear programs that allow their nonparametric estimation following the approach of Data Envelopment Analysis. To allow a convenient comparison, define a base linear program from which the two measures can be drawn as Define R, the solution to program "R" defrned by the base program (4) augmented by the constraint that R, -E; E;. Similarly, define program "S" as the base program augmented by the constraints that S; = E; ^d E; = 1. . Solution of progftlm S results in ,Srl ç6nsistent with

(3). Banker and Morey graphically compared the solutions (Rr,Srl) for the case where xl is a scalar. For the general case where xl is a vector, they noted that any optimal solution to program R is a feasible solution to program S. Their proposition is as follows.

Proposition I [START_REF] Banker | Efficiency Analysis for Exogenously Fixed Inputs and Outputs[END_REF] For any j, Si < R; where Si is the solution to program S for firmT and iR, is the solution to program R.

Formal proof follows directly by reference to the underlying input requirements sets derived from equation (1). In the absence of a constraint on xj , program R was derived from equation

(2). For xl fixed, program S follows from equation (3). To compare the solutions to program R and program S, write equation (2) interms of the constrained level of xo, say tl: R,(xi,rl ,!1)=min{Rr:R,xl el'(R,rl ,11),R, .&} t5l Given that R, e & , free disposal (assumed by P3) implies R,IÏ < rl and r 1np-l ,t) c I" (î.1 ,!1)

Simply stated, the constraint set for program R is contained within that of program S. It follows that: R,(x,,!) > SrY(xj ,r', ,!) t6l t4l

By similar logic, define program 9-1 where anarbitrary element of xu is fixed, e.g.x! .

Program Su-1 can be written as: sit{xj',*l,rT,!i):-infi-':si-'xi-1 et"-t1r!,ri,!),sl-' .&} t7l However, program I can be rewritten:

{(".ï-' ,r! ,fl ,t) = minfi:si *i-' . r'(qr| ,1,}),{ .4-} tsl Given that Sir! <rl ,itfollows that: /*l(S;t! ,xl ,l) c I'-t1x! ,rl ,!) and Si > S;-t .

These results provide the basis for the following proposition.

Proposition 2

The input based subvector efficiency of firmi varies systematically with the number of quasi- fixed factors which constrain its adjustment of variable factors. As v decreases from Mto 0, Sl = Ri > S/-'>...> S; >...) Sro = 0 .

t9l An Application to French Cereal Production Propositions 1 and 2 establish that estimates of radial technical efficiency will, in general, over estimate the efficiency of firms which face constraints on adjustment due to quasi-fixity of factor services. In this section, we present results which allow an empirical evaluation of the magnitude of such biases as well as an illustration of the magnification effect presented in Proposition 2. The application is based on a sample of 188 French cereal farms drawn from the Farm Agricultural Data Network (FADN) data set for 1990. On average, cereal production accounted for at least 70 percent of their total gross product. In all cases, the farms were specialized in soft wheat and maize, Vermersh et al. (1992). The sample caracterizes 45Yo of the French cereal farms sampled in the General Agricultural Census of 1988 and in this sense, the sample may be interpreted as representative of population of cereal producers in France.

To proceed, we estimate RTE and STE as alternative hypotheses concerning the fixed of factor services. Under a maintained hypothesis that we label as Hypothesis 1, we represent the product vector as incuding two outputs (cereal output (soft wheat and com), and other outputs (mainly, oilseeds); five variable inputs (fertilizer, pesticides, seeds, other intermediate inputs, and equipment); and three quasi-fixed factors (area cultivated under cereal, area cultivated in other crops, and labor (family and hired)). This represents our prior concerning the quasi-fixity of factors. 'We include labor in the vector of quasi-fixed factors given that family labor accounts for 95% of labor used in the sample. All variables except land and labor are measured in 1990 French francs3. Available data necessitated linear aggregation of valuesa. Land is measured in hectares, while labor is measured in annual worker units (AWU5). Equipment services were measured based on estimated value, a five year amortization, as well as maintenance expenses, rent, and machinery cooperative fees.

Equipment is viewed as variable due to the presence of active rental markets. Desuiptive statistics presented in Table 1 veriff the specialization in cereal production by the sampled farms. Cereal production accounted for over two-thirds of their revenues and occupied almost 75%o of their cultivated area. Fertilizers and pesticides dominated expenses.

Estimation of the technical efficiency scores under various maintained hypotheses of quasi-fixity were based on solution of the programs presented in equations ( 2) and (3).

Solutions were implemented using linear programming software available in SAS/OR.

Results are reported in Table 2 -4. Alternatives to Hypothesis 1 for the specification of the quasi-fixed factor vector include Hypothesis V where all factors are variable and Hypotheses 2-4 whichmaintain alternatives to Hypothesis I as defined in Table 3. First, Table 2 indicates that the number of firms identified as technically efficient (based on a technical efficiency score of unity and slack variables estimated as zero) under alternative hypotheses ranged betwen 88 and 93. This indicates robustness of the classification of efficient firms under the alternative hypotheses. Details on the charactersitics of these effrcient farms are available from the authors. However, similar robustness is not found across ranges of inefficiency.

For example, the number of firms found with scores below .85 varies from 71 under our preferred specification Hypothesis 1, to only 36 under Hypothesis V. This suggests that the distribution of the efficiency estimates is substantially altered by the efficiency measure used (RTE vs. STE) and by the specification of the quasi-fixity of inputs.

3 Economic data, whether expressed in volume or value, reflects both technical and allocative inefficiency. Past studies of technical efficiency have used both volume and value.

a Fâre and Primont (19S8) present results that indicate that use of value aggregated data will result in negatively biased estimates of technical efficiency. Thomas and Tauer have noted that measurement bias may occur when linear aggregation is used.

5One AWU equals 2200 hours of labor. indicate that the mean and standard deviation of scores varies across specification. On average, the estimated technical efficiency score under Hypothesis I (land and labor fixed)

was .8773 compared to .9319 under Hypothesis V (all factors variable), indicating that a substantial upward bias results from use of R, when quasi-fixed exists according to Hypothesis 1. Further, Table 3 indicates the extent of the magnification effect presented in equation ( 9) can be substantial in quantitative terms, depending on the factors that are fixed.

The differences between efficiency estimates based on Hypothesis V vs. Hypothesis I is also apparent from I to 37.8o compared to only 19J% under Hypothesis V.

Conclusions

These results have two important implications. First, where factors are quasi-fixed, both the level of efficiency and the distribution of effrciency among firms may be substantially over estimated when based on Hypothesis V. While econometric study of efficiency has necessarily recognized the importance of accurate specification of input variability and fixity, these results highlight the importance of this specificaiton decision for nonparametric studies of technical efficiency. Second, the under estimation will provide a misleading basis for recofirmendations to firms concerning existence of opportunities for input adjustments as well as for more global recommendations concerning the existence of generic opportunities for input adjustment.

  ,,, =nL,y,,âL,=L,Li> o vi :r,...,r] ,nl .&,Ei .4

  Summary of data for efficiency analysis(188 French cereal producers, 1990)

Table 3

 3 presents a different view of the distribution of efficiency scores. Results

Table 2 .

 2 Technical efficiency frequency

Table 3 .

 3 Technical efficiency measurement

		Hv		H1			TN	H3		H4
		nb	%	nb	%	nb	o/o	nb	%	nb	%
	:1	88	46.8	92	48.9	89	47.3	93	49.5	89	47.3
	between 0.95 and I	29	15.4	12	6.4	26	13.8	22	11.7	l7	9.0
	between 0.85 and 0.95	35	18.6	l3	6.9	JJ	17.6	3l	16.5	18	9.6
	<0.85	36	l9.l	7t	37.8	40	21.3	42	22.3	64	34.0
	Hypotheses	Mean		Std-Dev			Min			Max
	Hv	0.9319		0.0969			0.6223			I
	H1	0.8773		0.1552			0.5158			1
	H2	0.9262		0.1072			0.6006			1
	H3	0.9271		0.1047			0.6006			1
	H4	0.8915		0.1386			0.51s8			I
											10


C Broussolle

of products and regional competition in the agro food industry

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

ABSTRACT

This paper presents a study of relations between industrial structures and market equilibrium where demand is uncertain. In the model used, the number of firms and their respective sizes and production structures are endogenous to the model.

The approach used is to determine first of all the sector's natural configuration and supply function. This means identifying the number and respective sizes of production units that together give the best price conditions for adjusting industry supply to one demand characterized by a distribution of probabilities.

The analysis is then broadened, taking coalitions, product choice and the spatial factor into account so as to analyse the impact of these parameters on market structures.

In a study of industrial structures and market equilibrium (C. Broussolle, 1990), using concepts from contestable markets theory and a model in which the number of firms and their respective sizes and production structures were endogenous to the model, we were able to identify equilibrium industry configurations. These are effective when the number of f irms and the allocation of the vector of production among them minimize the total cost of production in the industry.

Corresponding market structures are termed "natural".

In this article, we shall f irst examine the case where both independent f irms and firms acting in collaboration are operating in the market. We shall then introduce plant localization and product variety as parameters. Besides the fact that it is useful to look at each of these separately in analysing industrial structures, it is also worth looking at both together, since geographical localization is considered an aspect of horizontal product differentiation. Regional competi tion and product competi tian thus enter into the same analytical framework. The aims are (a) to determine the natural configuration of a group of firms operating in different markets and (b) to assess the impact of plant localization and product choice on the structures of these markets.

I-THE MODEL

The group under consideration is that of firms competing in the markets where they sell their products. This group forms a structured network within which a flow of goods circulates.

Day by day analysis of orders placed wi th f irms shows irregularity in terms of in guantities ordered, time intervals separating two successive orders, and delivery times imposed by customers.

These observations imply that orders (whether in backlog or in production) should be treated as a variable of state of the 11 firms/market" system.

Manufacturing Production and demand of products were monitored day by day for a year. The process being monitored proved to be Markovian, or comparable to a Markovian process, Pi(n) being the probability of the system being in the state Ei (i=l,2, ... M) at date n . The group of probabilities Pi(n), which forms the group of vectors of state, describes the system for all future dates considered, probabilities Pi (0) being known and constituting the initial vector for the following equations of state:

M (p(n+l)) = [T)[p(n)J ~ Pi= 1 i=l
The model is constructed around the firms, and there are as The unit under consideration is quantity ~j of finished kle product delivered to one market and manufactured in one factory of one firm, from raw materials coming from one collection area. The corresponding "strategic segment" is described by four fields type of product, factory, firm, market. The product is manufactured in fixed-capacity plants, each f irm having a whole number of factories.

Output from a segment must meet uncertain demand , with delivery time constraints. Production capacity is such that the probability of demand not being met is below a certain threshold, this being reflected in excess production capacity.

The ratio of demand to total capacity, or plant utilization rate, expresses this relative difference. Supply adapts to dernand by bringing one or several more plants into use.

For each segment there are corresponding unit costs: collection, production, delivery. The aggregate cost is the surn of these costs.

At the outset, there is a group of segments qtie To reduce this target function, the mode! simulates transfers of production from one segment to another. Wi th a group of u segments, production can be transferred from a segment s to a segment t selected from arnong the (u-1) remaining segments. Segment t is selected as being the one that optimizes the target function. Thus, each segment has its corresponding production transfer strategy. These strategies (u in all) are compared.

Let the production transfer strategy for segment I be a switch to segment J. The esperance of the new demand in connection with J is equal to the sum of the esperances of initial demands I and J. We then calculate the number of plants of type J such that the probability of non-satisfaction of this new demand is below or equal to the chosen threshold.

For each segment qij kle the function F below: F{n,n,d) = a = upo with u in which this probability is expressed by = e-nd{l-n) . n denotes the number of plants, . d a period of days,

. n the plant utilization rate of factory 1 of firm k, (between o and 1) n = JL_ where Bis the esperance of dernand for the product, nµ.

• µ the unit capacity of one plant, po the probability that a customer will be kept waiting,

• u the probability that the wait will be longer than a given tirnelapse.

The factor pois expressed in the following equation:

po = n-1 k L (nn) + g k=O k! with g = n (nn) n! {1-n)
The nurnber of plants n of the new segment J is the srnallest whole nurnber such that the quantity produced is greater than or equal to the dernand. First the cost of production for J is deducted, then the new value for the target function, on the hypothesis that the transfer strategy will actually be carried out. This series of calculations allows classification of segments, and consequently of products, factories and firms, on cri teria of competi ti veness. This classification grades firms according ta their contribution to adjusting supply ta demand; from this classification, the natural market configuration can then be deterrnined.

II-INDUSTRIAL SUPPLY AND NATURAL MARKET CONFIGURATION

We will examine in turn the case where firms are independent and the case where bath independent firms and firms acting in collusion are operating in the market.

1-Independent firms

The natural configuration of a market is obtained from the industrial supply curve. This is the envelope of the marginal cost of substitution curves of the firms.

Let i be the product manufactured in factory 1 of firm k.

The model used enables us, as we saw, to grade uni ts of production according to their competitiveness in the market.

Supposing factory 1 of firm k lies in nth place, it is possible to find the minimal conditions (price, quantity produced, number of production units) that this type of factory must meet in order to gradually take the place of those above it (or below it), so as to meet the initial demand for the sector. In this way we can plot the marginal cost of substitution curve of product i manufactured in factory type I, compared to the same product obtained from other factories ranking n-1, n-2, ... 1 in the initial classification. An example of the curves obtained is Curve 1 in Graph 1. Following the same procedure for each firm and supposing -to simplify the presentation -that there are three firms, whose factories' sizes are µ1, µ2 and µ3 (µl <µ2<µ3), one gets curves 2 and 3.

The respective positions of these different curves are to be explained, in particular, by the trend of the esperance of the plant utilization rate as a function of demand, taking account of the sizes of the production units (Graph 2a). These utilization rates along with their corresponding unit costs (Graph 2b) show that theoretically less efficient plant is economically more interesting up to a certain threshold (BE), as its utilization rate implies a lower average cost of production than that recorded for larger-capacity plant (Broussolle 1989). As a corollary to this finding, we note that a (large) firm can only operate effectively in a market if the market is large enough. -output Q1 is supplied by firrn 1, n espérance mat..,,4::atique de la de=anè~ e$;>ére.nce ma t.1"1éma-t1:;;ue de la prcduc~~=~ -output Q2 -Q1 is supplied by firrn 2 and quantity Q-Q 2 by the third firm, knowing that these firms' respective numbers of factories n1, n2, n3, have been calculated using the model, and so may be different from those in the initial setup.

Graph 1 pria .,._,,d• .. ~ ' p -• •• --• • • --• -• • • • • ---: •• t •••--- , __ _ 1 • 1 : r" l • ................... , 1 : 1 ï:-------7-; 1 ! 1 1 : • 1 _______ Ji ' .......... ___ .,.
Of course, a certain number of firms and factories may be excluded from the natural configuration obtained. These are the ones with supply curves above the sectoral supply curve. Firms are excluded if their presence in the market is irrational owing to profit levels lower than those of firms in the natural configuration.

-Coalitions

In the most common case, there are firms operating separately in the market and also firms operating in coalitions. Coalitions may be restricted to simple cooperation, may go as far as merger, or may take a number of intermediate forms between these two extrernes. What consequences does such collusion have for the natural market configuration ? Let us take, as before, three firms E 1 , E2 and E3, each of which has one or more factories, and let us imagine collusion between firms E1 and E2, whose unit costs of production are higher than those of firm E 3 , and whose output is lower.

Apart from any other consideration, the collusion between firms E1 and E2 is reflected in the model by a pooling of the orders that were ini tially placed wi th each firm, and their distribution among the production units in such a way as to minimize manufacturing costs.

The three firms and the coalition E 12 are represented in

Graph 3b

COllt w,1CA1re da production

• 1 1 ' 1 1 1 1 1 ' ~S I Esp4r•nc• uthé- utique de i.. demande. E•p•rance . . ù>,- Datiqu• de l• production.
same Based on these results, it is possible to determine at the time the natural market configuration and the corresponding supply curve. As a first step, the resolution of the model allows us to classify coalitions and independent f irms according to their competi ti veness in the market. As a second step, we establish, as before, the marginal cost of substitution curves of the product manufactured in the factories of firms E1, E2 and E3 and coalition E12•

In fact, this gives three possible cases:

1-The natural configuration obtained is such that the coalition entirely replaces its member firms, and so provides the totality of their output. 2-The coalition considered is of no interest, in as much as its supply curve is above that of the natural market configuration.

3-The coalition is restricted to supplying a part of output (Qi2 Q 2 ), and member firms continue to supply separately a certain proportion of their respective initial outputs.

In the case shown here, i t can be seen (Graph 4) that segments of the supply curve of coalition E12 are intermediate wbetween those of firms E 2 and E 3 . As a result, to meet demand under the best price conditions, the natural market configuration is such that the quantities produced Q1, Q2-Q1, Q12-Q2 and Q-Q12 are supplied, respectively, by firms E1 and E2, coalition E12, and firm E3, and this in factories whose unit capacities are given, but whose respective numbers are supplied by the rnodel. . --------1 , :

: •••••..••.•..••.... -.: _-1-_-..1_ ~ ïP -
.,

.

1 : • : ---1' .J { 1 •------~-. . . ' 1 ;

Ouantit<ls

III-MARKET STRUCTURES AND REGIONAL COMPETITION

The industrial structures observed are the combined result of (a) firms adapting to existing technical and economic conditions and (b) strategies adopted by economic agents to modify their environment. From this standpoint, choice of plant localizationis as important as product choice. Both aspects involve the same set of problems in that product localization is one of the characteristics of horizontal product differentiation. Broadening the study to bring in the spatial factor therefore means extending the sphere of characteristics of each product to take account of the localization of the markets where i t is sold. The firms potentially concerned by these diversified demands will seek to appropriate market shares. They will integrate transport costs connected with geographical localization into their cost pr ice. Under these conditions, and knowing that for the first stage these localizations are gi ven, the problem is to determine those equilibrium configurations that integrate the spatial factor; as geographical space operates solely as the medium through which goods flow, and as an additional source of costs. In fact, even in this case, some further hypotheses have to be made about the localization of the production units to be created when determining the suppl y curve for a f irm that is gradually replacing the firms above it in the initial classification.

Let us consider a group of n firms manufacturing one product and able to supply m markets in part or in full. As a first step we determine the natural configuration of these n firms in each market. As a second step, the results are compared market by market, to show whether they are compatible and, if not, to identify the conditions under which they would become so.

In fact there are two possible cases. In the simpler and less likely case, one finds different firms in each of the configurations obtained. Under these conditions no regional competition problem arises. But this problem does arise if, in the natural configurations of some markets, the same firms are solicited for levels of output that are mutually incompatible.

To simplify the presentation, let us suppose that interfirm competition occurs in two markets, M1 and M 2 , and that a simultaneous search for natural configurations in these markets shas led to selection of the factories of firms E1 and E2 for the first market and those of firms E1, E2 and E5 for the second (Graph 5).

In market M1, supply from f irms in the natural configuration is shown by the stepped curve ABCDE which intersects supply curve D1 of market M 1 at coordinates P 1 and Q1. It can be seen that output AB is supplied by firm E1 and output CD by firm E2. Firrns E3, E 4 and E5 are excluded from the equilibrium configuration of market M 1 .

the supply curves for these configurations. Of course, a certain number of f irms rnay be excluded from the market for some or all of the varieties it supplies. The presence of a firm on one or several of these supply curves justifies, for given output levels, the manufacture of one or more varieties of the product under consideration. More broadly, this gi ves the product range for which each firm is competitive.

Although the methodology is identical, it should be noted that geographical localization regarded as an aspect of horizontal product differentiation is a cause of further constraints in establishing supply curves. Sale by several firms of different products in the same market is not equivalent to sale by several firms of a single product in different markets. In the latter case, if a firm cannot meet dernand in several markets simultaneously in the same niche of the supply curve, and if the firm concerned is not in a dominant position, choices will be "imposed" upon it. The same applies if horizontal product differentiation does not take account of the spatial factor.

2-Natural configurations and vertical product differentiation

The cost of production of a product obviously depends on its quality, and quality range will therefore be reflected in product prices. Consequently, while for a given firm there is only one demand function for a given product, there is on the other hand a supply function for each quality of this product. Under these conditions, a natural configuration exists for each quality, and the presence of a f irm on one or more of the suppl y curves for the se configurations indicates, for gi ven output levels, the range of qualities for which it is competent.

Of course, most products are supplied bath in a range of varieties and at different quality levels. Study of this cornmon case involves combining the results of the earlier analyses, and also raises some specific problems to be addressed in forthcoming research.

CONCLUSION

The concepts employed and the use of a model in which the nurnber of firms and their respective sizes and production structures are endogenous to the model enable selection, in a randorn universe, of the most efficient firms. Furthermore, by modifying, under certain conditions, the configuration so obtained, coalitions can in some cases change the nature of competition.

Of course, the industrial structures observed are the combined resul t of f irrns adapting to existing economic and technical conditions and of strategies adopted by econornic agents to modify their environment. Frorn this standpoint, and treated in the sarne way as product choice, plant localizationis one of the variables firms can manipulate to influence market structure. Bath aspects invol ve the sarne group of problerns insofar as product localization is one of the product differentiation factors that define viable firrns.

  . . . k ... s). Each firm k has one or more factories Uk = (Ukl Ukl Ukr) and supplies, from its plants Ikl = ( Ikll • .. Ikle ... Iklh), a range of products qkle = (qklel qklep ... qkler), which are sold on markets M = (1 ... j ... m) to meet demands characterized by distributions of probabilities.

  From the marginal cost of substitution curves for each type of factory, one can determine the sectoral supply curve. This stepped curve is made up from the elements of curves 1,2 and 3 the furthest to the right on Graph 1. The supply curve intersects the demand curve at coordinates P and Q and gives the natural market configuration, i.e. the industrial structure with which supply is in adjustment with demand under the best price conditions. The size of each horizontal segment measures the production capacity invested in one type of factory. Thus in the case in hand, the guantity Q of product soldat price P breaks down as follows:

  showing the trend in the mean plant utilization rate as a function of the esperance of demand, are associated with their corresponding unit costs (Graph 3b), one observes that coalition E 1 2 meets a greater demand than the sum of the demands that firms E 1 and E 2 could meet separately, and this at a lower unit cost than firm E3As a resul t, the demand-channelled to E 3 is, all else being equal-,--necessarily less than it was at the outset. Coalition E12 has thus proven

  . . . . . . . . . . . . . . .......... . --.. -.. .. !1; .. -•-• .. .

BIBLIOGRAPHY

BAUMOL W., PANZAR J. et WILLIG R., 1982


Bernard Jalabert

Alexis Fostier

Bernard Breton

Claudine Weil

Oocyte maturation in vertebrates

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


est destinée au dépôt

  The sexuality of cultured hermaphroditic fish species : analysis of morphological and endocrinological features in a protogynous hermaphrodite, Epinephelus microdon, as a basis for further research to control reproduction of the grouper L. Debas, Alexis Fostier, J. Fuchs, M. Weppe, G. Nédélec, A. Benett, . Aquacop, Chantal Cauty, Bernard Jalabert To cite this version: L. Debas, Alexis Fostier, J. Fuchs, M. Weppe, G. Nédélec, et al.. The sexuality of cultured hermaphroditic fish species : analysis of morphological and endocrinological features in a protogynous hermaphrodite, Epinephelus microdon, as a basis for further research to control reproduction of the grouper. Advances in Tropical Aquaculture : Colloque sur l'aquaculture tropicale, Tahiti, (French Polynesia ) February 20 -March 4, 1989, 9, 780 p., 1990, Actes de Colloques Ifremer. hal-02852364


Caroline Lavecchia
email: lavecchi@loria.fr

David Langlois
email: langlois@loria.fr

Kamel Smaïli
email: smaili@loria.fr

Phrase-Based Machine Translation based on Simulated Annealing

In this paper, we propose a new phrase-based translation model based on inter-lingual triggers. The originality of our method is double. First we identify common source phrases. Then we use inter-lingual triggers in order to retrieve their translations. Furthermore, we consider the way of extracting phrase translations as an optimization issue. For that we use simulated annealing algorithm to find out the best phrase translations among all those determined by inter-lingual triggers. The best phrases are those which improve the translation quality in terms of Bleu score. Tests are achieved on movie subtitle corpora. They show that our phrase-based machine translation (PBMT) system outperforms a state-of-the-art PBMT system by almost 7 points.

Introduction

Given a sentence in a source language, the goal of Machine Translation (MT) is to find out its translation in a target language. Different approaches exist to deal with this difficult challenge. Some approaches require a priori human knowledge in order to model both the source and target languages, and how to switch from one to another. The Systran MT system [START_REF] Varadi | New generation systran translation system[END_REF]) is based on this approach and proposes a translation model depending on transfer rules. The statistical approach follows a completely different direction.

The statistical MT does not require any external knowledge. It uses only parallel corpora to model the translation process. Such corpora are aligned at word or sentence level in order to link both source and target languages. The translation issue is treated as an optimization problem. Translating a sentence from English into French involves finding the best French target sentence f * which maximizes the probability of f given the English source sentence e. This translation model is based on the noisy channel model. The Bayes rule allows to formulate the probability P (f |e) as follows:

f * = argmax f P (f |e) = argmax f P (e|f ) * P (f ) (1)
Thus, the translation process consists of a language model P (f ) and a translation model P (e|f ). Language model parameters are trained from a target corpus, whereas parameters of the translation model are determined from the parallel corpus. Then, a decoder provides the best target sentence given the source sentence and the table translation parameters.

First statistical MT systems were word-based [START_REF] Brown | The mathematics of statistical machine translation: parameter estimation[END_REF]. Obviously, the human translation is a very complex process which is not only word based. Following this fact, recent researches showed that the use of phrase translation instead of word translation leads to better MT system quality. Dealing with phrases allows an explicit modeling of lexical units and captures easily local reordering. For example, without use of phrases, the translation of Pomme de terre gives apple of earth instead of potatoe. And in some situations, the use of phrases reduce the imprecision of reordering. For instance, without use of phrases, the translation of Tour Eiffel gives Tower Eiffel then reordering process may produce the correct English translation. By using phrases, we reduce the imprecision of translation and at least avoid some reordering problems. Probably, one of the most difficult issue is how to find out the best phrases in both source and target languages.

In order to retrieve phrases, several approaches have been proposed in the literature. Most of them require word-based alignments. For example, [START_REF] Och | Improved alignment models for statistical machine translation[END_REF] collected all phrase pairs that were consistent with the word alignment provided by Brown's models. Thus any contiguous source words must be the translation of any contiguous target words on the condition that words are aligned with each other. That means that retrieved phrases have not always linguistic motivation and could lead to noisy sequence of words.

In this paper, we propose an original idea based on interlingual triggers to build phrase translation without requiring word-based alignments. First we give an overview of inter-lingual triggers. Then we present the set up of our phrase-based machine translation system based on interlingual triggers. Finally, a description of the used corpora and the results are provided and discussed. We end with a conclusion which points out the strength of our method and gives some tracks about future work in our research group.

Machine Translation based on Inter-Lingual Triggers

We propose an original approach for SMT based on interlingual triggers. In the following, we present the notion of inter-lingual triggers and how to make good use of them in order to perform Machine Translation.

Review of inter-lingual triggers

Inter-lingual triggers are inspired by the concept of triggers used in statistical language modeling [START_REF] Tillmann | Word trigger and the EM algorithm[END_REF]. A trigger is a set composed of a word and its best correlated triggered words in terms of mutual information (MI). Trigger models are combined with n-gram models in order to enhance the probability of triggered words given a triggering word. Since classical triggers allow to establish a triggeringtriggered link between two events from the same language, we propose to determine correlations between words in a source language and words in a target language by using inter-lingual triggers. Therefore, an inter-lingual trigger is a set composed of a triggering source event and its best correlated triggered target events. We hope to find among the set of triggered target events, possible translations of the triggering source event. Inter-lingual triggers are determined on a parallel corpus according to the following formula:

M I(f, e) = P (f, e) * log( P (f, e) P (f ) * P (e) ) (2)
where f (respectively e) is a sequence of French (respectively English) words. M I(f, e) denote the mutual information assigned to e and f and P (e), P (f ) and P (f, e) are defined as follows:

P (X) = N (X) |Corpus| P (f, e) = N (f, e) |Corpus| (3)
where N (X) is the number of sentences where X occurs, N (e, f ) is the number of sentence pairs where e and f cooccur and |Corpus| is the number of sentence pairs in the training corpus.

For each French event f , we kept as inter-lingual triggers, the k English events e with higher MI values. In the following, an event is a word or a sequence of words and we differentiate two types of inter-lingual triggers:

• 1-To-1 triggers: one French word triggers one English word

• n-To-m triggers: a sequence of n French words triggers a sequence of m English words with n, m ∈ N.

Inter-lingual triggers have been used in [START_REF] Kim | Lexical triggers and latent semantic analysis for cross-lingual language model adaptation[END_REF] to enrich resource deficient languages from those which are considered as potentially important. Our purpose is to use them in order to perform statistical machine translation. To achieve that, we employ inter-lingual triggers to build translation tables required in the decoding process. To do that, we assign to each inter-lingual trigger a probability calculated as follows:

∀f, e i ∈ T rig(f

) P (e i |f ) = M I(e i , f ) e∈T rig(f ) M I(e, f ) (4
) where T rig(f ) is the set of k English events triggered by the French event f . In a previous work, we developed a Word-based Translation system based on 1-To-1 triggers (see section 2.2. for more details). In this paper, we extend inter-lingual triggers to carry out phrase-based Machine Translation. In the following, we present our method to build phrase translation table based on Simulated Annealing.

Word-based Translation with Inter-Lingual Triggers

In (Lavecchia et al., 2007b), we built a word-based Machine Translation (WBMT) system based on 1-To-1 triggers. First, we constructed a word translation table using the 50 best triggers for each French word. Then, we used the Pharaoh decoder to translate an English corpus into French. We showed that the performance of our system is similar to the ones achieved by a system based on IBM model 2 [START_REF] Brown | The mathematics of statistical machine translation: parameter estimation[END_REF], in terms of Bleu score [START_REF] Papineni | Bleu: a method for automatic evaluation of machine translation[END_REF].

In the light of this supporting results, we decided to investigate phrase-based Machine Translation (PBMT) based on inter-lingual triggers. As we have seen before, most of state-of-the-art methods collect phrase translation from word-based alignments. Our goal is to train a PBMT system without calling upon word alignment. We would like to learn phrase pairs only by taking advantage of inter-lingual triggers.

Method for learning phrase translation

Most of methods which use phrases in MT require wordbased alignments. For example, [START_REF] Och | Improved alignment models for statistical machine translation[END_REF] collected all phrase pairs that were consistent with the word alignment. In his method, any contiguous source words may be the translation of any contiguous target words on the condition that words are aligned with each other. That means phrases have no always linguistic motivation and retrieved translations could lead to noise. We are convinced that if we succeed in identifying common phrases in the source part of the training corpus, interlingual triggers will allow to retrieve its translations in the target part. This would generate less noise. Since source phrases are selected beforehand, our method does not require any word alignment. In the next sections, we detail how to extract source phrases. Then, we propose to use inter-lingual triggers in oder to find their potential translations in the target corpus. Finally, we present an adaptation of the Simulated Annealing algorithm in order to determine the best phrase translations among all those selected with inter-lingual triggers.

Phrase extraction

In the few last years we developed a statistical method to extract pertinent phrases [START_REF] Zitouni | Statistical language modeling based on variable-length sequences[END_REF] from large corpus. We use this method to rewrite source part of the training corpus in terms of phrases. To achieve that, an iterative process selects phrases by grouping words which have a high value of Mutual Information. Only the phrases which improve the perplexity are kept for the forthcoming steps. At the end of the process, we get a list of phrases and a source corpus rewritten in terms of phrases. With this source corpus expressed with pertinent phrases, we hope to find their potential phrase translations in the target corpus by using inter-lingual triggers.

Learning phrase translation

The source training corpus is henceforth rewritten in terms of phrases. Now, the question is how to find the potential translations of these source phrases in the target corpus. To achieve this, we propose to use inter-lingual triggers. In the following, we assume that each source phrase of l words can be translated by a sequence of j target words where j ∈ [l -∆l, l + ∆l]. At this step, no word alignment is performed. For this reason, we associate with each source phrase (2 * ∆l + 1) sets of its k best inter-lingual triggers. Thus, we allow a source phrase to be translated by different target sequences of variable sizes. guess that for short phrases ∆l is set to 1. Thus, for the cited example, we suppose that it could be translated by a sequence of at least one word and at most by a sequence of 3 words. For this reason, we associate it with its best 2-To-1, 2-To-2 and 2-To-3 inter-lingual triggers. In this example, we have selected 9 potential translations. Obviously, only press charges is a correct one. In the general case, we can have for each phrase of two words k potential translations.

That is why we propose to select those which are pertinent and discard the noisy ones.

All source phrases and their sets of inter-lingual triggers constitute the set of n-To-m inter-lingual triggers. Now, the issue is how to select the best n-To-m inter-lingual triggers. In other words, what are the pertinent phrases and their translations. To answer this question, we first compute Bleu score on a development corpus by using our wordbased system based on 1-To-1 inter-lingual triggers. This will constitute the baseline result. In a second step, we add randomly a subset of n-To-m triggers previously computed into the word-based system. With an adequate algorithm, we select the most relevant phrases, those which improve the Bleu score on a development corpus. The optimization algorithm we use is simulated annealing detailed in the next section.

An outline of retrieving the best phrase translations is given in Algorithm 1.

Algorithm 1 Method for learning and selecting the best phrase translations 1: Extract phrases from the source corpus 2: Determine n-To-m inter-lingual triggers which allow to associate each source phrase with the best target phrases of variable size 3: Compute the baseline Bleu score by using our wordbased system based on 1-To-1 inter-lingual triggers. 4: Select an optimal subset of n-To-m inter-lingual triggers on an iterative process handled by Simulated Annealing algorithm

Simulated Annealing tuning

Simulated Annealing (SA) algorithm is a technique applied to find an optimal solution to a combinatorial problem that becomes unmanageable using combinatorial methods.

The SA approach allows to solve such combinatorial problem while dealing with the local optimum problem.

The concept of SA is inspired from the physical annealing process of solids and is easily adaptable to solve large combinatorial optimization problems. In condensed matter physics, people are interested in obtaining low energy states of a solid. In other words, the issue is how to arrange the billions of particles in order to achieve a highly structured lattice with a low energy of the system.

Our aim is similar, in fact, the set of n-To-m triggers constitute a list of candidate phrase translations. We have to integrate a subset of this list in our MT system in order to increase the quality of translation. Naturally, it is unreasonable to try all possible combinations of translation. For this reason, we decided to use SA algorithm in order to select the ones which lead to the best performance. To achieve that, we start with a word-based MT system based on 1-To-1 inter-lingual triggers. Then we randomly add n-To-m triggers into the MT system until an optimal Bleu score is reached on the development corpus. The entire algorithm is given below:

Algorithm 2 Simulated Annealing algorithm 1: Start with a high temperature T.

2: With a temperature T and until the equilibrium is reached do From the current temperature T of the system and from the current state i which has an Energy E i , perturb the system which makes it moving from state i to j. The energy of state j is E j . If E j -E i >= 0 then state j is accepted as the current state; Otherwise, state j is accepted with a probability random(P ) < e(Ei -Ej)/T with P ∈ [0 -1] 3: Decrease the temperature and go to step 3 until the given low temperature is reached or until the energy stops increasing

It is necessary to define all the parameters of the algorithm in order to adapt it to our issue:

Initial temperature The temperature acts as a control parameter. Several values have been tested in our experiments for the initial temperature.

Initial configuration

The initial state is a word-based MT system based on 1-To-1 inter-lingual triggers.

System perturbation Agitate the system consists in randomly adding a subset of n-To-m inter-lingual triggers into the translation table of the current MT system.

Equilibrium State A each step of the SA algorithm, a whole decoding process is launched in order to evaluate the performance of the current MT system in terms of Bleu score. The equilibrium state is reached when the Bleu score stops increasing between two states.

The schedule annealing After each equilibrium state, the temperature has to be decreased carefully. For that, we choose a geometric series, which respects the progressive decreasing of the temperature.

Energy computing

The energy to be maximized is expressed by the Bleu score on a development corpus.

Stop criterion

The stop criterion of adding n-To-m interlingual triggers is reached when the Bleu score of the system converges.

At the end of the SA algorithm, only the n-To-m interlingual triggers which improve the performance of the initial word-based MT system are selected. In SA algorithm, skipping from one state to another guarantees to reach an optimal state in terms of the objective function. Consequently this algorithm increases necessarily the Bleu score.

In the next section, we present used corpus and conducted experiments to train and test our PBMT System based on inter-lingual triggers.

Results

Corpora

We present results on a subtitle parallel corpus built using Dynamic Time Wrapping algorithm (Lavecchia et al., 2007a). Subtitle corpora are very attractive due to the used spontaneous language which contains formal and informal words. We think that such corpus constitute a good challenge to go towards spontaneous speech translation system. Table 2 gives details about the used the parallel corpus. We use a train corpus to extract French phrases and to compute inter-lingual triggers (a study of few examples is given in section 3.2.). A development corpus is used to select the best phrase translations among all those determined by the set of inter-lingual triggers. Finally, we use a test corpus to validate our approach. ). All these elements account for weak Bleu scores reported in section (3.).

French English

In the following paragraphs, we present a study of few inter-lingual triggers.

Study of some inter-lingual triggers

Inter-lingual triggers are selected on a parallel training corpus. In our framework, training step leads to significant inter-lingual triggers as shown in Table 3. The first column presents French sequences of one or several words. Sequences that have more than one word are automatically picked up by the iterative process explained in section (2.3.1.). For each French sequence, the second column refers to the best correlated English sequences of one, two or three words in terms of MI. Finally the third column shows the MI value associated with each inter-lingual trigger. A qualitative analysis showed that our method leads to pertinent inter-lingual triggers. Thus, triggered sequences could often be considerated as potential translation of the triggering French sequence. Furthermore, inter-lingual triggers allow to retrieve synonyms as it is shown for the French word allumer which can be translated by light or turn on. Note also that they take into account the fact that n French words are not necessarily translated into n English words. Thus, bonjour is associated with good morning or petit déjeuner with breakfast. Finally, when a French sequence is translated into several English words, inter-lingual triggers will prefer the whole English sequence rather than subparts of it. This case is illustrated by the example porter plainte which is translated by press charges, the sequence which gets the highest MI value.

French

In the following sections, we evaluate our MT systems based on inter-lingual triggers. To achieve that, we use inter-lingual triggers to build translation tables required by the decoder Pharaoh [START_REF] Koehn | Pharaoh: A beam search decoder for phrase-based statistical machine translation models[END_REF] in order to translate an English source corpus into French. Bleu score allows us to evaluate the quality of the obtained French translation. In section (3.3.), we start with a Word-Based Machine Translation (WBMT) system based on 1-To-1 triggers. Then, we compare it with a state-of-the-art WBMT system based on IBM model 3. In section (3.4.), we design a Phrase-Based MT (PWMT) system based on n-To-m triggers and Simulated Annealing. Finally, we compare it with a stateof-the-art PBMT system based on the phrase-based model proposed by Och in [START_REF] Och | Statistical Machine Translation: From Single-Word models to Alignment Templates[END_REF].

Word-based Translation System

To build our WBMT system, we employ 1-To-1 triggers for which each French word is associated with a list of k English words. We hope to catch in this set of k English words potential translations of the French word. Several experiments showed that 10 is the optimal value for the parameter k. Then, we assign to each 1-To-1 trigger a probability calculated from M I as indicated in formula (4). This constitutes the word translation 4. The first line of the table reports performance of our WBMT system based on 1-To-1 triggers. The performance is compared to the one of a WBMT system based on IBM model 3 reported on the second line. For an optimal use of the decoder, the weights of the models involved in the decoding process are tuned on the development corpus1 for both systems. Results show that using 1-To-1 triggers leads to better translation quality. Indeed, the better performance of our system amounts to 12.49 in terms of Bleu score. While in an optimal use, the system based on IBM model 3 reaches only 12.39. Furthermore, this last model is trained in several iterations whereas training inter-lingual triggers needs only one iteration. In other words, with less time for training, our approach leads to better results than famous IBM models largely used in SMT.

Considering this very promising results for WBMT, we decided to make good use of inter-lingual triggers to process Phrase-based Machine Translation.

Phrase-based Translation System

French extracted phrases

To build our Phrase-based Machine Translation System, we extracted from the French part of the training corpus, a set of 15860 phrases which are composed of two or three words.

Only 2.20% (respectively 3.03%) of the phrases extracted from the training corpus were in the development (respectively test) corpus.

Candidate Phrase translations for SA algorithm

For each French unit (a word or a sequence) of l words, we select from the training corpus its 10 * (l ± ∆l) best interlingual triggers. For practical reasons, ∆l does not exceed 2. This means, for each potential English translation set among those containing l-∆l, l-∆l+1, . . . , l+∆l words, we kept the best 10 units. All this inter-lingual triggers make the set of candidate phrase translations (called n-Tom triggers) required by SA algorithm.

Results with Simulated Annealing

Different experiments have been conducted to optimize the parameters of the SA algorithm. In this section, we present the performance with the optimal set of parameters. We made several tests in order to determine the best value of the initial temperature. T = 10 -4 seems to be a convenient initial temperature. fulfilled a Bleu score of 14.14. In other words, by adding pertinent phrase translations, we achieved an improvement of more than 1.6 point in terms of Bleu compared to our word-based MT system.

In the next section, we compare our PBMT system based on inter-lingual triggers with a state-of-the-art PBMT in order to evaluate our approach.

Comparison with a state-of-the-art system

In order to validate our approach, we compare the performances of our PBMT system based on inter-lingual triggers with a state-of-the-art PBMT system (reference system). The phrase translation table of the reference system is acquired from a word-aligned parallel corpus by extracting all phrase-pairs that are consistent with the word alignment [START_REF] Och | Statistical Machine Translation: From Single-Word models to Alignment Templates[END_REF]. On the development corpus, as seen before, the use of pertinent n-To-m triggers improved the results achieved by 1-To-1 triggers by almost 13.21%. For the state-of-the-art methods, the use of phrases decreased the performance by 43.3% compared to the word-based method. Despite the few amount of training data, these first results show that the SA algorithm allowed to take off the phrase translations with no statistical significance. Overall phrase translations determined by n-To-m triggers, it selected only a subset which leads to an optimal translation quality. Furthermore, both WBMT and PBMT systems based on interlingual triggers leaded to better performances than the corresponding state-of-the-art systems.

Unfortunately, the lead of n-To-m triggers on 1-To-1 triggers is not corroborated on the test corpus. Indeed, the use of phrase translations decreased the Bleu score by 21%.

Over-fitting due to poor amount of data and used corpora can explained this under-achievement. Recall that the used corpora are subtitles of 36 different movies. Each movie is divided into three parts: one for the training corpus, one for the development corpus and one for the test corpus. And we have chosen movies without paying attention of the different cine styles. Consequently, talks and expressions are very disparate within each corpus. For example, the talks in a thriller will not be the same as in a comedy. By adding the poor amount of data, pertinent phrases chosen on the development corpus were not necessarily pertinent on the test corpus. Conversely, phrase translations unselected by the SA algorithm would maybe allow to improve performance on the test corpus. A good way to figure out this problem would be to classify movies according to their style (thriller, comedy, war movie, romantic comedy . . . ) and to set one PBMT system based on inter-lingual triggers by style. Hence, even if data are sparse, learned phrase translations would be proper to each movie style. And phrase translations selected during SA process on the development corpus would be the best ones on the test corpus too. Anyway, the impact of over-fitting are more important on the state-of-the-art systems. Indeed, on the development corpus, their performance decreases by 43.3% from WBMT to PBMT. Whereas for systems based on interlingual triggers, Bleu score increases by 13.9%. In the same manner, on the test corpus, state-of-the-art PBMT system decreases the Bleu score of the state-of-the-art WBMT system by 53.07%. While n-To-m triggers PBMT system lowers the performance of 1-To-1 triggers by only 21%.

Conclusion

In this paper, we presented our phrase-based Machine Translation system based on inter-lingual triggers. The latters allow to associate a triggering source phrase with its best triggered target phrases in terms of mutual information. We noticed that a triggered target phrase may often be assimilated to a potential translation of the source phrase. Thus, we decided to use inter-lingual triggers in order to set up our PBMT system. Most phrase-based translation models require word alignment on the parallel training corpus. Phrase translations extracted from this alignment are not always linguistically motivated and thus are not pertinent. In order to extract more relevant phrase translations and therefore improve the translation quality, we proposed an original method that does not need any word alignment. First we identified common source phrases by an iterative process. Then, we retrieved their potential translations by using inter-lingual triggers. And finally, we used simulated annealing algorithm to select the best phrase translations among all those determined by inter-lingual triggers. We trained and tested our PBMT system on a subtitle parallel corpus built using Dynamic Time Wrapping algorithm. This corpus constitutes a good challenge to go towards spontaneous speech translation system. Once phrase translation table were induced by inter-lingual triggers and simulated annealing algorithm, we used the decoder Pharaoh to translate text from English into French. We evaluated the translation quality with the Bleu metric. Results showed that our approach leaded to better translation quality compared to a state-of-the-art phrase-based approach that required word alignment. Indeed, our system based on interlingual triggers outperformed a state-of-the-art system by 7 points on a development corpus and by 4 points on a test corpus. Conducted experiments confirmed that phrase translations learned from word alignment can cause noise in the translation process. Identifying common source phrases and selecting their potential translations with inter-lingual triggers and SA algorithm allows to confine noise even on sparse data.

Our results are very encouraging and efforts are done in order to improve our model. The idea of using inter-lingual triggers seems to be very important. For the moment, we focus on word surface forms. However, considering interlingual triggers on syntactic features in order to integrate linguistic knowledge in the translation process may improve drastically the translation quality.

Figure 1 :

 1 Figure 1: Improvement of the Bleu score on a development corpus through the SA algorithm

Table 1 :

 1 Table1shows the potential translations of the source phrase porter plainte. In the following we Potential translations of the source phrase porter plainte

			n-To-m triggers
	source phrase	2-To-1	2-To-2	2-To-3
		press	press charges can press charges
	porter plainte	charges can press	not press charges
		easy	not press	you can press

Table 3 :

 3 Examples of English phrases triggered by French phrases

		English	M I × 10 -4
		press charges	6.92
	porter plainte charges	6.26
		press	5.29
		light	4.65
	allumer	to turn on	3.46
		turn on	2.88
		hi	32.43
	bonjour	hello	29.30
		good morning	19.55
		calm down	23.32
	calme toi	calm	21.99
		down	13.85
		breakfast	10.42
	petit déjeuner to breakfast	3.13
		say breakfast	3.138

Table 4 :

 4 table required by Pharaoh. Evaluation of WBMT systems Translation results in terms of Bleu on the development corpus are given in Table

	System	tm lm	d	w	Bleu
	1-To-1 Triggers 0.6 0.3 0.3 0	12.49
	IBM Model 3	0.8 0.6 0.6 0	12.39

  The initial configuration consists of the word translation table obtained in section (3.3.) with 1-To-1 triggers. This configuration leads as we have shown, to an initial energy of 12.49 in terms of Bleu score. Then, at each step of the SA algorithm, we agitate the current configuration by adding randomly phrase translations from the set of n-To-m triggers selected earlier on the training corpus. Conducted experiments showed that performance are optimal when we added randomly 10 potential translations of 10 French words or phrases. The improvement of the Bleu score obtained on the development corpus through SA algorithm is shown in Figure1. At the end of the SA process, our phrase-based MT system

		0.142
		0.14
		0.138
		0.136
	Bleu score	0.132 0.134
		0.13
		0.128
		0.126
		0.124
		0	100	200	300	400	500	600	700	800	900	1000
						Number of Iterations

Table 5

 5

		illustrates the performances of
	inter-lingual triggers	state of the art
	1-To-1	n-To-m	IBM3 reference
	Dev 12.49	14.14	12.39	7.02
	Test 13.63	10.77	14.00	6.57
	Table 5: System Evaluation in terms of Bleu score on the
	development (Dev) and the test (Test) corpora
	the different systems on both development and test corpora.
	The two first column denote the Bleu score achieved by our
	WBMT (1-To-1) and PBMT (n-To-m) systems based on
	inter-lingual triggers. Whereas the two last columns cor-
	respond to the performance of the state-of-the-art WBMT
	system based on IBM model 3 (IBM3) and the reference
	PBMT (reference).

tm (respectively lm, d) indicates the weight of the translation (respectively target language, distortion) model . The parameter w is for the word penalty. The target language model is a trigram model (Good-Turing smoothing)

Acknowledgments

This work is supported by EADS (European Aeronautic Defense and Space Company) foundation for the Speech-To-Speech translation Project.


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