Philippe Martin
email: philippe.martin@univ-paris1.fr

Thierry Mayer
email: tmayer@univ-paris1.fr

Florian Mayneris
email: mayneris@pse.ens.fr

PUBLIC SUPPORT TO CLUSTERS: A FIRM LEVEL STUDY OF FRENCH "LOCAL PRODUCTIVE SYSTEMS"

Keywords: JEL Classification: C23, R10, R11 and R12 Keywords: clusters, localization economies and public policies

This paper analyzes empirically a public policy promoting industrial clusters in France. Cluster policies have become popular in many countries but have not been extensively evaluated empirically. We use data on production and employment for firms that benefited from the policy and on firms that did not, both before and after the policy started. We first show that the policy selected firms in relative decline. Furthermore, our results suggest that the policy had no major effect on their productivity but may have helped them in terms of employment.

Introduction

Industrial clusters are popular among policy makers. Since the end of the 1980s, national and local governments in Germany, Brazil, Japan, South Korea, the Spanish Basque country, and France, inter alia, have attempted to foster their development. The work by Michael Porter (1998,2000), the leading figure of cluster strategies, has been very influential in this matter and is invariably used as a justification for cluster policies. There is however little macro or micro empirical analysis of their effect on firms performance. The present paper attempts to fill this gap by analyzing the effect, on individual firms, of a specific cluster policy in France.

A typical defense of cluster policies is that clusters bring economic gains and should therefore receive public support. Porter's definition of a cluster -"a geographically proximate group of interconnected companies and associated institutions in a particular field, linked by commonalities and complementarities" -is not very far from what economists call an agglomeration. The idea that clusters bring economic gains because firms perform better when located near other firms in the same sector is hardly new. In the late nineteenth century, Alfred Marshall identified several benefits of clusters or industrial districts. The different sources of agglomeration externalities, were first analyzed by Marshall and later rediscovered by Kenneth Arrow and Paul Romer. Those are 1) input externalities that save on transportation costs and make inputs purchases more efficient; 2) Labour market externalities that foster the creation of pools of specialized workers, who acquire cluster-specific skills valuable to the firms; 3) Knowledge externalities through which industrial clusters facilitate the exchange of information and knowledge.

Advocates of cluster policies need to address three questions:

1. How large are the gains from agglomeration? In particular, how much does the productivity of a firm increase when other firms from the same sector decide to locate nearby?

2. Do firms internalize these gains when making their location decisions? In particular, are "natural" clusters too small? 3. Can public policies that attempt to foster clusters affect positively the performance of the firms that belong to those clusters?

There is a large empirical literature that has attempted to answer the first question. The survey of Rosenthal and Strange (2004) reports that in the many empirical studies on agglomeration, the doubling of the size a cluster (generally measured as employment of a given sector in a given region) leads to a productivity gain between 3% and 8%. In another paper on French firm-level data, [START_REF] Martin | The dynamics of productivity in the telecommunications equipment industry[END_REF]), we estimate this elasticity to be around 4%. The starting point of those who defend cluster strategies is thus right: economic gains of clusters exist. Their enthusiasm should however be tamed; these effects are modest. In the same paper, we also find evidence that French firms internalize part of these productivity gains when they choose where to locate: the size of existing "natural" clusters is not very different from the size that would maximize productivity gains 1 . The observed size of clusters is slightly smaller than the one that would be optimal but the productivity gains that would be obtained by increasing the size of existing clusters are estimated to be small. Overall, the existing empirical work on clusters suggests that they produce economic gains and that these gains are not fully internalized by firms. Hence, the case for public intervention in favor of clusters can be made but there is no evidence that the expected gains should be large.

Finally, even if one assumes that there is a case for public intervention (gains from clusters exist and there are not entirely internalized by firms), there is little evidence on the answer to the third question, on which the present paper focuses. Can cluster policies actually help?

They could do so in two main dimensions. First, cluster policies could increase the size of existing clusters and thus improve the performance of firms if the cluster size is suboptimally small. Second, cluster policies could improve the workings of externalities (input markets externalities, labour markets externalities and technological externalities) inside clusters for a given size of the cluster. Both mechanisms could increase productivity of firms in the cluster.

In this paper, we exploit a rich French firm-level dataset to analyze the impact of a specific cluster policy that was implemented in 1999, by the Délégation Interministérielle à l'Aménagement et à la Compétitivité des Territoires (Diact, ex Datar), the French administration in charge of spatial planning and regional policy. The policy provided support to a group of firms, located in the same area and belonging to the same industry, called the "Local Productive Systems" (LPS). The main aim of the policy was to encourage cooperation among firms and to increase the competitiveness of firms in the cluster.

We assess the impact of public support to LPS on firms' total factor productivity (TFP) and on firms' employment. We use a difference in difference approach exploiting this detailed dataset that spans over the 1996 to 2004 period, during which a subsample of firms where selected to benefit from the policy.

We first analyze the characteristics of firms that were chosen by the authorities. This is interesting because it raises important political economy issues. Our results show clearly that the French LPS policy targeted firms located in backward regions and operating in declining industries. Hence, the policy turned out to be of a defensive type. The official objective was to promote agglomeration externalities and clusters dynamics and was supposed to mark a radical shift of the French regional policy, from traditional spatial equity to efficiency considerations. Our results suggest that the traditional equity objective was in reality still at play. 2 We also find that LPS firms receive more public subsidies in general than others. This is consistent with the study by [START_REF] Beason | Growth, Economies of Scale, and Targeting in Japan (1955-1990[END_REF] on Japan. They show that the reality of industrial policies implemented between 1955 and 1990 clashed with the official objective to help the growth of winners. Indeed, they find a negative correlation between the growth of a given industry and the intensity of the aid it received. Our results on the French cluster policy we study as well as those of [START_REF] Beason | Growth, Economies of Scale, and Targeting in Japan (1955-1990[END_REF] are consistent with two interpretations. One is that subsidies to declining industries reveal government political preferences [START_REF] Corden | Trade Policy and economic Welfare[END_REF], [START_REF] Krueger | Asymmetries in policy between exportables and import-competing goods[END_REF]). Another possible mechanism is provided by [START_REF] Baldwin | Entry and Asymmetric Lobbying: Why Governments Pick Losers[END_REF]. These authors show why governments often "pick losers" or more exactly why public subsidies are captured by firms in decline which have a greater incentive to lobby for subsidies.

We also find that the public support has been unable to reverse the relative decline of TFP at work for firms selected by the policy. However, it may have had a slight positive impact in terms of employment. Hence, it may have delayed the exit of declining firms, and in this case such policies could have had a negative impact on aggregate productivity.

This is also what [START_REF] Criscuolo | The effect of industrial policy on corporate performance: Evidence from Panel data[END_REF] conclude in a study that analyzes the impact of the Regional Selective Assistance in UK. They find that the policy, designed to subsidize firms in backward areas, has had a positive impact on firms' employment and investment but no effect on firms' productivity. By supporting less efficient firms, the authors judge that such a policy may slow down reallocations from less efficient plants and affect negatively aggregate productivity growth.

A more positive conclusion is reached by [START_REF] Branstetter | When Do Research Consortia Work Well and Why? Evidence From Japanese Panel Data[END_REF] who analyze Japanese R&D public policy and its effect on the patenting activity of firms involved in government-sponsored research consortia. They find a positive impact, though quite small when all controls are included. Their method, which consists in examining the relative patenting path of consortia firms the years after the inception of the consortium is close to ours.

A related literature has analyzed the effect of subsidies given to firms to locate in specific regions. [START_REF] Crozet | How do firms agglomerate? A study of FDI in France[END_REF] study for example the determinants of location choice by foreign investors in France over the period 1985-1995. They measure the impact of a French subsidy (the "Prime d'Aménagement du Territoire", PAT) and of European grants for regional policy on firms' location choice. They find a generally positive, but very weak and hardly significant effect of those policies. [START_REF] Head | Attracting foreign manufacturing: Investment promotion and agglomeration[END_REF] analyze the effect of state level policies in United States to attract Japanese firms and find that the probability to attract these firms increases with the subsidies. However, given that all states have such policies, the location of firms is not affected in equilibrium. Finally, [START_REF] Devereux | Firm location decisions, regional grants and agglomeration externalities[END_REF] study the effect of Regional Selective Assistance (RSA) grants3 on firms location in United Kingdom. They also find a positive but very weak effect of the policy.

The paper is structured as follows. We first describe in section 2 the Local Productive Systems policy and our data. We then lay out in section 2.5 our empirical strategy. We present our results in section 3 and some robustness checks in section 4. Section 5 concludes.

2 What are the "Local Productive Systems"?

The policy

The French agency in charge of regional policy (DIACT) issued in 1998 a tender intended to fund collaborative projects between firms of a given industry located in the same area. The purpose was clearly to promote agglomeration externalities and clusters dynamics. This policy corresponds to a quite radical shift in the objectives of French regional policy, from traditional spatial equity to taking more into account efficiency considerations in the geographic distribution of economic activities. One of the motivations was to replicate the alleged success of Italian industrial districts in the 1980's: the idea was to enhance, through public intervention, collaborations which developed "naturally" in Italy.

Around one hundred projects were submitted and around fifty of them received a subsidy in 1999. An additional fifty were funded in 2000, when the agency in charge issued a new tender. The tender was then transformed into a permanent one, and each year new or old propositions (only a handful of them now) are getting approved and funded by an ad hoc national commission.

The stated aim of the policy was to give a small monetary incentive (the average subsidy is around 37,500 euros) to set off or reinforce clusters. Conditions to receive this subsidy were not very restrictive at the beginning of the process. Conditions are now more demanding (established collaborations, credibility of the proposed action, knowledge of direct competitors etc.). Officially, the policy funds a project and not directly a group of firms. Very often, the official candidate organizing the project is a local public authority, and private firms join once the structure has secured the necessary funding. A wide range of actions can be funded: A study of feasibility for the development of a common brand, the creation of a grouping of employers or the implementation of collective actions in the field of exports for instance. The geographical scale of a LPS is generally the département or the employment area4 .

The LPS can be seen as the first cluster policy in France. A new policy, called "Competitiveness clusters" that started in 2005 is a much more ambitious and costly cluster policy than the one analyzed here (note however that a quarter of LPS projects have been transformed into competitiveness clusters).

The data and methodology

We use French annual business survey5 data, provided by the French ministry of Industry. We have information at both the firm and plant levels. This is restricted to firms with more than 20 employees and all the plants of those firms. Our data cover the period 1996-2004. At the firm level, we have all the balance-sheet data (in particular, production, value-added, employment, capital, exports, aggregate wages) and information about firm location, firm industry classification and firm structure (e.g. number of plants).

At the plant level, data are less exhaustive; they contain plant location, plant industry classification, plant number of employees and information about the firm the plant belongs to.

We obtained from the public authority in charge of the LPS policy, the DIACT, the list of LPS and the information about the subsidies obtained as well as the structure which administers. We contacted individually during the year 2006 around 90 LPS, to ask them the list of their adherents. Workable files were obtained for 56 of them, which represents 3,234 firms. We however lost information when we merged these firms with the annual business surveys to obtain data on production and employment. Many of the LPS reported the name and the address of firms, but not their national identification number. We consequently had to find out most firms in the annual business surveys thanks to their name and their zip code only. We merged successfully only 641 firms (the others are probably firms with less than 20 employees or with badly collected information), from 45 LPS created between 1999 and 2003.

From a geographic point of view, we dropped all firms located in Corsica and in overseas départements. Consequently, our sample covers the 94 continental French départements and 341 employment areas. From a sectoral point of view, we only retained firms belonging to manufacturing sectors6 . In particular, food-proceeding firms had to be dropped.

More observations are dropped: the ones for which value-added, employment or capital is missing, negative or null. We also dropped all the firms which changed geographical unit or industrial sector during the period, in order to use the geographical and sectoral dummies that control for spatial and industrial unobservable characteristics7 . We deflated value-added data by a branch price-index and capital data by a an investment price index valid for all industrial sectors. In the end, the sample is an unbalanced panel involving 417 firms which belong to a LPS. Eighty-eight 3-digit industrial sectors and thirty-nine LPS are represented.

For employment areas data, we use the "Atlas des zones d'emploi" published by the INSEE, the French institute for national statistics, in 1998.

Which industries are targeted by LPS?

Some simple descriptive statistics on the industries targeted by the LPS policy are useful. We distinguish the manufacturing industries which are not represented in the LPS (24 nontreated industries), the industries represented by less than 10 LPS firms (49 industries) and the industries represented in the LPS by at least 10 firms (14 industries). The average of several indicators for these three categories are presented in table 1. In 2004, the average labour productivity is lower in LPS industries than in the rest of manufacturing industries. This result is particularly clear for the industries where LPS are the most important. LPS industries are also much more labour intensive than the others.

Between 1996 and 2004, the employment loss for the average French non-LPS manufacturing industries is 10.68%. LPS intensive industries lost much less employment (1.04%). Their value added also increased more (28.96% vs 19.30%), but not proportionally to employment, so that labour productivity increased on average by 34.54% in non-LPS industries, and by only 32.64% in the 14 main LPS industries. Finally, LPS firms belong to industries that export less than the average but their exports grew faster over the period.

To summarize, LPS industries are on average much more labour intensive than the rest of manufacturing; they destroyed less employment than other industries in the 1996-2004 period but their productivity gains were also lower.

Who are LPS firms?

We now analyze the characteristics of firms that participated to one of the selected LPS. Table ?? presents summary statistics about the LPS firms of our sample. They are larger and less productive than non LPS firms. However, the standard deviation for all their characteristics is lower than for other firms. This suggests that the policy targeted firms with specific characteristics. To go further in this analysis, we estimate, with a logit model, the probability for a firm to become a LPS-firm. We take into account average firm-level characteristics prior their entrance in a LPS. We also control for characteristics of the employment areas where the firms are located. The way we compute firm-level average characteristics is not trivial. Our panel is unbalanced. Moreover, firms entered the LPS scheme in different years between 1999 and 2003. Hence, the number of years for which we can observe the firm characteristics prior their entrance in a LPS is not the same for all firms. If firms' characteristics are affected by annual common shocks, the computation of pre-LPS average characteristics could therefore be noisy; hence, we correct all individual observations for yearly trends. We then compute for each firm its average characteristics for the years before its "entry" in a LPS. For non-LPS firms and firms in LPS sustained in 2003, these average characteristics are computed with all the available de-trended observations from 1996 to 2003. We keep in the end 335 LPS firms in the sample.

The results are displayed in table 3. The index of TFP we use is obtained with an estimate of a production function, following an OLS approach. In the appendix, we show that our results are robust when we use a GMM estimation for TFP (see table 20). Column (1) presents results from a simple logit, where we control for the size (number of employees) of the firm, its TFP, the amount of subsidies (other than LPS) it receives and the number of workers in other firms of the same industry-area. It appears that in this very simple specification, LPS firms are bigger than the others and receive more public subsidies overall. These two characteristics of LPS are very robust. One interpretation is that LPS firms are important for local politicians because they are big employers and that they are good at lobbying for public subsidies.

The inclusion of industry-fixed effects does not change these results (regression (2)) but the inclusion of départements fixed effects (regression (3)) does: LPS are more productive than the other firms of their département. Given that the coefficient on TFP is close to zero and not significant when we do not control for département fixed effects, this means that LPS firms are located in less productive départements8 Note also that the number of employees in the other firms of the same industry-département, which is a proxy for potential localization economies, is close to zero and not significant in the simple OLS regression. This is surprising since the LPS policy is supposed to be a clusterpromotion policy. It may be partly due to the fact that LPS firms are bigger than the others, which mechanically reduces, for a given size of the industry in the département, the number of employees in surrounding firms. But the coefficient is strongly positive and significant when départements fixed effects are added. Hence, another non exclusive explanation would be that the LPS policy targeted clusters which are relevant at a local level, but not at a national level.

In regression ( 6), we include some characteristics of the employment areas where firms are located. The results are robust to this inclusion. Moreover, these regressions show that, relative to the average in the département, LPS firms are located in areas which are less dense, more dependent on industry, richer, and with less workers with vocational training. Note however, their average taxable income growth was smaller over the 1984-1994 period. We will use this regression and what it tells us about the observable characteristics of LPS firms to construct our sample for the matching approach when we analyze the impact of LPS status on firm performance.

Empirical methodology: a difference-in-difference approach

The stated objective of the LPS policy is to improve firms' competitiveness. To analyze whether the LPS was successful in this respect we quantify the impact of the LPS policy on firms' total factor productivity (TFP). We also analyze its impact on firm's employment. The approach we choose is based on the standard "difference-in-difference" method (DD) (see [START_REF] Bertrand | How Much Should We Trust Differences-in-Differences Estimates?[END_REF]).

y it is our dependent variable (firms' TFP or employment). The relation we bring to data is the following:

y it = γlps i + θlps in it + d t + it (1)
where lps i is a dummy variable which identifies LPS firms which at some point benefit from the LPS label. This dummy captures all time-invariant unobservable characteristics specific to firms targeted by the LPS policy. lps in it is a dummy which equals 1 for LPS firms the year it receives the subsidy and thereafter. d t is a time trend, common to all firms. If it is orthogonal to the regressors, θ is the DD estimator of the effect of LPS policy on firm's performance. It is indeed obtained by comparing the evolution of performance for a LPS firm before and after its entry in the LPS, to the evolution of performance for a non-LPS firm during the same period. Section 2.4 however showed that LPS firms had particular characteristics, especially in terms of location and industries, which both determined their probability of belonging to a LPS and their performance before. This suggests several sources of bias in our estimates of γ and θ. If the fact of being in a LPS is also correlated to specific shocks or to temporal trends (if it = u i + η it and if E(η it+1 -η it ) is different for LPS and non-LPS firms), our estimation will suffer from a simultaneity bias. The best way to control for both unobserved invariant characteristics and unobserved idiosyncratic shocks would be to instrument the LPS variables. There is however no obvious set of natural instruments that would be good predictors of entry into the LPS scheme, while being unrelated to the firm's performance. We will consequently address the invariant part of that endogeneity issue by adding different sets of fixed effects. This amounts to assuming that the biggest part of the problem mentioned is a correlation of LPS participation with u i rather than with η it . We combine this with a matching approach, which accounts for the fact that LPS firms are quite specific in the observable characteristics and identifies a group of non treated firms with the most similar set of such observables.

Results

LPS and Productivity

We first present our results on TFP. To estimate firm TFP, we regress firm value-added on employment and capital and keep the residuals; the estimated elasticities for employment and capital are respectively 0.85 and 0.15. In the appendix, we discuss the limitations of the OLS approach to TFP estimation and perform robustness checks where we estimate TFP with GMM. Results are very similar. In unreported investigations, we also tested our results using a [START_REF] Levinsohn | Estimating Production Functions Using Inputs to Control for Unobservables[END_REF] approach for the estimation of firms' TFP. Again, results (available upon request) are qualitatively robust. Figure 1: LPS firms and evolution of OLS TFP

A graphical exploration

We start with a graphical analysis of the evolution of productivity differential between LPS and non-LPS firms. We estimate the following four regressions:

tfp it = 5 j=-2 α j lps in ijt + d t + it (2) tfp it = fe z + 5 j=-2 α j lps in ijt + d t + it (3) tfp it = fe z + fe s + 5 j=-2 α j lps in ijt + d t + it (4) tfp it = fe i + 5 j=-2 α j lps in ijt + d t + it (5)
where lps in ijt equals 1 if j years separate time t from the moment when firm i will become (resp. has become) a LPS firm. The first regression simply estimates the difference of productivity between LPS and non-LPS firms according to the number of years which separate the LPS firm from the reception of the subsidy. We then add fixed effects with increasing levels of detail: region z, then region z/sector s, and finally firm-level i. Only the last regression actually yields a difference-in-difference estimator of the LPS effect. The four sets of results are presented in panels (a) to (d) of figure 1.

We first perform the estimation on the whole sample. The grey zone on each panel corresponds to the 5% confidence interval. According to the first estimation, without any control in panel (a), LPS firms are not significantly different from the others two years before their entry in a LPS, but a negative and significant productivity gap grows over time between both types of firms. With départements controls in panel (b), LPS firms appear more productive than the others before their entry in a LPS; this suggests that LPS firms are located in less productive départements. Nevertheless, LPS firms still seem to be on a declining path in terms of productivity, even though the differential with non-LPS firms for a given year is never significant at the 5% level. Those results clearly show that LPS firms are on a different temporal trend than the others. When firm fixed effects are controlled for in panel (d), the declining path seems to be stopped during two years after the entry in the LPS; but then, the decline with respect to other firms starts again.

Figure 2 presents the same results for single-plant firms. Indeed, the LPS policy is supposed to help firms better coordinate their strategies with firms nearby and more generally to enable firms to benefit more from the network of firms in the region. Multi-plant firms, which are also typically bigger, may be less dependent on their local environment and therefore respond less to the LPS policy. Moreover, and maybe more importantly, we do not have the information on the LPS status at the plant level. Hence, for multi-plant firms, the effect of the policy may be both weaker and mis-measured. Hence, we analyze the case of single plant firms (304 LPS firms) which do not suffer from those problems. Comments are roughly the same, except that with firm fixed effects, the declining path is not only stopped but reversed into a rising pattern; two years after the entry in a LPS, single plant firms are 3.4% more productive than the others (significant at 10%). But it is a very short run effect.

We now turn to proper difference-in-difference econometric analysis to investigate the robustness of those first results more systematically. In table 4, the simple OLS regression shows that the LPS firms are not "structurally" different from the others (with a negative but insignificant coefficient on the LPS dummy). However, they experience a negative and very significant productivity drop once they are in a LPS. When we introduce industry fixed effects, the coefficient of the variable "Being in a LPS" increases from -0.071 to -0.049, which suggests that LPS firms belong to declining industries. Interestingly, once départements fixed effects are taken into account (regression (3)), the coefficient on "LPS firms" becomes positive and significant; again this means that LPS firms are located in less productive départements. Nevertheless, "Being in a LPS" still has a negative and significant coefficient which persists when we introduce firm fixed effects in regression (6), though closer to zero. There are several possible interpretations of this rather pessimistic result on the LPS policy. One is that the LPS policy causes this negative effect. It is possible that the firms that receive the LPS label become more receptive to public pressure to postpone workers layouts. In this interpretation, firms may choose to forego labor saving productivity improvements. In section 3.2, we present results on employment which are consistent with this interpretation. Another interpretation -not exclusive of the first one -is that firms that enter a LPS do it when they face difficulties: η it and lps in it are certainly correlated and there would consequently be a simultaneity bias in the estimation of the causal impact of the policy on firms' TFP. The graphical analysis tends to corroborate that idea since even with firm and year fixed effects, LPS firms still appear different from the others before the implementation of the policy; it should not be the case in the absence of specific temporal trends. We try to address this issue in subsection 3.1.2. Results on single plant firms are presented in table 5. They confirm our main conclusions but some subtle differences emerge: LPS single-plant firms tend to be located in less productive Figure 2: LPS single-plant firms and evolution of OLS TFP départements and once they enter the LPS policy, their productivity stagnates. Overall, however, the LPS policy appears less negative or even slightly positive for the single plant firms, as we saw with the graphical analysis. But it is a very short effect and with the GMM TFP index, this positive effect cannot be detected.

LPS and temporal endogeneity

We saw that the entry into a LPS coincides with a decline in productivity and that this decline could be explained by sectoral and geographical determinants. The existence of such a correlation biases the estimation of the "causal" impact of the policy. We resort to two different strategies:

• we introduce département-time and industry-time fixed effects in order to purge the estimation from all the geographic and sectoral shocks, common to all firms over the period.

• we saw in subsection 2.4 that there was a clear selection of LPS firms on observable characteristics. If those characteristics are also correlated with the evolution of firms' TFP, we can improve the estimation of LPS policy impact by using a matching strategy.

Using the last regression of table 3 in subsection 2.4, we compute the probability for all firms to belong to a LPS. Note that in this regression, all the firms in industries or départements which are not represented in the LPS have already been eliminated . To comply more confidently to the common support condition of the matching approach, we drop from the sample those firms which have a probability above the 99th percentile for non-LPS firms and below the 5th percentile of LPS ones. 238 LPS firms remain in the sample, from which 178 are single plant firms.

In table 6, we see that for the whole sample, the introduction of département-time fixed effects does not affect the results with respect to a simple départements fixed effects estimation. On the contrary, the observed negative drop is larger with industry-time fixed effects than with industry fixed effects. This confirms that LPS firms operate in declining sectors. The LPS policy would have a static redistributive dimension (towards structurally less productive départements) and a more dynamic one (towards declining sectors). For single plant firms (table 7), as soon as industry-time fixed effects are controlled for, the drop after the entry in the LPS is not significant any more. Matching approach on both samples tends to reduce or cancel that drop too (tables 8 and 9). Our main results remain nevertheless unchanged: the LPS policy does not succeed in reversing the negative trend observed with simple OLS estimation.

Further issues

A possible defense of the LPS policy is that the absence of a measurable effect comes from the small size of the monetary subsidy involved. In other words, the policy is good but should receive more funds. To test this idea, we use for all single-plant firms9 involved in a LPS the information on the amount of the subsidy perceived by the LPS they belong to. The average subsidy for this group is 45,000 euros. In table 10, the significantly negative coefficient on the amount of the subsidy suggests that the strongest monetary support goes to weaker firms. This negative coefficient cannot be interpreted in causal terms since the subsidy variable has no impact once individual fixed-effect are introduced. We also tested the policy could have a different impact on firms of different size, on firms belonging to LPS of different size on firms belonging to LPS with different governance structure: no significant heterogeneity could be detected.

We also tested whether the LPS policy had an effect on the size of the clusters they targeted. Table 11 shows that LPS firms belong to pre-existing clusters: the number of employees from the same industry in the other firms of the département (the left hand side variable in the regressions of this table) is much higher for LPS firms once département fixed effects have been introduced. However, there is no indication that the cluster policy was attractive to other firms of the same sector. If anything, the years the LPS are implemented are years during which the size of the cluster to which these firms belong decreases. Since [START_REF] Martin | The dynamics of productivity in the telecommunications equipment industry[END_REF] have shown that the size of clusters has a positive impact on French firms productivity, this result may partly explain why we do not find productivity gains for LPS firms.

LPS and firms' labour demand

Up to now, the LPS policy, in spite of the official discourse presenting it as a clear break with policies in favor of regions and industries in difficulty, appears clearly as a defensive policy. If political economy factors are at the origin of the gap between the stated objectives and what we measure, we may be missing all the action when looking at the effect of the policy on productivity. The most important objective for national and local policy makers involved in the policy may in fact be employment of these firms. Preserving jobs rather than increasing productivity may have been the real objective. This is what [START_REF] Criscuolo | The effect of industrial policy on corporate performance: Evidence from Panel data[END_REF] conclude from the study of Regional Selective Assistance in the UK.

To look at this, we adopt the same strategy as for productivity and start with graphical analysis. It appears on figures 3 and 4 that LPS firms are "structurally" bigger than the others. Once individual fixed effects have been taken into account, LPS firms still appear bigger than the others. But they do before and after their entry in a LPS, without any clear change in the pattern of their size, so that it is difficult to identify a specific role of the policy.

We then concentrate on the econometric analysis. We regress firms' current employment Figure 3: LPS firms and evolution of employment on the two variables "LPS firm" and "Being in a LPS". We develop in appendix a more structural approach, which yields similar results. LPS firms are significantly bigger employers than the others, something we know from subsection 2.4. Interestingly, tables 12 and 13 for all firms and single plant firms respectively, show that LPS firms perform better in terms of employment once they are in the LPS. This is so once we control for firm fixed effects. The effect on employment is however quite modest, not specific to the period post-subsidy as suggested by the graphical analysis, and disappears completely with the matching estimations (tables 14 and 15).

Figure 4: LPS single-plant firms and evolution of employment

Robustness checks

We have conducted so far our analysis at the firm level. Two issues arise about this methodological choice:

1. Proponents of cluster policies often claim that these policies do not only affect the firms directly targeted but the whole sector in the region. In the presence of this type of externality, the estimation of the LPS policy at the firm level may underestimate its true economic impact.

2. There is possible measurement error in our sample of LPS firms: in our survey, it is possible that some LPS firms are identified as control firms. The reason is that we have to rely on partially incomplete information provided by managers in response to our survey.

To address both issues, we now present our analysis at the industry-département level rather than at the firm level. This allows to capture possible local spillover effects. This also reduces the measurement error since the geographical scale of the LPS policy is the département.

LPS and industry-départements' productivity and employment

We define the total factors productivity in industry s and département z at time t as a weighted sum of firms' TFP:

TFP szt = emp iszt emp szt × TFP iszt (6)
where emp iszt is the number of employees of firm i from industry s, in département z at time t and emp szt is the number of employees from industry s, in département z at time t.

We define an industry-département cell as being affected by the LPS policy when at least one firm from industry s and département z has been involved in LPS over the period.

Conclusions remain very similar to those obtained at firm-level except that when industrydépartement fixed effects are included, the negative productivity differential consecutive to the LPS treatment is not significant anymore10 . In unreported regressions, we use single-plant firms only to construct the indices of industry-département level productivity; all coefficients are insignificant, either with OLS index or GMM index.

From the point of view of employment growth, results are also very similar: the number of employees in LPS industry-département is structurally higher than the average and its growth is significantly higher when industry-département fixed effects are included. Once again, the positive employment differential after LPS treatment is not significant when we consider single-plant firms only.

We test heterogeneity in industry-département performance by distinguishing, as in section 2, "core" LPS industries (represented by at least 10 firms) and the others. This did not yield conclusive results.

To sum up, the analysis at a more aggregated level is consistent with the analysis at the firm level. This suggests that spillovers effects and measurement errors are not crucial.

LPS and firms' survival

In this subsection, we test the hypothesis that the LPS policy may have affected the probability of exit of firms. Indeed, in our political economy interpretation, this policy may have had a negative effect on productivity but may have helped firms to survive and therefore to maintain employment. We cannot test this hypothesis at the firm level because most of LPS managers gave us only the list of LPS firms still in activity in 2006. Hence, we cannot identify LPS firms which disappeared before our survey. This is why we conduct the analysis at a more aggregate level. For each industrydépartement, we compute the share of firms present in the sample in 1996 which are still alive in 2004, so that we have one observation per industry-département. The first regression shows that industry-départements targeted by the LPS policy are characterized by a higher survival rate between 1996 and 2004. We then control for the level of productivity and the average size of firms in the industry-département, which both affect positively the share of surviving firms between 1996 and 2004. It is well known that bigger and more productive firms are less likely to die (see for example [START_REF] Alvarez | Multinationals and Plant Exit: Evidence from Chile[END_REF]).

When controlling for industry and département fixed effects separately, the LPS policy appears associated with a higher survival rate; when both types of fixed effects are added, the coefficient on LPS policy is positive but not significant. We ran the same regressions for single-plant firms only and the coefficient on LPS industry-département remains positive and significant even with sectoral and geographic fixed effects.

These results suggest that the LPS policy is associated with higher survival rates of firms. Even if we cannot interpret these regressions as causal, they are again consistent with our political economy interpretation of the policy: targeted industry-départements were in decline in productivity terms but maintained employment both because employment increased at the firm level and because less firms disappeared. Contrary to the official statements, the objective and may be the impact of the LPS policy was employment and not productivity.

Conclusion

Our results on the first cluster policy implemented in France are not very positive. First, the policy targeted firms in regions and sectors that were experiencing difficult times in terms of productivity and therefore competitiveness. This was not its official objective and we can interpret the gap between the stated and revealed objectives results in political economy terms. The administration in charge of the policy, the DIACT, formerly the DATAR, was created to promote territorial equity and to help lagging regions. It appears that it was not able or willing to change in practice and our results point to bureaucratic continuity. Another possible interpretation of the gap between stated and revealed objectives is that the policy was captured by firms. Second, the policy did not succeed in reversing the relative decline in productivity for the targeted firms. We can only detect a very weak and transitory positive effect for single-plant firms. Third, the policy had a positive, even though modest and transitory, effect on the employment of firms involved in the LPS policy. This result is consistent with a political economy interpretation of our results: the revealed objective of the policy was to protect the employment of big firms (LPS firms are larger than average) in declining regions and sectors. Can we say good things about this policy? One could argue that this policy may have had no effect on firm level productivity but at least was not very costly. This low price tag does not apply to a more recent and ambitious cluster policy implemented in France, called competitiveness clusters, which cost 1,5 billion euros.

To our knowledge, our study is the first to analyze empirically, with firm level data, the impact of a cluster policy. It points to the apparent failure of the LPS policy to improve the performance of targeted firms through better cooperation and to increase the attractiveness of existing clusters. Obviously, our results cannot be generalized to other cluster policies which may have performed better. However, we interpret it as a cautionary tale for policy makers intending to commit large amounts of public money to such policies. The estimation of TFP

APPENDIX

To calculate firms' TFP, we estimate a production function. We use a Cobb-Douglas framework and we suppose that the value-added of firm i at time t, Y it , is:

Y it = A it K α it L β it (7)
where K it and L it are respectively the capital and the employees of the firm. After a log-transformation, the model we will estimate is:

y it = αk it + βl it + it (8)
The estimation of such a production function is not trivial. Indeed, some unobserved characteristics can both affect the amount of inputs and the level of output. If the entrepreneur is less risk-averse than the others, he might tend to adopt a particular labor-capital mix, he might have different innovation strategies and also might tend to seek less risky (and potentially less lucrative) markets. On the other hand, if the entrepreneur faces a positive productivity shock, he might produce more and hire more people in the same time. Here again, the estimates of inputs-elasticities may be spurious.

An important literature has developed about the estimation of production functions. We built on [START_REF] Griliches | Production functions: the search for identification[END_REF]Mairesse (1995), Olley and[START_REF] Martin | The dynamics of productivity in the telecommunications equipment industry[END_REF] and [START_REF] Petrin | Production function estimation in Stata using inputs to control for unobservables[END_REF] to calculate three estimates of inputs elasticities. For the GMM estimation, we firstdifference all the variables and we instrument inputs by their level at time t -2. It yields reasonable coefficients, with slightly increasing returns to scale (0.86 for labour and 0.21 for capital). The Levinsohn-Petrin (LP) method, on the contrary, exhibits a decreasing returns to scale production function, with rather credible coefficients (0.70 for labour and 0.15 for capital). With the Olley-Pakes approach, we obtain abnormally low coefficient on capital (0.76 for labour and 0.04 for capital).

We retain consequently the GMM. We calculate firms' TFP as the residuals of value added, once labour and capital have been taken into account and we use that TFP index as a dependent variable to measure the impact of the LPS policy on firms' productivity. Results are very similar to those obtained with a simple OLS TFP index. The estimation of labour demand Firm-level labour demand functions are usually estimated in the literature by using a dynamic model. Following [START_REF] Girma | Creating Jobs Through Public Subsidies: An Empirical Analysis[END_REF], we estimate the following log-linearized empirical model:

l it = αl it-1 + βy it + βw it + it (9)
where l it is labour demand, y it is value added and w it is the average wage of firm i at time t. We consider that firms are price-taker for wages, which seems to be a reasonable assumption given the low degree of variability of average wage across firms (in logarithm, mean(average wage)=3.06 and sd(average wage)=0.2811 . For symmetric reasons to those mentioned about the estimation of production functions, and for technical aspects of the estimation of dynamic models, l it-1 and y it are endogenous. Here again, we consequently use a GMM approach on first-differenced variables instrumented by their level at time t -2. All the coefficients have the expected sign (the current number of employees in a firm is positively affected by past level of employment and by current level of activity and negatively affected by current average wage) and the results are coherent with the literature.

We calculate the residuals of that regression and we use them to assess the impact of the LPS policy on firms' employment, once "core" determinants of employment have been taken into account.

  before/after the public support to the LPS of years before/after the public support to the LPS of years before/after the public support to the LPS (c) (d)

  before/after the public support to the LPS (c) (d)

  before/after the public support to the LPS (c) (d)

  before/after the public support to the LPS

Table 1 :

 1 Industry level summary statistics

	Variables		Non LPS-treated indus-	Industries with less than 10	Industries with at least 10
			tries		LPS-treated firms	LPS-treated firms
					Average level in 2004
	Labour productivity	59.72		56.89	50.45
	Capitalistic intensity	103.04		82.15	57.22
	Export share		0.37		0.36	0.28
				Evolution between 1996 and 2004 (in %)
	Employees		-10.68		-9.43	-1.04
	Value added		19.30		22.60	28.96
	Labour productivity	34.54		33.10	32.64
	Exports		23.19		50.06	58.08
	Note: Labour	productivity=value	added/employees,	capitalistic	intensity=capital
	stock/employees, export share=export value/sales. Values are in thousands of real
	euros.

Table 2 :

 2 Summary statistics about firms

			LPS firms		Non LPS firms
	Variable	Obs	Mean	Std. Dev.	Obs	Mean	Std. Dev.
	Value added	2701 10357.31 29727.32 146599 6038.23	42534.43
	Employees	2701	219.70	538.03	146599 123.79	651.70
	Capital Stock	2701 14466.74 55573.65 146599 7623.53	107991.50
	Labour productivity 2701	39.83	16.41	146599	41.24	19.22

Note: Value added, capital, capital intensity, labour productivity and exports are expressed in thousands of real euros

Table 3 :

 3 LPS determinants

	Dependent Variable: lps status

Note: Robust standard errors in parentheses a , b and c respectively denoting significance at the 1%, 5% and 10% levels. All regressions are clustered at the emloyment area level. Necessarily, t ≤ lps year

Table 4 :

 4 LPS and OLS TFP

	Dependent Variable: TFP

Table 5 :

 5 LPS and OLS TFP-Single plant firms Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: ln TFP

Table 6 :

 6 LPS and OLS TFP Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: TFP

Table 7 :

 7 LPS and OLS TFP-Single plant firms

	Dependent Variable: TFP

Note: Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

Table 8 :

 8 LPS and OLS TFP-Matching

	Dependent Variable: TFP

Note: Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

Table 9 :

 9 LPS and OLS TFP-Single plant firms, matching Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: TFP

Table 10 :

 10 LPS, OLS TFP and Subsidy

	Dependent Variable: ln TFP

Table 11 :

 11 LPS and localization economies

			Dependent Variable:
		ln (Employees, same industry-area, other firms) iszt
	Model :	(1)	(2)	(3)
	LPS firm	0.165 c	0.531 a	0.438 a
		(0.096)	(0.107)	(0.057)
	Being in a LPS	-0.173 b	-0.135 c	-0.118 c
		(0.075)	(0.078)	(0.070)
	Year fixed effects	yes	n.a	n.a
	Département fixed effects	no	yes	yes
	Industry-year fixed effects	no	no	yes
	Observations	100937	100937	100937
	R 2	0.00	0.00	0.33

Note: Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are clustered at the firm-level in regression (1), at the industry-time level in regressions (2) and (3).

Table 12 :

 12 LPS and firms' labour demand Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: ln Employees it

Table 13 :

 13 LPS and single plant firms' labour demand

	Dependent Variable: ln Employees it

Table 14 :

 14 LPS and firms' labour demand-Matching estimations Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: ln Employees it

Table 15 :

 15 LPS and single plant firms' labour demand-Matching estimations

	Dependent Variable: ln Employees it

Table 16 :

 16 LPS and OLS TFP-Industry/Département analysis Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account autocorrelation at the industry-département level.

	Dependent Variable: TFPszt

Table 17 :

 17 LPS and number of employees-Industry/Département analysis

	Dependent Variable: Employees szt

Table 18 :

 18 LPS and firms' survival-Industry/Département analysis

	Dependent Variable: Share of surviving firms 1996-2004

Table 19 :

 19 LPS and single plant firms' survival-Industry/Département analysis

	Dependent Variable: Share of surviving firms 1996-2004

Table 20 :

 20 LPS determinants-GMM TFP

	Dependent Variable: lps status

Note: Robust standard errors in parentheses a , b and c respectively denoting significance at the 1%, 5% and 10% levels. All regressions are clustered at the emloyment area level. Necessarily, t ≤ lps year

Table 21 :

 21 LPS and regional policiesNote: Standard errors in parentheses a , b and c respectively denoting significance at the 1%, 5% and 10% levels.

		LPS firm	PAT in the	Share of subsidized emp.
			dép. 2000-2006	in the dép. 2006
	LPS firm PAT in the dép. 2000-2006 Share of subsidized employment in the dép. 2006	1 0.0436 a 0.0255 a	1 0.5215 a	1

Table 22 :

 22 Summary statistics about single plant firms

			LPS firms		Non LPS firms
	Variable	Obs	Mean	Std. Dev.	Obs	Mean	Std. Dev.
	Value added	1837 3430.65	7931.83	102338 2830.81	7149.24
	Employees	1837	86.90	185.10	102338	68.21	131.19
	Capital Stock	1837 5839.06 33721.62 102338 3257.03	16707.36
	Labour productivity 1837	38.18	15.46	102338	39.88	18.34
	Note: Value added, capital, capital intensity, labour productivity and exports are expressed in
	thousands of real euros

Table 23 :

 23 LPS and GMM TFP Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: TFP

Table 24 :

 24 LPS and GMM TFP-Single plant firms Standard errors in parentheses. a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are corrected to take into account individual autocorrelation.

	Dependent Variable: ln TFP

Table 25 :

 25 LPS and GMM TFP

	Dependent Variable: TFP

Table 26 :

 26 LPS and GMM TFP-Single plant firms

	Dependent Variable: TFP

Table 29 :

 29 Labour demand Standard errors in parentheses a , b and c respectively denoting significance at the 1%, 5% and 10% levels. Standard errors are clustered at firm level.

		Dependent Variable: ln Employees it
	Model :	(1)
	ln Employees it-1	0.555 b
		(0.016)
	ln Value added it	0.187 b
		(0.027)
	ln average wage it	-0.406 b
		(0.03)
	N	91547
	Centered R 2	0.06
	Sargan-Hansen p-value	0.88
	Note:

The estimated positive elasticities in the literature, taken literally, would suggest that larger clusters are always better. In fact, we find that productivity gains first increase then decrease (due to congestion costs) with the size of clusters, allowing us to estimate an optimal size of the cluster.

An additional indication of that spatial equity objective is that the LPS projects are relatively evenly spread out on the national territory.

Which are very similar to the French PAT.

The départements are administrative areas. Employment areas are economic entities defined on the basis of workers' commuting. There are 94 départements and 341 employment areas in continental France

Called in French "Enquêtes Annuelles d'Entreprises".

In the French 2-digit classification, we kept sectors 17 to 36, sector 23 excluded.

We also dropped outliers, dropping 1% extremes for the following variables: average work productivity, capital intensity, yearly capital growth rate, yearly employment growth rate, yearly average work productivity growth rate, yearly average capital intensity growth rate. We also had to drop the "Weapons and ammunitions" industry, which is a clear outlier.

This is confirmed by the fact that LPS firm are located in départements which receive the "Prime d'Aménagement du territoire" (PAT), one of the main instruments of regional policy in France and which have a high share of subsidized employment: see table 21 in the Appendix).

Since we do not know which plant obtained the subsidy, we concentrate on single-plant firms. These are also the firms for which a positive effect, if it exists, should be best measured. We have the necessary information for 252 firms.

When using GMM index, results at firm level and industry-département level are also very similar, even with industry-département fixed effects

Treating average wage as endogenous yields a positive and significant coefficient, which seems rather improbable.

This Discussion Paper is issued under the auspices of the Centre's research programme in INTERNATIONAL TRADE AND REGIONAL ECONOMICS and PUBLIC POLICY. Any opinions expressed here are those of the author(s) and not those of the Centre for Economic Policy Research. Research disseminated by CEPR may include views on policy, but the Centre itself takes no institutional policy positions. The Centre for Economic Policy Research was established in 1983 as an educational charity, to promote independent analysis and public discussion of open economies and the relations among them. It is pluralist and nonpartisan, bringing economic research to bear on the analysis of medium-and long-run policy questions. no yes no yes no Département-time fixed effects no no yes no no Département fixed effects no no no yes no Industry-Département-time fixed effects no no no no Industry-time fixed effects no yes no yes no Département-time fixed effects no no yes no no Département fixed effects no no no yes no Industry-Département-time fixed effects no no no no yes no no Département fixed effects no no no yes no Industry-Département-time fixed effects no no no no no yes no yes no Département-time fixed effects no no yes no no Département fixed effects no no no yes no Industry-Département-time fixed effects no no no no


Boisbunon Aurélie
email: ❢aurelie.boisbunon@inria.fr

Josiane ✶❀✷

Zerubia

ESTIMATION OF THE WEIGHT PARAMETER WITH SAEM FOR MARKED POINT PROCESSES APPLIED TO OBJECT DETECTION

Keywords: Image processing, object detection, marked point process, Stochastic EM, Stochastic Approximation EM

We consider the problem of estimating one of the parameters of a marked point process, namely the tradeoff parameter between the data and prior energy terms defining the probabilit density of the process. In previous work, the Stochastic Expectation-Maximization (SEM) algorithm was used. However, SEM is well known for having bad convergence properties, which might also slow down the estimation time. Therefore, in this work, we consider an alternative to SEM: the Stochastic Approximation EM algorithm, which makes an efficient use of all the data simulated. We compare both approaches on high resolution satellite images where the objective is to detect boats in a harbor.

INTRODUCTION

Object detection is an important problem in several areas of image processing, such as monitoring of populations of animals or plants in ecology, monitoring of vehicles, boats, or cargos in highways/harbours, or detection of roads in remote sensing. When the number of objects is unknown and possibly important (several hundreds or thousands) and when the objects can be modelled by a simple parametric shape, it is interesting to use marked point processes [START_REF] Van Lieshout | Markov Point Processes And Their Applications[END_REF][START_REF] Descombes | Stochastic geometry for image analysis[END_REF]. This stochastic geometry methodology combines object-based methods with probabilistic approaches. It consists in defining the probability density of the configuration of objects from two terms: the data term using the knowledge given by the intensity of an image and assessing the quality of the estimated configuration for the image; the prior term introducing any preliminary knowledge on the shape of the objects or on the interaction between them (see Section 2). These two terms are weighted by a tradeoff parameter, which does not have a physical meaning in most cases and is usually difficult to compute analytically or numerically. Indeed, we are faced here with two issues: first, both the configuration and the parameter are unknown, resulting in an incomplete data case; second, the probability density of the configuration involves a normalizing constant that depends on the parameter, making the optimization problem harder to solve.

The problem of estimating this weight parameter has been studied rather scarsely, and very few works actually consider a valid method for that purpose, while finding a correct value is crucial for a good extraction. Among such works, Chatelain et al. (2009) [START_REF] Chatelain | Param[END_REF] and Ben Hadj et al. (2010) [4] dealt with the first issue by using the Stochastic Expectation-Maximization (SEM) algorithm developed by [START_REF] Celeux | The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem[END_REF] [START_REF] Celeux | The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem[END_REF], which consists in alternatively simulating the configuration and approximating its expected likelihood for a fixed value of the parameter, and maximizing the likelihood for a fixed configuration (see Section 3.1). The second issue was treated by replacing the likelihood by the pseudo-likelihood, whose normalizing constant is much easier to compute. However, it is well known that the SEM algorithm only converges in law, and not pointwisely [START_REF] Celeux | Stochastic versions of the EM algorithm: an experimental study in the mixture case[END_REF]. Therefore, it can lead to a prohibitive computational time for large images. This is the reason why, in this work, we investigate alternatives to SEM that have good convergence properties and can give an estimate of the parameters in a shorter time. In particular, we consider the Stochastic Approximation Expectation-Maximization (SAEM) algorithm proposed by [START_REF] Delyon | Convergence of a stochastic approximation version of the EM algorithm[END_REF] [START_REF] Delyon | Convergence of a stochastic approximation version of the EM algorithm[END_REF] (see Section 3.2). Section 4 shows numerical comparisons for the application on boat detection in harbours. Finally, we discuss in Section 5 of other possible methods that we intend to study in future works.

BASIC BACKGROUND ON MARKED POINT PROCESSES

Marked point processes add a mark on the point process representing the shape of the objects [START_REF] Van Lieshout | Markov Point Processes And Their Applications[END_REF][START_REF] Descombes | Stochastic geometry for image analysis[END_REF][START_REF] Stoyan | Stochastic geometry and its applications[END_REF]. This shape should be characterized by a few number of parameters for an easier analysis and a faster computation. The probability density of the configuration of objects is modelled by a Gibbs distribu-tion

❢ ✒ ❬❳•( ❂ ②( ❂ ✶ ❝✭✒✮ ❡①♣ ❢ ❯ ✒ ✭❳❀ ②✮❣ (1)
where ❳ is the configuration of objects, ② is the intensity of pixels in the image, ❝✭✒✮ is the normalizing constant and ❯ ✒ ✭❳❀ ②✮ is the energy of the model. The latter term itself is decomposed as

❯ ✒ ✭❳❀ ②✮ ❂ ✌ ❞ ❯ ❞ ✒ ✭❳❀ ②✮ ✰ ❯ ♣ ✒ ✭❳✮❀
where ❯ ❞ ✒ ✭❳❀ ②✮ is the data term assessing how well the estimated configuration fits the image, ❯ ♣ ✒ ✭❳✮ is the prior term including any preliminary information, and ✌ ❞ is parameter realizing the tradeoff between the two latter terms and is an element of ✒. Examples of ❯ ❞ are functions of the contrast between the intensity in the interior the object and the one in its border, measured by a distance. The prior term can take into account the specificities of the distribution of the objects in the space, such as orientation, alignment, overlapping ... Hence, the parameter ✒ includes not only the weight parameter ✌ ❞ , but also thresholds on the contrast and the measure of interactions.

In the numerical study of Section 4, we will focus on the ellipse model developed by Chatelain et al. (2009) [START_REF] Chatelain | Param[END_REF] and extended both by Ben Hadj et al. (2010) [4] and Craciun & Zerubia (2013) [START_REF] Craciun | Unsupervised marked point process model for boat extraction in harbors from high resolution optical remotely sensed images[END_REF] for boat detection in harbors. In this model, which was adapted from the one for tree detection by [START_REF] Perrin | Tree crown extraction using marked point processes[END_REF] [START_REF] Perrin | Tree crown extraction using marked point processes[END_REF], they used the Bhattacharya distance for the contrast, and Craciun & Zerubia (2013) [START_REF] Craciun | Unsupervised marked point process model for boat extraction in harbors from high resolution optical remotely sensed images[END_REF] considered only alignment and relaxed orientation from the former model in [START_REF] Chatelain | Param[END_REF]4], while using a more sophisticated model of ellipses.

ESTIMATION OF THE WEIGHT PARAMETER

The estimation of ✌ ❞ can be described by the following opti-

mization problem ♠❛① ✌✷| ✄ ✰ ❢• ✌ ✭❳❀ ②✮ ❂ ❢ ✒ ❬❳•( ❂ ②(❣❀ (2)
which corresponds to the maximization of the likelihood, that is, we look for the value of ✌ ❞ yielding the most probable configuration. As mentioned in the introduction, this problem is hard to solve for two reasons: (i) the problem corresponds to an incomplete data case since both ❳ and ✌ ❞ are unknown and (ii) the normalizing constant ❝✭✒✮ depends on all the parameters, including the weight ✌ ❞ . Also, it belongs to the set ✭✵❀ ✶✮ so that estimation from a grid or from trial and error cannot be efficient. Up to now, the answer to both problems has been given by the Stochastic EM algorithm applied to the pseudo-likelihood, which we develop in the next paragraph.

Note that the other parameters (thresholds on contrast, orientation, or weight between the different terms involved in the prior) have been chosen by trial and error sofar, but will also require full consideration in future works.

Pseudo-likelihood and Stochastic EM

The Stochastic Expectation-Maximization (SEM) algorithm was designed by [START_REF] Celeux | The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem[END_REF] [START_REF] Celeux | The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem[END_REF] as an alternative to EM for cases where the exact expected likelihood can be difficult to compute, either analytically or numerically. The expectation step is thus replaced by a simulation step followed by an approximation of the expectation based on the simulated data.

The algorithm cannot be applied directly here, since the normalizing constant in the likelihood is itself hard to compute. Therefore, Chatelain et al. (2009) [START_REF] Chatelain | Param[END_REF] proposed to replace it by the pseudo-likelihood as was done by Besag (1975) [START_REF] Besag | Statistical analysis of non-lattice data[END_REF]

P ▲ ✒ ✭①❀ ②✮ ❂ ❡①♣ ✚ ) ❳✂❑ ✕ ✒ ✭✉❀ ①❀ ②✮✄✭❞✉✮ ✛ ✂ ✭ ( ①✐✷① ✕ ✒ ✭① ✐ ❀ ①❀ ②✮ ✮ (3)
where ✕ ✒ ✭✉❀ ①❀ ②✮ ❂ ☞ ❡①♣❢ ✌ ❞ ❯ ❞ ✭✉✮ ❯ ♣ ✭✉✮❣ is the extended Papangelou intensity for one object and ✄ is the Poisson distribution. In other words, the pseudo-likelihood considers all objects to be independent. Note that the first term in Equation ( 3) represents the normalizaing constant of the pseudo-likelihood and its computation can be done numerically much more easily than the likelihood in Equation (1). More details can be found in [4].

The SEM algorithm hence gives an approximation of the maximum of pseudo-likelihood (MPL) estimator, as described in Algorithm 1. However, the SEM algorithm presents a major issue: it does not converge pointwisely, as mentioned by [START_REF] Celeux | Stochastic versions of the EM algorithm: an experimental study in the mixture case[END_REF] [START_REF] Celeux | Stochastic versions of the EM algorithm: an experimental study in the mixture case[END_REF].

Algorithm 1 SEM algorithm for Problem (2) Inputs -Initial value ✌ ✭✵✮ , image ② ❦ ❂ ✵ repeat (S) Simulate ❳ ✭❦✮ ✘ • ✌ ✭❦✮ ✭❳❀ ②✮ (E) Compute an approximation of the expected pseudo- likelihood ) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ ❂ •♦❣ P ▲ ✒ ✭① ✭❦✮ ❀ ②✮ (M) Update ✌ with ✌ ✭❦✰✶✮ ❂ ❛r❣ ♠❛① ✌ ) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ ❦ ✦ ❦ ✰ ✶ until •✌ ✭❦✰✶✮ ✌ ✭❦✮ • ✔ ✎ 3.2.
Improving on the convergence of SEM

Monte Carlo EM and Simulated Annealing

In order to overcome the issues of convergence of SEM, [START_REF] Celeux | Stochastic versions of the EM algorithm: an experimental study in the mixture case[END_REF] [START_REF] Celeux | Stochastic versions of the EM algorithm: an experimental study in the mixture case[END_REF] suggested averaging over the next iterations. This amounts to running the Monte Carlo Expectation-Maximization (MCEM) algorithm for the last step. Indeed, the SEM algorithm is actually a particular case of the MCEM algorithm, developped by Wei & Tanner (1990) [START_REF] Wei | A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithms[END_REF], where several observations ① ✶ ❀ ✿ ✿ ✿ ❀ ① ▼ are generated at the simulation step and the expectation step is obtained by averaging the log-likelihood over the ▼ observations. Hence, SEM is just MCEM with ▼ ❂ ✶. However, such a strategy raises the question of how many observations should we generate in order to obtain a satisfying value of ✌ ❞ . Especially, in our context where ① is the configuration of all the objects, each observation is obtained by running of a Multiple Birth and Death (MBD) algorithm on the current configuration [START_REF] Descombes | Stochastic geometry for image analysis[END_REF]. This means we have to run enough moves of the MBD algorithm for the new configuration to be different from the current one. Hence, a single simulated observation already results from a large number of simulated moves from MBD, and the computational cost of ♠ observations can be prohibitive even for a moderate number ▼.

On the other hand, Ben Hadj et al. (2010) [START_REF] Ben Hadj | Estimation des paramètres de modèles de processus ponctuels marqués pour l'extraction d'objets en imagerie spatiale et aérienne haute résolution[END_REF] and Descombes (2013) [2, Chapter 7] considered running a Simulated Annealing algorithm after few iterations of SEM in order to reach convergence. In other words, the maximization step is modified by

✌ ✭❦✰✶✮ ❂ ✭ ❛r❣ ♠❛① ✌ ) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ if ✉ ✔ ❡ ) ◗ ✭❦✰✶✮ ) ◗ ✭❦✰✶✮ ❚ ❦ ✌ ✭❦✮ otherwise❀
where ✉ is generated from the uniform distribution on ❬✵❀ ✶(,

) ◗ ✭❦✰✶✮ ❂ ❛r❣ ♠❛① ✌ ) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ and ❚ ❦ ❂ ❚ ✵ ❂ •♦❣✭❦ ✰ ✶✮
is the temperature of the Simulated Annealing process. In the sequel, we refer to that procedure as SEM-SA. Although this ensures a better estimate of ✌ ❞ , it also increases the computation time compared to SEM.

Stochastic Approximation EM

In view of the computational limitations of the marked point process we consider, we propose to deal with another option, namely the Stochastic Approximation Expectation-Maximization (SAEM) algorithm. The SAEM algorithm was proposed by Delyon et al. (1999) [START_REF] Delyon | Convergence of a stochastic approximation version of the EM algorithm[END_REF] as alternative to the Monte Carlo and the Stochastic EM algorithms. SAEM itself differs from both algorithms in the way the expectation is approximated. Indeed, it is based on the Stochastic Approximation method introduced by Robbins & Monro (1951) [START_REF] Robbins | A stochastic approximation method[END_REF], which in our context corresponds to

) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ ❂ ✭✶ ✰ ✜ ❦ ✮ ) ◗✭✌❀ ✌ ✭❦ ✶✮ ❀ ②✮ ✰ ✜ ❦ ▼✭❦✮ ▼✭❦✮ ❳ ✐❂✶ •♦❣ P ▲ ✒ ✭① ✭❦✮ ✐ ❀ ②✮❀
where ✜ ❦ is the step size, also called forgetting factor. Hence, it keeps memory of the past simulations through a convex combination of the previous approximated expectation and the one for the new simulated data governed by the stepsize. Jank (2006) [START_REF] Jank | Implementing and diagnosing the stochastic approximation EM algorithm[END_REF] argues that this stepsize is crucial for the algorithm to have good convergence properties. In particular, he considers the choice ✜ ❦ ❂ ❦ ☛ , with ☛ ✷ ✭✵✿✺❀ ✶✮.

In our case, we consider the case where ▼✭❦✮ ❂ ✶, just as in SEM, for the reason we mentioned before about computational cost in the simulation scheme.

Inputs -Initial value ✌ ✭✵✮ , image ② ❦ ❂ ✵ repeat (S) Simulate ❳ ✭❦✮ ♠ ✘ • ✌ ✭❦✮ ✭❳❀ ②✮, ♠ ❂ ✶❀ ✿ ✿ ✿ ❀ ▼✭❦✮ (E)
Compute an approximation of the expected pseudolikelihood

) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ ❂ ✭✶ ✰ ✜ ❦ ✮ ) ◗✭✌❀ ✌ ✭❦ ✶✮ ❀ ②✮ ✰ ✜ ❦ •♦❣ P ▲ ✒ ✭① ✭❦✮ ❀ ②✮ (M) Update ✌ with ✌ ✭❦✰✶✮ ❂ ❛r❣ ♠❛① ✌ ) ◗✭✌❀ ✌ ✭❦✮ ❀ ②✮ ❦ ✦ ❦ ✰ ✶ until •✌ ✭❦✰✶✮ ✌ ✭❦✮ • ✔ ✎ 4. NUMERICAL RESULTS
We applied the SAEM algorithm to the program developed from the works in [START_REF] Chatelain | Param[END_REF], [4] and [START_REF] Craciun | Unsupervised marked point process model for boat extraction in harbors from high resolution optical remotely sensed images[END_REF] for the detection of boats on high resolution satellite images. The program mainly consists of two parts: first, the estimation of the parameters, which also give a first configuration of the objets as a result of the iterative algorithm ; then, a simulated annealing procedure for the refinement of the configuration based on the estimated parameters.

Before running the algorithm and comparing the results to those obtained with SEM, we did our own ground truth from a part of the original image, shown on Figures 1 (top).

We then run the program 50 times for different values of the forgetting factor ☛ ✷ ❬✵✿✺❀✶✮, and computed several measures of performance. The first measure is the F-score which compares the quality of the extraction obtained at the end of the program compared to the ground truth. It is defined by

F-score ❂ ✷❚P ✷❚P ✰ ❋ P ✰ ❋ ◆ ❀
where ❚ P (true positive) is the number of pixels correctly estimated as part of a boat, ❋ P is the number of pixels that have been incorrectly estimated as part of a boat, and ❋ ◆ is the number of pixels that have been incorrectly estimated as part of the background. The F-score thus gives an estimation of the detection rate: the closer to 1, the better the detection is. The second measure is the time of computation of the estimation part: the program was run on Linux with a 2.4 GHz Intel processor and 4 GB of RAM. Note that the code is currently neither optimized nor parallelized and is mostly done in static programming.

Table 1 displays the F-scores and computational time of estimation averaged over the 50 runs, along with their standard deviations, for SAEM with different values of ☛ compared with the SEM-SA procedure described in Section 3.2.1. We can first notice that SAEM does not lead to a loss in accuracy of the extraction since the difference in average detection rate with SEM-SA is less than ✶✪ in the worst case (☛ ❂ ✵✿✽), and about ✵✿✸✪ in the best case (☛ ❂ ✵✿✾). When looking at the time for computing the estimation part, we clearly see a gain of approximately ✶✸✪ for ☛ ❂ ✵✿✺ or 0.9, whose average time is a little more than 7 minutes compared to the 8 minutes necessary for SEM-SA to give an estimate of ✌ ❞ . Hence, the advantage of using SAEM with ☛ ❂ ✵✿✾ is obvious here, as a similar detection rate is obtained in a shorter time. Note however that the gain might not be as important as expected. This is certainly due to the fact that the SEM-SA procedure is not programmed to reach convergence, but instead it is set to run the simulated annealing maximization step during only 10 iterations. The gain would probably be larger if this constraint was relaxed.

DISCUSSION

This work is a first step in the study of the estimation of the weight parameter. We proposed here to use the Stochastic Approximation EM algorithm instead of the procedure currently used for boat detection, which consists in an hybrid Stochastic EM algorithm where the maximization step is performed by Simulated Annealing. In practice, we were able to decrease the computational time for the estimation of the weight parameter at no loss in accuracy of the detection, which was our purpose. However, we believe that we can decrease even more the computational time with other methods. Therefore, in future work, we intend to study and compare methods such as Bayesian methods, Quasi Monte Carlo and Genetic Algorithms.

Finally, in the long run, it is important to check the validity and performances of these methods when we estimate more than one parameter, such as the weights for the different terms involved in the prior energy, or the thresholds on the contrast and the overlapping. This direction of research is however much more difficult to perform, as it is not clear how changing one parameter affects the others and as it can considerably increase the computational time.

Fig. 1 .

 1 Fig. 1. Detection of boats on a satellite image.

  Therefore, SAEM improves on both MCEM and SEM by taking the best of both worlds: good convergence properties of the approximation obtained from only one simulated observation. It should thus be faster than the SEM-SA procedure proposed in [2, Chapter 7] at little or no cost in accuracy of extraction.

Algorithm 2 SAEM algorithm for Problem

[START_REF] Descombes | Stochastic geometry for image analysis[END_REF]

Table 1 .

 1 Detection rate (%) and estimation time (min.) for different values of the forgetting factor in SAEM. The second and fourth columns correspond to mean values, and the third and fifth ones to standard deviations, all of them over 50 runs.eter estimation for marked point processes. application to object extraction from remote sensing images.," in Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), Bonn, Germany, August 2009.[4] S. Ben Hadj, F. Chatelain, X. Descombes, and J. Zerubia, "Parameter estimation for a marked point process within a framework of multidimensional shape extraction from remote sensing images," in Proc. ISPRS Technical Commission III Symposium on Photogrammetry Computer Vision and Image Analysis (PCV), Paris, France, September 2010.

	Forget factor	Detection rate	Estimation time
	☛		(%)		(min.)
	✵✿✺	✼✵✿✵✹	✭✵✿✾✸✶✮	✼✿✷✷ ✭✸✿✷✺✮
	✵✿✻	✻✾✿✾✹	✭✶✿✷✹✺✮	✼✿✺✺	✭✷✿✼✸✮
	✵✿✼	✻✾✿✾✻	✭✵✿✼✺✼✮	✽✿✷✸	✭✸✿✶✺✮
	✵✿✽	✻✾✿✾✶	✭✷✿✺✶✷✮	✼✿✼✸	✭✸✿✻✹✮
	✵✿✾	✼✵✿✶✹ ✭✵✿✾✽✾✮ ✼✿✸✶ ✭✷✿✺✷✮
	SEM-SA	✼✵✿✹✷	✭✶✿✶✾✾✮	✽✿✶✾	✭✷✿✵✶✮

The first author would like to thank the French Spatial Agency (CNES) for the financial support of her post-doc at INRIA. Both authors thank Alain Giros from CNES for fruitful discussions.