Stephen Bazen
email: stephen.bazen@univ-amu.fr

Kadija Charni

Do Earnings Really Decline for Older Workers ?

Keywords: age-earnings profile, older workers, labour supply, cohort effects. JEL classification : J.3,

Cross section data suggest that the relationship between age and hourly earnings is an inverted-U shape. Evidence from panel data does not necessarily confirm this finding suggesting that older workers may not experience a reduction in earnings at the end of their working life. In this paper we use panel data on males for Great Britain in order to examine why the two types of data provide conflicting conclusions. Concentrating on the over 50s, several hypotheses are examined: overlapping cohorts, job tenure, job-changing, labour supply behaviour and selectivity bias. Cohort and individual fixed effects partly explain the divergent conclusions. However, for fully, year-on-year employed individuals, there is no evidence of earnings decline at the end of working life. We find no role for selectivity due to retirement, although shorter working hours or partial retirement along with job-changing late in life do provide an explanation for why hourly earnings decline for certain older workers. We find no evidence that the process of ageing itself leads to lower earnings as suggested by the cross section profile.

The labour market for older workers has attracted increased interest in recent years.

From a policy-maker's point of view, the issues of pension finance and of the costs of providing health care in the face of an aging population have raised serious questions concerning the duration of working life. Most countries with contributionbased state pension systems have already increased the minimum retirement age and taken measures to deal with the problem of financing longer periods of retirement. In contrast, participation rates among the over 50s have significantly declined with age in the last thirty years, as individuals begin to leave the labour force and enter retirement earlier 1 . For example, in Great Britain in the 2000s, the participation rate falls from over 80% among 50 to 54 year olds to around 70% for the 55 to 59 age group, and less than 50% for those aged 60 to 64. The figures are even more striking for France (around 80%, less than 60% and around 15%, respectively). There is also evidence in some countries that older workers may be subject to discrimination in recruitment and retention. Each of these mechanisms will involve lower pension contributions thereby exacerbating the financing problem.

Related to these stylised facts is the cross section observation that average earnings are lower for individuals who are at the end of their working life -or put another way, that the age-earnings profile has an inverted U shape. There are various implications for policy if individuals' earnings do in fact decline as they become older. First there could be an incentive to reduce labour supply as leisure becomes relatively less expensive. This mechanism will also interact with the prospect of increased non-labour income through different forms of pension provision which are possible in certain countries. Second, if the monthly earnings of older workers decline, pension contributions (when set as a proportion of earnings) will follow. This paper addresses the issue of whether hourly earnings do in fact decline as a consequence of ageing. In a typical earnings equation, the age-earnings profile is specified as a smooth, concave function and usually reaches a maximum during a typical individual's working life -generally in the 40 to 50 age interval (see Figure 1a). This concave relation is predicted by human capital theory and is borne out in the cross section analysis of earnings differences 2 . Investments in human capital will normally decline towards the end of a working life since they are less profitable. Furthermore, investments made at an earlier age will be depreciating and 1 There is some evidence that this may be starting to change as a consequence of recent pension reforms.

2 Murphy and Welch (1996) explore various polynomial forms for this relation and categorically reject the quadratic specification that was derived and applied by [START_REF] Mincer | Schooling, experience and earnings[END_REF], and adopted widely since.

productivity will start to diminish. The consequence is a slowdown and eventual decrease in earnings with age, as depicted in Figure 1b. Several studies at the firm level have attempted to assess the importance of productivity decline among older workers (for example [START_REF] Aubert | Age, Wage and Productivity: Does Productivity Decline at the End of Workers Career?[END_REF]Crépon, 2003 for France, and[START_REF] Van Ours | Age, Wage and Productivity in Dutch Manufacturing[END_REF]Stoeldraijer, 2010 for the Netherlands).

Studies that use panel data on individuals however find that earnings do not necessarily decline with age at the individual level (see for example, [START_REF] Johnson | Wage Declines among Older Men[END_REF] for the US and Myck, 2010 for Great Britain and Germany). This is at odds with the predictions of human capital theory; productivity may not decline at the end of an individual's working life (see for example, Haegeland and Klette (1999)). This outcome is consistent with theories of incentive contracts which show that wage growth is possible in absence of human capital growth. For instance, Lazear's (1981) model or deferred compensation contract predicts that the earnings increase over a career with the same employer. Older workers receive higher wages in order to address the problem of moral hazard and imperfect information. In this model, firms use seniority wages to motivate workers to supply a higher level of effort and to avoid leaving the company as retirement approaches. Elsewhere, empirical studies suggest that older workers are appreciated in the world of work for their professional skills, experience and dedication. Several studies indicate that non cognitive or social skills are as important as cognitive skills and are associated with wage premia (Heckman and Rubinstein (2001), Postlewaite and Silverman (2006)). An individual's personality (behaviour, motivation, sociability, leadership) is valued by employers and contributes to raising individual earnings. All these conflicting elements leave open the question of how the cross-section regularity can be explained.

An alternative explanation may be related to the dynamics of the working population. Each cohort may have a monotonic upward-sloping earnings profile but overtakes the preceding cohort. A cross section will pick up members of different cohorts at each age rather trace the life-cycle earnings profile of a typical individual (as in Figure 1c). A further possibility is that earnings fall for a subset of older workers who change employer or job status in the latter part of their working life.

This may be planned and therefore voluntary, as suggested by [START_REF] Casanova | Wage and Earning Profiles at Older Ages[END_REF], or involuntary if firms use age as a criterion for lay-offs and displaced older workers subsequently take lower paid jobs. Finally, given the substantial decline in the participation rate from the age of 50 onwards noted above, there may be a selectivity bias in earnings regressions if those leaving the labour force are also those who tend to have higher earnings potential (see Figure 1d).

In this paper we explore these different possibilities using panel data on males for Great Britain. The first section describes the data source used and the definitions of earnings that are used in the subsequent analysis. The following section contains a descriptive analysis of earnings profiles by age and by cohort. The third section exploits the panel dimension of the data and looks at the age-earnings profile using a fixed effects specification. We examine the role of tenure, job mobility and partial retirement as possible explanations of the observed downturn in hourly earnings for older workers in cross section data. The penultimate section assesses the role of early retirement and other selection mechanisms that may affect the shape of the ageearnings profile, while the final section draws together our conclusions.

Earnings and the British Household Panel Survey

The data used are taken from the British Household Panel Survey (BHPS) for the period 1991 to 2007. We are primarily interested in older workers who are defined as those aged 50 or over, and so the main sample used are males born between 1936 and 1956. Each individual born between these two years will have been observed at some point or all their life in the survey while they are in the older workers category (aged 50 or over). The sample is truncated at the age of 65. Since this sample of the panel is not balanced, the numbers of individuals in each age interval will differ according to the cohorts that are followed through the period 1991 to 2007. An idea of the variability of the sample numbers is presented in Table 1 for all cohorts of males of working age. At the beginning of the panel survey in 1991, some 3,260 males are aged between 16 and 65, and this number declines over the period of the operation of the survey to around 1,860. Exits exceed entrants and the sample size diminishes.

However, given the dynamics of the population, the size of main sample used, which is the 50 to 65 age group is more stable and varies between 600 and 800 per year.

In cross section analysis, a snapshot is taken of an individual's labour market situation at particular point in his life-cycle. A significant proportion of individuals will not declare current earnings due to non participation or non employment, and this sub-group may not have the same observed or unobserved characteristics as the overall population of working age. In a panel survey, individuals may move between different types of labour status, and therefore will not provide data for a continuous earnings profile. Furthermore, many individuals are unable to declare an hourly wage or monthly earnings at the time of interview, but can declare earnings for the previous twelve month period. As in many surveys, respondents in the BHPS are asked to specify usual earnings.

Three concepts of gross hourly earnings 3 with variable response rates are used:

(a) current hourly earnings (b) usual hourly earnings if these differ from current earnings (c) hourly earnings imputed from annual earnings Since earnings are being compared over a long time span, the figures used are converted into real terms using a price deflator. The latter indicates that prices increased by 36% between 1991 and 2007, at an average rate of 1.85% per annum.

There is an interesting issue concerning education level in this context. The older section of labour force in the year 2000 will contain persons born before 1950 (figures by cohort are given in Table 2). At this time, and until the 1960s, only a minority of individuals would remain in education after the age of 16. The minimum school leaving age was 14 and the options for post-school training were relatively limited for these individuals. For those born prior to 1940, a majority had no formal qualifications, and less than 8% went to university. Of those born between 1950 and 1955, only a fifth had no formal educational qualifications, while 62% finished education with GCSE O-or A-levels, with 16% going to university. It therefore is difficult to examine differences in earnings between cohorts for a given education level across a period in which qualifications and access to education changed so dramatically.

Cross section and cohort age-earnings profiles

Figure 3 shows the cross sectional hourly earnings profiles for 1991 and 2007. The age-average hourly earnings relation in both cases has the typical concave inverted U-shape found in cross section data. Average earnings of those aged over 60 are around those earned by those in the 25 to 30 year age group. In the final year, 2007,

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In each case the respondent is asked to declare the amount earned and the corresponding number of hours worked. Hourly earnings are obtained by dividing total earnings by total hours declared. there appears to be more variation in the 40 to 50 age interval. For 1991, the maximum is situated on a plateau somewhere between the ages of 30 and 50, and average earnings appear to be lower for those aged 50 and over. For 2007, higher dispersion means that the profile is not very smooth and the observed maximum actually looks like an outlier and occurs at the age of 48. However it is not clear whether earnings in the age interval 45 to 50 are always higher than in the 40 to 45 interval. Since earnings are expressed in real terms, the positions of the profiles are comparable. The introduction of the national minimum wage in 1999 will have affected the starting point for the earnings of those entering the labour market from that year onwards.

The same age profiles are presented for real annual earnings in Figure 4. Obviously annual earnings include a labour supply dimension, and do not simply reflect the price of an individual's labour services. However, in consumption terms, total earnings are the relevant variable for decision-making and in terms of the well-being it is relevant to examine total rather than hourly earnings. The annual earnings profiles fall away in a more pronounced manner than for hourly earnings showing that labour supply is an important component. This observed inverted U-shaped profile could be the consequence of aging itself, as the human capital model suggests. However, it may simply be a statistical artefact found in cross section data and result from the tendency of successive generations or cohorts to earn more on average the preceding ones. This over-taking phenomenon can in theory generate a concave inverted U-shaped age-earnings profile in cross section data. Upward-sloping cohort age-earnings profiles are reproduced in a stylised manner in Figure 1c. In a cross section each cohort is observed at a particular point in its life-cycle, and the data will take the form of a series of clusters, one per cohort. When these are used to produce the cross section profile, in the case of overtaking there will be a concave inverted U-shape while the age-earnings profile for the cohort itself may never decline.

Using the same data source as for the cross section profiles in Figure 3, the cohort age-average earnings profiles are presented in Figures 5a and5b. These bear out the notion of overtaking, and the slope of the cohort earnings profile does not in general become negative from around the age of 50 as the cross section data would suggest.

In fact there is very little evidence to suggest that the slope becomes negativeexcept for the very oldest cohorts born before 1945, and then from around the age of 55. The age-earnings profiles by cohort present a very different picture from the cross section relationship. A fuller analysis using an age-period-cohort decomposition is undertaken in [START_REF] Charni | Is it better to work when we are older? An empirical comparison between France and Great Britain[END_REF].

Individual earnings of males -cross section versus panel data

The previous section suggests that cohort effects may be a key element in explaining the inverted U-shape of the age-earnings profile observed in cross section data. In this section, individual earnings profiles for males are examined using panel data methods. This form of data also allows an individual effect to be taken into account which will contain the effect of cohort membership plus other time invariant individual-specific characteristics:

  it it i t i it u A g x w         (3)
The equation for the logarithm of earnings ( it w ) contains individual effects ( i  ), time effects ( t  ), a vector of time invariant characteristics ( i x ) and a function of age ( it A ). In what follows only the part of the profile for the group aged 50 or over is considered. The sample used here is an unbalanced panel, due to absences at the time of the survey, non-declaration of earnings due to current labour market status, early retirement and most importantly in terms of the numbers concerned, truncation at the age of 65. For example, an individual born in 1940 will no longer be in the sample from 2006 onward.

One of the first studies of the age-earnings profile using a lengthy panel survey was undertaken by [START_REF] Johnson | Wage Declines among Older Men[END_REF]. In order to allow for the most flexible form of relation between the two variables, they define a dummy variable for each year of age such that

1  ait d if individual i is aged a years or older in year t and 0  ait d otherwise
These are used in a panel data earnings equation of the following form :

it ait a A a i t i it u d x w             (4)
The individual components ( i  ) are treated as fixed (rather than random) effects and, along with the vector of time invariant characteristics ( i x ), will be swept out when estimating the parameters of interest ( a  ). The coefficient on each dummy gives the marginal change in (log) earnings at that age. The sum of these coefficients up to a given age gives the total effect of age on earnings at that point in the life cycle. Since only the part of the profile for the group aged 50 or over is considered, the constant term along with the means other explanatory variables will determine the reference for someone aged 49. In order to have larger sample numbers per cell, the dummies are defined in two year steps -50 or over, 52 or over and so forth 4 .

To begin with, the pooled estimates (that is, ignoring the individual fixed effects) are presented. These equations are estimated using the three definitions of hourly earnings mentioned above and include controls for education, marital status, family and personal background and year dummies. The results, presented in Estimating the age-earnings profile using fixed effects amounts to replacing any time-invariant influence on earnings with an individual-specific constant. This has the advantage of bringing into the model any variable or influence that affects earnings that remains unchanged over time, including factors that have not or cannot be observed. This is achieved at the price of aggregating these influences into a single coefficient and not being able to identify their separate effects. Interestingly in the current context, any cohort effects will be included in the fixed effect. The results are presented in Table 4.

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This also enables the effect of age to be identified separately from that of cohort membership or period effects common to all cohorts -see [START_REF] Browning | The Age-Period-Cohort Problem: Set Identification and Point Identification[END_REF].

As in the pooled regressions, the age-earnings profiles corrected for fixed effects still indicate that earnings decline at the end of a working life. However, this occurs a little later -from 60 onwards compared to 58 according to the pooled estimatesexcept in the case of hourly earnings imputed from annual earnings, when the decline occurs from the age of 62. The age effects on earnings are however of much smaller magnitude. The extent of the decline is 3.7% from the age of 60 onwards when we use usual hourly earnings or the definition derived from annual earnings, and 4.7% for the current hourly earnings and does not decline any further thereafter.

Thus while the fixed effect will pick up cohort effects, there is still a downturn detected in hourly earnings in the typical age-earnings profile for males. It is worth noting that this effect is only just significant at the conventional 5% level.

Tenure and job-changing

The estimated decline in earnings for older workers could be the consequence of events that occur later in life rather than the effect of ageing itself, such job-changing -voluntary or not -and labour supply behaviour. [START_REF] Johnson | Wage Declines among Older Men[END_REF] for example compare three categories of employee: career, continuous and non-retirees.

The first group are those who are in the job with the longest tenure (with the same employer) in their career. Continuous workers are those who remain in employment (even though they may change employer) throughout the sample period or until the age of 65, whichever occurs first. These individuals are able to declare earnings in each year of the survey. Non-retirees are those who may not always be employed in each year over the sample period but who do not consider themselves to be retired when not in work.

Because the fact of not working during a certain period involves a loss in labour market experience and may affect the evolution of earnings, we re-estimate the equation on a subsample of 'continuous' workers who are defined as those who remain employed over the survey. The estimates are presented in column (2) of Table 5 for usual hourly earnings and are qualitatively similar as those obtained using fixed effects in column (1) except that there is no evidence of a decline in earnings after the age of 60 (the coefficient is smaller in absolute value and not significant even at the 10% level). The decrease in earnings at older ages may therefore be result of a bias due to certain individuals who do not work during all the period (this was illustrated in a stylised fashion in Figure 1d). The earnings profile for older workers who remain employed seems to be flat.

In columns 3, 4 and 5 of Table 5, we analyse the incidence of specific human capital on earnings by distinguishing between job stayers and job movers because the wage profile would be expected to be different. Because the transition between two jobs could be involuntary, we also differentiate two forms of job transition: voluntary and involuntary mobility, where the latter is defined as being a job change involving an intervening spell of unemployment.

Stayers -the effect of tenure on earnings

There may be a link to the role played by human capital on evolution of earnings, in particular the returns of specific human capital. Since the latter can be approximated by job tenure where human capital is accumulated through work experience with the same employer, and is not transferable between firms, there is a cost to switching employers due to the loss of the returns to this type of human capital. The earnings equation for a worker can be rewritten in the following way:

it it ait a A a i t i it u T d x w               (5)
where it T is tenure with the current employer. As expected, tenure has a positive effect on usual hourly earnings (Table 5, column 3). Staying one additional year with the same employer leads to an increase of 0.6% in the real hourly wage. While there is some debate about what this effect means, it is present. One possibility is that longer job tenure increases the stock of specific human capital which entails higher productivity to the firm and leads to higher earnings. [START_REF] Abraham | Job duration, seniority and earnings[END_REF] argue that longer tenures reflect high quality matches between employer and employee, producing rents that are shared in terms of earnings premia. In this equation there is still a significant negative coefficient on the dummy for those aged 60 or over suggesting that other things equal earnings decline by 3.5%, although there is some uncertainty concerning its importance (it is not significant at the 5% level).

Movers and the consequences of job mobility

An alternative way of looking at this is to examine the earnings consequences of changing job late in life. This could constitute an explanation for the decline in earnings observed in cross-section and picked up partially in panel data. Older workers may decide to change job/employer in order to work less (part-time work or less taxing work), or possibly for jobs with higher pay. They may also be laid-off and find alternative employment with lower pay. We define job mobility as any transition from one employer to another and we want to estimate the effects of changing employer on earnings. We use the information of past employment spells to construct a job mobility variable and distinguish two cases: voluntary mobility which is characterised by direct transition between jobs, and involuntary mobility which involves a spell of unemployment between jobs.

The earnings equation in this case takes the following form:

it it it ait a A a i t i it u U E d x w          2 1        (6)
where it E is a dummy variable for voluntary job change and it U a dummy when the transition involves a spell of unemployment which is considered to be involuntary mobility. Changing employer/firm after the age of 50 appears to have negative influence on earnings growth. Workers who voluntarily decide to change jobs/employers late in life earn 10.3% less than someone who does not (column 4).

One explanation could be the loss of returns to specific human capital accumulated through work experience (and which is not transferable between firms).

Alternatively, there may be a compensating differential due to non-wage characteristics of the new job obtained. Mobility between jobs has a negative impact on hourly earnings and this transition can explain why we observe an earnings decrease at the age of 60 in the column (1). In all, if we take into account all these facts, the earnings profile for older workers is relatively flat.

Job mobility involving a spell of unemployment entails substantial negative returns to hourly earnings. The fixed effects estimates show that involuntary job separations lead to earnings losses of 29% (column 4). We have also reported the impact of length of time spent unemployed (measured in months) which replaces the dummy variable it U in equation. Thus, for example, a worker who has been unemployment for a year has hourly earnings that are 12% lower than a worker who experienced no spell of unemployment (column 5). This could be associated with the loss of human capital or lack of motivation and drive caused by a spell of unemployment. In this situation, the worker is doubly punished by the market. Firstly, the wage is negatively affected because of the loss of returns to specific human capital and secondly, by the signal sent out by the fact of being out of work, which may result in lower productivity compared to an individual who is able switch directly between two jobs. The results show that the growth of earnings is positive and increases until the age of 60. In contrast to the previous case where we do not differentiate individuals who interrupt their career from others, a small decline in earnings is observed later in life and from the age of 60, although this is not significant at the 5% level.

Labour supply decisions

Another factor could be the impact of 'partial retirement'. There is evidence that 'partial retirement' plays a role in the decline of earnings at older age especially in the United States [START_REF] Gustmann | The Effect of Partial Retirement on Wage Profiles of Older Workers[END_REF][START_REF] Johnson | Wage Declines among Older Men[END_REF][START_REF] French | The Effects of Health, Wealth, and Wages on Labour Supply and Retirement Behaviour[END_REF][START_REF] Casanova | Wage and Earning Profiles at Older Ages[END_REF]. The hourly wage of a worker who transits into partial retirement, characterised by a decrease in the number of hours worked, declines. Indeed, older workers do not necessarily go directly into retirement but instead choose to switch to part-time work in order to prepare a smooth transition into retirement and such jobs are usually lower paid than full-time positions. The sample used in the results of the basic model (column (1)) includes observations from workers who switch into part-time work. So, declines in wages could be attributable to a shift into this status rather than being due to age. To see whether the downwardsloping hourly earnings profile is determined by the transition into part-time jobs or jobs less paid just before the retirement, we introduce a variable representing 'partial retirement' for workers who transit from full-time work to part-time work 5 . The fixed effects estimates are presented in column (5).

5

As a robustness check, we estimate the model using two definitions for 'partial retirement': employees who work i) less than 35 hours per week or ii) less than 30 hours per week. The results remain similar.

The estimated coefficients imply a 26% decline in the hourly wage for workers who reduce their labour supply. This is similar to the results of [START_REF] Casanova | Wage and Earning Profiles at Older Ages[END_REF] who finds a 32% decrease in hourly wages in the United States. At the same time, none of the age dummies are significantly negative and the profile remains flat after the age of 54. The negative impact of this move into partial retirement could therefore explain the decline associated with age identified in the basic specification. Once we control for this labour supply decision, there is no clear evidence of earnings decline at the end of working life as a result of the process of aging.

Selectivity effects in panel data

A final issue to be examined is the role of selectivity. In most analyses of earnings differences, the issue of bias due to endogeneity is a key concern. In the current case, the main issue for older workers will be the consequences of withdrawing completely from the labour force. While state pensions are paid in Great Britain from a minimum defined retirement age (for males 65), there exist other forms of replacement income from occupational pension funds, capital and housing wealth.

Choices concerning retirement mean that labour supply decisions may have an impact on the estimated age-earnings profile if those with higher earnings tend to move into retirement at a faster rate, leaving a non-random sample of earners from which econometric estimates can be obtained. This is illustrated in a stylised fashion in Figure 1d. The participation issue is a key facet of the labour force behaviour of older workers, since it is high (85%) at the age of 50 and declines steadily up to the age of 59 (60%), decreases noticeably around the age of 60 (to 50%) and trends downwards through to the state retirement age of 65 (see Figure 2). This selectivity issue in relation to retirement has recently been examined for the United States in unpublished papers by [START_REF] Casanova | Wage and Earning Profiles at Older Ages[END_REF] and [START_REF] Rupert | Revisiting wage, earnings, and hours profiles[END_REF]. The approach essentially involves estimating panel data earnings equations with the inclusion of a Heckman-type selectivity correction term obtained using a first stage probit model of participation to generate inverse Mills ratios. The procedure is not as straightforward as the cross section case since with panel data, individuals may move in and out of employment or the labour force, declaring a wage one year and not another. This complicates the selectivity correction procedure as [START_REF] Wooldridge | Selection Correction for Panel Data Models Under Conditional Mean Independence Assumptions[END_REF][START_REF] Wooldridge | Econometric analysis of cross section and panel data[END_REF] explains.

In line with [START_REF] Casanova | Wage and Earning Profiles at Older Ages[END_REF] and [START_REF] Rupert | Revisiting wage, earnings, and hours profiles[END_REF], we follow the Wooldridge approach to taking into account the potential impact of selection bias 6 .

The method tests and corrects for sample selection in fixed effects models estimated using panel data. To allow for the correlation between unobserved individual effects and explanatory variables, Wooldridge makes the assumption of 'Mean Conditional Independence'. Unobserved individual effects are specified as a linear projection on the observable variables and selection bias in the presence of individual fixed-effects can be controlled for.

The empirical specification of the log earnings equation is as follows :

it i it i it Z X w         1 1 * for T t n i ,....., 1 ,...., 1   and ( 7
)
where i refers to individuals and t to time periods.

it X 1 is a vector of time-varying regressors which includes age and additional over-identifying variables, i Z 1 is a vector of time-constant regressors, i  is an unobserved individual effect and it  is the idiosyncratic error term.

The participation selection equation takes the following form: There are other approaches to selectivity see for example [START_REF] Dustmann | Selection correction in panel data models: an application to the estimation of females' wage equations[END_REF].

it it it it v Z X s    2 2 1 2 *   for T t n i ,....
of the selection equation. The equation of interest is then augmented by the IMR and is estimated using a fixed effects specification to eliminated unobserved heterogeneity and potential correlation between unobserved individual specific effects and observables characteristics. Wooldridge shows that the significance of the coefficient associated to the Inverse Mills Ratio can be used to test the presence of selection bias.

The advantage of using Wooldridge's model over some approaches is that allows for the unobserved individual effects in both equations to be correlated with the observables (which are assumed to be exogenous). This approach does not require knowledge of the distribution of the error term in the equation of interest, and allows for time heteroscedasticity and autocorrelation in the error terms in both equations.

Normality of errors in the selection equation however is assumed. To allow for unobserved individual effects, it v in the selection equation to be possibly correlated with the observable variables it X 2 , Wooldridge assumes that unobserved individual effects are a linear projection on the (within) means of the regressors. The participation selection model for the both specifications is therefore rewritten as following:

it i it it it v Z Z X s     3 2 2 1 2 *    (9)
where it it Z v 2 has a normal distribution. The selection equation also includes i Z , a time average of all time-varying regressors which captures the correlation with observables.

The exclusion restrictions ( it Z 2 ) are variables that affect the probability of participation but do not directly influence hourly earnings. The restrictions used in most studies examining the employment selection issues involve family characteristics such as the number and ages of children. However, because the focus here is on individuals approaching retirement age, these instruments are less appropriate since the presence of young children in the household is unlikely. For this reason, the instruments used are non-labour income and three dummy variables.

The first is equal to one for those with health problems, the second is equal to one if the individual lives with his spouse and the third dummy is equal to one if his spouse is in employment. These are all likely to play a role in determining labour supply of older workers but should not have direct effects on potential hourly earnings.

First stage results

The first step consists in estimating the selection equations for the both definitions of participation : (a) employed or not and (b) declare annual earnings or not. In the former case, non-employed include the unemployed (Tables 6 and7). Both selection equations contain three types of explanatory variable that affect the decisions to participate in the labour market.

The first category includes time-varying regressors which in the current context are time dummies and age. As a key determinant in the decision of retirement, the decision to leave the labour market increases with age in the two specifications. The second set of variables included in the selection equation consists of time-constant regressors. The effects of these variables, which approximate the individual specific effects, are strongly significant in both specifications. Average age has a negative effect on participation (except for those who have worked during the last 12 months but no longer do so). Average age which approximates the cohort effect indicates that older cohorts (i.e. individual with a higher average age) have a higher probability of entering in retirement than younger cohorts at a given age. Average non labour income also has a strong negative effect on the probability of participation.

The effects of education are significant except for 'gcse'. Individuals with low levels of education tend to participate more than individuals with higher education. The coefficient for an individual with no qualifications is negative and significant at the 5% level. Individuals with no qualifications tend to continue working longer. College or university educated individuals are more likely to retire than individuals with A level at given age.

The final category of explanatory variables are variables representing the exclusion restrictions (excluded from the wage equation). They are all strongly significant bar one. The marital status variable is a dummy variable which is equal to one if the individual is married or living with a partner and zero otherwise. The estimated coefficient is negative but not significant. So living in a couple or not does not influence the probability of participating in the labour market per se. Whether the spouse is employed or not affects the decision to participate, but does not directly influence an individual's hourly wage. The situation of the spouse is an exclusion restriction variable because it captures the incentive of the individual to remain employed while his spouse is employed. Not unexpectedly, having an income other than labour income increases the probability of retirement. The health problems dummy negatively affects the decision to participate. An individual who has health problems is more likely with draw from the labour force than an individual in good health. The coefficients are similar for the both specifications.

Second-stage results

The results of the second stage of the Wooldridge are presented in columns 2 of Tables 6 and7. The coefficients associated with the Inverse Mills Ratio are negative and not statistically different from zero in the both specifications. The null hypothesis of no selection bias cannot be rejected in both cases. This means that the unobserved factors that affect the participation decision do not influence the earnings determining process. The age-earnings profile for the oldest workers is not different from that for those aged 50 and the age dummies indicate that there is no evidence of a decline in earnings.

We have found therefore that after taking into account the possibility of selection, there is no evidence that the workers earnings decline at older age whichever definition of hourly earnings used. The decline observed at end of career is not due to a bias related to unobserved factors that influence decisions to leave the labour force. This evidence is consistent with the results of [START_REF] Casanova | Wage and Earning Profiles at Older Ages[END_REF] who finds no selection into retirement and suggests that the wage process for older workers is best represented by a step function.

Conclusions

Cross-section studies suggest that the age-earnings profile has a concave, inverted U shape, which implies that older workers see their earnings decrease as a consequence of ageing, as they approach retirement. In this paper we have explored a number of possible explanations for the decline in hourly earnings observed for older workers in cross section data. Using panel data for Great Britain, we find that this decline does not appear to be due to age itself. Part of the phenomenon is due to cohort effects: younger cohorts tend to earn more than their predecessors. However, changes in labour market status late in life are also contributory factors.

jobs, reducing hours of work and spending time unemployed in later life all lead to lower earnings. There is no decline in earnings due to age for employees remaining in employment with same employer. Altogether these results provide evidence that on average the age-earnings profile remains flat at older age.

We have also investigated the role of selection on the decline inearnings. We found that selection bias out of employment cannot explain the decline observed in the cross section age-earnings profile. After controlling for selection, we do not find any evidence of negative wage growth after the age 50. The inverted U-shape observed in cross section data is not the typical earnings trajectory of a worker in the later years of their working life. While this may be reassuring for policy-makers, changes in labour market status late in life are found lead to earnings reductions. While there is a voluntary, labour supply dimension in certain cases, our findings indicate that involuntary separations in later life are associated with substantially lower hourly earnings. Older workers who are laid-off may experience labour market disadvantage in terms of finding jobs at a similar level, spending time in unemployment and being forced to accept lower paid jobs. If earnings do decline at older ages for certain individuals, it is the consequence of events which are linked to age, rather than due to the process of ageing itself. 1931-35 1936-40 1941-45 1946-50 1951-55 1956-60 Source: BHPS, authors' own calculations 1956-60 1961-65 1966-70 1971-75 1976-80 1981-85 Source: BHPS, authors' own calculations

  Figure 1a

  Figure 1c

Figure

  Figure 5bCohort-specific age-hourly earnings profiles[1991][1992][1993][1994][1995][1996][1997][1998][1999][2000][2001][2002][2003][2004][2005][2006][2007]

Table 3 ,

 3 indicate that there is evidence of declining earnings at the end of an individual's

career, but only from the age of 58. The estimates based on all definitions of hourly earnings show the same phenomenon: a statistically significant decline in average earnings that begins at the age of 58 and continues to decline thereafter. This is at odds with the cross section profile since any earnings decline occurs very late in working life.

Based on current and usual hourly earnings, someone aged 58 or over will earn 4.7% less than someone aged 56; an individual aged 60 or over, more than 11% less than one aged 59, and a further 7% from the age of 62. The estimates based on dividing annual earnings by annual hours of work are less clear-cut but still show that hourly earnings decline: the size of the decline is more pronounced (7% from the age of 58) and the coefficient on the dummy for aged 62 or over is not significant at the 5% level. The pooled estimates therefore indicate declining earnings for older workers after from the age of 58 onwards.

Table 1 :

 1 Cohort numbers by year (males)

	Year

of survey Cohort (Birth year) Total 1931-35 1936-40 1941-45 1946-50 1951-55 1956-60 1961-65 1966-70 1971-75 1976-80 1981-85

	1991	267	306	378	437	409	497	469	404	440	0	0	3,261
	1992	238	276	339	383	374	450	418	337	379	67	0	3,261
	1993	219	252	308	356	340	409	389	316	350	137	0	3,076
	1994	207	241	300	357	336	391	380	298	333	195	0	3,038
	1995	198	231	281	343	314	369	367	280	314	244	0	2,941
	1996	158	227	287	347	315	368	367	292	320	330	0	3,011
	1997	111	220	278	334	311	361	360	279	303	317	63	2,937
	1998	77	214	274	323	300	355	334	264	289	294	128	2,852
	1999	33	207	268	313	296	341	331	258	272	280	178	2,777
	2000	0	199	259	306	285	331	321	250	263	262	231	2,707
	2001	0	156	250	290	274	329	313	241	263	239	281	2,636
	2002	0	121	241	282	270	319	311	232	245	227	242	2,490
	2003	0	74	236	279	265	310	287	227	226	218	232	2,354
	2004	0	34	218	273	258	302	282	217	217	206	213	2,220
	2005	0	0	217	265	251	290	266	202	213	196	206	2,106
	2006	0	0	173	255	242	283	268	198	210	190	195	2,014
	2007	0	0	119	247	232	268	253	194	203	173	176	1,865

Table 2 :

 2 Summary statistics by cohort group (males)

	Variable

Cohort (Birth year) 1931-35 1936-40 1941-45 1946-50 1951-55 1956-60 1961-65 1966-70 1971-75 1976-80 1981-85

	Age	60.6	57.9	54.9	50.4	45.3	40.2	35.2	30.1	24.8	21.6	19.6
		(0.058) (0.069) (0.073) (0.069) (0.071) (0.066) (0.067) (0.075) (0.072) (0.067) (0.058)
	Married	0.860	0.857	0.863	0.848	0.786	0.842	0.757	0.643	0.367	0.198	0.086
		(0.009) (0.006) (0.005) (0.005) (0.005) (0.004) (0.005) (0.007) (0.007) (0.006) (0.006)
	University/	0. 062	0.074	0.110	0.127	0.156	0.200	0.162	0.193	0.124	0.119	0.067
	college	(0.006) (0.005) (0.005) (0.004) (0.005) (0.005) (0.005) (0.006) (0.005) (0.005) (0.005)
	A level	0.244	0.347	0. 425 0. 418 0. 495 0. 431 0. 441 0. 462 0. 475 0. 419 0. 364
		(0.011) (0.009) (0, 007) (0, 006) (0, 007) (0, 006) (0, 006) (0, 007) (0, 007) (0, 008) (0, 010)
	Gcse	0.113	0.147	0.116	0.173	0.122	0.163	0.193	0.179	0.232	0.273	0.258
		(0.008) (0.007) (0.005) (0.005) (0.004) (0.005) (0.005) (0.005) (0.006) (0.007) (0.009)
	No qualification 0.566	0.428	0.343	0.278	0.218	0.202	0.193	0.150	0.156	0.130	0.226
		(0.013) (0.009) (0.007) (0.006) (0.006) (0.005) (0.005) (0.005) (0.005) (0.005) (0.009)

Table 4 :

 4 Fixed effects estimates of earnings equations

		Dependent variable
		(1)	(2)	(3)
		log Real usual	log Real current	log Real hourly wage
		hourly wage	hourly wage	derived from annual
				labour income
	age≥52	0.003	0.002	-0.002
		(0.015)	(0.017)	(0.016)
	age≥54	0.029*	0.007	0.004
		(0.016)	(0.018)	(0.016)
	age≥56	0.013	0.016	0.012
		(0.015)	(0.016)	(0.018)
	age≥58	-0.000	0.004	-0.025
		(0.018)	(0.017)	(0.026)
	age≥60	-0.037**	-0.047**	0.004
		(0.019)	(0.022)	(0.035)
	age≥62	-0.019	-0.032	-0.063**
		(0.020)	(0.023)	(0.029)
	Observations	6323	5381	6377
	R 2	0.021	0.020	0.014

Note: Time dummies included. Standard errors, in brackets, are corrected for clustering, * indicates significance at 10%, ** 5%, *** at 1%.

Table 5 :

 5 Movers and stayersNote: Standard errors, in brackets, are corrected for clustering * indicates significance at 10%, ** 5%, *** at 1%. All regressions include time dummies

			Dependent variable Log Usual hourly wage
		(1)	(2)	(3)	(4)	(5)	(6)
		All	'Continuous'	'Stayers' 'Movers' 7 'Movers' 8 'Partially
		workers	workers				retired'
	age≥52	0.003	0.006	0.002	0.008	0.005	0.008
		(0.015)	(0.016)	(0.015)	(0.015)	(0.015)	(0.015)
	age≥54	0.029*	0.038**	0.031**	0.029*	0.031**	0.032**
		(0.016)	(0.018)	(0.016)	(0.015)	(0.016)	(0.016)
	age≥56	0.013	0.013	0.013	0.011	0.011	0.012
		(0.015)	(0.016)	(0.015)	(0.014)	(0.015)	(0.015)
	age≥58	-0.000	-0.024	-0.004	0.002	0.001	0.002
		(0.018)	(0.019)	(0.017)	(0.017)	(0.018)	(0.017)
	age≥60	-0.037**	-0.008	-0.035*	-0.036**	-0.035*	-0.024
		(0.019)	(0.021)	(0.019)	(0.018)	(0.019)	(0.019)
	age≥62	-0.019	-0.010	-0.018	-0.024	-0.021	-0.019
		(0.020)	(0.022)	(0.020)	(0.020)	(0.020)
	tenure			0.006***
				(0.001)
	voluntary				-0.098*** -0.089***
	mobility				(0.021)	(0.021)
	involuntary				-0.252*** -0.010**
	mobility				(0.041)	(0.003)
	partially						-0.231***
	retired						(0.061)
	Observations	6323	4010	6323	6323	6323	5754
	Adjusted R²	0.021	0.035	0.039	0.051	0.033	0.039

7

Involuntary mobility defined as a dummy variable 8 Involuntary mobility defined as the length of unemployment spell (in months) prior to current job

Table 6 :

 6 Wooldridge approach based on usual hourly wages

	(1) Pooled Probit	(2)Fixed Effects
	participation	Usual hourly wage

Note: Standard errors, in brackets, are corrected for clustering * p < 0.1, ** p < 0.05, *** p < 0.01 All regressions include time dummies

Table 7 :

 7 Wooldridge approach. Annual hourly wage

	(1) Pooled Probit	(2) Fixed Effects
	participation	Annual hourly wage

Note: Standard errors, in brackets, are corrected for clustering * p < 0.1, ** p < 0.05, *** p < 0.01 All regressions include time dummies

 1931-35 1936-40 1941-45 1946-50 1951-55 1956-60Source: BHPS, authors' own calculations


Pierre Héroux
email: pierre.heroux@univ-rouen.fr

Pierre Le Bodic
email: lebodic@gatech.edu

Sébastien Adam
email: sebastien.adam@univ-rouen.fr

Datasets for the Evaluation of Substitution-Tolerant Subgraph Isomorphism

Due to their representative power, structural descriptions have gained a great interest in the community working on graphics recognition. Indeed, graph based representations have successful been used for isolated symbol recognition. New challenges in this research field have focused on symbol recognition, symbol spotting or symbol based indexing of technical drawing.

When they are based on structural descriptions, these tasks can be expressed by means of a subgraph isomorphism search. Indeed, in consists in locating the instance of a pattern graph representing a symbol in a target graph representing the whole document image. However, there is a lack of publicly available datasets allowing to evaluate the performance of subgraph isomorphism approaches in presence of noisy data.

In this paper, we present three datasets that can be used to evaluate the performance of algorithms on several tasks involving subgraph isomorphism. Two of these datasets have been synthetically generated and allow to evaluate the search of a single instance of the pattern with or without perturbed labels. The third dataset corresponds to the structural description of architectural plans and allows to evaluate the search of multiple occurrences of the pattern. These datasets are made available for download. We also propose several measures to qualify each of the tasks.

I. INTRODUCTION

Graphs are data structures which have gain a great interest in the document analysis community during the last decade thanks to the computation power of nowadays computers. Indeed, the high computational complexity of algorithms which process graphs is now compensated by computer capacities. It is now possible to leverage the flexibility and the description power of this kind of data structure.

Many approaches in the literature on technical documents use region adjacency graphs in which vertices describe regions while edges express an attributed adjacency relationship [START_REF] Bodic | Symbol detection using region adjacency graphs and integer linear programming[END_REF]. In other representations, vertices is associated to primitive shapes (segments, arcs. . . ) while edges carry informations on the relative position of these shapes [START_REF] Qureshi | Spotting symbols in line drawing images using graph representations[END_REF], [START_REF] Locteau | Symbol spotting using full visibility graph representation[END_REF].

In the mean time, research on technical documents has shifted from recognition of isolated symbols to recognition in context, symbol spotting or indexing through visual words.

When using structural representations, recognize, locate or count the occurrences of a pattern symbol in an image turns into a subgraph isomorphism problem. Indeed, these tasks need to identify, locate or count the occurrences of the structural representation of the searched pattern in the structural representation of the whole document image.

As often in pattern recognition applications, noise may affect the structural representation, that is to say that there exist differences between the pattern graph and each of its searched occurrences. This implies that the subgraph isomorphism must tolerate these differences. This problem is known as errortolerant subgraph isomorphism.

The differences between the pattern and its occurrences in the target graph can be separated in two categories: 1) differences between labels may occur because the features which label vertices or edges are extracted with non sufficiently robust methods. 2) topological differences occur because of a non robust segmentation, that is to say that a region or a shape may be splitted or merged with an other one, resulting in splitted or merged vertices.

Consequently, the error-tolerant subgraph isomorphism problem may result from two distinct sources : a difference in the labeling or a difference in the topology. The substitutiontolerant subgraph isomorphism refers to the search of a subgraph isomorphism in the only presence of differences on label values, whereas topological differences are tackled by inexact subgraph isomorphism.

The communities working on graphics recognition or structural pattern analysis have always had concerns to propose databases that can be references for benchmarking or performance evaluation of an individual processing or a complete system. For example, the IAPR TC-10 and TC-15 provide several datasets among which we can cite the GREC symbol recognition contests [START_REF] Valveny | Report on the symbol recognition and spotting contest[END_REF], the IAM Graph dataset [START_REF] Riesen | Iam graph database repository for graph based pattern recognition and machine learning[END_REF]. However, there is very few data that can be used to test error-tolerant subgraph isomorphism. Moreover, to the best of our knowledge, those existing [START_REF] Foggia | A database of graphs for isomorphism and sub-graph isomorphism benchmarking[END_REF] only contain graphs with nominal labels or no label at all. Hence, there is a need for datasets for the evaluation of error-tolerant subgraph isomorphism methods which process graphs labelled with continuous values.

In this paper, we present several ground-truthed databases that we make available. These datasets are intended to evaluate methods for substitution-tolerant subgraph isomorphism when graphs are labeled with continuous values. We also propose the definition of performance measures to numerically qualify the detection of subgraphs. This paper is structured as follows. The problem statement and main definitions are presented in section II. Section III presents three datasets which are made available and the way the ground-truth is defined. Section IV proposes measures to evaluate the performance of subgraph isomorphism in the context of spotting and counting occurrences of pattern graphs. Finally, section V concludes the paper and draws some perspective to continue this initiative.

II. SUBSTITUTION-TOLERANT

SUBGRAPH ISOMORPHISM Definition 1. A directed attributed multigraph 1 G is a 4- tuple G = (V G , E G , µ G , ξ G ) where V G is the set of vertices of G, E G is a multiset of ordered pairs e = (v 1 , v 2 ) with v 1 ∈ V G and v 2 ∈ V G , i.e. edges of G. µ G : V G → L V is
a function assigning a label to a vertex, L V being the set of possible labels for vertices. ξ G : E G → L E is a function assigning a label to an edge, L E being the set of possible labels for edges.

Definition 2. Given a graph G = (V G , E G , µ G , ξ G ), a subgraph of G is a graph G = (V G , E G , µ G , ξ G ) such that V G ⊆ V , E G ⊆ E, ∀e = (v 1 , v 2 ) ∈ E G , v 1 ∈ V G , v 2 ∈ V G and µ G and ξ G are the restrictions of µ G and ξ G to V G and E G , i.e. µ G (v) = µ G (v) and ξ G (e) = ξ G (e) Definition 3. An injective function f : V S → V G is a subgraph isomorphism from a graph S = (V S , E S , µ S , ξ S ) to a graph G = (V G , E G , µ G , ξ G ) if there exists a subgraph G of G such that f is a graph isomorphism from S to G : • ∀v ∈ V G , f (v) = v ∈ V G , f -1 (v ) = v • for all e = (v 1 , v 2 ) ∈ E G , there exists a distinct edge e = (f (v 1 ), f (v 2 )) ∈ E G
Note that extra edges may exist in G between mapped vertices, i.e. a subgraph does not need to be induced.

In its exact formulation, the subgraph isomorphism must preserve the labelling, i.e. µ(v) = µ (v ) and ξ(e) = ξ (e ). In pattern recognition applications, where vertices and edges are labeled with measures which may be affected by noise, a substitution-tolerant formulation which allows differences between labels of mapped vertices and edges is mandatory. However, in order to take into account these differences, they are penalized by a non decreasing cost function. Finally, the total cost associated to the mapping between a graph

S = (V S , E S , µ S , ξ S ) and a subgraph of G = (V G , E G , µ G , ξ G )
is given by eq. ( 1).

C M (S, G) = i∈V S k∈V G c V (i, k) * x i,k + ij∈E S kl∈E G c E (ij, kl) * y ij,kl (1)
In this equation, c V (i, k) and c E (ij, kl) respectively denote the elementary cost for mapping a vertex i ∈ V S to a vertex k ∈ V G and the cost for mapping an edge ij ∈ E S to an edge kl ∈ E G . M represents a possible isomorphism f between S = (V S , E S , µ S , ξ S ) and a subgraph of

G = (V G , E G , µ G , ξ G ) as a set of binary variables x i,k and y ij,kl . x i,k is set to 1 if f (i) = k and equals 0 otherwise. Similarly, y ij,kl is set to 1 if ij ∈ V G is mapped to kl ∈ V S and is set to 0 otherwise.
Moreover, the binary variables in M must respect the following constraints in order to ensure that f is an isomorphism.

• Every vertex of V S must be matched to a unique vertex of V G :

k∈V G x i,k = 1 ∀i ∈ V S (2)
• Every edge of E S must be matched to a unique edge of E G :

kl∈E G y ij,kl = 1 ∀ij ∈ E S (3)
• Every vertex of V G must be matched to at most a vertex of E S :

i∈V S x i,k ≤ 1 ∀k ∈ V G (4)
• If two vertices are matched together, an edge originating the vertex of S must be matched with an edge originating the vertex of G:

kl∈E G y ij,kl = x i,k ∀k ∈ V G , ∀ij ∈ E S (5)
• If two vertices are matched together, an edge targeting the vertex of S must be matched with an edge targeting the vertex of G:

kl∈E G y ij,kl = x j,l ∀l ∈ V G , ∀ij ∈ E S (6)

III. DATASETS

In this section, we present the three datasets that are made available at http://litis-ilpiso.univ-rouen.fr2 . These datasets have been used to evaluate the substitution-tolerant subgraph isomorphism approach described in [START_REF] Bodic | An integer linear program for substitution-tolerant subgraph isomorphism and its use for symbol spotting in technical drawings[END_REF].

A. Exact synthetic dataset

The ILPIso_exact_synth dataset is a synthetic dataset which provides 180 pattern-target graph couples. The graph couples have been synthetically generated according to the following procedure. First, a random graph S is generated according to the Erdös-Rényi model [START_REF] Erdös | On random graphs[END_REF] whose parameters are n S , the number of vertices and p which is the probability that a directed edge between two distinct vertices exists. Vertices and edges are labeled with a random numerical value according a uniform probability distribution in [-100, 100]. Then, a graph G 0 is created as an exact copy of S. Finally, G 0 is completed to form a graph G with vertex and edge insertions (with the same random model for labels) according the Erdös-Rényi model until its size is n G .

The following parameters have been chosen:

• Size of G : |V G | = n G ∈ {50, 100, 250, 500} • Size of S : |V S | = n S ∈ {10, 25, 50}
• Probability that an edge connects two vertices : p ∈ {0.01, 0.05, 0.1}

The ILPIso_exact_synth dataset is composed of five instances of pattern-target graph couples for each combination of (n G , n S , p).

During this procedure, the mapping between the vertices of S and G is tracked to finally constitute the ground-truth. Even if new non-tracked isomorphisms can be added during the completion of G 0 to G, their cost cannot be lower than the ground-truth one whose cost is 0. Moreover, it has been experimentally checked that the isomorphism in the groundtruth information is the only one whose cost is 0. This dataset is mainly intended to check whether a unique instance of an exact subgraph isomorphism is successfully found with a tested algorithm.

B. Noisy synthetic dataset

The ILPIso_noisy_synth dataset is also a synthetic dataset. It also contains 180 pattern-target graph couples. It has been created in the same manner than the ILPIso_exact_synth dataset (with the same combination of values for (n G , n S , p)) but an additional step has been introduced. Before its completion to G, G 0 has been modified by editing vertices and edges labels. Each label has been added a random value according a gaussian distribution with m = 0 and σ 2 = 5.

This dataset is intended to evaluate the performance of a substitution-tolerant subgraph isomorphism search program where the mapping with the lower cost is searched. As for the ILPIso_exact_synth dataset, new non tracked isomorphisms are added when G 0 is completed to G, but it has been checked that the groundtruth corresponds to the isomorphism with the lowest cost.

C. ILPIso_real dataset

The ILPIso_real dataset is a dataset that contains structural representations of document images of architectural plans. It also contains graphs modeling 16 isolated graphical symbols. Complete plan images which lead to the building of ILPIso_real dataset are 200 images extracted from the floorplan section of the SESYD dataset [START_REF] Delalandre | Generation of synthetic documents for performance evaluation of symbol recognition; spotting systems[END_REF]: 20 first images for each of the 10 templates. Each symbol may occurs once or several times or not occur at all on each floorplan instance. Considering the structural representations, the problem of locating one or several instances of a symbol in a complete floorplan turns into the search of a subgraph isomorphism problem.

The structural representation for both complete plans and isolated symbol models are region adjacency graphs (RAG), which are extracted according to the following process. A vertex is created for each white connected component after a skeletonization process [START_REF] Di Baja | Skeltonization algorithm running on pathbased distance maps[END_REF] which has reduced the width of black strokes to 1 pixel. Two directed edges are created between the vertices which describe adjacent white regions. Vertices are labelled with a feature vector corresponding to a set of the 24 first Zernike Moments (ZM) [START_REF] Teague | Image analysis via the general theory of moments[END_REF] and edges are labelled with relative scale and relative distance. The algorithm used for extracting such RAGs is fully described in Figure 2 shows two examples of plans and the corresponding RAGs. Figure 3 represent the 16 symbol models.

Even if the document images have been synthetically generated, the graph dataset which has been produced can be considered as a real dataset. It can be used to evaluate the search of one or multiple instances of substitution-tolerant subgraph isomorphism. Indeed, since the images are synthetic there is not any problem with splitted or merged regions which would have generated topological differences between the pattern graph and its occurrences in the target graph. The only differences concern vertices and edges labels.

The ground-truth information associated to this dataset gives for each target graph, the number of occurrences for each of the 16 pattern graph. For each occurrence, the vertices identifier from the target graph involved are listed. The induced subgraph represents the subgraph of the target graph which is isomorphic with the pattern graph. However, due to symmetry phenomenons, there can not be an exact vertex-to-vertex mapping between a pattern graph and its isomorphic subgraph in the target. This impacts the definition of performance measures which can only be done at a subgraph level and not at the vertex level.

IV. PERFORMANCE MEASURES

In this section, we propose some performance measures which can be used with the databases presented in section III to evaluate substitution-tolerant subgraph method.

The ILPIso_exact_synth dataset can be used to evaluate the ability of an approximate algorithm to detect an exact subgraph isomorphism. As there is a unique instance of pattern-target matching for each graph couple, this measure can only be done at the database level. The most objective measure is the detection rate detection rate = #detected mapping #graph couples

The same measure can be used with the ILPIso_noisy_synth dataset for which the objective is to find the lowest cost mapping. For approximate search algorithm able to provide a ranked list of mapping, this measure can be extended to top n results.

top n detection rate = #detected mapping in the n first results #graph couples Finally, the most complex evaluation can be performed with the ILPIso_real dataset as it contains 0, 1 or several occurrences of the 16 pattern graphs in the 200 target graphs. This dataset can first be used to evaluate a structural information retrieval system where the objective is to select among the target graphs those containing occurrences of query patterns. As for classical IR techniques, the evaluation can be measured in terms of precision P and recall R.

P =

#relevant retrieved pattern graphs #retrieved pattern graphs R = #relevant retrieved pattern graphs #relevant pattern graphs Some algorithms are able to compute a numerical value (cost or probability) which can be thresholded to decide whether there is an occurrence of the query or not. Several values of the P/R trade-off can be obtained by varying the threshold resulting in a precision-recall curve.

But this dataset can also be used to evaluate a subgraph isomorphism as a spotting system. Indeed, in each structural representation of a complete floorplan, several sets of vertices have been identified. The induced subgraph for each vertex set is considered as an instance of the structural representation of a symbol model.

For a single query, the system under evaluation should return its results as a set that contains the identifiers of the vertices in the target graph that are involved in the searched instance. It may also be the case that the system decides that no instance of the pattern graph occurs in the pattern. When comparing the result with the groundtruth information, several configurations can happen :

• There can be a perfect mapping between the detected set of identifiers and the set given in the groundtruth (good detection).

• There can also be a decision of reject when to instance of the searched pattern exists in the target graph (good reject).

• We consider a false reject decision, when the system decides a reject whereas an instance of the pattern graph exists in the groundtruth.

• On the other hand, we denote as a false alarm the event of detecting a pattern graph instance whereas the groundtruth indicates that it does not exist.

Moreover, besides these classical definitions, we also consider :

• an erroneous detection when a pattern instance is detected but the corresponding vertex identifier set has an empty intersection with the one defined in the groundtruth.

• a partial detection when a pattern instance is detected but the corresponding vertex identifier set has a non empty intersection with the one defined in the groundtruth.

The ILPIso_real dataset contains 16 pattern graphs and 200 target graphs allowing to test a system according the criterions on 3200 single occurrence searches. As mentioned before, the reject decision may be taken according to a threshold on the cost of the mapping. As a consequence, several trade-offs between the measures can be achieved with different threshold values.

Finally, the ILPIso_real dataset can also be used to evaluate a system in its ability to find multiple occurrences of a pattern in the target graph. For this purpose, the measures detailed above can be used for the evaluation performance with some adjustments. Indeed, when a system tries to find the n th occurrence of a pattern graph, it should be considered that n-1 occurrences have already been, at least, partially found. So, the n th detected occurrence should be considered as a false alarm if and only if the real number of occurrences given by the groundtruth information is lower than n. Otherwise, it should be considered as a perfect, partial or erroneous detection. If the system returns a reject decision in the n th whereas not all occurrences have already been detected, it should be classified as a non detection. A good rejection decision is taken if the n th is a reject whereas all occurrences given in the groundtruth have been detected. For a complete evaluation, the number of searched instances should be greater than the maximum number of real instances of a pattern in target graph, which is 13.

V. CONCLUSION

In this paper, three datasets are presented that can be used for several tasks involving the search for subgraph isomorphisms. We have also proposed several measures allowing performance evaluation on different tasks.

Two synthetic datasets can be used to benchmark the search for a single instance of a pattern in a target graph. In the first one, no perturbation is brought to labels whereas the second one has numerical labels which have been modified with the add of a gaussian noise. The measure defined for this task quantify the detection rate for a subgraph isomorphism tool, and the second dataset serves at evaluating its robustness to noise on numerical labels.

The third dataset is a real dataset composed from the structural descriptions by means of attributed RAGs of architectural plans. These target graphs contain several instances of several pattern graph describing symbols occurring in the plans. Thanks to the associated measures, this dataset can be used to benchmark a retrieval system based on structural description or the search of multiple occurrences of pattern graph in a target in presence of perturbed numerical labels. This work could be extended in several directions. First, in order to have a better review of the robustness to label perturbation, it could be considered to offer several noise models with several levels. Otherwise, the real dataset is currently composed of attributed RAGs. Despite the important labeling effort this would require, it could interesting to propose alternative structural descriptions such as graphs of graphical primitives [START_REF] Qureshi | Spotting symbols in line drawing images using graph representations[END_REF].

In the longer term, the scope of evaluated tasks could be extended by the integration of error-tolerant subgraph isomorphism, where topological differences are allowed between the pattern and its occurrences in the target graph. But, this extension is not manifest since it raises some issues on the groundtruth definition. Indeed, defining if a vertex in the target graph belongs or not to the pattern instance is questionable. Moreover, this may require to define one-to-one, many-to-one, one-to-many and many-to-many mappings between vertices.

Figure 1

 1 Figure 1 illustrates an output produced by the synthetic data generation in the presence of noise.

  (a) A graph S (b) A graph G (c) Ground-truth for the mapping between S and G.

Fig. 1 .

 1 Fig. 1. Illustration of the generation of synthetic data for the substitutiontolerant subgraph isomorphism search with the following parameters : n S = 3, n G = 6, p = 0.3, labels in [-100,100] and a Gaussian noise (σ 2 = 5). The mapping cost is 24.93

Fig. 2 .

 2 Fig. 2. Examples of plans from the floorplan dataset with the corresponding RAGs

Fig. 3 .

 3 Fig. 3. Symbol models

In the remaining of the paper, the term graph denotes a directed attributed multigraph.

We also intend to propose theses datasets on the TC-10 and TC-15 websites.

The authors hope that the presented work could help the community. Any suggestion would be welcome and will be considered.