Carole Treibich

Aurélia

Aurélia Lépine

Estimating misreporting in condom use and its determinants among sex workers: Evidence from the list randomisation method

Keywords: list randomisation, condom use, female sex workers, Senegal

published or not. The documents may come

Estimating misreporting in condom use and its determinants among sex workers: Evidence from the list randomisation method

Introduction

Condom use is the main preventive tool available to limit the spread of sexually transmitted diseases (STIs) and human immunodeficiency virus infection/acquired immune deficiency syndrome (HIV/AIDS). Given that the consistent use of condoms is known as the most costeffective way to prevent HIV transmission [START_REF] Cohen | Comparing the cost-effectiveness of hiv prevention interventions[END_REF][START_REF] Creese | Cost-effectiveness of hiv/aids interventions in africa: a systematic review of the evidence[END_REF], condom use is the pillar of any HIV prevention strategy in most countries. The promotion of condom use is often based on multiple interventions such as awareness campaigns and the free provision of condoms. However, the evaluation of the effects of such policies as well as their value for money is difficult to determine due to the impossibility to directly observe sexual behaviours adopted by targeted groups. Researchers and policy makers have no other choice than relying on individuals' declarations. As a matter of fact, in a systematic review looking at the effect of interventions involving condom promotion, [START_REF] Foss | A systematic review of published evidence on intervention impact on condom use in sub-saharan africa and asia[END_REF] found that most of the evidence published on the effect of such interventions is based on self-reported condom use, despite the inherent bias of such a measure. Indeed, one may wonder whether direct elicitation of condom use would provide an accurate estimate and could be used to measure the impact and monetary value of condom-based interventions. This may be even more of a concern when considering stigmatised groups highly targeted by preventive services, such as female sex workers (FSWs).

Our paper aims to measure misreporting in condom use among FSWs in Senegal, a country particularly interesting for the study of FSWs. First, while HIV prevalence is 0.7% in the general population, FSWs in Senegal are up to 9 times more likely to be infected, with HIV/AIDS with an HIV/AIDS prevalence of 6.6% in 2015 [START_REF] Apaps | Enquête nationale de surveillance combinée des ist et du vih/sida (ensc)[END_REF]. Second, Senegal is the only African country where prostitution is legal and regulated by a public health policy.

In 1969, the Government of Senegal legalised prostitution and introduced a compulsory registration programme for FSWs to monitor the prevalence of STIs, and later on, the spread of the HIV/AIDS epidemic. As a consequence of the close monitoring of this population, registered FSWs are aware of the benefits provided by the use of condoms and receive condoms for free at their monthly routine health visits. Third, Senegal is a Muslim-dominated country where a woman's status is very low. As a result, despite having a legal status, FSWs face very high social stigma. In this context, we anticipate that condom use self-reported in face-to-face surveys is likely to be over-reported.

To estimate the amount of over-reporting and the determinants of condom use and of its overreporting, we used list randomisation. The list randomisation method or item count technique provides privacy to respondents and thus limits dishonest answers caused by social desirability bias in face-to-face interviews [START_REF] Holbrook | Social desirability bias in voter turnout reports: Tests using the item count technique[END_REF]. It has been applied to elicit vote preferences [START_REF] Corstange | Sensitive questions, truthful answers? modeling the list experiment with listit[END_REF][START_REF] Gonzalez-Ocantos | Vote buying and social desirability bias: Experimental evidence from nicaragua[END_REF][START_REF] Holbrook | Social desirability bias in voter turnout reports: Tests using the item count technique[END_REF], illegal migration [START_REF] Mckenzie | Eliciting illegal migration rates through list randomization[END_REF], the use of micro finance loans [START_REF] Karlan | List randomization for sensitive behavior: An application for measuring use of loan proceeds[END_REF] as well as opinions on topics such as same sex marriage [START_REF] Lax | Are survey respondents lying about their support for same-sex marriage? lessons from a recent list experiment[END_REF] and racism [START_REF] Blair | Statistical analysis of list experiments[END_REF][START_REF] Kuklinski | Racial attitudes and the 'new south[END_REF].

However, there are only a few studies that have applied the method in health research [START_REF] Chong | Do information technologies improve teenagers' sexual education? evidence from a randomized evaluation in colombia[END_REF][START_REF] Jamison | Mixed-method evaluation of a passive mhealth sexual information texting service in uganda[END_REF][START_REF] Labrie | Sexual risk behaviors and alcohol: Higher base rates revealed using the unmatched-count technique[END_REF][START_REF] Walsh | Self-reported alcohol consumption and sexual behavior in males and females: Using unmatched-count technique to examine reporting practices of socially sensitive subjects in a sample of university students[END_REF], and the method has never been used to measure over-reporting in condom use among a low educated stigmatised group. We believe that it is important to assess whether list randomisation could be introduced into national surveys to obtain a less biased estimation of condom use among high-risk groups in low-income countries. In addition to providing a better estimate of condom use, list randomisation allows the identification of subgroups for which condom use rates are lower. We used the parameters from the model by [START_REF] Geoffard | Rational epidemics and their public control[END_REF] to identify the main drivers of condom use. Finally, we compared the proportion of FSWs who openly declared using a condom with their last client with the proportion obtained through the indirect elicitation method to quantify the propensity to over-report condom use among subgroups. By doing so, we are able to test whether registered FSWs, who are by definition more exposed to HIV prevention services, are more likely to over-report condom use than their non-registered counterparts.

We found that 22% of FSWs did not use a condom in their last sexual intercourse with a client, which is significantly greater than the 3% obtained when asked directly. Hence, our results confirm that list randomisation could be a promising indirect method to elicit condom use in low-income countries. When estimating condom use among subgroups, we found that FSWs at a higher risk of HIV are significantly less likely to have used a condom with their last client. The results also show that increasing the knowledge of FSWs regarding HIV and the consequences of STIs as well as increasing the links with health facilities would be useful policies to increase condom use. Finally, we did not find that FSWs receiving more HIV prevention services are more likely to over-report condom use.

Methodology

List randomisation method and underlying hypotheses

The principle of list randomisation is to allocate respondents randomly into the following two different groups: a "control" and a "treatment" group. Individuals allocated into the "control" group are presented with a number of non-sensitive statements. They are not asked to say whether they agree with each of the statements but only with how many of them they agree with. The same statements are presented to the "treated" group; the difference is that a sensitive statement is added to the series of non-sensitive statements. Assuming that the two groups agree on average with the same number of non-sensitive statements, one can deduce the share of individuals in the "treated" group who agreed with the sensitive item by comparing the average number of agreed statements in each group (see [START_REF] Glynn | What can we learn with statistical truth serum? design and analysis of the list experiment[END_REF][START_REF] Holbrook | Social desirability bias in voter turnout reports: Tests using the item count technique[END_REF][START_REF] Kuklinski | Racial attitudes and the 'new south[END_REF].

The effectiveness of this methodology is based on the following three assumptions: (i) randomisation of the treatment, (ii) absence of any design effect, and (iii) honest answers. More precisely, individuals allocated to each group must be similar to ensure that they agree with the same number of non-sensitive items on average. Second, the addition of the sensitive item must not change the sum of the affirmative answers to the control items. Finally, as pointed out by [START_REF] Kuklinski | Racial attitudes and the 'new south[END_REF], the choice of the non-key items needs to be such that individuals are not urged to provide false answers. There are two different types of dishonest answers: those who honestly would answer "yes" to all the non-sensitive items and hence would no longer benefit from any privacy if they agree with the sensitive item (ceiling effects) and those who honestly would answer "no" to all non-sensitive items and hence would no longer benefit from any privacy if they disagree with the sensitive item (floor effects).

List randomisation implemented among FSWs in the Dakar region

In 2015, we interviewed 651 FSWs in the Dakar suburbs, which represents 15% of the total estimated number of FSWs in the Dakar region (APAPS, 2011(APAPS, -2012)). Ethical clearance was obtained from the London School of Hygiene & Tropical Medicine and from the national ethics committee in Senegal (reference number SEN15/15). Data were captured using electronic devices and all respondents were asked whether they used a condom with their last client. Then, we randomised the allocation of the participants into the "treatment" and "control" groups based on their identifying number to determine their elicited condom use.

In our survey, the "control" group was presented with the following question: I [the interviewer] will read three statements. I will then ask you how many of these statements you agree with. You should not tell me which specific statement you agree with but the number of statements you agree with. I will give you three marbles, and you have to hold them in your right hand. Keep both of your hands on your back side. For each of the statements, if you agree with it, please transfer one marble from your right hand to your left hand behind you.

If you do not agree with it, please do not transfer any marble. At the end, I would like to know the total number of statements you agreed with. This number should correspond to the number of marbles you have in your left hand. I will now read the statements.

1. It is safer to bring a client home than going to a hotel.

2. I prefer that the client pays me before intercourse.

3. Monday is the day I have the greatest number of clients.

FSWs from the "treatment" group received an additional marble and were presented the same statements plus the sensitive item that relates to condom use. Note that this sensitive item was presented in the second position in the treatment group list: 4. I used a condom during my last sexual intercourse with a client.

Theoretical framework

In this section, we present a theoretical framework for the potential mechanisms at play in the decision to engage or not engage in unprotected sex. To do so, we adapted the model by [START_REF] Geoffard | Rational epidemics and their public control[END_REF], which is a two-period model where individuals (in our case FSWs) decide whether to engage or not engage in an unprotected sex act in the period t and face the costs of being infected in the period t + 1. We added interdependence to the utility functions of FSWs and their clients [START_REF] Bergstrom | System of benevolent utility functions[END_REF] so that the utility function of (infected) FSWs u depends on their sexual partner's utility level v.

As it is commonly done in the literature for the compensating differential for unprotected sex [START_REF] Arunachalam | Compensated for life: Sex work and disease risk[END_REF][START_REF] Gertler | Sex work and infection: What's law enforcement got to do with it[END_REF][START_REF] Rao | Sex workers and the cost of safe sex: The compensating differential for condom use among calcutta prostitutes[END_REF], it is assumed that clients bear disutility from using condom. FSWs choose their behaviours to maximise their utility given their health state. The health status h of FSWs can take the following two values: susceptible (h = s) or infected with HIV (h = i). FSWs decide to adopt a behaviour b that can also take the following two values: protection against risk (b = p) or exposure to risk (b = e).

The utility derived by FSWs u(h, b) is a function of one's health and behaviour.

Ceteris paribus, protective activity (p) and infection (i) are both assumed to be costly as follows:

u(h, e) > u(h, p) and u(s, b) > u(i, b)
FSWs discount future utility at a discount rate δ(h), which is a function of their future health state, with δ(s) < δ(i).

The transition rate from state s to state i, which is conditional on exposure, is denoted as λ = π × P where P is the probability that a susceptible FSW matches with an infected client and π is the probability that an exposed activity between the FSW and her client will result in a new infection.

The underlying assumptions in the model are as follows:

P [h(t + 1) = i|(h(t), b) = (s, e)] = λ (1) P [h(t + 1) = i|(h(t), b) = (s, p)] = 0 (2) P [h(t + 1) = i|(h(t), b) = i] = 1 (3)
In other words, (1) the probability of getting infected if exposed for a susceptible FSW is λ;

(2) the probability of being infected under protection is zero and (3) the probability of remaining infected if already infected is one, whichever behaviour is adopted by the FSW.

FSWs engage in safe sex if and only if the cost of protection (the loss of current utility from protection) is below the expected future utility loss due to infection.

u(s, e) -u(s, p) ≤ π • P u(s) δ(s) - u(i) δ(i)
While in this model, infected FSWs have no reason to engage in safe sex as u(i, p) < u(i, e);

thus, we relax this assumption and assume interdependence in the utility functions of FSWs and their clients [START_REF] Bergstrom | System of benevolent utility functions[END_REF] so that the utility function of (infected) FSWs u depends on their sexual partner's utility level v, where

u(i, e) -u(i, p) ≤ π[u(i, v(s)) -u(i, v(i))] with v(s) > v(i)

Empirical strategy

Equation (4) shows that condom use estimated by list randomisation was estimated by regressing the number of statements the respondent agreed with (Y i ) on the allocation to the treatment group (T i ). The average condom use rate in the sample using list randomisation is then given by β and corresponds to the average difference in the number of statements between the control and the treatment group in the sample. All estimations compute robust standard errors to account for the difference in the variance of error term ε i between the treatment and control groups.

Y i = βT i + ε i (4)
We use list randomisation to further investigate the characteristics of FSWs who did not use a condom during their last sexual intercourse.1 Following Imai (2011), we investigated the relation between condom use and respondents' characteristics using a simple linear regression with the following interaction terms:

Y i = βT i + γS i + αS i × T i + ε i (5)
where S i is a characteristic of individual i that may be correlated with condom use. β

precisely reports the condom use rate among the subgroup for which S i = 0, while α is the difference in the condom use rate between the two subgroups (S i = 1 and S i = 0). The p-value of α indicates if the condom use rate is different between the subgroups.

To improve the statistical power, we added some variables that are assumed to be correlated with the non-sensitive statements:

Y i = βT i + γS i + αS i × T i + X i + ε i (6)
where X i is a set of sex worker characteristics potentially influencing the answer to the non-sensitive items (i.e., FSW preferences regarding the place where the sex act occurred or the proportion of the last four sexual intercourses for which payment was made after the sex act). X i also includes age, whether the FSW is divorced, whether the FSW lives with or next to her parents, the type of client (regular versus occasional) and whether clients are usually approached in a night club.

Data

Data were collected from 651 FSWs living in Dakar suburbs in June and July 2015, which represents 15% of the estimated total FSWs in the Dakar region (APAPS, 2011(APAPS, -2012)). Our sample contains an equal share of registered and non-registered FSWs. Given that sexual health services are integrated with reproductive health in Senegal, registered FSWs were recruited by the midwife in charge of their medical follow-up. All active registered FSWs from four (Pikine, Rufisque, Mbao and Sebikotane) out of the five STI health centres located in Dakar were contacted to participate in our study. Unregistered FSWs were recruited by NGO staffs and by peer FSWs using snowball sampling. FSWs were asked to come to the health centre and were interviewed at the health facility in private dedicated rooms. Survey participants received a CFAF 3,000 show-up fee that aimed to cover transport cost and the time spent at the health facility.

As part of the questionnaire, information on socio-economic and intrinsic characteristics, preferences (time preferences, risk aversion and altruism) and sex work activity (revenue, type of clients and type of sex acts) was collected from the respondents. Table 1 shows the different parameters from the model with their measurements from the data set and presents descriptive statistics of these variables.

6 Validation of list randomisation

Verification of the randomisation

We noted that randomisation ensured balance between the two groups with respect to their observable characteristics (Table A2, Appendix 2). The only significant difference observed was in the type of the sex worker's last client (p=0.06). However, given that we tested approximately 60 different treatment-control differences in this table, this unique significant difference is no more than what would be expected by chance. This was confirmed by the joint significance tests for a large share of the set of variables presented in Table A2.2

Absence of ceiling, floor and design effects

We also needed to ensure that the list of non-sensitive items provided enough privacy to respondents in the treated group (hypothesis 2) and that the addition of the sensitive item did not modify the answers regarding the non-sensitive statements (hypothesis 3). In Table A3 (Appendix 2), we estimated the proportion of FSWs in the control group who did not agree with any statement and answered "0" to the item count list question. If this proportion was high, it would encourage respondents in the treated group to report a positive value since answering "0" would reveal that they had unprotected sex. Since the proportion of individuals answering "0" in the control group was less than 3%, we did not face this issue. We also avoided the issue of ceiling effects because the proportion of respondents in the control group who answered "3" to the non-sensitive items was also very low (9%). This absence of ceiling and floor effects has been ensured thanks to the negative correlation between items 2 and 3. Some protection for the respondents is therefore built-in to allow them to honestly report their true behaviours towards condom use. Finally, the difference (Row 5) between the proportion of individuals in the treated group (Row 2) and in the control group (Row 4) who agree with at least j statements (j = 1, 2, 3, and4) is always positive, which provides evidence of the absence of design effects.

Results

Measuring misreporting in condom use

Using self-reported information, we found that 97.2% of FSWs declared to have used a condom with their last client. Table 2 presents the results from the list randomisation exercise. It appears that based on list randomisation, the condom use rate is equal to 79.7% when considering all female sex workers (Panel A) and 77.7% versus 97.2% when self-reported if we consider the sample of FSWs who answered to the self-declared question (Panel B). As a result, the selfreported condom use is overestimated by 19.5% points (95% CI [7.5; 31.6]).

The difference between self-reported condom use and the condom use elicited by list randomisation is statistically significant (p < 0.01). For the entire sample, i.e., when including the 10.6% who did not answer the direct question, the elicited condom use rate using list randomisation is 2 percentage points higher. This is an interesting result because it means that respondents who did not answer the self-reported question report slightly higher condom use (see Panel C). Notes: The proportion who used a condom is obtained by taking the difference between the "control" and "treated" means.

Measuring misreporting in condom use for subgroups

Table 3 displays the results obtained when performing the subgroup analysis based on the variables presented in Table 1.

Factors affecting the costs of protection We found that having high earnings3 is negatively correlated with the likelihood of using a condom (68.4% vs. 89.4%, p-value=0.07).

Nevertheless, FSWs who received free condoms are not more likely to use condoms than those who have to pay for condoms. In addition to these financial costs, FSWs were asked to compare their sexual pleasure with and without condoms. Those who declared that the use of condom does not decrease their sexual pleasure showed greater condom use, although the difference is not statistically significant (94.7% vs. 76.2%, p-value=0.17). No information regarding the type of violence experienced during the last sexual intercourse was collected. We attempted to overcome this issue (i) by comparing FSWs who suffered from violence by an occasional client in the last twelve months with those who did not and (ii) by looking at whether the last client negotiated the price of intercourse as a proxy for a FSW's low bargaining power. However, we did not find any significant differences in condom use between these subgroups.

Factors affecting the probability that a susceptible FSW matches with an infected client FSWs were asked about their last client characteristics. We found that FSWs are more likely to use a condom with clients perceived to be at high risk of HIV (90.4% vs. 77.4%), though this difference was not statistically significant.

Factors affecting the probability that an exposed activity between a FSW and a client will result in a new infection Having an occasional client reduces the probability of engaging in safe sex even if this difference is not statistically significant. FSWs who have more than three clients per week are significantly less likely to have used a condom (73.3% vs.

95.0%, p-value=0.08). FSWs who declare to be willing to take risks with their health tend to be less likely to use condoms, though the sample size of this subgroup does not enable us to detect a statistically significant difference. FSWs who have a better knowledge regarding HIV transmission modes are more likely to have used a condom with their last client (85.2% vs. 58.5%, p-value=0.06). For FSWs who think that condoms will not prevent against HIV infection, they have a lower condom use, but this difference is not statistically significant (62.0% vs. 82.9%, p-value=0.14).

Factors affecting the utility in the case of an infection We found that peer effects and social norms play a role in condom use. FSWs who declare that all the girls working in their area use condoms are more likely to have used a condom with their last client (98.8% vs.

63.2%, p-value=0.02). This is also the case for FSWs who entered sex work thanks to another sex worker (97.4% vs. 73.9%, p-value=0.06). Furthermore, we investigated the impact of fear of social stigma on condom use by looking at the following two different variables: those who would be ashamed if a neighbour learns about their sex work activity and those who fear the neighbour would repeat this to others. Both subgroups are more likely to use condoms (85.2% vs. 47.2%, p-value=0.02 and 84.6% vs. 42.7%, p-value=0.02 respectively).

As expected, perceived high STI consequences are positively associated with the decision to engage in safe sex. On one hand, FSWs who think they will lose more than 14 days of work in the case of an STI are more likely to have used a condom during their last paid intercourse (92.9% vs. 66.6%, p-value=0.02). Likewise, FSWs who expect to pay more than 15,000 CFAF in medical expenses in the case of a genital ulcer are more likely to have used a condom with their last client (85.2% vs. 65.7%, p-value=0.10). FSWs who are aware of the existence of antiretroviral treatment (ART) also displayed lower rate of condom use than FSWs who had never heard about ART, but this difference was not statistically significant (71.5% vs. 88.9%, p-value=0.13).

Factors reflecting the preference for health FSWs who visited a health centre in the past six months are more likely to have used a condom (86.0% vs. 60.9%, p-value=0.07). However, being a registered FSW and adhering to medical visits is not correlated with condom use.

Factors affecting the discount rate FSWs who declared that they have consumed alcohol or drugs before their last paid sexual intercourse are significantly less likely to have used a condom (40.7% vs. 81.4%, p-value=0.06).

Testing the disutility in using condoms once infected HIV positive FSWs (estimated via biological markers) are less likely to have used a condom than HIV negative FSWs (5.0% vs 80.0%, p-value=0.05). This negative relationship between HIV status and condom use was somehow confirmed for the entire sample when considering subjective expectations regarding HIV status. Additionally, FSWs who believe that they have another STI than HIV (also estimated via subjective expectations) are significantly less likely to have used a condom with their last client (57.1% vs 89.4%, p-value=0.01).

Interdependent utilities We found that altruistic FSWs are more likely to have used a condom (97.2% vs. 72.6%, p-value=0.05).

Table A4 (Appendix 3) shows that very similar results were found when introducing the set of covariates aiming at controlling for any sex worker's characteristics, which could influence the answers to the non-sensitive items or when restricting the sample to FSWs who answered the self-declared condom use question. Notes: † Condom inefficacy refers to a subjective probability higher than 80% of being infected after 100 protected intercourses. Sample is restricted to individuals who understood the subjective probabilities. "Obs" reports the total number of respondents for whom we have information on the variable. "Obs if Yes" presents the number of respondents who answered "Yes" to the related question. Columns (3) and ( 9) [Columns ( 4) and ( 10), respectively] display the proportion of FSWs who answered "Yes" ["No"] and who used a condom in their last sexual act.

Reading note: For the variable 'Registered with authorities', columns ( 7) to (10) can be read as follows : Among the 650 respondents, 325 of them are registered, 84.8% of legal sex workers used a condom with their last client, and 74.9% of illegal sex workers used a condom with their last client. Columns ( 5) and ( 11) correspond to the difference between (3) and ( 4) and between ( 9) and ( 10), respectively. Columns ( 6) and ( 12) give the robust standard errors of the estimated difference. ‡ List randomisation -estimation: equation ( 5):

Y i = βT i + γS i + αS i × T i + ε i .
Reported levels of significance are p<0.1, p<0.05 and p<0.01.

Using list randomisation, we found that FSWs in Dakar over-report condom use by 19.5 points.

The misreporting in condom use is higher than previously reported in the literature. Previous studies concluded that condom use was overestimated by 11 points among college students in the United States [START_REF] Labrie | Sexual risk behaviors and alcohol: Higher base rates revealed using the unmatched-count technique[END_REF] and by 14 points among young men in Uganda, but condom use was neither overestimated among young women [START_REF] Jamison | Mixed-method evaluation of a passive mhealth sexual information texting service in uganda[END_REF] nor among teenagers in Colombia [START_REF] Chong | Do information technologies improve teenagers' sexual education? evidence from a randomized evaluation in colombia[END_REF]. The high misreporting in our study is likely to be explained by the characteristics of the targeted population; given that sex work is morally condemned by the Senegalese society, FSWs fear to be looked down upon when disclosing socially unacceptable behaviours in a face-to-face survey. Hence, by guaranteeing anonymity, the list randomisation method seems particularly suited to this population. Nonetheless, we acknowledge that condom use may still be overestimated. While list randomisation guarantees privacy in response to survey participants, it cannot help with participants who do not want to reveal their true behaviours. For example, we found that among the respondents who did not answer the self-declared condom use question, only 8% did not use condoms with their last client according to list randomisation, which may indicate that condom use is still overestimated in this sub-sample (see Table 2, Panel C).

We empirically tested the role of the main parameters from [START_REF] Geoffard | Rational epidemics and their public control[END_REF]'s model. We found that one of the main reasons for not using condom comes from the existence of a premium for unprotected sex [START_REF] Gertler | Risky business: The market for unprotected commercial sex[END_REF][START_REF] Rao | Sex workers and the cost of safe sex: The compensating differential for condom use among calcutta prostitutes[END_REF]. By showing that more expensive sex acts were more likely to be unprotected, our results confirm the important role of clients in the AIDS epidemic. Policies that would aim to reduce the strong preference of clients for unprotected sex could be effective in limiting HIV transmission. While a main barrier to condom use is the fact that condom use is under the client's control [START_REF] Maia | Condom use, power and hiv/aids risk: Sexworkers bargain for survival in hillbrow/joubert park/berea, johannesburg[END_REF], our data show that only 7% of the 1,629 protected sexual intercourses contained in our data set were protected using a female condom. This suggests that the low bargaining power of FSWs may not be the main reason for not using condoms and justifies why FSWs with a lower bargaining power or FSWs exposed to physical violence from a client are not less likely to use condoms in our data set.

Our results provide some evidence on the role of factors affecting the utility in the case of infection upon condom use. First, we showed that FSWs who fear social stigma are more likely to use condoms. This indicates that while stigma reduction policies will certainly reduce social exclusion of FSWs, the absence of social sanctions could lead to greater risk-taking. Second, we found that condom use is significantly higher for FSWs who anticipate that infection will lead to important direct and indirect costs and a lower quality of life. For example, we found that women who have heard about ART are less likely to use condoms. There is some evidence in the literature that ART roll out is associated with greater risk taking [START_REF] Geoffard | Comportements sexuels risqués et incitations: L'impact des nouveaux traitements sur la prévention du vih[END_REF][START_REF] Gray | Stochastic simu-lation of the impact of antiretroviral therapy and hiv vaccines on hiv transmission; rakai, uganda[END_REF]. However, the negative relationship between ART knowledge and condom use could also come from the fact that FSWs on ART have a lower incentive to use protection, ceteris paribus. To test this, we excluded FSWs who believed that they were infected with HIV from our sample, which reinforced the negative relationship between ART knowledge and condom use. FSWs who knew about ART had a lower condom use by 25 points and this difference was statistically significant at 5%.

The model by [START_REF] Geoffard | Rational epidemics and their public control[END_REF] predicts that once infected, FSWs should stop using condoms because the benefits provided from protection is nil; this was confirmed empirically. This result is particularly alarming because it shows that the riskiest sexual intercourses are more likely to be unprotected compared with safer ones. In fact, we estimate that among the 4,225 sexual intercourses that occur weekly in our sample, 8% involved HIV positive FSWs. This is because HIV positive FSWs have on average 7 weekly clients (compared to 6.5 for HIV negative ones). Among those 329 sexual intercourses that are particularly at risk of infection, only 16 were protected according to the list randomisation results. However, the increased likelihood of adopting risky sexual behaviours once infected can be mitigated if we assume interdependent utilities. When testing this empirically, we find that altruistic HIV positive FSWs have a condom use rate that is 85 points greater than non-altruistic HIV positive FSWs, although the difference was not statistically significant given the small sample (p-value=0.12).

While our results are novel and in line with economic theory predictions, our study has several limitations. First, the small size of our sample prevents us from detecting moderate differences in condom use for a few subgroups. In addition to the issue of low statistical power, the small sample size also leads to higher uncertainty in the estimated proportion of condom use for several subgroups. For instance, while we estimate that the condom use by HIV positive FSWs (according to biological markers) is only 5%; this result may be attributable to the small HIV prevalence in our sample. In the treated group of HIV positive FSWs, the average number of true statements was 2.2, while it was 1.7 in the total sample, leading to an underestimation of condom use for this subgroup. When considering the number of true statements in the total sample, list randomisation concludes that condom use is 20 points lower for HIV positive FSWs. Despite the impossibility to investigate the causal effect of HIV status on condom use, our findings confirm that FSWs are an important vector of HIV transmission in Senegal. An important limitation however is that while we assume that clients bear disutility from using condoms, the data did not allow us to investigate the role of client preferences on condom use.

Future research on the use of the list randomisation method to elicit sexual behaviours could be conducted along three axes. First, additional methodological research that would provide guidance regarding the optimal design of list randomisation is required. While the number and choices of non-key items should not affect the results in theory, there is some empirical evidence that the choice of non-key items does matter [START_REF] Droitcour | The item-count technique as a method of indirect questioning: A review of its development and a case study application[END_REF]. Unlike previous papers using list randomisation to elicit condom use [START_REF] Chong | Do information technologies improve teenagers' sexual education? evidence from a randomized evaluation in colombia[END_REF][START_REF] Jamison | Mixed-method evaluation of a passive mhealth sexual information texting service in uganda[END_REF][START_REF] Labrie | Sexual risk behaviors and alcohol: Higher base rates revealed using the unmatched-count technique[END_REF][START_REF] Walsh | Self-reported alcohol consumption and sexual behavior in males and females: Using unmatched-count technique to examine reporting practices of socially sensitive subjects in a sample of university students[END_REF] and building on the agreement in the recent literature that it is better to select non-key items that relate to the topic of interest (see [START_REF] Imai | Using the predicted responses from list experiments as explanatory variables in regression models[END_REF][START_REF] Karlan | List randomization for sensitive behavior: An application for measuring use of loan proceeds[END_REF][START_REF] Wolter | The effectiveness of the item count technique in eliciting valid answers to sensitive questions. an evaluation in the context of self-reported delinquency[END_REF], our non-key items are related to sex work activity. Second, we showed that results obtained from list randomisation are to some extent imprecise, and, given the implementation challenges when performing a list randomisation, the method is often applied to small samples. When the length of the survey allows it, a double list randomisation where each group serves once as the control group and once as the treated group can increase precision [START_REF] Glynn | What can we learn with statistical truth serum? design and analysis of the list experiment[END_REF]. Finally, future research on condom use measurements should aim to test the validity of the results obtained with list randomisation. This could be performed by asking condom use question to clients in addition to FSWs since clients are less likely to over-report condom use [START_REF] Wilson | Sex worker, client sex behaviour and condom use in harare, zimbabwe[END_REF] or to compare the results obtained with the list randomisation to results obtained with another indirect elicitation method.

Conclusion

We implemented list randomisation on FSWs to test if condom use was over-reported. Our results are consistent with the fact that self-reported condom use leads to a large overestimation of condom use, which has direct implications when this measure is used to assess the impact and the value for money of condom-based interventions. When analysing the determinants of condom use, we provide some alarming evidence on the fact that sexual intercourses most at risk of infection are more likely to be unprotected than safer ones. We also highlight some important factors that affect the decision to engage in unprotected sex. While many of those factors have something to do with a FSW's personality and social norms, and hence are hardly changeable, our results also suggest that a mix of policies that consist of both educating FSWs and clients on the benefits of protected sex and reducing the costs associated with protected sex could be effective to increase condom use.

Table 1 :

 1 Determinants of condom use.

	Parameter	Description	Factors influencing the parameters	Expected
				effect	Obs Mean
	u(s, e) -u(s, p) Cost of protection	Revenue loss:	-
			Earned more than 12,500 CFAF for the last intercourse		0.499
			Beauty (≥ 5 out of 10)		0.776
			Condom price:	+
			Received free condoms		0.671
			Reduction in sexual pleasure:	+
			FSWs who declare having no reduction in sexual pleasure		0.195
			with condoms
			Violence:	-
			The last client negotiated the price		0.534
			Violence from a client in the past year		0.255
	P	Probability that a	Client at risk of HIV:	+
		susceptible	Last client was at risk of HIV		0.73
		individual matched with an
		infected one
	π	Probability that an	Risk taking:	-
		exposed activity	More than 3 clients a week		0.684
		between them will	Subjective risk taking in health domain (≥ 8 out of 10)		0.054
		result in a	Last client was an occasional client		0.440
		new infection	Performed anal sex during last intercourse		0.023
			HIV and STI knowledge:	+
			High HIV knowledge (≥ 6 out of 8)		0.796
			Poor trust towards the efficacy of condoms:	-
			Perceived condoms inefficacious to prevent HIV		0.254
			Agrees with 'One cannot avoid HIV by always using condoms'		0.171
	u(i, e)	Utility in the case of	Social exclusion:	+
		infection	Was introduced to the sex business by another SW		0.267
			Thinks that all girls in the same location use condoms		0.522
			Would be ashamed if a neighbour learns about her sex		0.844
			work activity
			Fears that a neighbour who learns about her sex activity		0.856
			will repeat it to others
			Medical and opportunity cost:	+
			Thinks that will lose more than 14 days of work if has an STI		0.486
			Expects to pay more than 15,000 CFAF in the case of a genital ulcer		0.364
			Quality of life if infected:	-
			Knows ART		0.504
	u(s)	Preference for health	Legal status:	+
			Registered with authorities		0.500
			Demand for prevention:	+
			Is affiliated with a health centre		0.731
			Does her monthly routine visit		0.729
			Visited a health centre in the last 6 months		0.750
			Had an HIV screening in the past year		0.806
	δ	Discount rate	Preference for present:	-
			Agrees with 'Instead of saving, I prefer to spend my money today'		0.788
			Alcohol or drug consumption during last intercourse		0.077
	u(i, p) < u(i, e) Disutility in using	HIV and STI status:	-
		condoms once	HIV positive according to medical records (biological test)		0.059
		infected	Subjective expectations about being HIV positive		0.065
			Subjective expectations about being STI positive		0.398
	u(v(s))	Interdependent	Altruism:	+
		utilities	Gave more than 40% of the amount received in a		0.289
			dictator game

Notes: All listed variables are binary variables.

Table 2 :

 2 Condom use with the last client estimated with the list experiment.

	Average number
	of agreed statements	Estimated
	Control Treatment condom use	CI 95%
	Panel A: All female sex workers (n=651)
	1.700	2.497	0.797	[68.52, 90.93]
	Panel B: Restricted to individuals who answered the self-declared question (n=582)
	1.685	2.462	0.777	[65.67, 89.75]
	Panel C: Restricted to individuals who did not answered the self-declared question (n=69)
	1.839	2.763	0.924	[64.10, 120.78]

Table 3 :

 3 Condom use by subgroups.

		Self-declared condom use				List randomisation ‡
		Obs if						Obs if
	Obs	Yes	Yes	No	Difference	SE	Obs	Yes	Yes	No	Difference	SE
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)

Table 3 :

 3 Condom use by subgroups (continued).

		Self-declared condom use				List randomisation ‡
		Obs if						Obs if
	Obs	Yes	Yes	No	Difference	SE	Obs	Yes	Yes	No	Difference	SE
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)

Table A2 :

 A2 Tests of randomisation (continued). Test of joint significance (when considering the variables indicated by ): F(32,612) = 0.63, p-value = 0.947 Test of joint significance (when considering the variables indicated by and ): F(39,581) = 0.76, p-value = 0.855 Notes: ‡ Experience in sex work = age -age at first paid sexual intercourse.

	Variables	Observations Control Treated p-value
		651	323	328
	Sex work activity
	Work mostly in bars or brothels	651	23.84	26.83	0.381
	Work mostly at home	651	28.48	29.57	0.760
	Experience in sex work ‡ (in years)	650	7.64	8.51	0.147
	Age at first sexual intercourse	650	17.24	17.28	0.891
	Age at first paid sexual intercourse	650	27.94	27.61	0.594
	Has only occasional clients	645	11.32	14.98	0.170
	Has only regular clients	645	33.02	32.42	0.871
	Last client was occasional	645	40.37	47.68	0.062
	Declared use of condom with last client	582	97.60	96.90	0.603
	Number of clients within a week	648	6.49	6.56	0.893
	Link with the authorities and the health system
	Legal sex worker (LSW)	650	47.68	52.29	0.240
	LSW since more than one year	650	37.46	38.53	0.779
	Thinks sex work is legal	610	60.30	64.22	0.315
	Police violence in the last 12 months	651	6.81	7.93	0.587
	LSW who go to her monthly visits	269	72.87	72.86	0.998
	Has received free condoms	641	65.41	68.73	0.372
	Is affiliated with a STD centre	648	72.36	74.01	0.637
	Came to a STD centre in the last month	651	56.97	56.10	0.824
	Had an HIV screeng in the past year	651	81.11	80.18	0.764
	HIV positive according to medical records	219	4.90	6.84	0.548
	Has got STI symptoms in the last month	646	20.67	23.55	0.383

The advantage of the list randomisation methodology lies in the possibility of conducting subgroup analysis (see Appendix 1 for a comparison of different techniques eliciting sensitive items).

More precisely, two tests of joint significance were performed and provided similar results. While the first one aimed to maximise the size of the sample considered (645 observations and

variables), the second one increased the number of variables included in the model (621 observations and 39 variables).

More than 12,500 CFAF for the sexual act, which corresponds to the median in our sample

An important concern is whether high-risk populations who are more exposed to intensive HIV prevention services tend to over-report more condom use. To measure the role of prevention on condom use over-reporting, we perform the following procedure. First, we computed the proportion of FSWs who declared using a condom with their last client depending on their registration status, assuming that registered FSWs are more exposed to HIV prevention than non-registered FSWs. Then, we estimated condom use for those groups using the list randomisation results presented in Table 3. Finally, we computed the difference in condom use obtained with the two methods as well as its associated standard error. Table 4 reports the propensity to over-report condom use depending on HIV prevention exposure. Overall, there is no evidence that FSWs receiving intensive HIV prevention services are more likely to over-report condom use. Conversely, we found that FSWs who have not visited a health centre in the last six months tend to be more likely to over-report their condom use when compared to FSWs who recently attended a health centre, though this result is not statistically significant. 2) correspond to the difference in condom use with the direct and indirect measures for the control group and the treated group, respectively. (3) is the difference between (1) and ( 2). ( 4) is the standard error of (3) and is equal to the square root of the sum of the squared standard errors of (1) and (2) (not reported in this table). The p-value indicates whether the difference is significantly different from zero and has been computed as follows: p-value = 2×normal(-abs((3)/(4)). None of the differences are statistically significant at the 10% level or lower. "Obs" reports the total number of respondents for whom we have information, i.e., 650 respondents. "Obs if Yes" presents the number of respondents who answered "Yes" to the question, i.e., 325 respondents are registered.

Appendices Appendix 1 Techniques eliciting sensitive items

Table A1 presents several techniques aiming at ensuring confidentiality and summarizes the strengths and weaknesses of each of them. We believe that given the low-literacy level of FSWs and the policy relevance of performing a sub-group analysis, the list randomisation methodology was the most appropriate method to estimate misreporting in condom use. † Condom inefficacy refers to a subjective probability higher than 80% to be infected after 100 protected intercourses. Sample is restricted to individuals who understood the subjective probabilities. "Obs" reports the total number of respondents for whom we have information on the variable. "Obs if Yes" presents the number of respondents who answered "Yes" to the related question. Columns (3) and (9) [Columns (4) and ( 10)] display the proportion of FSWs who answered "Yes" ["No"] and who used a condom in their last sexual act.

Appendix 2 Verification of the list randomisation hypothesis

Reading note: For the variable 'Registered with authorities', columns ( 7) to (10) can be read as follows: Among the 643 respondents, 322 of them are registered, 82.1% of legal sex workers used a condom with their last client, 76.2% of illegal sex workers did so. Columns ( 5) and ( 11) correspond to the difference between (3) and (4) and between ( 9) and (10) respectively. Columns ( 6) and ( 12) give the robust standard errors of the estimated difference. ‡ Equation (5): Y i = βT i + γS i + αS i × T i + ε i . ∓ Equation ( 6): Y i = βT i + γS i + αS i × T i + X i + ε i . X i include age, whether the FSW is divorced, whether the FSW lives with or next to her parents, whether last client was an occasional client, whether clients are usually approached in a night club, whether she had last paid sexual intercourse at home, proportion of the last four sexual intercourses for which payment was made after the sex act. Reported levels of significance are p<0.1, p<0.05 and p<0.01.


Jeremy Vatin
email: jeremy.vatin@centralesupelec.fr

Damien Rontani

Marc Sciamanna

Enhanced performance of a reservoir computer using polarization dynamics in VCSELs

Keywords: Neural networks, Vertical cavity surface emitting lasers, Optical data processing, Ultrafast technology, Semiconductor lasers

published or not. The documents may come

Enhanced performance of a reservoir computer using polarization dynamics in VCSELs

The growing amount of exchanged information in optical network makes data processing one of today's most critical issue [START_REF] Crutchfield | Introduction to focus issue: Intrinsic and designed computation: Information processing in dynamical systems-beyond the digital hegemony[END_REF]. Neuro-inspired architectures raise interest as a novel approach to process signal, including so-called reservoir computer. Reservoir computing is a technique of machine learning that trains artificial neural networks with recurrent connections by performing regression on a readout layer [START_REF] Jaeger | Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication[END_REF]. Several architectures have been demonstrated such as coupled photonic emitters [START_REF] Brunner | Photonic networks for neuromorphic computing[END_REF] or silicon photonic reservoir [4]. However having a fully implemented reservoir computer with a high number of nodes remains a technological challenge.

A reservoir architecture using a time-delayed feedback is a particularly promising solution to address this issue [START_REF] Appeltant | Information processing using a single dynamical node as complex system[END_REF]. The advantage is that the number of nodes can be easily increased since those are only virtually positioned along the feedback loop. This architecture has shown convincing performance whatever the physical nature of the processed information; i.e. opto-electronic [START_REF] Larger | Photonic information processing beyond turing: an optoelectronic implementation of reservoir computing[END_REF][START_REF] Paquot | Optoelectronic reservoir computing[END_REF] or optical [START_REF] Brunner | Parallel photonic information processing at gigabyte per second data rates using transient states[END_REF][START_REF] Nguimdo | Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics[END_REF][START_REF] Nakayama | Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal[END_REF]. The limitation of the time delay approach is its processing speed, due to the length of the delay line. Using a Vertical Cavity Surface Emitting Laser (VCSEL) can be a solution to that problem, while further improving the com-putational accuracy. When comparing with conventional edge-emitting lasers, VCSELs exhibit high-speed modulation capability, hence hopefully faster computation, but also the capability to emit coherent light along two coupled polarization modes, which leads to much coupled information generated in the same amount of time [START_REF] Muller | 1550-nm high-speed short-cavity vcsels[END_REF]. Added to that, VCSELs are already used in telecommunication transmission, that would ease the interface between the current network and our data processing device.

In this letter, we analyze numerically a photonic reservoir computer based on VCSEL and demonstrate that using the two-polarization modes dynamics of a VCSEL improves the performance in terms of speed and accuracy. Our new design is based on delayed optical feedback and optical injection. After showing its computational performance, we prove that it is also efficient in dealing with several benchmarking tasks including the very demanding channel equalization.

We consider an all-optical reservoir computer with timedelayed feedback. The main difference between our system and the ones considered in the literature so far is that our computation node strength relies on the two-mode polarization dynamics of a VCSEL (Fig. 1). he input (data to be processed) is optically injected thanks to a laser beam which is modulated through a Mach-Zehnder modulator. This modulated-beam is injected in the VCSEL along the main lasing polarization axis. The output of that laser is then sent to a τ -length delay-loop, before being reinjected in the VCSEL. Along the loop, the signal is rotated by π 2 thanks to a polarization controller (P.C.) in order to trigger lasing in the normally depressed lasing axis, and is attenuated in order to control the feedback strength. Because of that rotation, the depressed mode is injected in the main lasing polarization mode, and the main polarization mode is rotated and injected in the normally suppressed polarization mode. The output layer of our reservoir is composed of virtual nodes, which are positioned along the delay-loop. The time between each two nodes is here called θ. That definition leads us to the following relationship between θ and τ : τ = number of nodes × θ. As in [START_REF] Appeltant | Information processing using a single dynamical node as complex system[END_REF], the time discrete input u(k) is convoluted with a mask, which is composed of as many values as the number of nodes, randomly taken between either -1 or 1. Each value of the mask is hold during θ so that the total mask duration is τ .

In order to simulate our system, we consider the SFM model [START_REF] Miguel | Light-polarization dynamics in surface-emitting semiconductor lasers[END_REF][START_REF] Virte | Deterministic polarization chaos from a laser diode[END_REF] with a feedback term and an optical injection term:

Ėx = κ(1 + iα)[(N -1)E x + inE y ] -(γ a + iγ p )E x +Φ x (t) + κA inj (t)e (ωinj -ω0)t + F x (t), (1)
Ėy = κ(1 + iα)[(N -1)E y -inE x ] + (γ a + iγ p )E y +Φ y (t) + F y (t), (2)
Ṅ = -γ N [N -µ + N (|E x | 2 + |E y | 2 ) +in(E y E * x -E x E y * )], (3) ṅ
= -γ s n -γ N [n(|E x | 2 + |E y | 2 +iN (E y E * x -E x E * y )], (4)
where E x and E y are the values of the orthogonal linearly polarized optical field, N is the population difference between conduction and valence bands n the population difference between the carrier densities with positive and negative spin values and µ is the injection current normalized to threshold. κ is the field decay rate, α is the linewidth enhancement factor, γ N is the decay rate of the carrier population and γ s is the decay rate which influences the the mix-ing of career populations between the two different spins. ω 0 is the pulsation of the self-emitting slave laser, and η is the feedback strength.

A inj = P inj /2 × (1 + e iV /Vπ
) is the value of the injected optical field where P inj is the power of the tunable laser and V is the voltage relative to the input taken within [-πV π ; πV π ], ω inj is the pulsation of the master laser. We fix κ = 300 GHz, α = 3, γ a = -0.1 GHz, γ p = 6 GHz, γ N = 1 GHz, γ s = 50 GHz, and ω inj = ω 0 = 2πf c λ , where c is the speed of light and λ = 1550 nm is the wavelength. F x and F y are two Langevin noise sources modelling the spontaneous emission noise [START_REF] Martin-Regalado | Polarization properties of vertical-cavity surface-emitting lasers[END_REF]:

F x = βsp 2 √ N + nξ 1 + √ N -nξ 2 ,
and

F y = -i βsp 2 √ N + nξ 1 - √ N -nξ 2 ,
where ξ 1 and ξ 2 are complex Gaussian white noise, and β sp is the spontaneous emission factor. We first consider a deterministic case i.e. β sp = 0. Finally, Φ x (t) and Φ y (t) are the feedback terms. We are considering in this paper two different kinds of feedback, isotropic feedback (IF) and rotated feedback(RF) described as follows:

IF : Φ x (t) = ηE x (t-τ )e -iω0τ Φ y (t) = ηE y (t-τ )e -iω0τ , (5)
RF : Φ x (t) = -ηE y (t-τ )e -iω0τ Φ y (t) = ηE x (t-τ )e -iω0τ . (6)
To evaluate the reservoir performance, we analyze two task-independent indicators, which are the computational ability [START_REF] Legenstein | Edge of chaos and prediction of computational performance for neural circuit models[END_REF] and the memory capacity [START_REF] Jaeger | Tutorial on training recurrent neural networks, covering bppt, rtrl, ekf and the" echo state network" approach[END_REF]. The computational ability allows identifying the best laser operating point for achieving the highest performance. It corresponds to the capacity of the system to differentiate two different inputs and gather two identical inputs. More specifically it is defined as the difference between the ranks of two square matrices. Each row is the "value" of one node, taken here as the total power

|E| 2 = |E x | 2 + |E y | 2 .
Each column is the last time step of a different run of the reservoir. To fill the first matrix, we feed the reservoir with different random inputs for each run. For the second matrix we first feed the reservoir with different random inputs and after 90τ (time for transient to vanish), we feed it with arbitrary chosen inputs which are the same for each run. The computational ability is then a value between 0 and the number of nodes which we normalize between 0 and 1 by dividing by the number of nodes. The normalized quantity allows to compare different architecture with different number of nodes. To have a first idea of the computational ability of our VCSEL-based reservoir, we have run numerical simulations with 400 virtual nodes (Fig. 2) scanning four different parameters: the node interdelay θ, the injection current of the VCSEL µ, the power of the injecting laser P inj , and the feedback strength η.

The results are plotted in Fig. 2 in the θ-µ plane, keeping the same value of the injected power (0.1 mW) and of the feedback strength (10 GHz). Figure 2 shows there is a region of parameters in which the computational ability is close to 1, i.e. for θ in [0.01 ns, 0.04 ns] and µ in [1.2, 1.5]. We will then keep θ = 0.02 ns for the delay between two nodes, and µ = 1.3 as a value for the injected current for the following tests. We notice that with 400 nodes and θ = 0.02 ns, the delay line is of 8 ns. This is comparable to the shortest delay line that has been used in all-optical reservoir computer with the same number of nodes [START_REF] Nguimdo | Reducing the phase sensitivity of laser-based optical reservoir computing systems[END_REF].

We also test the computational ability as a function of the feedback strength and the injected power with the two different types of optical feedback described in Eqs. 5-6. This allows us to compare the cases where the VCSEL is emitting along only one polarization mode to the one where the VCSEL is emitting in both orthogonal polarization modes. Indeed, with isotropic feedback, the VCSEL is lasing only along its main polarization axis. The rotated feedback allows to trigger lasing along the depressed axis as well. This is why we will now refer to the isotropic feedback as single-mode reservoir, and to the rotated feedback as the dual-mode reservoir.

Figures 3(a) and (b) show the computational ability of the single-mode and dual-mode reservoir computer, respectively. The region of parameters for which our VCSELbased reservoir has its highest computational ability, regardless of the feedback configuration. The large region of injection and feedback strength for which the highest computational ability is achieved can be explained by a close inspection into the VCSEL dynamics (not shown here). The lowest performance (darkest blue) occurs when the VCSEL exhibits chaos, whereas the best performance is obtained when the system is in a steady state and for parameters just before the first bifurcation point to periodic dynamics -a condition sometimes referred as "edge of chaos" [START_REF] Bertschinger | Real-time computation at the edge of chaos in recurrent neural networks[END_REF]. The undamping of the relaxation oscillation depends on the injected power and the feedback rate. For instance, for P inj = 0.1 mW, the required feedback rate for such dynamics is 31.5 GHz.

The added-value of the dual-mode dynamics is even more evidenced when analyzing the memory capacity, which represents the number of steps in the past the system can remember. The memory capacity is mathematically defined as follows:

µ c = ∞ i=0 corr [y i (k), u(k -i)] , (7)
where y i (k) is a reconstruction of u(k -i), which means the reservoir computer is trained in order to get the i th previous input. Theoretically, i should range from 0 to infinity, however, it was shown that the memory of a timedelay reservoir cannot exceed its number of virtual nodes. The maps in Fig. 4 compare the memory capacity of the single-mode and dual-mode reservoir computer. Tests have been performed as previously with 400 nodes, taking for each node |E| 2 , i.e. the total optical power. We consider 800 samples for the training set and 3200 samples for the testing set. As in Fig. 3, we have scanned the injected power and the feedback strength and displayed the map of memory capacity for both single-mode and dual-mode reservoirs (Fig. 4). Results show in both configurations a range of parameters for which high memory capacity is achieved. This region of parameters coincides with that of high computational ability. The highest achievable memory depth is approximately 16 steps in the past for the dual-mode system [Fig. 4(b)], which is twice as much as the one reached for the single-mode reservoir (around 8 steps in the past) [Fig. 4(a)]. This is comparable to some of the time-delay all-optical reservoir architectures showing currently the best results in terms of memory [START_REF] Duport | All-optical reservoir computing[END_REF]. Moreover, considering only |E x | 2 for each node output in the dual-mode system yields the same memory capacity. This highlights that the improved memory capacity is well the result of the polarization mode dynamics induced by the rotating feedback. The previous results suggest an enhancement of the overall computing performance of a VCSEL-based reservoir using the polarization mode dynamics. A more explicit proof is provided when testing our reservoir on a practical task. To this end, we have chosen to test our reservoir with the channel equalization task (Ref. [START_REF] John Mathews | Adaptive algorithms for bilinear filtering[END_REF]) that has important implications in telecommunications. The objective is to reconstruct from observations u(i) an original signal d(i) after it has propagated through a nonlinear channel. The original signal d(i) is a random sequence of values taken in {-3; -1; 1; 3}. This sequence first goes through a linear channel, which links the output to the input as follows:

q(i) = 0.08d(i + 2) -0.12d(i + 1) + d(i) + 0.18d(i -1) -0.1d(i -2) + 0.091d(i -3) -0.05d(i -4) (8) +0.04d(i -5) + 0.03d(i -6) + 0.01d(i -7).
Then, the signal is modified using the non-linear function:

u(i) = q(i) + 0.026q(i) 2 -0.11q(i) 3 + v(i), (9)
where v(i) is a Gaussian white noise with an adjusted signal-to-noise ratio (SNR) from 12 to 32 dB. Equation [START_REF] Nguimdo | Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics[END_REF] gives us the observed data u(i) that are used to infer the corresponding d(i) using reservoir computing.

The reservoir performance on that task is evaluated by the Symbol Error Rate (SER), which is defined as the ratio of the number of false symbols and the total number of symbols. To perform the nonlinear channel equalization at high speed, we have decided to use only 32 virtual nodes, while keeping the same delay between each two nodes θ = 0.02 ns. The delay line is now 0.64 ns, which potentially allows processing speed at 1.5 GSymbols/s. We use as an operating point for our reservoir µ = 1.3, η = 23 GHz, and P inj = 0.08 mW, which corresponds to the highest value of computational ability and memory capacity in both optical feedback configurations. The training is realised with 10,000 samples using a mean square regression. The testing is then realised on 50,000 samples. For each SNR value, the resulting SER is taken as the mean value over 10 different runs. Figure 5 compares the performance of VCSEL-based reservoirs, with single-( ) and dual-mode ( ) polarization dynamics. In both cases the training considers for each node the total laser power.. The dual-mode reservoir computer achieves better performance compared to the single-mode one: its SER always remains lower than that of the single-mode system. In the highest SNR case, the dual-mode dynamics achieves SER = 10 -4 , hence a reduction by a factor 5 compared to the single-mode case. To maximize the performance, we have also trained the dual-mode reservoir using the power of each polarization mode for each node ( ). By doing this, we virtually double the number of node without impacting the length of the delay line, and thus the processing speed. We succesfully reach a SER = 10 -5 for the highest SNR case, hence another reduction by a factor 10. This is similar to the performance of an optoelectronic reservoir [START_REF] Paquot | Optoelectronic reservoir computing[END_REF], but with a 10000 times higher bit rate. It is also an order of magnitude better than the performance of other all-optical reservoirs [START_REF] Nguimdo | Reducing the phase sensitivity of laser-based optical reservoir computing systems[END_REF][START_REF] Duport | All-optical reservoir computing[END_REF]. show that the SER is unchanged. We have also tested the influence of other parameters on the performance of the reservoir. First, we consider the impact of the quantization noise, which is thought sometimes to lower performance on an all-optical reservoir [START_REF] Soriano | Optoelectronic reservoir computing: tackling noiseinduced performance degradation[END_REF]. Here however, accounting for a 8-bit quantization noise improves the memory capacity to 18, but the SER for channel equalization is slightly higher (3 × 10 -5 for the 32 dB SNR case). The results may also depend on the feedback phase. For example, the memory capacity reaches 19 with a feedback phase of π/4 and 18 with a feedback phase of π/2, but the performance on channel equalization remains the same. Finally, we have also tested the influence of the remaining SFM model parameters by varying e.g. γ s from 50 GHz to infinity and γ p from 50 to 200 GHz. The results remain unchanged. This confirms that the increase of performance is mostly attributed to the two mode polarization dynamics induced by the rotating feedback.

We have also tested our dual-mode reservoir with other benchmark tasks such as the Santa Fe prediction task. This test has been performed with the same parameters as the one used for channel equalization task: µ = 1.3, θ = 0.02 ns, P inj = 0.7 mW, η = 22 Ghz, 400 nodes and including the same value for β sp . We successfully reached a normalized mean-square error of 10 -3 . This is one order of magnitude lower than the numerical results obtained with a similar architecture but using a single-mode edgeemitting laser [START_REF] Hicke | Information processing using transient dynamics of semiconductor lasers subject to delayed feedback[END_REF].

In summary, we have studied a reservoir computer, which benefits from the rich and fast polarization dynamics of a VCSEL with optical feedback. We find that the dual-mode dynamics in VCSELs induced by polarization rotated feedback improves the performance of the reservoir computer: it can not only reach high computational performance, but also show a deeper level of memory compared to a single-mode reservoir. This allows the system to show particularly good results very demanding tasks. For example, our VCSEL-based reservoir successfully achieves channel equalization with a SER of 10 -5 and at a data rate of 1.5 GHz.

Figure 2 :

 2 Figure 2: (color online) Computational ability plot as a function of the injection current µ and the nodes inter-delay θ for an injection power of 0.1 mW and a feedback strength of 10 GHz.

Figure 3 :

 3 Figure 3: (color online) Computational ability depending on the injection power Pinj and the feedback strength η. (a) single-mode system. (b) dual-mode system

Figure 4 :

 4 Figure 4: (color online) Memory capacity depending on the injection power Pinj and the feedback strength η. (a) single-mode system. (b) dual-mode system.

Figure 5 :

 5 Figure 5: (color online) Channel equalization. (a) Symbol error rate for different signal-to-noise ratio. : single-mode system. : dual mode system considering both |E| 2 . : dual-mode system considering |Ex| 2 and |Ey| 2 . (b) Example of reconstruction of a signal with a 24 dB-SNR: the signal sent in the channel (dotted blue), the non-linearly modified signal at the output of the channel (grey), the reconstructed signal (red). The arrow points to the single error in this data sequence

  Figure 5(b) shows an example of signal reconstruction, where we highlight the only mis-reconstructed symbol. To evaluate the robustness to noise of our dual-mode reservoir, we have increased the spontaneous emission rate to the realistic value β sp = 4.5 × 10 -4 ns -1 . Our simulations

Acknowledgements

The authors thank the founders of the Chaire Photonique: Ministère de l'Enseignement Supérieur et de la Recherche, Région Grand-Est, Département Moselle, European Union (FEDER), Metz Métropole, Airbus GDI Simulation, Cen-traleSupélec, and Fondation CentraleSupélec.


Kevin Dalleau
email: kevin.dalleau@loria.fr

Miguel Couceiro
email: miguel.couceiro@loria.fr

Malika Smaïl-Tabbone
email: malika.smail@loria.fr

Malika Smail-Tabbone

Unsupervised extremely randomized trees

Keywords: Clustering, unsupervised classification, decision tree, extremely randomized trees, similarity measure, distance

published or not. The documents may come

Introduction and preliminaries

Many unsupervised learning algorithms rely on a metric to evaluate the pairwise distance between samples. Despite the large number of metrics already described in the literature [START_REF] Deza | Encyclopedia of distances[END_REF], in many applications, the set of available metrics is reduced by intrinsic characteristics of the data and of the chosen algorithm. The choice of a metric may strongly impact the quality of the resulting clustering, thus making this choice rather critical. Shi and Horvath [START_REF] Shi | Unsupervised learning with random forest predictors[END_REF] proposed a method to compute distances between instances in unsupervised settings using Random Forest (RF). RF [START_REF] Breiman | Random forests[END_REF] is a popular algorithm for supervised learning tasks, and has been used in various settings ( [START_REF] Percha | Discovery and explanation of drugdrug interactions via text mining[END_REF][START_REF]Random forest classifier for remote sensing classification[END_REF]). It is an ensemble method, combining decision trees in order to obtain better results in supervised learning tasks. Let L = {(x 1 , y 1 ), . . . , (x n , y n )} be a training set, where X = {x 1 , . . . , x n } is a list of samples (i.e., feature vectors) and Y = {y 1 , . . . , y n } is the list of corresponding class labels. The algorithm begins by creating several new training sets, each one being a bootstrap sample of elements from X. A decision tree is built on each training set, using a sample of m try features at each split. The prediction task is performed by performing a majority vote or by averaging the results of each tree, according to the problem at hand (classification or regression). This approach leads to better accuracy and generalization capacity of the model, and it reduces the variance of single decision trees [START_REF] Friedman | The elements of statistical learning[END_REF].

The method proposed by Shi and Horvath, called Unsupervised RF (URF), derives from the common RF algorithm. Once the forest has been constructed, the training data can be run down each tree. Since each leaf only contains a small number of objects, and all objects of the same leaf can be considered similar, it is possible to define a similarity measure from these trees: if two objects i and j are in the same leaf of a tree, the overall similarity between the two objects is increased by one. This similarity is then normalized by dividing by the number of trees in the forests. In doing so, the similarities lie in the interval [0, 1]. The use of this RF is made possible in the unsupervised case thanks to the generation of synthetic instances, enabling binary classification between the latter and the observed instances. Two methods for data generation are presented in [START_REF] Shi | Unsupervised learning with random forest predictors[END_REF], namely, addCl1 and addCl2.

In addCl1, the synthetic instances are obtained by a random sampling from the observed distributions of variables, whereas in addCl2 they are obtained by a random sampling in the hyper-rectangle containing the observed instances. The authors found that addCl1 usually leads to better results in practice. URF as a method for measuring dissimilarity presents several advantages. For instance, objects described by mixed types of variables as well as missing values can be handled. The method has already been successfully used in fields such as biology ( [START_REF] Kim | Using tumor markers to predict the survival of patients with metastatic renal cell carcinoma[END_REF][START_REF] Abba | Breast cancer molecular signatures as determined by sage: correlation with lymph node status[END_REF][START_REF] Rennard | Identification of five chronic obstructive pulmonary disease subgroups with different prognoses in the eclipse cohort using cluster analysis[END_REF]) and image processing [START_REF] Peerbhay | Random forests unsupervised classification: The detection and mapping of solanum mauritianum infestations in plantation forestry using hyperspectral data[END_REF].

However, the method, albeit its appealing character, suffers from some drawbacks. Firstly, the generation step is not computationally efficient. Since the obtained trees highly depend on the generated instances, it is necessary to construct many forests with different synthetic instances and average their results, leading to a computational burden. Secondly, the synthetic instances may bias the model being constructed to discriminate objects on specific features. For example, addCl1 leads to forests that focus on correlated features. P.Geurts, D.Ernst and L. Wehenkel [START_REF] Geurts | Extremely randomized trees[END_REF] presented a novel type of ensemble of trees method that they called Extremely Randomized Trees (or ExtraTrees, for short). This algorithm is very similar to RF in many ways. In RF, both the instance and feature samplings are performed during the construction of each tree. In ExtraTrees (ET) another layer of randomization is added. Indeed, whereas in RF the threshold of a feature split is selected according to some purity measure (the most popular ones being the entropy and the Gini impurity), in ET these thresholds are obtained totally or partially at random. In addition, instead of growing the trees from bootstrapped samples of the data, ET uses the whole training set. At each node, K attributes are randomly selected and a random split is performed on each one of them. The best split is kept and used to grow the tree. The ET algorithm is described in Figure 1.

Two parameters are of importance in this algorithm: K, that we already defined, and n min , that is the minimum sample size for a node to be split. Interestingly, the parameter K, that takes values in {1, ..., n f eatures }, influences the randomness of the trees. Indeed, for small values of K, the dependence of the constructed trees on the output variables gets weak. In the extreme case Fig. 1. The figure presets the ExtraTrees algorithm as extracted from [START_REF] Geurts | Extremely randomized trees[END_REF] where K is set to 1 (i.e., only one feature is selected and randomly split), the dependence of the trees on the observed label is removed.

Following the tracks of [START_REF] Shi | Unsupervised learning with random forest predictors[END_REF] on URF, we propose to use ET with a novel approach where synthetic case generation is no longer necessary. This approach, that we call addCl3, consists of a random labeling of each instance. Using properties of ET that we will discuss below, it is possible to compute a good similarity measure from a dataset where addCl3 is applied. The method outperforms URF in running time, while giving similar or better clusters. This paper is organized as follows. After a description of the method in Section 2, we focus on the empirical evaluation on real-world datasets in Section 3, before reviewing of the method and giving some perspectives in Section 4.

Unsupervised Extremely Randomized Trees

Two methods are used for the generation of synthetic data: addCl1 and addCl2. In these methods, the generation consists in sampling in the observed data. The synthetic data is assigned a label, while the observed data is assigned another one, enabling binary classification between observed and synthetic examples. The novel method we propose and evaluate in this work that we refer to as addCl3, does not focus on the generation of synthetic instances, but on the generation of labels instead. In addCl3, the label generation runs as follows:

1. Let n obs be the number of instances in the dataset. A list containing n obs 2 times the label 0 and n obs -n obs 2 times the label 1 is generated. 2. For each instance in the dataset, a label is randomly sampled without replacement from the aforementioned list.

This procedure ensures that the label distribution is balanced in the dataset. However, this leads to the same problem arising with addCl1 and addCl2 : the results are highly dependent on the generation step, as different realizations of the instance-label association or of the synthetic data may lead to completely different forests. To circumvent this issue, one solution is to run multiple forests on multiple generated datasets, and to average the results. Shi and Horvath found out that averaging the results from 5 forests, with a total of 5000 trees leads to robust results. Moreover, instead of running multiple forests on many generated datasets, it may be possible -and computationally more efficient -to run a single forest with a large amount of generated data, if some care is taken regarding the reweighting of each class. This workaround, proposed by a reviewer in [START_REF] Shi | Unsupervised learning with random forest predictors[END_REF], is easier to implement when our generation scheme is used. Indeed, since we do not add new instances, it is not necessary to reweight each class. Instead, we propose to duplicate the original dataset multiple times and apply addCl3 to obtain a balanced dataset. This approach is evaluated in Section 3.

With addCl3, the construction of the trees no longer depends on the structure of the data. Indeed, when addCl1 or addCl2 are used, the forests are trained to distinguish between observed and synthetic instances. In addCl3, the labels being assigned randomly, two similar instances may be labeled differently and may fall in different leaves. However, using ET with the number of features randomly selected at each node K = 1, the construction of the trees no longer depends on the class label, as described in the previous section. Hence, ET seems to be a suitable algorithm to use with addCl3. Algorithm 1 describes the Unsupervised Extremely Randomized Trees (UET) method. The algorithm Build an extra tree ensemble(D) is given in Figure 1. A few parameters can influence the results of UET:

1. The number of copies of the original dataset n copies before applying addCl3.

2. The number of trees n trees . 3. The minimum number of samples for a node to be split n min .

Empirical evaluation

In this section, we investigate the influence of the parameters introduced at the end of Section 2, as well as the performance of the method.

Optimization of parameters

For each evaluation presented in this subsection, the following process is repeated 10 times:

1. A similarity matrix is constructed using UET. 2. This similarity matrix is transformed into a distance matrix using the relation

DIS ij = 1 -SIM ij , used in [2].
3. An agglomerative clustering (with average linkage) is performed using this distance matrix, with the relevant number of clusters for the dataset.

For each clustering, Adjusted Rand Indices (ARI) are computed. This measure quantifies the agreement between two partitions of a dataset, adjusted for chance [START_REF] Rand | Objective criteria for the evaluation of clustering methods[END_REF][START_REF] Hubert | Comparing partitions[END_REF]. ARI takes values in [-1, 1], where a value of 1 indicates perfect agreement up to a permutation, while a value of 0 indicates a result no better than a random label assignment. Three datasets are used for this evaluation process: Iris [START_REF] Fisher | Iris data set. RA Fisher[END_REF], Wine [START_REF] Forina | An extendible package for data exploration, classification and correlation[END_REF] and Wisconsin breast cancer [START_REF] Mangasarian | Cancer diagnosis via linear programming[END_REF]. These datasets are described Table 1. Influence of the number of copies of the dataset

The use of addCl3 on a dataset leads to a balanced distribution of the labels. Instead of running k forests on as many datasets where addCl3 is applied k times, it is possible to run one forest on a dataset duplicated k times. We evaluate here the influence of this duplication process. The results are presented Figure 2. The ARI are compared using the Kruskal-Wallis test [START_REF] Kruskal | Use of ranks in one-criterion variance analysis[END_REF]. The results show that the ARI does not differ significantly in Wine, Iris and Wisconsin datasets (p = 0.26, p = 0.09 and p = 0.23, respectively).

Intuitively, as UET grows the tree without any consideration of the labels and without bootstrapping the samples, the results should stay relatively constant when number of duplications grows. This replication was needed in URF as the generation scheme could lead to significant differences in the output similarity. This intuition is confirmed with this experiment. Here, the randomness induced by the labeling step does not induce a difference in the construction of the trees. Indeed, since we set K = 1, trees are constructed totally at random. Any difference in the similarity matrix is rather related to the randomness induced by the choice of features to split at each node.

Influence of the number of trees

The influence of the number of trees n trees as also been studied in [START_REF] Geurts | Extremely randomized trees[END_REF], where this parameter is referred to as the averaging strength M . For randomized method such as RF and ET used in a supervised learning setting, the average error is a monotonically decreasing function of M [START_REF] Breiman | Random forests[END_REF]. In our experiments, we observed no substantial gain for n trees > 50. The difference in ARI are not significant (p > 0.1) for all three datasets. This observation confirms the one from Geurts et al. where values of n trees > 40 outperforms Tree Bagging. However, as the time to construct the ensemble grows linearly with the number of trees, it is a good option to choose small a value of n trees . We chose the value n trees = 50 by default. We noticed that this value is way below the overall number of trees recommended for URF, 5000. The results are presented Figure 3. Influence of the minimum number of samples to split ET tend to produce trees having 3 to 4 times the number of leaves than those of RF. As UET computes similarities by counting the number of times objects fall into the same leaf, the results are impacted by this increase in the number of leaves. It might be useful to stop the growth of the trees, in order to group similar instances in the same leaves more often. The minimum number of objects to split a node n min can control this growth. This parameter n min , also called the smoothing strength, has an impact on the bias and the variance. As stated by Geurts et al. [START_REF] Geurts | Extremely randomized trees[END_REF], the optimal value for this parameter depends on the level of noise in the dataset. They showed in [START_REF] Geurts | Extremely randomized trees[END_REF] that larger values of n min are needed when ET is applied to noisy data. In UET, the noise is maximal, as the labels are assigned randomly. The results of the evaluations performed varying n min are presented below. For n min = 2, we observe that the method fails to compute a similarity matrix leading to a good clustering. Values of n min between 20% and 30% of the data seem to give better results. The ARI variations for the three datasets according to n min are presented Figure 4.

Comparative evaluation of UET

In this section we first evaluate the relevance of clusterings obtained using UET by comparing the Normalized Mutual Information [START_REF] Strehl | Cluster ensembles-a knowledge reuse framework for combining multiple partitions[END_REF] (NMI) scores with the values presented in [START_REF] Elghazel | Feature selection for unsupervised learning using random cluster ensembles[END_REF]. This reference was chosen because results were provided for many well-known datasets. Then, we compare UET and URF, using another quality score presented previously, ARI. The ten datasets used in this section are available on the UCI website 1 and presented Table 2. UET are computed with n trees = 50 and n min = n samples 3

.

Comparative evaluation with results from the literature

For each dataset, UET was run 10 times, and the similarity matrices were averaged. The obtained matrix was then transformed into a distance matrix using the equation DIS ij = 1 -SIM ij , and an agglomerative clustering with 1 https://archive.ics.uci.edu/ml/index.php the relevant number of clusters was performed. The quality of the clustering was then evaluated with respect to NMI. This process is run 20 times, and we provide the mean and standard deviation of the quality metric. This evaluation was performed using scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] and our implementation of UET. This implementation will be available upon request.

In [START_REF] Elghazel | Feature selection for unsupervised learning using random cluster ensembles[END_REF], NMI obtained by running k-means 20 times and averaging the results are provided for each dataset. We compare our results to the ones obtained without feature selection, as none has been performed in our setting. The results are presented Table 3. They show that NMI scores obtained using UET are competitive in most cases. It is noteworthy that in some cases, UET alone without feature selection gives better results than the ones obtained by [START_REF] Elghazel | Feature selection for unsupervised learning using random cluster ensembles[END_REF] after feature selection. For instance, this is the case for Breast tissue dataset.

Comparison with URF

To compare UET and URF, we used the R implementation provided by Shi and Horvath2 , and compared the ARI obtained after running the partitioning around medoids (PAM) algorithm on the distance matrices obtained by both methods. 2000 trees and 100 forests are used for URF, with a value of m try = √ n f eatures 3 . We set UET parameters to n trees = 50 and n min = n samples 3

, and averaged the similarity matrices of 20 runs. These experiments were run on a computer with an Intel i7-6600U (2.6 Ghz) and 16 Go of 2133 MHz DDR4 RAM. We compared both ARI and time (in seconds) for each method. The results are presented Table 4. UET outperforms URF time-wise, while giving similar or better clusterings. Regarding the Isolet dataset, we manually terminated URF's computation as we weren't able to obtain results in an acceptable amount of time on our machine. However, we performed the computation on a more powerful machine, and were able to obtain an ARI of 28.39.

Conclusion and perspectives

In this preliminary work, we presented a novel method to perform unsupervised clustering using decision trees. This approach extends the unsupervised random forest method, by using extremely randomized trees as a base estimator. In the former method, the generation of synthetic instances was needed. This generation can be performed by two different approaches, AddCl1 or AddCl2. With the approach we proposed here, the generation of instances is no longer necessary. Indeed, for some parameter choices, extremely randomized trees can be made independent of the labels. We therefore present a way to bypass the need for instance generation, AddCl3, where a label is randomly associated with each observation, which results in a significant reduction in running time.

A performance evaluation of our method showed that essentially one parameter influenced the results, the smoothing parameter n min . This is explained by the fact that higher values of n min give better results in the presence of noise. In our case, the data is highly noisy, as the labeling is a random process. We found that a value of n samples 4

≤ n min ≤ n samples 3

gives good clusterings. Other parameters, such as the number of trees per forest n trees did not influence much the results of the procedure for values of n trees > 50, while increasing the time to perform the procedure. An interesting finding is that it is no longer necessary to duplicate the dataset multiple times to improve the results. However, due to the randomness of the procedure, it is still necessary to average the results of multiple UET to decrease the variance.

We compared the quality of the clustering between our method and (i) results found in the literature and (ii) results obtained by URF on multiple datasets. The quality is measured by normalized mutual information or adjusted rand index, according to the metric available in the literature. This empirical evaluation gave promising results, with overall similar or better NMI and ARI. The advantages of our method over URF are twofold. First, the generation of synthetic data is no longer necessary. Second, the method is 1.5 to more than 10 times faster than URF.

However, there is still room for improvement. We are aware that we only have tested our method on a few small datasets so far. A comparison with other metrics on large synthetic and real-world datasets would be interesting. Moreover, one of the major advantages of using a decision tree-based method to compute a distance is that (i) it enables the use of mixed-type variables and (ii) missing data can be handled. In fact, the latter were our original motivation and they constitute topics for future work.

Algorithm 1 :

 1 Unsupervised Extremely Randomized Trees Data: Observations O Result: Similarity matrix S D ←-addCl3(O); T ←-Build an extra tree ensemble(D, K) // Here K = 1; S = 0 n obs ,n obs // Initialization of a zero matrix of size n obs ; for d i ∈ D do for d j ∈ D do S i,j = number of times the samples d i and d j fall in the same leaf node in each tree of T = {t 1 , t 2 , ..., t M }; end end S i,j = Si,j M ;

Fig. 2 .

 2 Fig. 2. ARI performing UET+addCl3 and agglomerative clustering on Wine (a), Iris (b) and Wisconsin (c) datasets when the number of copies of the dataset increases.

Fig. 3 .

 3 Fig. 3. ARI performing UET+addCl3 and agglomerative clustering on Wine (a), Iris (b) and Wisconsin (c) datasets when the total number of trees varies. The ARI remains relatively constant.

Fig. 4 .

 4 Fig. 4. ARI performing UET+addCl3 and agglomerative clustering on Wine (a), Iris (b) and Wisconsin (c) datasets when the min. number of samples to split increases. Last value corresponds to 110% of the samples in a dataset.

Table 1 .

 1 Properties of used datasets

	Dataset # samples # features # labels
	Iris	150	4	3
	Wine	178	13	3
	Wisconsin 699	9	2

Table 2 .

 2 Datasets used for benchmarking

	Dataset	# samples # features # labels
	Iris	150	4	3
	Wine	178	13	3
	Wisconsin	699	9	2
	Lung	32	56	3
	Breast tissue 106	9	6
	Isolet	1559	617	26
	Pima	768	8	2
	Parkinson	195	22	2
	Ionosphere	351	34	2
	Segmentation 2310	19	7

Table 3 .

 3 Comparative evaluation with the results from[START_REF] Elghazel | Feature selection for unsupervised learning using random cluster ensembles[END_REF]. Best obtained values are indicated in boldface. In case of a tie, both values are in boldface. Time comparison was not performed in this case.

	Dataset	UET -NMI Literature -NMI
	Wisconsin	72.95 ± 4.94 73.61 ± 0.00
	Lung	28.89 ± 5.76 22.51 ± 5.58
	Breast tissue 59.59 ± 1.03 51.18 ± 1.38
	Isolet	69.95 ± 1.20 69.83 ± 1.74
	Parkinson	21.06 ± 5.33 23.35 ± 0.19
	Ionosphere	13.48 ± 3.25 12.62 ± 2.37
	Segmentation 69.31 ± 1.51 60.73 ± 1.71

Table 4 .

 4 Comparative evaluation between URF and UET

	Dataset	UET (ARI -Time (s)) URF (ARI -Time (s))
	Wisconsin	79.30 -823.36 s	81.36 -1267.82 s
	Lung	10.81 -7.45 s	8.16 -89.32 s
	Breast tissue 40.35 -25.25 s	39.05 -94.55 s
	Isolet	33.44 -4589.36 s	* -* s
	Parkinson	17.37 -66.91 s	13.44 -252.12 s
	Ionosphere 8.54 -184.97 s	7.59 -722.92 s

https://labs.genetics.ucla.edu/horvath/RFclustering/RFclustering.htm

mtry is the number of variables used at each node when a tree is grown in RF.

Acknowledgements Kevin Dalleau's PhD is funded by the RHU FIGHT-HF (ANR-15-RHUS-0004) and the Region Grand Est (France).