To publish or not to publish? On the aggregation and drivers of research performance Abstract This paper presents a methodology to aggregate multidimensional research output. Using a tailored version of the non-parametric Data Envelopment Analysis model, we account for the large heterogeneity in research output and the individual researcher preferences by endogenously weighting the various output dimensions. The approach offers three important advantages compared to the traditional approaches: (1) flexibility in the aggregation of different research outputs into an overall evaluation score; (2) a reduction of the impact of measurement errors and a-typical observations; and (3) a correction for the influences of a wide variety of factors outside the evaluated researcher's control. As a result, research evaluations are more effective representations of actual research performance. The methodology is illustrated on a data set of all faculty members at a large polytechnic university in Belgium. The sample includes questionnaire items on the motivation and perception of the researcher. This allows us to explore whether motivation and background characteristics (such as age, gender, retention, etc.,) of the researchers explain variations in measured research performance. Universities and colleges are increasingly interested in evaluating the performances of their academic staff, both in terms of teaching performance and of research performance. Given the growing attention to research, this paper will focus exclusively on research performance. In particular, this paper presents (1) a flexible tool to evaluate the multiple dimensions of research performance and (2) relates research performance to individual characteristics, motivation, and employment conditions. Current literature on research evaluation mainly employs single-criterion measures, such as reputational ratings gathered by polls or peer reviews, number of publications (eventually in conjunction with a journal quality index or a citation index, e.g., Van Leeuwen et al. 2003; van Raan 1996) in a predefined set of refereed journals (e.g., Zamarripa 1995; Sax et al. 2002) , or citation counts (e.g., McCain and Turner 1989; Nederhof et al. 1993; Lee and Bozeman 2005) . Recently, several opponents have criticized such simplistic measures doubting whether they are able to accurately convey research performance. In their opinion, the nature of research is by far too complex to be grasped by one single output criterion. For instance, Martin (1996) note that few would dispute that research is multi-dimensional in terms of its nature and outputs. Avital and Collopy (2001) agree arguing that research performance is broader than only one-dimensional measures. They suggest that (compared to single-criterion measures) multidimensional instruments are less sensitive to systematic measurement error and biases created by researchers who adjust their behavior in an attempt to improve evaluations (Avital and Collopy 2001, p. 53) . Hattie and Marsh (1996) also argued that weighted measures of research performance may be preferable to single-criteria numbers.. 1 In view of these considerations, a multi-criteria measure seems more appropriate in assessments of researchers' performances.. However, the construction of a multi-criteria Research Evaluation Score (RES-score) is an intricate matter with, amongst others, two important conceptual and methodological difficulties to overcome 2 : (1) How should one weight and aggregate the different output criteria? Or, stated differently, how important are the several research outputs in the overall performance evaluation? Is it legitimate to assign a uniform set of weights over the several output criteria (i.e., equal/fixed weights)? Also, is it legitimate to apply a uniform set of weights to all evaluated researchers? Some researchers are clearly specializing in writing international books, while other are specializing in attracting research funding. Using the same weights for all researchers, would be considered as unfair within a research unit.. 3 (2) How should the RES-scores be adjusted for the impact of exogenous characteristics which are (often) beyond the control of the researcher? There are numerous findings in the academic literature which suggest that some background characteristics (e.g., age, gender, 1 Note that the use of multidimensional measures has not only been suggested for evaluating individual researcher performances, but also for evaluating research departments. Vinkler (1998 Vinkler ( , 2006 , Ruiz et al. (2010) , and Bonaccorsi et al. (2006) , for instance, developed composite (i.e., multidimensional) indicators for evaluating the research performances of research institutes. 2 Another conceptual difficulty is the choice of academic output criteria that are deemed appropriate to be present in the performance evaluation. Selecting the relevant output criteria is the duty of faculty board members, the evaluated researchers and experts in evaluation methods (primarily scientometricians). 3 The question is even more prominent in the application at hand (see below). Similar to 'new' (polytechnic) universities in the UK and the colleges in the US, objectives of researchers in the application are more diverse than in 'traditional' research departments (although some diversity might be present there as well). Some researchers are specializing in writing international books, others in attracting research funding. rank/tenure, time spent on teaching, department policy, etc.,) may have a significant impact on the research performance of academic staff (Bellas and Toutkoushan 1999; Marsh 1996, 2002; Lee and Bozeman 2005; Maske et al. 2003; Ramsden 1994; Sax et al. 2002; Chen et al. 2006) . Intuitively, researchers realize some conditions are more beneficial to productive research while other conditions are more detrimental. Yet, traditional RES-scores do not account for differences in these uncontrollable conditions. Consequently, these scores are inherently biased towards researchers working under more favorable conditions. With this 'bias' in mind, several practitioners and researchers have claimed that uncorrected scores tend to be unfair as they give an advantage to those who work in more constructive conditions. The opposite reasoning holds true for academics who work under less favorable conditions. In their case, it is more difficult to obtain a good performance level (and, hence, a good RES-score). Thus, the remark of Emery et al. (2003, p. 44) made with respect to teacher evaluation tools, also applies to evaluation instruments for faculty research performances: ''Any system of faculty evaluation needs to be concerned about fairness, which often translates into a concern about comparability. Using the same evaluation system [without properly accounting for the differences in research conditions] for everyone almost guarantees that it will be unfair to everyone''. Stated differently, unadjusted RES-scores are potentially flawed and, therefore, unreliable as a measure of researcher performance. However, we are unaware of any study which corrects RES-scores for heterogeneity in (potentially) influential characteristics and conditions not under the control of the evaluated researchers. 4 The contributions of this paper are threefold. A first contribution of this paper is the proposal of a global RES-score. This study adds to the extant literature by outlining the weighting issue in the construction of a composite RES-score. In Section ''The weighting issue'' and ''Methodology'', we advocate a methodology to construct RES-scores which does address the weighting and correcting issues. In particular, we suggest a specially tailored version of the non-parametric Data Envelopment Analysis model (DEA; Charnes et al. 1978) . The so-called Benefit of the Doubt model (BoD) allows for the aggregation of various dimensions of research performance while incorporating the relative importance of these dimensions (e.g., a publication in an A-journal is more valued than a B-journal). The core idea is that output criteria on which the evaluated researcher performs well compared to his/her colleagues, should weight more heavily than the output criteria on which he performs relatively poor. The rationale for doing so is that a good (poor) relative performance is considered to be an indication of a high (low) attached importance by the evaluated researchers. Similar to 'new' (polytechnic) universities in the UK and the colleges in the US, researchers in our application are more diverse than in 'traditional' research departments (although large diversity might be present there as well). Some researchers are, e.g., specializing in writing international books, others in attracting research funding. The BoD model accounts for this by endogenously weighting the research outputs. Using endogenous weights, the 'benefit-of-the-doubt' (BoD) model allows each researcher for a certain degree of specialization. As such, it avoids the subjectivity of fixed weights.. The BoD has already been extensively described and studied in the JRC-OECD Handbook on Constructing Composite Indicators (OECD 2008) . It has also been used to construct composite indicators in wide ranging fields such as the economy (e.g., the Internal Market Index; Cherchye et al. 2007a) , human development (e.g., the Human Development Index; Despotis 2005), technological development (e.g., the Technology Achievement Index; Cherchye et al. 2008) , creative economy (Bowen et al. 2008) , and sustainable development (e.g., Sustainable Development Index; Cherchye and Kuosmanen 2006) . Recently, the European Commission has used the BoD technique to gauge member states' performance with regard to the Lisbon objectives (European Commission 2004, pp. 376-378). 5 Secondly, this paper attempts to fill the gap in (1) estimating the impact (in both size and direction) of background conditions on the measured research performances, and (2) correcting the RES-scores for the (un)favorable conditions in which the researcher works. In particular, we examine whether productivity in research can be related to a set of items describing individual researcher motivations and perceptions with respect to teaching and research (as well as the nexus between both activities), personal characteristics (e.g., age, gender), and working conditions (e.g., retention, teaching load, and time for research). From the point of view of university management, both types of information are useful. For instance, evaluation scores and rankings are particularly helpful in personnel decisions (e.g., recruitment, reappointment, promotion, retention, and dismissal, etc.,). The explanatory information, on the other hand, provides insights on the exact impact of working conditions on research performance can guide university management in attempts to facilitate an environment that is more conducive to creativity and productive research. Thirdly, to illustrate the practical usefulness of the approach, we apply the model on a dataset collected at department 'Business Administration' of the Hogeschool Universiteit Brussel (Belgium) in the academic years 2006-2007 and 2007-2008 . This university college resembles in many ways to the 'new' (polytechnic) universities in the UK and the colleges in the US. In particular, it used to be an educational institution with exclusive focus on teaching, but recently, thanks to the Bologna reforms (and an academization process initiated by the Flemish Government), it became increasingly research-oriented. The large resemblance with higher education institutions in other countries implicates that the university college under study is an excellent example to illustrate the usefulness of the presented ''Methodology''. The data set comprises output (research) data on all 81 researchers. We matched this data set with administrative and survey data. The administrative data contains information on age, gender, doctoral degree, tenure, (official) teaching load, and (official) time for research. The data are further enriched with a questionnaire on the researcher's opinions and perceptions on research satisfaction and personal goals.. The remainder of the paper is organized as follows. While the next section discusses the ''The weighting issue'' and the advantages of our methodology, in ''Methodology'' section we present the basic DEA model as well as its robust (extreme observations and/or data 5 Other approaches to construct composite indicators of complex phenomena are also discussed in the JRC-OECD Handbook (and in other reports such as Business Climate Indicators (DG ECFIN), Economic Sentiment Indicators (EU), Composite Leading Indicators (OECD), General Indicator for Science and Technology (NISTEP, Japan), or Composite Index of Technological Capabilities (Archibugi and Coco 2004) . 6 Nevertheless, the methodology does not examine the potential reverse causality among the variables (e.g., time for research may be endogenous to research output). Examining the causality of the variables requires besides the use of instrumental variables, a less flexible parametric framework. We consider this as scope for further research. measurement errors) and conditional (heterogeneity among researchers) extensions; while the subsequent sections report ''The data'' and the ''Results''. In the final section, we offer some concluding remarks and some avenues for further research. Given the increasing attention to research, universities and colleges are increasingly interested in evaluating the research performances of academics. Contrary to traditional single criterion measures, such as number of publications and citation counts, we suggested multi-criteria measures as they are more able to grasp the complex nature of research performances. In particular, we proposed a specially tailored version of the 'benefit-of-thedoubt' (BoD) model (which is rooted in the popular non-parametric Data Envelopment Analysis approach). The model is used to (1) integrate the performances of researchers on several research criteria into one overall Research Evaluation Score (RES-scores) while accounting for researcher characteristics and motivations as well as working conditions, and (2) non-parametrically analyze the impact (both in terms of direction and size) of these characteristics, motivations and working conditions on the RES-scores. In the context of constructing fair and robust RES-scores, this BoD approach has several advantages. First, for each individual researcher, weights for the output criteria are chosen such that the highest RES-score is realized. Secondly, the BoD model is flexible to incorporate the opinion of the faculty board and other stakeholders (including scientometricians) in the built-up of the RES-scores through a priori specified weight restrictions. Thirdly, the BoD model can be adjusted such that the resulting RES-scores are less sensitive to influences of outlying or extreme observations as well as potential measurement error in the data. Finally, the BoD model can be extended to account for several background influences (i.e., researcher characteristics and motivations as well as working conditions). Particularly with Kortelainen (2008) . The results indicate that, on average, higher RES-scores are given to researchers who (a) are female, (b) have a PhD, (c) are affiliated with one or more other universities (here, mainly KU Leuven), (d) get more satisfaction out of doing research, (e) perceive that timing is a major constraint to improve their research, (f) rate their ability as a researcher higher. On the other hand, alternative examined background characteristics (i.e., researcher age, retention, research time, salary increase, and collaboration within own department) did not significantly influence measured RES-scores.. From point of view of the university management, this information is potentially useful. For instance, knowing that being affiliated to other universities can have a positive influence on research productivity, faculty boards may motivate researchers to collaborate with academics outside the own university. In addition, the board might consider stimulating researchers without a doctoral degree to attend a PhD program as this may enhance their research productivity in the future. Management can also attempt to improve the personnel policy and the working environment with an eye towards increasing research satisfaction and, thus research productivity. Finally, although sometimes claimed differently, (intrinsic) motivation is more important than salary increases.. Without doubt, the impact of research characteristics, motivations and employment conditions on research productivity will vary with the research area as well as other particular circumstances and conditions. Therefore, an interesting agenda for future research would be to apply the proposed BoD methodology in other evaluation settings to check for recurring patterns in the results. Similarly, it would be interesting to use the nonparametric BoD method to the data of previous parametric studies to compare the results. In case of different results, at first sight, the results of BoD could be preferred as no a priori assumptions are made. Another suggestion would be to extend the analysis with other potentially influential characteristics. Moreover, we believe that proposed method is wellsuited to study the complex research-teaching nexus more profoundly. Finally, although not being a consideration of this paper, we stress the importance of studying the exact mechanisms by which aforementioned characteristics, motivations and conditions influence RES-scores more in detail. Before estimating the research performance of 73 researchers at HUB by the outlined conditional, robust BoD model, we present the RES-scores as they would be computed by To publish or not to publish? 671 the traditional methods in the literature (i.e., by an arithmetic or weighted average, and without accounting for the background). The resulting RES-scores, as presented in Table 3 , indicate an on average low relative performance. As one single person obtained 51 output items, the research performance of the others seems rather bleak. We observe from Table 3 that some researchers obtain a RES-score of 0. These researchers did not publish any paper during the examined period and hence receive the lowest possible evaluation score. Even if the output items are weighted by the weights determined at the Faculty Board, as presented in the second column, the performance of most researchers seems rather poor (with an average RES-score of 0.159). According to this weighted model, 75% of the researchers could improve his/her weighted RES-score by approximately 81% if he/she would work as efficient as the most efficient researcher in the sample. If the research performance of the faculty members would be evaluated by the use of a similar computation method (recall from a previous footnote that the 'most efficient' researcher published 27 very similar reports), the RES-score would not been taken seriously. Moreover, there would be huge resistance in using similar RES-scores as an incentive device (e.g., reducing teaching load, increasing wage).. By allowing for 'personalized' and 'optimal' weight restrictions, the BoD model is clearly more attractive to the individual researchers. To a certain extent (i.e., the weight bounds), researchers are given some leeway in their publication outlets. As such, the BoD model is less restrictive than the arithmetic or weighted average. Moreover, in its robust version, the BoD model accounts for outlying observations (e.g., the researcher with 27 similar publications) without losing information due to removing researchers from the data set. Summary statistics, as presented in Table 3 , indicate that 75% of the researchers could increase their publication performance by at least 13% if they would publish as efficient as the best performing researchers (i.e., third quantile RES-value of 0.87). Similar as before, researchers who did not publish anything during the evaluation period obtain the lowest possible RES-score of 0.. The evaluated researchers may still feel a significant reluctance against the RES-scores if they do not account for the background of the researcher (particularly given the research and practical evidence that background characteristics can have a considerable influence on the opportunities to do research). Therefore, in Model 1, we allow for heterogeneity among researchers by using the conditional and robust BoD model. In a 'relative competition' (e.g., for personnel decisions), by comparing comparable researchers, the RES-scores can Table 4 be considered as 'more fair'. Besides the employment conditions as retention, teaching load and research time, Model 1 accounts for some researcher background characteristics as gender, age, PhD and guest researcher at KU Leuven (i.e., HUB recently joined the KU Leuven and, as such, some researchers are affiliated with KU Leuven). 23 In other words, Model 1 controls for both truly exogenous factors (such as age, gender) or factors which are exogenous to the researcher as they are a university decision (e.g., hiring faculty without PhD, retention). Although this set of background variables is not exhaustive, it contains the variables that the faculty board at HUB (i.e., a mixture of policy makers and researchers) consider as appropriate (although one can have distinct views). For example, given the particular situation of HUB, the faculty board feels that having a PhD plays a role in explaining research performances as having the degree was not a requirement for faculty members at HUB (although recently, all appointments are only PhD's). A similar observation yields for affiliated researchers with KU Leuven, as the affiliation depends on the collaboration between two research groups (as such, if there is not a similar research group at KU Leuven, it is impossible to be affiliated). Also the age and gender is considered to play a role in research performance. Accounting for this set of background variables, the conditional RES estimates increase dramatically. A larger group of researchers (75%) becomes significant while the median researcher can improve its research performance by Table 5 23 Note again that we do not attempt to investigate the causality of the variables. There might arise (serious) endogeneity among the variables (in that, e.g., better performing researchers obtain a higher retention). The conditional BoD methodology only attempts to correct the RES-scores for background characteristics which are often considered by researchers to be crucial to their performance (and, therefore, should be included to make a 'fair' comparison of researchers). As such, the significance level of the background variables, as presented in Table 4 , indicates which of the background variables influence the RES-scores.. To publish or not to publish? 673 21%. Table 4 presents the direction (i.e., favorable or unfavorable to the RES-scores) and impact (i.e., if the impact is not significant, the RES-scores do not account for this variable) of the background characteristics. As a first class of variables, consider the impact of the researcher characteristics. First, observe that female researchers, on average, have better research performances (as measured by RES-scores). This observation contrasts to previous parametric findings in the literature which indicated that men perform in general better than females (e.g., Ramsden 1994; Maske et al. 2003; Sax et al. 2002) or that there is no significant difference in research performances of men and females (e.g., Lee and Bozeman 2005) . Secondly, note the favorable and significant impact of having a PhD on the RESscores. Among others, Fox and Milbourne (1999) and Rodgers and Neri (2007) reported similar results. Thirdly, researchers who are affiliated at KUL (and, thus cooperate with researchers at KUL) realize higher RES-scores compared to their non-affiliated colleagues. This result confirms the findings of some previous parametric studies which indicate that research collaboration has a favorable influence on research performances (e.g., Ramos et al. 2007; Ramsden 1994; Godin and Gingras 2000) . Finally, researcher age seems to have no significant impact on obtained RES-scores. This is in line with the findings of some previous parametric studies (e.g., Ramsden 1994; Gupta et al. 1999) . However, as presented in Table 1 , there are also other (parametric) studies which claimed that the relationship between the researcher's age and performance is on average negative (e.g., Sax et al. 2002; Levin and Stephan 1991) .. As a second class of background variables, consider the influence of the employment conditions. Firstly, we observe a positive and significant influence of retention on measured RES-scores. Stated differently, academics who work full-time at HUB realize better research performers relative to their colleagues who only work part-time at HUB. This result confirms the finding of several parametric studies (e.g., Feldman 1983; Chen et al. 2006; Fairweather 2002) . Further, we find that researchers who spent more time on research are more productive (i.e., higher RES-scores). This positive association is in line with the majority of the previous (parametric) studies in the literature. In fact, we found only two studies that claimed a non-significant relationship (i.e., Liddle et al. 1997) . The (official) time spent on teaching is not significantly related to RES-scores. Although this contrasts to the findings of several studies and general believes (see Table 1 ), we found a few studies which observed similar outcomes (i.e., Marsh 1996, 2002; Olsen and Simmons 1996) .. To explore the impact of motivation on research performance, we estimate in Model 2 the RES-scores while accounting for the individual background characteristics and a set of motivational variables (as obtained from the questionnaires). From the larger questionnaire we performed among the evaluated researchers, we deduced 5 relevant questionnaire items which are described in Table 5 . Some summary statistics are provided in Table 2 . Recall that a value of 5 denotes 'strongly agree', while a value of 1 denotes 'strongly disagree'. The results, as presented in Table 4 , indicate that once accounted for motivation, employment conditions (i.e., retention, research time, and teaching time) do not longer play a role in the RES-scores (i.e., the bandwidth of the employment conditions becomes very large such that the impact is faded out). However, we do observe that the research characteristics 'gender', 'having a PhD' and 'guest researcher at KU Leuven' still have a considerable positive influence on research performances (as measured by RES-scores). Moreover, we find that the more satisfaction a researcher obtains from doing research, the better his/her research performances are. Several (parametric) studies denoted a similar positive association (Fox 1992; Chen et al. 2006; Sax et al. 2002) . The researcher's perception as to whether time is a major constraint to improving his/her research productivity is positively and significantly related to RES-scores. This result is somewhat counterintuitive because it is usually thought that researchers, who experience a time constraint in the improvement of their research productivity, are less productive. Further, academics who rate their ability as a researcher (under ideal conditions) high, on average, perform better compared to colleagues who have a less positive view on their own research abilities. In other words, an optimistic self-image (with respect to one's own research abilities) contributes positively to the research performances. Vasil (1992 Vasil ( , 1996 and Marsh and Hattie (2002) reported similar findings. Lastly, both having a salary increase and having most of its collaboration partners within the own institution (here, HUB) have non-significant influences on RESscores. The 'benefit-of-the-doubt' model ''The weighting issue'' section discussed some advantages of the proposed BoD model. The BoD model relies on the non-parametric DEA approach, which is an efficiency measurement technique originally developed by Farrell (1957) and put into practice by Charnes et al. (1978) . In essence, DEA is a non-parametric model and, hence, does not require any a priori knowledge on the 'functional form' of the production function (i.e., the production function underlying the studied phenomenon). Obviously, this non-parametric feature is important in the evaluation of complex phenomena where objective knowledge on the underlying structure is usually lacking. In comparison to the traditional DEAproblem, the only difference is that the development of an overall RES-score only requires a look at the individual performances of researchers in the different research criteria i (with i ¼ 1; . . .; q) (thus, considering the outputs without explicitly taking into account the input dimension). 11 In this latter context, Melyn and Moesen (1991) alternatively labelled this method as the 'benefit-of-the-Doubt'-approach, a label that originates from one of the remarkable features of DEA: information on the appropriate weights can be retrieved from the observed data themselves (i.e., endogenous weighting).. Specifically, the core idea is that output criteria on which the evaluated researcher performs well compared to his/her colleagues in the reference set ! should weight more heavily than the output criteria on which he/she performs relatively poor. 12 The rationale for doing so is that a good (poor) relative performance is considered to be an indication of a high (low) attached importance by the evaluated researchers. 13 For example, if, in comparison to his/her colleagues (i.e., all observations y j,i in the reference set !), the researcher under evaluation published a high number of papers in international journals this reveals that the researcher considers such publication to be of high importance. Consequently, his/ her performances should weight more heavily on this criterion (i.e., high weight w c,i ). In other words, for each researcher separately, BoD looks for the weights that maximize (minimize) the impact of the criteria where the researcher performs relative good (poor) compared to the other researchers. Hence, BoD-weights w c,i are optimal and yield the maximal RES-score.. 14 This gives the following linear programming problem for each researcher under consideration c: . The objective function (2) reveals the 'benefit-of-the-doubt' interpretation: the BoD model lets the data speak for themselves and endogenously selects those weights w c,i which maximize the RES-scores. Any other weighting scheme than the one specified by the BoD model would worsen RES c (y). This data-orientation is justifiable in the context of evaluating research performances where there is usually a lack of agreement among stakeholders (i.e., policy makers, researchers, etc.,), and uncertainty about the proper importance values of the research output criteria. This perspective clearly deviates from the current practices of using single-criterion measures or multiple-criteria as in (1) with or without a correction for the perceived quality.. Notice that the standard BoD model as in (2)-(2b) grants evaluated researchers considerable leeway in the definition of their most favourable weights w c,i . More precisely, only two (rather minimal) constraints have to be satisfied. A first one is the 'normalization' constraint (2a) that ensures that all RES-scores computed with the evaluated researcher's most favourable weights w c,i , can at most be unity (or, equivalently, 100%). Thus, we obtain 0 B RES j B 1 (j ¼ 1; . . .; c; . . .; n) with higher values indicating better overall relative research performances. The second 'non-negativity' constraint limits weights to be non-negative (hence, w c,i C 0). Apart from these restrictions (2a) and (2b), weights can be chosen completely free to maximize the RES-score of the evaluated researcher vis-à-vis the other researchers. However, in some situations, it can allow a researcher to appear as a brilliant performer in a way that is difficult to justify. For instance, while having publications in several research output criteria, some researchers may prefer to only consider one of these criteria (i.e., the one in which the researcher performs best relative to their colleagues) in the built-up of their RES-scores (thus, assigning zero-weights to all other criteria) without violating the two basic restrictions. In such research evaluations, global RES-score reduce to the researchers' performances on one single dimension. Another concern is that chosen BoD-weights may too much deviate from what stakeholders (i.e., the faculty board, evaluated academics) believe is appropriate. Without doubt, opponents of research evaluations will claim that RES-scores based on improper weights are not meaningful.. Fortunately, BoD models are flexible to incorporate additional restrictions. Formally, this involves adding the general weight constraint (2c) to the standard BoD model:. with W e denoting the set of permissible weight values defined based upon the opinion of selected stakeholders e [ E. Especially with an eye towards practical evaluations, it is crucial for the credibility and acceptance of RES-scores to define such weight restrictions based on stakeholder opinions (if available). Formally, the complete ordinal ranking of the importance values of the nine research output criteria, as agreed upon by the stakeholders (see below and Table 1 ), is presented as follows: . From a technical perspective, we have to adjust these additional weight restrictions for the potential presence of zero values in the evaluation data. Indeed, in one or multiple output dimensions researchers may not have been able to produce any publication during the evaluation period (hence, the associated y c,i 's are equal to zero). The endogenous weighting procedure of BoD will automatically assign a zero weight to such output criteria. However, in our evaluation procedure (with the additional ordinal weight restrictions as specified above), this standard procedure may lead to infeasibilities. Kuosmanen (2002) and Cherchye and Kuosmanen (2006) proposed a simple modification of the weight restriction to prevent this infeasibility: multiply the constraints by the product of the corresponding y c,i 's. 15 Formally, . In this adjusted version of the additional weight restrictions, a standard weight w c,i = 0 for an output criterion i with y c,i = 0 does no longer enforce other weights to be equal to zero. In cases where one or both of the associated y c,i 's equals zero, the restriction becomes redundant and hence has no further influence on the other restrictions in (4). If none of the associated y c,i 's are equal zero, then the adjusted version of the weight restriction reduces to the original restriction as in (3). Formally, the introduction of these adjusted weight restrictions the basic BoD model entails replacing the general weight constraint (2c) for (4).

O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Classification via Minimum Incremental Coding Length Abstract. We present a simple new criterion for classification, based on principles from lossy data compression.The criterion assigns a test sample to the class that uses the minimum number of additional bits to code the test sample, subject to an allowable distortion. We demonstrate the asymptotic optimality of this criterion for Gaussian distributions and analyze its relationships to classical classifiers. The theoretical results clarify the connections between our approach and popular classifiers such as maximum a posteriori (MAP), regularized discriminant analysis (RDA), k-nearest neighbor (k-NN), and support vector machine (SVM), as well as unsupervised methods based on lossy coding. Our formulation induces several good effects on the resulting classifier. First, minimizing the lossy coding length induces a regularization effect which stabilizes the (implicit) density estimate in a small sample setting. Second, compression provides a uniform means of handling classes of varying dimension.The new criterion and its kernel and local versions perform competitively on synthetic examples, as well as on real imagery data such as handwritten digits and face images. On these problems, the performance of our simple classifier approaches the best reported results, without using domainspecific information. All MATLAB code and classification results are publicly available for peer evaluation at

B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O


Spherical Radon transform and the average of the condition number on certain Schubert subvarieties of a Grassmannian We study the average complexity of certain numerical algorithms when adapted to solving systems of multivariate polynomial equations whose coefficients belong to some fixed proper real subspace of the space of systems with complex coefficients. A particular motivation is the study of the case of systems of polynomial equations with real coefficients. Along these pages, we accept methods that compute either real or complex solutions of these input systems. This study leads to interesting problems in Integral Geometry: the question of giving estimates on the average of the normalized condition number along great circles that belong to a Schubert subvariety of the Grassmannian of great circles on a sphere. We prove that this average equals a closed formula in terms of the spherical Radon transform of the condition number along a totally geodesic submanifold of the sphere.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Simultaneous clustering and classification over cluster structure representation Two main tasks in pattern recognition area are clustering and classification. Owing to their different goals, traditionally these two tasks are treated separately. However, when label information is available, such separate treatment can not fully explore data information. First, classification is not favored by the data cluster structure. Second, clustering is not guided by valuable label information. Third, the relationship of clusters and classes is not revealed. Contrary to this separate learning treatment, simultaneous learning clustering and classification could benefit each other and overcomes these problems.Recently, a simultaneous learning framework SCC was proposed. Through modeling pðclass9clusterÞ classification and clustering mechanism in SCC depend only on cluster centroids. However, it produces severely nonlinear objective, thus has to use a heuristic searching method, modified Particle Swarm Optimization, to find the optimal solution. But it is very slow. Further, modeling pðclass9clusterÞ makes SCC hard to incorporate semi-supervised settings.In this paper, we propose an alternative framework SC 3 SR for simultaneous learning. Besides a classifier derived on the original data, another classifier on the newly-formed cluster structure representation is derived as well. Through this classifier, the clustering learning is guided by the label and classification learning is also favored by cluster structure of data. The final objective is continuously differentiable for which some principled optimization algorithms with convergence guaranteed exist. As a result, our algorithm is much faster than SCC. Further, we generalize this framework to semisupervised situation with the idea of manifold regularization and propose SemiSC 3 SR algorithm. Our experiments demonstrate the effectiveness of both SC 3 SR and SemiSC 3 SR. In traditional pattern recognition area, two main tasks are clustering and classification [11] . Clustering task is unsupervised in general. It groups similar instances into several meaningful clusters, and is usually used to find out data intrinsic structure. Classification task is usually supervised. It first learns a classifier from training data with labels, and uses the learned classifier to attach suitable labels to testing data. These two tasks are usually treated separately, because both goals are quite different. However, when label information is available, simultaneous learning clustering and classification tasks will benefit from each other [5] .. First, classification task will be favored when data intrinsic structure is considered. In pattern recognition area, well-known No Free Lunch theorem tells that no classifier is innately superior to any other classifier unless it incorporates the prior knowledge [9] . For classification, two kinds of prior knowledge are usually considered. One is on classification function, for example, the function is usually assumed to be smooth to avoid overfitting [10, 24] . The other is on data distribution, of which there are two types of popular assumptions, cluster and manifold assumptions [7] . Cluster assumption assumes that the points of each class tend to form clusters, and manifold assumption assumes that data lie on a low dimension manifold. This paper focuses on simultaneous learning clustering and classification tasks, hence cluster assumption is used as data prior knowledge. It suggests that when data intrinsic structure revealed by some clustering algorithm is incorporated into the classification task, better performance may be desired [5, 13, 25 ].. Second, clustering task will be guided by valuable label information. For example, when clustering faces, the Euclidean distance between the faces from the same person may be larger than the Euclidean distance between the ones from the different persons due to varying lighting [20] . However, one may want to put the faces from the same person into his(her) own clusters and push the faces from the different persons away even they are under similar lighting condition. When label information is available, the faces from the different persons will be forcibly pushed apart according to the guidance of label information, consequently faces from the same persons and under the same lighting condition will be grouped together.. Third, the relation between clusters and classes can be revealed, so some meaningful insight between clusters and classes could be caught. For example, we can know whether the cluster contains single class data or not, and by which clusters the class is formed.. Recently, a simultaneous learning framework for clustering and classification SCC was proposed [5] . In order to bridge the clustering and classification tasks, SCC proposed to model cluster posterior probabilities of classes pðclass9clusterÞ. Then by assuming that class is conditionally independent with data given cluster, the classification mechanism was broken down to pðclass9instanceÞ ¼ P pðclass9clusterÞpðcluster9instanceÞ according to Bayesian formula. As a result the classification and clustering tasks are naturally and seamlessly fused together, and the relation between class and cluster is revealed explicitly by pðclass9clusterÞ. In its implementation, SCC estimates pðclass9clusterÞ and pðcluster9instanceÞ only by cluster centroids and designs the objective with cluster centroids be sole argument. However, it is highly nonlinear [5] so hard to find the local optimal solution. In fact, SCC has to resort to a heuristic searching method, modified Particle Swarm Optimization method, to find the local optimal solution. But it is slow, sensitive to initial settings and has no guarantee of convergence. Furthermore, SCC is not capable of semi-supervised situation, because for unlabeled data, pðclass9clusterÞ cannot be estimated [5] .. In this paper, we propose an alternative framework called SC 3 SR (Simultaneous Clustering and Classification over Cluster Structure Representation) for simultaneous learning. The key point connecting clustering and classification tasks is classifier designed on cluster structure representation (CSR) rather than cluster posterior probabilities of classes pðclass9clusterÞ. Actually, besides the classifier designed on the original data, another classifier on its CSR is also jointly designed. For their consistence, their disagreement is minimized. Consequently, the classifier designed on the original data is influenced by the data cluster structure, while data clustering procedure is also guided by label information. The final objective is continuously differentiable with blockwise arguments, thus can be optimized through block coordinate descent algorithm which guarantees that the iteration procedure converges to a stationary point [2] . Experiments demonstrate that our algorithm is much faster than SCC. The difference with SCC is that our algorithm can implicitly rather than explicitly give out the relation between cluster and class pðclass9clusterÞ. But our framework is more flexible. Since SC 3 SR derives two classifiers from both the original data and the CSR, it could be easily generalized to deal with semi-supervised situation with the idea of manifold regularization [1] . Consequently we also develop incidentally SemiSC 3 SR algorithm in this paper and preliminarily examine its effectiveness by comparing with some popular semi-supervised algorithms. Furthermore, our framework can also be viewed as a dimension reduction method since the learned CSR can be also taken as a reduced-dimensionality representation of the original data. Finally, we enumerate the differences between SCC and SC 3 SR in Table 1 and give out our contributions as follows:. Propose a new simultaneously learning classification and clustering algorithm SC 3 SR which is much faster and more flexible than SCC.. Generalize SC 3 SR to semi-supervised situation and develop SemiSC 3 SR algorithm.. Examine their effectiveness and efficiency through experiments.. The rest of paper is organized as follows: in Section 2, some related works are reviewed. In Section 3, Both SC 3 SR and SemiSC 3 SR are proposed. Experimental results are presented in Section 4. And Section 5 concludes this paper. In this paper, a new simultaneous learning clustering and classification framework SC 3 SR is presented. In this framework, cluster structure representation is proposed to bridge the clustering and classification tasks, so the performances of clustering and classification tasks could benefit from each other. This is different with the SCC framework proposed by Cai et al. [5] which models cluster posterior probabilities of classes and bridges the clustering and classification tasks by Bayesian formula. Comparing with SCC, the formulation of SC 3 SR is easier to optimize. The block coordinate descent method is used to optimize SC 3 SR under guaranteed convergence. It is worth mentioning that extending SCC to semisupervised scenario is not straightforward [5] because for unlabeled data, pðclass9clusterÞ cannot be estimated, however extending SC 3 SR to semi-supervised scenario is relatively easier by the idea from manifold regularization framework [1] . Furthermore, SC 3 SR can also perform dimension reduction tasks. The learned low dimensional cluster structure representation can naturally reflect the original data structure. However, it is not so obvious for SCC to generate dimension reduced data representation. In the next step, we want to examine the influence of different CSRs on SC 3 SR framework. Consider the symmetry of W x and W h in the objective, the gradient and Hessian of W h can be derived by simply exchanging x with h in the W x 's equations. We omit the lengthy mathematical deduction here.. The gradient of h i for i¼ 1,y,N is defined as follows: has one more l 3 term. Thus by adding the gradients and Hessian matrices of the l 3 term to those of SC 3 SR, we get the corresponding SemiSC 3 SR's. We give out the gradients and Hessian matrices, marked by tilde, of the l 3 term here.. For notation convenience, we let logðpðc ¼ m9ðx=hÞ i ÞÞ ¼ q x=h mi and logðp x=h m: Þ ¼ ½q x=h m1 , . . . ,q x=h mN . W is the graph weight matrix and L ¼ TÀW is the Laplacian matrix where T is diagonal matrix with T ii ¼ P j W ij . Let t ¼ ½T 11 , Á ,T NN . L i is the ith column of Laplacian matrix L and e ¼ ½1, . . . ,1.. The gradient of l 3 term with respect to W x is. T withg w x m defined as follows: Much effort has been devoted to connecting clustering and classification tasks together, most of which treat clustering and classification in a sequential manner.. In popular 3-layer radial basis function neural networks (RBFNN) [18, 25] , clustering algorithms like k-means and fuzzy c-means (FCM) are first performed to help to determine the parameters of hidden layer, next the connection weights between the hidden layer and output layer are optimized by minimizing the mean square error between the target and actual outputs. Better generalization is obtained through incorporating clustering information [5] . However, clustering task does not benefit from label information [5] .. In fuzzy relational classifier (FRC) algorithm proposed by Setnes et al. [22] , the training data are first clustered by FCM, then a fuzzy relation between clusters and given classes is artificially established explicitly by means of a fuzzy composite operator. The whole procedure is illustrated in Fig. 1 [22] . Lately, Cai et al. [4] presented robust FRC(RFRC) algorithm to enhance the robustness of FRC from two aspects. First, the authors used the robust kernel FCM [8] as the clustering tool; second motivated by fuzzy k-nearest-neighbor algorithm [12] , they replaced hard class label with the soft class label to construct their fuzzy relation, as a result significantly boosting the robustness and accuracy of FRC. However, a common disadvantage in FRC and RFRC is that the fuzzy relation between clusters and classes is hard to optimize due to the indifferentiability and complexity of the composite operators and lacks the probabilistic interpretation thus fails to reveal the reliable relation between clusters and classes. Unlike the RBFNN, FRC and RFRC algorithms which conduct clustering task in unsupervised manner, some algorithms take supervised clustering algorithm to aid the classification tasks. VQþLVQ3 algorithm proposed by Kim et al. [13] utilizes clustering method to reduce the computation burden in nearest neighbor classifier while without sacrificing the classification performance. In the clustering procedure, LVQ3 algorithm which employs the label information is used to determine the cluster center and cluster labels. Similar to VQþ LVQ3, CCAS [26, 14] and its extension ECCAS [15] also employ a supervised clustering procedure to find out a set of prototypes. Both VQ þLVQ3 and (E)CCAS algorithms use nearest neighbor classifier in classification phrase, they actually do not have training phrases. The common idea is to find out the best prototypes as the class representatives for the subsequent nearest neighbor classifier.. Recently, Cai et al. [5] proposed a simultaneous learning clustering and classification algorithm named SCC. In SCC, the clustering and classification tasks are learned simultaneously in one framework. Through modeling cluster posterior probabilities of classes pðclass9clusterÞ, SCC succeeds in acquiring robust classification and clustering simultaneously and in revealing the underlying relation between clusters and classes. However, the objective of SCC is hard to optimize, thus SCC has to resort to a heuristic searching method, modified Particle Swarm Optimization method, to find the local optimal solution. Further, SCC cannot deal with semi-supervised situation because pðclass9 clusterÞ could not be estimated without labels.. The formulation of SCC combines the classification and clustering criteria into a single objective with a trade-off parameter, thus will make compromise between the clustering performance for the classification performance. Later Cai et al. [6] remedied this problem by optimizing the classification and clustering criteria together through multi-objective optimization method rather than the combined single one. However the algorithm, called MSCC, still inherits some flaws of SCC, despite achieving better performance. For example, it is hard to optimize thus has to resort to multi-objective particle swarm optimization method, and can not deal with semi-supervised situation. In this paper, we propose a new formulation to overcome these flaws. Our framework takes the single objective like SCC, thus comparing with MSCC is unfair to our framework because our framework does not take the more effective multi-objective framework. Yet, it can also be generalized into the multi-objective one, but this work is out of our paper's scope.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A modular network scheme for unsupervised 3D object recognition This paper presents an unsupervised learning scheme for recognizing 3D objects from their 2D projected images. The scheme consists of a mixture of nonlinear autoencoders which can compress various views of 3D objects into representations that indicate the view direction. We evaluate the performance of the proposed modular network scheme through simulations using 3D wire-frame objects and discuss its related issues on object representations in the primate visual cortex. The human visual system can recognize 3D (three-dimensional) objects from their 2D (two-dimensional) retinal images even though the images will signi"cantly vary as the viewpoint changes. An important problem in 3D object recognition is therefore how to accomplish view invariant recognition. The traditional approach is to recover the 3D shape and to construct structural descriptions of 3D volumetric primitives in the object-centered coordinate frames ( [14] , for example). Nevertheless, due to the complexity of 3D shape processing, the reliable construction of volumetric primitives is often di$cult and time-consuming. Alternatively, recent computational studies have explored view-based approaches, where 3D objects are recognized more directly from their 2D projected views ( [16, 18] , for example). In these approaches, a limited number of view examples are learned to achieve view invariant recognition through a generalization property of the trained systems. View-based models are attractive not only for their computational e$ciency and simplicity, but also for their biological relevance. Psychophysical studies have indicated that representations of 3D objects in the human visual system are viewpoint-speci"c [2, 5] . In addition, recent electrophysiological experiments on behaving monkeys suggest that the primate inferotemporal cortex (IT) employs viewer-centered object representations [12, 13] .. While most existing view-based models for 3D object recognition employ supervised learning, this paper focuses on unsupervised aspects of object recognition. In the real world, a given view is not always labeled explicitly as belonging to an object. Even when an identity for di!erent views of an object is provided through motion, only a fraction of all of these views is usually given at each temporal sequence. The human visual system seems to have a mechanism that can automatically cluster fragmental pieces of object views without using explicit object labels. Recent cognitive science studies have focused on the implicit learning ability of humans, where knowledge is automatically or unconsciously acquired without any explicit intention or instruction [25] . We therefore investigate a system that discovers object identities by itself solely from the input views of various 3D objects.. To achieve unsupervised object learning, we propose a modular network scheme which consists of a mixture of nonlinear autoencoders. Each autoencoder compresses a set of images of an object to form a manifold in a low-dimensional subspace. The compressed representations describe the degree-of-freedom of the input variations, which speci"cally represents the view direction or the pose of the 3D object. For the automatic clustering of various input views, we formulate an unsupervised modular learning algorithm, which is an extension of the adaptive mixture model [9}11] . The adaptive mixture model was devised for the competitive decomposition of a function approximation task based on maximum likelihood estimation. Unlike the original supervised formulation, the proposed unsupervised formulation does not incorporate a gating network which directly splits the input space.. There is a computational motivation for using a modular structure for unsupervised object classi"cation. When the input data for recognition is high-dimensional, it is essential to reduce the dimensionality and extract intrinsic information from the input variations. A number of techniques have been proposed for dimensionality reduction such as principal component analysis (PCA) [4] . In general, however, how the dimensionlity is reduced depends on the input data set to be used. For example, PCA representations are greatly altered if another set of data is included in the input data. In most data compression applications, all of the available data is simply used or a set of data is manually de"ned. The proposed scheme, on the other hand, performs clustering and compression at the same time, i.e., while the input space is divided into subspaces, the scheme performs dimensionality reduction within each subspace. Speci"cally, in the case of 3D object recognition, the scheme clusters various views of 3D objects into individual object classes and each module compresses the views in each object class to form a compact subspace. This paper is organized as follows. Section 2 describes details of the proposed unsupervised learning scheme. Section 3 examines the performance of the scheme through computer simulations. We used synthetic 3D wire-frame objects for the simulations, which have commonly been used in computational studies [18] , psychophysics [2, 5] , and electrophysiology [12, 13] . Section 4 discusses some related computational and biological issues and concludes this paper. The earlier versions of this research were presented in [1, 20, 21] . This paper expounds upon these preliminary results with additional simulations on the e!ect of varying the number of objects. One advantage of using a mixture of nonlinear autoencoders for recognition tasks is that the mixture scheme has the potential ability to deal with complex geometrical distributions of the data set in a high-dimensional input space. Speci"cally, in the case of 3D object recognition, we presume that a distribution of various views of a 3D object should be highly nonlinear in the input space so that the input views of di!erent 3D objects form manifolds that cannot be easily partitioned by hyperplanes. We investigated the linear separability of the view distributions of 3D objects to con"rm this presumption. Using the same data set employed in the simulations described in the previous section, we performed supervised training of a simple perceptron that contains no hidden layer. During the training, the binary teacher signal was provided to the output units of the network, which indicates an object identity of the input view. According to the convergence theorem of a simple perceptron, the network should converge to the correct data classi"cation within a "nite-learning period if the data sets are linearly separable [15] . Nonetheless, the simulation results showed that the classi"cation error did not approach zero, which suggests that the views of di!erent objects cannot be simply separated by hyperplanes. This observation indicates the e!ectiveness of the proposed modular scheme on complex clustering problems that are not linearly separable. We also note that any view of a 3D object can be described as a linear combination of a small number of other views [24] . This property, however, is only valid for x and y image coordinates under orthographic projection, while the proposed scheme can use di!erent types of input information (Section 2.2); in addition, there seems to be no straightforward way of using the linear combination property for unsupervised view clustering.. The proposed modular architecture may have some implications for the nature of object representations in the cortical areas of primates. Data from unit recording experiments have shown that cells in the anterior inferotemporal (IT) cortex selectively respond to moderately complex object features, and cells that respond to similar features cluster in a columnar region [6, 22, 23] . A recent physiological study using an optical imaging technique has revealed that in the primate inferotemporal cortex the activation spot gradually shifts as the viewing angle of a face changes; thus, representations of di!erent views of a face may cluster in a modular region [28] . Single-unit recording experiments have also suggested that 3D objects are represented in a viewer-centered coordinate frame in the IT area [12, 13] . These observations indicate that the extraction of intrinsic information on the object patterns and the clustering of extracted object representations to form multiple modular sub-regions occur in the anterior IT cortex.. However, it is still not clear how such intrinsic representations are extracted and self-organized into a modular structure in the IT area. The proposed network scheme may provide a useful framework for investigating neural mechanisms of modular organization in the cortex. In particular, e!ective clustering results of the proposed scheme suggest that competition among the modules as well as dimensionality reduction within each module may be possible underlying mechanisms of modular organization of 3D object representations in the cortex. In order to construct a more biologically feasible model, the proposed scheme should incorporate more elaborate functions in the future. Such functions include an automatic determination of the number of modules and the dimension of encoded information [3] , a representation of encoded information using population codes [29] , and a hierarchical construction of modules that encode local features in the lower level and object categories in the higher level.. To conclude, we have presented an unsupervised learning scheme for clustering 2D projected views of 3D objects without using any explicit identi"cation of the object classes. The scheme consists of a mixture of nonlinear autoencoders which extract view direction information from the object images. The results of simulations using 3D wire-frame objects demonstrated that the scheme is more e!ective in clustering 2D object views compared with the traditional K-means algorithm. The proposed modular scheme may thus provide a useful architecture in further investigations on complex pattern clustering and in studies on how object representations are acquired in the primate visual cortex.

O O O O O O B I I O O O O B I O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof Abstract-New measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states. One is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state's extension and takes the infimum over such extensions. Additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related. The second one is based on taking classical extensions. This scheme is generalized, which enables to construct measures of entanglement based on the mixed convex roof of a quantity, which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states. As one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key. The findings are finally related to analogous results in classical key agreement. Many bipartite entanglement measures are known, but few are known for multipartite states. Among the known bipartite entanglement measures, the squashed entanglement is distinct from other measures in its additivity and interesting construction [1] , [2] . It is based on quantum conditioning. In general, we do not know how to condition upon a quantum register. However if a function is built out of entropies, then there is a more or less magic way of conditioning, which is based on the fact that, accidentally, conditioning classical entropies can be represented by subtracting the conditioning entropy from the joint entropy -the entropy of the state on A conditioned by the state on B can be written formally as

B I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Local approximation schemes for topology control This paper presents a distributed algorithm for wireless adhoc networks that runs in polylogarithmic number of rounds in the size of the network and constructs a lightweight, linear size, (1 + ε)-spanner for any given ε > 0. A wireless network is modeled by a d-dimensional α-quasi unit ball graph (α-UBG), which is a higher dimensional generalization of the standard unit disk graph (UDG) model. The d-dimensional α-UBG model goes beyond the unrealistic "flat world" assumption of UDGs and also takes into account transmission errors, fading signal strength, and physical obstructions. The main result in the paper is this: for any fixed ε > 0, 0 < α ≤ 1, and d ≥ 2 there is a distributed algorithm running in O(log n·log * n) communication rounds on an n-node, d-dimensional α-UBG G that computes a (1 + ε)-spanner G of G with maximum degree Δ(G ) = O(1) and total weight w(G ) = O(w(MST (G) ). This result is motivated by the topology control problem in wireless ad-hoc networks and improves on existing topology control algorithms along several dimensions. The technical contributions of the paper include a new, sequential, greedy algorithm with relaxed edge ordering and lazy updating, and clustering techniques for filtering out unnecessary edges.ad-hoc wireless networks. In this paper we present a fast distributed algorithm for constructing a linear size, lightweight t-spanner of bounded degree for any given t > 1, on wireless networks. Below, we describe our result more precisely. Let G = (V, E) be a graph with edge weights w : E → R + . For t ≥ 1, a t-spanner of G is a spanning subgraph G of G such that for all pairs of vertices u, v ∈ V , the length of a shortest uv-path in G is at most t times the length of a shortest uv-path in G. The problem of constructing a sparse t-spanner, for small t, of a given graph G has been extensively studied by researchers in distributed computing and computational geometry and more recently by researchers in The results presented in this paper apply to α-UDGs embedded in constant-dimension Euclidean spaces, and do not directly generalize to doubling metric spaces. For low dimensional doubling metric spaces, we believe it possible to construct an O(log * n)-round distributed algorithm that produces a (1 + ε)-spanner with constant maximum degree. However, new techniques may be needed for lightweight spanners; the techniques presented in this paper use a key property (the leapfrog property) that does not seem to generalize to metrics of low doubling dimension.

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A second order algorithm for orthogonal projection onto curves and surfaces In this paper we analyze an algorithm which solves the point projection and the "inversion" problems for parametric curves and surfaces. It consists of a geometric second order iteration which converges faster than existing first order methods, and whose sensitivity to the choice of initial values is small. Applications include the ICP algorithm for shape registration. This paper investigates point projection and point inversion on parametric curves and surfaces by using curvature information. Experimental results show that the algorithms under consideration are robust and efficient. Applications to shape registration are discussed.

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Off-line test selection with test purposes for non-deterministic timed automata Abstract. This article proposes novel off-line test generation techniques from non-deterministic timed automata with inputs and outputs (TAIOs) in the formal framework of the tioco conformance theory. In this context, a first problem is the determinization of TAIOs, which is necessary to foresee next enabled actions after an observable trace, but is in general impossible because not all timed automata are determinizable. This problem is solved thanks to an approximate determinization using a game approach. The algorithm performs an io-abstraction which preserves the tioco conformance relation and thus guarantees the soundness of generated test cases. A second problem is the selection of test cases from a TAIO specification. The selection here relies on a precise description of timed behaviors to be tested which is carried out by expressive test purposes modeled by a generalization of TAIOs. Finally, an algorithm is described which generates test cases in the form of TAIOs equipped with verdicts, using a symbolic co-reachability analysis guided by the test purpose. Properties of test cases are then analyzed with respect to the precision of the approximate determinization: when determinization is exact, which is the case on known determinizable classes, in addition to soundness, properties characterizing the adequacy of test cases verdicts are also guaranteed. Conformance testing is the process of testing whether some implementation of a software system behaves correctly with respect to its specification. In this testing framework, implementations are considered as black boxes, i.e. the source code is unknown, only their interface with the environment is known and used to interact with the tester. In formal model-based conformance testing, models are used to describe testing artifacts (specifications, implementations, test cases, ...). Moreover, conformance is formally defined as a relation between implementations and specifications which reflects what are the correct behaviors of the implementation with respect to those of the specification. Defining such a relation requires the hypothesis that the implementation behaves as a model. Test cases with verdicts, which will be executed against the implementation in order to check conformance, are generated automatically from the specification. Test generation algorithms should then ensure properties relating verdicts of executions of test cases with the conformance relation (e.g. soundness), thus improving the quality of testing compared to manual writing of test cases.. For timed systems, model-based conformance testing has already been explored in the last decade, with different models and conformance relations (see e.g. [22] for a survey), and various test generation algorithms (e.g. [8, 18, 21] ). In this context, a very popular model is timed automata with inputs and outputs (TAIOs), a variant of timed automata (TAs) [1] , in which the alphabet of observable actions is partitioned into inputs and outputs. We consider here a very general model, partially observable and non-deterministic TAIOs with invariants for the modeling of urgency. We resort to the tioco conformance relation defined for TAIOs [17] , which is equivalent to the rtioco relation [19] . This relation compares the observable behaviors of timed systems, made of inputs, outputs and delays, restricting attention to what happens after specification traces. Intuitively, an implementation conforms to a specification if after any observable trace of the specification, outputs and delays observed on the implementation after this trace should be allowed by the specification.. One of the main difficulties encountered in test generation for those partially observable, non-deterministic TAIOs is determinization. In fact determinization is required in order to foresee the next enabled actions during execution, and thus to emit a correct verdict depending on whether actions observed on the implementation are allowed by the specification model after the current observable behavior. Unfortunately, TAs (and thus TAIOs) are not determinizable in general [1] : the class of deterministic TAs is a strict subclass of TAs. Two different approaches have been taken for test generation from timed models, which induce different treatments of non-determinism.. • In off-line test generation test cases are first generated as timed automata (or timed sequences, or timed transition systems) and subsequently executed on the implementation. One advantage is that test cases can be stored and further used e.g. for regression testing and serve for documentation. However, due to the non-determinizability of TAIOs, the approach has often been limited to deterministic or determinizable TAIOs (see e.g. [15, 21] ). A notable exception is [18] where the problem is solved by the use of an over-approximate determinization with fixed resources (number of clocks and maximal constant): a deterministic automaton with those resources is built, which simulates the behaviors of the non-deterministic one. Another one is [10] where winning strategies of timed games are used as test cases.. • In on-line test generation, test cases are generated during their execution. After the current observed trace, enabled actions after this trace are computed from the specification model and, either an allowed input is sent to the implementation, or a received output or an observed delay is checked. This technique can be applied to any TAIO, as possible observable actions are computed only along the current finite execution (the set of possible states of the specification model after a finite trace, and their enabled actions are finitely representable and computable), thus avoiding a complete determinization. On-line test generation is of particular interest to rapidly discover errors, can be applied to large and test purposes and prove some properties on generated test cases. Section 5 discusses some issues related to test case execution and test purposes and some related work. . Two OTAIOs are said equivalent if they have the same sets of traces. Let σ ∈ (R ≥0 Σ A obs ) * .R ≥0 be a trace, and s ∈ S A be a state,. = σ} denotes the set of states where A can stay after observing the trace σ.. = t} is the set of enabled delays in s with no observable action.. −→ A }) for the set of outputs and delays (respectively inputs) that can be observed from s. For S ⊆ S A , out(S ) = s∈S out(s) and in(S ) = s∈S in(s). Using these last definitions, we will later describe the set of possible outputs and delays after the trace σ by out(A after σ).. Notice that all notions introduced for OTAIOs apply to the subclass of TAIOs. In this article, we presented a complete formalization and operations for the automatic offline generation of test cases from non-deterministic timed automata with inputs and outputs (TAIOs). The model of TAIOs is general enough to take into account non-determinism, partial observation and urgency. One main contribution is the ability to tackle any TAIO, thanks to an original approximate determinization algorithm. Another main contribution is the selection of test cases with expressive test purposes described as OTAIOs having the ability to precisely select behaviors to be tested based on clocks and actions of the specification as well as proper clocks. Test cases are generated as TAIOs using a symbolic co-reachability analysis of the observable behaviors of the specification guided by the test purpose.. A first perspective of this work is to implement the approach in a test generation tool. Currently, the approximate determinization has been prototyped in Python thanks to a binding of the UPPAAL DBM library [25] . Other perspectives could be to combine this approach with the one of [14] for models with data, for the generation of test cases from models with both time and data in the spirit of [3] , but generalized to non-deterministic models.

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