Member, IEEE Francesco Bandiera

Senior Member, IEEE Olivier Besson

Member, IEEE Angelo Coluccia

Senior Member, IEEE Giuseppe Ricci

ABORT-Like Detectors: A Bayesian Approach

Keywords: Adaptive detection, Bayesian estimation, orthogonal rejection

In this paper, we deal with the problem of adaptive radar detection of point-like targets in presence of noise with unknown spectral properties. As customary, we assume that a set of data sharing the same properties of the noise in the cell under test is available. To cope with a limited number of training data, a Bayesian framework is adopted at the design stage. In order to come up with detectors with good rejection capabilities, the possible presence of a fictitious signal under the null hypothesis is modeled probabilistically, as opposite to the conventional ABORT-like approach. Several detectors are devised for the problem at hand, with different complexities. The performance assessment, conducted by means of Monte Carlo simulations, reveals that a good trade-off between detection power and selectivity can be achieved, even assuming a limited number of training data.

I. INTRODUCTION

R ESEARCH activity in the area of adaptive radar detec- tion of targets embedded in Gaussian or non-Gaussian disturbance has received a great attention in the last decades. Most of the published works follow the lead of the seminal paper by Kelly [START_REF] Kelly | An adaptive detection algorithm[END_REF], where the generalized likelihood ratio test (GLRT) is used to conceive an adaptive decision scheme capable of detecting coherent pulse trains in presence of Gaussian disturbance with unknown spectral properties. Other classical examples of detection algorithms include the adaptive matched filter (AMF) [START_REF] Robey | A CFAR adaptive matched filter detector[END_REF] and the adaptive normalized matched filter [START_REF] Conte | Asymptotically optimum radar detection in compound gaussian noise[END_REF], also known as adaptive coherence estimator [START_REF] Scharf | Adaptive matched subspace detectors and adaptive coherence estimators[END_REF]. More generally, such problem has been extensively studied and a large number of solutions are available in the open literature, see [START_REF] Gini | Selected list of references on radar signal processing[END_REF] for a list of references.

Since most of the previously quoted solutions suppose exact knowledge of the signal array response vector, they may experience a performance degradation when the actual steering vector is not aligned with the nominal one. Therefore, it is of interest to take into account possible steering vector mismatches at the The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Hongbin Li.

F. Bandiera, A. Coluccia, and G. Ricci are with the Dipartimento di Ingegneria dell'Innovazione, Università del Salento, 73100 Lecce, Italy (e-mail: francesco.bandiera@unisalento.it; angelo.coluccia@unisalento.it; giuseppe.ricci@unisalento.it).

O. Besson is with the Départment Electronique Optronique Signal, Université de Toulouse, Toulouse 31055, France (e-mail: olivier.besson@isae-supaero.fr). design stage of the detector. To this end, coherent returns can be roughly classified into mainlobe and sidelobe signals [START_REF] Pulsone | Adaptive beamformer orthogonal rejection test[END_REF], [START_REF] Fabrizio | Spatial adaptive subspace detection in OTH radar[END_REF] • a sidelobe signal (sidelobe interferer) is a coherent signal from a "direction significantly different" to that in which the radar system is steered: it can be due, for example, to a strong target located in a sidelobe direction or to the retransmission of a modulated signal (for the purpose of degrading the reception of the signal of interest); • a mainlobe target is instead a coherent signal backscattered from the nominal direction or a direction slightly different from the nominal one as a consequence of an imperfect modeling of the nominal steering vector, where the mismatch may be due to multipath propagation, array calibration uncertainties, beampointing errors, etc. Thus, the effectiveness of the detector depends on its ability to detect the presence of what is classified as mainlobe target, limiting the number of "false alarms" due to sidelobe interferers.

An effective approach to deal with this problem has been introduced in [START_REF] Pulsone | Adaptive beamformer orthogonal rejection test[END_REF], where the adaptive beamformer orthogonal rejection test (ABORT) is proposed; such a detector takes into account rejection capabilities at the design stage. The idea of the ABORT is to modify the null hypothesis, which usually states that data under test contains noise only, so that it possibly contains a fictitious signal which, in some way, is orthogonal to the assumed target's signature. Doing so, if a mismatched signal is present, the detector will be less inclined to declare a detection, as the null hypothesis will be more plausible than in the case where, under the null hypothesis, the test vector contains noise only. The extension of this idea to the case of signals belonging to known subspaces of the observables has been dealt with in [START_REF] Fabrizio | Spatial adaptive subspace detection in OTH radar[END_REF], as a possible means to maintain an acceptable detection loss for slightly mismatched mainlobe targets. It is important to observe that in the detector proposed in [START_REF] Pulsone | Adaptive beamformer orthogonal rejection test[END_REF] the fictitious signal is assumed to be orthogonal to the nominal steering vector in the "quasi-whitened" space, i.e., after whitening of the data through the sample covariance matrix computed over the secondary data set. The same approach is also proposed in [START_REF] Fabrizio | Spatial adaptive subspace detection in OTH radar[END_REF], where the quasi-whitening transformation is presented as a way to face with the absence of knowledge about the interference subspace. Following the above mentioned approach, in [START_REF] Bandiera | An ABORT-like detector with improved mismatched signals rejection capabilities[END_REF] it is assumed that the useful and the fictitious signals are orthogonal in the "truly" whitened observation space, i.e., after whitening with the true noise covariance matrix. The resulting detector, called the whitened-ABORT (W-ABORT), exhibits good selectivity properties at the price of a performance loss for matched signals.

Mainlobe and/or sidelobe targets have also been represented using the tools of subspace detection in [START_REF] Gini | Matched subspace CFAR detection of hovering helicopters[END_REF]- [START_REF] Kraut | Adaptive subspace detectors[END_REF] or constraining the corresponding steering vector to belong to a cone [START_REF] Maio | Robust adaptive radar detection in the presence of steering vector mismatches[END_REF]- [START_REF] Besson | Detection of a signal in linear subspace with bounded mismatch[END_REF].

Another class of receivers designed to deal with the problem of steering vector mismatches is that of the so-called two-stage detectors, i.e., detection architectures composed by cascading two detectors with different behaviors: the overall one declares the presence of a target if the cell under test survive both detection thresholdings, see [START_REF] Richmond | Performance of the adaptive sidelobe blanker detection algorithm in homogeneous clutter[END_REF]- [START_REF] Bandiera | Advanced radar detection schemes under mismatched signal models[END_REF]. Furthermore, the detector proposed in [START_REF] Kalson | An adaptive array detector with mismatched signal rejection[END_REF] is a special case of the more general class of tunable (possibly space-time) detectors which have been shown to be an effective means to attack detection of mainlobe targets or rejection of sidelobe ones [START_REF] Kalson | An adaptive array detector with mismatched signal rejection[END_REF], [START_REF] Bandiera | One-and two-stage tunable receivers*[END_REF].

Most of the above papers rely on the assumption that a set of secondary data, namely radar returns free of signals components, but sharing certain properties of the noise in the data under test, is available. Such secondary data are usually used to come up with fully-adaptive detection schemes. However, it has been evidenced that the homogeneous assumption for the secondary data is an idealized situation [START_REF] Klemm | Principles of Space-Time Adaptive Processing[END_REF], [START_REF] Melvin | Space-time adaptive radar performance in heterogeneous clutter[END_REF], and that nonhomogeneous environments are more commonly encountered. This fact calls, in turn, for the necessity to design adaptive algorithms capable of working with a number of secondary data as small as possible. To this aim, the tools of Bayesian estimation can be adopted, which allow to obtain general and flexible models, without very restrictive assumptions [START_REF] De Maio | Adaptive radar detection: A Bayesian approach[END_REF]- [START_REF] Hao | Bayesian radar detection with orthogonal rejection[END_REF]; however, the quoted papers mainly deal with the problem of modeling the noise characteristics rather than the signal ones. [START_REF] Coluccia | A tunable W-ABORT-like detector with improved detection vs. rejection capabilities trade-off[END_REF] is a first attempt to introduce randomness in the fictitious signal.

In this paper, we use Bayesian statistics to address the problem of adaptive radar detection of point-like targets under steering vector mismatches. More in detail, we extend the ABORT framework of [START_REF] Pulsone | Adaptive beamformer orthogonal rejection test[END_REF], [START_REF] Bandiera | An ABORT-like detector with improved mismatched signals rejection capabilities[END_REF] to the case where also the useful and the fictitious signals (in addition to the disturbance covariance matrix) are modeled as random quantities with some preassigned a priori distribution. We treat both the case where the useful and the interfering signals are orthogonal in the original space and the case where they are orthogonal in the whitened observation space. We propose ad hoc detection strategies based upon the likelihood ratio principle in which the unknown quantities are estimated according to several criteria. The performance assessment, carried out by Monte Carlo simulation, shows that some of the proposed solutions can provide good selectivity properties while retaining an optimal performance for matched signals. Remarkably, they are especially attractive when the number of secondary data is very small and the conventional detectors cannot be used.

The remainder of the paper is organized as follows. Section II is devoted to the problem formulation while Section III addresses the design of the proposed detectors. Section IV presents the performance assessment and, finally, Section V contains the conclusions of the work.

II. PROBLEM FORMULATION

We assume the radar is equipped with a linear array formed by antennas that collect samples from the range cell under test (CUT). The signal received from the CUT could be noise only or a noisy version of the signal backscattered by the target that we model as a coherent target echo. As customary, we suppose that a set of secondary data, free of signal components, but sharing the same statistical properties of the remaining disturbance in the CUT, is available. Secondary data are usually chosen as range cells surrounding the CUT in order to preserve the homogeneity assumption.

We denote by , , the vector containing the returns from the CUT and by , , the secondary data. The problem of detecting the possible presence of a coherent return from a given cell (in range, doppler, and azimuth) is formulated in terms of the following hypothesis testing problem [START_REF] Kelly | An adaptive detection algorithm[END_REF] where

• stands for the signal of interest (known steering vector of the target) and denotes its amplitude, modeled as a complex normal random variable, i.e., , with possibly very large. • and are conditionally independent and identically distributed random vectors drawn from a complex Gaussian distribution with zero mean and unknown, positive definite covariance matrix . More precisely,

. As to , it is assumed to follow a (complex) inverse Wishart distribution with ( ) degrees of freedom and mean , i.e.,

where and denote the determinant and the exponential of the trace of the matrix between braces, respectively, and is given by with denoting the Euler's Gamma function. Note that an inverse Wishart distribution for is a conjugate distribution with respect to the Gaussian model for , which will make derivation of posterior distributions relatively easy. Such a choice is also a convenient way to introduce colored loading in a Bayesian framework. In the following we assume however that , i.e., a noninformative prior, which corresponds to diagonal loading (DL). This has the advantage of not requiring knowledge of the mean of , and at the same time DL is known to be very effective in low sample support [START_REF] Abramovich | Modified GLRT and AMF framework for adaptive detectors[END_REF]. For brevity we will use the notation . • As anticipated in Section I, is introduced to enhance the selectivity of the detector. In fact, it makes the detector less inclined to declare the presence of the nominal signal ( hypothesis) in case of mismatches. is modeled as an unknown random vector taking on values either orthogonal to (

) or conditionally orthogonal to in the whitened space (i.e.,

), where denotes conjugate transpose.

III. DETECTOR DESIGNS

In this section, we propose several detectors that implement a ratio of conditional likelihoods where the conditioning (random) quantities are replaced by proper Bayesian estimates. We consider both orthogonal to and orthogonal to ; for the former case several alternatives are devised.

A. Case of Orthogonal to

Let us first consider the problem under and denote by an matrix whose columns form an orthonormal basis for the subspace orthogonal to , i.e., and , with the identity matrix of order . Since , it follows that for some . The joint likelihood function of , is thus given by

where . We assume that follows a Bernoulli-Gaussian distribution i.e., its probability density function (pdf) is given by (4) where is the Euclidean norm, is the Dirac delta, stands for the probability that , and is possibly very large. We denote this distribution as . An equivalent and possibly more convenient way to represent is to write where is a binary variable, and . From (4), follows a Bernoulli distribution with parameter , i.e., : we denote this distribution as . With some slight abuse, we will write , keeping in mind that is binary. Moreover, is independent of : the latter is Gaussian distributed, with mean zero and covariance matrix , i.e., . We will indifferently use the parametrization in terms of or in terms of , depending on which one is more convenient.

We will investigate three approaches. In the first one, joint estimation of (or and ) and will be conducted, while in a second approach, we will marginalize with respect to , leaving estimation of only. The third method entails an approach à la AMF where detection is carried out on only, assuming is known: an estimate of based on only is then substituted for in the detection statistic.

1) Joint Estimation of and : First observe that under the joint posterior distribution of , , and is given by [START_REF] Gini | Selected list of references on radar signal processing[END_REF] where denotes proportional to.

Let us now derive the conditional posterior distributions of , , and . Since [START_REF] Pulsone | Adaptive beamformer orthogonal rejection test[END_REF] with

(7a) (7b)
It ensues that the posterior distribution of is Gaussian (8) Similarly [START_REF] Gini | Matched subspace CFAR detection of hovering helicopters[END_REF] which implies that [START_REF] Sangston | Structures for radar detection in compound gaussian clutter[END_REF] with given in [START_REF] Gini | Vector subspace detection in compound-gaussian clutter. Part I: Survey and new results[END_REF], shown at the bottom of the page. Finally, we have [START_REF] Scharf | Matched subspace detectors[END_REF] with [START_REF] Kraut | Adaptive subspace detectors[END_REF] This conditional posterior distribution is a complex inverse Wishart distribution with degrees of freedom and parameter matrix .

(

) 11
Therefore, the conditional distributions , , and all belong to known families and are easy to simulate. This suggests the use of a Gibbs sampler which enables one to obtain the minimum mean-square error (MMSE) estimates of , , and . The Gibbs sampler is summarized in Table I, where it is shown how to sample , , and , thus obtaining the sequences , , and , respectively, using the previously introduced posterior distributions. The MMSE estimates can be obtained by averaging the samples of and after the so-called burning period of duration , i.e., and Under , the likelihood function is given by ( 14)

The posterior distribution of and is then [START_REF] Bandiera | Adaptive CFAR radar detection with conic rejection[END_REF] Since [START_REF] Besson | Detection of a signal in linear subspace with bounded mismatch[END_REF] it follows that [START_REF] Richmond | Performance of the adaptive sidelobe blanker detection algorithm in homogeneous clutter[END_REF] Accordingly, it is straightforward to show that [START_REF] Maio | Rao test for adaptive detection in gaussian interference with unknown covariance matrix[END_REF] where [START_REF] Greco | Radar detection and classification of jamming signals belonging to a cone class[END_REF] The above posterior distributions allow one to obtain the MMSE estimates of , under by resorting again to a Gibbs sampler; accordingly, the corresponding estimates are denoted by and . Based on the above computed estimates, the test is given by [START_REF] Richmond | The theoretical performance of a class of space-time adaptive detection and training strategies for airborne radar[END_REF] where is a threshold to be set according to the desired probability of false alarm (

). This detector will be referred to in the following as one-step detector with rejection in the orthogonal initial space (1S-OIS).

2) Marginalizing With Respect to : An alternative approach consists in marginalizing with respect to the covariance matrix, so as to leave estimation of only under and only under . To be more specific, let us marginalize the likelihood function in (3) with respect to ultimately obtaining [START_REF] Pulsone | A computationally-efficient twostep implementation of the GLRT[END_REF], shown at the bottom of the page, where is given by [START_REF] Greco | Radar detection and classification of jamming signals belonging to a cone class[END_REF]. A straightforward calculation shows that [START_REF] Richmond | Performance of a class of adaptive detection algorithms in nonhomogeneous environments[END_REF] (21) with . Therefore, [START_REF] Bandiera | Advanced radar detection schemes under mismatched signal models[END_REF] with [START_REF] Kalson | An adaptive array detector with mismatched signal rejection[END_REF] The posterior distribution of , is thus [START_REF] Bandiera | One-and two-stage tunable receivers*[END_REF] It ensues that the conditional posterior distribution of is still Bernoulli [START_REF] Klemm | Principles of Space-Time Adaptive Processing[END_REF] with

(27a) (27b) (27c)
As for the conditional posterior distribution of , we have from (25) [START_REF] De Maio | Adaptive radar detection: A Bayesian approach[END_REF] As expected, when , follows its prior Gaussian distribution. However, when , one has a product of Gaussian and Student distributions. In order to obtain the MMSE estimates of and , one could again investigate a Gibbs sampler. For the sampling of , we suggest a Metropolis-Hastings [START_REF] Robert | Monte Carlo statistical methods[END_REF] approach with a proposal distribution that would be a Student distribution, which amounts, in fact, to ignoring the exponential term in [START_REF] De Maio | Adaptive radar detection: A Bayesian approach[END_REF].

Let us now turn to the hypothesis. From ( 14), we can infer that [START_REF] De Maio | Adaptive radar detection: A Bayesian approach[END_REF] with given by ( 19), (30a) (30b) and . The posterior distribution of is thus [START_REF] Bidon | A Bayesian approach to adaptive detection in non-homogeneous environments[END_REF] If one assumes a flat prior for , i.e., , then follows a Student distribution and the MMSE estimate of is simply . On the other hand, with a Gaussian prior for , the posterior is a product of Gaussian and Student distributions. The MMSE estimator cannot be obtained in closed-form but Markov chain Monte Carlo (MCMC) methods have to be looked for, with a Metropolis-Hastings strategy to draw samples from [START_REF] Bidon | A Bayesian approach to adaptive detection in non-homogeneous environments[END_REF]. For the sake of simplicity, we assume that is very large and we simply approximate the MMSE estimate of by .

Finally, the above estimates can be used in the following test statistic [START_REF] Besson | Covariance matrix estimation with heterogeneous samples[END_REF] where indicates equivalence of the statistics for the hypothesis testing problem, and the last line follows from [START_REF] Pulsone | A computationally-efficient twostep implementation of the GLRT[END_REF] and [START_REF] De Maio | Adaptive radar detection: A Bayesian approach[END_REF]. This detector will be referred to in the following as marginalized one-step detector with rejection in the orthogonal initial space (M-1S-OIS).

3) A Two-Step Approach: A third possibility which is worth investigating is a 2-step approach, where is estimated from the secondary data only. This estimate is then substituted for in a test statistic build on only. More precisely, since , it follows that the MMSE estimate of from is simply [START_REF] Bandiera | Knowledge-aided covariance matrix estimation and adaptive detection in compound-gaussian noise[END_REF] which corresponds to a diagonally loaded version of the sample covariance matrix. Let us consider the posterior distribution of . From using ( 4) and ( 6), it follows that [START_REF] Bandiera | Adaptive detection of distributed targets in compound-gaussian noise without secondary data: A Bayesian approach[END_REF] where

(35a) (35b)
Therefore, follows a Bernoulli-Gaussian distribution, i.e., [START_REF] Coluccia | A tunable W-ABORT-like detector with improved detection vs. rejection capabilities trade-off[END_REF]

With

given by ( 37), shown at the bottom of the page. As a consequence, the MMSE estimate of , conditioned on , is [START_REF] Robert | Monte Carlo statistical methods[END_REF] The MMSE estimate of (conditioned on ) is, see [START_REF] Richmond | Performance of the adaptive sidelobe blanker detection algorithm in homogeneous clutter[END_REF] (39)

These estimates can now be used to construct the following test statistic [START_REF] Besson | Adaptive processing with signal contaminated training samples[END_REF] In contrast to the two previous methods, no MCMC simulation method is required here and the resulting detection scheme remains rather simple. This detector will be referred to in the following as two-step detector with rejection in the orthogonal initial space (2S-OIS).

B. Case of

Orthogonal to Let us now consider the alternative hypothesis that is orthogonal to in the whitened space. Since the subspace spanned by is the orthogonal complement of the one spanned by , it follows that where . This case is going to differ from the previous one since, as will be shown shortly: the likelihood function under will include terms of the form , i.e., a combination of terms that appear in both Wishart and inverse Wishart distributions, thus precluding the implementation of a fully Bayesian approach based on the whole data , .

Similarly to the previous section, we again assume that follows a Bernoulli-Gaussian distribution. Under , the likelihood of is thus given by with the prior distributions still given by and ( 4). The posterior distribution of , is hence (41) with given by [START_REF] Greco | Radar detection and classification of jamming signals belonging to a cone class[END_REF]. In this case, we have In contrast to the case , the posterior distribution does no longer belong to a known family (due to the terms in and in the trace) and therefore it does not seem feasible to draw samples from it. Accordingly, marginalization with respect to is not feasible.

In order to circumvent this problem, we again turn to a 2-step approach à la AMF, i.e., i) assume is known and devise a test from only and ii) replace by its MMSE estimate from the training samples. The latter is still given by [START_REF] Bandiera | Knowledge-aided covariance matrix estimation and adaptive detection in compound-gaussian noise[END_REF].

Let us now turn to the first step which consists of deriving a test based on only, assuming that is known. Under , is given by (44). It follows that the MMSE estimate of , which we denote as , is simply

Under , recalling that , the posterior distribution of is still given by ( 17) and consequently the MMSE of is simply

For the sake of simplicity, we will set in the previous equation, which amounts to a flat prior for . The decision statistic takes the following form (50)

We would like to emphasize that this last approach is quite simple, as the MMSE estimate of (based on only) and the MMSE estimates of or (conditioned on ) are given in closed-form. This detector will be referred to in the following as two-step detector with rejection in the orthogonal whitened space (2S-OWS).

IV. PERFORMANCE ANALYSIS

The performance analysis is conducted by Monte Carlo simulation. We resort to independent trials to evaluate the thresholds necessary to ensure a preassigned value of and to independent trials to compute the probabilities to decide for when a useful signal is present . We consider a scenario with and . We generate , as independent random vectors ruled by a complex Gaussian distribution with zero mean and (deterministic) covariance matrix

, where the clutter covariance matrix is Gaussian-shaped, namely it is Toeplitz with first row ; the power of the thermal noise is , so that the clutter-to-noise ratio (CNR) is 10 dB.

The steering vector of the target assumes a normalized Doppler frequency , a velocity such that the target competes with noise. The amplitude of the target is generated as a complex normal random variable; the signal to noise ratio is defined as and simulation results are reported as a function of the SNR.

The thresholds of the detectors are set to guarantee a chosen by considering a noise-only hypothesis ( ), since we are interested in assessing the performance of the receivers in terms of detection power (for matched steering vector) as well as their selectivity (for mismatched steering vector). The values of the main parameters used to implement the detectors are reported in the Table II.

For comparison purposes we consider some natural competitors for the problem at hand. Obvious references are the Kelly's detector [START_REF] Kelly | An adaptive detection algorithm[END_REF] and the W-ABORT [START_REF] Bandiera | An ABORT-like detector with improved mismatched signals rejection capabilities[END_REF]. Furthermore, we consider also a generalization of the W-ABORT, hereafter referred to as TW-ABORT, which exhibits improved detection power vs selectivity trade-off, provided its tuning parameter is chosen in the appropriate range (see [START_REF] Coluccia | A tunable W-ABORT-like detector with improved detection vs. rejection capabilities trade-off[END_REF]).

The case is shown in Fig. 1 for two representative values of the parameter in its lower and higher region (so as to assess the performance in the whole range omitting redundant intermediate values). As mentioned, for the sake of comparison we report also the Kelly's detector, the W-ABORT, and the TW-ABORT (with ). From Fig. 1 it is apparent that the Bayesian detectors are generally very powerful with superior performance especially at low to intermediate SNR; the 2S-OWS is always better than all other competitors, including the Kelly's detector even at high SNR; the M-1S-OIS conversely shows a performance degradation, more severe for higher . The comparison confirms the smaller loss of TW-ABORT with respect to the Kelly's detector compared to the W-ABORT.

Lower values of are critical since for the matrix becomes singular hence all detectors involving such a matrix cannot be computed; in fact, as approaches the performances of the competitors severely degrade. Fig. 2 reports the at the onset of the critical region, i.e., for and : it is apparent that, although the classical detectors are still computable, 1 their performance are not acceptable when the number of secondary data is small. It is therefore quite interesting that the performances of the proposed detectors are conversely satisfactory also when a limited number of training data is available, and show a certain robustness to variations of .

To assess the selectivity of the proposed detectors, we simulated a target with a mismatched steering vector having normalized Doppler frequency , where (so the matched case analyzed above is trivially obtained for ). Fig. 3 reports the case : it is apparent that Bayesian detectors, besides being very powerful, can also be more selective than the Kelly's detector; a clear ranking can be identified, with M-1S-OIS the most selective (among the proposed detectors) and 2S-OWS the least selective, comparable to the Kelly's detector. In general, however, none of the detectors is as selective as the W-ABORT. The TW-ABORT confirms an improved power vs selectivity trade-off compared to W-ABORT and Kelly's detector. The relative performances for other values of the parameters are similar, hence figures are omitted. For and these considerations are confirmed, 1 Just the Kelly's detector and W-ABORT are reported in order not to burden too much the figure. except that the performance of the M-1S-OIS slightly worsens, as shown in Fig. 4 (the curves for are very similar hence omitted).

It is worth noticing that the superior detection power of the 2S-OWS is traded-off with less selectivity compared to the Kelly's detector, while the other proposed detectors are just slightly worse in (compared to the 2S-OWS) but much better in selectivity. As a whole, the analysis suggests that a palette of different trade-offs is possible, with the class of detectors based on the Bayesian approach exhibiting superior performance compared to the classical approaches; this is especially remarkable in case of a limited number of secondary data, when non-Bayesian approaches fail.

We investigated also the sensitivity of the detectors to errors in the parameters assumed for setting the threshold. In particular, we analyzed how the changes when assuming under a covariance matrix different from the real one. In fact, the superior performance of the proposed detectors could result, in principle, from a trade-off with the constant false alarm rate (CFAR) property, while the Kelly's detector and the W-ABORT are CFAR. To investigate this point, we set the threshold by using , where denotes a chosen one-lag correlation coefficient. Results for and are shown in Fig. 5, where the value of the true one-lag correlation of the clutter is reported in abscissa. It is apparent that the proposed detectors are quasi-CFAR (with respect to the one-lag correlation coefficient) for , with fairly limited departure of the curves from the straight line in for both sides of the mismatch, i.e., an assumed correlation higher or lower than the true one, respectively. As reduces, this property becomes weaker for all proposed detectors but for 1S-OIS, which is somewhat more robust to these variations; the case is reported in Fig. 6. Finally, for the sake of completeness, some results in terms of , selectivity, and sensitivity are reported in Fig. 7 for the case , i.e., when the number of training data is very limited. The simulations basically confirm the previous considerations, except for some degradation in the "CFAR property", which as already said is traded-off for whenever conventional approaches are not applicable.

V. CONCLUSIONS

In this paper, we have considered the problem of adaptive radar detection of point-like targets in presence of noise with unknown spectral properties. To cope with a limited number of training data, a Bayesian framework has been adopted at the design stage, modeling the noise covariance matrix according to a complex inverse Wishart distribution. In order to come up with detectors with good rejection capabilities, the possible presence of a fictitious signal under the null hypothesis has been modeled probabilistically, as opposite to the conventional ABORT-like approach. We have devised several detectors based on the ratio of conditional distributions under the two hypothesis, where the unknown random parameters are replaced by MMSE estimates based on one-step or two-step procedures. Both the case orthogonal to and orthogonal to have been addressed, obtaining four new detectors with different complexity and different behavior in terms of detection power vs selectivity trade-off. Simulation results have shown that satisfactory performance can be obtained even assuming a limited number of training data. Thus, the proposed detectors seem good candidates to conceive augmented schemes that also select training data by discarding possible contaminated returns. The selection processing might be based on the results contained in [START_REF] Aubry | Covariance matrix estimation via geometric barycenters and its application to radar training data selection[END_REF] or specifically designed within the Bayesian framework by following the lead of [START_REF] Besson | Adaptive processing with signal contaminated training samples[END_REF].

  conditional posterior distribution of , one has (47)

Fig. 1 .

 1 Fig. 1. Probability of detection for

Fig. 2 .

 2 Fig. 2. Probability of detection at the onset of the critical region, for and ;.

Fig. 3 .

 3 Fig. 3. Probability of detection under mismatched conditions for . (a) ; (b) .

Fig. 4 .

 4 Fig. 4. Probability of detection under mismatched conditions for and .

Fig

  Fig. 5. under mismatched conditions for and . (a) ; (b) .

Fig. 6 .

 6 Fig. 6.under mismatched conditions for , and .

Fig. 7 .

 7 Fig. 7. Probability of detection and sensitivity analysis for

TABLE I GIBBS

 I SAMPLER FOR ESTIMATION OF , , AND

TABLE II MAIN

 II PARAMETERS USED IN THE IMPLEMENTATION OF THE DETECTORS and


Gabin Personeni

Marie-Dominique Devignes

Malika Smaïl-Tabbone

Philippe Jonveaux

Céline Bonnet

Adrien Coulet

Cooperation of bio-ontologies for the classification of genetic intellectual disabilities : a diseasome approach

Keywords: Semantic similarity, Bio-ontologies, Genetic Intellectual Disabilities, Diseasome

Bio-ontologies are widely used to annotate and characterize biological objects or situations, enabling the use of shared or similar features in classification tasks. It may appear beneficial to make two or more bio-ontologies cooperate for building more complete descriptions, and therefore more accurate classifications of biological objects. This hypothesis is evaluated here for the classification of an heterogeneous set of 374 Genetic Intellectual Disabilities (GIDs), using a diseasome approach. These GIDs are annotated with classes of the Human Phenotype Ontology (HPO) and their causal genes with the three aspects of the Gene Ontology (GO). We test two semantic similarity measures, and different combinations of ontologies, to connect semantically similar diseases. We then evaluate how well these ontologies, and their combinations, are exploited by the similarity measures to classify GIDs in accordance with an expert classification. Results show that combining the three aspects of GO achieves very good overall performance, and that, for each GID class, a particular combination of 2 or 3 GO aspects and occasionally HPO yields the best performance. These results illustrate how bio-ontologies can cooperate in a classification by refining the characterization of biological objects.

Introduction

Bio-ontologies, such as the Gene Ontology (GO) [START_REF] Ashburner | Gene Ontology: tool for the unification of biology[END_REF] or the Human Phenotype Ontology (HPO) [START_REF] Köhler | The human phenotype ontology project: linking molecular biology and disease through phenotype data[END_REF] are used to annotate biological objects such as gene products or diseases, enabling their semantic comparison. In particular, there exists a wide collection of semantic similarity measures, allowing to quantify the similarity of objects with regard to their annotations [START_REF] Pesquita | Evaluating go-based semantic similarity measures[END_REF][START_REF] Benabderrahmane | Intelligo: a new vector-based semantic similarity measure including annotation origin[END_REF]. We investigate in this article how several bio-ontologies can be used conjointly to cooperate to improve classification of a heterogeneous set of Genetic Intellectual Disabilities (GID).

Numerous studies report on the hypothesis that analyzing disease networks, here named diseasomes, may be a mean to discover new knowledge on mechanisms or treatments of diseases [START_REF] Barabási | Network medicine: a network-based approach to human disease[END_REF]. Various methods for diseasome building have been described in the literature. For instance, two diseases can be associated if they share one causing gene [START_REF] Goh | The human disease network[END_REF], phenotype [START_REF] Hidalgo | A dynamic network approach for the study of human phenotypes[END_REF] or are linked through a chain of protein-protein interactions [START_REF] Guney | Network-based in silico drug efficacy screening[END_REF]. Hoehndorf et al. [START_REF] Hoehndorf | Analysis of the human diseasome using phenotype similarity between common, genetic, and infectious diseases[END_REF] proposed a diseasome that associates diseases with respect to their phenotypic similarity. They assembled a dataset, extracted from the literature, of about 6, 000 diseases annotated with their associated phenotypes, using classes of the Monarch Disease Ontology (MonDO) [START_REF] Mungall | The monarch initiative: an integrative data and analytic platform connecting phenotypes to genotypes across species[END_REF]. The similarity of diseases with regard to their annotation were subsequently computed with the SimGIC function [START_REF] Pesquita | Evaluating go-based semantic similarity measures[END_REF]. We propose here to extend this approach by conjointly using annotations taken from several ontologies, and by assessing their respective contribution to a disease classification task.

The hypothesis that several bio-ontologies can somehow cooperate to refine a diseasome is evaluated here within the task of classifying GIDs. The classification of GIDs is of particular interest, and challenging for experts, because these diseases are very heterogeneous both in terms of causal genes and clinical outcomes. We focused on a set of 374 GIDs for which causal genes are known and used for genetic diagnosis. We manually classified these diseases with experts into five groups, on the basis of the biological mechanisms disturbed in the disease: regulation, regulation of genetic expression, metabolic, synaptic, neurogenesis. We detail in this article a diseasome approach based on semantic similarity of GIDs at both the phenotype and genetic level, and study how it can match an expert GID classification.

Material and Methods

Data and Ontologies

A dataset of 374 GIDs was built for this study on the basis of a list of 312 genes associated with GIDs derived from the work of Gilissen et al. [START_REF] Gilissen | Genome sequencing identifies major causes of severe intellectual disability[END_REF] who initially compiled two lists of GID genes: a list of 528 "known" GID genes and a list of 628 "candidates" genes, based on the number of reported patients in which a mutation or variant of the gene is observed. The 312 genes retained here (230 "known" genes and 82 "candidates") are those that are found associated with a genetic disease in OMIM database (Online Mendelian Inheritance in Man, http://omim.org) and used for diagnosis in the Genetics Laboratory of Nancy Hospital. Four distinct ontologies were used in this study: Human Phenotype Ontology (HPO) [START_REF] Köhler | The human phenotype ontology in 2017[END_REF] and the three aspects of Gene Ontology (GO) [START_REF] Ashburner | Gene Ontology: tool for the unification of biology[END_REF] named here BP for Biological Process, CC for Cellular Component and MF for Molecular Function. These three aspects of GO are organized into independent hierarchies of classes related by the subsumption relation, and are here considered as separate ontologies. HPO annotations of GIDs were collected from the HPO database (http://hpo.jax.org). BP, CC and MF annotations were collected from the GOA database [START_REF] Huntley | The goa database: gene ontology annotation updates for 2015[END_REF] at the European Bioinformatics Institute (https://www.ebi.ac.uk/GOA) for all UniProtKB proteins encoded by the genes associated to GIDs and transferred to the corresponding GID. The average number of HPO classes associated per GID was 22.4 ± 17.5, whereas the average numbers of BP, CC and MF classes per GID were 14.8 ± 16.4, 5.8 ± 4.3 and 5.2 ± 4, respectively. All GIDs could be associated with at least one HPO class and one BP class. Only 27 GIDs were found lacking one or the other aspect of GO annotation, mostly CC.

Expert classification of GIDs

GID diversity and heterogeneity renders their classification difficult. Our manual classification is an attempt to integrate the state-of-the-art knowledge about GIDs [START_REF] Inlow | Molecular and comparative genetics of mental retardation[END_REF][START_REF] Chelly | Genetics and pathophysiology of mental retardation[END_REF][START_REF] Kaufman | The genetic basis of non-syndromic intellectual disability: a review[END_REF][START_REF] Van Karnebeek | Treatable inborn errors of metabolism causing intellectual disability: a systematic literature review[END_REF] into the definition of five classes. The "Metabolic" class represents diseases affecting synthesis or degradation of metabolites, leading to metabolite deficiency or accumulation with deleterious consequences. The "Synaptic" class represents diseases affecting the structure and the function of synapses. The "Neurogenesis" class represents diseases affecting neuronal migration or proper development of central nervous system. The "Regulation of genetic expression" class represents diseases in which genetic expression (chromatin structure, transcription and its regulation, translation and post-translational modifications) is affected. The "Regulation" class is for all other diseases in which control of biological processes other than genetic expression is affected (for instance transport of proteins or energetic balance of the cell). Our dataset of 374 GIDs with their 312 responsible genes was manually distributed into these five classes by expert inspection of their OMIM notices (disease and gene ones). The resulting classification likely relies on several subjective arbitrary statements, but it appeared sufficient for the methodology used in this study. Table 1 quantitatively describes the composition of each class of GIDs. The broad definition of each class leads to possible assignment of the same disease and gene to two different classes. This is the case of 3, 5 and 10 GIDs of the Metabolic, Synaptic and Neurogenesis classes, respectively that are also classified in the Regulation class. One additional GID from the Neurogenesis class is also classified as Regulation of genetic expression. This GID (OMIM #613454: Rett syndrome, congenital form) illustrates the difficulty to classify GIDs, as it is described as a severe neurodevelopmental disorder and therefore classified in the Neurogenesis class, whereas its responsible gene is the FOXG1 gene, which codes for a repressor of the forked-head transcription factor family, pointing to the Regulation of genetic expression class.

Semantic similarity measures

Semantic similarity measures quantify the proximity of two ontology classes, or objects described by a set of classes, from an ontology. Such similarity measures may be used to build a diseasome, using disease annotations linked to an ontology [START_REF] Hoehndorf | Analysis of the human diseasome using phenotype similarity between common, genetic, and infectious diseases[END_REF]. We applied such similarity-based diseasome approach to build a diseasome of GIDs, on the basis of both their phenotypic and causal gene product annotations. These annotations are expressed as classes from four ontologies or ontologies fragments : HPO and the 3 aspects of Gene Ontology considered separately -BP, CC and MF.

We aim at assessing the contribution of several ontologies to a diseasome, but we also use two semantic similarity measures, to compare how different measures behave with ontology combinations. First, we use a node-based semantic similarity measure: SimGIC [START_REF] Pesquita | Evaluating go-based semantic similarity measures[END_REF], which computes the ratio of common classes among the ontology classes of two diseases, weighted by the information content of each class, and considering all ancestors of each disease annotations. The information content of a class, with respect to a dataset of annotations, is computed as IC(x) = -log 2 (P (x)), where P (x) is the probability that an object is annotated with the class x. Higher values of IC denotes higher specificity of the class. SimGIC was first introduced to compute similarity of genes annotated by GO classes, and has been successfully used with MonDO to build a diseasome based on phenotypic similarity [START_REF] Hoehndorf | Analysis of the human diseasome using phenotype similarity between common, genetic, and infectious diseases[END_REF]. Second, we use the edge-based similarity measure IntelliGO [START_REF] Benabderrahmane | Intelligo: a new vector-based semantic similarity measure including annotation origin[END_REF], which permits to compare two biological objects by first computing distances in the hierarchy between pairs of classes annotating the objects, and then aggregating each pairwise similarities into a single-object similarity score. As SimGIC, this aggregation step takes into account the information content of the compared classes to weight their contribution to the similarity.

To assess the contribution of each ontology, we build several similarity functions, using every possible combination of ontologies among BP, CC, MF and HPO, combined with both IntelliGO and SimGIC. For this purpose and in a first approach, we simply average the similarities computed separately with each ontology. We thus test 15 possible combinations of one or more ontologies, in turn combined with IntelliGO or SimGIC, resulting in 30 similarity functions.

Evaluation of similarity functions with respect to a reference classification

Hoehndorf et al. described in [START_REF] Hoehndorf | Analysis of the human diseasome using phenotype similarity between common, genetic, and infectious diseases[END_REF] a methodology to evaluate the accuracy of a similarity function with respect to a classification of diseases. This methodology aims at verifying that the similarity function gives higher similarity scores to pairs of diseases that belong to a same class of disease. This evaluation is based on a Receiver Operating Characteristic (ROC) analysis, quantifying the accuracy of a binary classification model at varying degrees of sensibility. In particular, a ranking of disease pairs, based on their similarity, serves as a classification model whose sensibility can be adjusted by defining a threshold of similarity above which pairs of diseases are labeled as positive by the model. The ROC curve represents the true positives rate as a function of the false positive rate. The ROC Area Under the Curve or ROCAUC can be computed from such a curve, and represents the probability for a random pair of diseases from the positive class to have a higher similarity than a random pair of diseases from the negative class. We note ROCAU C(R, P ) the function that computes the ROCAUC given a ranking R, and the set of positive elements P , describing which elements of R are to be considered positive for the purpose of the evaluation.

The ROCAUC-based evaluation can be conducted either for each single class of diseases or for the entire classification of diseases. This evaluation can also be performed with a classification of diseases in which a disease can belong to more than one class. The Algorithm 1 describes how the evaluation is performed globally on all disease classes, for an arbitrary similarity function sim. For each disease d of the dataset, we compute the ranking of the other diseases using sim, from the most similar to the least similar. Here, we want high-ranking diseases to share a disease class with d, thus, we consider the positive class to be the set of diseases sharing a GID class with d. The ranking is then evaluated by computing the ROCAUC for the prediction on that positive class. ROCAUCs for each disease in the dataset are then averaged to obtain a global evaluation score.

The Algorithm 2 describes how a similarity function sim is evaluated with respect to a single disease class, noted C. For each disease d of the GID class C, we compute a ranking of the other diseases using sim, and we define the positive class to be the set of diseases that belong to C. Again, the ranking is then evaluated by computing the ROCAUC for the prediction on that positive class. ROCAUCs for each disease of the class are then averaged to obtain a score representing how well the similarity function reflects this class.

Results

We apply the methodology described previously to our set of 374 GID. This set of diseases has phenotypic annotations expressed as HPO classes and the genetic annotations of their causing genes, expressed as GO classes and split into the three aspects of GO considered here as three independent ontologies. Indeed, as their class hierarchy are separate, semantic similarity measures such as IntelliGO and SimGIC cannot compare classes from these different aspects.

We computed all pairwise similarities on our set of GIDs, for all 30 possible similarity functions resulting from the combinations of the four ontologies (BP, CC, MF and HPO) and the two semantic similarity measures (IntelliGO and SimGIC). Each of these similarity functions was then evaluated by computing the average ROCAUC for each of the 5 GID classes, as described in Algorithm 2, and for all classes considered together, as described in Algorithm 1.

Tables 2 and3 present a selection of the results of theses evaluations for different similarity functions based on different combinations of ontologies, computed using IntelliGO and SimGIC respectively. Results are given for 6 classification tasks, one task for each GID class evaluated separately, as described in Algorithm 2, and a sixth task evaluating the performance of the similarity function on all 5 GID classes, as described in Algorithm 1. For each task, we tested every possible combination of ontologies from BP, CC, MF and HPO, but both Tables report only results for combinations of interest: single ontologies, all GO aspects, all ontologies, and all combinations that provide the best performance on any task for either SimGIC or IntelliGO.

Unsurprisingly, the results are highly variable depending on which ontologies are considered, and which similarity measure is used, and they also vary across different classification tasks. In particular, we observe that HPO does not positively contribute to the performance when combined with other ontologies, with the exception of the classification task of the Neurogenesis class. We also note that similarity functions using only HPO have poor performance compared with those using a single GO aspect in most cases, although such a function performs better than a random classifier. This suggests that, for classes other than Neurogenesis, HPO does not bring more information compared to GO than noise. However, combining several GO aspects produces a great increase in performance, which is notably visible for the Regulation of Genetic Expression class: IntelliGO performance increases from 0.740 in the best case with only one GO aspect to 0.803 with all three of them, and SimGIC performance increases from 0.905 to 0.936.

Table 2. ROCAUC obtained using IntelliGO to classify GIDs into 5 classes (Regulation, Regulation of Genetic Expression, Metabolic, Synaptic, Neurogenesis) with different combinations of ontologies among the three aspects of GO (BP, CC and MF) and the phenotype ontology HPO. For each GID class, an average ROCAUC is also computed as described in the Algorithm 2. An average ROCAUC is computed across all classes as described in the Algorithm 1. Values in bold denote the maximum average ROCAUC for each evaluation task (each single class or all classes). For each task, the Table presents the threshold at which the performance score is significant (p < 0.01), which was computed empirically using random similarity functions.

Classes

Reg If combining all three aspects of GO provides the best overall performance, we nonetheless observe that this is not necessarily the best combination for the classification task on each individual GID class :

-The Regulation class is poorly predicted by most similarity functions. In particular, similarity functions based on HPO alone are not performing better than a random classification. -The Regulation of Genetic Expression class is best when computed using SimGIC (0.936) and considering the three aspects of GO conjointly. However, in this case, considering HPO on top of GO does not offer any increase in performance. Similarly, IntelliGO performs best on this classification task when considering the three aspects of GO, but shows a decrease in performance when also considering HPO. -The Metabolic class is best predicted using IntelliGO (0.862) when using all three aspects of GO. However, we observe that SimGIC (0.785) obtains its best performance score on this classification task when considering only BP and MF. However, this divergence between the two measures is small, as IntelliGO performs almost as well with only BP and MF (0.856). -The Synaptic class is best predicted when using only the BP and CC aspects of GO for both SimGIC (0.819) and IntelliGO (0.772). -The Neurogenesis class is the only class for which we obtain better results when adding HPO to the aspects of GO, for both IntelliGO and SimGIC.

In particular, IntelliGO gets a performance score of 0.657 when using MF and HPO (compared to 0.610 with MF alone, or 0.590 with HPO alone), while SimGIC gets a performance score of 0.633 (compared to 0.561 with CC alone, or 0.594 with HPO alone).

Discussion and Conclusion

We study in this article how different ontologies can cooperate to improve how a diseasome can reflect the expert knowledge of a classification of GID. We evaluate here two semantic similarity measures, IntelliGO and SimGIC on a classification task realized with 5 GID classes, using different combinations of phenotypic and genetic ontologies. The results show that phenotypic annotations from HPO are not sufficient to reflect our expert classification, while genetic annotations from each aspect of GO can offer better performance. Furthermore, combining several aspects of GO further improve performance and, for the Neurogenesis GID class, combining a GO aspect with HPO offer the best performance. This illustrates that the cooperation of several ontologies is suitable for such a classification task and can improve a diseasome approach. However, the relevance of each ontology greatly depends on individual disease classes, and considering too many ontologies may have a negative effect. This constitutes a limitation of the use of semantic similarity measures, in that they are sensitive to the quality of annotations, as well to annotations irrelevant to class to predict. The results obtained show that overall, considering only the 3 aspects of GO yields often slightly better classification performance than considering both HPO and GO. Here, the contribution of HPO to the classification performance may be limited by the GID dataset we used, as these diseases may be too difficult to distinguish based on their phenotypes alone, as other studies show that HPO is suitable to classify more phenotypically heterogeneous sets of diseases [START_REF] Hoehndorf | Analysis of the human diseasome using phenotype similarity between common, genetic, and infectious diseases[END_REF]. It seems that in such cases, deciding on which ontologies to consider requires an iterative empirical approach that consider all possible combinations of ontologies. In fact, because performances do not necessarily increase with the number of ontologies, proposing a non-exhaustive strategy to select the best combination of ontologies for a particular task is not trivial.

Furthermore, it may be necessary to develop more sophisticated methods for aggregating similarities based on different ontologies. Here, we used an unweighted average of similarities each using a different ontology. Weighting the contribution of each ontology to the aggregated similarity could be done in several ways, for instance, by considering the number of annotations in each ontology for the compared disease, or by empirically determining an appropriate weighting scheme, rather than including or excluding ontologies. Moreover, as we observe that different settings and therefore different diseasome models are optimal only for certain classes, ways to aggregate different models could be explored, such as boosting algorithms [START_REF] Freund | Experiments with a new boosting algorithm[END_REF] or bagging predictors [START_REF] Breiman | Bagging predictors[END_REF].

Diseasomes based on semantic similarity are able to reflect an expert classification of diseases, as illustrated in the different classification experiments presented in this article. Such a diseasome can be used to classify new diseases by simple propagation of the neighboring diseases classes. Here, the cooperation of several biomedical ontology was shown to be relevant in many cases, however selecting the right ontologies to consider for a particular task require some trial and error. We note that both semantic similarity measures, IntelliGO and SimGIC, have varying performance on the different classification tasks presented in this article: some GID classes seem to be better predicted by one of these two measures. However, these differences do not permit to conclude that one of these measures perform strictly better than the other, as their overall performances are very similar. In summary, semantic similarity measures with various combinations of ontologies allow to propose a diseasome as a model synthesizing descriptions of GIDs in regards with several ontologies, in good agreement with an expert classification of such diseases. Such cooperation of bio-ontologies could also be explored with other machine learning methods.

Data: 1 : 2 :

 12 The set of diseases D, a similarity between diseases sim : D × D → R + Result: Average ROCAUC for all diseases begin ROCAU C avg = 0 foreach disease d ∈ D do ranking ← x ∈ (D -{d}) ranked in descending order of sim(d, x) pos ← {x ∈ D | d and x share a disease class} ROCAU C avg ← ROCAU C avg + ROCAU C(ranking, pos) end return ROCAU C avg /|D| end Algorithm Evaluation algorithm for a similarity function sim on a classification task with several overlapping classes of diseases. Data: The set of diseases D, a disease class C, a similarity between diseases sim : D × D → R + Result: Average ROCAUC for diseases of C begin ROCAU C avg = 0 pos ← {x ∈ D | x has class C} foreach disease d in class C do ranking ← x ∈ (D -{d}) ranked in descending order of sim(d, x) ROCAU C avg ← ROCAU C avg + ROCAU C(ranking, pos) end return ROCAU C avg /|D C | end Algorithm Evaluation algorithm for a similarity function sim on a classification task with respect to a single class of diseases C.

Table 1 .

 1 Distribution of our GID dataset in the five classes of our manual classification. Numbers of known genes refers to the list of Gilissen et al.[START_REF] Gilissen | Genome sequencing identifies major causes of severe intellectual disability[END_REF].

	Class	GIDs Genes Known genes
	Regulation	154	128	90(70%)
	Regulation of genetic expression 70	57	38(67%)
	Metabolic	105	96	78(81%)
	Synaptic	31	25	17(68%)
	Neurogenesis	33	21	17(81%)
	All	374	312	230(74%)

Table 3 .

 3 ROCAUC obtained using SimGIC to classify GIDs into 5 classes (see Table2for details).

	Classes	Reg. Reg. GE Metab. Synap. Neuro. All
	Class size	154	70	105	31	33	374
	Threshold (p<0.01) 0.507 0.514 0.511 0.533 0.531 0.506
	BP	0.523 0.905 0.785 0.758 0.568 0.691
	CC	0.515 0.846 0.659 0.787 0.561 0.641
	MF	0.507 0.830 0.735 0.588 0.560 0.641
	HPO	0.502 0.574 0.574 0.559 0.594 0.548
	BP, CC, MF	0.527 0.936 0.764 0.7911 0.581 0.695
	BP, CC, MF, HPO 0.524 0.933 0.748 0.768 0.628 0.691
	BP, CC	0.525 0.927 0.750 0.819 0.578 0.690
	BP, MF	0.522 0.916 0.785 0.711 0.571 0.688
	CC, HPO	0.514 0.817 0.662 0.757 0.633 0.640
	MF, HPO	0.513 0.813 0.643 0.625 0.616 0.622
	Maximum	0.527 0.936 0.785 0.819 0.633 0.695


domain: math shs

text:
Teresa F Blanco
email: teref.blanco@usc.es

Pablo G Sequeiros
email: pablo.gonzalez.sequeiros@usc.es

Cristina Núñez

María Salgado
email: maria.salgado@usc.es

A scale on cognitive configurations for tasks requiring visualization and spatial reasoning

Keywords: Visualization, spatial ability, preservice teacher, onto-semiotic approach

Introduction

We present a preliminary study focused on a scale on cognitive configurations for tasks requiring visualization and spatial reasoning. The main notion used is that of cognitive configuration from the onto-semiotic approach [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF]. The results show that there may be several configurations at each level and these levels depend on both certain conditions of the task and the visualization skills involved.

Theoretical framework

From the onto-semiotic approach (OSA), the analysis of the mathematical activity, of objects and processes taking part in it, focuses on the practices done by people implied in the solution of certain mathematical problems [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF]. The enforcement of this approach to visualization leads to distinguish between visual practices and non-visual or symbolic/analytic ones (Godino, Cajaraville, Fernández and Gonzato, 2011). In order to make these principles operative, the OSA poses as one of its tools the 'onto-semiotic configuration', i.e., the network of objects and processes involved in the fulfillment of a mathematical practice [START_REF] Font | The emergence objects from mathematical practices[END_REF]. These configurations can be socio-epistemic (networks of institutional objects) or cognitive (networks of personal objects). Here we will focus on the second ones.

Research problem and method

Can we set a scale on cognitive configurations associated to tasks requiring visualization and spatial reasoning (VSR)? To answer this question, three spatial tasks requiring counting, folding/unfolding and composing/decomposing have been selected. These tasks have been proposed to a total of 400 preservice teachers and the answers have been analyzed using the cognitive configurations (CC) proposed by the OSA (Fernández, [START_REF] Blanco | Razonamiento geométrico y visualización espacial desde el punto de vista ontosemiótico[END_REF].

Results

A variety of CC associated to each of the tasks has been found. This allows us to describe skill levels in VSR (Table 1) subject to certain conditions directly depending on the characteristics of the task, the visualization skills [START_REF] Del Grande | Spatial sense[END_REF] required and the synergy between visual and analytical languages of each configuration [START_REF] Godino | Synergy between visual and analytical languages in mathematical thinking[END_REF].

Conclusions

In general the ratio of students expressing high level CC is significantly below than of those exhibiting low level. The analysis shows that students mobilized variety and quantity of visual objects and processes. However, they do not reach the solution successfully. This fact might be due to students are not used to working with these objects and visual processes.

Table 1 : Scale of configuration levels

 1

	Task		Levels			Rank	%
	Truncated	Level 1 Level 2 Level 3 Level 4 Level 5 High: levels 4 and 5 10,25
	cube	CC2	CC1	CC3	CC6	CC5	Intermediate: level 3 11,50
			CC4	CC8	CC7		Low: levels 1 and 2	45
	Folding/	Level 1	Level 2		Level 3	High: level 3	8,5
	unfolding	CC1		CC2		CC3	Intermediate: level 2	14
				CC5		CC4	Low: level 1	52,50
	Perforated	Level 1	Level 2	Level 3	Level 4	High: levels 3 and 4 49,25
	cube	CC2	CC4		CC3	CC1	Intermediate: level 2 0,75
			CC5			CC6	Low: level 1	24,25

Proceedings of CERME10


text:
Chrysi Papadaki
email: chrysi@uni-bremen.de

Hide and think: Limiting the obvious and challenging the mind in geometry

Keywords: DGS, argumentation, visualization

In this paper I will present the implementation and some preliminary results of a teaching experiment conducted in a 9th grade geometry class within the framework of my doctoral studies. For the teaching experiment I designed geometry tasks in a Dynamic Geometry Software (DGS), with specific characteristics: they involve transitions from 2D to 3D geometric objects (and vice versa), they are "Black Box" tasks and they are designed to cause surprise, uncertainty or cognitive conflict to the students. The main foci of the paper are the description of the tasks, examples of strategies that a pair of students followed while dealing with them and the influence that the characteristics of the tasks had on students ' strategies, visualization and argumentation.

Introduction

In the lesson of geometry students are often asked to solve a problem, which is usually accompanied by a drawing. In many cases the drawing acts as an obstacle for students' argumentation, as "since they are able to see results on the drawings, since they can work easily on it, mathematical proof seems to be useless" (Mithalal, 2009, p. 796).

The idea behind the study presented in this paper could be expressed by the following question: What would happen if we asked students to solve geometry tasks designed in a DGS, in which the drawing of a geometrical solid that they would have to identify was actually invisible to them? This is something that one couldn't do with physical objects, hence the designing of the tasks in a DGS. My hypothesis is, that when challenging the students with such a task, they will reach a point at which they will realize that when naive visualization is limited or fails, they have to turn to more efficient strategies in order to prove a conjecture or justify an answer. In this hypothesis the strategies are based on the use of geometrical properties of the figure. The aim of the study is to examine how students' argumentation can be promoted by using tasks with specific characteristics and how visualization and argumentation are interweaved in 3D-DGS tasks.

I will show how these two parts of the geometrical activity are linked theoretically and then present the teaching experiment that was conducted and some preliminary results.

Theoretical background

The interplay between visualization and argumentation in the frame of tasks designed in DGS

The notion of visualization has constituted a topic of research interest both for psychologists and mathematics education researchers. Nevertheless, until today researchers have not yet agreed on one unique definition regarding this notion. In this problem-solving context it is the cognitive part of visualization that will be taken into account, as it is related to argumentation. [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF] classifies visualization into two categories, iconic and non-iconic visualization. In iconic visualization "the drawing is a true physical object, and its shape is a graphic icon that cannot Thematic Working Group 04 Proceedings of CERME10 be modified. All its properties are related to this shape, and so it seems to be very difficult to work on the constitutive parts of it" (Mithalal, 2009, p. 797). In non-iconic visualization the figure is analysed as a theoretical object represented by the drawing, using three main processes: Instrumental deconstruction: in order to find how to build the representation with given instruments. Heuristic breaking down of the shapes: the shape is split up into subparts, as if it was a puzzle. Dimensional deconstruction: the figure is broken down into figural unitslower dimension units that figures are composed of -, and the links between these units are the geometrical properties. It is an axiomatic reconstruction of the figures, based on hypothetico-deductive reasoning (ibid, p. 797) It follows that argumentation isolated or visualization isolated, is not enough for students to solve a problem or prove a statement. There needs to be a continuous interplay between the two, for students' geometric reasoning to progress and evolve. And it is through the use of tasks designed in a DGS that the interplay between visualization and argumentation is made stronger.

Several studies that have been conducted show that the use of DGS can promote students' visualization skills (see for instance [START_REF] Christou | Developing the 3DMath dynamic geometry software: Theoretical perspectives on design[END_REF]. During the last three decades DGS, like GeoGebra and Cabri, are being used more and more in the teaching of mathematics in secondary education. In this paper, the focus regarding DGS will be on their use in geometry teaching.

As Laborde characteristically writes: "DGS contain within them the seeds for a geometry of relations as opposed to the paper and pencil geometry of unrelated facts" (2000, p. 158). [START_REF] Healy | Software tools for geometrical problem solving: Potentials and pitfalls[END_REF] and [START_REF] Jones | Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations[END_REF] argue that DGS could play an important role in supporting students formulating deductive explanations and therefore also in the development of students' deductive reasoning, as they appear to have the potential to provide students with direct experience of geometrical theory and thereby break down what can be an unfortunate separation between geometrical construction and deduction (Jones, 2000, p. 81) Before I proceed to the presentation of the study and the tasks, I would like to give some more insight regarding the theoretical perspectives on which the task-design was based.

Theoretical perspectives behind the task design in this study

The design (the characteristics) of the tasks, aims at challenging the students to produce conjectures and examine their validity using strategies that go beyond iconic visualization and naïve empirical justifications, engaging them naturally in mathematical activity that involves non-conic visualizations and argumentation.

Characteristic No 1 -D-transitional tasks

According to [START_REF] Markopoulos | Teaching and learning of solids with the use of techological tools[END_REF], understanding the properties of a solid is equivalent to understanding the characteristic parts of a 3D shape, the comparative relations between the same or different structural parts and how the elements of the solid are interrelated. That is an idea which is very close to what [START_REF] Duval | What kind of activity is required for understanding and solving problem in geometry? A M.R.I. of geometrical activity[END_REF] calls "figural units" of a figure. That means that the properties of the component parts (figural units) of a 3D geometrical object are also properties of the 3D geometrical object itself. This correlation of properties between shapes and geometric objects of different dimensions is vividly present in tasks that involve both 2-dimensional (2D) and 3-dimensional (3D) geometric objects. This thought generated the idea of what I call D-transitional tasks. These are tasks involving transitions from 2D to 3D (and vice versa) geometric objects. Such tasks provide students with the opportunity to think about an object through the manipulation of another, and thus combine their properties and identify the relationships, connections and dependencies between their properties. Furthermore, contrary to other studies which also use 3D DGS geometry tasks (see for instance [START_REF] Hattermann | The drag-mode in three-dimensional dynamic geometry environments-two studies[END_REF], the students are not asked to see 2D units in 3D shapes, but to identify 3D shapes by studying 2D sub-figures of them.

Laborde argues that:

If properties of figures are not conceived as dependent, a deductive reasoning has no meaning. The question of the validity of a property conditional on the validity of other properties would not arise in a world of unrelated properties (2000, p. 157)

As Pittalis and Christou explain:

Students should understand that each paradigm of a 3D shape has a number of invariant and variant geometrical properties based on the properties of the isolated component parts and its own properties as a unified structure. The invariant properties constitute the criteria that the 3D shape should meet in order to represent an example of a class of 3D shapes (2010, p. 194)

Characteristic No 2 -Cognitive conflict, uncertainty and surprise

As cited in Hadas et al.: Goldenberg, Cuoco, and Mark (1998) stated that: "A proof, especially for beginners, might need to be motivated by the uncertainties that remain without the proof, or by a need for an explanation of why a phenomenon occurs. Proof of the too obvious would likely feel ritualistic and empty" (p. 6) They concluded that DGS may provide opportunities for the creation of uncertainties, leading students to seek for explanations (2000, p.128).

But uncertainty is not the only possible motivation for students to seek for a proof of the validity of their conjecture. According to [START_REF] Healy | Software tools for geometrical problem solving: Potentials and pitfalls[END_REF], and [START_REF] Laborde | Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving[END_REF] students feel the need for explanation when what they observe on the computer screen gives them a feedback that is surprising or is in conflict with what they expected.

Characteristic No 3 -Black Box: A way to create uncertainty and surprise

Black Box activities were designed by [START_REF] Laborde | Visual phenomena in the teaching/learning of geometry in a computer-based environment[END_REF] in the context of geometry teaching. Such activities have also been used by [START_REF] Knipping | Schwarze Kisten-Mit Black boxes Zusammenhänge erkunden[END_REF] during a research in geometry teaching using Cabri Geometry. In such a task, a construction is already offered to the students but the properties and rules on which this construction is based are hidden. The Black Box activities give to students the opportunity of interesting and productive explorations. "When students' predictions turn out to be wrong, this is a good opportunity for asking 'Why is it so?' and calling for an explanation or even proof" (Laborde, 2002, p. 311).

The teaching experiment

The teaching experiment was designed by the researcher and implemented by the participating teacher as part of his geometry lesson. The participants were 24 students of a 9 th grade class. Five Dtransitional tasks were designed in the 3D Graphics environment of GeoGebra 5. Before the implementation of the teaching experiment, a 90' session was dedicated to introduce the software to the students. During the teaching experiment the students worked in pairs on the computers.

For the purposes of the data collection there were used 3 cameras. Each camera was focused on one pair of students. For the three pairs of students that were video recorded, a screen-recording program was used to record their work in GeoGebra. The analysis of the data is based on the transcribed video of the students' discussions, the screen recordings and their notes on the worksheet they were provided with for each task.

Description of the tasks

In each task the GeoGebra window was divided into two sub-windows. On the right sub-window (3D Graphics) there was a 3D coordinate system in which a solid was designed, and a blue plane defined by axes x and y. On the left sub-window (Graphics) there were three sliders (h for height, t for tilt and s for spin), which the students could manipulate in order to move the solid, and a 2D depiction of the cross-section that was created when the solid intersected with the blue plane. The decision for the use of this representation was made, based on the prior knowledge of the students, who had already worked on intersections of solids in previous geometry lessons. In Task 1 (see Figure 1), the solid (a cylinder) was visible. The students were first asked to experiment with the three sliders and describe what the function of each slider was and how it affected the position of the solid in space. Subsequently, they were asked to examine the shapes of the cross-sections that were created in some different (h, t, s) positions of the sliders.

In Tasks 2, 3A, 3B and 3C (see Figure 2) the solid was hidden. The question set to the students in each of these tasks was "Which solid do you think this could be, judging from its cross-sections?".

The teaching experiment was conducted in two phases. During each phase the students worked as shown in Before each phase, the teacher explained the lesson procedure to the students. He explained to them that the focus of these tasks should be the justification of their answers. That was both linked to the didactical contract and the tasks. The students were asked to be as precise and explicit as possible in their explanations, to use mathematical arguments in order to support their answers to the rest of the classroom, during the classroom discussions that would follow.

During the time the students worked on the tasks, the teacher and the researcher were only observers of the situation. During the classroom discussions the teacher acted as a facilitator of the discussion.

Preliminary results

As the analysis of the collected data is still in progress, I will here present an example from the data as part of the preliminary results of this study. I will present the analysis of some excerpts from Gabriel and Elbert's discussion while working together on Task 3C (see Table 2). The students started by exploring freely the situation in the task, without using the worksheet that has been given to them as a helper. The first sign of surprise is seen in the reaction of Gabriel, when he starts exploring the cross-sections performed, by the blue plane, on the solid when moving it with the sliders The students' first conjecture, based on their visual perception, is that the hidden solid is a cube. Nevertheless, as soon as they find some unexpected cross-sections like pentagons and hexagons, their naïve visualization proves insufficient and they start questioning their initial conjecture. They start following a more organized exploration, by identifying some figural units of the hidden solid. They now start looking at the properties of the cross-sections (the 2D geometrical object), that is of the figural units, of the solid and relate and transfer them to the hidden solid (the 3D geometrical object). At this point, they discuss that according to the characteristics and properties of the cross-sections they have seen until now, they can only argue that the solid is a cuboid and not a cube yet. Although they have moved from iconic visualization to identifying figural units of the solid and some of their properties, they haven't yet reached the level of dimensional deconstruction.

Dialogue

[…] E: We know that the height and the width are equal. We know that it is square. G: Yes, as we proved, the face is square. We have proved that. But

The students end up to the conclusion that their initial conjecture was true, basing their justification on deductive arguments linking together and relating the properties and the characteristics of both the visible cross-sections and the invisible solid. This process resulted to the reconstruction of the hidden solid by first achieving the dimensional deconstruction of its cross-sections.

the length, the width and the height are equal, we have proved, because we changed the tilt and the object was always a square at 90, 180, 270 and 360 and 0. Just turned round the axis.

Conclusion

Gabriel and Elbert wanted to come up with a justification of their conjecture that would be satisfying for them. The nature of the task and the students' initially "unorganized" exploration soon caused them surprise and cognitive conflict. The students started doubting their conjecture and they turned to a deeper exploration moving from iconic visualization to identifying figural units of the solid and their properties. They based their reasoning on the properties of both the cross-sections and the solid and also the dependencies between them. They achieved the reconstruction of a hidden solid, judging by its 2D cross-section and subunits. At the end they correctly identified the hidden 3D shape as a cube by studying its 2D sub-figures. They stopped and accepted their conjecture as true only when they had produced what for them constituted a valid argumentation.

There is still a long way to go until I can present some more general results from my study.

Nevertheless, examples like the one presented here show that it is possible to support students' visualization, argumentation and deductive reasoning in geometry, by using tasks whose design is based on the idea of the interplay between visualization, argumentation and which have the characteristics presented in this paper.

Figure

  Figure 1: Snapshot of Task 1 Figure 2: Snapshot of Task 3C

  ****! What is this sh**? It should be symmetric! E: It could also be a cube. Change the height again. Make it zero (the tilt) and change the height. (They have set h=0 and d=0 and they change the tilt) E: Ay ay ay! G: We have here a diagonal crosssection, that is no... E: No cube (he laughs) G: No cube… Right?

  a cube. But it should be a cube. G: Could be one (cube). E: The other face is definitely square. We have seen that. G: Mhm.. (affirmative) E: The top face is also square. So, it could be a cube, it could -G: Wait, wait, wait. Spin 90, 90. So, 45. So, now is this thing perfectly oriented. E: Or it could be awhat do you want to see? G: More than one, more than one. I have more than one, therefore lets try the tilt. Now it is straight, right? I have simply oriented the thing (the solid) on the coordinate system E: That would work, if it is a cube, but it could also be a, a thingy. How is it called again? Prism? [...] G: That would be a cuboid.

Table 1 : Phases of the teaching experiment

 1 Table 1 below.

	Phases	Steps of the Phases	Duration
	1 st Phase -Students work in pairs	Task 1 -Visible Cylinder	22'
	(1 st day)	 Classroom discussion	18'
		Task 2 -Invisible Sphere	15'
		 Classroom discussion	5'
	2 nd Phase -Work in parallel. Students	Task 3A -Invisible Cone
	work in pairs	Task 3B -Invisible Pyramid	50'
	(2 nd day)	Task 3C -Invisible Cube
		 Classroom discussion	30'

Table 2 : Excerpts from the collaborative work of Gabriel and Elbert on task 3C

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