Daleen Badenhorst

Vincent Herbreteau

Yannick Chaval

Marie Pages

Terence J Robinson

Worawut Rerkamnuaychoke

Serge Morand

Jean-Pierre Hugot

Gauthier Dobigny
email: gauthier.dobigny@ird.fr

New karyotypic data for Asian rodents (Rodentia, Muridae) with the first report of B-chromosomes in the genus Mus

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The non-ambiguous identification of species is critical for fundamental as well as applied research, as it is the departure point for many aspects of comparative biology including systematics (e.g., Sites & Marshall, 2003, and references therein), biodiversity assessments, conservation [START_REF] Crandall | Considering evolutionary processes in conservation biology[END_REF]Moritz, 2000), biogeography (e.g., [START_REF] Avise | Phylogeography: retrospect and prospect[END_REF], community ecology and pests control (e.g., [START_REF] Hoffman | Plasmodium , human and Anopheles genomics and malaria[END_REF]. In particular, the lack of accurate species-specific identification of hosts is an important consideration for current epidemiological surveys since 1) numerous pathogens are carried by a restricted range of hosts, and 2) each rodent species possesses its own life traits. These two aspects are pivotal since they may lead to important differences in the role a host species can play in circulation and transmission of pathogens. For instance, a species may be an important reservoir for a human pathogen while its close, and sometimes morphologically similar relatives, are not (e.g., the two cryptic species Mastomys natalensis and M. erythroleucus, and their association with the Lassa Fever arenavirus in Africa; [START_REF] Lecompte | Mastomys natalensis and Lassa fever, West Africa[END_REF]. Additionally, some species-specific ecological characteristics (such as commensality and hence the propensity to transport with humans) and evolutionary properties (population structure and connectivity) are also central to epidemiological processes (reviewed in [START_REF] Mills | Ecologic studies of rodent reservoirs: their relevance for human health[END_REF]. In such a context, karyotypic data has been shown to be valuable, especially for identifying rodents that belong to sibling species complexes.

We carried out a survey of various rodent species found in different localities in Thailand as part of an epidemiological survey in this region. In particular, we focused on murids (rats and mice) that are suspected of being important reservoirs for several human pathogens (i.e., those responsible for hantavirus-mediated hemorrhagic fevers, typhus, leptospirosis, among others).

in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref

The first goal was to develop identification "tools" for use at the species-level. That this approach has application is illustrated by the Asian Hantaviruses (responsible for lethal haemorrhagic fevers) / rodents association found in Cambodian Rattus specimens that could not be discriminated at the species level [START_REF] Reynes | Evidence of the presence of Seoul virus in Cambodia[END_REF].

Asia is the area of origin for the Rattus sensu lato complex (i.e., Rattus sensu stricto and allied genera), a complex that includes the biological important index species, R. norvegicus (Aplin et al., 2003). Surprisingly, given their epidemiological importance, karyotypic data on species in this complex are limited. Even more importantly, numerous systematic modifications (reviewed in [START_REF] Wilson | Mammal species of the World: a taxonomic and geographic reference[END_REF] have been proposed since the pioneering cytotaxonomic work published in the late 1960s to the 1970s (see [START_REF] Marshall | Family Muridae: Rats and mice[END_REF] references therein), further complicating interpretations of correspondence between karyotype and host species. For example, many (if not most) of the taxa that were originally considered as different R. rattus subspecies (e.g., tanezumi, argentiventer, rattus) are now regarded as synonyms or have been elevated to distinct species. In the same manner, many congeneric species have been split into different genera (e.g., Berylmys, Maxomys, Leopoldamys, Niviventer). Consequently, species-specific karyotypes are often difficult to delimit, thus making a cytotaxonomic approach to species identification problematic.

In an attempt at redress, we performed a conventional cytogenetic analysis of 18 rodent species: Menetes berdmorei (Sciuridae), Mus caroli, M. cervicolor and M. cookii, Chiropodomys gliroides, Hapalomys delacouri, as well as representatives of most of the lineages within the Rattini tribe (sensu Lecompte et al., 2008), i.e. Rattus exulans, R. losea, R. tanezumi, Leopoldamys edwardsi, L. neilli, Maxomys surifer, Niviventer cf. langbianis, in : Journal of Zoology, vol. 279, n° 1 .p.44-56 CemOA : archive ouverte d'Irstea / Cemagref Berylmys berdmorei, B. bowersi, Bandicota indica, B. savilei. Our aim was to re-assess the reliability of karyotype-based diagnoses of Thai rodents, in an updated taxonomic framework. and Kalasin (16°26N 103°31E) using locally made locally made cage-traps. Except where otherwise detailed, taxonomy follows Carleton & Musser (2005). Species identifications were based on external measurements, morphological features, and, in females, mammae formula [START_REF] Marshall | Family Muridae: Rats and mice[END_REF][START_REF] Corbet | Mammals of the Indomalayan Region: a systematic review[END_REF]Aplin et al., 2003). Moreover, taking into account the ambiguous status of some of these diagnostic characters, most of the murine specimens were also investigated using molecular analyses (Tab. 1) involving several mitochondrial (cytochrome b, CO1, and 16S rDNA) and nuclear (IRBP) gene fragments. The DNA-based analyses will be presented elsewhere as a separate phylogenetic study (Pagès et al., in prep.).

MATERIAL and METHODS

Thousands

Importantly, however, and pivotal to the present work, our species-specific identifications relied on correspondence in both morphological and molecular data.

The majority of chromosome preparations were done in the field using the air-drying method on femoral bone marrow following yeast stimulation [START_REF] Evans | An air-drying method for meiotic preparations from mammalian testes[END_REF][START_REF] Lee | Yeast stimulation of bone marrow mitosis for cytogenetic investigations[END_REF].

In eleven instances (Tab. 1), intercostal tissue was sampled to establish fibroblast cell cultures using conventional techniques. Freshly prepared hypotonic (0.75 M KCl) and fixative (acetic acid-methanol 1:3) solutions were routinely used. Cell suspensions were stored in fixative and in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref transported to the laboratory where conventional staining, G-and C-banding [START_REF] Seabright | A rapid banding technique for human chromosome[END_REF][START_REF] Sumner | A simple technique for demonstrating centromeric heterochromatin[END_REF]) and image capture using the Genus software (Applied Imaging), were performed. A minimum of five complete metaphases were analysed for each specimen. Diploid (2n), fundamental (NF; i.e., total numbers of chromosomal arms in one cell) and autosomal fundamental (NFa; i.e., NF excluding sex chromosomes) numbers were determined, and karyotypes were arranged according to chromosomes morphology and size (conventional staining) and/or banding patterns. Chromosome nomenclature for Rattus followed Yosida (1976Yosida ( , 1977a)).

RESULTS and DISCUSSION

All specimens were analyzed using conventional staining, thus allowing an overall description of the species-specific karyotypes (Tab. 1). In addition, G-banded karyotypes were constructed where feasible. However, in several instances, slight variation and/or ambiguities in NF/NFa values were detected through conventional staining (Giemsa only) reflecting the difficulty in categorizing acrocentric/subtelocentric chromosomes where chromosomal contraction can be a confounding factor. Although not uncommon in mammalian cytogenetics, subtle differences in chromosome morphology (such as telocentric vs. subtelocentric) has been considered important in murine cytotaxonomy where it has been used to underpin the species identification of several taxa (e.g., Marshall, 1977, p. 407-408). In an attempt to constrain ambiguities introduced by terminology, we have chosen to view all chromosomes that have no visible short arm as telocentric and those that possess minute arms as acrocentric elements (subtelocentric chromosomes in some schemes) for NF/NFa assessment. If the short arms were more pronounced and clearly and repeatedly observed, chromosomes were considered to be as bi-armed (submetacentric and metacentric). The only squirrel (which is a sciurid, not a murid rodent) investigated in this study had a 2n=38 chromosome number. Most of the chromosomes were metacentric or submetacentric except for the five smallest pairs that were either telocentric or acrocentric (NF=76, NFa=72). This is in agreement with data previously published by Viegas-Péquignot et al. (1986) and Richard et al. (2003) on this species, and also specimens analysed by [START_REF] Marshall | Family Muridae: Rats and mice[END_REF] from Thailand. As a result the karyotype is not repeated here.

Rattus exulans (Peale, 1848), R. losea (Swinhoe, 1871) and R. tanezumi (Temmink, 1844) In total 26 specimens of Rattus sensu stricto were karyotyped (Tab. 1; data not shown). These included R. losea (n=5), R. tanezumi (n=13) and R. exulans (n=8). All but one specimen were determined to have a diploid number of 42 with NF values ranging from 56-61, 58-60 and 54-59 in R. exulans, R. losea and R. tanezumi, respectively. The 2n=43 R. tanezumi specimen was characterized by the presence of a G-negative and C-positive B-chromosome (data not shown), as has previously been documented in this taxon (Yosida, 1976[START_REF] Yosida | Supernumerary chromosomes in the black rat (Rattus rattus) and their distribution in three geographic variants[END_REF][START_REF] Gadi | Cytogenetic relationships in Rattus, Cremnomys, Millardia, Nesokia and Bandicota[END_REF]. The largely overlapping NF values are attributable to variation in the short arms (p arms) of several pairs resulting in their classification as acrocentric and/or submetacentric homologs (see Table 2). The polymorphisms most likely correspond to variation in pairs 1, 9 and 13 of Rattus r. tanezumi (now named R. tanezumi; see [START_REF] Wilson | Mammal species of the World: a taxonomic and geographic reference[END_REF] sensu Yosida and colleagues (1971aYosida and colleagues ( , 1971bYosida and colleagues ( , 1976Yosida and colleagues ( and 1977a)). This variation in centromere position has been noted in several Rattus species and subspecies (Yosida et al., 1971a(Yosida et al., , 1971b[START_REF] Yosida | Cytogenetical survey of black rats, Rattus rattus, in Southwest and Central Asia, with special regard to the evolutionary relationship between three geographical types[END_REF][START_REF] Yosida | Evolution of karyotypes and differentiation in 13 Rattus species[END_REF]Yosida, , 1977a;;Yosida & Sagai, 1973[START_REF] Yosida | Variation in C-bands in the chromosomes of several subspecies of Rattus rattus[END_REF]Gadi & Sharma, in : Journal of Zoology, vol. 279, n° 1 .p.44-56 CemOA : archive ouverte d'Irstea / Cemagref 1983; [START_REF] Motokawa | New records of the Polynesian rat Rattus exulans (Mammalia: Rodentia) from Taiwan and the Ryukyus[END_REF] and is interpreted to result from pericentric inversion (Yosida et al., 1971a(Yosida et al., , 1971b;;Yosida, 1976Yosida, , 1977a) ) and/or heterochromatin amplification/deletion [START_REF] Yosida | Variation in C-bands in the chromosomes of several subspecies of Rattus rattus[END_REF]Baskevich & Kuznetsov, 1998). However, the chromosomes and/or the segments involved are small and thus difficult to interpret, even when using G-banding, and the mechanism(s) underpinning this variation remains moot.

Importantly, [START_REF] Marshall | Family Muridae: Rats and mice[END_REF] proposed a number of apparently fixed (diagnostic) features among R. exulans, R. losea and R. tanezumi (which is synonymous with R. r. thai sensu Marshall; see [START_REF] Wilson | Mammal species of the World: a taxonomic and geographic reference[END_REF] in Thailand. On the basis of conventional staining, Marshall argues (pp. 407-408) that: i) pair 1 is acrocentric in R. tanezumi and submetacentric in R. exulans and R. losea, ii) pair 9 is submetacentric in R. losea but acrocentric in R. exulans and R. tanezumi, and iii) pair 11 (most probably corresponding to pair 13 sensu Yosida et al.) is submetacentric in R. exulans, whereas it is acrocentric in R. losea and R. tanezumi.

However, taking other studies into account (see above) and our own data (Table 2), it appears that none of these chromosomal morphs (1, 9 or 13) can be considered as species-specific. Moreover, heterozygotes for each of these three pairs were found in R. exulans and R. tanezumi (our study, Tab. 2), suggesting that these chromosomal polymorphisms were present in their last common ancestor and are therefore ancestral to the Rattus sensu stricto clade (Yosida et al., 1971a(Yosida et al., , 1971b;;[START_REF] Yosida | Variation in C-bands in the chromosomes of several subspecies of Rattus rattus[END_REF][START_REF] Yosida | Frequencies of chromosome polymorphism in pairs n°1, 9 and 13 in three geographical variants of black rats, Rattus rattus[END_REF] and date back ~2.2 My [START_REF] Robins | Dating of divergences within the Rattus genus phylogeny using whole mitochondrial genomes[END_REF].

Leopoldamys edwardsi (Thomas, 1882) and L. neilli (Marshall Jr, 1976) The five L. edwardsi specimens investigated in this study all display identical karyotypes (Fig. 1) with 2n=42 chromosomes. Most of the autosomes are acrocentric, except for four of in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref the five largest autosomal pairs (pairs 1, 2, 3 and 5 in Fig. 1) which possess short arms, as well as the three smallest ones which are metacentric (pairs 18, 19 and 20). The X is a medium size acrocentric chromosome, whereas the Y is C-positive and dot-like (and most likely acrocentric) element (NFa=54). Clearly observable blocks of C-positive heterochromatin were detected in several pericentromeric and telomeric regions, as well as in an intercalary position on one of the largest pairs (Fig. 1b; pair 2 on Fig. 1a).

Although L. edwardsi (from an unknown locality) was recently proposed to have 2n=44 chromosomes (Sablina et al., unpubl., quoted in the "Atlas of Mammalian Chromosomes", ed. S.J. O'Brien et al., 2007), our karyotypic results are in agreement with other reports (Yong, 1970;Rickart & Musser, 1993, and references therein). However, there is also a difference in the morphology of the Y chromosomes: the Y identified by Yong (1970) is a normal sized acrocentric, while our Y clearly appears as a dot like element (Fig 1b), suggesting there may be more fundamental differences underlying these observations. A single Leopoldamys female trapped in the vicinity of limestone karsts was referred to as L.

neilli on the basis of morphology as well as unambiguous molecular results (Pagès et al., in prep.). Even though our preparation was of poor quality (data not shown), making G-banding analysis problematic, the specimen unambiguously displayed 2n=44 chromosomes, with two large submetacentric and two small metacentric pairs (NF=52). These findings are in good agreement with previous reports (NF=52 in [START_REF] Marshall | Family Muridae: Rats and mice[END_REF] but differ from the NF=50 noted by Rickart & Musser (1983).

It would appear, therefore, that the two Leopoldamys species can be distinguished by karyotypes: L. edwardsi with 2n=42/NF=56 and L. neilli with 2n=44/NF=52. A third species, in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref L. sabanus, which is morphologically similar to L. edwardsi, has also been described from Thailand [START_REF] Wilson | Mammal species of the World: a taxonomic and geographic reference[END_REF]. The L. sabanus karyotype differs slightly from that of L. edwardsi since [START_REF] Marshall | Family Muridae: Rats and mice[END_REF] recorded a 2n=42/NF=54 complement for this species.

Unfortunately, no material was available to us to address the delimitation of the L. edwardsi / L. sabanus species. However, we anticipate that karyotypic differences between these two taxa are subtle, therefore unreliable for diagnostic purposes.

Niviventer fulvescens

Taxonomy within Niviventer is in need of revision [START_REF] Wilson | Mammal species of the World: a taxonomic and geographic reference[END_REF]. The results of our preliminary molecular (Pagès et al., in prep.) and morphological (Chaval, unpubl.) investigations are concordant and show that the three specimens analysed karyotypically by us form a natural grouping and are referred to herein as N. fulvescens (Pagès et al., in prep.).

Of these three specimens, a male was used to establish a fibroblast culture but the resulting lines appear to have undergone de novo rearrangement in culture. Nonetheless, the rearranged karyotype corresponds relatively well to the bone marrow karyotypes of the other two specimens which were both characterized by 2n=46 (NFa=52; Fig. 2). All chromosomes are acrocentric except for the submetacentric pair 2, as well as the three of the shortest autosomal pairs which are metacentric. The X is a large acrocentric chromosome, and the Y is a tiny acrocentric element which is entirely C-positive (data not shown). As far as we know, the Gbanding karyotype of N. fulvescens (Fig. 2) is presented here for the first time.

There are few published karyotype reports on subspecies/species of Niviventer and these are limited to using conventional staining, the only exception being [START_REF] Wang | Chromosomal polymorphisms due to heterochromatin growth and pericentric inversions in white-bellied rat, Niviventer confucianus, from China[END_REF] on N. confucianus. Except for tiny differences in centromeric placement, N. rapit, N. bukit, N.

in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref hinpoon [START_REF] Yong | Karyotypes of Malayan rats (Rodentia, Muridae, genus Rattus Fischer)[END_REF][START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF][START_REF] Marshall | Family Muridae: Rats and mice[END_REF], N. langbianis (Baskevich & Kuznetsov, 2000), N. fulvescens (Duncan et al., 1970[START_REF] Duncan | Karyotypes of five taxa of Rattus (Rodentia, Muridae) from Indonesia[END_REF] and N. cremoriventer [START_REF] Yong | Karyotypes of Malayan rats (Rodentia, Muridae, genus Rattus Fischer)[END_REF] are all invariant and closely resemble the cytotype that we present here for N. fulvescens. It is important to note that N. rapit and N. bukit from Thailand have since then been collapsed into N. fulvescens (see [START_REF] Musser | Notes on systematics of Indo-Malayan murid rodents, and descriptions of new genera and species from Ceylon, Sulawesi, and the Philippines[END_REF]Abe, 1983). The cytogenetic criteria proposed to discriminate all these Niviventer species are based on subtle shape differences of some pairs (see Table 2 in [START_REF] Wang | Chromosomal polymorphisms due to heterochromatin growth and pericentric inversions in white-bellied rat, Niviventer confucianus, from China[END_REF] which are most probably due to variation in heterochromatin, as observed between N. tenaster, N. fulvescens and N. langbianis (all from Vietnam) following C-banding protocols (Baskevich & Kuznetsov, 2000). Although we cannot state with certainty, it is highly probable that karyotypic variations within the genus will be too obscure to provide reliable cytotaxonomic identification.

Maxomys surifer (Miller, 1900) Five males and one female of M. surifer were analysed and the G-banded karyotype is presented for the first time (Fig. 3). All specimens displayed 2n=52, with seven metacentric (pairs 4, 5 and 14 in Fig. 3) or submetacentric (pairs 10, 12, 18 and 24) chromosomal pairs (NF=66). The Y chromosome is the smallest element, is entirely heterochromatic (Fig. 3b), and appears acrocentric in morphology. This is in agreement with available data for M. surifer from South East Asia (see [START_REF] Rickart | Philippine rodents: chromosomal characteristics and their significance for phylogenetic inference among 13 species (Rodentia, Muridae, Murinae)[END_REF], and references therein) including Thailand [START_REF] Marshall | Family Muridae: Rats and mice[END_REF]. However, some differences exist between these studies and the results of the present report especially with respect to the identification of the X chromosome.

The X was described as a metacentric in specimens collected from localities in Thailand [START_REF] Marshall | Family Muridae: Rats and mice[END_REF], Vietnam and Malaysia (see Rickart & Musser, 1993, and references therein), resulting in 2n=52/NFa=62. In sharp contrast, the X chromosome is one of the in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref largest chromosomal elements in specimens analysed by us, and is clearly acrocentric. Zoo-FISH analysis using a R. norvegicus X painting probe (Badenhorst et al., unpubl.) allowed us to confirm this result. Our M. surifer specimens are consequently characterized by NFa=64, and not NFa=62. When one considers our findings on the morphology of the X chromosome, the difference in NFa (NFa=62 vs 64) may be due to (i) a misidentification of the X in the past, (ii) a polymorphism that would involve the X chromosome, or (iii) the co-existence of different but cryptic Maxomys species that differ in the morphology of the X chromosome.

Further sampling and karyotyping will be required to address this point.

Berylmys berdmorei (Anderson, 1879) and B. bowersi (Blyth, 1851) A single B. berdmorei female, as well as one male and one female of B. bowersi were karyotyped. All three specimens had 2n=40. The karyotype comprised two large (pairs 1 and 2 in Fig. 4 and5), four medium-size (pairs 5, 6, 9 and 11) and one small submetacentric pairs (pair 18), one large (pair 3), five medium-size (pairs 12-16) and one small metacentric pair (pair 17), and three medium-size (pairs 4, 7, 8 and 10) and one short acrocentric (pair 19) pairs of chromosomes (NFa=64; see Fig. 4 and5). The X is a medium-size acrocentric chromosome and the B. bowersi Y is a small acrocentric element. Our results are in good agreement with previous reports (reviewed in Marshall and Newcomb, 1983) that suggested 2n=40 and NFa=64 for both species. An additional female B. berdmorei specimen was found to have 2n=41 due to the presence of an extra C-negative metacentric element that did not match with any of the other elements of the karyotype using G-banding comparison (Fig. 4) suggesting that this may be a B-chromosome. To our knowledge, this is the first report of Bchromosomes in this genus. Our results show some differences to those of [START_REF] Yong | Karyotype of four Malayan rats (Muridae, genus Rattus Fischer)[END_REF] and [START_REF] Yosida | Evolution of karyotypes and differentiation in 13 Rattus species[END_REF] who identified six pairs of acrocentric, eight pairs of metacentric and five pairs of subtelocentric autosomes; they also show a medium sized acrocentric X chromosome and a small acrocentric Y in specimens of B. bowersi. This mirrors other conflicting reports regarding the B. berdmorei karyotype. [START_REF] Marshall | Family Muridae: Rats and mice[END_REF] described six telocentric, six subtelocentric and seven metacentric pairs of autosomes in B. bowersi from Thailand, whereas [START_REF] Yong | Karyotype of four Malayan rats (Muridae, genus Rattus Fischer)[END_REF] recorded eight pairs of metacentrics, nine pairs of subtelocentrics, two pairs of acrocentric autosomes and a medium sized subtelocentric X in specimens from Malaysia. Conversely, [START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF] and Duncan et al. (1970) identified five pairs of subtelocentrics, eight pairs of metacentrics and six pairs of acrocentrics in this species in Thailand and Vietnam. The X and Y chromosomes of B. bowersi described by [START_REF] Yong | Karyotype of four Malayan rats (Muridae, genus Rattus Fischer)[END_REF] and [START_REF] Yosida | Evolution of karyotypes and differentiation in 13 Rattus species[END_REF] are comparable to our own. Conversely the B. berdmorei X has previously been reported as subtelocentric [START_REF] Yong | Karyotype of four Malayan rats (Muridae, genus Rattus Fischer)[END_REF], while our specimens clearly possessed an acrocentric X, in agreement with both Duncan et al. (1970) and [START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF]. Furthermore, we were unable to retrieve the two speciesspecific differences that were proposed to differentiate the karyotypes, i.e. a subtelocentric vs. acrocentric pair 6, and an acrocentric vs. subtelocentric pair 11 in B. berdmorei and B. bowersi, respectively (see [START_REF] Marshall | Family Muridae: Rats and mice[END_REF]. Given that our C-banding data show the presence of several C-positive short arms (Fig. 4 and5) which may be responsible for variations in chromosome morphology (i.e., the interpretation of acrocentric vs. subtelocentric), it seems prudent to conclude that the two Berylmys species cannot be distinguished cytogenetically with any confidence. Unfortunately, well resolved G-and C-banded chromosomes could not be produced due to the poor quality of our preparations. That said, the 2n=45 male clearly possessed 26 bi-armed chromosomes (NF=71) including a moderately large submetacentric X. The rest of the karyotype was composed of acrocentric elements, most probably including the Y (which would result in NFa=68) although its unambiguous identification was not possible from our preparations. The 2n=44 female was characterized by the same karyotypic morphology, except that it possessed one more acrocentric but one less bi-armed chromosome (data not shown). Although the difference between the 2n=44 and 2n=45 karyotypes may suggest a Robertsonian polymorphism in B. indica, based on our data we are unable to reach a definitive conclusion. Consequently, other explanations (e.g., X-autosome translocations, B chromosomes) cannot be excluded at this stage. For example, B. indica has been previously reported to contain 1-3 B chromosomes (Gadi et al., 1982;[START_REF] Gadi | Cytogenetic relationships in Rattus, Cremnomys, Millardia, Nesokia and Bandicota[END_REF].

Importantly, however, our results are slightly different to those previously published for this species collected from Thailand (2n=46; [START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF][START_REF] Marshall | Family Muridae: Rats and mice[END_REF], but are consistent with data on specimens from India (2n=42 for B. indica indica, and 2n=44 for B. indica nemorivaga; Ranjini, 1967;Avirachan et al., 1971;Gadi and Raman, 1977;[START_REF] Gadi | Cytogenetic relationships in Rattus, Cremnomys, Millardia, Nesokia and Bandicota[END_REF]. When taken together, the variation in 2n suggests a more complex systematic situation than currently thought within B. indica, and/or that there is a chromosomal polymorphism that remains to be described.

in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref Three males and two females of B. savilei were characterised by 2n=43 (Fig. 6). The diploid number of an additional male specimen was determined to be 2n=45. Unfortunately, as for B. indica, our preparations were of very poor quality thus greatly limiting the description of these karyotypes and our interpretation of their differences. The fundamental number could not be determined confidently from these five specimens. Nevertheless, we were able to establish a cell line for one female allowing for a more detailed investigation of the B. savilei karyotype. This specimen had 2n=43 comprising 23 meta-and submetacentric elements (NF=64; Fig. 6). Among the bi-armed chromosomes two correspond to the large metacentric X pair (as confirmed by Zoo-FISH using a Rattus norvegicus X paint; Badenhorst et al., unpubl.), and one was a B chromosome (Fig. 6a). C-banding showed that the X long arms as well as several acrocentric and metacentric centromeric regions were C-positive (Fig. 6b).

More surprisingly, the B-chromosome did not appear strongly stained following C-band (Fig. 6b). We are not aware of previous reports of B-chromosomes in B. savilei.

The karyotype described above differs from previous reports for specimens of this species collected in Thailand (2n=44, NFa=58; [START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF]. [START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF] described a karyotype consisting of six pairs of small metacentrics and two pairs of subtelocentrics and thirteen pairs of acrocentrics. Also, the largest autosomal pair was clearly submetacentric. Furthermore, the X of B. savilei was previously identified as a medium-sized acrocentric which differentiates it from other lesser bandicoot rats [START_REF] Markvong | Chromosomes of rats and mice of Thailand[END_REF][START_REF] Gadi | Cytogenetic relationships in Rattus, Cremnomys, Millardia, Nesokia and Bandicota[END_REF]. In absence of comparable data, we are unable to explain such differences. However, in our specimen the X was identified unambiguously by Zoo-FISH analysis (Badenhorst et al., unpubl.) as a large submetacentric chromosome. Therefore, males of B. savilei should have a 2n=43+1B karyotype (i.e., XY1Y2 males). Clearly, this requires confirmation by further cytogenetic investigations.

in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref

Hapalomys delacouri Thomas, 1927

A fibroblast cell culture was established from a male H. delacouri specimen caught in the vicinity of Loei, and our data are the first for this species from Thailand. Its karyotype was characterized by 2n=48 chromosomes and an NFa=92 (Fig. 7). All the autosomes were biarmed (metacentric or submetacentric). The metacentric X and the acrocentric Y were easily recognizable since they were respectively the largest and the smallest elements in the karyotype. Our findings differ significantly from published data on H. longicaudatus [START_REF] Yong | Karyotypes of Hapalomys and Pithecheir (Rodentia, Muridae) from Peninsular Malaysia[END_REF] in that these authors describe specimens from Malaysia as having 2n=50, with the X and Y chromosomes having similar morphology; all the autosomes were reported as acrocentric. This suggests that cytotaxonomy may be a valuable diagnostic tool within the genus. Our results constitute the first karyotypic data for this very poorly documented species.

Interestingly, two classes of C-positive material were identified on staining intensity suggesting a different repeat composition for this heterochromatic material (Fig. 7b). The genome appears to be particularly rich in heterochromatin, with some chromosome pairs displaying entirely heterochromatic long arms.

Chiropodomys gliroides (Blyth, 1856)

A fibroblast cell culture was established from a male C. gliroides specimen that possessed an entirely acrocentric chromosomal complement (2n=42, NF=42), including the X and Y chromosomes (NFa=40). Unfortunately, the preparations were of poor quality and construction of a G-banded karyotype was not possible. Nevertheless, strongly positive Cbands could be clearly observed in all pericentromeric regions, whereas the Y chromosome in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref was entirely heterochromatic, although less heteropycnotic that the pericentromeric material of the autosomes (Fig. 8). These results are in partial agreement with previous data concerning this very poorly documented species. [START_REF] Yong | Chromosomes of the pencil-tailed tree-mouse, Chiropodomys gliroides (Rodentia, Muridae)[END_REF][START_REF] Yong | Heterochromatin blocks in the karyotype of the pencil-tailed tree-mouse, Chiropodomys gliroides (Rodentia, Muridae)[END_REF] reported that eight Malayan specimens were characterized by 2n=42 with most chromosomes being acrocentric in morphology with C-positive pericentromeric regions [START_REF] Yong | Heterochromatin blocks in the karyotype of the pencil-tailed tree-mouse, Chiropodomys gliroides (Rodentia, Muridae)[END_REF]. Additionally there was one metacentric chromosome pair, a submetacentric chromosome pair, and the sex chromosomes were bi-armed. These bi-armed elements were all clearly absent in our single specimen of this species (Fig. 8). At this point, we cannot distinguish whether the differences are intraspecific (i.e. represent polymorphisms), or that they underpin the recognition of apparently cryptic species within C. gliroides.

Mus caroli (Bonhote, 1902), M. cervicolor Hodgson, 1845 andM. cookii Ryley, 1914 Seven male and seven female mice, trapped close to or within pineapple fields in Loei, were characterized by an identical 2n=40 acrocentric karyotype (NFa=38). The karyotypes of the Thailand species M. musculus castaneus, M. caroli, M. cervicolor, M. cookii, M. shortridgei and M. fragilicauda (see Auffray et al., 2003, and references therein) all have the same invariant karyotype [START_REF] Auffray | Phylogenetic position and description of a new species of subgenus Mus (Rodentia, Mammalia) from Thailand[END_REF] indistinguishable from that evidenced by our specimens. Molecular typing using rDNA16S and rDNA28S nucleotide sequences of the 14 specimens analysed by us allowed the identification of three different species in our sample, M. caroli, M. cervicolor and M. cookii (A. Orth, pers. comm.). Several of these animals were caught in the same trap lines, often only meters apart, thus representing another report of true syntopy as has been similarly suggested for M. caroli, M. cervicolor and M. fragilicauda in Thailand [START_REF] Auffray | Phylogenetic position and description of a new species of subgenus Mus (Rodentia, Mammalia) from Thailand[END_REF].

in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref

In addition, two female specimens, identified as M. cookii on nucleotide sequences (A. Orth, pers. comm.), had 2n=41 chromosomes (Fig. 8). The two karyotypes each comprised 40 acrocentric chromosomes, in addition to one unpaired medium-size metacentric and one unpaired acrocentric element respectively. These two unpaired chromosomes are G-negative (not shown) and C-positive (Fig. 8b) and are considered by us to be B-chromosomes. To our knowledge, this is the first report of B-chromosomes in the genus Mus. Although several hypotheses have been proposed to explain some aspects of the biology of these extra, non-Mendelian chromosomes (reviewed in [START_REF] Camacho | B-chromosome evolution[END_REF], little is known about their origin and their impact on the genomes, especially in mammals. Keeping in mind the importance of the mouse as a biological model, this observation clearly merits further investigation.

Conclusion: chromosomal evolution in Thai rodents and epidemiological implications

Despite a central role as reservoirs and/or vectors of several human diseases, reliable systematic data necessary to underpin accurate species-specific geographical, ecological and genetic information on Asian rodents are limited, and many taxonomic associations are unclear. Cytotaxonomy has been of great help in similar epidemiological investigations on other continents i.e., Africa), frequently providing unambiguous identification of species including those that are morphologically cryptic (e.g., [START_REF] Duplantier | Chromosomal characterization of three species of the genus Mastomys in Senegal[END_REF][START_REF] Volobouev | Chromosomal characterisation of Arvicanthis species (Rodentia: Murinae) from Western and Central Africa: implications on taxonomy[END_REF][START_REF] Dobigny | A new sibling species of Taterillus (Rodentia, Gerbillinae) from West Africa[END_REF]Granjon & Dobigny, 2003;[START_REF] Rambau | Molecular genetics of Rhabdomys pumilio subspecies boundaries: mtDNA phylogeography and karyotypic analysis by fluorescence in situ hybridization (FISH)[END_REF][START_REF] Veyrunes | CemOA : archive ouverte d'Irstea / Cemagref African pygmy mice, subgenus Nannomys (Murinae; Mus)[END_REF]. However, apart from Hapalomys, the preliminary results presented here suggest that this does not hold for the Asian species surveyed in our study. Although most genera can routinely be discriminated on karyotype (as they often can on morphology), few speciesspecific chromosomal characters could be identified within genera investigated herein. For in : Journal of Zoology,vol. 279, CemOA : archive ouverte d'Irstea / Cemagref instance, although L. neilli and L. edwardsi could be delimited cytogenetically, the two Berylmys, three Mus, three Rattus species analysed by us are indistinguishable on karyotype.

Although it remains to be confirmed, a similar situation may exist within Niviventer, as well as between L. edwardsi and L. sabanus. Karyotypic criteria that have previously been claimed to be diagnostic appear in our analyses to simply represent ancestral polymorphisms that are still undergoing idiosyncratic lineage sorting (i.e., hemiplasic characters sensu Avise & Robinson, 2008;[START_REF] Robinson | Hemiplasy and homoplasy in the karyotypic phylogenies of mammals[END_REF], thus impacting on the usefulness of these as diagnostic markers in species identification. Our findings suggest that new diagnostic features should be sought as a matter of urgency for use in these epidemiologically important rodent groups. DNA-based methods such as primer-specific analysis that have been used successfully in Africa (e.g., [START_REF] Lecompte | Molecular identification of four cryptic species of Mastomys (Rodentia, Murinae)[END_REF] and Europe (e.g., [START_REF] Michaux | Molecular identification of three sympatric species of wood mice (Apodemus sylvaticus, A. flavicollis, A. alpicola) in Western Europe (Muridae, Rodentia)[END_REF] are compelling candidates that should be prioritised for testing in Asian murids.

Table and Figure Legends

Table 1: Specimens investigated in the present study. Those that were diagnosed using DNA sequencing are noted as "+" in the Id. DNA column. Individual numbers refer to the CBGP Asian rodent collection (resp. Y. Chaval, Montpellier, France). Sex, origin, diploid (2n) and autosomal fundamental (NFa) numbers, type of biological material (bm: bone marrow; cc: cell culture), sex chromosomes morphology (X/Y) as well as number of B-chromosomes (Bs) are provided for each animal.

  of rodents have been trapped and are currently under investigation as part of a long-term epidemiological survey of rodent-borne pathogens in South Eastern Asia. Among these, sixty five specimens were collected in 2006 and 2007 in Northern and North-Eastern Thailand in the vicinity of the settlements Loei (17°29N 101°43E), Phrae (18°09N 100°08E)

  in : Journal ofZoology, vol. 279, CemOA : archive ouverte d'Irstea / Cemagref Menetes berdmorei (Blyth, 1849) (Sciuridae)

  in : Journal ofZoology, vol. 279, CemOA : archive ouverte d'Irstea / Cemagref

  Bandicota indica(Bechtein, 1800) and B. savilei Thomas, 1916 in : Journal ofZoology, vol. 279, CemOA : archive ouverte d'Irstea / Cemagref Eight bandicoot-rats, including two B. indica (one male, one female) and six B. savilei (four males, two females) specimens were karyotyped.The B. indica specimens were characterized by 2n=44 and 45, respectively (data not shown).

Fig. 1 :

 1 Fig. 1: G-banded karyotype (a) and C-banded metaphase (b) of a male Leopolodamys

Fig. 2 :

 2 Fig. 2: G-banded karyotype (a) and C-banded metaphase (b) of a female Niviventer fulvescens

Fig. 3 :

 3 Fig. 3: G-banded karyotype (a) and C-banded metaphase (b) of a male Maxomys surifer

Fig. 4 :

 4 Fig. 4: G-banded karyotype (a) and C-banded metaphase (b) of a female Berylmys berdmorei

Fig. 5 :

 5 Fig. 5: G-banded karyotype (a) and C-banded metaphase (b) of a male Berylmys bowersi

Fig. 6 :

 6 Fig. 6: G-banded karyotype (a) and C-banded metaphase (b) of a female Bandicota savilei

Fig. 7 :

 7 Fig. 7: G-banded karyotype (a) and C-banded metaphase (b) of a male Hapalomys delacouri

Fig. 8 :

 8 Fig. 8: C-banded metaphase of a male Chiropodomys gliroides (n°R5238).

Fig. 9 :

 9 Fig. 9: Conventionally stained (a) and C-banded (b) metaphases of one female Mus cooki

  in : Journal ofZoology, vol. 279, CemOA : archive ouverte d'Irstea / Cemagref

  Fig. 1

  Fig. 3

  Fig. 5

  Fig. 7

Fig. 9 in:

 9 Fig. 9

Table 2 :

 2 Chromosomal polymorphism observed for pairs 1, 9 and 13 (sensu Yosida and colleagues; see text for details) in the Rattus specimens investigated here. M and A correspond to meta-/submetacentric and acrocentric morphs, respectively. Like for Table1, individual numbers refer to the CBGP Asian rodent collection (resp. Y. Chaval, Montpellier,

	France).

  : Journal of Zoology, vol. 279,

	651	Table 2
	652
		Species	Individuals	Polymorphism in Rattus species
				pair 1	pair 9	pair 13
		Rattus losea	R4203	MM	MM	AA
			R4229	MM	MM	AA
			R4230	MM	MM	AA
			R4260	MM	MM	AA
			R4724	MM	AA	AA
		Rattus tanezumi	R4113	AA	AA	AA
			R4481	AA	AM	AA
			R4003	AM	AM	AM
			R4008	AA	AM	AA
			R4016	AA	AA	AA
			R4096	AA	AA	AA
			R4145	AA	AA	AA
			R4182	AA	MM	AA
			R4436	AA	?	?
			R4447	AM	AA	AA
			R4450	AM	AM	AA
			R4542	AA	MM	AA
			R4075	AM	MM	AA
		Rattus exulans	R4033	MM	MM	AA
			R4035	MM	MM	AA
			R4100	MM	MM	AM
			R4103	MM	MM	AA
			R4104	?	?	?
			R4112	MM	AA	AA
			R4140	?	?	?
			R4216	MM	AA	MM

in CemOA : archive ouverte d'Irstea / Cemagref

CemOA : archive ouverte d'Irstea / Cemagref

Acknowledgements

This study is part of the « Biodiversité Comparée des Roboviroses à Hantavirus en Europe et en Asie du sud-est » programme (ANR Project n°00121-05, J.P. Hugot). Daleen Badenhorst research is supported by a NRF bursary and a bilateral France/South Africa Protea grant to GD and TJR. We are grateful to Annie Orth and Josette Catalan for their help in the lab, Jo and Kaï (drivers), Kittiponk Chaisiri, Khomson Satchasataporn, the staff and students from the Faculty of Veterinary Technology, Kasetsart University, as well as the villagers in Loei and Phrae for their help in preparing and assisting the fieldwork. Cell culture experiments were performed using facilities provided by the "Evolutionary consequences of inversion polymorphism in Mammals" programme (ANR Project n°ANR05-JC05-48631, G. Dobigny) and the Evolutionary Genomics Group, University of Stellenbosch.

in : Journal of Zoology, vol. 279,


Rubén Ibáñez Pinillo

Francisco Chinesta
email: francisco.chinesta@ensam.eu

Emmanuelle Abisset-Chavanne
email: emmanuelle.abisset-chavanne@ec-nantes.fr

Erik Abenius

Antonio Huerta
email: antonio.huerta@upc.edu

Chapter 12 Radars in Transport Applications

Keywords: ADAS, Data-driven, Dimensionality reduction, Big-data

In the recent years, automotive car industry is evolving towards a new generation of autonomous vehicles, where decision making is not fully perform by the driver but it partially relies on the technology of the car itself. Indeed, a CPU inside the car will process all information coming from the sensors, distinguishing different scenarios appearing in the real life and ultimately allowing decision making. Since the CPU will be confronted with plenty of information, tools like machine learning or bigdata analysis will be a useful ally to separate data from information. These existing machine learning techniques, such as kernel Principal Component Analysis (k-PCA), Locally Linear Embedding (LLE) among many other techniques, are useful to unveil the latent parameters defining a given scenario. Indeed, these algorithms have been already used to perform real-time classification of signals appearing throughout the road. Selecting the modeling of the electromagnetic response of the radar plays an important role to achieve real time constraints. Even though Helmholtz equation represents accurately the physics, the computational cost of such simulation is not affordable for real-time applications due to high radar operating frequencies, resulting into a very fine finite element mesh. On the other hand, far field approaches are not so

Introduction

Automotive car industry is moving towards a new generation of autonomous vehicles, where the driver is not the main responsible of decision making but the technology of the car itself. Indeed, a CPU inside the car will process all information coming from the sensors, allowing to distinguish different scenarios appearing in the real life. Since the CPU will be confronted with plenty of information, tools like machine learning or big-data analysis will be useful to separate data from information. Thus, Sect. 12.2 provides an state of art of existing machine learning techniques Roweis and Saul [START_REF] Roweis | Nonlinear dimensionality reduction by locally linear embedding[END_REF]; Scholkopf et al. [START_REF] Scholkopf | Kernel principal component analysis[END_REF]; Lee and Verleysen [START_REF] Lee | Real-time, image-based, simulation of multiscale models based on locally linear embedding[END_REF] which are meant to be useful when unveiling the latent parameters definining a given scenario.

Regarding the modeling of the electromagnetic response of the radar, even though Helmholtz equation represents accurately the physics, the computational cost of such simulation is not affordable for real-time applications due to high radar operating frequencies, resulting into a very fine finite element mesh. On the other hand, far field approaches are not so accurate when the objects are very close due to plane wave assumption. That is the reason why, the Geometrical Optics method is investigated in Sect. 12.3 to fulfill both real-time and accuracy constraints. The main hypothesis under such model is that waves are treated as straight lines constrained to optical reflection laws. Therefore, there is no need to mesh the interior of the domain. However, the accuracy such approach is compromised when the size of the objects inside the domain are comparable to the wave lengths or in the vicinity of angular points.

Section 12.4 consists of the application of manifold learning and big data analysis into a data set of precomputed scenarios. Indeed, the identification of an unknown scenario from electromagnetic signals is purchased. Nevertheless, in our future work we will try to give an answer to questions such as how many receptors do we need to identify unequivocally the scenario, where to locate the receptors, or which parts of the scenario have a negligible impact in the electromagnetic response.

State of Art Manifold Learning

In this section, and for the sake of completeness, two widely employed manifold learning techniques are revisited: Principal Component Analysis (PCA) and the Locally Linear Embedding (LLE) Roweis and Saul [START_REF] Roweis | Nonlinear dimensionality reduction by locally linear embedding[END_REF]; Lopez et al. [2].

Principal Component Analysis

Let us consider D observed variables defining the vector y ∈ R D . These are commonly referred as the snapshots of the system: nodal values of the essential field throughout time in usual finite element modeling. We assume that these variables are therefore not uncorrelated and, notably, that there exists a linear transformation W defining the vector ξ ∈ R d , where d < D represents the unknown so-called latent variables, according to y = W ξ.

(12.1)

The transformation W, D × d, is assumed to verify the orthogonality condition W T W = I d , where I d represents the d × d-identity matrix (WW T is not necessarily I D ). The existence of such a transformation is precisely at the origin of PCA methods.

We assume the existence of M different snapshots y 1 , . . . , y M , that can be stored in the columns of the D × M matrix Y. The associated d × M reduced matrix contains the associated vectors ξ i , i = 1, . . . , M.

We assume that both observed and latent variables are centered, that is

M i=1 y i = 0 M i=1 ξ i = 0 . (12.2)
If it is not the case, prior to proceed, observed variables must be centered by removing the expectation of E{y} to each observation y i , i = 1, . . . , M. Since the exact expectation is unknown, one commonly accepted procedure is to substitute it by the sample mean.

PCA is able to calculate both d-the necessary number of members in the basis of the reduced-order subspace-and the transformation matrix W. PCA proceeds by guaranteeing maximal preserved variance and decorrelation in the latent variable set ξ. From a statistical point of view, therefore, it can be assumed that the latent variables in ξ are uncorrelated (no linear dependencies among them) or mutually orthogonal, thus constituting a basis. In practice, this means that the covariance matrix of ξ, defined as

C ξξ = E{ T }, (12.3) is diagonal.
However, the observed variables are expected to be correlated. The goal of PCA is then to extract the d uncorrelated latent variables in υ, according to

C yy = E{YY T } = E{W T W T } = WE{ T }W T = WC ξξ W T , (12.4)
that by pre-multiplying and post-multiplying by W T and W respectively, and taking into account that W T W = I, leads to:

C ξξ = W T C yy W.
(12.5)

The covariance matrix C yy can then be factorized by applying the singular value decomposition, C yy = V V T , (12.6) with V containing the orthonormal eigenvectors and the diagonal matrix containing the eigenvalues assumed in descending order. Substituting the factorized expression of the covariance matrix (12.2) into Eq. (12.1) it results This shows that the eigenvalues in correspond to the variances of the latent variables (the diagonal entries of C ξξ ).

C ξξ = W T V V T W. ( 12
In real situations, some noise may corrupt the observed variables. As a consequence, all eigenvalues of C ξξ are larger than zero, and the choice of d columns in V becomes more difficult. Assuming that the latent variables have larger variances than the noise, it suffices to choose the eigenvectors associated with the largest eigenvalues. This is the common practice in finite element model order reduction procedures. A number of columns of V are kept so as to preserve a chosen amount of the energy of the system.

From a geometrical point of view, the columns of V indicate the directions in R D that span the subspace of the latent variables ξ. The name PCA then arises naturally from the fact of keeping the components associated with the largest variance.

PCA constitutes a polyvalent method, developed, discovered and re-discovered many times in different branches of applied science and engineering. It determines data dimensionality, builds an embedding accordingly, and extracts the latent variables. However, PCA is still based upon one critical assumption: the linear dependency expressed by Eq. (12.1) between observed and latent variables (in other words, between the reduced-order and full-order models).

From a geometrical point of view, the columns of V indicate the directions in R D that span the subspace of the latent variables. We illustrate this interpretation in Fig. 12.1 where at left we can appreciate points that apparently belongs to R 2 , however, it is easy to see that all these points belong to a slow one-dimensional manifold. PCA find an alternative coordinate system given by V (axes in red) in which all these points are described from a single coordinate.

Frequently, latent variables posses a manifold structure, and therefore it simply does not exist a basis able to construct a projection such as that in Eq. (12.1). This is the case, for instance, in non-linear, large strain solid dynamics. Nonlinear methods are often more powerful than linear ones, because the connection between the latent variables and the observed ones may be much richer than a simple matrix multiplication. This situation is sketched in Fig. 12.2 where it can be noticed that no-rotation allows to extract the one-dimensional slow manifold. Thus, PCA seems informing that the different points belongs to a two-dimensional space, with the risk of concluding that the closest point (using the 2D euclidean distance) to the red point is in fact one that is very far from it when using the more appropriate geodesic distance on the one-dimensional slow manifold. Thus, the extraction of the slow manifold is compulsory and PCA is unable to accomplish the job.

Local-PCA-lPCA-applies standard PCA locally, that is, at each data-point and its closest neighbors. It is sketched in Fig. 12.3. The main issue related to its practical

Locally Linear Embedding

We consider the different points y i ∈ R D , i = 1, . . . , M, and proceed in two steps: 1. Each point y i , i = 1, . . . , M is linearly interpolated from its K nearest neighbors ("locally linear"). In principle K should be greater that the expected dimension d of the underlying manifold and the neighbors should be close enough so as to ensure the validity of linear approximation. In general, a small but enough number of neighbors K and a large-enough sampling M ensures a satisfactory reconstruction. For each point y i we can write the locally linear data reconstruction as:

y i = j∈S i W i j y j , (12.10)
where W i j are the unknown weights and S i the set of the K -nearest neighbors of y i . If we perform this locally linear interpolation for every data point in the high dimensional space, the set of weights that best approximates the manifold structure of the data will be obtained by minimizing the functional

F(W) = M i=1 y i - M j=1 W i j y j 2 , (12.11)
where W i j is zero if y j does not belong to the set of K -nearest neighbors of y i . 2. We assume now that each linear patch around y i , ∀i, is mapped onto a lower dimensional embedding space of dimension d ≪ D. To maintain the neighborhood structure of the set weights are assumed to remain unchanged in the low-dimensional, embedding space. The problem thus becomes the determination of the coordinates of each point y i in the low dimensional embedding space,

ξ i ∈ R d . For this purpose a new functional G is introduced, that depends on the searched coordinates ξ 1 , . . . , ξ M G(ξ 1 , . . . , ξ M ) = M i=1 ξ i - M j=1 W i j ξ j 2 , (12.12)
where now the weights are known and the reduced coordinates ξ i are unknown.

The minimization of functional G results in a M × M eigenvalue problem whose d-bottom non-zero eigenvalues define the set of orthogonal coordinates in which the manifold is mapped.

It is important to note that functional G(ξ 1 , . . . , ξ M ), with the different coordinates ξ i already calculated as just described, offers an error estimator on the locally linear embedding capacity, and even a local estimator can be derived by considering

E(ξ i ) = ξ i - M j=1 W i j ξ j . (12.13)
Thus, if we consider the introduction of a new point ξ in the embedding space R d after identifying its neighbors set S(ξ) and calculating the locally linear approximation weights, we can come back to R D and reconstruct y from its neighbors y i , i ∈ S(ξ).

Geometrical Optics

In this part, the basic concepts behind geometrical optics phylosophy are revisited. This approach combines optical reflection plus a balance of energy based on a loss of energy each time a reflection is produced. Figure 12.4 represents how a laser is reflected inside a closed surface (∂Ω), the reflections will stop when the energy carried by the ray becomes zero. Each time a reflection occurs, a given amount of energy is lost according to the absorption coefficient of the surface. Therefore, if a given laser is thrown with a initial energy (E ext ) from a initial point (x 0 ), such energy will be distributed along the surface (E int ) according to the geometry and the absorption coefficient distribution, as shown in Eq. (12.14).

E ext (x 0 ) = E int (x), x ∈ ∂Ω (12.14)
Hence, our main goal is to build a system of equations able to give us the spatial distribution of the absorbed energy E int (x) for a given source located at the position x 0 with an intensity that may depend on the angle of departure (θ d ), so E ext (x 0 , α d ).

Figure 12.5 shows the convention for the angle of departure, it is measured from the local normal direction. A counterclockwise rotation of the normal direction will When discretizing the element source x 0 into D-angular directions where the received energy could be stored, it will give an energy vector whose size is [D].

E s = ⎡ ⎢ ⎣ E α1 x0 . . . E αD x0 ⎤ ⎥ ⎦ (12.15)
where E α j x0 is the energy absorbed in x 0 , with an angle of arrival α j . The balance of energy between the energy going out from the source (x 0 ) and the energy coming back to the same point, could be written as a system of Eq. (12.16).

E s = Mf

(12.16)

where (i, j) component of the matrix M is the energy coming back to the source with an discretized arrival angle α a = i∆α, which has been thrown from an angle Figure 12.6 exemplifies how a column of matrix M is constructed. Imagine that the source has five discretized angles. Equation (12.17) describes the balance between a source located at x 0 with angle of departure α 4 and the received energy at the same element with an angle α 2 . The r coefficient takes into account the energy absorptions due to reflection that the ray has suffered till it arrives to the same departure point.

E α 4 ext = r E α 2 x0
(12.17)

If the same process is repeated for any possible angle of departure, the system of equations (12.16) is obtained. Moreover, any possible source could be described as the addition of unitary impulsional sources multiplied by their magnitude since the problem is linear. (i.e. the absorption coefficient does not depend on the magnitude of the incident wave.)

It is important to recall that all reflections that occur while the ray is traveling along the domain are analytical, thus the only kind of discretization error is the angular discretization of the source. That is the reason why, a convergence analysis in terms of angular discretization is crucial in this case.

Convergence Results

Figure 12.7 shows the domain where the algorithm is tested. There are four obstacles which have an absorption coefficient of 0.5, whereas the external wall has an absorp- Equation (12.18) defines the total energy coming back to the radar from the source, it is a constant scalar quantity. For this convergence analysis, the source and the receptor will be located at the same region.

E T = π 2 -π 2 π 2 -π 2 E(α d , α a )dα d dα a (12.18)
where α d and α a are the departure and arrival angles, respectively. On the other hand, quantities giving a glimpse of directional energy such as Eqs. (12.19), (12.20), (12.21) are interesting, since the total energy (E T ) could be a poor descriptor sometimes. Figure 12.8 shows the E D A function for the tested scenario. As it can be seen in the E D A , there is a primal reflection occuring at an angle of 40 • , caused by the blue car. Furthermore, there are more secondary reflections coming from the interaction with the other cars. Functional E D A is the indicator that contains more information since it keeps track of both departure and arrival angles. Indeed, if we have E D A , building the other two indicators is just a mere integration into a given direction. Equation (12.22) defines the energetic error. Where E H T is the total energy coming back to the radar for the most refined solution and E L T is the total energy coming back to the radar for a less refined solution.

E D (α d ) = π 2 -π 2 E(α d , α a )dα a (12.19) E A (α a ) = π 2 -π 2 E(α d , α a )dα d (12.20)
E = ||E H T -E L T || E H T (12.22)
Figure 12.9 shows the energetic error in logarithmic scale varying the angular discretization. The reference solution is constructed with 8000 discretized angles. As it can be seen, there is a fast convergence at the beginning. However, beyond 500 discretized angles, the solution starts to saturate, giving a relatively slow convergence

Black-Boxing the Scenario

The convergence results shown so far are based on relatively easy obstacles. Indeed, only four straight sides were required to replicate a square-like obstacle. Let's imagine for a single moment that our object in the scenario is hard to replicate i.e. a real car, an extrange pedestrian, etc. Therefore, a lot of elements will be required to represent accurately the real geometry of the obstacle, compromising the computational cost of the algorithm.

Black-Boxing consists of computing the response of an obstacle off-line, generating a library with all possible objects and defining a transfer function for each one of them. Then, if we know the obstacles inside our scenario, we will not have to reproduce the exact geometry but replace it with a Black-Box having a transfer function characteristic of the object inside. Figure 12.10 schematizes such procedure.

This transfer function consists of knowing three variables: the ratio of dissipated energy and both the location and angular direction of the outgoing ray. These three ingredients are functions of both the location and angular direction of the incoming ray. Figure 12.11 shows how a circle is black-boxed. As it can be seen, only two parameter, the box arclength (χ) and the income angular direction (α in ), suffices to characterize the three variables defining the transfer function.

Figure 12.11 shows the transfer function generated with a circle. As it can be seen, the energy ratio subplot gives information of how many reflection the ray has done before going out of the domain. Indeed, the yellow region corresponds with a ray that has not touched the circle, whereas the blue region corresponds with a single reflection since the absorption coefficient was set to 0.5. The other two subplots Figure 12.12 shows the transfer function generated with a star. It is important to note that the energy ratio is smaller than 0.5 since primary, secondary and terciary reflections occurs between the star peaks.

The Scenario Manifold

Imagine that a lot of different scenarios are precomputed a priori (off-line) and then they are placed into a dataset as shown in Fig. 12.13, which constitutes the scenario manifold M S . Hence, each electromagnetic response (E) depends on the variables to identify our scenario i.e. geometrical parameters defining both position and orientation of the obstacles. For instance, if a 2D square needs three variables to be identified (i.e. the vector position and the orientation angle), each point in the scenario manifold with N obstacles will be composed of p = 3 × N coordinates. Comparing scenarios involving different number of obstacles is possible by means of a nested representation i.e. defining the maximum number of obstacles and completing the coordinates of a scenario with less obstacles. Furthermore, if angles and positions have to be compared, the angle should be a dimensionalized with respect to a characteristic obstacle length.

Once the scenario manifold is generated, inferring the obstacle positions and orientations knowing a new electromagnetic response constitutes the main industrial interest. Equation (12.23) sets a minimization problem on the data set such that a similar electromagnetic response on the manifold is seeked. where subindex n makes reference to a new electromagnetic response whose associated scenario is unknown. However, the consideration of diverse scenarios may lead to a drastic increment on the cardinality of the parametric space (p). At this point, manifold learning techniques like the ones explained in Sect. 12.2 will be of crucial importance to define a reduce set of parameters (ξ), where the physics of the problem take place as shown in Eq. (12.24).

argmin ||E(α; p n ) -E(α; p i )|| ∀ p i ∈ M S (12.23)

argmin ||E(α; ξ

n ) -E(α; ξ i )|| ∀ ξ i ∈ M S (12.24)
Another point of critical importance is how to define the electromagnetic response in such a way that the scenario is identified unequivocally. The first inference is done just by knowing the electromagnetic response at one receptor which is located at the same position than the radar, both of them being located at the middle point of the bottom side of the square. Multiple solutions were found to the minimization problem (12.23). Indeed, there are several scenarios that share the same electromagnetic response, being different from a geometrical point of view. Figure 12.14 (left) shows the a priori unknown scenario (top-left) together with 5 other candidates sharing the same electromagnetic response composed with one receptor. As it can be seen, blue, yellow and purple obstacles varies along the different candidates, whereas the orange location and orientation is consistent along the candidates. Indeed, if a correlation of the obstacles taking into account the different candidates is done, the orange obstacle is the one presenting the most correlated value.

The same procedure is repeated but using three receptors instead of one. All three receptors are placed at the bottom side of the square between point 1 and 2. Once again, the minimization procedure gives several candidates which are shown in Fig. 12.14 (right). As it can be seen, the position of the orange obstacle is identified properly since this obstacle is the one causing most of the electromagnetic signal. Moreover, the blue obstacle correlation has slightly increased with respect to the

Conclusions

Fast scenario identification is a topic with a huge impact from an industrial point of view. Three different ways of reducing the computational time are being investigated. The first one regards the set of equations employed to solve the problem. Geometrical optics has been investigated as an alternative route to Helmholtz/Maxwell equations to handle high frequency radar signal simulations. The second procedure consists of generating off-line the transfer functions associated to different obstacles, speeding up the on-line computations. The third methodology makes use of manifold learning techniques to extract the relevant information defining the scenario from a electromagnetic point of view. Questions like the number and position of the sensors to identify in the best possible way the scenario constitutes a current research line.

  .7) This equality holds only when the d columns of W are taken collinear with d columns of V. If the PCA model is fully respected, then only the first d eigenvalues in are strictly larger than zero; the other ones are zero. The eigenvectors associated with these d nonzero eigenvalues must be kept: W = VI D×d , (12.8) yielding C ξξ = I d×D I D×d .(12.9)

Fig. 12. 1

 1 Fig. 12.1 Geometrical interpretation of PCA

Fig. 12. 3

 3 Fig. 12.3 Sketch of local-PCA

Fig. 12. 4 0 Fig. 12. 5

 405 Fig. 12.4 Closed surface (∂Ω), laser ray (red) thrown from x 0

Fig. 12. 6

 6 Fig. 12.6 Balance of energy for one ray

Fig. 12. 7

 7 Fig. 12.7 Scenario used to test the convergence results

Fig. 12. 8

 8 Fig. 12.8 Energy coming back to the radar as a function of departure and arrival angles, E D A

Fig. 12. 9

 9 Fig. 12.9 Convergence analysis in terms of the angular discretization

Fig. 12 .

 12 Fig. 12.10 Black-Boxing procedure

Fig. 12 .

 12 Fig. 12.12 Three variables definining the circle transfer function of a star obstacle

Fig. 12. 13

 13 Fig. 12.13 Scenario manifold

Fig. 12 .

 12 Fig. 12.14 Different candidate scenarios sharing one receptor (left) and three receptor (right) electromagnetic response