Vitalijs Zubkovs

Frédéric Jamme

Slavka Kascakova

Franck Chiappini

François Le Naour

Matthieu Réfrégiers

Single vs. two photon microscopy for label free intrinsic tissue studies in UV light region

Fibrillar distribution in rat tail tendon and mice liver can be measured with optical methods. Two-photon excitation gives easy assessment of fibrotic collagen type I and II. Single photon deep ultraviolet (DUV) excitation imaging highlights 10 all collagen types without discrimination. Their combination on the same tissue area provides a better overview of collagens in fibrillar diseases.

Introduction 15

Two-photon microscopy is a fast developing tool for cell biology and tissue diagnosis. [1][2][3][4] Its efficiency for imaging tissues in depth has been demonstrated and well recognized. 4 Due to its intrinsic non-linear nature it shines two type of signals, i) second harmonic generated (SHG) diffusion that brings easy localization of the 20 non-centrosymmetric structures present in biological materials with specific information concerning their orientation; ii) using near infrared excitation, it allows to image the ultraviolet excited autofluorescence of biological samples 6 so called two-photon excitation fluorescence (TPEF). It is one of the possible methods 25 to image label free biological samples with a great selectivity. SHG imaging can be applied for visualization of collagen 7 , which is a key component of load-bearing tissues (bones, tendons, etc.) and the most abundant protein in human body (30% of proteins present in body mass 1 ). Abnormalities in collagen development in several organs are often caused by a progression of inflammatory diseases and can be used as a pathological signature. SHG microscopy can be used to grade fibrosis in biological tissues. It should be highlighted that only fibrillar collagen types might be detected by SHG microscopy. Therefore globular collagen types do not give any SHG signal. 1,3 On the other hand, ultraviolet excitable autofluorescent compounds can be imaged using single photon ultraviolet (UV) light in cells and tissues 8,9 . UV fluorescence microscopy in deep ultraviolet (DUV) range (200 -300 nm) is an excellent tool for non-destructive qualitative specimen analysis. Two-Photon Excitation Fluorescence (TPEF) microscopy can also be applied for autofluorescent imaging of endogenous compounds. But this technique cannot compete with monophotonic microscopies in discreteness of a recorded spectrum, despite same range lateral resolution. 10 We describe for the first time the complementarities and synergies between ultraviolet single photon and two-photon excitation of autofluorescent compounds in tissues and compare the resolution and penetration depth of both modalities. Rat-tail 50 as positive control and mice liver as unknown sample were studied and the potential of the combination was explored.

Methods and Materials

DUV Synchrotron light imaging

The full field synchrotron DUV imaging set-up is build around a 55 Zeiss Axio Observer (Carl Zeiss, France) inverted microscope constructed with quartz-only optics. The white beam of DISCO beamline at Synchrotron SOLEIL 11 is monochromatized by an iHR320 (Jobin-Yvon Horiba, Longjumeau, France) before coupling with the entrance of a modified Zeiss Axio Observer Z1 60 (Carl Zeiss, Germany), monochromatic beam was chosen in order to determine composition of collagen in biological tissues exciting close to the 290 nm maximum, tryptophan and tyrosine were excited with the same excitation wavelength. A sharp dichroic mirror transmitting only above 300 nm (Omega Optical, 65 Brattleboro, Vermont) reflected the incident light that was further focused onto the sample through a Zeiss Ultrafluar 100x (N.A. 1.25, glycerine immersion). 8 Emission was recorded with a Pixis 1024-BUV (Princeton Instruments, USA) camera after passing through a series of bandpass filters (Semrock, Rochester, USA). 70 Fluorescence images were typically recorded in few seconds exposure (Table 1; DUV1 and DUV2). For bead's stack imaging, z scanning was performed with a PInano Z piezo slide scanner stage (PI, Germany). The whole system was controlled via µManager. 12 75

Two-photon imaging microscopy

The samples were analyzed with a Nikon A1MP+ confocal microscope (NIKON, France). A tuneable Maï Taï XF Ti:Saphire mode-locked laser (SPECTRA PHYSICS, France) excited the sample between 710 and 920 nm through a 40x water immersion 80 objective (N.A. 1.25, water immersion). Reflected second harmonic generated signal were detected at half the excitation wavelength with ad-hoc filters in front of the NDD GaAsP detectors. Excitation wavelength, acquisition time, and filters sets were chosen in order to maximize resolution and signal intensity 85 in each channel (Table 1; rows IR1 to IR2). Monodisperse 220 nm polystyrene beads from the polybeads sampler kit III (Polysciences, France) were deposited and dried on a quartz coverslip before observation. They give a very convenient size distribution standard (220 ± 20 nm) for ultraviolet fluorescence microscopy due to the natural autofluorescence of polystyrene around 390 nm under DUV excitation. Rat tail tendon Rat tendons have been studied with different methods (SHG microscopy, Scanning Electron Microscopy (SEM), spectroscopy, chemical methods, etc.) and biochemical 10 composition and their fibre organization is well characterized. [13][14][15] This tissue is rich in collagen type I fibrils making it an excellent sample for comparison of microscopy techniques. Rat-tail tendons were taken from Sprague-Dawley rats following EU 1986 procedures. Tendons were deposited on slides after physiological serum rinsing.

Mice liver histological sections

The study has been focused on liver samples from a mouse model fed on chow diet. The mice were grown in the animal facility of Institut André Lwoff (Villejuif, France). Mice were sacrificed 20 after 5 weeks of breeding. For SHG and DUV analysis fresh liver samples were fixed in formaldehyde and dehydrated in alcohol of increasing concentration up to absolute alcohol and finally immersed into toluene. Thereafter, the tissue was embedded in liquid paraffin, solidified and sectioned at 10, 20 µm with the Slices were mounted on quartz coverslips (thickness 0.17 mm) by addition of a drop of distilled water. In the standard sample preparation protocol, the human serum albumin is usually used. However using the UV excitation the addition of human serum 30 albumin can lead to parasite signal. Therefore we excluded the albumin from preparation protocol. The tissues were well preserved but presented several cracks formed during the segmentation procedure.

Results and Discussion

Comparing resolutions

The usual way to compare the resolution of microscopes is to use calibrated nanobeads with fluorescent probes excited in the energy range of interest. In our case, for ultraviolet excitation, the best compromise we found was to use the natural fluorescence of 40 polystyrene beads of known size. As seen on Fig. 1, the observed beads size after classic maximum likelihood estimation deconvolution is very close with both setups. Indeed, the natural confocal excitation provided by TPEF is compensated by the very low (290 nm) excitation wavelength 50 of widefield DUV fluorescence microscopy. Therefore, for all the following images, we can directly compare the information content of TPEF and DUV images.

UV imaging of intrinsic tissues

Rat-tail 55 Before studying liver tissues samples, we wanted to compare the different modalities with a simple tissue, namely rat tail tendon which is known to be rich in type I collagen. 15 The imaging area was chosen randomly during the first acquisition. It should be mentioned that both imaging modalities are non-invasive for 60 biological tissues and can therefore be applied in random order. The abundance of types I, II collagen in rat tail tendon is highlighted by SHG microscopy (figure 2, middle). In the tendon collagen architecture, two types of features can be highlighted: 70 elongated, wave-like fibres and curved interwoven fibrils. 14 TPEF signal can be recorded simultaneously with SHG signal using higher wavelength bandpass filter sets, however, compared to single photon DUV excited fluorescence, TPEF excitation profile is less structured and selective. 16 It represents NADH and FAD 75 distribution (table 1). The same sample was transferred onto the DUV microscope for examination of collagen. On the figure 2, similarities in organization of collagen bundles can be noticed between DUV fluorescence and SHG images. Separate collagen fibres bundles are well distinguishable, however, in cellular regions DUV fluorescence shows several other collagen fibres types which results in higher concentration (around 10% of all collagen types 15 in rat tail tendon).

Mice liver microscopy

As shown before on rat-tail tendon, collagen localization can be highlighted with DUV microscopy (Fig. 2) but cannot be typed. We observe collagen type I and II surrounding the central vein but also cells in DUV while in SHG only around the vein (Fig. 3).

Ultraviolet excited fluorescence (either two-photon or single photon, yellow in Fig. 3) permits also to visualize hematocytes that remained trapped inside the vein, only DUV microscopy shows the network of fibres surrounding those cells. DUV microscopy illustrates the distribution of endogenous 20 compounds in the liver tissue. The filters have been chosen close to regions of fluorescence maxima of tryptophan, collagen and elastin. The band pass filter at 327 -353 nm (excitation at 290 nm) was used to visualize tryptophan occurrence in the biological sample (Fig. 3). It reveals protein distribution in the liver with 25 different pathologies. The differences in signal intensities between two channels (red filters at 327 -353 nm and green at 412 -438 nm) allow the detection of collagen abundance.

Collagen type IV forms a structural basis of cell membranes.

Because it is a globular protein, it can be imaged by DUV 30 microscopy only, not by SHG. 3 Therefore, DUV fluorescence imaging provides deeper knowledge concerning collagen repartition in tissues (Fig. 3).

Collagen fibres develop mostly in vascular regions. Collagen fibres have spring-like structures that are randomly oriented 35 surrounding cell membranes. But close to veins or arteries collagen fibres are co-directed around the tubular objects (Fig. 3).

Conclusions

Two-photon microscopy found a broad range of applications in biology, for example, multi-photon excitation imaging of neuron 40 activity, cancer research studies of angiogenesis, metastasis and embryo development visualization. 10 On the other hand, Deep UV light (200-300 nm) and near UV light (300-400 nm) are used for excitation of many endogenous compounds, which are responsible for native tissue fluorescence. The endogenous 45 fluorophores present in most biological tissues are: NADH coenzymes which is present in mitochondria, chloroplasts, peroxisomes, cytosol and is involved in cellular metabolism, presents an excitation maxima at 260 and 345 nm 9,[START_REF] Kasimova | [END_REF] , aromatic autofluorescent amino acids: tryptophan (excitation maxima 220 50 and 277 nm), tyrosine (274 nm), phenylalanine (257 nm) compose many proteins. Furthermore, flavins, collagen, elastin, FAD and lipopigments are endogenous chromophores in tissues under UV light excitation. 9,18 Development of new methods for liver pathological tests is on 55 front line of actuality in many known researches. SHG microscopy has high potential for characterization of fibrotic stages in liver and was approved the Fibrosis-Metavir test 3 . Moreover, it has been shown that collagen fibres detection in liver by non-linear microscopy provides better sensitivity 60 compared to conventional staining methods. 19 Combination of SHG and synchrotron light excitation autofluorescence microscopies for biological tissue analysis has not been reported in any other study. Therefore, this research work has a significant value not only as a tool for the characterization of pathological 65 disorders, but also demonstrates the compatibility of these techniques for analyzing the same sample. While SHG permits a fine localization and quantification of fibrillar collagen type I and II, DUV fluorescence microscopy images all collagen crosslinks without discrimination. While this loss in selectivity could be 70 considered as a disadvantage, it may also be seen as an advantage because comparison between DUV and SHG images permits to localize collagens of other types than I and II. Due to the potential presented for liver tissues fibrillar characterization, the mentioned microscopy methods could be 75 applied to study pathological disorder caused by non-alcoholic fatty liver disease (NAFLD). Those disorders are expressed in an unordinary accumulation of lipid droplets in the liver of nonalcoholic patients. 20 NAFLD is classified between simple steatosis (fatty liver) and non-alcoholic steatohepatitis (NASH).

80

The histopathological spectrum of NAFLD extends from steatosis through NASH to advanced fibrosis, cirrhosis and hepatocellular carcinoma (HCC). While steatosis refers to lipids accumulation, the key factor concurs to favour NASH are oxidative stress culminating with liver injury and inflammation. 21 It has been

4
shown that the simple hepatic steatosis confers a long-term prognosis. It is a process reversible, considered innocuous in its pure form. Indeed patients without evidence of NASH rarely progressed to cirrhosis 22,23 , whereas the disease progression occurs in 43% of patients with NASH, and progression to cirrhosis may occur in 28% of these cases 23,24 . In addition, HCC has long been described to arise on a cirrhotic liver. Recent data however show that some patients with NASH can progress to HCC bypassing the stage of cirrhosis 25 . Therefore, NASH is now emerging as a leading risk factor owing to the epidemic of obesity and type 2 diabetes mellitus becoming major public health problem worldwide. Despite the major public health concern of NAFLD, it is currently impossible by the usual histological methods to identify at early stage patients that will progress from steatosis to NASH. This justifies the need of new methods to quantitatively assess the early biochemical changes related to this pathology. At present, the gold standard for the diagnosis of liver pathology is liver biopsy. These liver probes are analyzed mainly by visual inspection after appropriate staining protocols. Although rich on 20 information, from the clinical point of view, histopathology of liver is prone to intra-and inter-observer variability, which can yield poor reproducibility even when performed by experts. In the study of El-Badry et al. 26 , 4 pathologists assessed the features of NAFLD/NASH and a strong disagreement was found for all parameters including the overall diagnosis. Evaluation of NAFLD is therefore strongly observer-dependent and seems weakly reproducible. The incapacity of objectively assess the NASH and identify at an early stage patients that will progress from steatosis to NASH justifies the need of new methods to quantitatively 30 assess the biochemical changes related to this pathology. Recently, joint application of time-of-flight Secondary Ion Mass Spectrometry, DUV Microspectroscopy and infrared microspectroscopy for lipid, protein, sugar and nucleic acid mapping on liver samples has been developed. 1,27 Early studies showing SHG microscopy as a tool for liver-pathology examination, were reported. 28,29 Since then, several research groups assessed the method as a perspective technique bringing more sensitive results than conventional fibrosis scoring methods 3,9 and proposed a quantification method. 20,30 Because 40 SHG microscopy presents perspectives as a fast, non-destructive, quantitative technique, which eliminates human factors in liver analysis procedure, its combination with DUV single photon fluorescence microscopy looks even more promising.

Notes and references

Figure 1 .

 1 Figure 1. 220 nm polystyrene beads fluorescence imaging showing the resolving power of each microscopy technique. (left) 390 nm fluorescence after DUV excitation at 290 nm excitation. (right) TPEF image at 376 nm after IR 710 nm excitation. Scale bars 0.5 µm.

Figure 2 .

 2 Figure 2. Rat-tail tendon multimodal imaging: Co-localized region of rattail tendon showing separate collagen fibres magnified. Collagen excitation at 290 nm in DUV, emission 412 -438 nm. (upper) and 65

Figure 3 .

 3 Figure 3. DUV fluorescence image (left), collagen is in green, amino acids in red and SHG image (green) merged with TPEF (yellow) (right) of healthy mice liver (excitation: 330 nm; emission: 412 -438 nm; targeting regions of collagen localization). Scale bar 20 µm.

45a

  DISCO Beamline, Synchrotron SOLEIL, F-91192 Gif sur Yvette, France. Fax: 33 1 69359456; Tel: +33 1 69359655; E-mail: matthieu.refregiers@synchrotron-soleil.fr b actual adress: EPFL / STI -IBI -STI / LBP -CH -1015 Lausanne, Switzerland 50 c Inserm U785, F-94800 Villejuif, France d Univ. Paris-Sud 11, UMR-S785, F-94800 Villejuif, France

Table 1 .

 1 Microscopes acquisition parameters.

	Number	Excitation wavelength (nm)	Emission filter (nm)	Image acquisition time (s)	Targeted compounds	Co-excited compounds
	DUV1	290	327-353	5	Tryptophan Tyrosine,
						Pyridoxine,
						Collagen
	DUV2	290	412-438	5	Collagen	Elastin,
						NADH,
						Pyridoxine
	IR1	810	399-414	32	Collagen (SHG)
	IR2	810	447-472	32	NADH, FAD (TPEF)
	Polystyrene beads

Acknowledgments

Vitalijs Zubkovs is supported by ERASMUS MUNDUS student grant. We acknowledge Synchrotron SOLEIL support through projects


Marion Pilté

Silvère Bonnabel

Frédéric Barbaresco

Tracking the Frenet-Serret frame associated to a highly maneuvering target in 3D

In this paper we consider the problem of tracking a maneuvering aircraft. The dynamical model used is based on assumptions of nearly constant tangential velocity, curvature, and torsion, of the target trajectory. Using the Frenet-Serret frame, that is, an element of the special orthogonal group SO(3) we cast the tracking problem into a filtering on Lie groups framework. We then use an invariant extended Kalman filter to estimate the various quantities involved. The resulting filter is simple to implement, and shown to gracefully accomodate some realistic target trajectories.

I. INTRODUCTION

In radar applications, track maintenance is one essential component of the process. Mathematically, it boils down to a filtering problem, where one must filter the current position of the aircraft as well as its velocity and possibly higher order derivatives, from noisy position measurements. We will refer to this problem simply as "target tracking". When the aircraft is maneuvering, the problem is difficult due to the unpredictable nature of the motion. This area has been the object of extensive research over the four past decades, see [START_REF] Bar-Shalom | Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software[END_REF]. The main degrees of freedom for tracking are 1the dynamical model describing the motion of the target, and 2-the (statistical) filter used. As concerns dynamical models, there are many possibilities, but linear models are often used in industrial applications, the most famous being the Singer model [START_REF] Singer | Estimating optimal tracking filter performance for manned maneuvering targets[END_REF]. As concerns the filters, a simple robust solution is the noise process-adaptative Kalman filter of Castella [START_REF] Castella | An adaptive two-dimensional Kalman tracking filter[END_REF]. More modern approaches include particle filters [START_REF] Daum | Nonlinear filters: beyond the kalman filter[END_REF] and the reference filter for tracking which is the interacting multiple model (IMM) filter, see e.g. [START_REF] Bar-Shalom | Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software[END_REF]. The latter filter runs banks of (extended) Kalman filters in parrallel based on various models and assess weights to each model by evaluating likelihood of the measured outputs. This allows accomodating the various types of targets and degrees of maneuverability a single radar can be confronted with. The acadamic community has now largely turned to the challenges of multi-target tracking, with joint applications in video, see [START_REF] Mahler | Phd filters of higher order in target number[END_REF].

Nowadays, the radar air defense industry is facing novel challenges with ever increasingly maneuvering targets. Some targets can reach Mach 7 velocities with 15g accelerations. A way to inject some structure through a motion model marion.pilte@mines-paristech.fr S. Bonnabel is with Mines ParisTech, PSL Research University, Center for Robotics, silvere.bonnabel@mines-paristech.fr

F.
Barbaresco is with Thales Air System, frederic.barbaresco@thalesgroup.com into a trajectory that is deliberately trying to make the radar lose track of it, is to resort to physical considerations: The changes in aerodynamic lift and thrust-drag accelerations are limited, and those accelerations can in fact be expected to be piecewise nearly constant.

In the present paper, we propose to study the problem of tracking a highly maneuvering target in the 3D space. We propose as a very simple geometric model to use the Frenet-Serret formulas to describe the motion and assuming nearly constant curvature and torsion. This model includes helical motions that are particularly challenging to track. Our model can be related to [START_REF] Bullock | Maneuver detection and tracking with a nonlinear target model[END_REF], [START_REF] Bishop | Nonlinear approach to aircraft tracking problem[END_REF] and more recently [START_REF] Bunch | Dynamical models for tracking with the variable rate particle filter[END_REF], see Section II-B for more information. The position x is a 3D vector and the Frenet-Serret frame a rotation matrix R of SO(3), so (x, R) can be viewed as an element of the Lie group SE [START_REF] Castella | An adaptive two-dimensional Kalman tracking filter[END_REF]. Using this fact, we cast the tracking problem (partially) into a filtering problem on Lie groups. Slightly extending the Invariant Extended Kalman Filter (IEKF), introduced in [START_REF] Bonnabel | Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem[END_REF] and [START_REF] Bonnabel | Left-invariant extended kalman filter and attitude estimation[END_REF], we obtain a novel tracking algorithm. The IEKF is indeed a recent methodology that accounts for the geometric nature of the state space, and comes with convergence properties [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF]. It can also be related to the discrete EKF on Lie groups of [START_REF] Bourmaud | Continuous-discrete extended Kalman filter on matrix lie groups using concentrated gaussian distributions[END_REF], or to the generalized multiplicative EKF of [START_REF] Martin | Generalized multiplicative extended kalman filter for aided attitude and heading reference system[END_REF], see also [START_REF] Forbes | Continuous-time norm-constrained kalman filtering[END_REF].

The paper is organized as follows. In Section II we derive a stochastic evolution model from the Frenet-Serret equations and discuss the design of estimation algorithms. In Section III, we explain how the theory of [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] can be adapted when the state space consists of a Lie group and of additional quantities, and we describe the IEKF equations for the tracking problem. Finally, in Section IV, simulations illustrate the relevance of the filter.

II. PROBLEM AND PROPOSED MODELING

Consider a vehicle, say, an aircraft, evolving in the 3D space. It seems natural to describe the motion using piecewise constant control commands in a frame attached to the aircraft, that is, using intrinsic coordinates, as proposed in e.g., [START_REF] Godsill | Variable rate particle filters for tracking applications[END_REF]. As rotations of the aircraft around its velocity vector are unobservable from position measurements only, we propose to model its trajectory using Frenet-Serret formulas and to assume the tangential velocity, curvature, and torsion to be nearly constant. Denoting by x t ∈ R 3 the position of the target, the Frenet-Serret formulas write

d dt x t = uT (1)
d dt T = uκN, d dt N = u(-κT + τB), d dt B = -u τN (2)
where u ≥ 0 will be referred to as the (norm of the) tangential velocity, κ is the curvature, τ is the torsion, and T, N, B are all unit vectors perpendicular to each other. Those equations are also known as Frenet-Serret theorem and using a concise matrix notation, equation ( 2) can be re-written

d dt T N B = u T N B   0 -κ 0 κ 0 - τ 0 τ 0   (3)
Defining γ and τ as γ = uκ, τ = u τ we finally get:

d dt T N B = T N B   0 -γ 0 γ 0 -τ 0 τ 0   (4)
In the sequel, the parameters γ and τ will be referred to as curvature and torsion with a slight abuse of language, but in the application section, we will come back to κ and τ.

A. Proposed model for the target

Using Frenet-Serret formulas, we let R t ∈ SO(3) be the matrix R t = T N B . We then assume nearly constant tangential velocity, curvature, and torsion, leading to the following model for the dynamics:

d dt x t = R t v t + w x t , d dt R t = R t (ω t + w ω t ) × , d dt γ t = w γ t , d dt τ t = w τ t , d dt v t = w u t (5)
where the curvature γ t , the torsion τ t and velocity norm u t are unknown parameters, ω t = (τ t 0 γ t ) T , ν t = (u t 0 0) T , and w ω t , w x t , w γ t , w τ t , w u t are white noises that account for small changes over time in the motion parameters. Moreover, we let (a) × ∈ R 3×3 denote the skew symmetric matrix associated with cross product with the vector a ∈ R 3 .

Remark 1: The model ( 5) can be easily extended to accommodate nearly constant accelerations of the Singer model type [START_REF] Singer | Estimating optimal tracking filter performance for manned maneuvering targets[END_REF], by replacing d dt v t = w u t with d dt v t = a t and d dt a t = -αa t + w u t . The filter below can easily be adapted. The observations are the noisy positions at times t 1 ,t 2 , ...:

Y n = x t n +V n (6)
where the V n 's are Gaussian independent observation noises in R 3 . It is also common to express the radar position measurements in spherical coordinates of range, elevation and azimuth angles which leads to a nonlinear measurement model. Even if it poses additional difficulties in its own right, it is not the main object of the present paper and the technique presented here can be adapted linearizing the output map as in the EKF methodology. There are also techniques to transform (more precisely estimate without bias) the measurements from spherical to Cartesian, see [START_REF] Longbin | Unbiased converted measurements for tracking[END_REF].

B. Associated difficulties and links with prior literature

The proposed model ( 5) seems well suited to the tracking of highly maneuvering targets. Indeed it is closely related to constant commands in the body frame, albeit different. Devising a filter to track equations ( 1)-( 2), or more precisely the retained model [START_REF] Mahler | Phd filters of higher order in target number[END_REF], from position measurements is difficult for various reasons. First, because there are many quantities to be estimated only from position measurements and which appear nonlinearly in the equations. Then, because of the constraint of T, N, B forming an orthonormal base of the 3D space. It is not trivial to encode such nonlinear constraints in an extended Kalman filter. Using the model ( 5), we see that the pair (x t , R t ) defines an element of the Lie group SE(3). This indicates (part of) the state could be estimated by bringing to bear the Lie group structure underlying the state space. There has been extensive work on observers on Lie groups over the past years. However, for target tracking applications it is important that the filter output a covariance matrix, preventing the use of deterministic observers. Indeed, radar tracking involves the non trivial task of associating radar reports (so called "plots") with targets. This task (which is not considered in the present paper that is only concerned with tracking) is critical and requires an associate window based on the estimation error dispersion (covariance). This is why we resort to a Lie group based extended Kalman filter -the invariant EKF of [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] -for the tracking task.

There has been various prior attempts to use intrinsic coordinates to describe the target motion. The most prominent work in this direction is, to our best knowledge, the pioneering work of Antoulas and Bishop, see e.g., [START_REF] Bishop | Nonlinear approach to aircraft tracking problem[END_REF], [START_REF] Nabaa | Validation and comparison of coordinated turn aircraft maneuver models[END_REF]. If we assume the acceleration of the target is large with respect to the gravity vector field g, which is the case for highly maneuvering targets the NCS-CT model of [START_REF] Bishop | Nonlinear approach to aircraft tracking problem[END_REF] writes ...

x t = ẋt × ẍt ẋt 2 × ẍt ( 7
)
This equation is obtained assuming the kinematic acceleration ẍt , when projected onto the body frame, is constant.

Although the motivations for this model is akin to our motivations, the obtained equations are slightly different. Note indeed that, first the velocity of the target ẋt must always be different from zero for the model to be valid, which is not the case with equations [START_REF] Mahler | Phd filters of higher order in target number[END_REF]. Moreover, the model ( 7) is based on a zero torsion assumption. As a result quoting [START_REF] Nabaa | Validation and comparison of coordinated turn aircraft maneuver models[END_REF] "necessary condition for the model to be an accurate representation of actual trajectories is for the real trajectories to have small torsions". It is easily seen that the retained model [START_REF] Bishop | Nonlinear approach to aircraft tracking problem[END_REF] does not lend itself very well to extended Kalman filtering due to its strong nonlinearities. The authors thus propose a geometric filter, which is in fact a deterministic observer (of the Luenberger type [START_REF] Luenberger | Observing the state of a linear system[END_REF]), but for a nonlinear model, see [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF].

Another work that is akin to the proposed Frenet-Serret model ( 1)-(2) or equivalently [START_REF] Mahler | Phd filters of higher order in target number[END_REF] in state space form, is [START_REF] Bunch | Dynamical models for tracking with the variable rate particle filter[END_REF]. The proposed model is based on constant tangential and normal accelerations and constant-plane motion, and differs from [START_REF] Bar-Shalom | Tracking a maneuvering target using input estimation versus the interacting multiple model algorithm[END_REF], [START_REF] Bar-Shalom | Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software[END_REF] only in the way the noise enters the system. The authors assume indeed the system to be deterministic, and the acceleration to jump from time to time. They then use a particle filter to select the most likely accelerations. This method is fundamentally different from the Kalman filtering approach adopted in the present paper, and way more costly than an EKF computationally speaking.

III. PROPOSED INVARIANT EKF

In the present section we propose a novel extended Kalman filter for the tracking problem [START_REF] Mahler | Phd filters of higher order in target number[END_REF]. Since the state space is not a Lie group, the invariant EKF theory cannot be readily applied. It is however easy to generalize to this problem, as we explain in the following paragraph.

A. Extending the Invariant EKF methodology

The early theory of the Invariant extended Kalman filter [START_REF] Bonnabel | Left-invariant extended kalman filter and attitude estimation[END_REF], [START_REF] Bonnabel | Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem[END_REF] was concerned with invariant dynamics on Lie groups. In [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF], the theory was generalized to account for a broader class of dynamical systems on Lie groups and convergence properties were obtained. Indeed, it suffices that the dynamics, defined on a matrix Lie group G ⊂ R N×N with state χ t ∈ G and input u t ∈ U ⊂ R l , satisfy:

d dt χ t = f u t (χ t ) + χ t w t (8)
where w t is a continuous white noise, with additional condition

f u (ab) = a f u (b) + f u (a)b -a f -u(Id)b (9)
for all (u, a, b) ∈ U × G × G. As early noticed in [START_REF] Barczyk | Invariant observer design for a helicopter UAV aided inertial navigation system[END_REF], when the state space consists of a Lie group element χ t and some additional vector z t ∈ R k , and the dynamics write

d dt χ t = f u t (χ t , z t ) + χ t w χ t , (10)
d dt z t = g(z t ) + w z t , (11)
an Invariant EKF can still be devised formally by merely using an invariant error for the Lie group variable χ t and a usual linear error for the vector variable z t . This comes at a price, though, as convergence properties are lost. However, the structure of the filter which embraces the partial Lie group structure of the state space is relevant, and in practice very good performances can be expected. Finally note that, a trivial Lie group structure can be defined on such an extended state space, by multiplying the χ's and adding the z's. This route was recently pursued in [START_REF] Bourmaud | Continuous-discrete extended Kalman filter on matrix lie groups using concentrated gaussian distributions[END_REF], to estimate pose and velocities of a body in space from pose measurements. Unfortunately, this trivial extension does not make the overall dynamics for the extended state (χ t , z t ) satisfy condition [START_REF] Bonnabel | Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem[END_REF], so convergence properties are lost no matter what, and the end filter is very similar anyway.

B. Proposed filter's structure

In the present paper we propose to partition the state space using the following Lie group embedding in the special Euclidean group SE(3) using the homogeneous matrices:

χ t = R t x t 0 1,3 1 , w t = (w ω t ) × w x t 0 1,3 0
χ t is an element of SE(3), and letting z t = (γ t , τ t , u t ) T , the system (5) can be rewritten in the form ( 10)-( 11), with

f ω,u : R x 0 1,3 1 → R(ω) × Rv 0 1,3 0 , g(z) = 0 3,1
and the observation ( 6) can be re-written as:

Y n = χ t n 0 3,1 1 + V n 0 ( 12
)
The reader can easily verify condition [START_REF] Bonnabel | Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem[END_REF]. To linearize the equations and tune the gains, we propose to use a state estimation error of the form (χ -1 t χt , ẑtz t ), that is, the error consists of a left-invariant error on SE(3) as concerns the Lie group part of the state space, and a mere difference as concerns the vector part. The error is thus defined as follows:

η =       η R t η x t η γ t η τ t η u t       =       R T t Rt R T t ( xt -x t ) γt -γ t τt -τ t ût -u t       (13)
As the conventional EKF, the IEKF relies on two steps. During the propagation step it is a mere copy of the system's equations ( 5), using the compact notation ( 10)-( 11):

d dt χt = f ω t ,u t ( χt ), d dt ẑt = g(ẑ t ) (14)
Whenever a measurement is available, i.e., at t = t n , the state is updated

χ+ t n = χt n exp(L χ n ( χ-1 t n Y n )), ẑ+ t n = z t n + L z n ( χ-1 t n Y n ) (15)
where exp denote the exponential map of SE(3) (see equation [START_REF] Nabaa | Validation and comparison of coordinated turn aircraft maneuver models[END_REF] for the detailed formula). In other words, the update of χ t follows directly from [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF], whereas the update of z t mimics the one of the conventional EKF. Indeed, the innovation χ-1 t n Y n was defined as in the standard IEKF methodology.

Back to the problem's original variables of System (5), the propagation step writes:

d dt Rt = Rt ( ωt ) × , d dt xt = Rt vt , d dt γt = 0, d dt τt = 0, d dt vt = 0 ( 16
) where ωt = ( τt 0 γt ) T , νt = ( ût 0 0) T , and the update writes:

      R+ t n x+ t n γ+ t n τ+ t n û+ t n       =       Rt n exp m [(δ ω ) × ] Rt n xt n + B(δ ω )δ x γt n + δ γ τt n + δ τ ût n + δ u       (17)
where

δ ω δ x δ γ δ τ δ u T = L n ( RT t n (Y n -xt n ))
, exp m denotes the matrix exponential map in R 3×3 , and

B(δ ω) = I 3 + 1 -cos δ ω δ ω 2 (δ ω ) × + δ ω -sin δ ω δ ω 3 [(δ ω ) × ] 2 .
The gain matrix L n ∈ R 9×3 is computed through a Riccati equation based on linearizations around the estimated trajectory, as in the conventional EKF theory.

C. Linearization of the error system and gain computation

To tune the gains, we need to compute how the error (13) evolves during the propagation, and to write the update step, up to first order terms in the error (linearization). As in the standard EKF, the linear Kalman theory will then allow to estimate the error and its dispersion (covariance matrix) through a Riccati equation.

1) Propagation step: During the propagation step, the system's equations are [START_REF] Mahler | Phd filters of higher order in target number[END_REF], and the filters' are [START_REF] Longbin | Unbiased converted measurements for tracking[END_REF]. As a result, the time derivative of the error ( 13) is

d dt η t =       -(ω t + w ω t ) × η R t + η R t ( ωt ) × -(ω t + w ω t ) × η x t + v t + w u t -η R t vt -w γ t -w τ t -w u t       (18)
To linearize this equation, following the IEKF methodology, see [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF], we let η R t ≈ I 3 + (ξ R t ) × . This means that ξ R t ∈ R 3 is a small instantaneous rotation (random Gaussian) vector that encodes the discrepancy in orientation between the true and the estimated vehicles. We also let

ξ x t = η x t , ξ ω t = η ω t , ξ τ t = η τ t , ξ u t = η u t .
The rationale, akin to the standard EKF methodology (in the presence of non-additive noise), is then to neglect all second order terms in the components of ξ as well as terms of order ξ w , that is the noise when it enters multiplicatively the linearized error equation. First we use

(ξ R t ) × ( ωt ) × -( ωt ) × (ξ R t ) × = (ξ R t × ωt ) × .
This allows to identify the term d dt ξ R t we are seeking using that (a

) × = (b) × implies a = b. Let ξ =       ξ R t ξ x t ξ γ t ξ τ t ξ u t       ∈ R 9 ( 19
)
During the propagation step we have d dt ξ t = A t ξ t + w t with

A t =               0 -γt 0 0 0 0 0 -1 0 γt 0 -τt 0 0 0 0 0 0 0 τt 0 0 0 0 -1 0 0 0 0 0 0 -γt 0 0 0 -1 0 0 -ût γt 0 -τt 0 0 0 0 ût 0 0 τt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0              
(20) and with w t = ((w ω t ) T , (w x t ) T , (w γ t ) T , (w τ t ) T , (w u t ) T ) T 2) Update step: Here we must compute the impact of the update step [START_REF] Nabaa | Validation and comparison of coordinated turn aircraft maneuver models[END_REF] on the linearized error [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF]. To do so, we first note that the innovation as we define it in [START_REF] Godsill | Variable rate particle filters for tracking applications[END_REF], that is χ-1 [START_REF] Nabaa | Validation and comparison of coordinated turn aircraft maneuver models[END_REF] and using the fact that the first order approximations of B and exp m be the identity, we easily derive the effect of the update (17) on the error [START_REF] Martin | Generalized multiplicative extended kalman filter for aided attitude and heading reference system[END_REF], and its linear approximation in ξ , see [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF], writes

t n Y n = RT t n (Y n -xt n ), verifies RT t n (Y n -xt n ) = RT t n (x t n -xt n ) + RT t n V n = -(η R t n ) -1 η x t n + RT t n V n and thus since η x t = ξ x t and (ξ R t ) × ξ x t is second order, we have RT t n (Y n -xt n ) ≈ -Hξ + RT t n V n where H ∈ R 3×9 is defined by H = 0 3,3 I 3 0 3,3 . Looking at
ξ + t n = ξ t n -L n 0 3,3 I 3 0 3,3 ξ t n -RT t n V n (21)
3) Resulting gain: Let Q t be the trusted covariance of the process noise w t and N n the trusted covariance of the measurement noise V n . Considering the linearized system obtained with propagation d dt ξ t = A t ξ t + w t and update ξ + t n = ξ t n -L n Hξ t n -RT t n V n the corresponding Kalman gain is obtained through the standard following Riccati equations:

d dt P t = A t P t + P t A T t + Q t S n = HP t n H T + RT t n N n Rt n L n = P t n H T S -1 P + t n = (I 9 -L n H)P t n D.
Discussion on the filter's expected stability Unfortunately, as previously underlined, the obtained EKF does not possess convergence guarantees. However, we have the following result, that indicates some stability properties, at least in practice, should be inherited from the IEKF structure of the filter:

Proposition 1: Consider model ( 5) with noise turned off, measurements with noise turned off too, and assume that the quantities u t , γ t , ω t are all known, and may be freely varying inside an interval (α, β ) with α, β > 0. Assume also that the lowest eigenvalues of Q t and N n are lower bounded by some ε > 0. Then the IEKF proposed above when reduced to the task of estimating x t and R t , locally converges around any trajectory of the system.

Proof: The reduced system can be viewed as a nonholonomic car in 3D, that is, evolving in SE(3) instead of SE [START_REF] Singer | Estimating optimal tracking filter performance for manned maneuvering targets[END_REF]. The result appears then to be a straightforward extension of the simplified car example of [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF].

The result above indicates that if the tangential velocity, curvature, and torsion, were known at all times, and if the IEKF were used as a nonlinear observer for the corresponding tracking problem, then this observer would converge. Of course those quantities are not known, so the result above is only partially satisfying. However, it is reassuring as it would not even be provable for an (M)EKF (see [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] indeed).

IV. SIMULATIONS

In the previous section, we derived the filter equations for the tracking problem. In the present section, we test this filtering method on two different trajectories with motions close to real tracking problems.

A. Aircraft trajectory

The first trajectory tested has been simulated using a trajectory simulation program designed by the company Thales Air Systems. It is inspired by data from a real fighter flight experiment. The trajectory is presented on Figure 1, along with the noisy Cartesian measurements. The results are presented on Figure 2. In all Figures, the red curve corresponds to the estimation and the blue one to the real data. Remember that we had defined γ = uκ and τ = u τ. So we have estimated γ and τ. Here we directly plot κ = γ/u and τ = τ/u, the real curvature and the real torsion. For this trajectory, the position, norm of velocity and curvature are very well estimated by the filter. We see the curvature is in fact piecewise constant, and not nearly constant, but the process noise we added allows the filter to track it very well. The torsion curve shows some peaks. These peaks are filtered out by any algorithm since the quantity measured (the position) is obtained through three integrations of the torsion. More generally, it is hopeless to track the torsion correctly unless it is fairly large and fairly constant over time.

B. Helical trajectory

To simulate the second trajectory, we describe the evolution of a maneuvering target by [START_REF] Mahler | Phd filters of higher order in target number[END_REF]. We add some measurement Gaussian noise. The noise added is greater along the third coordinate, z (as it is the case in practice). The reference trajectory and the noisy measured trajectory are presented on Figure 3. We have simulated two consecutive maneuvers. The first one switches from a constant velocity straight line motion to a motion with non-zero curvature and torsion (they remain constant during the motion) along with a change in the norm of the velocity. The second maneuver switches from this last motion to a motion with another constant curvature and norm of velocity and no more torsion. So the target undergoes first a straight line motion, then a helical motion with a change in speed, and then a circle in a plane, with again a change of speed. Such trajectories can be encountered in current air defense applications, and are very challenging to track. We then perform filtering with the IEKF presented in the previous section. The estimations of the position x, of the curvature κ, of the torsion τ, and of the norm of the velocity u are displayed on Figure 4. We have poorly initialized all parameters to see the behaviour of the filter when confronted to high initial errors (in practice, the position is relatively well-known, but neither are the velocity nor the curvature). The norm of the velocity and the position are the parts of the state that converge the fastest. The curvature is also quite well estimated, even though it converges slower than the norm of the velocity. The torsion is again the most difficult parameter to estimate, although it eventually converges, it is not well estimated by the filter. This stems from the fact that it is weakly observable since it is small. This is problematic, as even very small values of the torsion lead non negligible helical motions, due to the three integrations over time. However, if it is relatively well-known initially, then the initial covariance can be adjusted so that the initial part suits more to the real data, after a jump, it also converges to the true value, if we wait for a sufficiently long time.

C. Comparison with other filtering algorithms

As explained in Section II, few filtering algorithms perform well on this kind of trajectories. Indeed, most of the time the models are expressed in two dimensions. The wellknown IMM designed by Bar-Shalom in [START_REF] Bar-Shalom | Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software[END_REF] is itself in 2D, so is the one presented in [START_REF] Boers | Interacting multiple model particle filter[END_REF]. It is also difficult to build an Extended Kalman Filter from the Frenet-Serret equations above, because there are orthogonality constraints that cannot easily be dealt with the standard EKF formulation.

The Variable Rate Particle Filter, introduced in [8] might work well, but it involves high computation cost, and the comparison thus seems unfair.

In industrial radars, a well-known trick is to use a linear model (constant acceleration) and adapt the process noise as in the work of Castella [START_REF] Castella | An adaptive two-dimensional Kalman tracking filter[END_REF]. Just to have an idea of what it might yield we have compared the root mean square errors (RMSE) of both filters in Table I. The noises have been tuned manually to get the best possible results with both filters. The Castella seems to perform better for the third coordinate z. It is due to the fact that the noise along this coordinate is higher, and the estimations of the Castella tend to follow the measurement and do not filter the noises enough, especially when it is high.

V. CONCLUSIONS

We have considered a tracking problem, modeled the trajectory assuming constant norm of the velocity, curvature, and torsion, and cast the model into a Lie group setting. Then, the theory of the invariant EKF, originally devoted to pure Lie groups was adapted to allow the estimation of some extra parameters, namely the norm of the velocity, and the curvature and torsion.

Two challenging tracking examples, close to real industrial problems, have been simulated. The position, the curvature, and the norm of velocity are very well tracked by the filter although they undergo various jumps over the experiment. This is satisfying, as the position and velocity vectors are the only two quantities that need be output with great accuracy. In future work, we plan to integrate our IEKF as a building brick into an IMM or a Rao-Blackwellized particle filter to accommodate jumps and multi-model estimation.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Reference and noisy aircraft trajectory inspired by real flight data

Fig. 3 .

 3 Fig. 3. Reference and noisy trajectory simulated.

Fig. 4 .

 4 Fig. 4. Estimation of the position (top left) and of parameters u t (top right), γ t (bottom left), τ t (bottom right)

TABLE I COMPARISON

 I OF THE RMSE FOR THE POSITION AND THE VELOCITY FOR THE IEKF AND CASTELLA'S KALMAN FILTER

	Parameter IEKF	Castella
	x	3.15	4.94
	y	3.15	4.91
	z	5.42	2.84
	ẋ	0.76	1.51
	ẏ	0.71	1.47
	ż	0.49	0.50

VI. ACKNOWLEDGEMENTS

This work is partly supported by a DGA-MRIS scholarship and THALES AIR SYSTEMS.


Anne Sarah Briand

Etienne Come

Martin Trepanier

Latifa Oukhellou

Smart Card clustering to extract

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Smart Card clustering to extract typical temporal passenger habits in Transit network. Two case studies: Rennes in France and Gatineau in Canada Anne--Sarah Briand 1 , Etienne Côme 1 , Martin Trépanier 2 , Latifa Oukhellou 1 1 Université Paris--Est, French Institute of Science and Technology for Transport, Development and Network (IFSTTAR), 14--20 Bd Newton 77420 Champs--sur--Marne, France 2 Université de Montréal, Pavillon André--Aisenstadt, CIRRELT, CP 6128, Succursale Centre--ville, Montréal QC H3C 3J7 Canada Smart card data are increasingly used for the analysis of urban mobility in particular multimodal and/or intermodal mobility. Their richness from spatial, temporal and quantitative point of view made them good material for mobility analysis. Indeed, they can be used to predict passenger flows, as well as to extract relevant information on transport network operation. They can also be used to better understand passenger behaviours. In the latter, statistical models are built to automatically extract a reduced set of clusters, each cluster contains a subset of passengers sharing the same temporal habits in terms of public transport usage.

In this paper we will investigate a statistical modeling to perform the clustering of passengers based on their ticketing logs in the public transport. The aim is partition the passengers into groups on the basis of their travel hours. The clustering is proposed to be performed in unsupervised way by using advanced data partitioning tools, that is dedicated Gaussian mixture models. Doing so, we will be able to extract typical patterns describing different types of transport usage, namely sporadic usage, typical home-work commute behavior, scholar usage etc.

Considering two case studies of the city of Rennes in France and the city of Gatineau in Canada, three main contributions will be illustrated in this paper:

-The first contribution is related to the statistical modeling based on a two-level generative model. Unlike different clustering approaches proposed in the litterature that consider a discrete representation of time (e.g. trips are aggregated over pre-defined time slots), the proposed model allows to take into account the continuous nature of timestamps. The obtained distributions offer a direct way to interpret the results in terms of times of typical transport usage as well as in terms of variance around these time peaks of usage. -The second contribution is related to the way to deal with the type of the day in the statistical model. It will be shown that this choice is closely linked to the public transport usage in the city. A categorization weekday/weekend is enough sufficient in Gatineau whereas a model accounting for all the days of the week is more appropriate to highlight the passenger habits in Rennes. This is mainly due to the temporal organization of scholars in France most of which do not study on wednesday afternoon. Moreover, this is also due to the usage of public transport for nighlife activities or for leisure activities in the weekend. -The third contribution is related to the analysis of different temporal activities that could be revealed by such method and link those activities with the city characteristics. By crossing the clustering results with spatial organization of both cities, it is indeed possible to have a better understanding of passenger activity on the transportation network. It will be shown that passengers, whom city is a centre of activity (Rennes) will be more likely to go out at night or during the weekend than passengers, whom city is located in the suburbs, which is the case of Gatineau. The results also highlight cultural differences in the use of public transportation between France and Canada.

Experiments are carried out on two real datasets. The first data set was collected on the public transport network of Rennes Metropole (France) and provided by the STAR network (Service de Transport en Commun de l'Agglomération Rennaise). The network is composed of one metro line (ligne a) and around 135 bus lines. 43 municipalities containing more than 400,000 inhabitants are serviced by this network.

The second dataset was provided by the Société de Transport de l'Outaouais (STO) based in Gatineau, Canada. As a medium-sized public transit authority, the STO operates 310 buses and services 291,000 inhabitants. The STO has been operating its smart card system since 2001.

The obtained results exhibits different cluster profiles that can be used to differentiate between passengers with regular travel hours and those with diffuse travel hours.

The approach is also able to detect subtle trends such as the presence of nightlife activity. The results provided by our method could help the transport operators to have a better knowledge of their passengers' behaviour patterns and adapt their offer to suite the different demands they portray (e.g. specific card type for passengers with regular temporal pattern). Moreover a more precise view of passengers' activity peak than classic aggregate time clustering is provided.


Patrick J Burns
email: patrick.j.burns@nyu.edu

Measuring and Mapping Intergeneric Allusion in Latin Poetry using Tesserae

Keywords: allusion, intertextuality, generic enrichment, Lucan, Latin epic, Latin love elegy, Tesserae

Most intertextuality in classical poetry is unmarked, that is, it lacks objective signposts to make readers aware of the presence of references to existing texts. Intergeneric relationships can pose a particular problem as scholarship has long privileged intertextual relationships between works of the same genre. This paper treats the influence of Latin love elegy on Lucan's epic poem, Bellum Civile, by looking at two features of unmarked intertextuality: frequency and distribution. I use the Tesserae project to generate a dataset of potential intertexts between Lucan's epic and the elegies of Tibullus, Propertius, and Ovid, which are then aggregrated and mapped in Lucan's text. This study draws two conclusions: 1. measurement of intertextual frequency shows that the elegists contribute fewer intertexts than, for example, another epic poem (Virgil's Aeneid), though far more than the scholarly record on elegiac influence in Lucan would suggest; and 2. mapping the distribution of intertexts confirms previous scholarship on the influence of elegy on the Bellum Civile by showing concentrations of matches, for example, in Pompey and Cornelia's meeting before or during the affair between . By looking at both frequency and proportion, we can demonstrate systematically the generic enrichment of Lucan's Bellum Civile with respect to Latin love elegy.

I INTRODUCTION

"There is no better way to penetrate the secrets of Lucan's workshop, to observe how the poem crystallized in his mind, than to examine passages where he borrows from, adopts, or echoes his predecessors." So wrote [Bruère, 1951 p. 222]. Bruère's contributions to the study of Lucan's allusive practice, especially his two articles co-authored with [START_REF] Thompson | Lucan's use of Virgilian reminiscence[END_REF][START_REF] Thompson | The Virgilian background of Lucan's fourth book[END_REF]), brought a heightened awareness of the intertextual nature of the Bellum Civile, and in particular, its relationship to Virgil's Aeneid. The presence of intertextual influence from genres other than epic, however, has received far less attention. 1 Recent work by [START_REF] Sannicandro | Personaggi Femminili Del Bellum Civile Di Lucano[END_REF][START_REF] Caston | Lucan's elegiac moments[END_REF][START_REF] Mccune | Lucan's Militia amoris: Elegiac expectations in the Bellum civile[END_REF] has sought to remedy this imbalance. These studies, however, have tended to * This paper developed from dissertation research completed at Fordham University under advisor Matthew

McGowan and an early version of this work was presented at the Institute for the Study of the Ancient World in the fall of 2015. I would like to thank Sebastian Heath, David Ratzan, Neil Coffee, James Gawley, Caitlin Diddams, and Neil Bernstein for their helpful feedback and would also like to thank the anonymous referees for their comments and suggestions. 1 So, for example, [Groß, 2013 p. 279] writes in arguing for the intertextual influence of Horace on Lucan's poetry: "To date, the reception of Horace in Lucan has received sufficient attention neither in Lucan philology nor in Horatian scholarship. Because it was assumed that the reception of texts would only occur within the framework of their own genre, the Horatian odes and epodes were only seen as intertexts of lyrics, but never of epic poems. Accordingly, the analysis of Lucan's references to other poets was limited to other epics or tragedy, but lyrical predecessors were not considered." With respect to elegy specifically, note the comment of [Caston, 2011 p. 134], who writes that in looking at Lucan's "elegiac moments" she is able to "to highlight a genre that has been overlooked" in studies of Lucan's intertextuality. emphasize localized readings, either referring to a limited number of elegiac source texts or treating a select group of episodes in Lucan's poem. In this paper, I use datasets drawn from the Tesserae Project at the University at Buffalo ([http1]) to systematically compare potential intertexts between Lucan's epic as a whole with reference to the complete works of the Latin love elegists Tibullus, Propertius, and Ovid.

II BACKGROUND

Systematic collections of elegiac references, no less analysis of these references, in epic poetry remain a desideratum in Latin literary criticism. Even within the epic genre, there are two works of traditional philological research which stand out for treating influence in a systematic and comprehensive manner, namely [START_REF] Knauer | Die Aeneis und Homer: Studien Zur Poetischen Technik Vergils, Mit Listen Der Homerzitate in Der Aeneis[END_REF] (with its subtitle "mit Listen der Homerzitate in der Aeneis") and [START_REF] Nelis | Vergil's Aeneid and the Argonautica of Apollonius Rhodius[END_REF]. [Farrell, 2005 p. 107] has written that the "mind recoils from the thought of a library full of books entitled, 'The Aeneid and Homer,' 'The Aeneid and Apollonius,' 'The Aeneid and Ennius,' and so forth." At the same time, having access to this kind of reference material would undoubtedly be useful. The existence of a book called "The Bellum Civile and Latin Love Elegy" would certainly appear in the bibliography of this study if it existed. [START_REF] Coffee | Intertextuality in the digital age[END_REF] remarks that traditional scholarly methods have avoided these kinds of comprehensive treatments of intertextuality because of the massive scholarly labor involved. Software is now available, however, to greatly reduce the procedural difficulty to which Coffee refers. Lists of potential intertexts can be compiled much more easily using Tesserae's web based tool ([http1]), which allows for the quick gathering of evidence for potential intertextuality between two texts, shifting scholarly labor from detection to analysis.

Problem of unmarked intertextuality

In his study of the intertextual relationship between Horace and Lucan, [START_REF] Groß | Plenus Litteris Lucanus: Zur Rezeption der Horazischen Oden Und Epoden in Lucans Bellum Civile[END_REF] observes that almost all intertextuality in classical poetry in unmarked, that is, it is not characterized by explicit signposts, but rather through implicit markers. Here, Groß follows the definition of unmarked intertextuality from [START_REF] Helbig | Intertextualität und Markierung[END_REF], who includes as two of the implicit markers 1. the "frequency" (Frequenz) of intertexts in the later text, and 2. the "distribution" (Distribution) of these intertexts, that is their location and relative density, throughout the text. This definition finds sympathy in two literary critical approaches to the problem of unmarked intertextuality, namely the "allusive system" discussed by [START_REF] Farrell | Intention and intertext[END_REF] and the "code model" discussed by [START_REF] Conte | The Rhetoric of Imitation: Genre and Poetic Memory in Virgil and Other Latin Poets[END_REF]. Both Farrell and Conte argue that the relationship between two texts can be drawn to some degree by the volume of potential intertexts and their consistent presence throughout a target text. A collocation tool like Tesserae, by algorithmically determining and reporting a complete collection of correspondences, offers a formalization of Farrell's system and Conte's model. Moreover, the data collected from Tesserae results can be used to formalize Helbig's observation about frequency and distribution as implicit signposts for unmarked intertextuality. The analysis of Tesserae results can measure frequency by showing the number of times similarity in word use triggers a match and can measure distribution by showing which parts of the Bellum Civile show a greater or lesser number of matches.

Literature Review

In recent years, researchers at Tesserae have published a series of papers testing the assumptions of traditional Latin literary criticism against their algorithmic model ( [START_REF] Coffee | Intertextuality in the digital age[END_REF]Coffee et al, 2013;[START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF]). These papers have used the first book of Lucan's Bellum Civile as their target text and Virgil's Aeneid as their source text, evaluating the results of the automated tool against philological commentaries by assigning them, following [START_REF] Thomas | Virgil's Georgics and the art of reference[END_REF], values of "meaningful" and "not meaningful," as well as "interpretable" and "not interpretable." [START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF] reports that scores assigned by the Tesserae algorithm correlate well with supervised assignments of meaning and interpretability.

The Tesserae publications have confirmed the traditional scholarly view that Lucan's poetic diction draws significantly on Virgil. That said, this research has consistently pointed the way towards wider applicability of algorithmically based methods for the study of intertextuality: [START_REF] Coffee | Intertextuality in the digital age[END_REF] suggests that systematic collection and measurement of textual similarities using a tool like Tesserae can build an "intertextual 'fingerprint'," that can be used to make meaningful comparisons between the poetic practices of different authors.

Important work on testing Tesserae search results is also being done by [START_REF] Bernstein | Using Tesserae to extend the philological commentary[END_REF][START_REF] Gervais | Flavian intertextuality: A digital approach[END_REF][START_REF] Bernstein | Comparative rates of text reuse in classical Latin hexameter poetry[END_REF], who have concentrated on the platform's "macrophilological applications," that is ways in which the complete collection of search results for a given genre, author, or work can be used to draw conclusions, not about specific intertexts, but rather about larger patterns of intertextuality. [START_REF] Bernstein | Comparative rates of text reuse in classical Latin hexameter poetry[END_REF], in particular, in a study that looks at intertextual relationships in Latin hexameter poetry as a whole, argues that Tesserae can be used to generate an unlabeled dataset which captures the intertextual relationship between multiple Latin texts and can then be used as the basis for further analysis and interpretation.

III DATA

Texts

This study uses the following texts available from the Tesserae Github repository ([http2)].

The following editions of Latin epic poetry are used:

• Virgil, Aeneid: Greenough, J. B., ed. 1900. Bucolics, Aeneid, and Georgics of Vergil. Boston: Ginn & Co. • Lucan, Bellum Civile: Weise, H., ed. 1835. M. Annaei Lucani Pharsaliae Libri X.

Leipzig: G. Bassus.

The following editions of Latin love elegy are used: 2

• Tibullus: Postgate, J.P., ed. 1915. Tibulli aliorumque Carminum libri Tres. Oxford: Oxford University Press. • Propertius: Mueller, L., ed. 1898. Sex. Propertii Elegiae. Leipzig: Teubner.

2 The collection listed above has been decided upon in order to align this work with that of Tesserae. It is obviously not the only arrangement available. [START_REF] Pichon | De sermone amatorio apud Latinos elegiarum scriptores[END_REF], for example, defined his sample as follows: the canonical works of Latin elegy mentioned in Ovid Tristia 4.10.53-54 and Quintilian Institutiones 10.1.93, to which he adds (or qualifies the inclusion of) Catullus, the Corpus Tibullianum, all of the Heroides regardless of authenticity, and certain poems from Ovid's Tristia and Epistulae ex Ponto. For Ovid, I use Ehwald's editorial decision to define the subset of Ovid's elegiac work which qualifies as erotic. Accordingly, the Fasti, Ibis, Tristia and Epistulae ex Ponto will not be used in this study. Along similar lines, I have based my decision to include the Corpus Tibullianum on Postgate's editorial decision and Tesserae's use of this edition.

• Ovid: Ehwald, R., ed. 1907. Amores, Epistulae, Medicamina faciei femineae, Ars amatoria, Remedia amoris. Leipzig: Teubner.

I have used the volumes listed above so that meaningful comparisons can be made with Tesserae studies which have already been published as well as those being conducted by other researchers. This follows the recommendation of [START_REF] Mcgillivray | Methods in Latin Computational Linguistics[END_REF], who argues that it is methodologically critical to work within a "collaborative research paradigm," that is to work from a common set of texts and to build directly upon existing tools and frameworks in an effort to maintain replicability in literary research. I have published the data set and the code used to generate the tables and figures on Github [http3].

Tesserae Search Results

Tesserae describes itself as a framework for "detecting allusions" in Latin poetry. More precisely, it is a search tool designed: 1. to compare the texts of two authors by looking for shared words, and 2. to return a list of similar passages scored for significance on a scale from 2 to 10. All results require that the units of text under consideration contain a minimum of two shared words. Once this requirement is met, matches are scored algorithmically based on two factors: word frequency and phrase density. Descriptions of the scoring algorithm can be found in [Forstall et al, 2015 p. 504]. 3 Word frequency refers to how common or uncommon a matched word is within the two texts; phrase density refers to the number of interstitial words separating the matched words. These parameters are designed to make explicit the formal criteria that scholars have traditionally applied implicitly when identifying an allusion. Accordingly, less common words which are adjacent receive higher scores than more common words which are separated by gap of several words. For example, the adjacent collocation of the rare words livor edax (Ovid Amores 1.15.1: Quid mihi Livor edax, ignavos obicis annos… ~ Lucan Bellum Civile 1.288: Livor edax tibi cuncta negat: gentesque subactas) receives a score of 10, while a separated collocation of the very common words quod and te (e.g., Ov. Amores 2.9b.47 ~ Lucan Bellum Civile 9.854) receives a score of 3.

On the one hand, by its nature, the Tesserae algorithm collects matches in an unrestricted manner. That is, it returns as many matches as fit its criteria-the text analytic equivalent of trawl fishing or strip mining. The result is a high number of false positives or dubious, semantically empty connections, especially due to ambiguity in lemmatization ( [START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF]). These matches correlating with the low end of the Tesserae scoring scale will be largely ignored in this study. On the other hand, [START_REF] Coffee | Intertextuality in the digital age[END_REF][START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF] have shown that the high scores (that is, scores 10 and 9) generated by the Tesserae scoring algorithm correlate with meaningful and interpretable results, and that the next tier of scores (scores 8 and 7) correlate with meaningful results. Accordingly, the high scores will be the focus of this study.

The data used for the study of intertextual frequency and distribution was gathered using version 3 of the Tesserae search interface. 4 The following parameters were used for these searches: Defaults were used where possible to ensure to the greatest degree possible comparability between this study and other Tesserae-based studies. One exception is "line" as the unit for this study; the maps of intertextual distribution use line numbers for the x-axis and the number of matches per line for the y-axis.

• unit = line • feature = stem • stopsize = 20 • stbasis = corpus • stopwords = qui
Tesserae results yield the following information:

• Result number • Target Location 6 • Target Text • Source Location • Source Text • Shared words • Score
Here is an example of a record from the .csv file returned by a Tesserae search between the Bellum Civile and Ovid's Amores:

these are the settings as reported in the "Tesserae V3 results" when exported as csv, txt, or xml. "Feature" determines how words are treated by the algorithm for making comparisions; under the setting "stem" (called "Lemma" in the Tesserae interface), the program "returns sets of parallels between texts where the matched words have the same dictionary headwords," that is, amor in one text matches amoris, amori, amorem, etc. in another. "Stopwords" is the default Tesserae stoplist, that is, the list of words ignored in this study; the basis for determining the stopword list ("stbasis") is the corpus of all works available in Tesserae. Accordingly, the algorithm ignores the 20 most commonly appearing words in this corpus. "Max_dist" refers to the maximum distance that the algorithm uses for its window for matches; that is words in a text must be between two and ten words apart from each other, counting inclusively, to yield a result. "Dibasis" refers to the manner in which the algorithm accounts for distance between words; according to [http4], with the setting "freq," Tesserae "attempts to zero in on the most relevant words in an allusion, measuring the distance only between the phrase's two most infrequent words." "Cutoff" refers to the minimum score returned by Tesserae; I have set it at zero to gather the full range of Tesserae results, although scores below 7 will be dropped for most parts of this study as noted below.

6 The terms source text and target text are used in the study of intertextuality, in Latin literature and elsewhere, to refer to the relationship of texts in the "traditional citation of parallel passages," ( [Fowler 1997, p. 14]) most commonly with a passage in an earlier text (the source) corresponding to a passage in a later text (the target).

IV METHODS

Following [START_REF] Helbig | Intertextualität und Markierung[END_REF], I have designed this study to measure two of his criteria for unmarked intertextuality: frequency and distribution.

Frequency is measured by aggregating the count of matches by author or work. Tesserae returns search results as a .csv and these results are then converted into dataframes for processing with the Python Pandas module [START_REF] Mckinney | Python for Data Analysis Data Wrangling with Pandas, NumPy, and IPython[END_REF]). Raw counts are normalized to matches per 100 lines to account for differing text lengths. Lengths were determined using the Tesserae texts given in the section "2.1 Texts" above. Tesserae scores below 7 are discarded to reduce "noise" and to restrict the analysis to scores which have been shown to yield the most significant results ( [START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF][START_REF] Bernstein | Comparative rates of text reuse in classical Latin hexameter poetry[END_REF]).

Distribution is measured by using Pandas to aggregate the total number of Tesserae matches from the elegists above score 7 for each line in the Bellum Civile. Lines from the Bellum Civile with no matches from Tesserae are assigned a count of zero. Because the goal of studying the distribution of unmarked intertextuality is to see how the feature appears throughout the entire work, rather than in any given line, the counts of matches per line are smoothed by taking the running average of scores within a specific window, here 25 lines. This reduces the line-by-line variability in counts while providing a better sense of potential intertextual density in different sections of the target text. The smoothed scores are then mapped by book using line numbers for the x-axis and the smoothed counts of Tesserae matches for the y-axis.

V DISCUSSION

Measuring Frequency

Table 1 shows the total number of Tesserae matches with Lucan's Bellum Civile as the target text for each of the elegists and their works, with Virgil's Aeneid included for comparison. The raw counts, however, are not sufficient to compare the authors/works, because they are of greatly varying lengths. For example, Propertius's four books of elegies have a total of 3,982 lines compared to the 9,896 lines of Virgil's Aeneid. Accordingly, it is necessary to normalize these scores so that they can be compared more usefully. By normalizing the counts of Tesserae results to matches per 100 lines, we get a different picture than the received view of intertextuality in Lucan, as shown in Table 2. The number of matches between Lucan and Virgil is much higher that the number between Lucan and any of the elegists. When normalized, however, the elegists show on average only a 17% difference, and in the case of Ovid's Heroides, the matches per 100 lines is slightly higher (469.67 in the Heroides versus 468.72 in the Aeneid).

Figure 1 shows the total number of matches by score for Virgil. As noted above, research on Tesserae results shows that results scoring 7 or above correlate best with meaningful results. Accordingly, in order to reduce the signal-to-noise ratio and limit the remainder of this part of the study to only the most meaningful results, scores below 7 will not be considered below.

Figure 2 shows the total number of matches by score above this threshold for Virgil.

By combining the strategies listed above of normalizing the scores and concentrating on the scores above a certain threshold, we can now compare the elegists to each other, again with the Aeneid included as a baseline. Looking at Figure 3, we can see that the shared diction between Virgil and Lucan as measured by the Tesserae algorithm, that is with consideration of word frequency and phrase density, is higher than for any of the elegists. This is hardly surprising considering that the history of scholarship on Lucan has privileged this relationship far beyond that of any other poetic predecessor. Rather the conclusion that can be drawn from this chart is that the minimal scholarly attention that has been paid to the influence of the elegists on Lucan, based on the Tesserae data, is out of proportion with the attention paid to the Aeneid.

The score most likely to be meaningful and interpretable, that is score 10, is exceptionally rare in all authors, consistently appearing less than one time per 100 lines. For score 9, the elegists show roughly half as many matches per 100 lines (average 2.77 per 100 lines; 48.7% of Virgil's 5.70). The count for Ovid's Heroides is notable for standing out as being much higher than the others (4.25 matches per 100 lines; 90.4% of Virgil's 5.70), suggesting that the density of specific allusions noted between these poems and the Bellum Civile by previous scholars (e.g. [START_REF] Bruère | Lucan's Cornelia[END_REF][START_REF] Sannicandro | Personaggi Femminili Del Bellum Civile Di Lucano[END_REF]) is supported by the evidence.

We find an example of a meaningful and interpretable match at line 7.590 of the Bellum Civile, when the poet addresses Brutus on the battlefield at Pharsalus, encouraging him not to enter the fray and kill Julius Caesar: "Ne rue per medios nimium temerarius hostis, Do not rush too rashly out through enemy lines." The collocation nimium temerarius appears in only two other passages of classical Latin literature, both in love elegy. Propertius uses it at 2.8.13 to chastise himself for being "too rash" in supporting an unfaithful lover, and Ovid uses the same language at Ars amatoria 2.83 to describe Icarus's insolence in disobeying his father. Lucan adopts elegiac diction at a critical moment to undermine Brutus's potential for epic glory. 7 The rarity of this collocation and the fact that the words are adjacent within their lines lead to Tesserae scores of 9 between the Bellum Civile and both elegiac works, and so here we see the correlation of high scores between the elegists and Lucan with respect to lines from the former which are, to use the description of [Hinds, 2007 p. 119], "thematically grounded" in the latter.

With the meaningful and non-interpretable scores (8 and 7), that is those which contribute to the general elegiac texture of the work and show evidence of elegy as a code model for the Bellum Civile, we see a similar pattern. Virgil's Aeneid again shows a higher number of matches per 100 lines, but not by the overwhelming margin that scholarly attention between the two genres would suggest: for score 8, the elegists show a little more than two-thirds as many matches per 100 lines (average 14.95 per 100 lines; 70.0% of Virgil's 21.38), and for score 7, more than three-quarters as many matches (average 68.0 per 100 lines; 77.5% of Virgil's 87.76).

Mapping Distribution

In addition to data about frequency, Tesserae searches for the elegists and the Bellum Civile also provide us with data about the location of the matches. Figure 4 shows a plot of the distribution of aggregated Tesserae matches per line for the complete set of elegiac texts as the source text and book 1 of the Bellum Civile as the target text. We learn from this text map that the matches are randomly distributed throughout the book, suggesting that elegy is functioning as a code model for the Bellum Civile. The line with the highest number of matches ( 15) is Lucan 1.61 (…inque vicem gens omnis amet: Pax missa…), likely reflecting the presence of "love" (amet) in this line, the central theme of the elegiac poems under consideration; the average number of matches per line is 1.69, and 65% (449 out of 695) of lines in book 1 have at least one match.

That said, the elegiac weight of any given line gives us information that is perhaps too localized. In order to get a better sense of which episodes show a sustained interaction with elegy, it is preferable to plot the running average of matches within a certain window of lines.

Figure 5 shows a plot of this running average for each book of the Bellum Civile using a window of 25 lines.

As compared to the line-by-line mapping of counts from the Tesserae results, plotting by window confirms once again that matches are distributed consistently throughout the poem. It also, and much more usefully as a prompt for further investigation, provides a clearer idea of where in the Bellum Civile additional research into elegiac influence would be most fruitful. Certain peaks in these text maps corroborate the work of previous scholarship on elegiac influence in Lucan. For example, we see pronounced upticks in the average number of matches at the end of book 5 for Pompey and Cornelia's meeting before Pharsalus (Lucan 5.722-815; see [START_REF] Sannicandro | Personaggi Femminili Del Bellum Civile Di Lucano[END_REF][START_REF] Bruère | Lucan's Cornelia[END_REF]), and at the beginning of book 10 for the appearance of Cleopatra (Lucan 10.53-106, 172-192;see [McCune, 2014;[START_REF] Groß | Plenus Litteris Lucanus: Zur Rezeption der Horazischen Oden Und Epoden in Lucans Bellum Civile[END_REF]).

Another interesting takeaway from this view of the distribution of elegiac language in the Bellum Civile, is the presence of peaks in parts of the epic where scholarly research into the interaction of these two genres has not been previously focused, as, for example, in the book 1 proem (1.1-32) or Cato's speech to his troops in book 9 (9.222-283).

VI Conclusion

The automated detection and measurement of intertextuality, in particular the definition of "allusion" formalized by Tesserae, offers the insight into the "poet's workshop" that Bruère described. It allows researchers to systematize and quantify the intertextual readings brought out in traditional, qualitative literary analysis of Latin poetry. This study uses the evidence generated by Tesserae to support Helbig's conception of frequency and distribution as implicit indicators of unmarked intertextuality. It also provides data to the idea that an "allusive system" can be deduced from a mass of textual similarities. The large number of scores 7 and higher for all three Latin love elegists, and the fact that these results are not confined to a small number of locations in the text suggest that this genre acts to some degree as a "code model" for the Bellum Civile. [Coffee et al, 2013 p. 227] showed how Tesserae could be used to corroborate the dominant scholarly opinion about Lucan's reliance on Virgil "for the basic idiom of epic." What is less in line with the traditional view of Lucan's allusive practice however is to what extent the results given above demonstrates elegy's contribution to his epic "idiom." The frequency of intertextual correspondence may be less than for Virgil's Aeneid and their distribution more diffuse, but this study was not meant to disprove Virgil's authority or deny his influence on the later epic poet. Rather it is meant to show that another genre can compete for some space in Lucan's attention to poetic predecessors. Scholarly consideration of the effect of elegy on the Bellum Civile is miniscule compared to the massive amount of research on Lucan's poetic relationship to, or even dependence, on Virgil as a poetic model. Yet, as we see here, the gap in the evidence is not as wide as the record of scholarship suggests.

This distant reading approach to verifying the allusive system and the code model is a starting point. [START_REF] Bernstein | Comparative rates of text reuse in classical Latin hexameter poetry[END_REF] comments that a limitation of the Tesserae platform is that it does not yet offer the "sensitive assessment of significance" that is taken for granted in traditional forms of Latin literary criticism. Along these lines, the test for frequency in this study does not address whether individual results are in fact meaningful or interpretable, and the test for distribution, while pointing out episodes worthy of further scrutiny, has not evaluted them. [START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF] reminds users that Tesserae data must still be interpreted within the established scholarly conversation concerning intertextuality in Latin poetry.

Examination of the peaks and troughs of the text maps of allusive distribution forms an excellent starting point for a more "sensitive assessment" of the intertextual relationship between Lucan and the love elegists in the future and also provides a more empirical method for investigating the "generic enrichment" ( [START_REF] Harrison | Generic Enrichment in Vergil and Horace[END_REF][START_REF] Harrison | Generic Interfaces in Latin Literature: Encounters, Interactions and Transformations[END_REF]) found in Latin poetry. 3.

VII TABLES AND FIGURES

Tables

  quis sum et in is hic non ego ut cum tu ad ille quod ab si atque neque the terms used in this list are available at the Tesserae site [http4].5

  Figure 1. Number of Tesserae matches by score for Virgil's Aeneid.

  Figure 1. Number of Tesserae matches by score for Virgil's Aeneid.

Figure 2 .

 2 Figure 2. Number of Tesserae matches by score for Virgil's Aeneid at or above a threshold of 7.

Figure 3 .

 3 Figure 3. Chart showing the number of Tesserae matches by score (at or above a threshold of 7) for the Latin love elegists and Virgil's Aeneid normalized to count per 100 lines as noted in Table3.

Figure 4 .

 4 Figure 4. Total count of Tesserae matches per line of book 1 of Lucan's Bellum Civile for the complete set of elegiac texts.

Figure 5 .

 5 Figure 5. Average count of Tesserae matches for the elegists in the ten books of the Bellum Civile, smoothed using a window of 25 lines.

Table 1 .

 1 Total number of Tesserae matches for Lucan's Bellum Civile as the target text and the Latin love elegists and Virgil's Aeneid as source texts.

		Work	Tesserae
			Matches
	Tibullus	Elegies	6,860
	Propertius	Elegies	13,812
	Ovid	Amores	9,966
		Ars Amatoria	9,235
		Heroides	18,646
		Medicamina	285
		Remedia Amoris 3,711
	Virgil	Aeneid	46,385
		Work	Lines	Tesserae	Count per
				Matches	100 Lines
	Tibullus	Elegies	1,929	6,860	355.62
	Propertius	Elegies	3,982	13,812	346.86
	Ovid	Amores	2,445	9,966	407.61
		Ars Amatoria	2,330	9,235	396.35
		Heroides	3,970	18,646	469.67
		Medicamina	100	285	285.00
		Remedia Amoris 814	3,711	455.90
	Virgil	Aeneid	9,896	46,385	468.72

Table 2 .

 2 Number of Tesserae matches normalized per 100 lines for Lucan's Bellum Civile as the target text and the Latin love elegists and Virgil's Aeneid as source texts.

		Work	Score 10	Score 9	Score 8	Score 7
			(per 100)	(per 100)	(per 100)	(per 100)
	Tibullus	Elegies	0.31	2.64	18.56	76.31
	Propertius	Elegies	0.33	3.16	15.77	67.88
	Ovid	Amores	0.49	2.74	18.45	75.83
		Ars Amatoria	0.13	2.79	14.98	72.27
		Heroides	0.13	4.26	18.49	79.07
		Medicamina	1.00	2.00	6.00	33.00
		Remedia Amoris 0.25	1.84	12.41	71.62
	Virgil	Aeneid	0.87	5.70	21.38	87.76

Table 3 .

 3 Number of Tesserae matches by score (at or above a threshold of 7) normalized per 100 lines for Lucan's Bellum Civile as the target text and the Latin love elegists and Virgil's Aeneid as source texts.

http://jdmdh.episciences.org

[START_REF] Forstall | Modeling the scholars: Detecting intertextuality through enhanced word-level N-gram matching[END_REF] report the equation for the Tesserae scoring system as follows:where "f(t) is the frequency of each matching term in the target phrase; f(s) is the frequency of each matching term in the source phrase; d t is the distance in the target; d s is the distance in the source."

Journal of Data Mining and Digital Humanities http://jdmdh.episciences.org

The Tesserae searches used in this study were run between September 13-16, 2015.

The following comments should be helpful in getting a quick understanding of the Tesserae settings. Note that

As[Gorman, 2001 p. 270] writes: "In a most anti-epic stance, Lucan actually admonishes Brutus to avoid the fight, in direct contravention to what a hero ought to be doing. Courage, the defining quality of the epic warrior, has become the fault of rashness (nimium temerarius)."