Marie Duffiet

Maxime Blangero

Pierre-Etienne Cabelguen

Claude Delmas

Dany Carlier
email: dany.carlier@icmcb.cnrs.fr

Influence of the Initial Li/Co Ratio in LiCoO 2 on the High Voltage Phase Transitions Mechanisms

The influence of the initial Li/Co stoichiometry in LiCoO2 (LCO) (1.00 ≤ Li/Co ≤ 1.05) on the phase transition mechanisms occurring at high voltage during lithium de-intercalation (V > 4.5 vs. Li + /Li) was investigated by in situ X-ray diffraction. Even if the excess Li + in Li1.024Co0.976O1.976 doesn't hinder the formation of the H1-3 and O1 phases, the latter are obtained at higher voltages and exhibit larger c parameters compared to their analogues formed from Li1.00CoO2. We also showed that for the stoichiometric Li1.00CoO2, the de-intercalation process is more complex than already reported, with the formation of an intermediate structure between H1-3 and O1.

Layered LiCoO2 (LCO) crystallizing in the R-3m space group has been extensively studied for years as positive electrode material for Li-ion batteries (LiBs) since its discovery by Goodenough [START_REF] Mizushima | LixCoO2 (0 < x < 1): A New Cathode Material for Batteries of High Energy Density[END_REF] . Still today, it is one of the most common materials in commercial LiBs used in portable electronics, thanks to its great ability to reversibly de-intercalate and re-intercalate lithium over thousands of cycles up to 4.3 V vs. Li + /Li (equivalent to 4.2 V vs. graphite). Up to this voltage, approximately x = 0.62 Li can be extracted corresponding to a charge capacity of about 174 mAh/g vs. Li + /Li.

Li + ions removal is accompanied by an insulator -metal transition 2 occurring at 3.95 V leading to a biphasic domain followed by a reversible structural change from a rhombohedral to a monoclinic phase (O3 → O'3) observed at x = 0.5 due to Li + /vacancy ordering [START_REF] Shao-Horn | Probing Lithium and Vacancy Ordering in O3 Layered LixCoO2 ( x ≈ 0.5 ) An Electron Diffraction Study[END_REF][START_REF] Reimers | Electrochemical and In Situ X-Ray Diffraction Studies of Lithium Intercalation in LixCoO2[END_REF] . Both phenomena can be clearly identified on the LiCoO2//Li electrochemical cycling curve and have contributed to make it a very typical signature of the stoichiometric LiCoO2 (noted st-LCO in the following). In fact, as shown by Levasseur, st-LCO is the end member of a Li1[LitCo1-t]O2-t solid solution where extra lithium ions occupy the cobalt site; the electrical neutrality being insured by oxygen vacancies [START_REF] Levasseur | Oxygen Vacancies and Intermediate Spin Trivalent Cobalt Ions in Lithium-Overstoichiometric LiCoO2[END_REF] . The material prepared with Li/Co > 1.00 will be called overlithiated LiCoO2 (noted overl-LCO in the following) (Supporting information-S1). Even if the solid solution range is very narrow t < 0.04, the effect on the electrochemical behavior is significant vs. st-LCO; as a result of the Li-excess defects and entropy increase, the two-phase domain and the monoclinic distortion disappear while cycling at room temperature. The presence of oxygen vacancies leads to the formation of intermediate spin Co 3+ which are indirectly detected by [START_REF] Myung | Effects of Al Doping on the Microstructure of LiCoO2 Cathode Materials[END_REF] Li NMR, making this technique the best way to characterize Li-overstoichiometry [START_REF] Levasseur | Oxygen Vacancies and Intermediate Spin Trivalent Cobalt Ions in Lithium-Overstoichiometric LiCoO2[END_REF][START_REF] Ménétrier | Really" Stoichiometric LiCoO2[END_REF] .

Increasing the charge voltage higher than 4.3 V vs. Li + /Li would enable the extraction of more Li + ions and reach charge capacities higher than 174 mAh/g. In this state of charge range, a dramatic degradation of cycling performance prevents industrial applications. In particular, structural transitions and electronic evolutions at such high voltages are believed to be root causes of material degradation and cell failure (electrolyte oxidation and oxygen loss on the particle surface, cobalt dissolution resulting in destabilization of anode SEI, particle fracturing and loss of electrical contact…). In the last years, many strategies have been explored to overcome these issues, by either: i) slightly tuning the initial chemical composition of LCO with the substitution of some Co 3+ ions with many different dopants such as Mg, Ti or Al [START_REF] Myung | Effects of Al Doping on the Microstructure of LiCoO2 Cathode Materials[END_REF][START_REF] Yu | The Investigation of Ti-Modified LiCoO2 Materials for Lithium Ion Battery[END_REF][START_REF] Prahasini | Synthesis and Characterization of Cu Doped LiCoO2 Cathode Material for Lithium Batteries Using Microwave Assisted Sol-Gel Synthesis[END_REF][START_REF] Liu | Synthesis of Mg and Mn Doped LiCoO2 and Effects on High Voltage Cycling[END_REF] , or ii) changing the surface composition of the LCO particles, coating metal oxides [START_REF] Kim | Suppression of Cobalt Dissolution from the LiCoO2 Cathodes with Various Metal-Oxide Coatings[END_REF][START_REF] Cho | Zero-Strain Intercalation Cathode for Rechargeable Li-Ion Cell[END_REF] , fluorides [START_REF] Sun | Role of AlF3 Coating on LiCoO2 Particles during Cycling to Cutoff Voltage above 4.5 V[END_REF][START_REF] Lee | Interface Characterization of MgF2-Coated LiCoO2 Thin Films[END_REF] , phosphates [START_REF] Kim | Controlled Nanoparticle Metal Phosphates (Metal = Al , Fe, Ce, and Sr) Coatings on LiCoO2 Cathode Materials[END_REF] and carbonates [START_REF] Dai | Extending the High-Voltage Capacity of LiCoO2 Cathode by Direct Coating of the Composite Electrode with Li2CO3 via Magnetron Sputtering[END_REF] . In the meantime, least attention has been given to a more fundamental knowledge about the mechanisms of structural changes in the LCO system. Even if a few articles [START_REF] Ohzuku | Solid-State Redox Reactions of LiCoO2 (R3̅ m) for 4 Volt Secondary Lithium Cells[END_REF][START_REF] Amatucci | The End Member of the LixCoO2 Solid Solution[END_REF][START_REF] Ven | First-Principles Evidence for Stage Ordering in LixCoO2[END_REF][START_REF] Chen | Staging Phase Transitions in LixCoO2[END_REF] have identified two major topotactic phase transitions from O3 → H1-3 → O1 structures occurring above 4.4 V vs. Li + /Li, the possible influence of the initial Li/Co stoichiometry on those mechanisms has not been investigated yet. In this paper, we aim to better evidence the structural transitions occurring at high voltage depending on the Li/Co stoichiometry of the pristine LCO phase and particle size. [START_REF] Myung | Effects of Al Doping on the Microstructure of LiCoO2 Cathode Materials[END_REF] Li magic angle spinning nuclear magnetic resonance spectroscopy ( 7 Li MAS NMR) was used to properly assess the Li/Co stoichiometry of the samples. In situ operando X-ray diffraction was used to characterize the structural transitions occurring at high voltage (up to 5 V).

In a preliminary study we demonstrated that the particle size of the LCO materials in the µm range (2 -4 µm vs 10 -30 µm) did not affect the phase transition mechanisms occurring at high voltage (Supporting information S2). We will thus focus the discussion here on the impact of the initial lithium content x0 (correlated to the initial Li/Co stoichiometry, as Li/Co = (1+t)/(1-t) = x0 / (1-t)). We selected two samples: (i) a stoichiometric LiCoO2 (denoted as "st-LCO") with Li/Co = 0.99 and D50 = 34 µm, (ii) an overlithiated Li1.024Co0.976O1.976 (denoted as "overl-LCO") with Li/Co = 1.05 and D50 = 15 µm. General characterization of the two samples is provided in Supporting Information (Figure S3). The layered structure of st-LCO and overl-LCO is confirmed by the XRD patterns (both indexed in the R-3m space group). Additionally, traces of Co3O4 are detected for st-LCO, as expected for a sample prepared with an initial Li/Co ratio of 0.99. No clear change of cell parameters can be observed depending on the initial Li/Co stoichiometry of the two samples. Substantial differences between st-LCO and overl-LCO can only be found in their [START_REF] Myung | Effects of Al Doping on the Microstructure of LiCoO2 Cathode Materials[END_REF] Li MAS NMR spectra. Indeed, the [START_REF] Myung | Effects of Al Doping on the Microstructure of LiCoO2 Cathode Materials[END_REF] Li MAS NMR spectrum for st-LCO shows a single diamagnetic signal centered at -0.4 ppm, while several more signals are detected for overl-LCO. This is a clear confirmation that st-LCO is, indeed, stoichiometric, as its only signal arises from the presence of a low spin state diamagnetic (LS) Co 3+ environment. The additional signals observed for overl-LCO -whose exact assignation is still unknown -are though confirming the presence of paramagnetic intermediate spin IS-Co 3+ cobalt ions formed due to the presence of Li in the Co site associated with an O vacancy [START_REF] Levasseur | Oxygen Vacancies and Intermediate Spin Trivalent Cobalt Ions in Lithium-Overstoichiometric LiCoO2[END_REF][START_REF] Ménétrier | Really" Stoichiometric LiCoO2[END_REF] . As these differences have already proven to significantly change the electrochemical behavior of LCO samples, the corresponding electrochemical cycling curves obtained using the two LCO samples vs. Li in coin cells cycled in the 3.0 -4.7 V and 3.0 -5.0 V voltage ranges are provided in Figure 1. As already reported, the two samples exhibit different shape in the low voltage domain (no plateau at 3.95 V and no phase transition at x = 0.5 for overl-LCO). In the high voltage domain up to 4.7 V the profiles differ: i) upon charge, a small plateau can be observed at 4.53 V for st-LCO, well evidenced by the narrow and intense peak on the dx/dV curve, whereas a broader contribution is observed for such potential for overl-LCO; ii) upon discharge from 4.7 V, st-LCO sample does clearly exhibit two successive plateaus at 4.52 V and 4.48 V, whereas overl-LCO exhibits only one plateau at 4.45 V. A larger irreversible capacity loss is systematically observed for overl-LCO independently of the applied cut off voltage. Last and for both samples, a large polarization is observed when the batteries are cycled up to 5 V. In order to better understand those phenomena, we recorded in situ operando XRD patterns in the high voltage range for both LCO samples. In Figure 2, we plotted the patterns collected for the most intense (0 0 l) diffraction line upon charge in two relevant voltage domains for more clarity: 4.20 -4.60 V (a, b) and 4.55 -5.0 V (c,d), respectively corresponding to the successive O3 -H1-3 and H1-3 -O1 phase transitions already reported in the literature for supposedly Li-stoichiometric samples only [START_REF] Chen | Staging Phase Transitions in LixCoO2[END_REF][START_REF] Xia | Phase Transitions and High-Voltage Electrochemical Behavior of LiCoO2 Thin Films Grown by Pulsed Laser Deposition[END_REF] . For a better understanding of the mechanisms, we also plotted the evolution of the d(0 0 l ) interslab spacing as a function of the voltage in Figure 3. At ~ 4.20 V vs Li + /Li, the two compounds exhibit a single sharp intense diffraction line, corresponding to the (0 0 3) diffraction peak of a O3-type LixCoO2 structure with a similar interslab distance for the two materials (d(0 0 3) = 4.81 Å), higher than the one of the pristine materials (d(0 0 3) = 4.68 Å). At 4.20 V vs Li + /Li, approximately 0.4 Li + ions are remaining in both LCO compounds. A schematic representation of the O3 structure is given in Figure 3. As we slowly further de-intercalate Li + ions up to 4.55 V, the interlayer distance d(0 0 3) decreases as revealed by the shift of peak position on Fig. 2a and2b. This signal also undergoes a clear broadening assigned to the competition between nucleation and growth of O1-type stacking faults into the O3-type matrix. No other significant change is observed for this (0 0 3)O3 peak before the appearance of a new peak at 4.53 V and at 4.57 V for st-LCO and overl-LCO, respectively. This peak corresponds to the (0 0 6) diffraction line of the H1-3-type structure (depicted in Figure 3). For st-LCO, the associated average interlayer distance d(0 0 6) is 4.51 Å, meaning that the corresponding chex.. parameter is approximately 27.06 Å, in good agreement with the value reported by Dahn's group [START_REF] Chen | Staging Phase Transitions in LixCoO2[END_REF]

(27.07 Å).

Despite the presence of defects in the pristine overl-LCO, the H1-3 phase is formed during Li + deintercalation but appears at slightly higher voltage and does exhibit a higher interlayer distance In addition, it can be noticed that the final O1 structures obtained from both st-LiCoO2 and overl-LiCoO2 show significant differences regarding their respective interslab spacing. Indeed, a lower value for st-LCO (d001 = 4.23 Å) sample is obtained, as compared to overl-LCO (d001 = 4.31 Å) sample. This could be related to the presence of vacancies in the Co1-tO2-t slabs. Following the conclusions on the effect of the initial Li/Co stoichiometry on Li + de-intercalation process at low voltage (below 4.4 V) for LCO materials, we showed that the initial Li/Co stoichiometry also affect the mechanisms involved at high voltage. The successive structural transitions from O3 to H1-3 and O1 phase are observed for both samples, but these phases appear at higher voltage and does exhibit different cell parameters for an overlithiated compound. Note that less volume changes are obtained for the overlithiated compound despite a higher irreversible capacity, that may therefore be due to some structural reorganization, stronger electrolyte oxidation and cobalt dissolution. We also showed that for the stoichiometric LCO, the de-intercalation process occurring at high voltage is more complex than already reported, with the formation of an intermediate phase between H1-3 and O1. Further investigation using synchrotron XRD will be required to better understand those phenomena.

Experimental methods

Different LCO samples were provided by Umicore with various Li/Co stoichiometries and particle sizes.

Powder X-ray diffraction (XRD) patterns were collected on a PANalitycal X'pert PRO MPD diffractometer in Bragg-Brentano θ-θ geometry equipped with a Fe filter, a spinner and X'Celerator multi-strip detector. Each measurement was made within an angular range of 2θ = 10 -120° and lasted for 15 hours with 0.016° intervals. The Co-Kα radiation was generated at 35 kV and 30 mA (λ(Kα1) = 1.789 Å; λ(Kα2) = 1.793 Å). [START_REF] Myung | Effects of Al Doping on the Microstructure of LiCoO2 Cathode Materials[END_REF] Li MAS NMR spectra were recorded on a Bruker 300 Avance spectrometer at 116 MHz (7.05 T magnet), with a standard 2.5 mm Bruker MAS probe. A Hahn echo sequence [tπ/2-τ1-tπ-τ2] synchronized with one period of rotor rotation was used for a 30 kHz spinning frequency. The 90° pulse duration was equal to tπ/2 = 2.0 µs was determined using a LiCl 1 M solution. A recycle time of D0 = 40 s was used for st-LCO, whereas a shorter D0 = 2s was enough for the overl-LCO sample, to avoid T1 saturation effects.

Scanning electron micrographs (SEM) were taken using a Hitachi Model S-4500 microscope after metallizing the powders with gold.

For BET (Brunauer, Emmett et Teller) measurements, we used Micromeritics, TriStar II 3020 equipment. A nitrogen gas was applied for sample flushing at 300℃ for 1 h and helium gas for measurement at 25℃. Inductively Coupled Plasma (ICP) measurements were carried out on Agilent ICP-720ES equipment after sample digestion using hotplate heating in concentrated HCl solution. A "4-point" method was used. LCO:C:PVDF electrodes (90:5:5 %wt) with the two above-mentioned LCO active materials were prepared from a slurry using N-methyl-pyrrolidone (NMP) as solvent casted onto 6.5µmthick-aluminum circles for the in situ XRD experiments. Typical active material loading and diameter for an electrode used during in situ XRD experiments were respectively 15 mg/cm² and 20 mm; 10mg/cm² and 15mm for tests in coin cells. The electrodes were dried overnight under vacuum at T = 120 °C and stored in an argon-filled glovebox, the homemade in situ cells were assembled using pure lithium as counter-electrode and 1 M LiPF6 in EC:DEC:DMC as electrolyte.

Coin cells for preliminary tests were assembled the same way.

For the in situ XRD experiments, electrodes were pre-charged at C/30 up to 4.2 V without collecting any XRD pattern, since the goal of this experiment was the investigation of the phase transitions occurring at higher voltage. After reaching E = 4.2 V vs. Li + /Li, the charge proceeded at a lower C rate C/100 up to 5.0 V. XRD patterns were collected in operando on a PANalytical X'Pert PRO MPD diffractometer, using the Cu Kα1 radiation. Special attention was given to the (0 0 l) diffraction peak evolution, that allows to directly follow the changes in the interslab distance which is characteristic of the metal and lithium layer stacking modifications. Therefore, XRD acquisition was recorded every hour between 18 and 21.6°. Due to large exposure time at high voltages leading to substantial electrolyte degradation, we chose to discuss the data versus the cell voltage in the following and not versus state of charge expressed by the lithium amount (x). As a composition guide line, the formula of the ideal H1-3 phase is Li0.167CoO2 while the O1 phase is expected for the CoO2 composition.

Figure 1 .

 1 Figure 1. 1 st cycle for st-LCO (x0 = 1.00) and overl-LCO (x0 = 1.024) at C/30 as positive electrode in

  (d(0 0 6) = 4.55 Å). The slab distance range observed for the intermediate H1-3 phase is also different for the two compounds: very narrow for st-LCO (d varies between 4.51 Å and 4.46 Å) and larger for overl-LCO (d varies between 4.55 to 4.45 Å).

Figure 2 .

 2 Figure 2. [2 columns] Cumulated XRD patterns recorded in situ operando during the charge of st-LCO and

Figure 3 .

 3 Figure 3. [2 columns] Plot of the average interlayer distance d(0 0 l) for each phase formed during the

Acknowledgements

This work beneficiates from financial support from Umicore within the framework of Conventions Industrielles de Formation par la REcherche (CIFRE), handled by the Association Nationale Recherche Technologie (ANRT). We would like to thank them to this end. We are also grateful to Laurent Gautier and Stéphane Levasseur (Umicore) for fruitful discussions. Michel Ménétrier and Cathy Denage's involvements in the 7 Li MAS NMR and SEM were greatly appreciated too.

b) and in the 4.55 (or 4.60 V) -5.00 V (c, d) range, respectively corresponding to the successive O3 -H1-3 and H1-3 -O1 phase transitions. The black arrows show the evolution of the patterns towards the charge. Upon further charge up to 5 V, the mechanism observed for the two materials are clearly different (Figure 2c andd). For st-LCO, whereas a single transition from H1-3 to O1 structural type is expected from literature [START_REF] Amatucci | The End Member of the LixCoO2 Solid Solution[END_REF][START_REF] Chen | Staging Phase Transitions in LixCoO2[END_REF][START_REF] Xia | Phase Transitions and High-Voltage Electrochemical Behavior of LiCoO2 Thin Films Grown by Pulsed Laser Deposition[END_REF] , we first observe a gradual broadening and asymmetry of the (0 0 6)H1-3 diffraction line towards the lower d values, then the appearance of two new "diffraction lines" corresponding to d = 4.36 Å and 4.23 Å, both still very broad. The line at 4.23 Å can be assigned to the (0 0 1) diffraction line of the O1 structure type already reported in the 1990's by Amatucci et. al [START_REF] Amatucci | The End Member of the LixCoO2 Solid Solution[END_REF] , but the line located at 4.36 Å was not reported so far and might result from intergrowth between H1-3 and O1 structures. The appearance of the new peak at 4.36 Å is better seen on the XRD pattern, after we applied a potentiostatic step at E = 5.0 V for several hours following the charge of the compound and recorded a few more patterns (Figure 2e). Since it is located for intermediate d-values between H1-3 and O1 phases, we propose that the intergrowth between these structures is not completely random, thus forming an intermediate stacking. Its formation could be a way to minimize the internal constraints due to the strong d-interslab space diminution from H1-3 to O1 for the system. Note that even after a long floating at 5 V, the diffraction lines for the three phases (H1-3, intermediate and O1) are still observed (Fig. 2e). In any case, from their peak broadening, those structures may contain a high number of stacking faults resulting in overall structural disorder. For overl-LCO, the mechanism is different, no intermediate peak is observed: the H1-3 structure gradually transform into an O1 structure with the formation of several intermediate stacking faulted structures, with a continuous process.

Supporting information:

The Supporting Information is available free of charge on the ACS Publications website.

Schematic structures of the st-LCO and overl-LCO materials are providing in S1. In S2, data for st-LCO with different particles size are compared and in S3 complementary data for st-LCO and overl-LCO are provided.


Daniel E Clark

Flávio De Melo

A linear-complexity second-order multi-object filter via factorial cumulants

Keywords: Multi-target tracking, point processes, factorial cumulants

Multi-target tracking solutions with low computational complexity are required in order to address large-scale tracking problems. Solutions based on statistics determined from point processes, such as the PHD filter, CPHD filter, and newer second-order PHD filter are some examples of these algorithms. There are few solutions of linear complexity in the number of targets and number of measurements, with the PHD filter being one exception. However, the trade-off is that it is unable to propagate beyond first-order moment statistics. In this paper, a new filter is proposed with the same complexity as the PHD filter that also propagates second-order information via the second-order factorial cumulant. The results show that the algorithm is more robust than the PHD filter in challenging clutter environments.

I. INTRODUCTION

Due to the practical demand and complexities of multiple target tracking, there have been many different approaches to addressing the problem. Methods for multiple hypothesis tracking have attracted a great deal of attention since they provide an intuitively appealing basis for designing an engineering system, eg. [START_REF] Reid | An algorithm for tracking multiple targets[END_REF]- [START_REF] Blackman | Design and analysis of modern tracking systems[END_REF]. Methods for managing joint multi-target probabilities [START_REF] Fortmann | Sonar tracking of multiple targets using joint probabilistic data association[END_REF]- [START_REF] Kastella | Discrimination gain for sensor management in multitarget detection and tracking[END_REF] represent a different way of dealing with data association ambiguities. Methods based on nearest neighbour assignments [START_REF] Blair | Nnjpda for tracking closely spaced rayleigh targets with possibly merged measurements[END_REF], [START_REF] Maskell | Statistical methods for target tracking[END_REF] provide pragmatic solutions to dealing with data association assignment. More recently, closed-form solutions to a class of multi-target tracking problems have been developed [START_REF] Williams | Hybrid poisson and multi-bernoulli filters[END_REF]- [START_REF] Vo | Labeled random finite sets and multi-object conjugate priors[END_REF], which are gaining attention for scenarios where there is a prioritisation of estimation accuracy over computational complexity.

Methods based on point processes [START_REF] Washburn | A random point process approach to multiobject tracking[END_REF]- [START_REF] Streit | The pointillist family of multitarget tracking filters[END_REF] and random finite sets [START_REF] Goodman | Mathematics of data fusion[END_REF] have also attracted a lot of attention. In particular, the applications of methodology developed in the context of random finite sets have proved to be popular [START_REF] Mahler | Multitarget Bayes filtering via first-order multitarget moments[END_REF]- [START_REF] Mullane | A random-finite-set approach to bayesian SLAM[END_REF]. Specifically, methods based on propagation of first-order moment statistics of a point process, or intensity function, have been developed for addressing tracking problems, eg. [START_REF] Mahler | Multitarget Bayes filtering via first-order multitarget moments[END_REF].

Methods for propagating higher-order information have been proposed [START_REF]PHD filters of higher order in target number[END_REF], [START_REF] Mahler | PHD filters of second order in target number[END_REF], [START_REF] Singh | Filters for Spatial Point Processes[END_REF]. It has been shown that the popular CPHD filter [START_REF]PHD filters of higher order in target number[END_REF] can have undesirable behaviour [START_REF] Fränken | Spooky Action at a Distance in the Cardinalized Probability Hypothesis Density Filter[END_REF], which has been shown to be due to strong negative correlations Daniel Clark is with Telecom SudParis, France. Email: daniel.clark@telecom-sudparis.eu and Flávio de Melo is with Heriot-Watt University, UK. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) [Grant Number EP/K014227/1]; and the MOD University Defence Research Collaboration (UDRC) in Signal Processing.

being induced after Bayes' rule [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF]. More recently, secondorder filtering solutions have been proposed that propagate the mean and variance in target number [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF].

In the current paper, a new linear-complexity multi-target filter is derived and implemented that propagates second-order factorial cumulants. It is shown that the filter is more robust than the PHD filter while maintaining the same run-time. This is achieved by making a different approximation on the joint target-measurement process before applying Bayes' rule.

The paper is structured as follows: Point processes are described in the next section in terms of functionals, namely the probability generating functional the factorial cumulant generating functional, and statistics in terms of factorial moments and factorial cumulants and their connections. Section III describes point process models, including the Poisson process and the negative binomial point process via the Laplace-Stieltjes transform, and the extension to the Panjer point process. Section IV presents the algorithm description in terms of propagation of the first two factorial cumulants. Section V shows some experimental scenarios comparing with the PHD filter. The paper concludes in Section VI. The appendix presents the mathematical proofs of the algorithm specification. Algorithm 1 presents the algorithm pseudo-code of a Gaussian mixture implementation based on a similar approach to the Gaussian mixture PHD filter [START_REF] Vo | The gaussian mixture probability hypothesis density filter[END_REF].

II. POINT PROCESSES

In this section we describe point processes [START_REF] Daley | An introduction to the theory of point processes[END_REF], [START_REF]The general theory of stochastic population processes[END_REF] and their statistical descriptors, including factorial moments and factorial cumulants. We denote by X ⊆ R dx the d xdimensional state space describing the state of an individual object. A point process Φ on X is a random variable on the process space X = n≥0 X n , i.e., the space of finite sequences of points in X . A realisation of Φ is a sequence ϕ = (x 1 , . . . , x n ) ∈ X n , representing a population of n objects with states x i ∈ X . Point processes can be described using their probability distribution P Φ on the measurable space (X, B(X)), where B(X) denotes the Borel σ-algebra of the process space X [START_REF] Stoyan | Stochastic geometry and its applications[END_REF]. The projection measure P (n) Φ of the probability distribution P Φ on X n , n ≥ 0, describes the realisations of Φ with n elements; the projection measures of a point process are always defined as symmetrical functions, so that the permutations of a realisation ϕ are equally probable.

A. Generating functional representations

The probability generating functional (p.g.fl.) G Φ of a point process Φ is defined by

G Φ (h) = n≥0 n i=1 h(x i ) P (n) Φ (dx 1:n ), (1)
respectively for test function h : X → [0, 1]. The factorial cumulant generating functional (f.g.fl.), is defined by

C Φ (h) = ln (G Φ (h)) . (2)
We can construct the f.g.fl. through the definition of Khinchin measures as follows. Let {K

Φ } n≥1 be a sequence of measures and consider a point process defined through the p.g.fl.

G Φ (h) = exp (C Φ (h)) (3)
where

C Φ (h) = -K (0) Φ + n≥1 h(x 1 ) . . . h(x n )K (n) Φ (dx 1:n ), (4)
and

K (0) Φ = n≥1 K (n) Φ (dx 1:n ) (5)
ensures that G Φ (1) = 1 and therefore G Φ is a p.g.fl.. The Khinchin measures were recently used to determine results for Bayesian estimation of multi-object systems with independently identically distributed correlations [START_REF] Houssineau | Bayesian estimation of multi-object systems with independently identically distributed correlations[END_REF] by considering a bivariate version defined in an analogous way. We shall use the bivariate Khinchin process to determine the key result in the paper via application of the Laplace-Stieltjes transform.

B. Factorial moments and factorial cumulants

The n-th order factorial moment measure ν

(n) Φ
of a point process Φ are the measures on X n such that [START_REF] Stoyan | Stochastic geometry and its applications[END_REF]

ν (n) Φ (B 1 × • • • ×B n ) = E = x1,...,xn∈Φ 1 B1 (x 1 ) . . . 1 Bn (x n ) , (6)
for any regions B i ∈ B(X ), 1 ≤ i ≤ n. The notation 1 B denotes the indicator function, i.e., 1 B (x) = 1 if x ∈ B, and zero otherwise. Factorial moments and cumulants can be determined from the following derivatives using derivatives as described in [START_REF] Clark | Faa di Bruno's formula for chain differentials[END_REF].

ν (n) Φ (B 1 × • • • ×B n ) = δ n G Φ (h; 1 B1 , . . . , 1 Bn )| h=1 . (7) c (n) Φ (B 1 × • • • ×B n ) = δ n C Φ (h; 1 B1 , . . . , 1 Bn )| h=1 . (8)
Setting the directions to be indicator functions 1 B1 , . . . , 1 Bn and evaluating at h = 1, noting that G Φ (1) = 1, we find expression for the factorial cumulants in terms of factorial moments,

c (n) Φ (B 1 × • • • ×B n ) = δ n (ln •G Φ )(h; 1 B1 , . . . , 1 Bn )| h=1 = π∈Π(B1,...,Bn) (-1) |π|-1 (|π| -1)! ω∈π ν (|ω|) Φ (ω), (9)
where ν

(|ω|) Φ

(ω) is taken to mean that the factorial moment is evaluated on the product of terms in ω, which are subsets of partition π, in the set of all partitions of the Π(B 1 , . . . , B n ) of elements B 1 , . . . , B n . This result was given in [30, p202] though without explicitly determining via differentiation. Hence, we have the relation for n th -order factorial cumulants in terms of factorial moments determined via derivatives of the generating functional. This will be used to determine the results for different point process parameterisations. The inverse relation is found to be [START_REF] Williams | Hybrid poisson and multi-bernoulli filters[END_REF] where C Φ (1) = 0. Note that the second-order cumulant is related to the mean and variance with

ν (n) Φ (B 1 × • • • ×B n ) = π∈Π(B1,...,Bn) ω∈π c (|ω|) Φ (ω),
c (2) Φ (B, B) = var Φ (B) -µ Φ (B), (11)
which will be important for filter assessment.

III. POINT PROCESS MODELS

This section describes the models that will be used to form approximations in the point process prior used before the update. In particular, we describe the Poisson point process and determine the over-dispersed negative-binomial point process via application of the Laplace-Stieljes transform. We then describe the relation between the negative-binomial process to the Panjer point process [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF] which encompasses the negativebinomial, Poisson and binomial processes within the same model.

A. Poisson point process

A Poisson point process with parameter λ and spatial distribution s is a process with spatial distribution s, whose size is Poisson distributed with rate λ. Its Probability Generating Functional (p.g.fl.) is given by

G Poisson (h) = exp [h(x) -1] µ(dx) , (12)
where the intensity measure µ of the process is such that µ(dx) = λs(dx). It can be shown that the first-order factorial moment and the variance of a Poisson process are equal when evaluated in any region B ∈ B(X ), i.e., µ Φ (B) = var Φ (B).

In other words, the random variable describing the number of objects within B has its mean equal to its variance. The factorial cumulant generating functional (f.g.fl.) is then

C Poisson (h) = (h(x) -1) µ(dx) (13)
This is a linear, functional so computing cumulants are zero for orders greater than 1, and the first-order cumulant is equal to the first-order factorial moment, i.e.

c

(1)

Φ (B) = µ(B). (14)
This section follows the work by Bates and Neyman [START_REF] Bates | Contributions to the Theory of Accident Proneness[END_REF] on accident proneness, with adaptation from probability generating functions to probability generating functionals. Let us consider the multi-variate probability generating functional describing a mixture of n Poisson processes that are conditionally independent on the same parameter λ, i.e.

G (h 1 , . . . , h n |λ) = exp λ n i=1 a i [h i (x) -1] s i (dx) , (15)
where, for instance, n i=1 a i = 1. Then the unconditional process after margninalisation over random variable Λ is found via the following expectation

G (h 1 , . . . , h n ) = E [G (h 1 , . . . , h n |Λ)] (16)
= ∞ 0 exp λ n i=1 a i [h i (x) -1] s i (dx) dF (λ) = L * n i=1 a i [h i (x) -1] s i (dx)
where the Laplace-Stieljes transform of a distribution F (λ) is defined with

L * (t) = ∞ 0 e -tλ dF (λ). (17)
If we take Λ to be Gamma distributed with α, β > 0, i.e.

p Λ (x) = β α Γ(α) x α-1 e -βx , (18)
then the unconditional process becomes

G (h 1 , . . . , h n ) = 1 + 1 β n i=1 a i [1 -h i (x)] s i (dx) -α . (19)
This can be viewed as a kind of multi-variate negative binomial distribution. Univariate and bivariate instances of this formula can be found by restricting the number of terms in the summation to be one or two. The reason for introducing this approach here is that the same reasoning will be applied when considering a bivariate p.g.fl. which has an exponential form constructed with Poisson processes to determine an alternative bivariate p.g.fl. that is able to retain second-order information.

The following section describes the Panjer point process [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF], based on the Panjer distribution [START_REF] Fackler | Panjer class united -one formula for the Poisson, Binomial, and Negative Binomial distribution[END_REF], which extends the negative binomial by considering negative α and β.

C. Panjer process

A Panjer point process is a process whose size is Panjer distributed with parameters α and β with spatial distribution s [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF], [START_REF] Fackler | Panjer class united -one formula for the Poisson, Binomial, and Negative Binomial distribution[END_REF]. For finite and positive α and β, the Panjer distribution describes a negative binomial distribution. For finite and negative α and β we obtain a binomial distribution. The limit case α, β → ∞ with constant ratio λ = α β yields a

Poisson process with parameter λ [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF], [START_REF] Klugman | Loss Models: From Data to Decisions[END_REF]. The p.g.fl. of a Panjer process with parameters α, β is given by [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF] G

Panjer (h) = 1 + 1 β [1 -h(x)]s(dx) -α . (20)
The f.g.fl. of the Panjer process becomes,

C Panjer (h) = -α ln 1 + 1 β [1 -h(x)]s(dx) , (21)
which takes a very similar form to the f.g.fl. of the Bernoulli process. Hence, the cumulants become

c (n) Panjer (B 1 × • • • ×B n ) = (n -1)! α β n s(B 1 ) . . . s(B n ). ( 22
)
Thus when

B 1 = B 2 = X , c (1)
Panjer = α β , c (2)
Panjer = α β 2 , (23)
and hence

α = c (1) Panjer 2 c (2) Panjer , β = c (1) Panjer c (2) Panjer . ( 24
)
IV. ALGORITHM SPECIFICATION In this section, the new linear-complexity filter and it assumptions are presented. The multi-target model and general assumptions are based on the work [START_REF] Mahler | Multitarget Bayes filtering via first-order multitarget moments[END_REF]. The key difference is that correlations are maintained into the bivariate probability generating functional through the insight in Bates and Neyman [START_REF] Bates | Contributions to the Theory of Accident Proneness[END_REF] that dependencies can be introduced in functionals that take an exponential form through the application of the Laplace-Stieltjes transform. The proofs are given in the Appendix and the algorithm pseudo-code is described in Algorithm 1.

A. Time prediction step (time k)

In the time prediction step, the posterior target process Φ k-1 is predicted to Φ k|k-1 based on prior knowledge on the dynamical behaviour of the targets. The assumptions of the time prediction step can be stated as follows:

Assumptions IV.1. . (a) The targets evolve independently from each other; (b) A target with state x ∈ X at time k -1 survived to the current time k with probability p s,k (x); if it did so, its state evolved according to a Markov transition kernel t k|k-1 (•|x); (c) New targets entered the scene between time k -1 and k, independently of the existing targets and described by a newborn point process Φ b,k with p.g.fl. G b,k .

The following theorem describes the prediction of the n thorder cumulant.

Theorem IV.2 (Factorial cumulant prediction). Under Assumptions IV.1, the n th -order factorial cumulant of the predicted target process Φ k|k-1 is given by

c (n) k|k-1 (B 1 ×. . .×B n ) = c (n) b,k (B 1 ×. . .×B n )+c (n) s,k (B 1 ×. . .×B n ), (25)
in any B i ∈ B(X ), where c

(n)
s,k is the n th -order factorial cumulant of the process describing the surviving targets

c (n) s,k (B 1 × . . . × B n ) = (26) n i=1 p s,k (x i )t k|k-1 (B i |x i )c (n) k-1 (d(x 1 , . . . , x n )).
In the algorithm factorial cumulants are computed as follows. The first-order factorial cumulant density c

(1) k|k-1 (x) is the same as prediction of the first-order factorial moment density as calculated in [START_REF] Mahler | Multitarget Bayes filtering via first-order multitarget moments[END_REF]. The predicted second-order cumulant is a scalar computed over the whole state space, i.e. c

(2)

k-1 (X × X ). B. Data update step (time k)
In the data update step, the predicted process Φ k|k-1 is updated to Φ k given the current measurement set Z k , collected from the sensor. This is achieved through the following assumptions:

Assumptions IV.3. . (a)
The bivariate target-measurement process is approximated by a Panjer process determined via the Laplace-Stieltjes transform of a bivariate Khinchin process linear in both arguments. (b) A target with state x ∈ X is detected with probability p d,k (x); if so, it produces a measurement whose state is distributed according to a likelihood l k (•|x). (c) The clutter process is described by intensity function λ c,k (z), and second-order cumulant c

c,k (Z, Z). Theorem IV.4 (Factorial cumulant update). Under Assumptions IV.3, the n th -order cumulant of the updated target process Φ k is given by

c (n) k (B 1 × . . . × B n ) = ( 27
) (n -1)! (α k|k-1 + |Z|)µ φ k (B 1 ) . . . µ φ k (B n ) α k|k-1 + µ d k (X , Z) + λ c,k (Z) n + (-1) n-1 z∈Z µ z k (B 1 ) . . . µ z k (B n ) X p d (x)l(z|x)c (1) k|k-1 (dx) + λ c,k (z) n   in any B 1 , . . . , B n ∈ B(X )
, where the missed detection term µ φ k is given by

µ φ k (B) = B (1 -p d,k (x))c (1)
k|k-1 (dx).

(

) 28
and where we have the following terms relating to detection statistics

µ z k (B) = B p d,k (x)l(z|x)c (1) k|k-1 (dx) (29)
µ d k (X , Z) = X Z p d,k (x)l(dz|x)c (1)
k|k-1 (dx)

and the parameter α k|k-1 is computed with

α k|k-1 = µ d k|k-1 (X , Z) + λ c,k (Z) 2 c
(2)

k|k-1 + c (2) c,k . (31)
Similar to the prediction step, the first-order cumulant density c

(1) k (x) is calculated along with the second-order factorial cumulant computed over the whole space, c

(2) k (X × X ).

Implementation issues

A closed-form solution to the algorithm developed is presented in Algorithm 1 in the appendix. It is based on the solutions developed for PHD filter [START_REF] Mahler | Multitarget Bayes filtering via first-order multitarget moments[END_REF], CPHD filter [START_REF]PHD filters of higher order in target number[END_REF], and second-order PHD filter [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF], eg. [START_REF] Vo | The gaussian mixture probability hypothesis density filter[END_REF], [START_REF] Vo | Analytic implementations of the cardinalized probability hypothesis density filter[END_REF], [START_REF] Schlangen | A second-order phd filter with mean and variance in target number[END_REF]. The filter is the same complexity as the PHD filter with target-number variance [START_REF] Delande | Regional Variance for Multi-Object Filtering[END_REF], if the variance is computed over the entire state space. Compared with the Gaussian mixture PHD filter [START_REF] Vo | The gaussian mixture probability hypothesis density filter[END_REF], with linear-complexity analysis presented in Section III.C of that work, in the prediction step, there are the following additional calculation to compute the predicted second-order cumulant,

c (2) k|k-1 = p 2 s,k c (2)
k-1 + c (2) b,k (32)
α k|k-1 = (c (1)
k|k-1 + λ c,k ) 2 /(c (2)
k|k-1 + c (2) c,k ) (33)
and in the update, there is additional calculation of the following terms defined in Algorithm 1, c

k = (µ φ k ) 2 l 2 (φ) - z∈Z k µ z k µ z k + λ c,k (z) 2 , (2)
1 (φ) = (α k|k-1 + |Z|)/(α k|k-1 + µ d k ) (34)
2 (φ) = (α k|k-1 + |Z|)/(α k|k-1 + µ d k + λ c,k ) 2 . (35)
V. NUMERICAL EXPERIMENTS

In this section, we present experimental results for an illustrative example, under parametric evaluations that vary either the clutter model, number of targets, average number of false alarms per frame, or probability of detection. These parametric evaluations intend to show differences of performance of the PHD, CPHD and linear-complexity cumulant-based (LC-Cumulant) filters for different ranges of settings.

Let us consider a two-dimensional scenario within the region [-1000, +1000] × [-1000, +1000] (m × m). Each target is described by its state vector x = (p x , p y , v x , v y ) T , where (p x , p y ) is a pair that specifies a position in Cartesian coordinates and (v x , v y ) is the pair specifying velocity in the same coordinates. Each target moves at nearly-constant velocity, with transition matrix and state process covariance matrix given respectively by

F = I 2 I 2 ∆t 0 2 I 2 , Q = I 2 ∆t 3 /3 I 2 ∆t 2 /2 I 2 ∆t 2 /2 I 2 ∆t σ 2 q ,
where I 2 and 0 2 are the identity and zero matrices with dimensions 2 × 2, ∆t = 1 s is the sampling period, and the standard deviation of velocity increments is characterized by

σ q = 1 m/s 3 2
. Probability of survival is set as p s = 0.99. A sensor collects Cartesian position measurements characterized by the output matrix and measurement noise covariance matrix, H = ( I 2 0 2 ) and R = I 2 σ 2 r , respectively, where σ r = 10 m is the standard deviation of the measured positions. False alarms can happen according to a Poisson point-process with intensity λ c,k (z) = λ c • s c (z), where λ c is the average number of false alarms per scan, and s c (z) is the spatial distribution of clutter, assumed uniform in the surveillance region with "volume" V = 2000 2 m 2 . The example is simulated for T = 100 s. Targets appear in batches at positions uniformly sampled within the area [-800, +800]×[-800, +800] (m×m), with random velocities uniformly sampled within the ranges [-10, +10] × [-10, +10] (m/s × m/s).

By denoting N t as the total number of targets that appear in the scene, the batches of target appearance are set as follows:

• 0.25N t targets are already in the scene at t = 0 and will remain up to t = 100 s with exception of 5 targets that are set to disappear at t = 80 s, • 0.25N t targets appear at t = 20 s and remain, • 0.25N t of targetsappear at t = 40 s and remain, • 0.25N t of targets appear at t = 60 s and remain,

• from N t targets in the scene, 5 targets disappear at t = 80 s and the remaining N t -5 stay up to t = 100 s. The birth process is a Poisson point-process with intensity function

µ b (x) = 4 i=1 w (i) b N (x; m (i) b , P (i) b ), where w (i) b = N t /(4T /∆t), m (1) b = (-500, -500, 0, 0) T , m (2) b = (-500, +500, 0, 0) T , m (3) b = (+500, -500, 0, 0) T , m (4) b = (+500, +500, 0, 0) T , P (i) b = diag(500 2 I 2 , 10 2 I 2 ), for i = [1..4],
where diag(A, B) is a block diagnoal matrix formed whose blocks are the matrices A and B.

For all filters, pruning of Gaussian components is based on the weight threshold τ prn = 10 -5 , merging is performed with threshold of τ mrg = 4 m, and the number of maintained components is limited at J max = 100 (see [START_REF] Vo | The gaussian mixture probability hypothesis density filter[END_REF] for details on the pruning and merging procedure). Measurements are gated with gate-size probability of p gate = 0.999. The cardinality distribution for the CPHD filter is estimated to a maximum of n max = 2N t terms. This maximum number of cardinality terms has been chosen to keep the CPHD filter computational effort competitive in relation to the other filters for difficult scenarios. The LC Cumulant filter is evaluated in comparison to the PHD and CPHD filters for four different cases. For each case, 200 Monte Carlo (MC) runs are performed, each with independently generated clutter, and independently generated (target-originated) measurements for each trial. For all filters, performance is evaluated in terms of:

• mean Optimal Subpattern Assignment (OSPA) metric for cut-off c OSPA = 100 and norm order p OSPA = 1, • root-mean-square error (RMSE) of the estimated number of targets, and • computation time (per time step). All indexes are averaged over time steps and consolidated for all values of the varying parameters.

A. Results

Case 1 : For this case, Figures 1a-3b present, the mean OSPA over time, and the cardinality mean and standard deviation over time for the PHD, CPHD and LC Cumulant filters, where we can perceive the advantage of estimating second-order information on the target number. The LC Cumulant filter maintains second-order information about the target number via the second-order factorial cumulant.

It is clear from the figures that, the performance of the LC Cumulant filter is similar to that of the CPHD filter, but at a computational cost that is practically the same as that of the PHD filter. In all three subcases, with Poisson, binomial and negative-binomial clutter models, the PHD filter underestimates the correct number of targets due to the difficulties imposed by a relatively low probability of detection p d = 0.80, non-Poisson clutter, and closely spaced targets.

Case 2 : The consolidated performance indexes for case 2, averaged over all time steps for different numbers of targets, N t ∈ {10, 20, 30, 40, 50}, are shown in Figures 456. In general, the performance of the LC Cumulant filter is not much different from that of the CPHD filter, but at a lower computational complexity. In terms of average mean OSPA metric, the LC Cumulant filter shows a performance that approaches that of the CPHD filter as the number of targets increases owing to that the LC Cumulant filter seems less sensitive (on average) to the increase in target number. The average cardinality RMSE of the CPHD and LC Cumulant filter seems to increase sub-exponentially with the number of targets, at a small rate than that of the PHD, which is more sensitive to the scenario complexity.

Case 3 : The consolidated performance indexes for case 3, averaged over time for p d ∈ {0.60, 0.70, 0.80, 0.90, 0.99}, are presented in Figures 789. Once again, the performance of the LC Cumulant filter is very similar to that of the CPHD filter in terms of average mean OSPA and average cardinality RMSE, but its computational cost is much smaller than that of the CPHD filter, being rather comparable to that of the PHD filter. As expected, the error indexes and cardinality variance decrease as the probability of detection increases, for all filters. filter overall, with marginally better performance for higher numbers of false alarms. In this case, all filters take almost the same time to perform the computations, but it remains clear that the LC Cumulant filter requires less computational effort than the standard CPHD filter implementation, presenting runtimes that are comparable to the PHD filter. Note that the average cardinality RMSE of all filters seem to increase sub-exponentially with the number of false alarms per frame, suggesting a dependency of the signal-to-noise ratio that is polynomial in the number of measurements.

APPENDIX: PROOFS B. Proof of Thm. IV.2

In the following, we denote by G s,k the p.g.fl. of the point process describing the evolution of a target from the previous time step, which might have survived (or not) to the present time step. The p.g.fl. G k|k-1 of the predicted target process

G k|k-1 (h) = G b (h)G k-1 (G s (h|•)), (37)
hence the c.g.fl. C k|k-1 of the predicted target process is

C k|k-1 (h) = ln (G b (h)G k-1 (G s (h|•))) (38) = ln (G b (h)) + ln (G k-1 (G s (h|•))) = C b (h) + C k-1 (G s (h|•)).
The n th -order factorial cumulant requires the n th -order derivative of C k|k-1 (h), i.e.

c

(2)

k|k-1 (B 1 × . . . × B n ) = δ n C k|k-1 (h; 1 B 1 , . . . , 1 Bn ) h=1 . (39)
The survival process for a target with state x at the previous time step can be described with a Bernoulli point process with The n th -order derivative δ n G s (h|•; 1 B 1 , . . . , 1 B n ) = 0 for second-order and above since G s (h) is linear in h. Hence the result follows by Faà di Bruno's formula [START_REF] Clark | Faa di Bruno's formula for chain differentials[END_REF].

C. Proof of Thm.IV.4

Now consider the f.g.fl. of a bivariate Khinchin process [START_REF] Houssineau | Bayesian estimation of multi-object systems with independently identically distributed correlations[END_REF],

C J,k (g, h), i.e. C J,k (g, h) = C J,k (0, 0) + C J,k (0, h) + C J,k (g, 0)+ (42) n≥1,m≥1 m j g(z j ) n i h(x i )K (n,m) J,k (dx 1:n , dz 1:m ),
where the zero terms are calculated with

C J,k (0, 0) = - n,m≥0|n+m≥1 K (n,m) J,k (dx 1:n , dz 1:m ), (43)
C J,k (0, h) = m≥1 h(x 1 ) . . . h(x n )K (n,0) J,k (dx 1:n ), (44)
We can determine an alternative bivariate p.g.fl. through the Laplace-Stieltjes transform i.e.

G J,k (g, h) = L * C J,k (g, h) C J,k (0, 0)

= 1 -C J,k (g, h) β k|k-1 C J,k (0, 0) -α k|k-1 , The p.g.fl. of the updated target process Φ k is obtained from the differentiation of the joint p.g.fl. using Bayes' rule [START_REF] Mahler | Multitarget Bayes filtering via first-order multitarget moments[END_REF] with the following

G k (h) = δ |Z k | G J,k (g, h; (δ z ) z∈Z k )| g=0 δ |Z k | G J,k (g, 1; (δ z ) z∈Z k ))| g=0 . ( 47
)
Noting that we can set α k|k-1 = -β k|k-1 C J,k (0, 0), and given the assumption that C(g, h) is a linear functional in g which means that there is only one remaining partition after application of Faà di Bruno's formula [START_REF] Clark | Faa di Bruno's formula for chain differentials[END_REF], the updated p.g.fl. becomes G k (h) = α k|k-1 + C J,k (0, h) α k|k-1 + C J,k (0, 1)

-(α k|k-1 +|Z|) z∈Z δC J,k (0, h; δ z ) δC J,k (0, 1; δ z )

and consequently, the f.g.fl. becomes C k (h) = -(α k|k-1 + |Z|) ln α k|k-1 + C J,k (0, h) α k|k-1 + C J,k (0, 1)

+ z∈Z ln δC J,k (0, h; δ z ) δC J,k (0, 1; δ z ) .

Noting that the joint p.g.fl. in [19, p1174] is a bivariate Khinchin process [START_REF] Houssineau | Bayesian estimation of multi-object systems with independently identically distributed correlations[END_REF] linear in g and h, with , where α k|k-1 is defined with the intensity and second-order cumulant of the joint process,

α k|k-1 = µ d k|k-1 (X , Z) + λ c,k (Z) 2 c (2)
k|k-1 + c

(2) c,k .

Differentiating the f.g.fl. with respect to h and setting the argument to be equal to 1, leads to the factorial cumulants as stated.

Output

Posterior target process: {w

(i) k , m (i)
k , P

(i) k } N k i=1 , c (2) k
Algorithm 1: Data update, second part: Update component weights.

Case 1 : 2 :

 12 For N t = 50 targets that appear in the scene, p d = 0.80, λ c = 10 number of false alarms per scan on average, all filters are tested for three different clutter models: a) Poisson process with µ c = λ c = 10 and var c = 10, b) binomial process with µ c = λ c = 10 and var c = λ c /20 = 0.5, c) negative binomial process with µ c = λ c = 10 and var c = 20λ c = 200. Case For p d = 0.90, λ c = 10 false alarms per scan on average (Poisson distribution), filters are tested for different numbers of targets that appear in the scene, N t ∈ {10, 20, 30, 40, 50}. Case 3: For N t = 20 targets, λ c = 10 false alarms per scan on average (Poisson distribution), filters are tested for different probabilities of detection, p d ∈ {0.60, 0.70, 0.80, 0.90, 0.99}. Case 4: For N t = 20 targets, p d = 0.90, filters are tested for different numbers of false alarms per frame (Poisson distribution), λ c ∈ {10, 20, 30, 40, 50}.

Case 4 :Fig. 1 :

 41 Fig. 1: Case 1a: MOSPA and cardinality, Poisson clutter

Fig. 2 :

 2 Fig. 2: Case 1b: MOSPA and cardinality, binomial clutter

Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :Fig. 9 :Fig. 10 :Fig. 11 :

 34567891011 Fig. 3: Case 1c: MOSPA and cardinality, negative-binomial clutter

C

  J,k (g, 0) = n≥1 g(z 1 ) . . . g(z m )K (0,m) J,k (dz 1:m ).

C

  J,k (g, h) = Z [g(z) -1] λ c,k (dz) x) (p d (x)l(dz|x)g(z) + q d (x)) -1] µ k|k-1 (dx),we can use this directly in the derivation. Set λ = -C J,k (0, 0) = µ k|k-1 (X ) + λ c,k (Z) in the Laplace-Stieltjes transform. and hence the f.g.fl. becomes-(α k|k-1 + |Z|) (51) × ln α k|k-1 + X [h(x)q d (x) -1]µ k|k-1 (dx) + λ c,k (Z) α k|k-1 + X p d (x)µ k|k-1 (dx) + λ c,k (Z) + z∈Z ln X h(x)p d (x)l k (z|x)µ k|k-1 (dx) + λ c,k(z)X p d (x)l(z|x)µ k|k-1 (dx) + λ c,k (z)

Algorithm 1 : 2 for 1 ≤

 121 Data update, first part: Compute single targetsingle measurement updates and corrective terms 1 ,Missed detection termsfor 1 ≤ i ≤ N k|k-1 do w j ≤ M k do w (i•n k|k-1 +j) k N k = N k|k-1 + N k|k-1 M k c k = (µ φ k ) 2 l 2 (φ) -z∈Z k µ z k µ z k +λ c,k(z) 2

  G s (h|x) = 1 -p s (x) + p s (x) h(y)t(dy|x).

			(40)
			It follows that
			δ(G s (h|x); 1 B ) h=1 = p s (x) δ(hy; 1 B )t(dy|x) h=1 (41)
			= p s (x) 1 B (y)h(y)t(dy|x) h=1 ,
			= p s (x)t(B|x).
		10 0
			PHD
	Computation time / time step [s]	10 -1	CPHD LC-Cumulant

Fig. 12: Case 4: average runtime vs. number of false alarms parameter p s (x) and spatial distribution t(•|x),

(φ) := (α k|k-1 + |Z|)/(α k|k-1 + µ d k + λ c,k )

(φ) := (α k|k-1 + |Z|)/(α k|k-1 + µ d k + λ c,k ) 2

Input

Posterior target process: {w

Newborn process: {w

Survival process c

(1)

Newborn process

Predicted process: {w

, c

(2) k|k-1

Current measurements:

Clutter process:

Panjer parameters c

(1)

c,k ) Missed detection and measurement terms for