Laetitia Gerossier

Anaëlle Dubois

Alexia Paturel

Nadim Fares

Damien Cohen

Philippe Merle

Joel Lachuer

Anne Wierinckx

Pierre Saintigny

Brigitte Bancel

Phillippe Merle

Janick Selves

Anne Schnitzler

Bérengère Ouine

Aurélie Cartier

Leanne De Koning

Vincent Puard

Ivan Bieche

Hector Hernandez-Vargas

Janet Hall

Isabelle Chemin
email: isabelle.chemin@inserm.fr

PARP inhibitors and radiation potentiate liver cell death in vitro. Do hepatocellular carcinomas have an achilles' heel?

Keywords: Liver cancer, hepatitis B virus (HBV) X protein, gammaH2AX, DNA damage, SMC5/6, Talazoparib, Veliparib

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

Abstract

Background

A promising avenue for cancer treatment is exacerbating the deregulation of the DNA repair machinery that would normally protect the genome. To address the applicability of poly(ADP-ribose) polymerase (PARP) inhibitors (PARPi) combined with radiotherapy for the treatment of hepatocellular carcinoma (HCC) two approaches were used: firstly, the in vitro sensitivity to the PARPi Veliparib and Talazoparib +/-radiation exposure was determined in liver cell lines and the impact of the HBV X protein (HBx) that deregulates cellular DNA damage repair via SMC5/6 degradation was investigated. Secondly, PARP expression profiles and DNA damage levels using the surrogate marker gammaH2AX were assessed in a panel of control liver vs HCC tissues.

Methods

Cell cytotoxicity was measured by clonogenic survival or relative cell growth and the DNA damage response using immunological-based techniques in Hep3B, PLC/PRF/5, HepG2-and HepaRGderived models. Transcriptome changes due to HBx expression vs SMC6 loss were assessed by RNA sequencing in HepaRG-derived models. PARP and PARG transcripts (qPCR) and PARP1, H2AX and gammaH2AX protein levels (RPPA) were compared in control liver vs HBV-, HCV-, alcohol-and non-alcoholic steatohepatitis-associated HCC (tumor/peritumor) tissues.

Results

PARPi cytotoxicity was significantly enhanced when combined with X-rays (2Gy) with Talazoparib having a greater impact than Veliparib in most in vitro models. HBx expression significantly lowered survival, probably driven by SMC5/6 loss based on the transcriptome analysis and higher DNA damage levels. PARP1 and PARP2 transcript levels were significantly higher in tumor than peritumor and control tissues. The HBV/HCV/alcohol-associated tumor tissues studied had reduced H2AX but higher gammaH2AX protein levels compared to peritumor and control tissues providing evidence of increased DNA damage during liver disease progression.

Conclusions

These proof-of-concept experiments support PARPi alone or combined with radiotherapy for HCC treatment, particularly for HBV-associated tumors, that warrant further investigation.

List of abbreviations

HCC (Hepatocellular carcinoma); RT (radiotherapy); HBV (hepatitis B virus); SBRT (sterotactic body RT); PARP (poly(ADP-ribose) polymerases); PARPi (PARP inhibitors); IR (ionizing radiation); HBV (hepatitis B virus); HBx (hepatitis B virus X protein); SMC5/6 (structural maintenance of chromosomes complex 5 and 6); HR (homologous recombination); DSBs (double strand breaks); PARG (poly(ADP-ribose) glycohydrolase); gammaH2AX (serine 139 phosphorylated Histone H2AX); PT (peritumor); T (tumor); HepaRG TR (HepaRG cells containing a plasmid with the tetracycline promotor); HepaRG TRX (HepaRG cells expressing the HBx protein under a tetracycline inducible promotor); HepaRG TRX shscramble (cells expressing an in-efficient shRNA construct in a lentiviral GFP vector); HepaRG TRX sh-SMC6 (cells expressing a 29mer shRNA construct targeting the SMC6 transcript); HU (hydroxyurea); DAPI (4,6-diamidino-2-phenylindole); AU (arbitrary units); D37 (mean lethal dose); RPPA (Reverse Phase Protein Array); FPKM (Fragments Per Kilobases of exon per Million mapped reads); DDR (DNA damage response).

Introduction

Over 90% of primary liver cancers are hepatocellular carcinomas (HCC) and an often too late diagnosis and few treatment options can explain their poor prognosis. Treatment with systemic chemotherapy are ineffective in HCC and many of the main oncogenic HCC drivers, such as the TERT promoter, TP53 and CTNNB1 [START_REF] Calderaro | Histological subtypes of hepatocellular carcinoma are related to gene mutations and molecular tumour classification[END_REF], have not proven to be targetable. In contrast to many other cancer sites, ionising external beam radiotherapy (RT) has not been widely used in HCC management outside the palliative setting [START_REF] Ohri | Radiotherapy for Hepatocellular Carcinoma: New Indications and Directions for Future Study[END_REF]. However, with recent technological advances in radiation delivery, hypofractionated sterotactic body RT (SBRT) used as a monotherapy or in combination with other liver-directed therapies (reviewed in [START_REF] Ohri | Radiotherapy for Hepatocellular Carcinoma: New Indications and Directions for Future Study[END_REF][START_REF] Chino | The role of external beam radiotherapy in the treatment of hepatocellular cancer[END_REF]) or with radiosensitising drugs such as poly(ADP-ribose) polymerases (PARP) inhibitors (PARPi) [START_REF] Guillot | Update on hepatocellular carcinoma breakthroughs: Poly(ADP-ribose) polymerase inhibitors as a promising therapeutic strategy[END_REF] could provide promising local control and support their use for HCC.

In order to explore the potential of PARPi for HCC treatment, we have expanded an in vitro pilot study [START_REF] Guillot | PARP inhibition and the radiosensitizing effects of the PARP inhibitor ABT-888 in in vitrohepatocellular carcinoma models[END_REF] showing that concomitant Veliparib and ionizing radiation (IR) treatment lowered liver cell survival to assess a second PARPi Talazoparib and the impact of the expression of hepatitis B virus (HBV) encoded X (HBx) protein on this response. HBx degrades the cell's structural maintenance of chromosomes complex 5 and 6 (SMC5/6) to allow the completion of the HBV lifecycle [START_REF] Decorsière | Hepatitis B virus X protein identifies the Smc5/6 complex as a host restriction factor[END_REF]. The SMC5/6 complex participates in homologous recombination (HR), one of two major DNA double strand break (DSB) repair pathways and promotes sister-chromatid HR during replication. This complex also contributes to stalled replication forks' integrity and replication recombination intermediates' removal (see [START_REF] Aragón | The Smc5/6 Complex: New and Old Functions of the Enigmatic Long-Distance Relative[END_REF] for review), all cellular processes that can modulate the response to PARPi [START_REF] Mégnin-Chanet | Targeting poly(ADP-ribose) polymerase activity for cancer therapy[END_REF]. In parallel to this in vitro approach, the expression profiles of key PARP genes in a panel of control liver and HCC peritumor (PT) and tumor (T) tissues was determined. Indeed, PARP expression is a prerequisite for the use of PARPi and to investigate whether targeting PARP activity is an option for HCC therapy the transcript profiles of three PARP genes together with that of poly(ADP-ribose) glycohydrolase (PARG) that degrades the polymers and the protein levels of the major PARP protein PARP1 were also evaluated. In addition, as the cellular consequences of PARP inhibition can depend on cell proliferation rates [START_REF] Mégnin-Chanet | Targeting poly(ADP-ribose) polymerase activity for cancer therapy[END_REF] the transcript and protein expression profiles of the proliferation marker MKI67 were determined together with the protein levels of the surrogate DNA damage marker gammaH2AX (also known as serine 139 phosphorylated Histone H2AX) [START_REF] Bonner | γH2AX and cancer[END_REF] in order to assess DNA damage levels during liver disease progression.

Materials and methods

Cell lines

The human hepatoma cell line HepG2 (ATCC HB-8065) and its derivatives HepG2 2.2.15 carrying the HBV wildtype genome and HepG2 K6 carrying a mutant form of HBV with no HBx protein [START_REF] Sells | Replicative intermediates of hepatitis B virus in HepG2 cells that produce infectious virions[END_REF][START_REF] Lucifora | Hepatitis B virus X protein is essential to initiate and maintain virus replication after infection[END_REF], Non-differentiated HepaRG cells expressing the tetracycline inducible promotor (HepaRG TR) or the HBx protein under the tetracycline inducible promotor (HepaRG TRX) [START_REF] Lucifora | Hepatitis B virus X protein is essential to initiate and maintain virus replication after infection[END_REF], HepaRG TRX shscramble expressing an in-efficient shRNA construct in a lentiviral GFP vector (Origene) and HepaRG TRX sh-SMC6 cells expressing a 29mer shRNA construct targeting the SMC6 transcript (Origene), Hep3B (ATCC HB-8064) and PLC/PRF/5 (ATCC CRL-8024) cells were grown in a humidified 37°C incubator under 5% CO2.. All cell lines were regularly checked for mycoplasma contamination.

Cytotoxicity measurements

Cell survival was assessed using the colony formation assay for all cell models except the nondifferentiated HepaRG TRX-based lines. Cells were seeded (1000 -2000/25cm 2 flask) and left to adhere overnight before treatment. The PARPi Talazoparib (DMSO, 10uM) or Veliparib (PBS, 4mM) (Selleckchem, USA) were added directly to cell culture medium to achieve the final concentrations required. To assess the impact of PARP inhibition combined with IR, PARPi were present in the culture medium for 24h with the radiation delivered 1h after their addition. Cells were irradiated using a 6-MeV γ-ray clinical irradiator (SL 15 Phillips) at the Anti-Cancer Centre Léon-Bérard (Lyon, France) with a dose rate of 6Gy min -1 to give the required dose.

To assess hydroxurea's (HU) impact on cell survival, HU (Sigma, 0.2M in PBS) was added directly to culture medium (final concentration 2mM) for a period of 24h.

After drug removal and replacement of culture medium, colonies were grown under standard culture conditions for 10 days, then fixed in methanol and stained with 0.05% Coomassie brilliant blue in 3:1:6 ethanol/acetic acid/water. Colonies with more than 50 cells were counted and the surviving fraction (the colony count relative to mock-treated cells) determined.

Cell growth rates in non-differentiated HepaRG TRX-based lines were assessed by plating cells at a fixed density and allowing them to grow for the required time after treatment before counting viable cell numbers.

Western Blot

Cells (2.10 6 ) were lysed in RIPA buffer (50mM Tris pH7.4, 250mM NaCl, 0.1% SDS, 2mM DTT, 0.5% NP40) containing a protease inhibitor cocktail (Halt™) (30min, 4°C) or in 2% SDS in water containing Halt™ and sonicated for 10x 30s and centrifuged (15min, 4°C, 13,000g). Protein concentrations were estimated using the BCA Protein Assay (Pierce™). Samples were diluted in Laemmli 5X, boiled and 20μg protein extract loaded per lane. Proteins were separated on 8% (SMC6 detection) or 12% acrylamide gels and after transfer to nitrocellulose membranes were incubated with primary antibodies overnight at 4°C (1:1000 anti-gammaH2AX (Abgent ab18311), 1:1000 anti-H2AX (Cell Signalling 2595), 1:500 anti-Smc6 (Abgent AT3956a), 1:1000 pADPR (Abcam, clone ab14459)) except for anti-β-actin (1:10000 anti-β-actin (Sigma-Aldrich A1978) 1h, room temperature.

Membranes were then incubated (1h, room temperature) with Horseradish-peroxidase-conjugated secondary antibodies (1:10000). Detection used ECL (BioRad), ChemiDoc and Image Lab software (Bio-Rad).

Flow cytometry

To assess the impact of HU treatment on cell cycle progression HepG2, HepG2 2.2.15, HepG2 K6 and HepaRG TRX (non-differentiated, induced or not) (2x10 6 cells/25cm 2 flask) were seeded and left to attach overnight before treating with 2mM HU for varying times. Cells were then processed using the BD Cycletest™ Plus DNA reagent kit (Becton Dickinson) containing propidium iodide according to the manufacturer's protocol. Flow cytometric analysis was performed using a FACSCalibur cytofluorometer (Becton-Dickinson Biosciences). At least 10 4 events were recorded and data analysis was done with CellQuest Pro software (Becton-Dickinson Biosciences).

Immunofluorescence protein detection

The presence of gammaH2AX foci in irradiated cells was assessed by immunofluorescence. Cells were plated on coverslips (after induction in the case of the non-differentiated HepaRG TRX cells) and allowed to adhere overnight before irradiation as described above. Cells were returned to the incubator to allow DNA repair to proceed and then fixed (15min, room temperature, 4% formaldehyde), washed twice in PBS and permeabilized in ice cold lysis solution (0.5% Triton, 10.5% Sucrose, 0.06% MgCl2, 20mM Hepes pH7, 50mM NaCl, 3min) before incubation (40min, 37°C) with anti-gammaH2AX (Abgent ab18311) (1:1000) in blocking buffer (PBS-BSA 3%) and incubation with an IgG secondary antibody (1:200), Alexa 488 (Fluor®) (20min, 37°C). After washing, coverslips were mounted with medium containing 4,6-diamidino-2-phenylindole (DAPI; Duolink®). Immunofluorescence images were acquired using an upright microscope Nikon NIE. At least 200 cells were analysed per experimental condition. The number of foci was quantified in arbitrary units (AU) by a macro using Image J software developed by J. Jacquemetton (CRCL, Lyon).

Expression shSMC6/Shcontrol for RNA seq analysis

Non-differentiated HepaRG TR, TRX, TRX+shscramble (control) and TRX+shSMC6 (construct C) cells (1x10 6 ) were seeded in 6-well plates and left to attach before tetracycline induction (1µg/ml) or not, for 4 days. SMC6 depletion was validated by western blot as previously described [START_REF] Decorsière | Hepatitis B virus X protein identifies the Smc5/6 complex as a host restriction factor[END_REF] and RNA extraction performed using RNeasy TM following manufacturer instructions. Three independent experiments were carried out.

RNA samples were sequenced using Illumina's NextSeq following standard procedures. Resulting base called sequences were demultiplexed with Bcl2fastq v2.17.1.14 using 0 mismatch, followed by trimming with cutadapt v1.9.1 and quality check with fastQC. Reads were mapped against hg19 genome (GRCH37, Feb. 2009) using TopHat v2.1.0 with default parameters (bowtie 2.2.9), and signals were normalized using cufflinks v2.1.1 which uses the FPKM (Fragments Per Kilobases of exon per Million mapped reads) method.

After preprocessing, FPKM values, corresponding isoform annotations, and phenotype data, were imported into R for further analysis with R/Bioconductor packages. Boxplots of normalized signals, hierarchical clustering, and multidimensional scaling were used for data inspection. Based on this, one outlier on each experimental group was removed at this step.

To define differentially expressed genes, we modelled experimental conditions as categorical variables in a linear regression using an empirical Bayesian approach [START_REF] Gordon | Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments[END_REF] Pairwise comparisons with a False Discovery Rate (FDR)-adjusted P value below 0.05 and a fold-change of at least two, were considered statistically significant. Overlaps between gene sets were performed with the VennDiagram packages.

Patients and liver samples

RNA extraction and cDNA preparation

Total RNA was extracted from frozen tissues with Extract-all (Eurobio) according to the manufacturer's instructions. One μg was treated with DNAse-I (Roche) and then retro-transcribed (RT) using the SuperScript III First-Strand Synthesis System (Thermofischer), according to the manufacturer's instructions.

Quantitative real-time PCR

Primers for GUS used an endogenous control were designed using Oligo 5 (National Biosciences, Plymouth, MN, USA) and their specificity checked by searches in dbEST and nr databases. Primers for PARP1, PARP2, PARP3, PARG and MKI67 and the PCR reaction conditions are described elsewhere [START_REF] Bieche | Variations in the mRNA expression of poly(ADP-ribose) polymerases, poly(ADP-ribose) glycohydrolase and ADP-ribosylhydrolase 3 in breast tumors and impact on clinical outcome[END_REF]. Quantitative values for the different transcripts were obtained from the cell cycle number (Ct value) at which the fluorescent signal became exponential. Results, were expressed as target gene expression relative to the GUS gene.

Protein extraction and Reverse Phase Protein Array (RPPA) analysis

Protein extracts were prepared from frozen liver sections as previously described [START_REF] Bonnin | Toll-like receptor 3 downregulation is an escape mechanism from apoptosis during hepatocarcinogenesis[END_REF] and stored at -80° until further processing from 93 tissues from HCC patients (46 T, 47 PT) of the following aetiologies; 29 alcohol (13 T, 16 PT), 32 HBV (17 T, 15 PT) and 32 HCV (16T,16 PT). 8 surgically resected normal liver tissues served as controls.

Protein concentration was determined (Pierce BCA reducing agent compatible kit, ref 23252) and extracts printed onto nitrocellulose covered slides (Supernova, Grace Biolabs) using a dedicated arrayer (2470 arrayer, Aushon Biosystems) in five serial dilutions (1500 to 93.75 µg/ml) and three replicates per dilution. Arrays were labeled with 4 specific antibodies (H2AX 1:2000 (CST 2595); Phospho-Histone H2AX (ser139) 1:1000 (Abcam ab2893); PARP (uncleaved) 1:1000 (Abcam ab32378); Ki67 1:250 (DAKO M7240) as described in [START_REF] Meseure | Prognostic value of a newly identified MALAT1 alternatively spliced transcript in breast cancer[END_REF]. Read-out was done using IRDye 800CW (LiCOR) on an Innoscan 710-AL scanner (Innopsys). For staining of total protein, arrays were incubated 30 min in Super G blocking buffer (Grace Biolabs), rinsed in water, incubated 5min in 0.000005% Fast green FCF (Sigma) and rinsed again in water. Raw data were normalized using Normacurve [START_REF] Troncale | NormaCurve: A SuperCurve-Based Method That Simultaneously Quantifies and Normalizes Reverse Phase Protein Array Data[END_REF].

Statistics

The distributions of the gene mRNA levels were characterised by their median values and ranges using GraphPad Prism software (version 7.05.237). For the unsupervised hierarchical clustering of the RPPA data, Pearson was used as a distance measure and Ward as the clustering algorithm. Paired ttests, Mann Whitney test, Kruskal Wallis test, Fisher's exact, Wilcoxon matched-pairs signed rank test and Spearman rank correlation test were used as appropriate. Differences were considered significant at confidence levels greater than 95% (P ≤ 0.05)

Other materials and methods

Additional information regarding cell culture conditions, inducible lentiviral vectors and online data for HCC livers, are provided as Supplementary Materials and Methods.

Results

In vitro assessment of sensitivity to the PARP inhibition alone and in combination with IR

Our previous in vitro studies showed variability in PARPi sensitivity with a short (2h) exposure to Veliparib (10µM) reducing clonogenic survival in HepG2 but not PLC/PRF/5 or Hep3B liver cells [START_REF] Guillot | PARP inhibition and the radiosensitizing effects of the PARP inhibitor ABT-888 in in vitrohepatocellular carcinoma models[END_REF]. However, in both HepG2 and the more radiation resistant PLC/PRF/5 cells, PARP inhibition lowered IR-induced cell survival based on the mean lethal dose (D37) [START_REF] Guillot | PARP inhibition and the radiosensitizing effects of the PARP inhibitor ABT-888 in in vitrohepatocellular carcinoma models[END_REF]. We demonstrate here that a longer Veliparib (10µM) exposure (24h) significantly decreased cell survival in all three cell models (Figure 1). In addition, exposure to a second PARPi, Talazoparib, chosen because it is a more potent PARPi than Veliparib in terms of induced cytotoxicity [START_REF] Murai | Trapping of PARP1 and PARP2 by Clinical PARP Inhibitors[END_REF] at a dose (50nM/24h) that inhibited PARP activity as assessed by polyADPribose levels after doxorubicin treatment (Supplementary Figure 1B), significantly lowered cell survival compared to untreated cells. Under these experimental conditions a differential impact between the two PARPi on cell survival was only seen in HepG2 cells (54.2 ± 12.7% Veliparib vs 33 ± 7.9% Talazoparib, P ≤ 0.001) (Figure 1A).

When combined with a fixed 2Gy X-ray dose Veliparib (10µM/24h) exposure significantly attenuated the radiation sensitivity of HepG2 (Figure 1A) and Hep3B (Figure 1C) but not PLC/PRF/5 cells (Figure 1B). In contrast, the combined treatment of Talazoparib (50nM/24h) and X-rays (2Gy) significantly lowered cell survival in all three models compared to radiation treatment alone, with the lowest survival seen in HepG2 cells (9.5 ± 4.2%) (Figure 1A). As Hep3B cells are reported to be both HR and NHEJ repair competent (see for instance [START_REF] Zhang | Harmine suppresses homologous recombination repair and inhibits proliferation of hepatoma cells[END_REF]), the impact of 10-fold higher inhibitor doses was also investigated. These higher inhibitor doses significantly reduced cell survival with Talazoparib alone and in combination with radiation having a significantly greater impact than either Veliparib +/-radiation exposure (Figure 1C).

Impact of HBx viral protein on cell survival after IR and PARP inhibition

As HBx expression leads to the degradation of the cell's SMC5/6 complex that is involved in processes that could modulate the impact of PARPi we next examined its impact on cell survival using HepG2-derived cell lines differing in HBx expression: HepG2 2.2.15 cells express the whole HBV genome [START_REF] Sells | Replicative intermediates of hepatitis B virus in HepG2 cells that produce infectious virions[END_REF] and HepG2 K6 cells carry a mutant HBV that encodes all HBV proteins except HBx [START_REF] Lucifora | Hepatitis B virus X protein is essential to initiate and maintain virus replication after infection[END_REF]. In the absence of an efficient antibody against HBx, its expression was routinely deduced from SMC6 degradation in cellular protein extracts using western blotting (Supplementary Figure 1Ai).

Under the experimental conditions tested, the clonogenic survival after treatment with either Veliparib (10µM/24h) +/-X-rays (2Gy) was significantly lower in the presence of HBx (Figure 2A). A second non-tumoral cell model (HepaRG TRX) was also investigated in which HBx is transiently induced for 5 days thus reducing longer-term selection effects or adaption to HBx expression. In the HepaRG TRX model HBx expression had no effect on relative cell numbers after Veliparib exposure (24h/10µM) but cell growth was significantly reduced after Talazoparib (24h/50nM) (Figure 2B) or IR (4Gy X-rays) exposure. The combination of Talazoparib and IR significantly attenuated relative cell growth compared to radiation exposure alone. These response profiles probably reflect both the higher efficiency of PARP1 trapping by Talazoparib and the involvement of the SMC5/6 complex in the removal of such complexes in replicating cells. DNA damage levels were next assessed in the HBx-expressing cells after X-ray exposure (4Gy) using the formation and persistence of gamma-H2AX foci as a surrogate DNA DSB marker. A rapid increase in gammaH2AX foci/cell was seen after radiation exposure with a subsequent decline in foci numbers in the HepG2 parental and HepG2 K6 cells starting 30min/1h post-irradiation. In contrast, in HepG2 2.2.15 foci levels remained at the high levels seen initially after treatment over the first 2h post-irradiation after which a decline in numbers was noted. However, foci numbers remained substantially higher than in HepG2 and HepG2 K6 cells at later time points studied suggestive of altered DNA repair capacity (Figure 2C). Similarly, in the HepaRG model a delayed repair of DNA damage was observed in the HBx-expressing cells with higher numbers of foci persisting until 4h post-irradiation after which a decline in numbers was observed (Figure 2D) in support of the persistence of higher levels of DNA damage in the presence of HBx and associated with SMC5/6 complex loss.

HBx viral protein affects survival under replicative stress

As the SMC5/6 complex is also involved in a late step of cellular DNA replication, the sensitivity of HBx-expressing cells to HU induced cell killing was next assessed. This potent inhibitor of ribonucleotide reductase depletes the cellular dNTP pool resulting in stalled replication forks that, after a prolonged HU treatment (24h/2mM), collapse into DSBs resulting in S-phase cell cycle checkpoint activation. Under these exposure conditions HBx-expressing HepG2 2.2.15 cells had a significantly lower survival than either the parental or HepG2 K6 cells (Figure 3Ai). A significant reduction in relative cell growth was also seen in HBx-expressing HepaRG cells (Figure 3Bi).

The presence of blocked replication forks and the generation of DSBs caused by HU treatment will activate the intra-S cell cycle checkpoint to block the exit from S-phase of cells allowing time for DNA repair to occur. Whilst no HBx-dependent difference in the proportion of cells in S-phase were seen after HU treatment in either cell model (Figure 3Aii/Bii), HBx-expressing cells had higher levels of gammaH2AX 24h after HU treatment indicative of higher persisting levels of DNA DSBs (Figure 3Aiii/Biii). These profiles would suggest that the sensitivity to the cell killing effects of HU associated with HBx expression was due to an impairment in the processing of DNA strand breaks and not a defect in cell cycle checkpoint activation.

Is the HBx-associated cell sensitivity due to the degradation of SMC6 or other transcriptional changes?

As a first approach to address whether the sensitivity to IR, HU and PARPi in HBx-expressing cells is solely due to SMC5/6 complex degradation or whether HBx modulates other DNA Damage response (DDR) components that in-turn impacts on sensitivity to these agents we compared the transcriptome of proliferating HepaRG tetracycline-induced HBx cells vs cells depleted in SMC6 using an shtargeting approach. Most differentially expressed genes were unique to each of these two conditions with only 24 total overlapping genes (Supplementary Figure 2). Of note no significant changes in the transcript levels of the PARP transcripts (PARP1, PARP2 and PARP3) or PARG were noted when either HBx was induced or SMC6 depleted.

Next, we focused on those genes commonly up-or down-regulated by HBx induction and SMC6 depletion, with the assumption that these genes are surrogate markers of SMC6 loss under HBx expression. Ten genes displayed such behaviour (Table 1 and Supplementary Table 1). Three were significantly down-regulated (PARPBP (also known as PARI), PBLD and LRRCC1) of which PARPBP has a potential DDR role and thus could contribute to the increased sensitivity seen in HBx-expressing cells to the panel of DNA damaging agents tested. Seven genes showed an up-regulation (SRPK2, YAF2, PRR19, HSPB3, CA12, SP3 and SRPRA).

In addition to these overlapping genes, 27 genes were down-and 33 up-regulated when HBx was expressed and 89 genes were down-and 31 up-regulated when SMC6 was depleted (Supplementary Figure 2, Supplementary Tables 2 and3). KCNV1 and ZNF showed alternative splicing when HBx was expressed. DDR genes did not appear to be over-represented in these panels. For instance, no changes in BRCA1 or RAD51 transcript levels that can influence PARP inhibitor sensitivity [START_REF] Murai | Trapping of PARP1 and PARP2 by Clinical PARP Inhibitors[END_REF] were noted. However, BABAM2 and RAD52 expression, both encoding genes involved in DSB repair, were increased in the presence of HBx. This might be a compensation mechanism for the loss of SMC6 and clearly needs further investigation.

Profile of PARP gene expression and DNA damage levels in HCC tissues

To determine the expression profile of PARP genes in HCC tissues, we assessed PARP1, PARP2 and PARP3 transcript levels together with those of PARG and MKI67 in PT and matched T tissues from a cohort of individuals with HCV-, HBV-, NASH-and alcohol-associated HCCs with or without cirrhosis (cohort characteristics are detailed in Supplementary Table 4) compared to a panel of control liver tissues. A significant increase in all the transcript studied was observed in T compared to PT tissues (Figure 4A). Cirrhosis in PT tissues was associated with significantly higher transcript levels of PARG, PARP2, PARP3 and MKI67 compared to non-cirrhotic PT tissues but only for PARP2 were levels significantly higher than control tissues (Figure 4B). Significantly higher levels of all the transcripts were found in T tissues compared to PT or control tissues irrespective of cirrhosis status.

In terms of aetiology-specific transcript profiles, only PT tissues from HCV-associated HCCs had significantly higher levels of PARP2 and MKI67. This difference may in part reflect that HCVassociated PT tissues had the highest proportion of tissues with cirrhosis. For all 4 aetiologies PARP1, PARP2 and MKI67 transcript levels were significantly higher in T than corresponding PT tissues, with increases in PARG seen in T compared to PT tissues from HCV-, HBV-and alcoholassociated HCCs and for PARP3 in HBV-and alcohol-associated HCCs (Figure 4C).

Protein extracts available from a subset of HCV, HBV and alcohol-associated HCC tissues were used to assess the protein expression levels of PARP1, MKI67, gammaH2AX and H2AX. RPPA validated antibodies were not available for the other genes investigated. First, we assessed if PARP1 or MKI67 mRNA and protein levels were correlated. Weak but significant correlations for PARP1 (r=0.31, P=0.0387), and MKI67 (r=0.45, P=0.0005) mRNA/protein levels (Supplementary Figure 3A) were found in T tissues but not in control or PT tissues (data not presented). Next, we assessed the variation in protein levels during disease progression. A non-supervised hierarchical clustering of the results (Supplementary Figure 3B) suggests the presence of at least two major subgroups of samples, cluster A being enriched in T samples, while cluster B contains more PT and controls (Chi-square test P=0.0003), suggesting that T and PT samples differ in their expression levels for the measured proteins. Yet, despite higher transcript levels in T tissues compared to control and PT tissues, no significant increases in protein levels were found for PARP1 and MKI67 (P=0.382, P=0.078 respectively) (Figure 5A). However, lower H2AX levels were associated with disease progression from control to PT to T tissues, whereas gammaH2AX levels increased, resulting in a significant increase in the ratio of gamma-H2AX/H2AX (Figure 5B). Gamma-H2AX is an integrated marker of DNA damage and repair levels and an extremely sensitive organismal stress indicator [START_REF] Bonner | γH2AX and cancer[END_REF]. Increasing gamma-H2AX levels during HCC tumorigenesis are thus indicative of increased stress and DNA damage and might contribute to the vulnerability to PARP inhibition.

Discussion

This study has shown that in vitro HCC models are sensitive to the cell killing effects of PARPi when used as a single agent and that PARPi significantly attenuated radiation sensitivity in all the liver models tested. Our data is also in agreement with the literature that Talazoparib is a more potent PARPi than Veliparib in terms of induced cytotoxicity [START_REF] Murai | Trapping of PARP1 and PARP2 by Clinical PARP Inhibitors[END_REF]. Whilst the cytotoxic activities of PARPi were initially related to their ability to block DNA single strand break repair, it is now recognized that additional cytotoxic mechanisms exist including the generation of PARP-DNA complexes which block replication [START_REF] Murai | Trapping of PARP1 and PARP2 by Clinical PARP Inhibitors[END_REF]. Indeed, stalled replication forks are a major source of genome instability in proliferating cells and strategies that exacerbate replication stress are a promising avenue to improve anticancer therapies (see [START_REF] Liao | Mechanisms for stalled replication fork stabilization: new targets for synthetic lethality strategies in cancer treatments[END_REF]). The presence of the HBx protein increased this cytotoxicity which can in part be explained by the increased replicative stress concomitant to the HBx-induced loss of the SMC5/6 complex. Of the genes modulated in HBx-induced and SMC5/6 depleted cells, whilst no changes in the PARP transcripts studied were noted, the reduction in PARPBP transcript levels may contribute to these increases. Indeed, PARPBP depletion has been shown to enhance replication stress and DNA-damage accumulation [START_REF] Nicolae | PARI (PARPBP) suppresses replication stress-induced myeloid differentiation in leukemia cells[END_REF] which is concordant with the in vitro cellular phenotype of the HBx-expressing cells. It has been recently reported that, based on The Cancer Genome Atlas (TCGA) and Human Protein Atlas databases, PARPBP was significantly upregulated in HCC tissues compared with normal liver tissues (P ≤ 0.05) and correlated with worse overall survival and recurrence-free survival [START_REF] Yu | Overexpression of PARPBP Correlates with Tumor Progression and Poor Prognosis in Hepatocellular Carcinoma[END_REF]. Of the other genes down-regulated in HBx-expressing/SMC6 depleted cells PBLD was reported down-regulated in HCC tissues [START_REF] Long | Glutamine synthetase as an early marker for hepatocellular carcinoma based on proteomic analysis of resected small hepatocellular carcinomas[END_REF] , [START_REF] Li | Decreased expression of PBLD correlates with poor prognosis and functions as a tumor suppressor in human hepatocellular carcinoma[END_REF], suggesting that it's expression can be modulated by HBx-independent mechanisms, with this deletion found to be an independent poor prognosis predictor in HCC patients [START_REF] Li | Decreased expression of PBLD correlates with poor prognosis and functions as a tumor suppressor in human hepatocellular carcinoma[END_REF].

In addition to transcriptional modulation, it cannot be excluded that the effect of HBx on in vitro cell survival and DNA damage levels may be related to protein/protein interactions with a partner implicated in DDR pathways as seen for the HBx-induced loss of the SMC5/6 complex. Whilst a comprehensive proteomics evaluation of the impact of HBx expression was outside the scope of this present study, altered DDR protein expression have been reported in HBx-expressing models that potentially could impact on cell survival. For instance, in HBx-transgenic mice decreased expression of Mre11 and Rad51 expression has been noted [START_REF] Ahodantin | Hepatitis B virus X protein promotes DNA damage propagation through disruption of liver polyploidization and enhances hepatocellular carcinoma initiation[END_REF]. HBx can also bind the tumour suppressor protein p53 that can modulate both DNA strand break repair pathways [START_REF] Menon | Involvement of p53 in the repair of DNA double strand breaks: multifaceted Roles of p53 in homologous recombination repair (HRR) and non-homologous end joining (NHEJ)[END_REF] and nucleotide excision repair [START_REF] Becker | Hepatitis B virus X protein interferes with cellular DNA repair[END_REF].

Gene expression analysis carries the potential to dissect the molecular heterogeneity of cancers and can distinguish on this basis sub-classes for certain cancer types, including HCC [START_REF] Calderaro | Molecular and histological correlations in liver cancer[END_REF]. Unfortunately, this increasing knowledge has not yet resulted in biomarker discovery. Thus, improved clinical care and identifying the most relevant characteristics of HCC and personalised medicine remains a major challenge. A comprehensive review of the expression profiles of 450 human DDR genes in different cancers [START_REF] Pearl | Therapeutic opportunities within the DNA damage response[END_REF] highlighted the therapeutic opportunities for targeting DDR, however, for HCC data was only available for 18 T samples. Subsequent bioinformatics analyses of a panel of 59 DNA repair genes in the HCC data from the TCGA to define a co-regulated gene cluster would suggest that DNA repair stratification could be useful for predicting prognosis and designing clinical trials for targeted therapy [START_REF] Lin | Prognostic value of DNA repair based stratification of hepatocellular carcinoma[END_REF]. A recent analysis of the response to 31 anticancer agents in a large panel of liver cancer cell lines whose protein, RNA and mutational signatures resembled that of the aggressive proliferation class of HCC, identified genetic alterations and gene expression patterns associated with response to these agents [START_REF] Caruso | Analysis of Liver Cancer Cell Lines Identifies Agents With Likely Efficacy Against Hepatocellular Carcinoma and Markers of Response[END_REF]. This information might be used to select patients for clinical trials but did not extend to DNA repair inhibitors such as PARPi that are proving promising for other cancer types. In addition, based on the impact of HBx observed here it would be of interest to determines its expression profile in HBV-associated PT and T tissues to determine whether it can impact on therapeutic responses.

There is some limited information on PARP family gene and protein expression profiles in HCC. In agreement with our data, increased PARP1 mRNA expression was reported based on the online TCGA HCC dataset [START_REF] Li | PARP-1 serves as a novel molecular marker for hepatocellular carcinoma in a Southern Chinese Zhuang population[END_REF] (and Supplementary Figure 4) and in Chinese populations [START_REF] Li | PARP-1 serves as a novel molecular marker for hepatocellular carcinoma in a Southern Chinese Zhuang population[END_REF][START_REF] Qi | The long noncoding RNA lncPARP1 contributes to progression of hepatocellular carcinoma through up-regulation of PARP1[END_REF] but not found in a Polish population [START_REF] Krupa | The Relationship Between Single-Nucleotide Polymorphisms, the Expression of DNA Damage Response Genes, and Hepatocellular Carcinoma in a Polish Population[END_REF]. In our study, no correlation was found between mRNA and protein levels and no increase in PARP1 protein levels was observed in T tissues relative to control and PT tissues. This is in contrast to the data of Li et al [START_REF] Li | PARP-1 serves as a novel molecular marker for hepatocellular carcinoma in a Southern Chinese Zhuang population[END_REF] and Xu et al [START_REF] Xu | Global proteomic profiling in multistep hepatocarcinogenesis and identification of PARP1 as a novel molecular marker in hepatocellular carcinoma[END_REF] who reported higher PARP1 protein levels in HCC T tissues. In agreement with our data PARP2 and PARG mRNA expression was elevated in T samples compared to the matched non-T samples in the on-line TCGA HCC dataset, however, no increases were seen for PARP3. The reasons for these discrepancies remain to be established but may be due to technical issues relating to antibody choice and tissues used for comparative purposes and reflect differences in the compositions of tissue panels and the impact of etiology on expression profiles. The elevated gammaH2AX seen in both this study and Evert et al [START_REF] Evert | Deregulation of DNA-dependent protein kinase catalytic subunit contributes to human hepatocarcinogenesis development and has a putative prognostic value[END_REF] are indicative of a higher DNA damage burden and/or replication stress that could be interpreted as an imbalance in DSB repair in HCC tissues and has been associated with cancer progression in many cancer types (see for example [START_REF] Bartkova | DNA damage response as a candidate anti-cancer barrier in early human tumorigenesis[END_REF]). Intriguingly this increase is accompanied here by a reduction in H2AX levels during progression from control to T tissue. These changes parallel observations by Grusso and colleagues in triple negative breast cancer patients where chronic oxidative stress promotes H2AX degradation [START_REF] Gruosso | Chronic oxidative stress promotes H2AX protein degradation and enhances chemosensitivity in breast cancer patients[END_REF]. HCC is considered an inflammation-associated cancer, thus it is tempting to speculate that oxidative stress may be the underlying cause of the reduction in H2AX levels associated with liver disease progression. Grusso and colleagues also noted that a ROS-mediated H2AX decrease was observed after chemotherapy and was an indicator of the therapeutic efficiency and survival in Triple Negative Breast Cancer patients [START_REF] Gruosso | Chronic oxidative stress promotes H2AX protein degradation and enhances chemosensitivity in breast cancer patients[END_REF]. Whether combining PARP inhibitors that would block repair, with radiotherapy that would not only generate oxidative stress and thus further reduce H2AX levels, but also cytotoxic DNA damage, would enhance HCC tumour cytotoxicity and thus be beneficial in a therapeutic setting or lead to increased genomic instability and its consequences would need to be investigated in clinical trials.

Conclusion

In conclusion, the in vitro potentiation of cell death by PARPi alone or in combination with radiation exposure, taken together with the observations of elevated DNA damage levels in HCC T tissues, may represent a vulnerability that can be exploited for therapeutic benefit. Indeed, whilst RT has not been widely used for HCC treatment, it is a choice that needs further clinical evaluation [2] [3]. In addition to their direct impact on the DDR response, PARPi may have additional therapeutic benefits. For instance, the combination of IR and PARP inhibition can improve the vasculature leading to the reoxygenation of hypoxic T tissues, thus bypassing hypoxia-induced radioresistance (see [START_REF] Guillot | PARP inhibition and the radiosensitizing effects of the PARP inhibitor ABT-888 in in vitrohepatocellular carcinoma models[END_REF] and references therein). Additionally, PARP inhibition can trigger the STING-dependent immune response and enhance the therapeutic efficacy of immune checkpoint blockade independent of BRCAness [START_REF] Shen | PARPi Triggers the STING-Dependent Immune Response and Enhances the Therapeutic Efficacy of Immune Checkpoint Blockade Independent of BRCAness[END_REF]. As the synergy between RT and the immune response is well documented the question arises if combining PARP inhibition, radiation, and immunotherapy could be a possible treatment strategy (see review [START_REF] Césaire | Combining PARP Inhibition, Radiation, and Immunotherapy: A Possible Strategy to Improve the Treatment of Cancer?[END_REF]) and in particular for HCC where there is a growing body of evidence for each individually as therapeutic options.

  180 HCC patients treated by surgical resection, included between 1996 and 2014 from three French clinical centres (Lyon Croix Rousse and Toulouse Hospitals and the French National Biological Resource Centres) were retrospectively studied (Agreement DC-2008-235). RNA samples were available from matched T and PT (at least 2 cm distant from the T) tissues for 150 patients and for the other 30 only T RNA samples were analysed. Protein extracts were available from 93 tissues (46 T, 47 PT) of the following aetiologies; 29 alcohol (13 T, 16 PT), 32 HBV (17 T, 15 PT) and 32 HCV (16 T,16 PT). RNA samples from 11 and protein extracts from 8 surgically resected normal liver tissues surrounding colorectal adenocarcinoma metastasis (Centre Léon Bérard Biological Resource Centre, ministerial agreements #AC-2013-1871 and DC-2013-1870) were used as control samples respectively. Signed informed written consents in compliance with the requirements of the local ethical committee were obtained from patients before surgery.

Fig. 1 .

 1 Fig.1. Clonogenic cell survival after PARP inhibition and/or IR exposure. Cell survival assessed by colony formation in (A) HepG2, (B) PLC/PRF/5 and (C) Hep3B cells after exposure to the PARP inhibitors Veliparib or Talazoparib (24h) and/or IR. (2Gy). Data represents the mean +/-SD of ≥ 3 independent experiments carried out in triplicate. *P ≤ 0.05, **P ≤ 0.01, ***P ≤ 0.001, ****P ≤ 0.0001, ns = non-significant. (Mann Witney U-test).

Fig. 2 .

 2 Fig. 2. Impact of HBx expression on cell survival and DNA damage levels. (A) Mean +/-SD survival in HepG2 2.2.15 (expressing HBx) or HepG2 K6 (no HBx) after Veliparib (24h) +/-IR (2Gy). (B) Mean +/-SD cell survival of HepaRG-TRX cells expressing HBx (+ tetracycline) or not after Veliparib or Talazoparib +/-IR. *P ≤ 0.05, **P ≤ 0.01, ***P ≤ 0.001, ****P ≤ 0.0001, ns = non-significant (Mann Witney U-test, data from ≥ 3 independent experiments). (C) GammaH2AX foci numbers after IR (4Gy) in HepG2-derived cells (one representative experiment, ≥ 200 cells/condition). (D) Mean +/-SD GammaH2AX foci numbers in HepaRG-TRX cells +/-HBx after IR (4Gy) (3 independent experiments, ≥ 200 cells/condition).

Fig. 3 .Fig. 4 .

 34 Fig. 3. Impact of HU on cell survival and cell cycle progression in HepG2 and HepaRG TRXderived cells. (Ai and Bi) cell survival (mean +/-SD, ≥ 3 independent experiments) after HU (2mM/24h) exposure. (Aii and Bii) % of cells in the S phase of the cell cycle 24h after treatment with HU (2mM) (mean +/-SD, ≥ 3 independent experiments). (Aiii and Biii) Level of gammaH2AX in cell protein extracts after exposure to HU (2mM/24h). One representative experiment shown. **P ≤ 0.01, ***P ≤ 0.001, ****P ≤ 0.0001, ns = non-significant (Mann-Whitney U-test.)

Fig. 5 .

 5 Fig. 5. Comparisons of protein levels in control, PT and T HCC tissues. (A) Comparison of normalised PARP1 and Ki67 protein levels and (B) normalised H2AX and gammaH2AX protein levels and their ratio in comparison in control (CON) (8), PT (47) and T tissues (46) from HCV, HBV and NASH-associated tumors. *P ≤ 0.05, **P ≤ 0.01, ***P ≤ 0.001, ****P ≤ 0.0001, ns = nonsignificant (Kruskal Wallis test).

Table 1 : Genes co-regulated by HBx expression and SMC6 loss †

 1 Genes that are co-regulated by HBX expression and SMC6 loss are shown in bold in the Venn diagram (Supplementary Figure2). ‡ log fold change in transcript expression in presence of HBx or loss of SMC6 (shSMC6), -represents a reduction in expression.

	Gene	Transcript	logFC Hbx ‡ adj.P.Value §	logFC shSMC6 ‡	adj.P.Value §
	PBLD	uc001jns.1	-10.03	9.15E-03	-8.78	1.59E-02
	PARPBP uc001tjk.3	-8.23	4.19E-04	-8.86	3.14E-04
	LRRCC1 uc010lzz.2	-2.76	4.79E-02	-8.50	1.35E-04
	HSPB3	uc003jph.2	5.84	5.89E-03	6.58	3.33E-03
	SRPRA	uc010sbm.2	6.02	4.32E-02	8.75	5.80E-03
	SP3	uc002uig.3	6.31	3.35E-02	8.09	8.33E-03
	CA12	uc002ame.3	6.99	8.15E-03	7.59	5.37E-03
	YAF2	uc010sko.2	7.89	7.68E-05	7.69	1.08E-04
	PRR19	uc002oth.1	8.05	7.43E-04	6.83	1.92E-03
	SRPK2	uc003vct.4	8.49	2.71E-05	10.10	2.62E-05
	†

§ P values are false discovery rate corrected.

Acknowledgments

The authors would like to thank Hien Luong Nguyen, Magali Jacquet, Loïc Peyrot and Lydie Lefrancois who contributed to early stages of this project, David Cox, Agnes Tissier and Virginie Petrilli (CRCL, Lyon) for helpful discussions, J. Jacquemetton (CRCL, Lyon) for the development of macros for counting foci and Adeline Granzotto (CRCL, Lyon) for organising the irradiation of cells.

The authors thank Pr. M. Rivoire (Centre Léon Bérard, Lyon) for providing human liver tissues.

This work was supported by funding from La Ligue Nationale Contre le Cancer, the ANRS (grant number 12357 to D.C.) and La Ligue Contre le Cancer Rhone Alpes.


Zaineb Smida
email: zaineb.smida@umontpellier.fr

Lionel Cucala
email: lionel.cucala@umontpellier.fr

Ali Gannoun
email: ali.gannoun@umontpellier.fr

Ghislain Durif
email: ghislain.durif@umontpellier.fr

A Wilcoxon-Mann-Whitney spatial scan statistic for functional data

Keywords: Cluster detection, Functional data, Hilbert space, Spatial Scan statistic, Wilcoxon-Mann-Whitney test. 2000 MSC: 46E20, 62G10, 62H11, 62R10

A nonparametric scan method for functional data indexed in space is introduced. The associated scan statistic is derived from the Wilcoxon-Mann-Whitney test statistic defined for infinite dimensional data. It is completely nonparametric as it does not assume any distribution concerning the functional marks. Whatever the clustering scenario, this scan test seems to be efficient to detect and locate the cluster. This method is applied to a data set for extracting features in Spanish province population growth. A significant spatial cluster of low demographic evolution rates is found, exhibiting a specific phenomenon in the North-West of Spain.

Introduction

Spatial cluster detection has become a fruitful area of statistics that has particulary expanded in recent decades. It is used to identify aggregations of events in a specific area, see [START_REF] Lawson | Spatial cluster modelling[END_REF] for a thorough review. One of the most popular cluster detection technique is the scan statistic which was firstly introduced by [START_REF] Naus | Clustering of random points in the line and plane[END_REF]. It was defined as the maximum number of events observed within a window with constant size, known as the scanning window, as it moves continuously over the studied region. Knowing the distributions of these scan statistics [START_REF] Alm | On the Distributions of Scan Statistics of a Two-Dimensional Poisson Process[END_REF] helps to decide whether exceptional or not observing a cluster of events. The field of spatial scan statistics was highly enhanced by the article written by [START_REF] Kulldorff | A spatial scan statistic[END_REF]: he proposed scanning the study area with variable size circular windows and selecting the most likely cluster as the one maximizing a likelihood ratio test. He used either Bernoulli or Poisson model and estimated the clusters' statistical significance via a Monte-Carlo procedure. These innovations gave birth to several works in which researchers adapted the spatial scan statistics to different types of data, using different probability models: exponential [START_REF] Huang | A spatial scan statistic for survival data[END_REF], normal [START_REF] Kulldorff | A scan statistic for continuous data based on the normal probability model[END_REF], multivariate Gaussian [START_REF] Cucala | A Multivariate Gaussian scan statistic for spatial data[END_REF], etc. All these spatial scan methods are developed for univariate and multivariate data indexed in space. However, the development of sensoring and computing tools brings more and more access to data of functional type coming from various fields of applications such as environmetrics, biometrics, medecine and econometrics [START_REF] Ramsay | Functional Data Analysis[END_REF]. These data are not real random variables or vectors but they are a sample of random curves where each element is considered as a function. Moreover, these functional data are often indexed in space [START_REF] Delicado | Statistics for spatial functional data: some recent contributions[END_REF] and, even if a few studies have been conducted on modelling [START_REF] Cronie | Functional marked point processes -A natural structure to unify spatio-temporal frameworks and to analyse dependent functional data[END_REF] or clustering [START_REF] Gaetan | Spatial clustering of curves with an application of satellite data[END_REF] such data, to our knowledge, there is no spatial cluster detection method designed for this kind of data yet. In the present work, we develop a scan statistic for functional data indexed in space: thanks to this statistic, we are able to detect spatial clusters in which the observations of a functional random variable are different than elsewhere, and also to compute the significance of these differences. Since no likelihood is associated with functional random variables [START_REF] Ferraty | Recent advances in functional data analysis and related topics[END_REF], maximising a likelihood ratio test is not possible here. Thus, we follow the idea by [START_REF] Cucala | Variable Window Scan Statistics: Alternatives to Generalized Likelihood Ratio Tests[END_REF] that any test for equality of two distributions can give birth to a scan statistic. The rest of this paper is organized as follows. In section 2, we build a nonparametric spatial scan statistic for functional data based on the Wilcoxon-Mann-Whitney statistic proposed by [START_REF] Chakraborty | A Wilcoxon-Mann-Whitney type test for infinite-dimensional data[END_REF] and we evaluate its statistical significance using random permutations. In section 3, first, the spatial scan statistic is compared to other methods on simulated datasets. Then, we apply it to a real dataset illustrating the demographic evolution over time in Spanish provinces and we exhibit a specific behaviour in the North-West of Spain in the last twenty-two years. We conclude with a discussion and a brief scope for future work.

2. A nonparametric spatial scan statistic for functional data

Introducing the statistic

Consider a random variable X taking values in a functional space χ. For sake of simplicity, we will suppose that χ is an Hilbert space such as L 2 ([0, 1], R). Let X 1 , . . . , X n be observations of X at n different spatial locations s 1 , . . . , s n included in D ⊂ R 2 . Following the terminology of point process theory, D is the observation domain and X i is the mark associated with location s i , for all i = 1, . . . , n. Our goal is to detect a cluster of unusual marks, i.e. a spatial zone Z ⊂ D in which the functional marks exhibit a different behaviour than elsewhere. In order to do that, we aim to set up a scan statistic, which is usually defined as the maximum of a concentration index observed in a collection of variable size potential clusters [START_REF] Nagarwalla | A scan statistic with a variable window[END_REF]. Concerning the potential clusters, two main possibilities have been proposed in the literature. In the first one, the windows have known geometric shapes: rectangular [START_REF] Loader | Large-deviation approximations to the distribution of scan statistics[END_REF][START_REF] Chen | Approximations for Two-Dimensional Variable Window Scan Statistics[END_REF], circular [START_REF] Kulldorff | Spatial disease clusters: detection and inference[END_REF][START_REF] Kulldorff | A spatial scan statistic[END_REF], elliptic [START_REF] Kulldorff | Tests of spatial randomness adjusted for an inhomogeneity[END_REF] or any other shape. In the second one, the windows have irregular shapes and the procedure to identify them is based only on pairwise distances [START_REF] Demattei | Arbitrarily shaped multiple spatial cluster detection for case event data[END_REF][START_REF] Assunção | Fast detection of arbitrarily shaped disease clusters[END_REF][START_REF] Duczmal | A simulated annealing strategy for the detection of arbitrarily shaped spatial clusters[END_REF]. In this work, without loss of generality, we consider the circular clusters introduced by [START_REF] Kulldorff | A spatial scan statistic[END_REF]. Hence, the set of potential clusters S is defined as follows:

S = {D i,j , 1 ≤ i ≤ n, 1 ≤ j ≤ n},
where D i,j is the disc centred on s i and passing through s j . We remark that, since i might be equal to j, the number of potential clusters is n 2 . Following the initial work by [START_REF] Kulldorff | A spatial scan statistic[END_REF], the spatial scan statistics designed for univariate or multivariate marks are most often based on a concentration index derived from a likelihood ratio. This likelihood ratio relies on assuming a specific probability distribution for the marks and testing the null hypothesis H 0 (absence of a cluster) against an alternative one H 1,Z (presence of a cluster in Z) for every potential cluster Z ∈ S. However, for functional random variables, even if approximations have been proposed [START_REF] Jacques | Funclust: A curves clustering method using functional random variables density approximation[END_REF], the notion of probability density generally does not exist. Thus, our clustering index will rely on a nonparametric test for equality of distributions, as proposed by [START_REF] Jung | A nonparametric spatial scan statistic for continuous data[END_REF] and [START_REF] Cucala | A Multivariate nonparametric scan statistic for spatial data[END_REF] in the univariate and multivariate settings respectively. Hereinafter, we suppose that X 1 , . . . , X n are independent observations of the functional random variable X (this is a classical assumption in scan statistics). Let Z ∈ S be any potential cluster of size n Z , where n Z = ∑ n i=1 1(s i ∈ Z) and Z c its complement of size n Z c = n -n Z . Assume that the marks in Z and Z c respectively follow probability measures P Z and P Z c on χ. We suppose that P Z and P Z c differ by a shift ∆ Z ∈ χ. For testing the hypothesis H 0 : ∆ Z = 0 (equality of distributions) against H 1,Z : ∆ Z ̸ = 0, a Wilcoxon-Mann-Whitney test statistic in such space is defined by [START_REF] Chakraborty | A Wilcoxon-Mann-Whitney type test for infinite-dimensional data[END_REF] as:

T WMW (Z) = 1 n Z n Z c ∑ {i:s i ∈Z} ∑ {j:s j ∈Z c } X j -X i ∥X j -X i ∥ χ ,
where ∥.∥ χ stands for a norm on χ. Remark that this statistic is a natural extension to the functional setting of the well-known statistics introduced by [START_REF] Wilcoxon | Individual comparaisons by ranking methods[END_REF] and [START_REF] Mann | On a test of whether one of two random variables is stochastically larger than the order[END_REF] in the univariate setting, since every element of the first sample is compared to every element of the second one. The statistic T WMW (Z), taking values in χ, cannot be used directly as a concentration index since its distribution highly depends on n Z , the size of the potential cluster Z. Thus, as recomended by [START_REF] Cucala | Variable Window Scan Statistics: Alternatives to Generalized Likelihood Ratio Tests[END_REF], we introduce the standardized concentration index

U(Z) := √ n z n Z c n T WMW (Z)
which is designed to compare potential clusters having different population sizes, as claimed by the following lemma.

Lemma 1. The null limiting distribution of U (Z) is the same for any potential cluster Z ∈ S.

The proof comes directly from the convergence theorem by [START_REF] Chakraborty | A Wilcoxon-Mann-Whitney type test for infinite-dimensional data[END_REF] stating that, under

H 0 , if n Z /n → γ ∈ [0, 1] as n Z , n Z c → ∞, (n Z n Z c /n) 1/2 T WMW (Z) converges weakly to G(0, Γ), (1)
where G(m, C) is the distribution of a Gaussian random element in χ with mean m ∈ χ and covariance C. Since the covariance operator Γ does not depend on n Z and n Z c , the result holds. Thus, the scan statistic can be defined as the maximum of the norm of this concentration index on the set of potential clusters S which has been previously defined. The Wilcoxon-Mann-Whitney functional scan statistic (WMWFSS) is

Λ WMWFSS = max Z∈S ∥U(Z)∥ χ
and the potential cluster detected, for which Λ WMWFSS is obtained, is

C ˆ= arg max Z∈S ∥U(Z)∥ χ .
This latter is called the most likely cluster.

Computing the scan statistic

• The computation of the scan statistic Λ WMWFSS involves the computation of the concentration index U (Z) for every potential cluster Z ∈ S and, since this index is issued from a sum of n Z × n Z c terms, a naive computation can be very time-consuming. However, here are two computational tricks to address this problem:

all concentration indices U (Z), for every Z ∈ S, rely on the computation of

R i,j = X j -X i ∥X j -X i ∥ χ
for every 1 ≤ i < j ≤ n. Thus, these (n-1) 2 /2 terms must be calculated at the very beginning of the process and stored. Remark that the computation of the R i,j 's will be different whether the functional marks X 1 , . . . , X n are known explicitly (for example by their decomposition on a certain basis) or only partially observed. See [START_REF] Ramsay | Functional Data Analysis[END_REF] for more details.

-In order to optimize the computation process, we decided to calculate the indices U (Z) in a very specific order. Here is an example: let Z and Z ′ be any potential clusters such that Z ′ = Z ∪ s k .

Then, the concentration index for Z ′ can be obtained from

(n Z ′ n Z ′c n) 1/2 U(Z ′ ) = ∑ {i:s i ∈Z ′ } ∑ {j:s j ∈Z ′c } X j -X i ∥X j -X i ∥ χ = ∑ {i:s i ∈Z} ∑ {j:s j ∈Z c } X j -X i ∥X j -X i ∥ χ + ∑ {j:s j ∈Z c } X j -X k ∥X j -X k ∥ χ - ∑ {i:s i ∈Z} X k -X i ∥X k -X i ∥ χ = (n Z n Z c n) 1/2 U(Z) + ∑ {j:s j ∈Z c } X j -X k ∥X j -X k ∥ χ - ∑ {i:s i ∈Z} X k -X i ∥X k -X i ∥ χ
This set-up requires to iterate over only n elements instead of (n Z -1) × (n -n Z + 1) and dramatically decreases the computational cost.

• When χ = L 2 ([a, b], R) where a, b ∈ R and a < b, the algorithm used to derive the WMWFSS and its associated most likely cluster C ˆis as follows:

Algorithm 1 Computing the WMWFSS and the most likely cluster MLC 1: Data: {(s 1 , X 1 ), . . . , (s n , X n )} 2: For all i, j ∈ {1, . . . , n} compute

R i,j = X j -X i ∥X j -X i ∥ L 2
and let R = {R i,j } i,j∈{1,...,n} .

3: For all i, j ∈ {1, . . . , n} compute the distance d i,j between locations s i and s j and let d = {d i,j } i,j∈{1,...,n} . 4: function TWMW (computing the WMW test statistic) 5: Initialization: c ˜= -∞, i ˜= 0 and j ˜= 0.

Input: R, A ⊆ {1, • • • , n}, B ⊆ {1, • • • , n}

21:

for i = 1 to n do

22:

O = ORDER({d i,j } j∈{1,...,n} )

23:

for j = 1 to (n -1) do 24:

v in = {O k } k∈{1,...,j} and v out = {O k } k∈{j+1,...,n} 25: c = √ j(n-j) n × ∥TWMW(R, v in , v out )∥ L 2 26: if c > c ˜then 27: c ˜= c, i ˜= i, j ˜= j 28:
O ˜= ORDER({d i ˜,j } j∈{1,...,n} )

29:

MLC = {O ˜k} k∈{1,...,j ˜}

Computing the statistical significance

After computing the scan statistic Λ WMWFSS and the most likely cluster C ˆ, it is necessary to evaluate its significance. However, the distribution, under H 0 , of a variable window scan statistic has no analytical form. To overcome this problem, [START_REF] Dwass | Modified randomization tests for nonparametric hypotheses[END_REF] proposed a test procedure based on Monte-Carlo simulations allowing to give an approximation of the null distribution. This method was subsequently extended by Bernard (1963) and [START_REF] Hope | A simplified monte carlo significance test procedure[END_REF]. It relies on comparing the observed scan statistic to scan statistics issued from datasets simulated under H 0 . Here, since no assumption is made on the distribution of the functional marks, the only way to obtain such datasets is by running a method called random labelling [START_REF] Cucala | A distribution-free spatial scan statistic for marked point processes[END_REF]: a simulated dataset is obtained by randomly associating the functional marks X i to the spatial locations s i . Based on T random permutations, let

Λ (1) WMWFSS , . . . , Λ (T ) WMWFSS
be the observations of the scan statistics associated with the simulated datasets. Then, as stated by [START_REF] Dwass | Modified randomization tests for nonparametric hypotheses[END_REF], the p-value of the scan statistic Λ WMWFSS , observed in the initial sample, is given by

p value = 1 + T ∑ i=1 1 {Λ (i) WMWFSS >Λ WMWFSS } T + 1 .
Of course, the larger the number of permutations T , the better the estimation of the p-value of the scan statistic. However, since the computational cost cannot be neglected, one needs to find a trade-off between the two aspects. The most likely cluster C ˆis said to be significant if p value is less than the type I error α.

Applications

Simulation study

We decided to run a simulation study to evaluate the performance of the functional scan statistic Λ WMWFSS proposed in the previous section. We generated artificial datasets using the geographic locations of the administrative centers of the 94 french administrative areas named as départements. The simulated true cluster, denoted by C, is defined as a set of départements in the Parisian area according to two configurations: (i) 8 départements and (ii) 10 départements. Maps of the simulated clusters are given in Fig. 1. The functional marks associated with each location take values in χ = L 2 ([0, 1], R) and are defined as follows:

∀i = 1, . . . , 94, X i (t) = ∞ ∑ k=1 Z i,k e k (t) + ∆(t)1 {s i ∈C} ,
where for all k ⩾ 1, e k (t) = √ 2 sin(t/σ k ) is an orthonormal basis of χ, σ k = ((k -0.5)π) -1 and Z i,k 's are independent random variables which correspond to the projection of X i on the Karhunen-Loève basis [START_REF] Karhunen | Uber lineare methoden in der wahrscheinlichkeitsrechnung[END_REF][START_REF] Lévy | Processus stochastiques et mouvement brownien[END_REF]. The decomposition of the functional marks above is based in the Karhunen-Loève expansion which is widely used in several issues related to image processing and functional data analysis [START_REF] Ahmed | Binary Functional Linear Models under Choice-Based Sampling[END_REF].

We have investigated two different cases, namely a standard Brownian motion (sBm) process: Z i,k /σ k having a N (0, 1) distribution and a centered Student-t process with five degrees of freedom: Z i,k /σ k having a t(5) distribution. The probability measures of the functional marks inside and outside the cluster C differ by a shift ∆. Three types of shifts are studied: and∆ 3 (t) = c sin(2πt), c > 0 for all t ∈ [0, 1]. The parameter c is called the cluster intensity: remark that, since the functional marks are independent, this parameter totally controls their level of spatial heterogeneity. Different values of this parameter were considered for each ∆. The range of ∆ 2 being smaller than the ranges of ∆ 1 and ∆ 3 , it is combined with larger values of c.

∆ 1 (t) = ct, ∆ 2 (t) = ct(1 -t)
Since, as already said in the Introduction, we do not know any other cluster detection method dedicated to functional data indexed in space, we decided to compare the Wilcoxon-Mann-Whitney functional scan statistic to two univariate scan statistics applied to summaries of the functional marks:

• the first scan method relies on the mean values of the marks

X ¯i = ∫ 1 0 X i (t)dt, i = 1, . . . , n.
Each mean value is associated with its location and the univariate Wilcoxon-Mann-Whitney scan statistic introduced by Cucala ( 2016) is computed, using the same set of potential clusters and same random permutations than the functional one. This mean-based univariate scan statistic is denoted by Λ MBUSS .

• the second one, inspired from the LISA function defined by [START_REF] Mateu | Detecting Features in Spatial Point Processes with Clutter via Local Indicators of Spatial Association[END_REF], relies on the deviations from the mean function of the marks

D i = ∫ 1 0 (X i (t) -X ¯(t)) 2 dt, i = 1, . . . , n,
where

X ¯(t) = 1 n n ∑ i=1 X i (t)
is the mean function of the observed functional marks. Each deviation is associated with its location and the univariate Wilcoxon-Mann-Whitney scan statistic is computed. This deviation-based univariate scan statistic is denoted by Λ DBUSS .

To compare the three scan methods, we generated 100 simulated datasets for each distribution of the marks and each value of the cluster intensity c and we computed three distinct criteria for each method: the alarm rate (AR), the True Positive (TP) rate (also called the sensitivity) and the False Positive (FP) rate. These three criteria were calculated as follows:

-The alarm rate (AR) was defined as the proportion of datasets exhibiting a significant cluster with a type I error equal to 0.05 and based on T = 99 random permutations.

-The TP rate, denoted by %TP, was defined as the mean proportion of the True Positive (TP) départements over all simulated datasets. It was calculated as the number of départements included both in the significant cluster C ˆand in the true cluster C divided by the number of départements included in C.

-The calculation of the FP rate, denoted by %FP, is similar to the TP one. It was defined as the average proportion of the False Positive (FP) départements i.e, the number of départements included in the most significant cluster C ˆbut not in the true cluster C divided by the number of départements not included in C.

The whole results of this simulation study are given in Appendix B but they are summarized in Table 1 and Table 2 below. From Table 1 andTable 2, the sizes of the different methods (i.e. the alarm rates when c=0) are close to the correct type I error which is equal to 0.05, regardless of the distribution of the marks. As expected, the performances of all scan statistics tend to increase with high cluster intensity c and we can remark that the alarm rate of Λ WMWFSS is higher than Λ MBUSS and Λ DBUSS in all different cases: this is expected as the first one relies on the whole information of the curves, the second one is only based on their mean value and the third one is derived from the distances between each curve and the mean curve. It should be noted that:

-When c increases, the alarm rate of all scan methods increases whatever the shift ∆ and the size of the true cluster C. However, when the process is Student-t distributed, the alarm rate increases more slowly than when it is normally distributed. This difference can be explained by the fact that the Student-t distribution is more heavy-tailed than the Gaussian one. The relation between the alarm rate and the cluster intensity c seems to be the following: the alarm rate slowly increases when c is small but then, when c reaches a certain threshold, the slope gets steeper and the alarm rate very quickly gets close to 1. Since this threshold is different depending on the distribution of the functional marks, the discrepancy between the alarm rates of Normal and Studentt distributions is far from being constant. Remark also that, for equal values of the cluster intensity c, the alarm rate is larger when the size of the cluster goes from 8 to 10: it is always easier to detect a larger cluster.

The difference in alarm rates between Λ WMWFSS and Λ MBUSS is slight when the shift between the marks inside and outside the cluster is linear, but it increases when this shift is quadratic and moreover when it is sinusoidal (see Table B.7 and Table B.10 in Appendix B): we can see that the alarm rate of Λ MBUSS does not exceed 10% (is close to the nominal level 5%) using ∆ 3 whatever the size of the true cluster and the distribution of the processes, since the sinusoidal shift has absolutely no consequence on the mean value of the process. The deviation-based scan statistic Λ DBUSS is more adapted to this sinusoidal shift but still displays lower alarm rates than Λ WMWFSS .

-The true positive and false positive rates also improve when the cluster intensity c increases (increasing for %TP and decreasing for %FP). As for the alarm rate, the recovering of the location of the cluster is harder when the process is Student-t distributed than normally distributed but the size of the cluster has no great impact on %TP and %FP. The whole information included in the functional marks is as useful for detecting the presence of a cluster than for recovering its exact location. Thus, unsurprisingly, the %TP and %FP rates obtained by the functional method Λ WMWFSS are globally better than the ones obtained by the univariate methods. The difference is more obvious concerning the false positive rates: more often than the functional one, the univariate methods tend to exhibit clusters larger than the true cluster C.

Application to real data

Here, we give an example of the use of our scan statistic to extract features in Spanish province population growth, as presented by [START_REF] Cronie | Functional marked point processes -A natural structure to unify spatio-temporal frameworks and to analyse dependent functional data[END_REF]. In order to study the structure of the Spanish population, we considered one of the most important population characteristics which is the demographic evolution. This latter can change over time because of factors like birth and death rates, immigration rate or economical situations. The Spanish province population is provided by the Spanish Institute of Statistics (www.ine.es) and the boundary and centre coordinates of the 47 provinces of Spain (see Fig. 2) by the R package raster [START_REF] Hijmans | raster: Geographic data analysis and modelling[END_REF]. For geographical reasons, we decided to exclude from the study Baleares and Canarias islands as well as the Spanish autonomous cities (Melilla and Ceuta) which are located on the Northwest coast of Africa and sharing a border with Morocco. To each point (centre) i, for i = 1, . . . , 47, we associated the functional mark X i , i.e. the demographic evolution over time, for 22 distinct years starting from 1998 to 2019 (see Fig. 3). The demographic evolution in each province was defined as the total population over the years 1998 to 2019 divided by the total population in 1998.

Analysis of the real dataset

Our objective here is to detect a spatial area where the demographic evolution would be significantly different. In order to identify such a cluster, we computed the functional scan statistic on this dataset: Λ WMWFSS = 2.72025. Remark that here the computation of the scan statistic is slightly different from what is done in the simulation study since it is estimated from 22 observation points. Based on T = 999 permutations, the value of the statistic is highly significant (p value = 0.001) and the most likely cluster C ˆis plotted in Fig. 4. This cluster includes 13 provinces in the west of Spain (Asturias, Galicia, Extremadura and the west of Castilla y León) in which the marks are significantly lower than in the rest of the observation domain. In the west part of Castilla y Leòn, the most likely cluster includes the región leonesa and the west of the Castilla la Vieja (Ávila, Palencia and Valladolid ). We can see the demographic evolution curves associated with the most likely cluster in Fig. 5. We can see that this cluster includes the provinces which have the lowest demographic evolution compared to the rest of Spain. This can be explained by the increase in mortality rate and the decrease in birth rate in these regions. Between years 2006 and 2018, according to the Spanish Institute of Statistics, the 4 autonomous communities detected in the cluster are the territories which have the lowest birth rates (per 1000 inhabitants) compared to the other autonomous communities in Spain. In particular, the last 2 provinces with the lowest birth rate (per 1000 inhabitants) are Ourense (6.12 in 2006 and 4.82 in 2018) and Zamora (6.08 in 2006 and 5.13 in 2018). Moreover, the mortality rate (per 1000 inhabitants) is higher in the provinces belonging to the detected cluster and in particular Zamora has the highest mortality rate (12.46 in 2006 and 15.75 in 2018). This explains why Zamora has the lowest evolution demographic (see Fig. 5) and is close to becoming a demographic desert. Such a demographic decrease can be explained by the emigration in the last years of the youngest population abroad and to other regions of Spain like Cataluña and Madrid where the average hourly wage is higher and the unemployment rate is lower than the autonomous communities detected by the functional scan statistic (for more details, see the website of the Spanish Institute of Statistics).

Then, using the same dataset, we also computed the univariate scan statistics Λ MBUSS and Λ DBUSS and their p-values and we recorded the computation time. The results are given in Table 3. Contrary to the deviation-based univariate scan statistic, the mean-based univariate scan statistic detects a very significant cluster which is very similar to the one detected by Λ WMWFSS (see Fig. 6). This is not surprising since the main difference between the curves inside this cluster and the curves outside is their mean level rather than their shape. Moreover, the cluster detected by Λ MBUSS is larger than the one detected by Λ WMWFSS . As said in the simulation study, contrary to the functional scan method, the univariate scan methods tend to exhibit larger clusters than the true one, as noticed by the %FP rate. Thus, we believe that the most likely cluster detected by our functional scan method, based on the analysis of the curves on the whole time period, should be investigated first. Concerning the computation time, we remark that the functional scan statistic, even if it takes advantage of the whole information of the data, is not that time-consuming compared to the univariate ones. This performance was achieved thanks to the use of the fonction NPFSS from the R package HDSpatialScan introduced very recently by [START_REF] Frévent | HDSpatialScan: Multivariate and Functional Spatial Scan Statistics[END_REF].

Finally, after identifying the most likely cluster, we have tested the presence of a secondary cluster, following the method by Zhang et al. (2010): once a significant cluster is found, remove the data included in that cluster and restart the analysis. However, on this dataset and using the functional scan statistic, the secondary cluster is not significant since its p-value equals 0.282, using T = 999 permutations.

Analysis of the sensitivity of the method

Secondly, we decided to add noise to the preceding real dataset in order to test the sensitivity of the proposed method. We also investigated the choice of the number of permutations. We considered the noisy data . . . , 47, ∀t ∈ [1998, 2019],

X ′ i (t) = X i (t) + αϵ i (t), ∀i = 1,
where the X i 's are the initial functional marks (the demographic evolution of the Spanish population measured in each province), the ϵ i 's are independent centered sBm processes and α is the parameter controlling the variance of the added noise. We simulated 100 noisy datasets with different levels of variance α and computed the functional scan statistic and its p-value based on different numbers of random permutations T . As in section 3.1, Table 4 presents the alarm rates, TP and FP rates we obtained. In this case, the TP and FP rates are not computed based on the true cluster (which is unknown) but on the most likely cluster obtained without noise (see Fig. 4). As expected, when the level of noise added to the initial data increases, the alarm rate of the test decreases since the presence of a significant cluster becomes less and less obvious. Moreover, for moderate level of noise, the clusters detected are not that different from the most likely cluster obtained without noise (the TP rate is close to 1 and the FP rate close to 0) but there is an evolution when α increases. It seems that, as already described by [START_REF] Mcdonough | Detection of signals in noise[END_REF], when the noise level α is small, the signal to noise ratio is large enough so that the scan method still works. However, when α reaches a certain threshold around 0.25, the signal to noise ratio becomes too small and the scan method fails. On the other hand, we can remark that the influence of the number of permutations T is quite limited: we may just mention that choosing T = 29 random permutations leads to less accurate p-values, so that the alarm rate obtained with that value of T might be slightly different from the others.

Discussion

Nowadays and with the development of modern technology, scientists often observe functional data instead of univariate or multivariate ones. As a consequence, there is a need for testing procedures adapted to these infinite dimensional data. To this end, this paper proposes a nonparametric spatial scan statistic based on the Wilcoxon-Mann-Whitney two-sample test for functional data introduced by [START_REF] Chakraborty | A Wilcoxon-Mann-Whitney type test for infinite-dimensional data[END_REF]. As shown in the application to simulated and real data, this scan procedure is much more suitable for functional data than existing ones, and its implementation in the R package HDSpatialScan makes it easy and quick to compute. For sake of simplicity, we decided to focus on functional data belonging to an Hilbert space. We must mention that extending this work to data belonging to a more general Banach space is straightforward since the Wilcoxon-Mann-Whitney statistic of [START_REF] Chakraborty | A Wilcoxon-Mann-Whitney type test for infinite-dimensional data[END_REF] can be generalized to such a space. This scan statistic allows to detect clusters using functional data indexed by space without assuming anything about their distribution. Another functional spatial scan statistic could be proposed using any other two-sample test statistic for functional data [START_REF] Zhang | Statistical inferences for functional data[END_REF]Zhang et al., 2010) as long as its asymptotic distribution is known. In a preprint, [START_REF] Frévent | Detecting spatial clusters on functional data: new scan statistic approaches[END_REF] recently proposed a parametric spatial scan statistic which is derived from the functional ANOVA test introduced by [START_REF] Cuevas | An anova test for functional data[END_REF]. In their work, they compared our scan statistic Λ WMWFSS with their statistic. They conclude, with simulation studies, that our nonparametric method performs better against non Gaussian data. R codes of this parametric extension are also available in the package HDSpatialScan [START_REF] Frévent | HDSpatialScan: Multivariate and Functional Spatial Scan Statistics[END_REF]. The scan method we propose allows to detect multiple clusters. If two "opposite" clusters (for example one exhibiting higher rates than expected and the other lower rates than expected) exist in two disjoint areas of the observation domain D, the scan method first computes the concentration index for both of them and decides which one is the most significant one. Secondly, the sequential procedure may exhibit the other cluster. Actually, two "opposite" clusters can cancel out with each other only if the intersection of their areas is not null but this seems very unlikely to happen. When the functional marks associated with the spatial locations are time series, another possibility would be considering spatio-temporal cluster detection such as [START_REF] Kulldorff | A space-time permutation scan statistic for the early detection of disease outbreaks[END_REF]. Our approach is completely different since each functional mark is taken as a whole and cannot be split: the goal is to highlight the functional marks exhibiting a different behaviour on the entire temporal observation domain. Our work is based on the frequent assumption in the literature of spatial scan statistics that the observations in different spatial locations are independent. One should be aware that this sometimes unrealistic assumption is just a means to introduce mathematical tools that can be applied even if data are spatially correlated, as explained by [START_REF] Glaz | Research on probability models for cluster of points before the year 1960[END_REF]. However, taking into account this spatial correlation in our method could be envisaged, as [START_REF] Loh | Accounting for spatial correlation in the scan statistic[END_REF] did in the univariate case. Finally, since we may observe different curves in each spatial location (for example the temporal variation of different atmospheric pollutants), another perspective would be to develop a functional extension of the multivariate Gaussian scan statistic introduced by [START_REF] Cucala | A Multivariate Gaussian scan statistic for spatial data[END_REF].

  = permutation vector of (1, . . . , n) 15: for k = 1 to n do 16: p k = order of value v k in v following ascending order 17: function WMWFSS (computing the WMWFSS scan statistic) value), MLC (most likely cluster) 20:

Figure 1 :

 1 Figure 1: The 94 French départements. In red: simulated clusters (8 and 10 départements).

Figure 2 :

 2 Figure 2: The 47 Spanish provinces and their geometrical centres.

Figure 3 :

 3 Figure 3: Demographic evolution in the 47 provinces from 1998 to 2019.

Figure 4 :

 4 Figure 4: The most likely cluster detected by the functional scan statistic Λ WMWFSS .

Figure 5 :

 5 Figure 5: The demographic evolution curves (from 1998 to 2019) in each province are presented. Curves in red correspond to provinces inside the cluster, curves in black correspond to provinces outside the cluster and the curve in green corresponds to Zamora which is inside the cluster too.

Figure 6 :

 6 Figure 6: The most likely cluster detected by Λ MBUSS and demographic evolution curves associated.

Figure

  Figure A.7: An example of the simulated data for the sBm process with ∆ 1 (t) = t (left panel) and ∆ 1 (t) = 3t (right panel). Curves in red correspond to the observations in the cluster.

Figure A. 8 :

 8 Figure A.8: An example of the simulated data for the sBm process with ∆ 2 (t) = 4t(1t) (left panel) and ∆ 2 (t) = 7t(1 -t) (right panel). Curves in red correspond to the observations in the cluster.

Figure

  Figure A.9: An example of the simulated data for the sBm process with ∆ 3 (t) = sin(2πt) (left panel) and ∆ 3 (t) = 2.5 sin(2πt) (right panel). Curves in red correspond to the observations in the cluster.

Table 1 :

 1 Simulation study-AR, %TP and %FP results of Λ WMWFSS , Λ MBUSS and Λ DBUSS when ∆ 1 = ct, ∆ 2 = ct(1 -t) and ∆ 3 = c sin (2πt) using two distributions: Normal and Student-t. The true cluster contains 8 départements. Bold values indicate the best performance in each line.

		Normal distribution			Student-t distribution
	c		Λ WMWFSS Λ MBUSS Λ DBUSS	c		Λ WMWFSS Λ MBUSS Λ DBUSS
	0.0	AR	0.070	0.050	0.060	0.0	AR	0.060	0.060	0.020
		%TP	0.554	0.875	1.000		%TP	0.479	0.479	1.000
		%FP	0.442	0.553	0.589		%FP	0.444	0.446	0.645
					∆ 1 (t) = ct
	1.5	AR	0.380	0.340	0.150	1.5	AR	0.240	0.220	0.090
		%TP	0.908	0.882	1.000		%TP	0.885	0.847	1.000
		%FP	0.110	0.165	0.148		%FP	0.098	0.164	0.472
	2.0	AR	0.730	0.660	0.300	2.0	AR	0.600	0.510	0.180
		%TP	0.967	0.966	0.933		%TP	0.935	0.939	0.993
		%FP	0.049	0.087	0.073		%FP	0.095	0.142	0.228
	2.5	AR	0.920	0.890	0.570	2.5	AR	0.790	0.730	0.390
		%TP	0.978	0.961	0.879		%TP	0.949	0.938	0.978
		%FP	0.056	0.070	0.077		%FP	0.045	0.063	0.137
				∆ 2 = ct(1 -t)
	4.5	AR	0.460	0.320	0.160	4.5	AR	0.360	0.310	0.130
		%TP	0.853	0.844	0.938		%TP	0.760	0.706	0.904
		%FP	0.101	0.139	0.262		%FP	0.121	0.144	0.286
	5.5	AR	0.700	0.530	0.260	5.5	AR	0.450	0.380	0.150
		%TP	0.950	0.934	0.923		%TP	0.908	0.898	1.000
		%FP	0.042	0.077	0.178		%FP	0.070	0.130	0.261
	6.5	AR	0.870	0.760	0.460	6.5	AR	0.610	0.470	0.200
		%TP	0.991	0.984	0.929		%TP	0.932	0.910	0.988
		%FP	0.041	0.068	0.091		%FP	0.067	0.097	0.153
				∆ 3 (t) = c sin(2πt)
	1.0	AR	0.310	0.080	0.170	1.0	AR	0.170	0.070	0.140
		%TP	0.895	0.531	0.882		%TP	0.772	0.571	0.938
		%FP	0.156	0.552	0.347		%FP	0.126	0.150	0.273
	1.25 AR	0.660	0.040	0.350	1.25 AR	0.390	0.060	0.200
		%TP	0.981	0.781	0.979		%TP	0.949	0.667	0.938
		%FP	0.037	0.573	0.250		%FP	0.109	0.455	0.251
	1.5	AR	0.960	0.060	0.660	1.5	AR	0.820	0.050	0.310
		%TP	0.988	0.833	0.981		%TP	0.970	0.425	0.960
		%FP	0.010	0.271	0.071		%FP	0.053	0.490	0.199

Table 2 :

 2 Simulation study-AR, %TP and %FP results of Λ WMWFSS , Λ MBUSS and Λ DBUSS when ∆ 1 , ∆ 2 and ∆ 3 using two distributions: Normal and Student-t. The true cluster contains 10 départements. Bold values indicate the best performance in each line.

		Normal distribution			Student-t distribution
	c		Λ WMWFSS Λ MBUSS Λ DBUSS	c		Λ WMWFSS Λ MBUSS Λ DBUSS
	0.0	AR	0.060	0.050	0.030	0.0	AR	0.060	0.060	0.030
		%TP	0.317	0.480	1.000		%TP	0.200	0.550	0.667
		%FP	0.534	0.545	0.635		%FP	0.204	0.206	0.544
					∆ 1 (t) = ct
	1.5	AR	0.650	0.540	0.160	1.5	AR	0.420	0.360	0.120
		%TP	0.926	0.913	0.963		%TP	0.883	0.872	1.000
		%FP	0.096	0.156	0.263		%FP	0.136	0.147	0.313
	2.0	AR	0.900	0.860	0.400	2.0	AR	0.710	0.660	0.200
		%TP	0.960	0.956	0.913		%TP	0.956	0.933	0.950
		%FP	0.051	0.076	0.119		%FP	0.074	0.091	0.123
	2.5	AR	1.000	0.980	0.600	2.5	AR	0.960	0.910	0.400
		%TP	0.988	0.979	0.963		%TP	0.980	0.968	0.950
		%FP	0.029	0.041	0.083		%FP	0.040	0.049	0.085
				∆ 2 = ct(1 -t)
	4.5	AR	0.660	0.520	0.170	4.5	AR	0.420	0.380	0.120
		%TP	0.950	0.933	0.918		%TP	0.874	0.871	0.942
		%FP	0.090	0.139	0.218		%FP	0.118	0.129	0.289
	5.5	AR	0.930	0.740	0.230	5.5	AR	0.650	0.530	0.170
		%TP	0.974	0.972	1.000		%TP	0.906	0.898	0.994
		%FP	0.040	0.051	0.139		%FP	0.058	0.097	0.345
	6.5	AR	0.990	0.900	0.600	6.5	AR	0.900	0.820	0.360
		%TP	0.980	0.973	0.960		%TP	0.956	0.952	0.975
		%FP	0.026	0.049	0.091		%FP	0.035	0.055	0.162
				∆ 3 (t) = c sin(2πt)
	1.0	AR	0.690	0.070	0.250	1.0	AR	0.340	0.050	0.120
		%TP	0.948	0.757	0.912		%TP	0.953	0.800	1.000
		%FP	0.052	0.388	0.259		%FP	0.096	0.429	0.348
	1.25 AR	0.960	0.040	0.480	1.25 AR	0.760	0.020	0.230
		%TP	0.993	0.950	0.975		%TP	0.963	0.500	0.874
		%FP	0.015	0.393	0.143		%FP	0.042	0.369	0.151
	1.5	AR	1.000	0.060	0.720	1.5	AR	0.950	0.100	0.350
		%TP	1.000	0.983	0.994		%TP	0.984	0.840	0.971
		%FP	0.004	0.274	0.056		%FP	0.014	0.411	0.148

Table 3 :

 3 The p-values and computation time (in seconds) of the different scan methods using different number of permutations.

		T=99	T=999
	Method	p value time p value time
	Λ WMWFSS	0.01 1.14 0.001 10.92
	Λ MBUSS	0.01 0.62 0.003 7.20
	Λ DBUSS	0.56 0.34 0.527 6.83

Table 4 :

 4 Real data plus noise -Alarm rate, %TP and %FP results of the functional scan statistic Λ WMWFSS for different variance level α and number of permutations T .

	α		T=29 T=59 T=99 T=999
	0.05 AR	0.990	1.000	1.000	1.000
		%TP 1.000	1.000	1.000	1.000
		%FP 0.000	0.000	0.000	0.000
	0.1	AR	0.980	1.000	1.000	1.000
		%TP 0.998	0.988	0.984	0.985
		%FP 0.013	0.031	0.038	0.035
	0.15 AR	0.940	1.000	1.000	1.000
		%TP 0.980	0.948	0.949	0.943
		%FP 0.096	0.100	0.130	0.129
	0.2	AR	0.870	0.970	0.990	0.990
		%TP 0.893	0.898	0.904	0.890
		%FP 0.229	0.248	0.266	0.266
	0.25 AR	0.560	0.630	0.680	0.690
		%TP 0.771	0.866	0.769	0.817
		%FP 0.380	0.350	0.407	0.383
	0.3	AR	0.220	0.270	0.290	0.290
		%TP 0.664	0.795	0.618	0.695
		%FP 0.418	0.416	0.444	0.449

Table B .

 B 5: Simulation study-AR , %TP and %FP results of the functional scan statistic Λ WMWFSS and the univariate ones Λ MBUSS and Λ DBUSS when ∆ 1 (t) = ct using two distributions: Normal and Student-t. The true cluster contains 8 départements.TableB.6: Simulation study-AR, %TP and %FP results of the functional scan statistic Λ WMWFSS and the univariate ones Λ MBUSS and Λ DBUSS when ∆ 2 (t) = ct(1 -t) using two distributions: Normal and Student-t. The true cluster contains 8 départements. Table B.7: Simulation study-AR, %TP and %FP results of the functional scan statistic Λ WMWFSS and the univariate ones Λ MBUSS and Λ DBUSS when ∆ 3 (t) = c sin(2πt) using two distributions: Normal and Student-t. The true cluster contains 8 départements. The following Table B.8, Table B.9 and Table B.10 give the results obtained in this simulation study. Bold values indicate the best performance in each line.

		Normal distribution Normal distribution Normal distribution		Student-t distribution Student-t distribution Student-t distribution
	c c c		Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS	c c c	Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS
	0.0 4.0 AR AR 1.0 AR	0.070 0.410 0.310	0.050 0.330 0.080	0.060 0.120 0.170	0.0 AR 4.5 AR 1.0 AR	0.060 0.360 0.170	0.060 0.310 0.070	0.020 0.130 0.140
		%TP %TP %TP	0.554 0.869 0.895	0.875 0.867 0.531	1.000 0.750 0.882	%TP %TP %TP	0.479 0.760 0.772	0.479 0.706 0.571	1.000 0.904 0.938
		%FP %FP %FP	0.442 0.193 0.156	0.553 0.243 0.552	0.589 0.214 0.347	%FP %FP %FP	0.444 0.121 0.126	0.446 0.144 0.150	0.645 0.286 0.273
	1.25 AR 4.5 AR 1.25 AR	0.310 0.460 0.660	0.260 0.320 0.040	0.110 0.160 0.350	1.5 AR 5.5 AR 1.25 AR	0.240 0.450 0.390	0.220 0.380 0.060	0.090 0.150 0.200
		%TP %TP %TP	0.887 0.853 0.981	0.880 0.844 0.781	1.000 0.938 0.979	%TP %TP %TP	0.885 0.908 0.949	0.847 0.898 0.667	1.000 1.000 0.938
		%FP %FP %FP	0.188 0.101 0.037	0.199 0.139 0.573	0.403 0.262 0.250	%FP %FP %FP	0.098 0.070 0.109	0.164 0.130 0.455	0.472 0.261 0.251
	1.5 5.0 AR AR 1.5 AR	0.380 0.560 0.960	0.340 0.470 0.060	0.150 0.210 0.660	2.0 AR 6.5 AR 1.5 AR	0.600 0.610 0.820	0.510 0.470 0.050	0.180 0.200 0.310
		%TP %TP %TP	0.908 0.944 0.988	0.882 0.910 0.833	1.000 0.946 0.981	%TP %TP %TP	0.935 0.932 0.970	0.939 0.910 0.425	0.993 0.988 0.960
		%FP %FP %FP	0.110 0.077 0.010	0.165 0.111 0.271	0.148 0.162 0.071	%FP %FP %FP	0.095 0.067 0.053	0.142 0.097 0.490	0.228 0.153 0.199
	1.75 AR 5.5 AR 1.75 AR	0.590 0.700 1.000	0.450 0.530 0.070	0.160 0.260 0.940	2.5 AR 7.5 AR 1.75 AR	0.790 0.850 0.880	0.730 0.760 0.030	0.390 0.340 0.460
		%TP %TP %TP	0.962 0.950 1.000	0.956 0.934 0.911	0.953 0.923 0.899	%TP %TP %TP	0.949 0.951 0.972	0.938 0.950 0.833	0.978 0.901 0.959
		%FP %FP %FP	0.074 0.042 0.009	0.085 0.077 0.400	0.197 0.178 0.058	%FP %FP %FP	0.045 0.065 0.015	0.063 0.099 0.217	0.137 0.119 0.096
	2.0 6.0 AR AR 2.0 AR	0.730 0.830 1.000	0.660 0.590 0.060	0.300 0.290 1.000	3.0 AR 8.5 AR 2.0 AR	0.920 0.960 0.990	0.870 0.820 0.070	0.520 0.570 0.760
		%TP %TP %TP	0.967 0.973 1.000	0.966 0.958 1.000	0.933 0.957 0.993	%TP %TP %TP	0.967 0.990 0.996	0.945 0.988 0.893	0.964 0.982 0.991
		%FP %FP %FP	0.049 0.034 0.007	0.087 0.046 0496	0.073 0.126 0.041	%FP %FP %FP	0.035 0.023 0.009	0.036 0.040 0.387	0.094 0.074 0.070
	2.5 6.5 AR AR 2.25 AR	0.920 0.870 1.000	0.890 0.760 0.020	0.570 0.460 1.000 2.25 AR 3.5 AR 9.5 AR	0.980 0.990 1.000	0.940 0.910 0.070	0.800 0.730 0.890
		%TP %TP %TP	0.978 0.991 1.000	0.961 0.984 1.000	0.879 0.929 0.984	%TP %TP %TP	0.974 0.991 1.000	0.973 0.984 1.000	0.956 0.945 0.997
		%FP %FP %FP	0.056 0.041 0.005	0.070 0.068 0.052	0.077 0.091 0.029	%FP %FP %FP	0.035 0.020 0.003	0.050 0.035 0.561	0.048 0.073 0.063
	3.0 7.0 AR AR 2.5 AR	1.000 0.960 1.000	1.000 0.810 0.050	0.870 0.530 1.000	4.0 AR 10.5 AR 2.5 AR	0.990 0.990 1.000	0.980 0.930 0.040	0.920 0.890 0.950
		%TP %TP %TP	0.996 0.992 1.000	0.986 0.986 1.000	0.951 0.981 0.995	%TP %TP %TP	0.990 0.997 1.000	0.980 0.995 1.000	0.942 0.980 0.996
		%FP %FP %FP	0.019 0.026 0.003	0.027 0.047 0.481	0.057 0.075 0.027	%FP %FP %FP	0.021 0.015 0.002	0.031 0.027 0.311	0.044 0.055 0.046
	3.5	AR	1.000	1.000	0.910	4.5 AR	1.000	0.990	0.980
		%TP	1.000	1.000	0.968	%TP	0.995	0.990	0.941
		%FP	0.012	0.022	0.032	%FP	0.013	0.021	0.029

• When the true cluster is a set of 10 départements:

Table B .

 B 8: Simulation study-AR, %TP and %FP results of the functional scan statistic Λ WMWFSS and the univariate ones Λ MBUSS and Λ DBUSS when ∆ 1 (t) = ct using two distributions: Normal and Student-t. The true cluster contains 10 départements.TableB.9: Simulation study-AR, %TP and %FP results of the functional scan statistic Λ WMWFSS and the univariate ones Λ MBUSS and Λ DBUSS when ∆ 2 (t) = ct(1 -t) using two distributions: Normal and Student-t. The true cluster contains 10 départements.TableB.10: Simulation study-AR, %TP and %FP results of the functional scan statistic Λ WMWFSS and the univariate ones Λ MBUSS and Λ DBUSS when ∆ 3 (t) = c sin(2πt) using two distributions: Normal and Student-t. The true cluster contains 10 départements.

		Normal distribution Normal distribution Normal distribution				Student-t distribution Student-t distribution Student-t distribution
	c c c		Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS	c	c c		Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS Λ WMWFSS Λ MBUSS Λ DBUSS
	0.0 4.0 AR AR 0.5 AR	0.060 0.440 0.120	0.050 0.410 0.100	0.030 0.100 0.090	0.0 4.5 0.5	AR AR AR	0.060 0.420 0.110	0.060 0.380 0.080	0.030 0.120 0.090
		%TP %TP %TP	0.317 0.911 0.442	0.480 0.907 0.650	1.000 0.900 0.989			%TP %TP %TP	0.200 0.874 0.755	0.550 0.871 0.725	0.667 0.942 0.822
		%FP %FP %FP	0.534 0.115 0.356	0.545 0.155 0.430	0.635 0.217 0.413			%FP %FP %FP	0.204 0.118 0.443	0.206 0.129 0.571	0.544 0.289 0.500
	1.0 4.5 AR AR 0.75 AR	0.210 0.660 0.260	0.200 0.520 0.060	0.050 0.170 0.100	1.0 5.5 0.75 AR AR AR	0.210 0.650 0.110	0.190 0.530 0.050	0.060 0.170 0.100
		%TP %TP %TP	0.795 0.950 0.877	0.785 0.933 0.333	1.000 0.918 1.000			%TP %TP %TP	0.786 0.906 0.855	0.774 0.898 0.820	1.000 0.994 1.000
		%FP %FP %FP	0.185 0.090 0.138	0.230 0.139 0.343	0.362 0.218 0.239			%FP %FP %FP	0.186 0.058 0.134	0.200 0.097 0.238	0.514 0.345 0.307
	1.25 AR 5.0 AR 1.0 AR	0.360 0.800 0.690	0.300 0.550 0.070	0.050 0.180 0.250	1.25 AR 6.5 AR 1.0 AR	0.310 0.900 0.340	0.270 0.820 0.050	0.080 0.360 0.120
		%TP %TP %TP	0.922 0.956 0.948	0.903 0.956 0.757	1.000 0.983 0.912			%TP %TP %TP	0.771 0.956 0.953	0.744 0.952 0.800	0.850 0.975 1.000
		%FP %FP %FP	0.218 0.044 0.052	0.254 0.085 0.388	0.355 0.097 0.259			%FP %FP %FP	0.171 0.035 0.096	0.173 0.055 0.429	0.405 0.162 0.348
	1.5 5.5 AR AR 1.25 AR	0.650 0.930 0.960	0.540 0.740 0.040	0.160 0.230 0.480	1.5 7.5 1.25 AR AR AR	0.420 0.960 0.760	0.360 0.840 0.020	0.120 0.530 0.230
		%TP %TP %TP	0.926 0.974 0.993	0.913 0.972 0.950	0.963 1.000 0.975			%TP %TP %TP	0.883 0.974 0.963	0.872 0.964 0.500	1.000 0.958 0.874
		%FP %FP %FP	0.096 0.040 0.015	0.156 0.051 0.393	0.263 0.139 0.143			%FP %FP %FP	0.136 0.023 0.042	0.147 0.058 0.369	0.313 0.120 0.151
	1.75 AR 6.0 AR 1.5 AR	0.750 0.980 1.000	0.630 0.880 0.060	0.260 0.440 0.720	1.75 AR 8.5 AR 1.5 AR	0.580 0.990 0.950	0.470 0.940 0.100	0.150 0.700 0.350
		%TP %TP %TP	0.933 0.977 1.000	0.922 0.973 0.983	0.950 0.939 0.994			%TP %TP %TP	0.328 0.989 0.984	0.298 0.983 0.840	1.000 0.986 0.971
		%FP %FP %FP	0.064 0.028 0.004	0.071 0.050 0.274	0.280 0.122 0.056			%FP %FP %FP	0.072 0.019 0.014	0.106 0.056 0.411	0.316 0.076 0.148
	2.0 6.5 AR AR 1.75 AR	0.900 0.990 1.000	0.860 0.900 0.060	0.400 0.600 0.990	2.0 9.5 1.75 AR AR AR	0.710 1.000 0.990	0.660 0.980 0.050	0.200 0.880 0.770
		%TP %TP %TP	0.960 0.980 1.000	0.956 0.973 1.000	0.913 0.960 1.000			%TP %TP %TP	0.956 0.990 0.989	0.933 0.990 0.900	0.950 0.958 0.973
		%FP %FP %FP	0.051 0.026 0.003	0.076 0.049 0.294	0.119 0.091 0.050			%FP %FP %FP	0.074 0.016 0.013	0.091 0.030 0.560	0.123 0.068 0.122
	2.25 AR 7.0 AR 2.0 AR	0.950 0.990 1.000	0.870 0.950 0.070	0.480 0.700 1.000	2.25 AR 10.5 AR 2.0 AR	0.860 1.000 1.000	0.750 0.990 0.070	0.330 0.980 0.920
		%TP %TP %TP	0.960 0.996 1.000	0.960 0.992 1.000	0.944 0.990 1.000			%TP %TP %TP	0.968 0.996 1.000	0.968 0.993 1.000	0.918 0.981 0.997
		%FP %FP %FP	0.047 0.020 0.002	0.061 0.032 0.332	0.089 0.078 0.031			%FP %FP %FP	0.064 0.012 0.006	0.069 0.024 0.425	0.131 0.055 0.057
	2.5	AR	1.000	0.980	0.600	2.5	AR	0.960	0.910	0.400
		%TP	0.988	0.979	0.963			%TP	0.980	0.968	0.950
		%FP	0.029	0.041	0.083			%FP	0.040	0.049	0.085

• When the true cluster is a set of 8 départements:

The following