MD Sylvain Lescuyer
email: lescuyer.s@gmail.com

MD Aurore Meyer

MD Jean-Christophe Weber

A spontaneous hematoma in a healthy young woman

Keywords: scurvy, purpura, corkscrew hair, spontaneous hematoma

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

Case description

A 40-year-old woman presented herself with a large hematoma of her left lower limb which appeared spontaneously three weeks prior to her visit. The hematoma was responsible for a symptomatic anemia, so she was transfused with two packed red blood cells. Her past medical history was marked only by a depressive syndrome. She was currently taking neither medication nor drugs. She had visited a doctor during the past week for this hematoma, a purpura involving both of her legs for six months, and a possible diagnosis of erythema nodosum. She had reported occasional epistaxis. Nothing else had been noted during the clinical examination. Laboratory results had come back with a normal blood count except for mild anemia, normal coagulation tests, normal renal and hepatic functions, no evidence for an auto-immune disease or a viral infection. A skin biopsy of the purpura had been made but the results were still in progress at the time of hospitalization. Our examination revealed a perifollicular distribution of purpura and hair dystrophies (figure). What is the diagnosis? Diagnosis A spontaneous hematoma alongside normal coagulation tests and those findings were consistent with scurvy. We thought that past erythema nodosum were misdiagnosed spontaneous hematomas. Detailed questioning revealed a diet restricted to pasta. Dental hygiene was poor without hemorrhagic gingivitis. A blood sample for vitamin C serum level dosing was taken and the patient immediately treated with ascorbic acid infusions followed by oral supplementation one week after. Nutritional evaluation brought out B9 vitamin and iron deficiencies. Skin biopsy analysis showed perifollicular red blood cells extravasation consistent with scurvy as well. Vitamin C serum level was low (0,6mg/l; N: 5-17). At week three, purpura has vanished and hematoma was improving. Scurvy's dermatological manifestations are due to collagen alteration [1]. Corkscrew hair dystrophy is well known. Swan-neck hair dystrophy is less known but just as evocative. Oral involvement is characteristic but not systematic because of its late onset. Diagnosis of scurvy is clinical, thought histological analysis of a skin biopsy may help. Asymptomatic low vitamin C level is frequent [2]. Risk factors for scurvy are not always obvious, as in our patient: clinicians should be aware of it and openly ask for diet restrictions. Clinical manifestations need to be known to avoid such serious complications. Treatment is simple and risk-free. Other deficiencies must be screened for, especially iron, because of vitamin C implications in its metabolism [2].

  H, Grandière-Perez L, Caumes E. Scurvy, a re-emerging disease. Rev Med Interne. 2005;26(11):885-90. 2: Colacci M, Gold WL, Shah R. Modern-day scurvy. CMAJ. 2020;192(4):E96.

Figure :

 : Figure: Left leg. Perifollicular purpura (panel A). Hair dystrophies (panel B): corkscrew (arrows) and swan-neck (arrow heads) dystrophies.


Rémi Pereira

De Oliveira

Pierrick Lucas

Amélie Chastagner

Claire De Boisseson

Laurence Vial

Marie-Frédérique Le Potier

Yannick Blanchard
email: yannick.blanchard@anses.fr

Evaluation of un-methylated DNA enrichment in sequencing of African Swine Fever Virus complete genome

African swine fever is a febrile hemorrhagic fever disease that is caused by the African swine fever virus (ASFV) and is lethal for domestic pigs and wild boar. ASFV also infects soft ticks of the genus Ornithodoros, some species of which can act as a vector for ASFV. Whole genome sequencing of ASFV is a challenge because, due to the size difference of the host genome versus the viral genome, the higher proportion of host versus virus DNA fragments renders the virus sequencing poorly efficient. A novel approach of DNA enrichment, based on the separation of methylated and unmethylated DNA, has been reported but without an evaluation of its efficacy. In this study, the efficiency of the un-methylated DNA enrichment protocol was evaluated for pig and tick samples infected by ASFV. As expected, fewer reads corresponding to ASFV were found in the methylated fraction compared to the un-methylated fraction. However, the sequencing coverage of the unmethylated fraction was not improved compared to the untreated DNA. In our hands, the ASFV DNA enrichment was inefficient for tick samples and very limited for pig samples. This enrichment process

represents extra work and cost without a significant improvement of ASFV genome coverage. The efficiency of this enrichment approach and the cost/benefit ratio are discussed.

Keywords: African swine fever virus, NGS, enrichment, methylation

1-Introduction

African swine fever (ASF) is a highly contagious hemorrhagic fever in Suidae that is lethal for domestic pigs and wild boar. The African swine fever virus (ASFV), a large enveloped DNA virus with a genome size between 170 and 190 kb, is the etiologic agent of ASF and the only member of Asfarviridae family. ASFV can infect two different hosts, Suidae and soft ticks of the genus Ornithodoros, the last being a possible vector of ASFV [START_REF] Boinas | The Persistence of African Swine Fever Virus in Field-Infected Ornithodoros erraticus during the ASF Endemic Period in Portugal[END_REF][START_REF] Plowright | The epizootiology of African swine fever in Africa[END_REF].

Although ASFV is a large DNA virus, with a genome of 170 to 190 kb, this genome is far smaller than the genomes of its two hosts. Pigs have a genome of 2.8 billion kb (Groenen et al., 2012) while ticks have a genome estimated at 1 billion kb [START_REF] Geraci | Variation in genome size of argasid and ixodid ticks[END_REF]. Standard DNA extraction methods, like commercial kits using an affinity column or phenol chloroform, do not separate the host and viral DNA genomes, leading to a heavily unbalanced output of sequences in favor of the host's genome, even after cell culture amplification of the virus prior to sequencing. This is particularly critical for samples that have low viral titers, such as tick samples, making the sequencing of the virus from these samples highly challenging.

The first complete ASFV sequence, published in 1995, was of the non-virulent isolate BA71V, a Vero cell culture adapted strain [START_REF] Yáñez | Analysis of the complete nucleotide sequence of African swine fever virus[END_REF]. This first full ASFV genome sequence, 170 kb long, was obtained by Sanger sequencing after viral DNA fragmentation and cloning of the whole genome into several plasmids. Despite the subsequent emergence and use of Next Generation Sequencing (NGS), the sequencing of ASFV genomes has remained challenging, and ASFV sequences have had to be completed or confirmed by PCR amplification of some DNA fragments followed by Sanger sequencing [START_REF] Chapman | Genomic analysis of highly virulent Georgia 2007/1 isolate of African swine fever virus[END_REF][START_REF] De Villiers | Phylogenomic analysis of 11 complete African swine fever virus genome sequences[END_REF][START_REF] Olesen | Complete genome sequence of an African swine fever virus (ASFV POL/2015/Podlaskie) determined directly from pig erythrocyte-associated nucleic acid[END_REF][START_REF] Zani | Deletion at the 5'-end of Estonian ASFV strains associated with an attenuated phenotype[END_REF].

Different strategies for viral genome enrichment have been used, including centrifugation/ultracentrifugation, filtration, and DNAse and RNAse treatments [START_REF] Hall | Evaluation of rapid and simple techniques for the enrichment of viruses prior to metagenomic virus discovery[END_REF].

Genome capture, a more sophisticated technique based on a library of probes specifically designed to selectively hybridize to a target genome, also has been used for the sequencing from clinical samples of viruses that are difficult to sequence like Herpes Simplex Virus [START_REF] Briese | Virome Capture Sequencing Enables Sensitive Viral Diagnosis and Comprehensive Virome Analysis[END_REF][START_REF] Gaudin | Hybrid Capture-Based Next Generation Sequencing and Its Application to Human Infectious Diseases[END_REF][START_REF] Greninger | Ultrasensitive Capture of Human Herpes Simplex Virus Genomes Directly from Clinical Samples Reveals Extraordinarily Limited Evolution in Cell Culture[END_REF]. While very efficient, capture techniques are expensive, and the preliminary knowledge of the viral sequences needed for the design of the probes also limits their use for quasispecies studies.

In eukaryotes, DNA methylation is an epigenetic modification targeting CpG dinucleotides, extremely stable DNA sequences frequently found in CpG islands that are involved in gene regulation [START_REF] Dor | Principles of DNA methylation and their implications for biology and medicine[END_REF]. DNA methylation is also present in some large DNA viruses, such as Herpesviruses and Iridoviruses, which replicate in the nucleus [START_REF] Hoelzer | Presence and role of cytosine methylation in DNA viruses of animals[END_REF]. Interestingly, contrary to most DNA viruses, the viral replication of ASFV occurs exclusively in the cytoplasm of infected host cells [START_REF] Dixon | African swine fever virus replication and genomics[END_REF], whereas the DNA methylase is exclusively present in the nucleus of the cell. The ASFV genome thus should avoid CpG methylation, as suggested by a recent study demonstrating, on a small part of the genome, that ASFV DNA was un-methylated [START_REF] Weber | Intracellular African swine fever virus DNA remains unmethylated in infected Vero cells[END_REF]. Recently, new complete genome sequences of African ASFV strains were obtained after an enrichment step based on the separation of methylated and un-methylated DNA [START_REF] Masembe | Genome Sequences of Five African Swine Fever Virus Genotype IX Isolates from Domestic Pigs in Uganda[END_REF][START_REF] Ndlovu | Genome Sequences of Three African Swine Fever Viruses of Genotypes I, III, and XXII from South Africa and Zambia, Isolated from Ornithodoros Soft Ticks[END_REF]. However, the efficiency of this approach on the different samples used (high or low quantity of ASFV DNA) was not evaluated.

In this study, we investigated the efficiency of this un-methylated DNA enrichment strategy for ASFV sequencing. Two types of samples infected with the Georgia2007/1 ASFV strain were tested: 1) blood of viremic pigs, representing samples with a high quantity of ASFV DNA, and 2) infected Ornithodoros erraticus, a European soft tick from Portugal, representing samples with a low quantity of ASFV DNA. If efficient, sample enrichment based on methylated and un-methylated DNA discrimination should separate host DNA from viral DNA. For comparison purposes, we sequenced the untreated DNA (UD), the methylated fraction (MF) and the un-methylated fraction (UMF) resulting from the enrichment process.

2-Materials and methods

2.1-Infection of SPF pigs and soft tick Ornithodoros erraticus

The ASFV strain Georgia2007/1 was used for this experiment. This strain, member of the ASFV genotype II group, was initially isolated from domestic pigs in Georgia in 2007 and was kindly provided by Dr. Linda Dixon (OIE reference Laboratory, Pirbright Institute, UK). This strain has a genome size of 189 344 bp coding for 269 ORFs [START_REF] Forth | A Deep-Sequencing Workflow for the Fast and Efficient Generation of High-Quality African Swine Fever Virus Whole-Genome Sequences[END_REF]. Five specific-pathogen-free (SPF) pigs (889Geo, 936Geo, 941Geo, 6547Geo, 6518Geo) were inoculated intra-muscularly with 10 4 50% hemadsorbing dose (HAD50) of Georgia2007/1 ASFV strain. Heparin blood samples were collected on viremia at day 3 post-inoculation. Ornithodoros erraticus (E geo F1-6, E geo M1-2) were infected by feeding on Georgia2007/1 infected pigs and were frozen three months after the blood meal.

2.2-DNA extraction

DNA extractions were performed with the High Pure PCR Template Preparation Kit (Roche Diagnostics, Meylan, France). For pigs, DNA extraction was realized from 400 µL of heparin blood sample. Ticks, at 3 months post-infection, were washed 5 secs on sodium hypochlorite 2.6 % then 3 times in sterile Phosphate Buffered Saline (PBS). After washing, ticks were crushed in 200 µL of sterile PBS with two steel beads of 3 mm and 4 mm at 25 Hz during 3 min with VWR Star-Beater (VWR International bvba, Leuven, Belgium). Then 800 µL of sterile PBS were added and the whole solution was clarified by centrifugation at 5000 g during 5 min. Supernatants were filtered on 0.22 µm Sterile Millex filter (Merck Milipore, Carrigtwohill, Ireland) for E geo F1-2, E geo M1 and 0.45 µm filter for E geo F3-6 and E geo M2. Filtered supernatants were centrifuged at 13000 g during 5 min and 400 µL of the supernatants were used for DNA extraction.

2.3-Real-time PCR

Each DNA extract was tested by quantitative real-time PCR for ASFV VP-72 and beta-actin genes. The primers and probe used for ASFV VP-72 gene were previously described [START_REF] Tignon | Development and inter-laboratory validation study of an improved new real-time PCR assay with internal control for detection and laboratory diagnosis of African swine fever virus[END_REF]. The beta-actin primers and probes used in our study were also described for swine by Tignon et al.. For the tick samples, the primers described by [START_REF] Duron | Tick-Bacteria Mutualism Depends on B Vitamin Synthesis Pathways[END_REF] were used, however, instead of SYBR Green, a Taqman probe was designed for our duplex real-time PCR Hex-5'-CGAGAGGAAGTACTCCGTCTGG-3'-BHQ1.

2.4-Quantification of DNA, sample enrichment, library preparation and sequencing

DNA samples were assayed with Qubit Fluorometer (Invitrogen, Paisley, UK). For the pig samples, DNA quantification was performed with 1 µL of DNA; for the tick samples, DNA quantification was performed with 10 µL of DNA. The quantities of DNA obtained for each initial sample are presented in Table 1.

After purification, all initial DNA samples were submitted to an enrichment step by separating methylated DNA and non-methylated DNA with NEBNext Microbiome DNA Enrichment Kit (NEW ENGLAND Biolabs, Evry, France) according to the manufacturer's protocol. More precisely, for the pig samples, 100 ng of DNA were used for the NE library and the leftover (around 550 ng) for the enrichment procedure. For the tick samples, 100 ng of DNA were also used for the NE library and the leftover DNA, which varied between 260 ng and 450 ng, for the enrichment procedure. After the enrichment procedure, each initial sample was then split into three fractions: 1) fraction 1, the initial DNA sample, corresponding to an aliquot before the enrichment process (UD), 2) fraction 2, the enriched methylated DNA sample (MF), and 3) fraction 3, the depleted methylated DNA sample (UMF). DNA quantification of the MF and UMF fractions was performed after the enrichment process. For library preparation, Ion Xpress Plus Fragment Library Kit (Thermo Fischer Scientific, Frederick, Maryland, USA) was used and DNA fragments between 250 bp and 290 bp were sizeselected with Ion Xpress Barcode Adapters 1-96 kit (Thermo Fischer Scientific, Frederick, Maryland, USA). For the DNA purification steps, magnetic beads from Agencourt AMPure XP Kit (Beckman Coulter, Villepinte, France) were used.

All samples were sequenced with Proton Ion Torrent technology (Thermofischer Scientific, Frederick, Maryland, USA). For the sequencing, the different libraries (UD, UMF, MF) were multiplexed and simultaneously sequenced on a P1 chip for Proton sequencing. The P1 chip allows the production of up to 100 M reads. The Phred Quality Score ≥20 was calculated for generated bases and expressed as a percentage (%≥Q20). Row data can be found in the GenBank SRA under accession no. PRJNA643370.

2.5-Bioinformatic analysis

The reads were cleaned with Trimmomatic 0.36 software using the following parameters: ILLUMINACLIP:oligos.fasta: 2:30:5:1: true; LEADING: 3; TRAILING: 3; MAXINFO: 40:0.2; MINLEN: 36 [START_REF] Bolger | Trimmomatic: a flexible trimmer for Illumina sequence data[END_REF]). An alignment then was performed using Bowtie2 (version 2.2.5) [START_REF] Langmead | Fast gapped-read alignment with Bowtie 2[END_REF] set on very fast (-D 5 -R 1 -N O -L 22 -i s,0,2.50) with cleaned down-sampled reads on a local ncbi nucleotide database. This very fast alignment detected only the highly similar sequences (>99% homology) and provided a snapshot picture of the data. The bam files were then converted to a blast output readable by Krona taxonomic viewer [START_REF] Ondov | Interactive metagenomic visualization in a Web browser[END_REF] to visualize the gross organisms distribution in the different samples. For an accurate count of ASFV reads, the sequence of strain Georgia2007/1, Genbank accession number FR682468.1 [START_REF] Chapman | Genomic analysis of highly virulent Georgia 2007/1 isolate of African swine fever virus[END_REF], was used as reference for a Bowtie2 alignment (command line option -al) versus all cleaned reads.

2.7-Statistical analysis

Statistical analyses were performed with RStudio software (version 1.1.463). For all data analyses, three groups were considered: 1) the pool of UD fractions, 2) the pool of UMF fractions and 3) the pool of MF fractions. The Wilxocon test was used for the statistical analysis of these three groups. Our analysis focused on the impact that the enrichment process had on the NGS results, and more specifically on ASFV genome enrichment and coverage between each group.

3-Results

3.1-Detection and relative quantification of viral and host DNA in initial samples

A duplex quantitative real-time PCR directed against ASFV and beta-actin was performed for all samples and the ASFV/beta-actin ratios of the PCR Ct were calculated for the estimation of the relative quantity of the genomes of ASFV and the hosts (Table 1). Assuming a similar efficiency of our PCR reactions, a ratio of less than 1 indicates a higher level (by genome copy number) of ASFV genomes than host genomes. Such a ratio (less than 1) was found for pig samples. For tick samples, the ratio was superior to 1, excepting the E geo F1 and E geo M1 samples for which the ratio was respectively 0.99 and 0.96.

3.2-Quality analysis of generated data

After the enrichment process, for pig samples, 158 ng to 205 ng of total DNA were obtained for UMF and 208 ng to 325 ng for MF. For tick samples, 153 ng to 391 ng of total DNA were obtained after enrichment for UMF and 9 ng to 24 ng for MF (Table 2).

The generated bases, reads and Phred quality scores are shown in Table 3. Briefly, for pig samples, a mean number of 6 635 376 reads were generated for UD, 2 510 484 for UMF and 1 963 487 for MF. For tick samples, a mean number of 4 820 610 reads were generated for UD, 7 317 952 for UMF and 1 733 646 for MF.

The Phred quality scores were equivalent between all of the libraries, for both pig and tick samples.

3.3-Metagenomic analysis

Due to a high variability in the number of reads obtained by samples, from 15 000 to 21x10 6 reads (Table 3), and for computing time efficiency, the metagenomic classification of the reads was performed on subsets of the raw data ranging from 5 to 100% of the reads according to samples (Table 4). The output of taxonomical assignment of the reads with the fast bowtie alignment varied greatly according to the samples, with lower outputs for tick samples due to a poor representation of this species in the databases. As far as possible, the order of magnitude in the number of reads analyzed was kept in a range of hundreds of thousands of reads to be representative of the samples, with the exception of the MF fraction of the 889Geo sample, which displays only 15 736 reads (Table 4).

The taxonomic classification of reads obtained from pig samples was divided into three main groups: pig genome, ASFV genome and other sequences, which represent reads that are not assigned to a specific taxon. For UD pig samples, 93% of detected reads belonged to the pig genome, 0.18% to the ASFV genome and 6.82% to other sequences (Fig. 1A). For UMF pig samples, 83% of reads belonged to the pig genome, 0.75% to the ASFV genome and 16.25% to other sequences (Fig. 1B).

For MF pig samples, 92.20% of reads belonged to the pig genome, 0.04% to the ASFV genome and 7.76% to other sequences (Fig. 1C). A statistical difference was found only for the proportion of ASFV reads obtained in UD and UMF (p-value = 0.034).

The taxonomic classification of reads obtained from tick samples was divided into five groups: tick, pig, ASFV, bacteria and other sequences. For UD tick samples, 38.63% were identified as tick, 27.71% as pig, 0.03% as ASFV, 3.85% as bacteria and 32.78% as other sequences (Fig. 1D). For UMF tick samples, 35.38% corresponded to tick, 25.71% to pig, 0.04% to ASFV, 4.48% to bacteria and 34.39% to other sequences (Fig. 1E). For MF samples, 36.63% corresponded to tick, 24.35% to pig, 0% to ASFV, 1.34% to bacteria and 37.68% to other sequences (Fig. 1F). No statistical difference was found between UD, UMF and MF samples.

3.4-Number and percentage of ASFV reads in all samples

For all samples, the number of reads, the percentage and the mean coverage of reads aligned on the ASFV reference are shown in Table 5.

For pig samples, the quantity of ASFV reads varied between 3000 and 71 286 for UD, between 95 and 40 620 for UMF, and between 45 and 1220 for MF, representing 0.29%, 0.35% and 0.098% of total reads obtained for UD, UMF and MF of pig samples, respectively. The slight difference in the number of ASFV reads between the UD and MF fractions was statistically significant (p-value = 0.024).

For tick samples, the quantity of ASFV reads varied between 0 and 604 for UD, between 0 and 813 for UMF, and between 0 and 57 for MF. ASFV reads represented, for the best fraction, 0.002% of the total reads in tick samples (Table 5). No statistical difference was found between the UD, UMF and MF samples.

ASFV genome sequencing coverages are also shown in Table 5. For pig samples, the mean coverage was 15.11 for UD, 9.63 for UMF and 0.25 for MF fractions. For tick samples, the mean coverage was 0.11 for UD, 0.12 for UMF and 0.005 for MF fractions.

4-Discussion and conclusion

In this study, we evaluated a new strategy of DNA enrichment for high-throughput sequencing of the ASFV genome. The strategy, already used in two studies [START_REF] Masembe | Genome Sequences of Five African Swine Fever Virus Genotype IX Isolates from Domestic Pigs in Uganda[END_REF][START_REF] Ndlovu | Genome Sequences of Three African Swine Fever Viruses of Genotypes I, III, and XXII from South Africa and Zambia, Isolated from Ornithodoros Soft Ticks[END_REF] but without an evaluation of its efficacy, is based on a specificity of the ASFV DNA genome which is not methylated, contrary to the host genome. Separation of methylated and un-methylated DNA should result in an enrichment in ASFV reads in un-methylated samples. This strategy was tested against samples corresponding to the different ASFV hosts, i.e., pig and tick samples.

Before the enrichment process, ratios in genome copy number of ASFV and hosts were roughly estimated (by PCR ratio) to be 1 in tick samples, and around 0.7 in pig samples. After mapping the reads on the different genomes, the ASFV DNA indeed represented less than 0.4% and 0.01% of the reads obtained in pig and tick samples respectively. For all pig samples, in the UMF fractions (depleted methylated DNA), the percentage of ASFV genome reads increased from 0.18% to 0.75% and, conversely, the percentage of pig genome reads decreased by 10% compared with the UD fractions. For tick samples, the proportion of reads corresponding to the host genome decreased by 3% between UMF and UD, indicating a low efficiency of the enrichment process. The decrease between the UD and UMF factions was more extensive for pig genome reads (10%) than the decrease for tick genome reads (3%). These observations can be explained by the difference in DNA methylation between invertebrates and vertebrates. Indeed, the DNA of invertebrates is less methylated than the DNA of vertebrates [START_REF] Bird | Variable patterns of total DNA and rDNA methylation in animals[END_REF][START_REF] Tweedie | Methylation of genomes and genes at the invertebrate-vertebrate boundary[END_REF]. However, the sequence of the whole Ornithodoros erraticus genome is unknown, so the percentage of tick reads is probably underestimated. Moreover, the proportion of ASFV reads in tick samples only increased in two samples out of eight and remained lower than 0.01%, which in our case is insufficient to obtain the whole sequence of the ASFV genome, with a mean coverage of less than 0.61.

In comparison with another enrichment approach for ASFV whole genome sequencing, DNA extraction from erythrocytes of a viremic pig showed 0.84% of ASFV reads in their total reads with a mean coverage of 103 reads per nucleotide [START_REF] Olesen | Complete genome sequence of an African swine fever virus (ASFV POL/2015/Podlaskie) determined directly from pig erythrocyte-associated nucleic acid[END_REF]. This approach could be a good strategy for fresh isolated erythrocytes but is not suitable for frozen whole blood.

In 2019, a workflow for efficient ASFV sequencing was proposed. This consisted of a target enrichment approach by ASFV hybridization with probes fixed on magnetic beads followed by the use of NGS technologies (either Illumina alone, or a combination of Illumina and Nanopore sequencing). This target enrichment approach allows better coverage compared to the no enrichment approach, and a smaller amount of data is generated during sequencing [START_REF] Forth | A Deep-Sequencing Workflow for the Fast and Efficient Generation of High-Quality African Swine Fever Virus Whole-Genome Sequences[END_REF]. However, this approach has only been tested on the reference genome used for the design of the probes and a strain close to this reference. Knowing that the core genome was constituted of 102 ortholog genes on the 301 ortholog genes described in ASFV pan-genome [START_REF] Wang | Pan-Genomic Analysis of African Swine Fever Virus[END_REF], the efficiency of enrichment by ASFV hybridization should be validated on other genotypes before being applied to unknown strains.

Our study is the first to investigate the efficiency and the utility of the separation of methylated and un-methylated DNA for the whole genome sequencing of the ASFV genome. Our results suggest a low enrichment of ASFV DNA using this approach. The cost of the enrichment process by separation of methylated and un-methylated DNA is around $35 per sample. An increase in the deepness of sequencing would most probably give similar results as EMD samples, with less extra work on the samples and may be cheaper than the enrichment process. In comparison, the target enrichment strategy, which costs approximately $225 per sample, is most effective, resulting in a significant reduction of data generated. However, the reduction of the sequencing costs does not compensate for the extra cost of the enrichment process; the real advantage of this strategy is the high quality of the sequence obtained. Cov. = sequencing coverage in average number of reads per nucleotide position.

Fig. 1 :

 1 Fig. 1: Relative abundance of reads generated in the different samples classified according the

Table 1 : Relative quantification of ASFV genome, host beta-actin and total DNA in pig and tick 389

 1

	390	samples.
		Sample	Host	VP-72 ASFV PCR (Ct*)	Beta-actin PCR (Ct*)	DNA quantity (ng)	ASFV PCR/Beta-actin PCR ratio
		889Geo		17.07	22.64	2610	0.74
		936Geo		17.07	22.27	2180	0.77
		941Geo	Pig	16.12	22.43	2130	0.72
		6518Geo		15.88	22.11	2060	0.72
		6547Geo		16.39	22.66	1540	0.72
		E geo F1		27.2	27.1	554.4	0.99
		E geo F2		33.95	29.33	392.4	1.16
		E geo M1		27.13	28.35	594	0.96
		E geo F3 E geo F4	Tick	35.51 38.77	28.5 28.85	374.2 357.5	1.24 1.34
		E geo F5		33.79	28.39	330.8	1.19
		E geo F6		31.29	28.26	410.8	1.10
		E geo M2		31.11	28.66	407.3	1.08
	391	*Ct = Cycle Threshold.

Table 2 : DNA quantity used for enrichment process and DNA quantity obtained after enrichment. 392

 2

					DNA after	DNA after
	Sample	Host	DNA for UD fraction (ng)	DNA for enrichment (ng)	enrichment for UMF fraction	enrichment for MF fraction	% DNA recovery (UMF+MF)
					(ng)(% input)	(ng)(% input)
	889Geo		99	550	178 (32)	282 (51)	84
	936Geo		100	550	203 (37)	209 (38)	75
	941Geo	Pig	98	550	162 (29)	275 (50)	79
	6518Geo		99	550	205 (30)	290 (53)	90
	6547Geo		98	550	158 (29)	326 (59)	88
	E geo F1		100	454	368 (81)	24 (5)	86
	E geo F2		100	292	210 (72)	9 (3)	75
	E geo M1		100	494	347 (71)	23(4)	75
	E geo F3 E geo F4	Tick	100 100	274 257	202 (74) 172 (67)	14(5) 9 (3)	79 70
	E geo F5		100	230	153 (67)	9 (4)	70
	E geo F6		100	310	392 (126)	13 (4)	131
	E geo M2		100	307	246 (80)	10 (3)	83
	393

Table 3 : Quantity of total bases generated, quality score of bases, quantity of total reads and mean 394 read length per sample.

 3 Quality of bases was analyzed by Phred Quality Scores ≥20 (%≥Q20) and 395 expressed in percentage. 396

	Fraction Samples	Generated bases	%≥Q20	Quantity of total reads	Mean read length (bp)
	UD	pigs	889Geo	229 542 433	80.79	1 682 871
			936Geo	385 414 722	80.86	2 749 366
			941Geo	410 776 191	76.63	3 124 738
			6518Geo	452 263 244	81.64	3 792 151
			6547Geo	2 935 297 256	78.98	21 827 756
			mean for pigs	882 658 769	79.78	6 635 376
		ticks E geo F1	1 213 297 206	82.27	9 285 185
			E geo F2	1 013 734 715	84.43	8 726 201
			E geo M1	855 764 828	87.38	7 243 049
			E geo F3	404 865 960	82.73	3 132 828
			E geo F4	468 405 245	83.33	3 618 451
			E geo F5	319 338 365	84.07	2 578 862
			E geo F6	177 777 191	84.00	1 437 348
			E geo M2	305 292 664	83.29	2 542 952
			mean for ticks	594 809 522	83.94	4 820 610
			global mean	705 520 771	82.34	5 518 597
	UMF	pigs	889Geo	285 321 178	80.59	2 246 476
			936Geo	222 820 790	80.98	1 751 451
			941Geo	17 393 456	82.28	150 956
			6518Geo	986 698 698	83.10	6 468 942
			6547Geo	248 717 004	80.28	1 934 593
			mean for pigs	352 190 225	81.45	2 510 484
		ticks E geo F1	963 624 225	83.76	7 817 144
			E geo F2	940 491 921	83.99	8 008 483
			E geo M1	1 118 167 936	86.58	9 220 704
			E geo F3	152 407 597	82.92	1 203 993
			E geo F4	244 449 864	84.03	2 013 848
			E geo F5	324 824 629	84.41	2 728 169
			E geo F6	287 311 527	82.70	2 170 620
			E geo M2	148 976 803	84.68	1 380 652
			mean for ticks	522 531 813	84.13	4 317 952
			global mean	457 015 818	83.10	3 622 772
	MF	pigs	889Geo	2 151 395	80.31	15 736
			936Geo	56 192 863	76.39	414 846
			941Geo	104 536 025	79.83	754 918
			6518Geo	1 139 572 253	82.56	7 757 735
			6547Geo	126 458 347	80.45	874 201
			mean for pigs	285 782 177	79.91	1 963 487
		ticks E geo F1	885 423 136	85.64	8 165 392
			E geo F2	254 288 416	84.29	2 169 858
			E geo M1	109 382 507	84.16	977 897
			E geo F3	53 307 894	85.23	445 062
			E geo F4	16 326 955	83.85	149 147
			E geo F5	17 295 136	82.75	135 737
			E geo F6	127 530 447	84.14	1 048 408
			E geo M2	80 531 098	84.62	777 670
			mean for ticks	193 010 699	84.34	1 733 646
			global mean	228 692 036	82.63	1 822 047

Table 4 : Number and percentage of the total reads used for metagenomics analysis. 399

 4

	Fraction	Samples		Number of reads used for metagenomics analysis	Percentage of reads used for metagenomics analysis
	UD	pigs	889Geo	504	30%
			936Geo	824	30%
			941Geo	937	30%
			6518Geo	1 137	30%
			6547Geo	1 091	5%
			mean for pigs	899	25%
		ticks	E geo F1	464	5%
			E geo F2	436	5%
			E geo M1	724	10%
			E geo F3	939	30%
			E geo F4	1 085	30%
			E geo F5	773	30%
			E geo F6	431	30%
			E geo M2	762	30%
			mean for ticks	702	21.25%
			global mean	778	23.12%
	UMF	pigs	889Geo	673	30%
			936Geo	525	30%
			941Geo	150	100%
			6518Geo	646	10%
			6547Geo	580	30%
			mean for pigs	515	40%
		ticks	E geo F1	390	5%
			E geo F2	400	5%
			E geo M1	922	10%
			E geo F3	361	30%
			E geo F4	604	30%
			E geo F5	818	30%
			E geo F6	651	30%
			E geo M2	414	30%
			mean for ticks	551	21.25%
			global mean	534	30.62%
	MF	pigs	889Geo	15	100%
			936Geo	414	100%
			941Geo	452	60%
			6518Geo	775	10%
			6547Geo	524	60%
			mean for pigs	436	66%
		ticks	E geo F1	408	5%
			E geo F2	216	10%
			E geo M1	391	40%
			E geo F3	356	80%
			E geo F4	149	100%
			E geo F5	135	100%
			E geo F6	314	30%
			E geo M2	466	60%
			mean for ticks	304	53.12%
			global mean	355	59.56%
	400
	401

Table 5 : Number and percentage of ASFV reads per sample and the sequencing coverage of the 402 reference strain Georgia2007/1. 403

 5

	Samples	UD			UMF		MF
		ASFV reads	%	Cov.	ASFV reads	%	Cov.	ASFV reads	%	Cov.
	889Geo	3000	0.18	2.20	5866	0.26	3.42		0.43	0.05
	936Geo	5544	0.2	4.25	4594	0.26	3.25		0.01	0.03
	941Geo	10760	0.34	7.68	95	0.06	0.06		0.01	0.06
	6518Geo	14222	0.38	9.19	40620	0.63	33.96		0.02	0.99
	6547Geo	71286	0.33	52.21	10491	0.54	7.46		0.02	0.1
	Mean	20962	0.29	15.11	12333	0.35	9.63		0.098	0.25
	E geo F1	604	0.007	0.47	391	0.005	0.29		0.001	0.04
	E geo F2	12	0	0.01	6	0	0		0	0
	E geo M1	546	0.008	0.39	813	0.009	0.61		0.001	0
	E geo F3	1	0	0	0	0	0		0	0
	E geo F4	0	0	0	0	0	0		0	0
	E geo F5	2	0	0	0	0	0		0	0
	E geo F6	5	0	0	26	0.001	0.02		0	0
	E geo M2	28	0.001	0.02	10	0.001	0.01		0	0
	Mean	150	0.002	0.11	156	0.002	0.12	0.00025	0.005

Acknowledgements

The authors are grateful to Jean-Marie Guionnet and Angélique Moro for animal care and sampling at ANSES. Thanks to the Vectopole Sud for funding the insectary where the Ornithodoros erraticus were raised, the Direction Générale de l'Alimentation for the financial support provided CIRAD, and the European Union Horizon 2020 program 503 COMPARE (no. 643476) for partially financing this work. We are thankful to CIRAD and ANSES for funding the PhD grant of Rémi Pereira De Oliveira. Thanks to Grace Delobel for English editing.

Funding

This work was partially funded by the European Union Horizon 2020 program 503 COMPARE (Collaborative Management Platform for detection and Analyses of [Re-]emerging and foodborne outbreak in Europe; no. 643476).

Declaration of competing Interest

The authors declare that they have no competing interests.


Faheem Aslam

Saqib Aziz
email: saqib.aziz@rennes-sb.com

Duc K Nguyen

Khurrum S Mughal

Maaz Khan

Forex Market Efficiency in times of the COVID-19 On the Efficiency of Foreign Exchange Markets in times of the COVID-19 Pandemic

Keywords: COVID-19 pandemic, forex market, MF-DFA, high frequency, efficiency JEL Classifications: C10, C32, G10, G15

We employ multifractal detrended fluctuation analysis (MF-DFA) to provide a first look at the efficiency of forex markets during the initial period of the ongoing coronavirus disease 2019 (COVID-19), which has disrupted the global financial markets. We use high-frequency (5-min interval) data of six major currencies traded in forex markets during the period October 1, 2019 to 31 March 31, 2020. Before applying MF-DFA, we examine the inner dynamics of multifractality through seasonal and trend decompositions using loess. Overall, the results confirm the presence of multifractality in forex markets, which demonstrates, in particular, (i) a decline in the efficiency of forex markets during the COVID-19 outbreak and (ii) heterogeneous effects on the strength of multifractality of exchange rate returns under investigation. The largest effect is observed for the Australian dollar, which shows the highest (lowest) efficiency before (during) the COVID-19 pandemic, assessed in terms of low (high) multifractality. The Canadian dollar and the Swiss Franc exhibit the highest efficiency during the COVID-19 outbreak. Our findings may help policymakers shape a comprehensive response to improve forex market efficiency during such a black swan event.

1

Introduction:

The COVID-19 pandemic has jolted the global economy within a short span of time. Many countries around the world have been obligated to impose, among others, travel restrictions, border shutdowns, lockdowns, and social distancing in order to control the pandemic. These measures have severe impacts on supply channels, economic activities, and international trade at all scales. In its April 2020 World Economic Outlook, the International Monetary Fund predicted that the global economy might contract by 3% in 2020. 1 Financial markets are also witnessing a massive disruption with a very aggressive revaluation and adjustment process across the world amid this global pandemic. For instance, the Dow Jones Industrial Average dropped by 12.9%, and the S&P 500 index lost nearly 12% in a single day, on March 16, 2020. It was the worst percentage drop since the infamous "Black Monday" crash of 1987.

Stock markets are in turmoil as the pandemic has severely restricted economic activity due to protection measures and suspension of major events.

It is now well known that sudden "large" shocks, such as the global financial crisis of 2008, cause structural changes in both commodity and financial markets, which can then have potential asymmetric effects on market efficiency, portfolio allocation, and volatility spillovers (e.g., Rapach and Strauss, 2008;Managi and Okimoto, 2013;Mensi et al., 2015).

Since the onset of the COVID-19 pandemic, the foreign exchange (forex) market, which is by far the largest financial market 2 , has also witnessed unprecedented movements, and is thus subject to close watch by global portfolio investors, market regulators, and policymakers.

Most central banks scrambled to adjust the monetary frameworks to address the feedback loop between exchange rate movements and capital outflows in a bid to weather the financial setbacks from the COVID-19 outbreak. Policymakers' concerns stem from the simultaneous 1 https://www.imf.org/en/Publications/WEO/Issues/2020/04/14/weo-april-2020 2 The forex market is the largest financial market in terms of average daily trading volumes (with a record of US$ 6.6 trillion in April 2019, compared to US$ 5.1 trillion recorded previously in April 2016) and geographical presence. https://www.bloomberg.com/news/articles/2019-09-16/global-currency-trading-surges-to-6-6-trilliona-day-market decrease in aggregate and foreign demand, which could put pressure on the currencies with the flow of foreign aid across countries from international donor agencies and financial institutions. Both corporate and institutional investors are worried about the eventual fall of their portfolio investments and seek possible safe havens for their financial assets as the dollar funding cost in the forex market has demonstrated a sharp rise during this turmoil. While the current aggregate volatility levels are atypical, the COVID-19 impact may vary among currencies according to their market standing, risk sensitivities, and the government's policy response. For instance, the Australian dollar (AUD) hit a 17-year low of $0.59215, and the New Zealand dollar an 11-year low of $0.5850 cents. On the other hand, the currencies viewed as safe havens, including the Japanese yen (JPY), traded at 107.42 yen per US$ while the Swiss Franc (CHF) rose to US$ 0.9598 per CHF.

Concerning the COVID-19 impact on forex markets, a critical aspect that needs to be addressed is the efficiency of the forex market. The extant research shows that forex market efficiency is difficult to detect [START_REF] Katusiime | Foreign exchange market efficiency and profitability of trading rules: evidence from a developing country[END_REF], and the market efficiency of exchange rates changes over time, in particular, during crisis-like situations [START_REF] Levich | Measuring excess-predictability of asset returns and market efficiency over time[END_REF].

Inefficiency in the forex market generates different puzzling anomalies and delayed overshooting [START_REF] Li | Foreign exchange market inefficiency and exchange rate anomalies[END_REF]. Forex market efficiency is estimated through various approaches such as pairwise co-integration tests [START_REF] Layton | Multivariate cointegration testing of the efficiency of Australia's spot forex market[END_REF], linear unit root tests [START_REF] Giannellis | Testing for efficiency in selected developing foreign exchange markets: an equilibrium-based approach[END_REF], correlation functions [START_REF] Podobnik | Stochastic processes with power-law stability and a crossover in power-law correlations[END_REF], network analysis [START_REF] Jeong | The large-scale organization of metabolic networks[END_REF], and Pedroni's panel co-integration method [START_REF] Makovský | Modern approaches to efficient market hypothesis of FOREX-the central European case[END_REF].

The mainstream literature on financial markets is mainly based on the fundamental assumption that stock prices follow a normal distribution and on Bachelier's (1900) random walk hypothesis (RWH). However, the econophysics literature rejects the RWH hypothesis, arguing that asset prices have different fundamental properties [START_REF] Mandelbrot | The variation of some other speculative prices[END_REF][START_REF] Mandelbrot | When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models[END_REF][START_REF] Mandelbrot | The variation of the prices of cotton, wheat, and railroad stocks, and of some financial rates[END_REF], including fat tail [START_REF] Gopikrishnan | Price fluctuations and market activity[END_REF], long-term correlation [START_REF] Alvarez-Ramirez | Short-term predictability of crude oil markets: a detrended fluctuation analysis approach[END_REF], volatility clustering [START_REF] Kim | Long-term memory and volatility clustering in high-frequency price changes[END_REF], fractals and multifractals [START_REF] He | The empirical analysis for fractal features and long-run memory mechanism in petroleum pricing systems[END_REF], and chaos [START_REF] Adrangi | Chaos in oil prices? Evidence from futures markets[END_REF]. Therefore, fractal analysis has widely been applied in the financial market research. By using R/S analysis, [START_REF] Peters | Fractal market analysis: applying chaos theory to investment and economics[END_REF] and [START_REF] Edgar | Chaos and order in the capital markets, a new view of cycles, prices and market volatility[END_REF] demonstrated monofractal properties in several financial markets. However, monofractals cannot describe the height-height correlation function [START_REF] Barabási | Multifractality of self-affine fractals[END_REF], while multifractal detrended fluctuation analysis (MF-DFA) can overcome this weakness [START_REF] Kantelhardt | Multifractal detrended fluctuation analysis of nonstationary time series[END_REF][START_REF] Alvarez-Ramirez | Short-term predictability of crude oil markets: a detrended fluctuation analysis approach[END_REF]. The MF-DFA method has been employed in finance literature by, among others, [START_REF] Podobnik | Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series[END_REF], [START_REF] Wang | Cross-correlations between Chinese A-share and B-share markets[END_REF], [START_REF] Mandelbrot | A multifractal model of asset returns[END_REF], [START_REF] Kumar | Multifractal properties of the Indian financial market[END_REF][START_REF] Kumar | Multifractal properties of the Indian financial market[END_REF][START_REF] Oh | Multifractal analysis of Korean stock market[END_REF].

In the context of forex market efficiency, Ning et al. (2018) investigate the multifractal properties of both British Pound (GBP) and Euro (EUR) exchange rates from 2015 to 2017 through the MF-DFA method. Their results reveal significant nonlinear multifractal properties. [START_REF] Han | Comparative analysis of the multifractality and efficiency of exchange markets: evidence from exchange rates dynamics of major world currencies[END_REF] show, by examining the exchange rates of four significant currencies (EUR, GBP, CAD and JPY), that their fat-tail distributions and long-range correlation cause their multifractal properties. A recent work [START_REF] Shahzad | Intraday return inefficiency and long memory in the volatilities of forex markets and the role of trading volume[END_REF] confirms that JPY is a highly efficient currency while GBP is the least efficient. While it is too early to estimate the exact magnitude of economic and financial impacts of the COVID-19 pandemic, this paper provides a first look at the forex market using a sophisticated MF-DFA approach to understand the patterns of large movements in major currencies and the extent of forex market risk due to the pandemic.

This study contributes to the literature in three distinct ways. First, we use five-minute high-frequency exchange rate data for six major currencies (AUD, CAD, CHF, EUR, GBP, and JPY) against the US dollar to draw new and useful conclusions about the impacts of COVID-19 on forex market efficiency. Second, we decompose intraday exchange rate returns by seasonal-trend decomposition using loess (STL). The STL method is advantageous in that it enables us to unearth the inner dynamics of asset returns in addition to improving information reliability and decomposition by removing seasonal components (Laib et al., 2018a). Third, we use the robust multifractal MF-DFA technique for an in-depth comparison of the multifractal behavior of sample currencies before and during the COVID-19 outbreak.

The MF-DFA, a generalization of the DFA approach [START_REF] Kantelhardt | Multifractal detrended fluctuation analysis of nonstationary time series[END_REF], allows estimation of multiple scaling exponents within time series.

Our main findings show evidence of a significant change in the strength of multifractality, pointing to a general decline in forex market efficiency during the COVID-19 period compared to the pre-pandemic level. The efficiency difference might have its roots in how investors perceive currencies-as assets-and in the fundamentals that determine their underlying value. In any case, the price efficiency of any asset is based on the premise that its prices have incorporated all relevant information. Considering various channels through which prices adjust in the forex markets, the difference in the speed of price adjustment may reflect overshooting. Dornbush (1976) shows that an unanticipated change in money supply leads to exchange rate overshooting because consumer prices cannot move immediately to reflect the change in money supply. Therefore, a temporary disequilibrium in forex markets may represent the adjustment of prices based on information received through a relatively faster channel.

The COVID-19 outbreak has led to an immediate response from investors while policymakers' interventions may take time. Forex markets tend to become inefficient under such volatile conditions. Against this backdrop, our study provides policymakers and regulators with well-coordinated and gradual responses to the turbulent conditions in the forex markets in order to ensure their efficiency and the stability of the respective currencies.

The remainder of the study is organized as follows. Section 2 presents the data and methodology. Section 3 reports and discusses the empirical results. Section 4 provides some concluding remarks.

Data and Methodology

Data description

We use high-frequency data (five-minute intervals) to reveal the inner dynamics of forex market efficiency under the COVID-19 impact. The high-frequency data more accurately accounts for long memory and structural changes due to improved continuity in the distribution function. The high-frequency data sample consists of six major currencies traded in forex markets: AUD, CAD, CHF, EUR, GBP, and JPY. All prices are expressed in indirect quotes against US dollars (i.e., amounts of foreign currency per unit of the US dollar). Data from October 1, 2019 to March 31, 2020 are collected from TrueFx and converted to fiveminute windows using the "high-frequency" toolkit in R, developed by Boudt et al. (2020). 3 COVID-19 was first reported by Chinese authorities on December 31, 2019 to the World Health Organization (WHO). Based on this date, the intraday exchange rate data are divided into two three-month trading periods [START_REF] Zhu | A novel coronavirus from patients with pneumonia in China[END_REF]: October 1, 2019 to December 31, 2019 (before COVID-19) and January 1, 2020 to March 31, 2020 (during COVID-19).

The intraday five-minute exchange rate fluctuations are presented in Figure 1. We calculate the intraday forex return as in Eq. ( 1).

= - -∆ (1)
where, denotes the closing price of currency i at time t, ∆ is the time interval ∆ (five minutes).

******* Table 1 and Figure 1 about Here******* Table 1 shows descriptive statistics of exchange rate returns for the six currencies used in this study. The average five-minute returns are nearly zero before COVID-19. However, the AUD, CAD, EUR, and GBP returns turn negative while the JPY and CHF remain positive during the COVID-19 outbreak. Before COVID-19, the GBP exhibit the highest volatility and fluctuates within a range of 2 to -0.6 basis points, while the AUD return series remains the most volatile during the COVID-19 outbreak fluctuating between 1.1 to -1.6 basis points. All exchange rates were positively skewed before COVID-19, but the AUD, CAD, and GBP have become negative during the pandemic. The kurtosis coefficient of all return series is greater than 3, indicating their fat-tailed behavior. Figure 1 shows all exchange rate trends before and during COVID-19. A consistent movement with small fluctuations can be noticed before COVID-19, while the first quarter of 2020 exhibits relatively large fluctuations.

Methodology

We apply multifractal detrended fluctuation analysis (MF-DFA) to measure the efficiency of exchange rates. Before doing so, we employ the STL method to decompose the time series [START_REF] Cleveland | STL: A seasonal-trend decomposition[END_REF]. Since we use five-minute returns, the STL method helps handle any type of seasonality because it is robust to outliers and flexible enough to allow seasonal component variations over time. In addition to improved reliability and decomposition through removing seasonal components, it also reveals the inner dynamics of asset returns (Laib et al., 2018a). From a technical perspective, the STL method breaks down each exchange rate return time series into a deterministic trend (Ti), a seasonal (Si) component, and a stochastic remainder (Ri) component (Laib et al., 2018b;[START_REF] Miloş | Multifractal detrended fluctuation analysis (MF-DFA) of stock market indexes: empirical evidence from seven Central and Eastern European markets[END_REF] as follows:

= + + (2)
To carry out the STL decomposition, we use the R Stats package (Laib et al., 2018a[START_REF] Shiskin | The X-11 variant of the census method II seasonal adjustment program[END_REF][START_REF] Miloş | Multifractal detrended fluctuation analysis (MF-DFA) of stock market indexes: empirical evidence from seven Central and Eastern European markets[END_REF]. 4 For our 24-hour trading data with five-minute frequency, we set the span of the loess window for seasonal extraction equal to 288, which is the number of trading prices in one day. The results of the STL decomposition of intraday exchange rates are shown in Figure 2. For example, four AUD graphs in panel A present the original time series of daily returns (first row), the seasonal component (second row), the trend component (third row), and the remainder component (fourth row). The AUD exhibits a seasonal pattern consistent with previous findings (De [START_REF] De Bondt | Further evidence on investor overreaction and stock market seasonality[END_REF]Thaler, 1987, Miloş et al., 2020). However,

the

Multifractal Detrended Fluctuation Analysis

Once STL decompositions are obtained, we apply the MF-DFA technique to compute multifractality in the time series following a five-step procedure. Formally, let , ,…….., be a possible non-stationary time series and N the length of the series (number of observations). Then, the MF-DFA method can be summarized as follows:

Step 1: Construct the profile for MF-DFA estimation as follows:

= ∑ -̅ (3)
where, = 1,2,3. . . . . . , $ and ̅ = ∑ represents the average of the whole time series.

Step 2: Divide the profile into $ % = & $ ' ( ) non-overlapping boxes of the same length s. While the length N of the series is usually not a multiple of the considered time scale s, it disregards a short part of the profile at the end. It is not necessary to disregard this part of the series, and the same procedure will be repeated from the opposite side. Thus, 2$ % segments are obtained in total. [START_REF] Peng | Mosaic organization of DNA nucleotides[END_REF] initiated the process with 10 < ' < $' 5 ( .

Step 3: Calculate the local trend for each of the 2$ % parts by kth-order polynomial fit.

After that, the variance is given by

-. ', / = 1 ' 01 2 / -1 ' + 34 -5 6 3 7 . % 8 (4)
for the segments between m = 1 and m = Ns and by

-. ', / = 1 ' 01 2$ -/ -$ % ' + 34 -5 6 3 7 . % 8 (5)
for the segments between / = $ %9 and / = 2$ % .

Step 4: In Equations ( 4) and ( 5), 5 6 8 represents the polynomial fit of the segments.

Averaging yields the qth-order fluctuation function, given by

-: % = ; 1 2$ % 02-. ', / 4 . < = :/. ? /: (6)
for any q ≠ 0, and by

-G % = H5 I 1 4$ % 0 2-. ', / 4 . < = K (7)
for L = 0.

The parameter q allows us to distinguish between segments with large and small fluctuations. Its negative and positive values show small and large fluctuations, respectively.

The DFA case is given for q=2. Note that -: % is the increasing function of s.

Step 5: Compute the relationship between -: % and s. The task consists of establishing the scaling exponent of the fluctuation function for any fixed q. If -: % is a power law, the time series are in the log-log scale for that particular q.

M N O ~QR N (8)
Through the least-square fit, the slope of ln -: ' is the generalized Hurst exponent h(q), which describes the fractal structure of the time series. An increase in the s scale shows how fast -: ' of local fluctuations grows. If h(q) is constant at different values of q, then the series is monofractal. Otherwise, the series is said to be multifractal. For multifractal series, h(q) describes the scaling behavior of segments with large (small) fluctuations when q is positive (negative). Usually, the multifractal series comes with smaller h(q) at positive q values. The range of h(q), which indicates the level at which the series is multifractal, can be estimated by ∆ℎ = ℎ :6 T -ℎ :6UV . Specifically, it is called the multifractal spectrum width, which represents the interval between the maximum and minimum probabilities and, thus, measures the degree of the multifractality property. The h(q) value declines as q increases [START_REF] Zunino | A multifractal approach for stock market inefficiency[END_REF], and a higher ∆ℎ indicates the presence of higher multifractality in time series [START_REF] Cajueiro | Does financial market liberalization increase the degree of market efficiency? The case of the Athens stock exchange[END_REF][START_REF] Anagnostidis | Has the 2008 financial crisis affected stock market efficiency? The case of Eurozone[END_REF]. The richer the multifractality, the higher is the degree of market inefficiency because of long-range autocorrelation properties and fat-tail characteristics. Since multifractality is negatively correlated with market efficiency, the higher the width of the multipole spectrum, the lower is the degree of market efficiency [START_REF] Caraiani | Evidence of multifractality from emerging European stock markets[END_REF][START_REF] Domino | The use of the Hurst exponent to predict changes in trends on the Warsaw Stock Exchange[END_REF]. The MF-DFA turns into the DFA at q = 2. More broadly, the Hurst exponent describes the fractal structure of the time series. When h(q) = 0.5, the fluctuation related to q is said to follow a random walk. It is persistent when h(q) > 0.5 and anti-persistent when h(q) < 0.5.

The h(q) value estimated from the MF-DFA can also be presented as a function of Renyi exponent τ(q), as follows:

W N = NR N -X (9)
The singularity strength h(q) and the singularity spectrum D(q) can be calculated via Legendre transform, or related to h(q), through the following equations:

ℎ L = YZ L YL = ℎL -Lℎ′ L (10) \ L = LℎL -Z L = 1 + L2] -ℎ L 4 (11)
The multifractal spectrum D(q) describes the fractal dimension of the ensemble formed by all the points that share the same singularity exponent h(q). The width of the multifractal spectrum is the difference between h(q)max and h(q)min, representing the maximum and minimum probability, respectively. A wider multipole spectrum potentially indicates a lower efficiency level in the forex market [START_REF] Domino | The use of the Hurst exponent to predict changes in trends on the Warsaw Stock Exchange[END_REF][START_REF] Caraiani | Evidence of multifractality from emerging European stock markets[END_REF]. We use the R package MF-DFA, developed by Laib et al. (2018a) and Laib et al. (2018b), for the MF-DFA analysis. 5 We used time scales ranging from 10 to N/5 intraday (five-minute exchange rate) observations.

Empirical Results and Discussions

Figure 3 shows the MF-DFA findings for the remainder component of all exchange rates for 2019 (before COVID-19) and 2020 (during COVID-19) under consideration. It indicates the fluctuation function log2 -: ' versus log2(s) plot (q = -10, q = 0, and q = 10), the value of the generalized Hurst exponent h(q) over the range q ∊ [-10, 10], the Renyi exponent τ(q), and the multifractal spectrum _ `. log2 -: versus log2(s) plot at q = -10, q = 0, and q = 10, which is well-shaped and appears as a straight line. We calculate the slope of the generalized Hurst exponents for both the short and the long term. The results for 2019 show that h(q) reaches its highest value of 0.58 at q = -10, declines to 0.53 at q = 0, and drops to its lowest value of 0.45 at q = 10. Likewise, from its highest value of 0.71 in 2020, at q = -10, it declines to 0.49 at q = 0 and then to 0.28, its lowest value, at q = 10. This declining trend of the generalized Hurst exponent h(q) confirms its dependence on q, suggesting the existence of multifractality in the time fluctuations of the remainder component for the AUD in 2019. We obtain similar results and patterns for the series of exchange rate returns of the remaining five currencies. Their results are presented in Figure 3, from 3-B to 3-L.

Furthermore, the Hurst component for 2019 and 2020, h(q) = 0.52 and h(q) = 0.46, respectively, at the q = 2 scaling component (the setting for stationarity of return series)

indicates high persistence in 2019 and low persistence in 2020 in the remaining components. -10, 10]. The strength of multiple spectra in Figure 3 also demonstrates the comparative strength of multifractality across exchange rates before and during the COVID-19 pandemic. The larger the range, the more the multifractality dwells in the series [START_REF] Kantelhardt | Multifractal detrended fluctuation analysis of nonstationary time series[END_REF]. The finding evidences a declining pattern of the Hurst exponent h(q) for all six exchange rates we consider in 2019 and 2020, confirming the time fluctuations of multifractality in the remainder components (Laib et al., 2018a).

******* Table 2 about Among the six intraday exchange rate returns, the GBP and CAD currencies witness the highest degree of multifractality overall, while the AUD and JPY exhibit the lowest degree before COVID-19.

COVID-19 caused significant movements and affected the strength of intraday multifractality in the first three months of 2020 with marked changes in the strength of multifractality of currencies. More specifically, during the COVID-19 period, we observe the highest multifractality for the AUD (∆h = 0.42) followed by the JPY (∆h = 0.40), while the CAD (∆h = 0.25) and CHF (∆h=0.30) have the lowest multifractality. The multifractal properties are indicative of the financial markets' efficiency [START_REF] Anagnostidis | Has the 2008 financial crisis affected stock market efficiency? The case of Eurozone[END_REF].

Therefore, based on the multifractal properties observed, the efficiency levels of AUD, JPY, and EUR decreased while those of CAD, CHF, and GBP improved during the COVID-19 period. The AUD showed the most extreme behavior among the six selected currencies; it was the most efficient currency before COVID-19 but became less efficient with the outbreak of the pandemic.

Similarly, in terms of relative efficiency among the six currencies, the JPY was the second-most efficient currency before the COVID-19 period, but was severely affected during the outbreak and became one of the inefficient currencies. Overall, the increase in multifractality confirms that COVID-19 has adversely affected the forex market as it touches a lower level of inefficiency. Our findings are broadly in line with [START_REF] Shahzad | Intraday return inefficiency and long memory in the volatilities of forex markets and the role of trading volume[END_REF].

The persistence of the return series is an important determinant of multifractality. The classical Hurst exponent (q = 2) shows persistent behavior (positive autocorrelation) in 2019

for the JPY (h(q) = 0.54), the AUD (h(q) = 0.52) and the GBP (h(q) = 0.50), which means that positive (negative) values in the previous period would most probably be chased by positive (negative) values in the subsequent period. The remaining three currencies, namely, CAD (h(q)=0.48), CHF (h(q)=0.47), and EUR (h(q)=0.46), exhibit anti-persistent behavior with negative autocorrelation. This evidence implies that any change (positive/negative) in previous time periods would probably be followed by an opposite (negative/positive) change in subsequent times.

The influence of COVID-19 can be noticed in the persistence behavior of forex markets. Three out of six currencies analyzed in the COVID-19 period show changes in their persistence pattern. In 2020, the CHF (h(q) = 0.52) and EUR (h(q) = 0.51) shifted from antipersistent to persistent behavior. The Hurst exponent (q = 2) for the AUD declined from h(q) = 0.52 in 2019 to h(q) = 0.49 in 2020. We observe no change in the persistence levels for the GBP and the JPY before and during COVID-19. Similarly, the CAD was found to be antipersistent before and during the COVID-19 period. Overall, the persistence level has increased among these exchange rates.

The AUD and JPY have shown a most peculiar behavior during the COVID-19 pandemic. Both currencies demonstrated a high degree of inefficiency, in relative terms, during the COVID-19 period as opposed to increasing efficiency before. On the other hand, the CAD was at the lowest end in terms of efficiency before COVID-19, but demonstrated the highest degree of efficiency during the pandemic. These differences might have their roots in how these currencies are perceived by investors-as financial assets-and in the fundamentals that determine their underlying value. The AUD is known as a commodity currency because it is sensitive to fluctuations in commodity prices [START_REF] Lodewijks | Global economy and the Australian dollar[END_REF]. There is also evidence that it is highly correlated with Chinese economic conditions because of Australia's raw material exports to Chinese industries. 6 With the economic slowdown in the Chinese economy, the demand for the AUD currency decreased.

Moreover, as COVID-19 cases rose in China and other countries, the AUD depreciated against the USD in the forex market. The decrease in interest rates decided by the Reserve Bank of Australia due to lower inflation expectations will prompt investors to shift to markets offering higher returns.

The recovery of the AUD relative to the USD towards the end of March 2020 represents the risk appetite of investors. 7 Our analysis shows that the AUD became more inefficient after the COVID-19 outbreak, which suggests profitable investment opportunities.

In the meanwhile, the JPY currency, which generally is considered a safe haven, lost its value amid rising COVID cases in neighboring China. 8 For its part, the CAD depreciated against the USD due to COVID-19 outbreak fears, and the Canadian central bank left room for interest rate cuts if required 9 . We cautiously explain it as an outcome of the central bank's timely intervention to provide room for economic recovery and stability that, in turn, boosted investor confidence in the Canadian economy and its currency. Thus, we see lower volatility in CAD/USD during COVID-19 compared to the prior period.

6 https://www.dailyfx.com/forex/fundamental/forecast/weekly/aud/2020/03/07/Australian-Dollar-at-Mercy-of-COVID-19-as-Chinas-Economy-Slows.html 7 https://www.thestar.com.my/business/business-news/2020/04/08/dollar-drops-aussie-sterling-gain-as-risk-appetite-increases 8 https://www.fxstreet.com/news/usd-jpy-the-japanese-yen-losing-its-safe-haven-status-rabobank-202002201011 9 https://www.theglobeandmail.com/investing/markets/inside-the-market/market-news/article-canadian-dollar-falls-ahead-of-poloz-speech-as-coronavirus-worries/

In a nutshell, the varying impacts of COVID-19 on the multifractality and efficiency of the forex markets under consideration can be explained by the structure of these markets and the behavior of traders during the COVID-19 pandemic. A significant portion of international news headlines evoked uncertainty, negative sentiments, and fear and sadness (Aslam et al., 2020a). In such a turbulent environment, investors tend to shy away from trading activity to avoid financial losses, which adversely affects the financial markets [START_REF] Mukerji | Ambiguity aversion and incompleteness of financial markets[END_REF]Tallon, 2001, Levy and[START_REF] Levy | Terror and trade of individual investors[END_REF]. COVID-19 affects not only the trading behavior but also the multifractal behavior of financial markets with the considerable uncertainty (Aslam et al., 2020b) and complex issues associated with a low degree of information determinacy. These factors potentially explain the differences in the overall effects of COVID-19 on forex market efficiency, with the heterogeneous expectations of investors and traders about the fundamentals underlying a currency as well as strong trade ties with a severely affected country as with the AUD and reversal of safe haven perception as in the case of the JPY.

Forex market efficiency also depends on the policy response from governments on both fiscal and monetary aspects, which are subject to several factors linked to the prevailing economic and political environment, among others. For example, an unanticipated change in money supply may lead to exchange rate overshooting as consumer prices cannot move immediately to reflect the money supply change (Dornbush, 1976). Building on this analogy, we can argue that a temporary disequilibrium in the forex markets may represent the adjustment of prices to information received through a relatively faster channel. The COVID-19 pandemic emerging from China received a quick response from investors while adjustments by policymakers may take time due to the rather measured interventions such as the recent deployment of central bank dollar swap lines.

Conclusion

This paper provides a first look at forex market efficiency in the context of the COVID-19 pandemic as its economic and social costs are a matter of great concern to the society, policymakers, market operators, and individual investors. For this purpose, we conducted a comparative analysis of the efficiency level of six major currencies traded in forex markets, namely, AUD, CAD, CHF, EUR, GBP, and the JPY. Our study is thus vital as it provides an insightful analysis of fractals in the six major international forex markets. The intraday data of five-minute frequency from October 1, 2019 to March 31, 2020 is divided into before and during the COVID-19 outbreak, with a three-month trading window for each. The robust MF-DFA and STL methods are employed to capture the presence of multifractality in the sample exchange rates against the UD dollar. These methods provide insight into the long-range dependence of these exchange rates in terms of their multifractal structures.

Market efficiency is pivotal in both resource allocation and capital formation, which lead to economic development and stability [START_REF] Rizvi | How does crisis affect efficiency? An empirical study of East Asian markets[END_REF]. The findings of this study show varying degrees of forex market efficiency before and during the COVID-19 outbreak.

Investors in the forex market can structure their investment and risk management strategies to exploit market inefficiencies [START_REF] Dragotă | Market efficiency of the Post Communist East European stock markets[END_REF]. In this scheme of things, the estimated Hurst coefficient h(q) provides insight into the dynamics and trending characteristic of forex markets, which are useful for investors in aligning their strategies to market conditions. Against the backdrop of COVID-19, both AUD and CAD exchange rates exhibit anti-persistent behavior (0 < H < 0.5), suggesting mean-reverting characteristics in the two currencies. This implies that, for a time series, a higher financial asset value in the prior periods may decline in the subsequent periods through a mean-reversion process. For the remaining four currencies, a high value of the H exponent (0.5 < H < 1) is indicative of long memory in the time series, where the future value of a series depends partially on its values in prior periods.

In non-technical terms, this evidence suggests a lack of efficiency in the sample currency markets both before and during the COVID-19 pandemic. The AUD and JPY demonstrate the highest degree of efficiency before whereas the CAD and CHF do so during COVID-19.

While only three (CAD, CHF, and GBP) of the six major currencies improved their efficiency level during COVID-19, most currency returns generally tend to be more persistent.

Since the COVID-19 pandemic is an ongoing crisis, we believe that no single policy response can be considered successful in the short term. The extant literature suggests minimum intervention in forex markets, while effective management of internal risks in the forex markets [START_REF] Han | Nonlinearity and efficiency dynamics of foreign exchange markets: evidence from multifractality and volatility of major exchange rates[END_REF] and financial liberalization [START_REF] Kawakatsu | Financial liberalization and stock market efficiency: an empirical examination of nine emerging market countries[END_REF] improve market efficiency. The changes in the market efficiency levels of the considered forex markets may be driven by investors' herd behavior due to the crisis and the fear prevailing in the market regarding the underlying fundamentals of the economy. Such behavior may lead to higher autocorrelation, and hence a decrease in market efficiency. Policymakers may potentially address such behavioral anomalies by ensuring stability in the underlying macroeconomic fundamentals. Market participants would do well to reassess their investment and risk management framework in order to mitigate the new and somewhat higher risk levels of their exposures during the current tumultuous times.

While we acknowledge that the findings of this study should be viewed with caution given the sample period under analysis, it does pave the way for future research regarding the COVID-19 impact on forex markets in the short and long run, the effectiveness of policy responses in maintaining their stability, and volatility transmissions, using larger samples and analysis windows. The changes in investor behavior across two time horizons will potentially transform market linkages and correlations. Thus, the fractal models can be used to further explore the fractal contagion effects caused by the COVID-19 pandemic. Moreover, investigating the primary source of the increase in multifractality to understand the psychological aspects of investment behaviors is potentially an interesting direction for future research.

Declaration of interest: none

Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 1 for explanation of the currency codes).

  trend component shows a decline after mid-March 2020. As to the remainder component, it does not follow any specific pattern. The STL decomposition results for other exchange rate returns are reported in Figure 2, panels B to L. ******* Figure 2 about here*******

*

  ****** Figure 3 About Here******* For illustration, Figure 3-A shows the standard MF-DFA findings for the remainder components of AUD for 2019. The top-left panel of Figure 3-A shows the fluctuation function

Figures 3 -

 3 Figures 3-A and 3-B (bottom left) show the Renyi exponent, τ(q), which is nonlinear for

  here******* More concretely, the GBP shows the highest multifractality in 2019, with the widest generalized Hurst exponent (∆h = 0.49), followed by the CAD (∆h = 0.41), thus indicating the highest level of unevenness in local fluctuations in the GBP and CAD exchange rate return series. By contrast, the AUD and JPY have the lowest degree of multifractality, with ∆h = 0.13 and ∆h = 0.22 (Hurst exponent width), respectively, corresponding to the highest level of evenness in local fluctuations in the AUD and JPY exchange rate returns. The EUR (∆h = 0.31) and CHF (∆h = 0.36) stay in the middle in terms of multifractality before COVID-19.

FigureFigure 1 :

 1 Figure 1-A). AUD

Figure

  Figure 2-A). STL decomposition AUD-2019

Figure 2 :

 2 Figure 2: STL decomposition of intraday (five-minute) exchange rate returns for 2019 (blue) and 2020 (red). Row 1: Original five-minute time series; row 2: seasonal component; row 3: trend component; row 4: remainder. STL = seasonal-trend decomposition using loess; seeTable 1 for explanation of the currency codes.

FigureFigure 3 :

 3 Figure 3-A). MF-DFA AUD-2019 Figure 3-B). MF-DFA AUD-2020

Table 2

 2

	reports the width of the generalized Hurst exponent ∆h, which indicates the
	strength of multifractality, for 2019 (before COVID-19) and 2020 (during COVID-19) over
	the range q ∊ [

Table 1 . List of currencies with summary statistics of their exchange rate returns against the US dollar (5-minute interval)

 1

	Currency	Australian dollar-AUD	Canadian dollar-CAD	Swiss Franc-CHF	Euro-EUR	British Pound-GBP	Japanese Yen-JPY
	Summary Statistics of Exchange rate returns
	2019* (Before COVID-19)
	Mean	0.000002	0.000001	0.000002	0.000002	0.000004	0.000000
	Std. Dev.	0.000229	0.000164	0.000197	0.000172	0.000379	0.000175
	Minimum	-0.003934	-0.004323	-0.003187	-0.003072	-0.006215	-0.002948
	Maximum	0.003485	0.005343	0.003490	0.003598	0.020735	0.004551
	Skewness	0.408098	0.818122	0.429145	0.792372	9.226872	0.360279
	Kurtosis	22.960322	114.113335	26.032817	37.991999	501.914380	53.750461
	Observations	18950	18950	18950	18950	18950	18950
	2020** (During COVID-19)
	Mean	-0.000007	-0.000004	0.000000	-0.000001	-0.000003	0.000001
	Variance	0.000720	0.000399	0.000399	0.000393	0.000535	0.000476
	Minimum	-0.016646	-0.015399	-0.004651	-0.004142	-0.009830	-0.005915
	Maximum	0.011148	0.004472	0.012390	0.007217	0.009860	0.014995
	Skewness	-1.367940	-3.624304	1.678191	0.821631	-0.600611	3.188010
	Kurtosis	61.144564	143.700572	65.516308	25.499180	35.421532	96.912052
	Observations	18362	18362	18362	18362	18362	18362
	* 2019: October 1, 2019 to December 31, 2019
	** 2020: January 1, 2020 to March 31, 2020

Table 2 .

 2 Generalized Hurst exponents for six currency exchange rates against the US dollar for 2019 and 2020 and their range over q ∊ [-10, 10].

	Order q AUD-19 CAD-19 CHF-19 EUR-19 GBP-19 JPY-19 AUD-20 CAD-20 CHF-20 EUR-20 GBP-20 JPY-20
	-10	0.5825	0.6915	0.6586	0.6410	0.7076	0.6315	0.7065	0.5907	0.6988	0.7164	0.7854	0.7405
	-8	0.5695	0.6748	0.6424	0.6238	0.6930	0.6173	0.6921	0.5752	0.6814	0.6990	0.7707	0.7243
	-6	0.5542	0.6531	0.6214	0.6012	0.6739	0.5998	0.6732	0.5546	0.6580	0.6743	0.7506	0.7016
	-4	0.5397	0.6259	0.5949	0.5724	0.6496	0.5814	0.6493	0.5290	0.6279	0.6397	0.7226	0.6705
	-2	0.5311	0.5922	0.5631	0.5387	0.6209	0.5697	0.6183	0.5036	0.5935	0.5976	0.6829	0.6308
	0	0.5280	0.5456	0.5243	0.5014	0.5817	0.5632	0.5696	0.4816	0.5577	0.5564	0.6298	0.5825
	2	0.5205	0.4787	0.4742	0.4606	0.5032	0.5382	0.4903	0.4541	0.5201	0.5173	0.5676	0.5229
	4	0.5025	0.4003	0.4156	0.4182	0.3819	0.4968	0.4026	0.4161	0.4808	0.4793	0.5078	0.4538
	6	0.4810	0.3407	0.3632	0.3808	0.2972	0.4589	0.3423	0.3819	0.4462	0.4469	0.4631	0.3972
	8	0.4627	0.3035	0.3252	0.3527	0.2492	0.4313	0.3056	0.3570	0.4205	0.4226	0.4331	0.3603
	10	0.4485	0.2796	0.2991	0.3325	0.2200	0.4118	0.2821	0.3392	0.4021	0.4048	0.4127	0.3363
	∆h	0.1340	0.4119	0.3595	0.3085	0.4876	0.2197	0.4244	0.2515	0.2967	0.3116	0.3727	0.4042

Note: See Table

1

for explanation of the currency codes.

  Table 1 for explanation of the currency codes.

The detailed documentation is available at https://cran.r-project.org/web/packages/highfrequency/highfrequency.pdf/.

The details of STL are available at https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/stl/.

The detailed documentation is available at https://www.rdocumentation.org/packages/MF-DFA/versions/1.1/topics/MF-DFA/.


Phan Viet

Hoa Vu

Frédéric Brigui

Arnaud Breloy

Yajing Yan

Guillaume Ginolhac

A New Phase Linking Algorithm for Multi-temporal InSAR based on the Maximum Likelihood Estimator

Keywords: Multi-temporal InSAR, Surface Displacement Monitoring, Phase Linking, Maximum Likelihood Estimator, Block Coordinate Descent algorithm

This paper presents a new algorithm for improving the estimation of interferometric SAR (InSAR) phases in the context of time series and phase linking approach. Based on maximum likelihood estimator of a multivariate Gaussian model, the estimation of the InSAR phases is solved using a Block Coordinate Descent algorithm. Compared to the state-of-the-art approaches, the main improvement lies on the joint estimation of the covariance matrix and the InSAR phases instead of using a plug-in coherence estimate obtained from the sample covariance of the data or the modeling of the temporal decorrelation of the target under observation. Results of synthetic simulations confirm the improvement brought by the proposed estimator.

INTRODUCTION

Since the last decade, the research community has been paying more attention to multi-temporal InSAR techniques, as satellite SAR images allow for the monitoring of large deforming areas with sub-centimeters accuracy [START_REF] Prébet | A data-adaptive EOF-based method for displacement signal retrieval from InSAR displacement measurement time series for decorrelating targets[END_REF][START_REF] Tong Minh | Radar interferometry: 20 years of development in time series techniques and future perspectives[END_REF]. Recently, with the systematic acquisitions of Sentinel-1 A/B images, regular and operational monitoring of displacement with multi-temporal InSAR approaches constitutes a prevalent subject in numerous studies.

Multi-temporal InSAR approaches have been developed based on how the signal decorrelation over time can be accounted for. For this, analysing the backscattering properties of SAR images is needed to define different scatterers. In the literature, two kinds of scatterers, Permanent Scatterer (PS) and Distributed Scatterer (DS) are distinguished. The current state-of-the-art multi-temporal InSAR approaches rely on 1) processing with point-wise time coherent PS, namely PS Interferometry [START_REF] Ferretti | Permanent scatterers in SAR interferometry[END_REF], 2) the construction of redundant interferogram networks in DS Interferometry [START_REF] Lanari | A small-baseline approach for investigating deformations on full-resolution differential SAR interferograms[END_REF]; 3) the combination of PSI and DSI [START_REF] Ferretti | A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR[END_REF]. PSI approaches have widely been deployed for urban area monitoring, but its application to natural areas is often limited due to the weak PS points density. In DSI, Small BAseline Subset (SBAS) approaches use only small temporal and spatial baselines SAR image pairs in the interferogram network, with the objective to minimize signal decorrelation. With these strategies, phase bias has been observed in the displacement estimation in case of long time series [START_REF] Ansari | Study of systematic bias in measuring surface deformation with SAR interferometry[END_REF]. Another important DSI approach corresponds to the Phase Linking (PL) approach. The main idea of this approach is to use all the N ×(N -1)/2 interferograms generated from a time series of N SAR images to yield the best estimate of N -1 single referenced phase difference [START_REF] Monti-Guarnieri | On the exploitation of target statistics for SAR interferometry applications[END_REF]. This approach allows for a full exploitation of all possible combinations of a SAR image stack by formally taking the impact of the temporal decorrelation into account. Most recent advances in multitemporal InSAR, e.g. EMI [START_REF] Ansari | Efficient phase estimation for interferogram stacks[END_REF] and sequential estimator [START_REF] Ansari | Sequential estimator: Toward efficient InSAR time series analysis[END_REF], have been developed from this baseline approach.

The Maximum Likelihood Estimator (MLE) based PL approaches (namely MLE-PL for the sake of brevity) present a statistically optimal estimator for the parameters of interest (e.g. DEM error, displacement rate). Besides properly weighting in a ML sense all the interferograms to limit spatial and temporal decorrelation, another advantage of MLE-PL approaches is that the estimates are asymptotically unbiased with a minimum variance by virtue of the properties of the MLE. In general, the MLE-PL approaches require prior and reliable information on the coherence to drive the estimation algorithm. Therefore, the performance of MLE-PL approaches strongly depends on the reliability of the prior coherence information.

In the state-of-the-art, MLE-PL approaches [START_REF] Monti-Guarnieri | On the exploitation of target statistics for SAR interferometry applications[END_REF][START_REF] Ferretti | A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR[END_REF][START_REF] Cao | Mathematical framework for phasetriangulation algorithms in distributed-scatterer interferometry[END_REF] do not solve the exact MLE, as they rely on a plug-in estimate of the coherence matrix. In this paper, we propose to develop the MLE-PL by jointly solving the estimation of all parameters (phases and covariance matrix). The proposed approach is based on the block coordinate descent (BCD). Additionally, we derive a Majorization-Minimization algorithm to solve the PL step. Simulations illustrate the interest of this approach. A time series of N SAR images are stacked along the temporal and spatial dimensions into a cube (Figure 1). Sliding window x i (colored in gray) contains a local observation for N snapshots:

SAR TIME SERIES MODEL

x i = [x 0 i , ..., x N -1 i ] ⊤ (1)
Within the scope of this study, we assume that the set

{x i } L i=1 with x i ∈ C N , ∀i ∈ [1, L] are distributed scatterers which are spatially homogeneous over L adjacent pixels. {x i } L i=1
is thus a set of independent and identically distributed (i.i.d.) vectors. As in the current literature, we assume in this paper that x i follows a zero-mean complex circular Gaussian distribution with the probability density function (PDF) of

f (x, C) = 1 π N Det(C) exp -x H C -1 x (2)
The second moment of x relates to interferograms which is given

∀(k, l) ∈ [0, N -1] 2 by E[x l (x k ) * ] = γ k,l σ k σ l exp(j(θ l -θ k )), (3)
where:

• σ 2 n = E[x n (x n ) H ] is the variance of x n . The vector of standard deviation is denoted as σ = [σ 0 , ..., σ N -1 ] • γ k,l ∈ [-1, 1] is the correlation coefficient between x k
and x l . We denote Γ the matrix with entries

[Γ] k,l = γ k,l . [Γ] l,l = 1, ∀l ∈ [0, N -1] • θ n is the phase of image n. θ = [θ 0 , ..., θ N -1 ].
Equation 3 can be rewritten in matrix form as

E[xx H ] ≜ C = ediag(θ) ((σσ ⊤ ) • Γ) Σ ediag(θ) H (4)
where Σ denotes the covariance matrix and with

ediag(θ) =    exp(jθ 0 ) 0 . . . 0 exp(jθ N -1 )    (5)
The matrix C fully characterises the InSAR phases and coherence. The InSAR principle is then to estimate all the elements of C. Since Σ is unknown in practice, several papers [START_REF] Monti-Guarnieri | On the exploitation of target statistics for SAR interferometry applications[END_REF][START_REF] Ferretti | A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR[END_REF][START_REF] Cao | Mathematical framework for phasetriangulation algorithms in distributed-scatterer interferometry[END_REF] considered using a plug-in estimate to solve for the MLE of θ (assuming known Σ). This estimate is usually |S|, i.e., the entry-wise modulus of the the sample covariance matrix (SCM). This two-step approach yields good results in practice, however it is know to be sub-optimal in terms of MSE. We propose in this paper to jointly estimate the matrix C and the phase difference by solving the MLE with an iterative approach.

NEW PHASE LINKING BASED ON THE MAXIMUM LIKELIHOOD ESTIMATOR

We reparameterize the equation ( 4) as

C(Σ, θ) = ediag(θ)Σ ediag(θ) H (6)
The maximum likelihood estimator corresponds to the solutions of the (negative log likelihood) minimization problem minimize

Σ,θ L (C (Σ, θ)) subject to Σ real symmetric θ 0 = 0 (7)
where L is the log-likelihood associated with model [START_REF] Tong Minh | Radar interferometry: 20 years of development in time series techniques and future perspectives[END_REF]. We propose a Block Coordinate Descent (BCD) algorithm to compute the MLE. This corresponds to an algorithm that iteratively minimizes the objective w.r.t to each variable (Σ or θ) while keeping the other fixed.

Update Σ

Let us update the variable Σ by minimizing L with fixed θ. The problem becomes

minimize Σ log |Σ| + Tr Σ -1 ediag(θ) H Sediag(θ)
subject to Σ real symmetric (8) where we used the two following relations

C -1 (Σ, θ) = ediag(θ)Σediag(θ) H -1 = ediag(θ)Σ -1 ediag(θ) H (9) log |C (Σ, θ) | = log |Σ| (10)
and dropped the multiplicative constant L. The minimizer is then obtained as the real part of the modified sample covariance matrix

Σ ⋆ = real(ediag(θ) H Sediag(θ)) (11)

Update θ

By fixing Σ, the problem is referred to as phase-linking [START_REF] Monti-Guarnieri | On the exploitation of target statistics for SAR interferometry applications[END_REF] or phase triangulation [START_REF] Cao | Mathematical framework for phasetriangulation algorithms in distributed-scatterer interferometry[END_REF], which reads

minimize θ Tr ediag(θ)Σ -1 ediag(θ) H S subject to θ 0 = 0 (12)
Let us denote w θ = e jθ0 , • • • , e jθ N -1 T . We then have the relation

Tr ediag(θ)Σ -1 ediag(θ) H S = w H θ (Σ -1 • S)w θ . ( 13
)
Let us also denote the set of vector with entry-wise unit norm as

U N = {w ∈ C N | |[w] i | = 1, ∀i ∈ [[0, N -1]]}. ( 14
)
From these notations, we need to know how to solve the generic problem

minimize w∈U N w H Mw (15)
that allows us to obtain a solution θ ⋆ for (12) from the phases of a solution w ⋆ of (15) (when plugging M = Σ -1 • S). If we restrict w to the constrained set U N , we have the relation

w H (M -λ M max I)w = w H Mw -N λ M max const. (16)
Hence, optimizing either the objective in (15) or ( 16) on U N will lead to the same solution. The quadratic form w H (Mλ M max I)w is concave, thus it can be majorized at point w t by its first order Taylor expansion

g(w|w t ) = 2Re{w H (M -λ M max I)w t -wt } + const. ( 17
)
with equality achieved at w t . Minimizing this surrogate corresponds to the problem

maximize w∈U N 2Re{w H wt } ( 18
)
whose solution is w ⋆ = P U N { wt }, where P U N {•} is the operator that project each entry of a vector on the unit complex circle (i.e., entry-wise normalization). Hence, we have a majorization-minimization algorithm [START_REF] Sun | Majorization-minimization algorithms in signal processing, communications, and machine learning[END_REF] to solve for (15) (that looks like a modified power method). The algorithm is summed up in the box Algorithm 1. Notice that this algorithm can also be used to solve the standard PL formulations, where M is constructed from a plug-in estimate of Σ [START_REF] Monti-Guarnieri | On the exploitation of target statistics for SAR interferometry applications[END_REF][START_REF] Ferretti | A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR[END_REF][START_REF] Cao | Mathematical framework for phasetriangulation algorithms in distributed-scatterer interferometry[END_REF].

Final algorithm

The final algorithm is shown in the box Algorithm 2 and is denoted as MLE-PL.

SIMULATION

To assess the performance of the proposed algorithm, we generate a matrix C from (4). The covariance matrix Σ is chosen as a Toeplitz matrix with coherence coefficient ρ ranging from 0.5 to 0.9. The N SAR phases are generated as random values in (-π, π), yielding θ = [-1.13, 0.25, 2.37, -1.78, -0.67] for all Monte-Carlo runs. At last, L i.i.d samples are then simulated from the distribution (2). A spatial window of L pixels is ranging from 6 to 100 with N = 5 dates and 1000 number of Monte Carlo simulations.

For comparison, other methods are also tested: 2p-InSAR is a common InSAR processing in which InSAR phases are only estimated in spatial dimension and with the use of a multilooking window. PL is the conventional 2-step phase linking [START_REF] Cao | Mathematical framework for phasetriangulation algorithms in distributed-scatterer interferometry[END_REF]: M = Σ -1 • S is constructed with the plug-in estimate Σmod = |S| (entry-wise modulus), then problem (15) is solved to obtain the phases estimates. In our setting, we use Algorithm 1 to solve (15).

We first test the convergence of the MLE-PL method in Figure 2 for L = 20 and ρ = 0.5. The cost function related to ( 7) is decreasing and reaches convergence after 7 iterations. In the following part of this study, the number of iterations is set to 10 to ensure the convergence of the MLE-PL. Different values of the coherence coefficient are studied, from high coherence (ρ = 0.9) to medium coherence (ρ = 0.7) to low coherence (ρ = 0.5). The MSE of each method decreases with the number of available samples which is expected. Interestingly, we observe that MLE-PL allows for an improvement of the accuracy of the phase estimation compared to PL or 2p-InSAR in all the considered setups.

Low coherence ρ = 0.5 Medium coherence ρ = 0.7

High coherence ρ = 0.9

CONCLUSION

In this paper, a new phase linking algorithm based on MLE was proposed and applied to simulated data. This MLE-PL algorithm was obtained by jointly solving the estimation of all parameters (phases and covariance matrix). Additionally, we derived a Majorization-Minimization algorithm to solve the PL step that is also suitable to others standard PL approaches. Simulations illustrated that this new method offers some improvement in terms of MSE for various degree of temporal coherence.

Fig. 1 .

 1 Fig. 1. SAR time series datacube representation. Sliding window (in gray) x on dataset of the single-look-complex SAR acquisitions.

Algorithm 1 1 :: repeat 3 : 4 : 1 : 5 :

 113415 MM algorithm for Phase-linking problem (12) Entry: M ∈ C N ×N , w 0 ∈ U N 2Compute wt = (λ M max I -M)w t Update w t = P U N { wt } 5: t = t + 1 6: until Convergence 7: Output: w ∈ U N Algorithm 2 BCD algorithm for MLE problem (7) Entry: SCM S with M = Σ -1 • S Update θ from the output of Algorithm 1 6: until Convergence 7: Output: MLEs Σ, θ and C(Σ, θ)

Fig. 2 .

 2 Fig. 2. Cost function of the MLE-PL method with L = 20, N = 5, ρ = 0.5.

For

  one true InSAR phase, Figure 3 shows the histogram of the estimates obtained by different methods with N = 5, L = 20 and ρ = 0.7. All the methods give estimated values around the simulated InSAR value. The empirical variance of the MLE-PL and the PL estimates appear lower compared to the 2p-InSAR one.

Fig. 3 .

 3 Fig. 3. Histogram of InSAR phase estimations obtained by M LE -P L, P L and 2p-InSAR. L = 20, N = 5, ρ = 0.7.

Figure 4

 4 Figure4presents the Mean Squared Error (MSE) of estimated InSAR phases for the different methods and in terms of the number of pixels within a spatial window of L pixels. Different values of the coherence coefficient are studied, from high coherence (ρ = 0.9) to medium coherence (ρ = 0.7) to low coherence (ρ = 0.5). The MSE of each method decreases with the number of available samples which is expected. Interestingly, we observe that MLE-PL allows for an improvement of the accuracy of the phase estimation compared to PL or 2p-InSAR in all the considered setups.

Fig. 4 .

 4 Fig. 4. MSE of InSAR estimates using 2p-InSAR, PL and MLE-PL with Gaussian distributed input data. N = 5, 1000 Monte Carlo trials.