Karolina Andersson Sundell
email: karolina.a.sundell@nhv.se

MSocSc Mika Gissler

PhD Max Petzold

MD Margda Waern

Karolina Andersson

PhD Sundell Mpharmsc

Antidepressant utilization patterns and mortality in Swedish men and women aged 20-34 years

Keywords: utilization, antidepressants, young adults, mood stabilisers, mortality, Swedish Prescribed Drug Register

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

Introduction

By the year 2020, depression is expected to constitute the second largest source of global burden of disease after heart disease [START_REF] Murray | Alternative projections of mortality and disability by cause 1990-2020: Global Burden of Disease Study[END_REF]. Consequences of depression for the individual and for the society include lowered quality of life, social isolation, decreased intellectual capacity, inability to carry out activities of daily living, increased health care costs, increased need for institutionalised care and increased risk of suicide [START_REF] Pollack | Comorbid anxiety and depression[END_REF][START_REF] Simon | Generalized anxiety disorder and psychiatric comorbidities such as depression, bipolar disorder, and substance abuse[END_REF][START_REF] Sobocki | Health-related quality of life measured with EQ-5D in patients treated for depression in primary care[END_REF][5]. Full remission from depression with antidepressant treatment is associated with significant improvements in health-related quality of life, as well as significant and economically important reductions in health care costs [START_REF] Sobocki | The mission is remission: health economic consequences of achieving full remission with antidepressant treatment for depression[END_REF].

The reported prevalence of antidepressant use in the general population varies between 4% and 13% [START_REF] Beck | Antidepressant utilization in Canada[END_REF][START_REF] Henriksson | Infrequent use of antidepressants in depressed individuals (an interview and prescription database study in a defined Swedish population 2001-2002[END_REF][START_REF] Isacsson | Selective serotonin reuptake inhibitors have broadened the utilisation of antidepressant treatment in accordance with recommendations. Findings from a Swedish prescription database[END_REF][START_REF] Kivimaki | Social inequalities in antidepressant treatment and mortality: a longitudinal register study[END_REF][START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF]. Figures are higher among females than males and increase with age [START_REF] Beck | Antidepressant utilization in Canada[END_REF][START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF].

Gender differences in patterns of use have been reported, e.g. a larger proportion of the women received tricyclic antidepressants compared to men and women were more likely to continue treatment during at least six months [START_REF] Isacsson | Selective serotonin reuptake inhibitors have broadened the utilisation of antidepressant treatment in accordance with recommendations. Findings from a Swedish prescription database[END_REF]. Previous studies have reported that 15-35% of those initiating antidepressant treatments discontinue within one or two months.

Discontinuation was more common in groups with low education, low income, unemployment and foreign citizenship [START_REF] Isacsson | Selective serotonin reuptake inhibitors have broadened the utilisation of antidepressant treatment in accordance with recommendations. Findings from a Swedish prescription database[END_REF][START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF][START_REF] Hansen | Early discontinuation of antidepressants in general practice: association with patient and prescriber characteristics[END_REF][START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF]. Van Geffen et al. reported that the majority of those that discontinued antidepressant treatment did not inform their physician about their decision [START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF]. This indicates a considerable information gap between patients and physicians and substantial problems for these patients potentially leading to relapse. Regarding antidepressant use and mortality, results are mixed. Some report increased rates of mortality and suicide in particular, [START_REF] Kivimaki | Social inequalities in antidepressant treatment and mortality: a longitudinal register study[END_REF][START_REF] Osborn | Suicide and severe mental illnesses. Cohort study within the UK general practice research database[END_REF][START_REF] Valuck | Antidepressant discontinuation and risk of suicide attempt: a retrospective, nested case-control study[END_REF], among antidepressant users whereas others do not [START_REF] Haukka | Association between medication and risk of suicide, attempted suicide and death in nationwide cohort of suicidal patients with schizophrenia[END_REF][START_REF] Tiihonen | Antidepressants and the risk of suicide, attempted suicide, and overall mortality in a nationwide cohort[END_REF]. None of these studied mortality in young adults specifically.

The number of sick-listings and disability pensions related to mental ill-health has increased in Sweden, especially among young adults aged 35 years and less [START_REF] Andersson | Regional differences in disability pension and sickness absence with psychiatric diagnoses in Sweden and Norway 1980-2000: the influence of demography and access to psychiatric health care[END_REF]. We were therefore interested in investigating utilization patterns of antidepressants in this age group. The aim of this study was to compare utilization of antidepressants in 20-34 year old men and women in a total national database. Further, we wanted to examine mortality rates in relation to utilization pattern.

Methods

Study population and study period

The study population encompassed all Swedish residents aged 20-34 years in 2006 (n=1,691,356). The study period ranged from January 1, 2006 until December 31, 2006.

Data on purchased medicines

The Swedish Prescribed Drug Register, encompassing all prescribed medicines dispensed in Swedish pharmacies since July 1, 2005, was used to identify persons who filled at least one antidepressant prescription at a Swedish pharmacy between July 1, 2005 and December 31, 2006 [START_REF] Wettermark | The new Swedish Prescribed Drug Register--opportunities for pharmacoepidemiological research and experience from the first six months[END_REF]. The register contains information regarding each dispensed drug, the unique personal identification number of the person for whom the drug was prescribed and the date of purchase. Medicines are included in the register irrespective of whether they are encompassed by the Swedish Pharmaceutical Benefits Scheme or not, but the register does not cover medicines provided during in-patient care. Drugs are coded according to the Anatomical Therapeutic Chemical (ATC) system [20]. The Swedish Prescribed Drug Register includes information about substance, ATC-code, package size, number of packages and number of defined daily doses (DDD) for each dispensed drug. Antidepressants were defined as all drugs in the ATC-group N06A. The register also specifies whether the purchase concerned a standard prescription or a multi-dose dispensed drug. Within the Swedish Pharmaceutical Benefits Scheme, 90 days supply of prescription drugs can be dispensed at each occasion [21]. The database distinguishes between standard prescriptions and medicines dispensed within the multi-dose drug dispensary system. Users of multi-dose dispensed drugs collect their medicines more frequently, usually every second week.

Ethics approval, which was a prerequisite for data access, was obtained from the regional ethics board in Gothenburg.

Measures

For each individual, index date was defined as the date of the first purchase of an antidepressant in 2006. One-year prevalence was defined as the proportion of the total population aged 20-34 years (as of December 31, 2006) that purchased at least one antidepressant during 2006. We defined new users as individuals who purchased an antidepressant in 2006 who had filled no antidepressant prescriptions between July 1 and December 31, 2005.

The number of filled antidepressant prescriptions per person was investigated for the year 2006 for new users. New users were in this analysis restricted to those who filled their first antidepressant prescription between January 1 and June 30, 2006. Those utilising the multidose system were excluded since they show an artificially regular pattern. Purchases of start packages (≤32 tablets or capsules per package) at index date were also assessed. The number of doses per filled prescription was calculated for both all users and new users, based on number of tablets/capsules. Separate calculations were made for those utilizing the multi-dose system. Total number of doses dispensed during the study period was calculated for each individual.

Concurrent use of SSRIs, venlafaxine and mirtazapine was assessed among new users as was switching between antidepressant types. Concurrent use was defined as purchases of two or more of these within a 125 day period. Switching was defined as non-concurrent use of the specified antidepressants. The time frame of 125 days was chosen to correspond with the Swedish pharmaceutical benefits scheme [21], where in practice packages for 100 days often are dispensed instead of 90 days due to package sizes of 98 or 100 units. Allowing an adherence of at least 80 percent, one filled prescription would last for a maximum of 125 days.

Use of mood stabilizers was defined as one or more purchase of lithium, carbamazepine, lamotrigine or valproic acid during the study period. Concurrent use of mood stabilizers was assessed among all antidepressant users during the study period. Concurrent use of antipsychotics (N05A) was also assessed.

Data on death and migration

Information about date of death and migration in 2006 was obtained by record linkage to the Total Population Register using the unique personal identification number available for all Swedish citizens and permanent residents. Mortality rates were calculated per 10,000 inhabitants. In total, 166 persons died and 592 migrated during the study period (0.2% and 0.6% of the antidepressant users, respectively). These persons were omitted from the description of the number of filled antidepressant prescriptions per individual and switching and concurrent use of other antidepressants.

To assess whether the individual theoretically had enough antidepressant medication at the time of death assuming full adherence, the dispensing records of the last dispensed prescription were reviewed for those who died. The prescribed dose as stated in the dosage text was used as the consumption rate. If the dispensed amount was insufficient to cover the entire time period, the number of months that had elapsed from the last theoretical dose until date of death was recorded.

Statistical analysis

Data management and statistical analyses were performed using SAS version 9.1.3 (SAS Institute, NC) and Stata version 10.1 (StataCorp, TX). Pearson χ 2 -test and Fisher's exact test were used to analyse differences in proportions, and Student's t-test was used to analyse differences between means. Differences in proportions were calculated and presented as risk ratios, RR, with 95% confidence intervals. Risk ratios for mortality were calculated stratified by sex, but no age-adjustment was made due to the rather short age interval. The level for statistical significance was 0.05.

Results

During 2006, 94,239 Swedes aged 20-34 years (34,472 men and 59,750 women) purchased at least one antidepressant, corresponding to a one-year prevalence of 5.6%. Prevalence was higher among women compared to men and increased with age (p<0.001) (Table 1). Four out of ten were new users (Table 1). Altogether, 2.5% of the antidepressant users (n=2,335) had multi-dose dispensed drugs.

Multi-dose dispensed drugs was more frequent among male users than among female users (3.3% vs. 2.0% p<0.001).

Type of antidepressants purchased

Table 2 shows the distribution of different antidepressant substances at baseline for all users by sex. Selective serotonin inhibitors (SSRIs) dominated at baseline; citalopram (28%) and sertraline (24%) were the most common substances. Mirtazapine was more than twice as frequent in men compared to women (RR 2.22) whereas fluoxetine was almost twice as common among women than among men (RR 1.70) (Table 2).

The proportion of tricyclic antidepressants (TCA) increased by age for both men and women (men 4.7% for 20-24 years, 6.2% for 25-29 years, 8.5% for 30-34 years, p<0.001; women 3.9%, 5.3% and 7.5% respectively, p<0.001) whereas the proportion of SSRIs declined by age (men 74.6%, 71.3% and 70.6% respectively, p<0.001; women 81.4%, 79.3% and 76.6% respectively, p<0.001). This was consistent in both men and women and also among new users. No such patterns were found for monoamino-oxidase A inhibitors. For the group other antidepressants, there was a slightly increasing trend by age for men (0.7%, 0.8% and 1.0% respectively, p=0.014), but not for women.

Amounts of antidepressants purchased

The average number of doses purchased per filled prescription (tablets/capsules excluding solution) was lower among new users compared to those who purchased antidepressants in both 2005 and 2006 (63.4 vs. 79.8, p<0.001). The average number of doses purchased was similar for male and female users. After exclusion of multi-dose users, the average number of doses increased from 77.5 (median 98) to 85.4 (median 98).

Among new users who purchased their first antidepressant in the first six months of 2006 (n=21,199), the mean number of filled prescriptions was 3.8 (median 3.0) during the study period for those purchasing prescriptions. There were no relevant differences between men and women in average number of filled prescriptions. Over one-fifth (22.3%) filled only one prescription; this proportion was somewhat higher among males than females (24.1% vs. 21.4%, p<0.001) (Table 3). A lower proportion of those filling only one prescription purchased a start package at index date compared to those that filled several prescriptions (48.2% vs. 62.3%, p<0.001), this difference was consistent among both men and women when analysed separately. Among new users, the average number of doses purchased from index date until end of study period was higher among those who did not purchase a start package compared to those who did (298 vs. 250, p <0.001).

Among those who filled one prescription only, a higher proportion of the men had purchased a start package compared to the women (51.1% vs. 46.3%, p=0.001). Furthermore, the proportion of start packages differed between the different types of antidepressants (TCA 21.5%; SSRI 49.0%; other antidepressants 72.7%, p<0.001). This was consistent among both men and women when analysed separately. The proportion of individuals that filled only one prescription differed by substance at index date and sex (Table 4). The proportion was higher among men than women for escitalopram, whereas the reverse was observed for duloxetine.

Mortality rates

Table 5 shows mortality rates and risk ratios by utilization pattern. The mortality rate for all antidepressant users was significantly higher among men than women (23.5 vs. 14.2 per 10,000 inhabitants, p=0.009). The risk ratio was higher among antidepressant users than among non-users. This was also the case for those utilizing the multi-dose system compared to those with prescriptions (Table 5).

Mortality was increased among antidepressant users who purchased carbamazepine, lamotrigine or valproic acid compared to those who did not (RR 2.95 (95% CI 1.59-5.47)).

This was not the case for lithium. The mortality rate was also elevated among antidepressant users who filled at least one antipsychotic prescription during the study period (RR 2.83 (95% CI 2.20-3.63)). We observed somewhat more pronounced differences in risk ratios among women for several of the comparisons whereas the differences were somewhat attenuated for men. There were, however, exceptions (Table 5).

Among those who died, a majority (72.3%) theoretically had antidepressants available at the date of death. This was consistent also in the subgroup who were new users (74.2%). Among new users who had filled one prescription only, 58.9% theoretically had antidepressants available at the time of death whereas the corresponding proportion for those filling two or more prescriptions was 79.1%. For those who theoretically had finished treatment before death, the median time that elapsed from the end of treatment until death was 1.75 months (inter-quartile range 4.25). In 47.7% of these, the time between theoretical end date and death was a month or less.

Discussion

One out of 18 adults aged 20-34 years in Sweden purchased antidepressants in 2006 and almost half of these were new users. About twice as many women as men aged 20-34 years filled at least one antidepressant prescription. The discontinuation rate was high and slightly elevated in men. Mortality was higher among users and further elevated among early discontinuers. Further, those who also used mood stabilizers or antipsychotics had elevated mortality. This was not found for those who concurrently used lithium. We observed that the mortality rate was higher among men than women, but the risk ratios associated with death were somewhat more elevated among women than men.

The one-year prevalence of purchased antidepressants in this study was slightly higher than earlier studies of younger adults in other settings and somewhat different age groups [START_REF] Beck | Antidepressant utilization in Canada[END_REF][START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF][START_REF] Rosholm | Are there differences in the use of selective serotonin reuptake inhibitors and tricyclic antidepressants? A prescription database study[END_REF]. Antidepressant sales have steadily increased over the last decades, thus the level of use varies with the time when the measurement was conducted [START_REF] Rosholm | Are there differences in the use of selective serotonin reuptake inhibitors and tricyclic antidepressants? A prescription database study[END_REF][START_REF] Rosholm | Changes in the pattern of antidepressant use upon the introduction of the new antidepressants: a prescription database study[END_REF].

Citalopram and sertraline were the most prominently prescribed substances at baseline followed by venlafaxine and mirtazapine. This finding differs somewhat from those reported from studies conducted in other settings [START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF][START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF][START_REF] Rosholm | Are there differences in the use of selective serotonin reuptake inhibitors and tricyclic antidepressants? A prescription database study[END_REF][START_REF] Zito | Antidepressant prevalence for youths: a multi-national comparison[END_REF][START_REF] Kairuz | Prescribing patterns of tricyclic and selective serotonin reuptake inhibitor antidepressants among a sample of adolescents and young adults[END_REF]. For example, recent Dutch and Italian studies have reported paroxetine to be the most commonly used substance followed by citalopram [START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF][START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF]. The black box warnings concerning antidepressants and emerging suicidality in young people might have affected the prescribing patterns in this age group. Sex differences in utilization patterns were in accordance with earlier research [START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF][START_REF] Kairuz | Prescribing patterns of tricyclic and selective serotonin reuptake inhibitor antidepressants among a sample of adolescents and young adults[END_REF].

Our finding that one-fifth filled only one prescription is in line with previously reported early discontinuation rates [START_REF] Isacsson | Selective serotonin reuptake inhibitors have broadened the utilisation of antidepressant treatment in accordance with recommendations. Findings from a Swedish prescription database[END_REF][START_REF] Percudani | Antidepressant drug use in Lombardy, Italy: a population-based study[END_REF][START_REF] Hansen | Early discontinuation of antidepressants in general practice: association with patient and prescriber characteristics[END_REF][START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF]. Previous studies have also reported variation in early discontinuation rates by antidepressant type [START_REF] Hansen | Early discontinuation of antidepressants in general practice: association with patient and prescriber characteristics[END_REF][START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF], although differences were even more pronounced in our study. This could be due to differences in size of the examined population, time windows as well as variations in prescription patterns. Furthermore, the results in this study suggested that the use of start packages might help to prevent early discontinuation. It is possible that those receiving start packages for the first filled prescription of antidepressants are more closely monitored or differ in other aspects. We also found that the total amount of purchased medication was lower among those who purchased a start package at index date which somewhat contradicts the notion that start packages might prevent early discontinuation. Our data do not allow for in depth investigation which would require assessment of each individual's theoretical pattern of use including time exposed taking indication into account during a longer observation period. In a six-month follow up of selfreported depression symptoms, van Geffen et al. reported that those that never filled their prescriptions had lower severity of depression at baseline [START_REF] Van Geffen | Characteristics and reasons associated with nonacceptance of selective serotonin-reuptake inhibitor treatment[END_REF]. Severity of depression decreased in both continuers and discontinuers over time in that study, which might reflect the natural course of the depression. There could be some situations in which the physician prescribed an antidepressant, without discussing treatment preferences with the patient which might contribute to non-adherence.

We found increased mortality rates for young adults in Sweden who purchased antidepressants compared to those who did not. Previous studies have shown conflicting results. Some report increased mortality among antidepressant users [START_REF] Kivimaki | Social inequalities in antidepressant treatment and mortality: a longitudinal register study[END_REF], whereas others report the opposite [START_REF] Haukka | Antidepressant use and mortality in Finland: a register-linkage study from a nationwide cohort[END_REF]. This study only encompasses data for one and a half years and a young population. The number of deaths was low, and our findings should be interpreted with caution. We did not have access to cause-of-death data, nor could we control for potential confounding factors such as co-morbidity and confounding by indication. There are many factors that contribute to increased mortality. Although we could not control for potential confounding; we have tried to include some aspects e.g. concurrent use of antipsychotics and mood stabilisers. Our study design did not allow any attributions of mortality, but our findings indicate that it is important to take utilization patterns such as length of treatment, type of antidepressant and concurrent drug use into account when analysing mortality risk, as has been shown earlier in relation to suicide [START_REF] Sondergard | Do antidepressants prevent suicide?[END_REF][START_REF] Sondergard | Continued antidepressant treatment and suicide in patients with depressive disorder[END_REF] and suicide attempt [START_REF] Valuck | Antidepressant discontinuation and risk of suicide attempt: a retrospective, nested case-control study[END_REF]. We found elevated mortality rates among new users filling one antidepressant prescription only; this is influenced by the fact that one cannot fill another prescription after death. Therefore, those dying in the study period are likely to have filled fewer prescriptions compared to the survivors due to shorter follow-up time.

To the best of our knowledge, this is the first study to demonstrate elevated mortality rates among young adults with concomitant use of anticonvulsant mood stabilizers. No increase in mortality could be shown for concurrent lithium users. Recent studies encompassing mixed age adult populations have reported higher mortality rates and slightly elevated suicide rates among users of anticonvulsant mood stabilizers compared to lithium users [START_REF] Smith | Association between consistent purchase of anticonvulsants or lithium and suicide risk: A longitudinal cohort study from Denmark, 1995-2001[END_REF][START_REF] Sondergard | Mood-stabilizing pharmacological treatment in bipolar disorders and risk of suicide[END_REF][START_REF] Patorno | Anticonvulsant medications and the risk of suicide, attempted suicide, or violent death[END_REF]. According to the authors, part of the difference was explained by the fact that anticonvulsant users were older than lithium users [START_REF] Smith | Association between consistent purchase of anticonvulsants or lithium and suicide risk: A longitudinal cohort study from Denmark, 1995-2001[END_REF]. Another consideration is that lithium users may be more closely monitored because of the potential for serious adverse side effects. Further studies with longer follow-up periods and more background variables and diagnostic information are needed to investigate mortality risks and patterns more closely.

Strengths and limitations

As mentioned earlier the Swedish Prescribed Drug Register includes all dispensed medicines from July 1, 2005 independent of reimbursement status [START_REF] Wettermark | The new Swedish Prescribed Drug Register--opportunities for pharmacoepidemiological research and experience from the first six months[END_REF]. While persons who filled a single antidepressant prescription prior to hospitalization would be misclassified as discontinuers, they would need new medicines after discharge, so this is unlikely to alter the results. A longer study period and linkage to the hospital inpatient register would be needed to shed further light on this. This would also allow for confounding by indication to some extent. In order to investigate whether there were differences in amounts between those purchasing a start package and those who did not, the total amount of antidepressants was calculated for each individual. The theoretical length of treatment periods, however, could not be assessed due to the short study period and the non-standardised dosage information. Another limitation concerns the lack of data regarding treatment setting. Persistence has been reported to be higher among patients that received their first antidepressant prescription from a psychiatrist compared to those that received their first prescription from a general practitioner [START_REF] Gardarsdottir | Does the length of the first antidepressant treatment episode influence risk and time to a second episode?[END_REF].

The short run-in period was necessitated by the fact that the Swedish Prescribed Drug Register was newly established when the study was initiated which might have affected the results. The comparably short run-in period, and older data as well as a young age group may help to explain the considerably higher proportion of new users found in this study compared to other studies [START_REF] Isacsson | Selective serotonin reuptake inhibitors have broadened the utilisation of antidepressant treatment in accordance with recommendations. Findings from a Swedish prescription database[END_REF]. Garđarsdóttir et al. [START_REF] Gardarsdottir | Potential bias in pharmacoepidemiological studies due to the length of the drug free period: a study on antidepressant drug use in adults in the Netherlands[END_REF], however, showed that the differences in incidence when the run-in period was increased from six to twelve months were smaller for the age group 18-30 years than for the older age groups.

A further limitation is that the register lacks standardised data on indication and dosage. Therefore, we could not distinguish between patients that received antidepressants for treatment of depression nor bipolar disorders or depression with psychotic characteristics versus treatment of other psychiatric conditions. This may have implications on recommended length of treatment period and also on expected therapeutic response, since bipolar disorder and major depression with psychotic symptoms might be recalcitrant to treatment. However, previous studies have reported that depression is the most common indication for antidepressants in outpatient care followed by anxiety/panic disorders [START_REF] Gardarsdottir | Indications for antidepressant drug prescribing in general practice in the Netherlands[END_REF][START_REF] Henriksson | Indications for and outcome of antidepressant medication in a general population: a prescription database and medical record study, in Jamtland county, Sweden[END_REF]. Finally, the register is based on purchases, and there is no information regarding adherence, nor is there any information regarding other forms of medical and psychological treatments.

Conclusion

Although twice as many women compared to men purchased antidepressants there were rather small differences in patterns of use. Mortality rates were higher among male users compared to female users and associated with treatment discontinuation, but the number of deaths was low. The finding that mortality rates were particularly high among the young adults in our study population who were on augmentation treatment with mood stabilizers but not elevated for those on lithium calls for further study. Discussion about patients' attitudes to medicine use and intensified support during treatment initiation should be considered in order to decrease the discontinuation rate.

Table 1 .

 1 One-year prevalence of antidepressant use (rate per 100 inhabitants) and proportion of new users by age and sex. Differences in rates per 100 inhabitants between men and women were tested with Pearson's χ 2 -test, and trend test was used to analyze trends over age groups, p-values are presented in the table.

	Age	Men

Table 2 .

 2 Distribution of substances at the first antidepressant purchase for all users in 2006 by sex, expressed as percentage of total number of users (34,472 men and 59,750 women). Substances purchased by less than 100 individuals are included in the 'Others' category. Risk ratios (RR) were calculated to investigate differences between sexes and presented with 95% confidence intervals (95% CI), women was the reference category.

	Substance	Men Women	Total
		%	%	%	RR	95% CI
	Tricyclic antidepressants	5.9	6.7	6.4	0.87 0.83-0.91
	Amitryptiline	3.8	5.1	4.6	0.74 0.69-0.79
	Clomipramine	1.8	1.5	1.6	1.19 1.07-1.31
	Trimipramine	0.3	0.1	0.2	2.49 1.80-3.44
	Selective serotonin re-uptake inhibitors	71.7	78.7	76.0	0.91 0.90-0.92
	Citalopram	28.0	30.0	29.3	0.93 0.91-0.95
	Escitalopram	7.1	6.7	6.9	1.05 1.00-1.10
	Fluoxetine	6.7	11.4	9.7	0.59 0.57-0.62
	Paroxetine	7.0	5.2	5.8	1.35 1.28-1.42
	Sertraline	22.8	25.3	24.4	0.90 0.88-0.93
	Other antidepressants	22.1	14.3	17.2	1.54 1.50-1.59
	Duloxetine	1.3	1.4	1.4	0.87 0.77-0.97
	Mianserin	0.8	0.7	0.8	1.21 1.04-1.41
	Mirtazapine	11.2	5.1	7.3	2.22 2.12-2.32
	Moclobemide	0.2	0.2	0.2	1.30 0.97-1.74
	Reboxetine	0.6	0.3	0.4	1.90 1.55-2.32
	Venlafaxine	8.2	6.8	7.3	1.21 1.15-1.26
	Others	0.2	0.2	0.2	1.16 0.87-1.53
	Total	100	100	100

Table 3 .

 3 Distribution of number of filled antidepressant prescriptions per individual for new users that purchased their first antidepressant between January and June 2006 (n=21,199). Risk ratios (RR) were calculated to investigate differences between sexes and presented with 95% confidence intervals (95% CI), women was the reference category. Differences in mean between sexes were tested with Student's t-test, pvalue presented in the table.

	Number of filled prescriptions Men Women Total RR 95% CI
		%	%	%
	1	24.1	21.4 22.4 1.12 1.07-1.19
	2	17.4	15.4 16.2 1.13 1.06-1.20
	3	15.9	16.4 16.3 0.97 0.91-1.03
	4	12.8	15.7 14.7 0.82 0.76-0.88
	5 or more	29.6	31.1 30.7 0.96 0.92-1.00
	Mean	3.7	3.9	3.8	P < 0.001
	Median	3.0	3.0	3.0

Table 4 .

 4 Proportion of individuals that filled one prescription only for new users by substance and sex (n=21,199). Substances bought by less than 100 individuals have been excluded. Risk ratios (RR) were calculated to investigate differences between sexes and presented with 95% confidence intervals (95% CI), women was the reference category. All users (n=94,239) vs. non-users (n=1,597,117) b Multi-dose users vs. users with prescriptions c Start 2005 or earlier vs. start 2006 d One filled prescription vs. more than one filled prescription e Tricyclic antidepressant vs. all other antidepressants f SSRI vs. all other antidepressants g Baseline antidepressant in the group other antidepressants vs. baseline antidepressant not in the group other antidepressants h Concurrent use of any mood stabilizer vs. no concurrent use of any mood stabilizer i Concurrent use of lithium vs. no concurrent use of lithium j Concurrent use of carbamazepine, lamotrigine or valproic acid vs. no concurrent use of carbamazepine, lamotrigine or valproic acid k Concurrent use of antipsychotic drugs vs. no concurrent use of antipsychotics

	Substance	Men Women RR 95% CI Total
		%	%	%
	Tri-cyclic antidepressants
	Amitryptiline	48.2 44.2	1.08 0.96-1.23 45.4
	Clomipramine	27.5 28.6	0.96 0.61-1.51 28.2**
	SSRIs
	Citalopram	22.2 20.6	1.08 0.98-1.18 21.2
	Escitalopram	20.2 14.7	1.39 1.10-1.74 16.9
	Fluoxetine	23.9 18.1	1.32 1.09-1.61 19.6
	Paroxetine	21.3 24.1	0.96 0.76-1.21 23.8
	Sertraline	23.2 24.2	1.19 1.06-1.34 18.5
	Other antidepressants
	Duloxetine	14.0 24.2	0.58 0.38-0.88 20.5
	Mianserin	52.4 45.2	1.15 0.84-1.59 47.9**
	Mirtazapine	29.0 25.1	1.20 0.40-3.56 27.4
	Venlafaxine	18.2 15.7	1.15 0.86-1.54 20.6
	** n<150

a l Carbamazepine, lamotrigine or valproic acid

Acknowledgements

Data collection was supported in part by a post doc scholarship from the Swedish Lundbeck foundation. The study was also financed with internal funds. The authors declare that they have no conflict of interest.

Concurrent use and switching

A higher proportion of those that started on SSRIs switched to or concurrently used mirtazapine (0.5%; 4.8%) compared to venlafaxine (0.2%; 3.1%) (p<0.001). A similar pattern was seen among those that started on mirtazapine, a higher proportion switched to or concurrently used SSRIs (2.7%; 18.0%) compared to venlafaxine (0.3%; 4.9%) (p<0.001).

Among those starting on venlafaxine, rate of switching to SSRIs and mirtazapine was similar (1.2% vs. 0.6%) (p=0.132). Among those starting with venlafaxine, the proportion of concurrent use with SSRI was higher compared to concurrent use with mirtazapine (8.2% vs.

5.0%) (p=0.006).

Concurrent use of mood stabilizing medicines and antipsychotics

About 1.1% of those that purchased an antidepressant during 2006 also filled a prescription for lithium (n=995). There was no sex difference (1.1% men vs. 1.0% women, p=0.643). In total, 3.8% of the antidepressant users (n=3,559) also purchased a mood stabilizer (carbamazepine, valproic acid or lamotrigine). The proportion was higher among men than women (4.1% vs. 3.6%, p<0.001).

In total, 9.6% of the antidepressant users (n=9,065) also used an antipsychotic agent. A mood stabilizer was used in 349 of the antidepressant users who also purchased an antipsychotic agent. The proportion concurrent use of antipsychotics was higher among men than women (12.7% vs. 7.9%, p<0.001).


Mustafa ; Gunes

N Balkan
email: balkan@essex.ac.uk

Engin Tiras

Sukru Ardali

Chantal Fontaine

Alexandre Arnoult

Alexandre ; Laas-Cnrs

Quantum lifetimes and momentum relaxation of electrons and holes in Ga 0.7 In 0.3 N 0.015 As 0.985 /GaAs Quantum Wells

Keywords: electrical transport, magnetoresistance, materials characterisation, mobility, semiconductors GaInNAs, effective mass, quantum lifetime

published or not. The documents may come

Quantum lifetimes and momentum relaxation of electrons and holes in Ga0.7In0.3N0.015As0.

Introduction

Dilute nitrides, particularly the quaternary material system of GaInNAs/GaAs, have been attracting a great deal of attention because of their potential applications in a variety of optoelectronic devices [1][2][3][4][5][6][7]. Most of the work reported in the literature is concerned with the optical properties. However, studies of electronic transport, particularly in GaNAs, have also been reported [8][9][10][11][12][13][14].

It is predicted, theoretically, that the electron effective mass in GaInNAs is increased compared to that in the host matrix, GaInAs, as a result of reduced parabolicty, associated with nitrogen incorporation. The hole effective mass, however, is expected to remain unchanged [15,16]. Previous experimental work has been concerned mainly with effective masses obtained indirectly from the analysis of transition energies [17][18][19][20][21], quantum confinement energy [START_REF] Jones | Proc. SPIE[END_REF][START_REF] Zhang | [END_REF], exciton diamagnetic shifts [START_REF] Jones | Proc. SPIE[END_REF], magnetophotoluminescence [24] and energy-loss spectroscopy [25]. There are only a few experimental studies reported in the literature where the electron effective mass is obtained experimentally by direct methods, for example by using optically detected cyclotron resonance technique [26]. There are only a few experimental studies concerning the determination of hole effective mass [27][28][29] In modulation doped 2-D III-V compounds, the Shubnikov-de Haas (SdH) effect has been proven to be a powerful tool for the determination of the in-plane effective mass m* and the quantum lifetime (τ q ) of electrons [30,31]. To our knowledge there is no such study in dilute nitrides to date. The aim of the current work is to determine directly the 2D in-plane electron and hole effective masses and quantum and transport lifetimes in modulation doped n and p type GaInNAs/GaAs quantum wells.

Experimental

The Ga 0.7 In 0. Hall-bar shaped samples as a function of both, temperature, in the range between T =1.8 and 32 K, and magnetic field, in the range between B= 0 and 11 T, in a cryogen free superconducting magnet system (Cryogenics Ltd., Model no J2414). A constant current source (Keithley 2400) and a nano-voltmeter (Keithley 2182A) were employed in the conventional dc measurements. The current flow was in the plane of the electron gas and the current through the sample was kept low enough to ensure ohmic conditions. Steady magnetic fields were applied perpendicular to the plane of the samples.

Results and discussion

Figure 1 shows the magneto-resistance R xx (B) measured at different temperatures for both n and p-type modulation-doped Ga 1-x In x N y As 1-y /GaAs quantum wells. SdH oscillations are clearly visible over the magnetic field range between B= 5 and 10 T. No higher harmonics are apparent in the oscillations and the oscillation amplitude is reduced with increasing temperature [START_REF] Seiler | Landau Level Spectroscopy[END_REF]. It is also evident that the oscillatory effect is superimposed on a monotonically increasing component, which occurs as a result of positive magneto-resistance in the barriers [START_REF] Lisesivdin | [END_REF]. This may affect the accuracy of the determination of oscillation amplitude, particularly at elevated temperatures. Therefore, in order to exclude the effects of the background magneto-resistance (R b ) and to extract the SdH oscillations we used the negative second derivative of the raw magnetoresistance data with respect to the magnetic field, i.e. (-δ 2 R/δB 2 ) [30][31][START_REF] Seiler | Landau Level Spectroscopy[END_REF][START_REF] Lisesivdin | [END_REF]. The oscillations in the second derivative of magneto-resistance have well defined envelopes and symmetrical about a horizontal line as shown at three different temperatures in figure 2.

The double-differentiation technique does not change either the peak position or the period of the oscillations [START_REF] Lisesivdin | [END_REF].

The period of the SdH oscillations has been obtained by plotting the reciprocal of magnetic field (1/B n ), at which the nth peak occurs, against the peak number (n). Provided that the carriers occupy only the lowest sub band, the plot of 1/B n versus n will be a straight line as we observe in figure 3. The slope of 1/B n versus n plot yields the oscillation period, ∆(1/B n ) and the 2D carrier density (N 2D ) of each sample can be calculated using [31,[START_REF] Lisesivdin | [END_REF] ( )

1 2 * 1 E E m e N e B F D - = =       ∆ h h π (1)
where E F -E 1 is the energy difference between the Fermi level and the bottom of the first subband. The results thus obtained for n and p type doped samples are n=2.68x10 12 cm -2 and p=1.32x10 12 cm -2 , respectively. The carrier density is found to be essentially independent of temperature in the range from T=1.8 to 32 K.

The in-plane effective mass m* can be extracted from the temperature dependence of the SdH amplitude at constant magnetic field using [31][START_REF] Seiler | Landau Level Spectroscopy[END_REF][START_REF] Lisesivdin | [END_REF].

                = n B n B n n eB T m k T eB T m k T B T A B T A h h * 2 sinh . * 2 sinh . ) , ( ) , ( 2 0 0 2 0 π π (2)
where A(T,B n ) and A(T o ,B n ) are the amplitudes of the oscillation peaks observed at a magnetic field B n and at temperatures T and T o , respectively. In the derivation of equation ( 2) the quantum lifetime of electrons is assumed to be independent of temperature [31,35]. The relative amplitudes, A(T,B n )/A(T o ,B n ), are plotted in figure 4 for B=7.38 Tesla and B=8.3 Tesla for n and p type samples respectively. The relative amplitude decreases with increasing temperature as a result of the thermal damping [30,36] . We obtained similar values for all the magnetic fields of measurements suggesting strongly that the in-plane effective mass is essentially independent of magnetic field. /GaAs quantum wells agree extremely well with those obtained for the same nitrogen mole fraction by Heroux et al. [6] and Tomic and O'Reilly [16]. As for the hole effective mass, our result for the p-type modulation-doped Ga 0.7 In 0.3 N 0.015 As 0.985 /GaAs quantum wells are higher than the theoretically estimated values by Wartak and Weetman [27,28] and our theoretical and experimental results that are obtained by using optical transition energies [34]. This might be the result of heavy and light hole band mixing in p-type material and the involvement of both heavy and light holes in the SdH oscillations. It is worth noting that the SdH oscillation peaks in p-modulation doped material are broader than those in the n-modulation doped sample which might be explained in terms of strain induced band mixing in the valence band. Indeed our observed effective hole mass is close to the range of values quoted for the heavy and light hole masses obtained by using time-resolved and magneto-photoluminescence spectroscopy by Polimeni et al. [29].

The Fermi energies for the samples obtained from the oscillation period using equation ( 1) together with the in-plane effective mass m* of 2D carriers are: E F -E 1 =75.6 meV and 36.0 meV for the n and p type samples respectively. The quantum lifetime (τ q ) can be determined from the magnetic-field dependence of the amplitude of the SdH oscillations (i.e. Dingle plots) at a constant temperature provided that the carrier effective mass is known [31,[START_REF] Lisesivdin | [END_REF]36]. Figure 5 shows typical plots for the Dingle plots for the samples investigated. There agreement is excellent between the experimental data and the straight line described by [31,36], i.e.

ln A T B B C m e B n n q n ( , ). .sinh * / -       = - 1 2 1 χ χ π τ , (3)
where

χ π ω = 2 2 k T B c / ( ) h , *) / ( m eB n c = ω
is the cyclotron frequency and C is a constant. The quantum lifetimes, obtained from the slope of the Dingle plot using equation (3) together with the measured values of m*, are τ q =0.051 ps and τ q =0.218 ps for electrons and holes, respectively. These values remain constant within 2 % in the whole temperature and magnetic-field ranges of the measurements. The quantum lifetime and ) in table 2.

Transport mobility in the n and p -modulation doped samples is determined, in the ohmic regime, from the Hall measurements at liquid helium temperatures for the n and p type doped samples, µ e = 1100 cm 2 /Vs and µ h = 3125 cm 2 / Vs respectively [34] are also shown in table 2. Using the experimentally determine effective mass together with the classical Hall mobility expression the transport lifetime τ tr for electrons and holes as 0.043 ps and 0.275 ps, respectively. It is worth noting that both the quantum and transport lifetimes of holes much higher than that of electrons. The transport -to-quantum lifetime ratios of electrons is τ tr /τ q ~ 0.8 ; while the transport-to-quantum lifetime ratios of holes is τ tr /τ q ~1.3 .Theoretical calculations relating the transport lifetime to the quantum lifetime predict, in the extreme quantum limit, a τ t /τ q ratio greater than unity for smallangle scattering and smaller than unity for large-angle scattering [31,34,37]. This implies that in our p-type sample where τ tr /τ q ~ 1.3 the scattering of holes are mainly forward displaced in momentum space, which is in agreement with the model where the Hall mobility at low temperatures is limited by the interface roughness scattering [34]. In the ntype sample where τ tr /τ q ~ 0.8 the dominant electron scattering is large-angle scattering. This may be expected from scattering of electrons with higher mass nitrogen complexes that occupy states below and above the conduction band [34]. Furthermore, it is evident from table 2 that both the low temperature Hall and the quantum mobilities of holes are significantly larger than electron mobilities suggesting that electrons in the conduction band experience an extra scattering compared to the holes, in accord with models based on scattering with nitrogen complexes occupying states below the conduction band [37,38].

In order to determine whether the observed difference in the electron and hole mobilities is intrinsic and associated with the disparity in their effective masses we carried out a detailed modelling of the Ga x In 1-x N y As 1-y conduction (CB) and valence band (VB) structures. The band structure of Ga x In 1-x N y As 1-y /GaAs QWs is calculated by solving the coupling between the GaInAs CB and the localized nitrogen levels. The eigenvalue problem is then solved by the transfer-matrix method, taking into account the interfacial discontinuity condition [39,40]. The valence band material parameters used for the calculations are the ones of GaInAs [34][35][36][37][38][39][40][41] ) [34]. However, the nitrogen incorporation has negligible effect on the valence band (VB), thus the calculated hole effective mass (

0 * 105 . 0 m m h =
) is independent of nitrogen in the composition [34]. In addition, the nitrogen induces the flattening of the lower sub-band, giving rise to an enhanced density of states of electron at CB edge. Therefore, the density of electrons is expected to be higher than that of holes in the GaInNAs QWs with same nitrogen concentration as depicted in table 2.

Conclusions

The carrier density (N 2D ), effective mass (m*), and quantum lifetime (τ q ) for electrons and holes in n and p-type modulation-doped Ga

T L =1.8 K T L =12 K T L =22 K -δ 2 R/δB 2 B (T) (a
L =2 K T L =5 K T L =10 K -δ 2 R/δB 2 B (T) (b)
T L =1.8 K T L =12 K T L =22 K 1/B (T -1 ) Peak number ∆(1/B)=0.018 T -1 N 2D =2.68x10 16 m -2 E 1 -E F =75.
T L =2 K T L =10 K T L =13 K 1/B (T -1 ) Peak number ∆(1/B)=0.037 T -1 N 2D =1.32x10 16 m -2 E 1 -E F =36 meV (b)

  3 N 0.015 As 0.985 /GaAs quantum well samples were grown by the Molecular Beam Epitaxy (MBE) technique onto semi-insulating GaAs substrates and did not undergo either rapid or slow thermal annealing. The samples have three 7 nm wide Ga 0.7 In 0.3 N 0.015 As 0.985 wells, separated by 50 nm wide GaAs barriers. The 40 nm central position of the barriers are doped (with Si for n-type, and with Be for p-type) and the on either side is undoped (spacer layers) as shown in Table1. For the transport measurements ohmic contacts were formed by using Au/Ge/In for the n-type and Zn/Au for the p-type material. Magneto-resistance measurements were carried on

  . The in-plane effective mass of 2D electrons and holes are then determined by fitting the experimental data for the temperature dependence of A(T,B n )/A(T o ,B n ) to equation (2

  mass for the n-type modulation-doped Ga 0.7 In 0.3 N 0.015 As 0.985

  electrons and holes are tabulated together with the transport lifetime and transport (Hall) mobility (

7 modified 8 -

 78 band k.p Lüttinger-Kohn Hamiltonian, including tetragonal strain and strong

  1-x In x N y As 1-y /GaAs quantum wells have been determined from SdH oscillations. The two-dimensional (2D) carrier density and the Fermi energy with respect to subband energy (E F -E 1 ) have been obtained from the periods of the SdH oscillations. The in-plane effective mass (m*) and the quantum lifetime (τ q ) of electrons have been extracted from the temperature and magnetic field dependencies of the SdH amplitude, respectively. The in-plane effective and holes at their measured Fermi energies are 0.084m 0 and 0.155m 0 respectively. The quantum life time for electrons and holes in our n and p-type modulation doped quantum wells are 0.051ps and 0.218ps respectively. A detailed analysis of the quantum and transport life times indicates strongly that the momentum relaxation of holes are forward displaced in k-space where large angle scattering is prominent for the electrons. This discrepancy is believed to be due to scattering of electrons with nitrogen complexes.

F

Figure 1 .

 1 Figure 1. Magneto-resistance (R xx ) as a function of magnetic field both (a) n and (b) ptype modulation-doped Ga 0.7 In 0.3 N 0.015 As 0.985 /GaAs quantum wells measured at different temperatures.

Figure 2 .

 2 Figure 2. The second derivative of magneto-resistance with respect to magnetic field plotted against the magnetic field. The double differentiation removes the background magneto-resistance without affecting the position and the amplitudes of the oscillatory component.

Figure 3 .

 3 Figure 3. The reciprocal magnetic field (1/B n ) plotted as a function of the oscillation peak number. The straight line is the least-squares fit to the experimental data and the circles, squares and triangles symbols correspond to the peak magnetic field data for (a) n and (b) p-type modulation-doped samples measured at three different temperatures.

Figure 4 .

 4 Figure 4. Temperature dependence of the normalized amplitude of the oscillation peak at a fixed magnetic field of (B n ) measured for (a) n and (b) p-type modulation-doped samples. The data points are represented by the full circles. The full curves are the best fits of equation (2) to the experimental data.

Figure 5 .

 5 Figure 5. Determination of the quantum lifetime for (a) n and (b) p-type modulationdoped samples. The data points are represented by the full squares. The straight lines are the least-squares fits of equation (3) to each set of the experimental data.

  Figure 1.

Figure 3 Page

 3 Figure 3

Figure 5 Page

 5 Figure 5

  . As far as the CB is concerned, the energy

	level of nitrogen ( N E ) is assumed to be constant relative to the vacuum level. Moreover,
	the coupling parameter in band anticrossing model (BAC), NM V	has been taken as
	dependent on both In and N compositions. The details of theoretical model used in this
	work is given elsewhere [34] and only calculated effective mass of carriers lying band
							* m e =	0	.	084	0 m	) than the nitrogen free material
	GaInAs (	* m e =	0	.	0	54	0 m

edge for e1 and hh1 level in the GaInNAs quantum well are tabulated in table 2. The calculation indicates that the presence of nitrogen induces strong perturbation mainly on the conduction states by mutual repulsion between the CB-edge state and localized N states leading to enhanced non-parabolicity of CB in GaInNAs. Therefore, the electron effective mass is much higher in GaInNAs (

Table 1 .

 1 The structure of the n and p-type modulation-doped Ga 1-x In x N y As 1-y /GaAs quantum wells sample used in the studies.

	o
	r
	P
	e e r
	R e v i e w
	O n
	l y

x3 Page 11 of 28 http://mc.manuscriptcentral.com/pm-pml

Table 2 .

 2 Effective mass, transport and quantum lifetimes and mobilities for holes and electrons

			Philosophical Magazine & Philosophical Magazine Letters		Page 12 of 28
	Sample	2D carrier	Fermi level	Effective	Effective	Quantum	Transport	Quantum	Transport
	(Doping)	density cm -2	(from	mass	mass	lifetime	lifetime	mobility	mobility
			The first	(Experiment)	(Theory)	τq	τtr	µq(cm 2 /Vs)	µtr(cm 2 /Vs)
			subband)
	1930: Ga 0.7 In 0.3 N 0.015	n=2.68x10 12	75.6 meV	* e m =0.084m 0	* e m =0.080m 0	0.051ps	0.043 ps	1067	1100
	As 0.985 (n)
	1931: Ga 0.7 In 0.3 N 0.015	p=1.32x10 12	36.0 meV	* h m = 0.155m 0	* h m =0.105m 0	0.218ps	0.275 ps	2473	3125
	As 0.985 (p)

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Acknowledgments

E. TIRAS is grateful to the Scientific and Technical Research Council of Turkey TUBITAK (Project no: 110T377) and Anadolu University (Project no: 091047) for their financial support. We also acknowledge the support of COST Action: MP0805 for providing the suitable platform for the successful execution of the research.


F Chatelain

S Achard

O Michel

C Gouy-Pailler

MULTIVARIATE APPROACH FOR BRAIN DECOMPOSABLE CONNECTIVITY NETWORKS

Keywords: functional MRI, brain connectivity, partial correlation, graphical Gaussian model, maximum likelihood, decomposable graphs

This paper deals with the analysis of brain functional network using fMRI data. It recapitulates the concept of decomposable connectivity graph. Graphs are a usual tool to represent complex systems behavior, although edge strength estimation issues have not yet received a universally adopted solution. In the framework of linear Gaussian instantaneous exchanges, the well known partial correlation is usually introduced. However its estimation remains a challenge for highly connected or dense systems. Here, we propose to combine a wavelet decomposition and a graphical Gaussian model approach relying on decomposable graphs. This is shown to improve the estimations of brain function networks in the presence of long range dependence; the results are compared to those obtained with classical partial correlation estimators.

INTRODUCTION

The observation and description of the living brain has attracted a lot of research during the last centuries. First, it has been shown that brain areas can be associated to a specific function, which is called phrenology. However, during the 19th century, a connectionist approach was introduced in order to model the brain as a global complex function. Since the development of functional Magnetic Resonance Imaging (fMRI), it is possible to observe the whole living brain during several minutes with a high spatial resolution (voxels up to one mm 3 ) and a temporal resolution of one or two seconds. Consequently, some authors [?] attempt to model and characterize brain functional connectivity networks.

Indeed, neuroimagery provides us with time series, associated with voxels, which correspond to the information processed by a brain area in time. These data have brought to light the dynamic nature of the brain, which follows complex time patterns. The complexity of the signals obtained from fMRI prevents from using classical statistical techniques to analyse the dependence between time series. Prior works [?, ?, ?] have shown that brain dynamics are typically long memory with fractal scaling properties, and that brain function networks have a complex, modular, small-world organization which is topologically conserved over multiple frequency scales.

The approach introduced in [?] consisted in using wavelets to take into account the long memory property of the fMRI time series after averaging the voxels for a given anatomical brain region. * partly funded by ANR 2010 JCJC 0302 01 Then, the brain function networks were extracted using pairwise correlations of the wavelet coefficients. Thanks to this approach, the fMRI time series can be assumed to be Gaussian, and we propose to combine a wavelet decomposition and a graphical Gaussian model approach [?] in order to improve the estimations of brain function networks.

Section 2 presents the concept of conditional dependence, and its relation to correlation and partial correlation estimators. In section 3, we recall the maximum likelihood estimator for graphical Gaussian models. Finally, in section 4, the two approaches are compared on simulated multivariate data with long memory properties and a fixed model of connections extracted from real fMRI data.

In the sequel of the paper, all the estimations will be based on the wavelet coefficients extracted at a given scale. We will denote (x 1 , . . . , x n ), the wavelet coefficients obtained at a specific scale. In the context of maximum likelihood estimations, (x 1 , . . . , x n ) is assumed to be independent which is a reasonable hypothesis since it has been shown that for long memory time series the wavelet coefficients are stationary and asymptotically decorrelated [?].

INSTANTANEOUS CONDITIONAL DEPENDENCE

Let x t i , t ∈ {1, . . . , n} be the time series associated to the voxel of index i, i ∈ V = {1, . . . , p}. Connectivity between voxels will be assessed by studying dependences between the associated time series. For practical issues, dependence analysis is often restricted to linear dependence between time series [?], and the analysis is summarized by the variance or covariance matrix of the p dimensional multivariate process x t = (x t 1 , . . . , x t p ) T . An appealing approach is to relate high correlation value (eventually normalized) to the existence of an 'information exchange' between the time series. This leads to a major pitfall: two series may seem dependent of each other and exhibit a high correlation, although they are not 'connected'. This situation appears if both voxels receive information from a common third one, whilst they are independent conditionally to that voxel. The importance of conditioning was already quoted in Granger's seminal work [?], and was recently revisited in a non Gaussian framework, for directional dependences [?].

In this paper, linear dependence model and Gaussian processes are considered. Therefore, covariance provides an exhaustive representation of the available information. Furthermore, only instantaneous dependence is considered here. All possible conditioning in the instantaneous dependence between two series (x t i , x t j ) is taken into account by evaluating the correlations between the innovations processes ε t i and ε t j defined below. Let

H V \{i,j} k = span x k l , k ∈ V \ {i, j}
be the Hilbert subspace spanned all the observations but x t i and x t j . The best (MS sense) linear estimator of x t i , given H

V \{i,j} t

writes

xt i = P x t i |H V \{i,j} t
where P(x|H) denotes the orthogonal projection of x on H. The corresponding innovation is given by ε

t i = xt i -x t i .
The partial correlation pi,j between the two series is defined as

ki,j = -cov[ε t i , ε t j ] pi,j = k i,j √ k i,i k j,j for all 1 ≤ i, j ≤ p (1)
Equation ( 1) is an important result from multivariate analysis. It states that the components of [P ]i,j = pi,j for the partial correlation matrix are directly derived from the ki,j

= [K]i,j = -[Σ -1 ]i,j for i = j, where Σ -1 is the inverse of the covariance matrix of x t [?].
Note that the partial correlation matrix P satisfies -1 ≤ pi,j ≤ 1, ∀{i, j} ∈ V 2 . Estimating P may be tackled in two ways; either it is inferred by computing the inverse of the sample covariance matrix Γ or it is directly estimated. In the former case, the estimator will be noted Pemp. This paper focuses on the latter approach, where a robust estimation of P based on decomposable graph representation is derived in the context of long range dependent process.

MAXIMUM LIKELIHOOD ESTIMATION FOR GRAPHICAL MODELS

Graphical models are classically used to characterise conditional independence constraints between random variables. Let G = (V, E) be an undirected graph, where V = {1, ..., p} denotes the set of nodes, while E is the set of undirected edges (i, j) ∈ V × V . Then, a multivariate model indexed by V is said to be Markov with respect to the graph G if for any edge (i, j) in E, the ith and jth variables are conditionally independent given all the other variables (see [?] for more details on graphical models).

Graphical Gaussian Models

Let x be a p-dimensional centered Gaussian vector Np(0, Σ) whose elements are indexed by V . Define K = Σ -1 to be the so-called precision matrix of x. A fundamental result about graphical Gaussian models is that the Markov property induced by the graph G is equivalent to impose some sparsity constraints on K = (ki,j) 1≤i,j≤p :

ki,j = 0 for all (i, j) / ∈ E.

(2)

Consider now a sample (x 1 , . . . , x n ) from the graphical Gaussian model associated with G. From the density expression of a Gaussian vector, the likelihood is expressed as

p(x 1 , . . . , x n ; K) ∝ det(K) n/2 exp - n 2 trace (KπG(Γ)) , (3)
where

Γ = 1 n n i=1 (x i )(x i ) T
is the sample covariance matrix, πG(•) denotes the projection operator associated with the topology of G:

πG(A) = aij if (i, j) ∈ E, 0 otherwise, (4)
and K belongs to the convex cone PG of symmetric definite matrix satisfying the linear constraints (2). Expression (3) shows that the graphical Gaussian model describes an exponential family with canonical parameter K ∈ PG and sufficient statistic with the incomplete sample covariance πG(Γ). As a consequence, the maximum likelihood estimator (MLE) K of the precision matrix K is well defined and exists with probability one for large enough sample size n.

Numerical iterative methods, such as Iterative Proportional Scaling [?, p. 134] have been derived to evaluate this MLE. More recently, covariance selection methods, with ℓ1 penalization to enforce sparsity, have received a great attention [?, ?, ?]. These last methods aim at estimating jointly the entries of the covariance (or precision) matrix and the conditional independence structure. However, even when this structure is known, it is not possible to derive a tractable analytic expression of the MLE in the general case.

Decomposable Graphical Gaussian Models

First, we recall that the cliques of a graph correspond to the maximal complete subgraphs, i.e. the fully connected subgraphs that are not included in another fully connected subgraphs. A graph is said to be decomposable if it admits a perfect ordering of its cliques (C1, . . . , C k ): for i = 2, . . . , n, it exists l < i such that

Si = Ci i-1 j=1 Cj ⊆ C l .
The sets Si are called the separators. In this case, the likelihood of the graphical Gaussian model specified by G can be factorized

p(x; K) = k i=1 p(xC i ; KC i ) k i=2 p(xS i ; KS i ) , (5)
where xA denotes the subvector of x corresponding to the elements indexed by A ⊂ V , while KA stands for the inverse of the covariance submatrix ΣA corresponding to the entries of Σ indexed by A × A ⊂ V × V . Note that a necessary and sufficient condition for decomposability is that the graph does not contain a cycle of length greater than or equal to four as an induced subgraph (decomposable graphs are also called triangulated or chordal graphs). This hypothesis is in agreement with the concepts of hierarchy, modularity and "nearly-decomposability" (strong interactions between the elements of a subsystem, here the elements of a clique, and weaker interactions among the subsystems, here conditional independence of the cliques given a set of separators) introduced by Simon [?] to characterize complex systems.

A consequence of property ( 5) is that the MLE of K admits the following closed form expression

KMLE = k i=1 Γ -1 C i 0 - k i=2 Γ -1 S i 0 , ( 6
)
where Γ is the sample covariance matrix, and [MA] 0 , for A ∈ V , denotes the completion on V of the submatrix MA with zero values. Equation ( 6) ensures that the MLE exists with probability one if and only if the sample size n is greater than or equal to the cardinal of the clique with maximal size nC = max 1≤i≤k |Ci|. Furthermore, based on the MLE expression, the minimal variance unbiased estimator (MVUE) has been recently derived in [?]

KMVUE = k i=1 n -|Ci| -1 n Γ -1 C i 0 - n -|Si| -1 n k i=2 Γ -1 S i 0 .
Henceforth, we will focus on the estimation of the partial correlation matrix (1). Partial correlations present the advantage to be invariant from some scale changes that occurs in fMRI experiments. By functional invariance principle, the MLE of P denoted as PMLE is obtained directly by plugging KMLE in (1).

We will also consider the plug-in MVUE denoted as PPMVUE derived by plugging KMVUE in (1). Note that this estimator of P may be biased, even if it is based on the MVUE of K.

SIMULATION RESULTS

Data generation

A surrogate data generation approach has been adopted to allow for realistic comparisons between the aforementioned estimators. First a real fMRI dataset is considered as a basis for computing a realistic sparse precision matrix K, hence a graph of dependencies. In order to achieve this step, the fMRI time series are first gathered into ninety (p = 90) distinct components according to spatial considerations. The sample correlation matrix is estimated on wavelet decomposition on the frequency band 0.02-0.04 Hz (see [?] for details).

The resulting observed sample correlation matrix is then thresholded (by keeping the highest ones among the absolute pairwise correlations) such that the precision matrix K only contains sparse nonzero entries. Among the p(p-1) 2 = 4005 available off-diagonal entries of the precision matrix, only 700 have been retained.

In the next step the artificial data generation is achieved by controlling three crucial characteristics of the p = 90 times series, namely the correlation matrix, the Hurst exponents, and the spectrum of the series. An R 1 [?] implementation of the algorithm described in [?] by the authors allowed the generation of series of arbitrary length, from 256 to 4096 points. The correlation matrix of the artificial multivariate series is set to the one extracted from real data. Hurst exponents of the p series were set to 0.7. The rationale behind this generation procedure is that clear long range memory effects (Hurst exponents strictly greater than 0.5) have been observed in real datasets, which results in biased correlation estimations mainly influenced by low-frequency components [?].

In order to alleviate long range dependence, a wavelet transform is used to decompose the signal into successive scale-specific components. In the sequel of the paper, only coefficients of the wavelet transform at scale three are retained. Therefore the resulting time series used for the estimation are composed of various length between 32 and 512 points.

Graph triangulation

The realistic precision matrix used in the simulation does not yield a decomposable graph. In this case, it is quite natural to embed it in a decomposable graph by adding some edges. The resulting graph is a so-called triangulation of the input graph: the induced cycles whose length are greater than 4 are triangulated by adding new edges. Finding the minimal width triangulation, i.e. that minimizes the number of edges to be added, is NP-hard. However suboptimal solution can be derived efficiently by node elimination heuristics. In this simulation, a minimum degree heuristic is used for the triangulation [?]. It results in a decomposable graph derived by adding 141 edges to the original graph. The triangulation and factorization tasks have been 1 Parts of the procedure are made available in the R package called brainwaver performed by using the java JPSGCS package [?]. The sparsity patterns of the corresponding precision matrices are depicted in fig. 1. It is important to note that even if a few conditional dependences are artificially added in the model, the estimates of their magnitude should be quite small. In this simulation, the maximal clique size of the decomposable graph is nc = 28. Therefore, the MLE derived from the decomposable graphical Gaussian model is defined with probability one when the sample size n is greater than or equal to 28.

Occipital

Estimation performance

In order to appreciate the performance of the considered partial correlation estimators, this section studies their mean square errors (MSEs). Several experiments have been conducted corresponding to different sizes of the wavelet coefficient time series, namely n = 32, 64, 128, 256 or 512. For each experiment, the number of Monte-Carlo runs is 1000.

In the simulations, the MSE are computed as

MSE ( P ) = E || P -P || 2 , (7)
where ||A|| = trace (AA T ) is the classical Frobenius norm. It is important to note that, by construction, the MSEs of the graphical estimators PMLE and PPMVUE equal zero for all the entries indexed by (i, j) / ∈ E. This is not the case for the regression estimator Pemp, for which there is no prior on the covariance structure of the model. Thus, to perform a fair comparison, only the errors corresponding to the non fixed to zero entries in the precision matrix K are taken into account. It leads to consider MSE πG( P ) . Finally the MSEs are normalized by dividing by the number of nonzero entries in the considered matrix. Thus, it corresponds to the average of the MSEs associated with each nonzero entry of the partial correlation estimator.

Fig. 2(a) displays the decimal logarithm of the normalized MSEs of πG( P ) as a function of the decimal logarithm of the sample size. It shows that all the estimators have similar performance for large enough samples. However, for small samples, the MLE appears well behaved and clearly outperforms the regression estimator when it exists, i.e. when n ≥ 90. More surprisingly, the MLE appears to be also more accurate than the one derived from the MVUE of the precision matrix.

Finally, fig. 2(b) displays the normalized MSEs associated with the set of edges T added during the triangulation step. It is interesting to note that the estimators behave quite similarly than on the whole graph G. This result also illustrates that although the constraint of equality to zero has been relaxed, the maximum likelihood estimates of the partial correlation associated with these edges are very close to zero.

CONCLUSION

In this paper, we illustrated the potential of ML approaches for estimating partial correlation on decomposable graphs. In the case of fMRI data exhibiting long range dependences, such graphs are constructed on wavelet transformed series. Although extracted graphs from real data sets do usually not satisfy the decomposability property, it may be forced by triangulation. It is shown that ML estimation of partial correlation on the forced decomposable graph exhibit good performances, even for small sample size. A future work will focus on model selection for brain connectivity networks, as suggested by Dempster in [?], with this new estimation scheme.

Fig. 1 .

 1 Fig.1. Top: Anatomical representation of the 700 selected connections where short, < 7.5cm, (resp. long) euclidean distances are grey (resp. black). Bottom: Pattern of the nonzero entries in the precision matrix. Marker •: before triangulation, marker ×: additional edges due to the triangulation.

  Average MSEs over the triangulated edges

Fig. 2 .

 2 Fig. 2. log 10 normalized MSEs of the partial correlation estimators vs the log 10 of the sample size. Red solid line: MLE, blue dotted line: PMVUE, black dashed line: Pemp Estimator.


Florent Chatelain

Aude Costard

Olivier Michel

Extraction de galaxies dans les données hyperspectrales MUSE : une approche objet non supervisée

Il s'agit de détecter de manière automatique les galaxies présentes dans les cubes de données hyperspectrales issues du spectroimageur MUSE. Un processus objet est utilisé pour représenter de manière parcimonieuse les champs de données massifs MUSE. L'estimation conjointe des paramètres du modèle et de la configuration des galaxies présentes dans la scène imagée est effectuée de manière automatique, dans un contexte entièrement bayésien. Le modèle utilisé satisfait à un principe d'invariance par transformation affine, permettant de correctement prendre en compte les grandes variations de dynamique au sein du cube. Les galaxies sont extraites sur des cubes de données "quasi-réelles". Leurs spectres ainsi que les propriétés spectrales du fond astrophysique sont également estimés.

Introduction

Le projet d'instrument MUSE (Multi Unit Spectroscopic Explorer [START_REF]Official website of the MUSE project[END_REF]), destiné à l'ESO (European Southern Observatory) vise à fournir aux astronomes un instrument de nouvelle génération afin d'observer le ciel profond. Les performances de MUSE devront permettre de détecter des galaxies extrêmement lointaines (et donc très jeunes) qui sont jusqu'à 100 fois moins lumineuses que celles actuellement détectables. Prévu pour être associé au VLT à l'horizon 2012, cet instrument mesurera un spectre en chacun des points du champ de vue de 60×60 arcsec. Il délivrera ainsi des "cubes de données" composés d'une pile d'images, ou feuillets, acquises à 3578 longueurs d'onde différentes variant de 430nm jusqu'à 930nm. Pour chaque feuillet de ce cube, i.e. pour chaque image "monochromatique", la résolution spatiale est de 0.2×0.2 arcsec. Par conséquent, chaque acquisition fournit un cube de données de 1570Mo stockant 3578 images de 300 × 300 pixels chacune, et pouvant contenir les réponses de plusieurs milliers de galaxies qui existent sur des plages de longueurs d'onde éventuellement différentes. L'exploitation de champs de données aussi massifs nécessite alors de définir des algorithmes dédiés afin d'extraire automatiquement les galaxies présentes. Dans ce contexte, des approches par processus "objet" tels que des processus ponctuels marqués semblent particulièrement bien adaptées.

Un processus ponctuel marqué est un processus aléatoire dont les réalisations sont des configurations aléatoires d'ob-jets. Chaque objet y est modélisé par un point (sa position), ainsi que par des marques qui correspondent à ses propriétés, ici géométriques ou spectrales. Ces processus fournissent donc une représentation naturellement parcimonieuse d'une configuration d'objets d'intérêt, présents dans une image. Ils permettent en effet de se détacher du modèle numérique induit par l'image, pour mieux se rapprocher du modèle physique. Les problèmes de complexité calculatoire inhérents aux approches pixeliques (et dus à la dimension ou à la taille des données) sont alors considérablement réduits : le nombre de paramètres ne dépend que du nombre d'objets effectivement présents dans la configuration. Pour ces raisons, ces processus ont été largement utilisés au cours de la dernière décennie, tout particulièrement en imagerie de télédétection [START_REF] Descamps | Automatic flamingo detection using a multiple birth and death process[END_REF][START_REF] Ortner | A marked point process of rectangles and segments for automatic analysis of digital elevation models[END_REF]. Cependant, un problème majeur des approches développées précédemment réside dans leur grande sensibilité aux choix de certains "hyperparamètres" qui contrôlent les termes d'attache aux données et les a priori du modèle. Ces paramètres sont alors classiquement ajustés manuellement à une valeur ad hoc pour chaque jeu de données.

L'approche proposée dans ce travail consiste à se placer dans un contexte entièrement bayésien. Ainsi des a priori non-informatifs, peuvant être introduits à un second niveau de hiérarchie dans le modèle bayésien, sont pris en compte afin de modéliser ces paramètres. La loi a posteriori du modèle bayésien hiérarchique obtenu peut alors être échantillonnée à l'aide de méthodes de Monte Carlo par chaînes de Markov à saut réversibles (RJMCMC) classiques. Il s'ensuit un algorithme général et robuste permettant d'estimer ces hyperparamètres sur la base des seules données observées. En outre, la détection de la configuration des objets d'intérêt, i.e. les galaxies, est effectuée conjointement à l'estimation de ces paramètres.

Dans la littérature, les galaxies sont couramment modélisées à l'aide de composantes elliptiques [START_REF] Simard | The deep groth strip survey. ii. hubble space telescope structural parameters of galaxies in the groth strip[END_REF][START_REF] Peng | Detailed Structural Decomposition of Galaxy Images[END_REF]. Aussi les galaxies sont-elles représentées comme des objets elliptiques dans notre modèle de processus ponctuels marqués.

Ce papier est organisé de la manière suivante. Le modèle bayésien de processus ponctuel marqué est présenté dans la partie 2, les grandes lignes de l'algorithme d'échantillonnage y étant également esquissées. Les résultats obtenus sur des données "quasi-réelles" sont présentés dans la partie 3.

Modèle bayésien

Dans la suite, on note y λ le vecteur M ×1 qui correspond au feuillet du cube hyperspectral associé à la longueur d'onde λ, représenté sous forme vectorisée. La configuration des objets est notée u = {u 1 , u 2 , . . .} et on désigne par n(u) le nombre d'objets présents dans cette configuration. Il est important de remarquer que la configuration des objets u est supposée commune à tous les feuillets du cube hyperspectral. Par conséquent, cette configuration ne dépend pas de longueur d'onde λ considérée.

Modèle d'observation monochromatique

Pour une longueur d'onde λ donnée, le modèle d'observation utilisé est défini comme :

y λ = s λ + 1m λ + n λ , (1)
où m λ > 0 représente la valeur moyenne du fond astrophysique, 1 = (1 . . . 1) T est le vecteur unitaire M ×1 et n λ est un vecteur M × 1 correspondant à un bruit blanc gaussien centré de variance σ 2 , qui reflète les fluctuations du fond astrophysique. Le vecteur s de taille M × 1 correspond à la réponse de l'instrument à une configuration de galaxies u dans la scène imagée. Le signal s est modélisé comme u) est le vecteur des intensités associées à chacun de ces objets à la longueur d'onde λ. Les équations précédentes illustrent le fait que le problème direct ainsi considéré correspond à un mélange linéaire de sources. Soulignons que la contrainte de positivité des intensités w i,λ n'est pas détaillée dans cet article.

s λ = X λ w λ , où X λ = x 1,λ . . . x n(u),λ ∈ R M ×n(u) est la matrice composée des réponses x i,λ du système d'imagerie associées à chaque objet u i (supposé d'intensité unitaire) et w λ = (w 1,λ , . . . , w n(u),λ ) T ∈ R n(
Sous l'hypothèse d'indépendance des observations associées aux différents feuillets, la vraisemblance s'obtient alors comme le produit des vraisemblances associées à chaque feuillet λ :

f (y λ |u, m λ , σ 2 λ ) = 1 2πσ 2 λ M 2 exp - Z T λ Z λ 2σ 2 λ , (2)
où Z λ = y λ -1m λ -X λ w λ . Pour des raisons de brièveté, la dépendance par rapport à λ sera omise dans le reste de l'article.

A priori sur les paramètres

Un a priori non-informatif impropre est choisi pour le couple de paramètres (m, σ 2 ) : p(m, σ 2 ) = 1 σ 2 1 (0,+∞) 2 (m, σ 2 ), un a priori gaussien de Zellner [START_REF] Zellner | Bayesian Inference and Decision Techniques : essays in honour of Bruno de Finetti, chapter On assessing prior distributions and Bayesian regression analysis with g-prior distribution[END_REF] étant choisi pour le vecteur des intensités :

w|u, σ 2 , g 2 ∼ N 0, g 2 σ 2 (X T X) -1 .
Il est important de noter que ces a priori satisfont à un principe d'invariance : toute transformation affine des mesures y, ou de la base de décomposition du signal X, transforme la distribution a posteriori et donc les estimées de la même façon. Les grandes variations de dynamique entre les différents feuillets observés pourront alors être correctement pris en compte. De plus, il est intéressant de remarquer que l'hyperparamètre g 2 représente le rapport signal à bruit (RSB) de la scène observée puisque

g 2 = E[s T s] E[n T n]
. Finalement, une structure bayésienne hiérarchique est introduite en choisissant un a priori conjugué pour l'hyperparamètre g 2 de la forme p(g 2 ) ∝ 1 (1+g 2 ) 1+α . Notons que ce dernier a priori ne satisfait plus au principe d'invariance. Néanmoins, le paramètre g 2 étant déjà une grandeur normalisée (un RSB), le choix de l'hyperparamètre α ne s'avère pas crucial quant à la robustesse du modèle (typiquement, des valeurs telle que α = 10 -3 garantissant une loi suffisamment vague sont suffisantes).

Distribution a posteriori conditionnellement à une configuration u

La distribution a posteriori des paramètres, conditionnellement à une configuration u, se déduit directement de la vraisemblance associée au problème direct ainsi que des a priori introduits au paragraphe précédent selon la formule de Bayes.

p(w, m, σ 2 , g 2 |u, y) ∝ 1 2πσ 2 M +n(u) 2 +1 1 g 2 n(u) 2 × exp - Z T Z + 1 g 2 w T X T Xw 2σ 2 × 1 (1 + g 2 ) α+1 1 (0,+∞) 2 (m, σ 2 ) (3)
Le modèle ainsi obtenu dépend néanmoins du vecteur w qui représente un très grand nombre de paramètres d'intensité par feuillet, ce qui s'avère notamment inefficace du point de vue échantillonnage. Par conséquent, ces paramètres sont par la suite considérés comme des paramètres de nuisance. Ceci conduit à intégrer la loi a posteriori par rapport à ces paramètres afin d'obtenir la loi a posteriori marginalisée suivante :

p(m, σ 2 , g 2 |y, u) ∝ 1 σ 2 M 2 +1 1 1 + g 2 n(u) 2 +α+1 × e - (y-1m) T I- g 2 1+g 2 X(X T X) -1 X T (y-1m) 2σ 2 × 1 (0,+∞) 2 (m, σ 2 ). (4)

A priori sur la configuration et distribution a posteriori

La configuration d'objets u = {u 1 , u 2 , . . .} correspondant aux galaxies est modélisée comme un processus ponctuel marqué [START_REF] Stoyan | Stochastic Geometry and Its Applications[END_REF] défini par référence à la mesure d'un processus de Poisson homogène d'intensité β défini sur l'espace produit des espaces des points et des marques. Un terme de pénalisation "hardcore" est introduit afin de régulariser la solution et d'éviter les détections multiples :

h(u) = 0 si ∃i = j tel que max s∈u i ∩u j x i,s s∈u i x i,s , s∈u i ∩u j x i,s s∈u j x i,s > t,
1 sinon, où s ∈ u k désigne l'ensemble des pixels appartenant au support spatial de l'objet k, x k,s est la réponse associée à cet objet au pixel s, et où t est fixé à la valeur déterministe t = 1 2 . Ainsi, les configuration où deux objets dont les réponses via l'instrument partagent plus de la moitié de leur puissance sur la même région sont interdites. De plus, un second niveau de hiérarchie est introduit en choisissant un a priori non-informatif impropre sur l'intensité β du processus : p(β) = 1/ √ β. Après marginalisation par rapport à ce paramètre, on obtient l'a priori suivant 3 Echantilloneur à saut réversible L'algorithme RJMCMC utilisé pour échantillonner conjointement la configuration des objets et les hyperparamètres fait intervenir plusieurs types de mouvements : 1) des mouvements simples pour mettre pour mettre à jour les hyperparamètres (mouvements de type Gibbs, ou Gibbs par bloc) ou les caractéristiques géométriques des objets de la configuration, et 2) des mouvements plus complexes qui modifient la dimension du modèle (naissance/mort,...). Ces différents mouvements sont ensuite aléatoirement sélectionnés dans la chaine.

p(u) ∝ Γ (n(u) + 1/2) h(u)µ(u), (5) où Γ(a)

Echantillonnage des paramètres

Les lois conditionnelles des paramètres, étant donné les autres, qui peuvent être obtenues directement à partir de (4), sont utilisés comme lois de proposition. Il s'ensuit des mouvements de Gibbs, les mises à jour des paramètres étant effectuées séquentiellement dans la dynamique de la chaîne.

Echantillonnage des paramètres du fond

(m, σ 2 ) On note ν = M -1, et W = I - g 2 1 + g 2 X(X T X) -1 X T , δ 2 = (1 T W 1) -1 , m = δ 2 1 T W y, s 2 = ν -1 δ 2 y T W y -δ 2 (1 T W y) 2 . (7)
A partir de (4) et [START_REF] Stoyan | Stochastic Geometry and Its Applications[END_REF], il vient que le paramètre de variance

σ 2 est distribué selon la loi Inverse-Gamma σ 2 |(m, g 2 , y, u) ∼ IG M 2 , 1 2δ 2 (νs 2 + (m -m) 2
) . En intégrant maintenant σ 2 , il vient que le paramètre m est conditionnellement distribué selon une loi de Student tronquée positive T + (ν, m, s) de degré de liberté ν, de paramètre de location m, et de paramètre d'échelle s :

p(m|g 2 , y, u) ∝ ν + m - m s 2 -ν+1 2 1 (0,+∞) (m).
Ceci conduit à considérer le mouvement de Gibbs par bloc suivant :

1. simulation de m|(λ 2 , y, u)

∼ T + (ν, m, s), 2. simulation de σ 2 |(m, λ 2 , y, u) ∼ IG( M 2 , 1 2δ 2 [νs 2 +(m- m) 2 ]).

Echantillonnage du paramètre de RSB g 2

A partir de (4), la loi a posteriori de g 2 s'obtient comme

p(g 2 |m, σ 2 , y, u) ∝ 1 1 + g 2 n(u) 2 +α+1 e - β 1+g 2 , où β = (y -1m) T X(X T X)X T (y -1m)/(2σ 2 ). Ceci mon- tre que la variable v = 1 + g 2 est distribuée selon une loi Gamma G (∞,+∞) ( n(u)
2 + α, β) tronquée sur (1, +∞) qui peut être simulée directement. La variable g 2 est alors mise à jour comme g 2 = v -1.

Mouvements sur la configuration

Afin de simuler la configuration des objets, des mouvements de naissance/mort sont effectués. Pour une configuration u donnée, on note p B (u) la probabilité de choisir le mouvement de naissance, p D (u) = 1-p B (u) étant la probabilité de choisir le mouvement réversible (mort) (en pratique, p B (u) = p D (u) = 1/2).

Dans le cas d'une naissance, le nouvel objet proposé v est tiré indépedemment de la configuration u selon le processus de Poisson de référence. Dans le cas d'une mort, l'objet proposé pour être supprimé est sélectionné uniformément dans la configuration courante, donc avec une probabilité p S (u i |u) = 1/n(u).

Finalement, pour une naissance, la configuration proposée est v = u ∪ {v} et le rapport de Metropolis-Hastings-Green a pour expression

r(u, v) = p D (v) p B (u)
p(v, θ|y) p(u, θ|y) p S (v|v).

Dans le cas d'une mort, la configuration proposée est v = u\{u i } et le rapport de Metropolis-Hastings-Green devient

r(u, v) = p B (v) p D (u)
p(v, θ|y) p(u, θ|y)

1 p S (u i |u)
.

Résultats

L'algorithme décrit précédemment a été appliqué sur un cube de données "quasi-réelles" synthétisé par le CRAL [START_REF]Official website of the CRAL laboratory[END_REF] dans le cadre de l'ANR Dahlia [9]. Ces données synthétiques reproduisent de manière extrêmement fidèle (d'un point de vue statistique, en terme de RSB, de densité d'objets, de contenu spectral,...) les données qui seront acquises par l'instrument MUSE.

Le cube ainsi étudié a été partiellement intégré par rapport à λ : au lieu de considérer les 3578 longueurs d'onde, de fines tranches provenant de l'intégration de 8 longueurs d'ondes consécutives sont prises en compte. Il en résulte un nouveau cube réduit de taille 300 × 300 × 448. Le profil des intensités associées à chaque ellipse u i est modélisé comme une densité gaussienne bivariée f 2D pour laquelle l'ellipse u i est une surface d'iso-probabilité au niveau α : (x,y)∈ui f 2D (x, y)dxdy = 1α. Une projection (intégration par rapport à λ) du cube débruité, où le bruit et le fond astrophysique estimés ont été soustraits, est représentée Fig. 1 (en niveau de gris inversés). Les contours des 384 galaxies détectées sont également représentés en rouge. Ces premiers résultats s'avèrent très encourageants. Différents exemples caractéristiques de spectres estimés parmi les galaxies détectées sont tracés sur la figure 2. La première ligne correspond à des spectres de galaxies brillantes, alors que sur la seconde sont représentées des galaxies faibles et probablement distantes. Ce sont ces dernières qui sont particulièrement intéressantes en vue de l'interprétation astrophysique.

Conclusions

Un processus ponctuel marqué bayésien a été proposé afin d'extraire et estimer automatiquement les galaxies observées dans les cubes de données MUSE. Différentes perspectives se dégagent parmi lesquelles la prise en compte d'un modèle réaliste de la fonction d'étalement du point de l'instrument, ou encore l'amélioration du modèle afin de mieux prendre en compte les grandes variations de dynamique entre les différents objets détectés.

= +∞ 0 t a- 1 e

 01 -t dt est la fonction Gamma classique et µ la mesure d'un processus de Poisson d'intensité unitaire. La loi a posteriori jointe de la configuration u et du vecteur de paramètre θ = (m, σ 2 , g 2 ) se déduit finalement comme p(u, θ|y) ∝ p(θ|u, y)p(u) (6) où p(θ|u, y) et p(u) représentent respectivement la loi a posteriori sur les hyperparamètres (4) et l'a priori sur la configuration (5).

FIGURE 1 -

 1 FIGURE 1 -Projection du cube débruité (niveaux de gris inversées) et contours (rouge) des 384 galaxies détectées

FIGURE 2 -

 2 FIGURE 2 -Exemples de spectres estimés sur la configuration détectée. Courbe bleue : galaxie, courbe noire : fond astrophysique, ligne haut : galaxies brillantes, ligne bas : galaxies faiblement brillantes