MD Eloi Marijon
email: eloi_marijon@yahoo.fr

PhD Muriel Tafflet

PhD, FRACP David S Celermajer

MD Florence Dumas

MSc Marie-Ce ´cile Perier

MD; Hazrije Mustafic

MD Jean-Franc ¸ois Toussaint

PhD Michel Desnos

MD Michel Rieu

MD Nordine Benameur

MD; Jean-Yves Le Heuzey

MD; Jean-Philippe Empana

PhD Xavier Jouven

Sports-Related Sudden Death in the General Population

Keywords: arrhythmia, cardiac arrest, athletes, cardiopulmonary resuscitation, defibrillation, bystander help

Background-Although such data are available for young competitive athletes, the prevalence, characteristics, and outcome of sports-related sudden death have not been assessed previously in the general population. Methods and Results-A prospective and comprehensive national survey was performed throughout France from 2005 to 2010, involving subjects 10 to 75 years of age. Case detection for sports-related sudden death, including resuscitated cardiac arrest, was undertaken via national ambulance service reporting and Web-based screening of media releases. The overall burden of sports-related sudden death was 4.6 cases per million population per year, with 6% of cases occurring in young competitive athletes. Sensitivity analyses used to address suspected underreporting demonstrated an incidence ranging from 5 to 17 new cases per million population per year. More than 90% of cases occurred in the context of recreational sports. The age of subjects was relatively young (mean±SD 46±15 years), with a predominance of men (95%). Although most cases were witnessed (93%), bystander cardiopulmonary resuscitation was only commenced in 30.7% of cases. Bystander cardiopulmonary resuscitation (odds ratio 3.73, 95% confidence interval 2.19 to 6.39, P<0.0001) and initial use of cardiac defibrillation (odds ratio 3.71, 95% confidence interval 2.07 to 6.64, P<0.0001) were the strongest independent predictors for survival to hospital discharge (15.7%, 95% confidence interval 13.2% to 18.2%). Conclusions-Sports-related sudden death in the general population is considerably more common than previously suspected. Most cases are witnessed, yet bystander cardiopulmonary resuscitation was only initiated in one third of cases. Given the often predictable setting of sports-related sudden death and that prompt interventions were significantly associated with improved survival, these data have implications for health services planning.

Clinical Perspective on p 681

Sports-related SDs in young competitive athletes are highly visible and usually attract considerable public and media attention. On the basis of relatively large studies of sports-related SDs in young athletes from the United States and Italy, [START_REF] Corrado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF][START_REF] Maron | Comparison of U.S. and Italian experiences with sudden cardiac deaths in young competitive athletes and implications for preparticipation screening strategies[END_REF] specific preventive measures have been suggested, and some have been introduced in those countries. [START_REF] Corrado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF][START_REF] Maron | Comparison of U.S. and Italian experiences with sudden cardiac deaths in young competitive athletes and implications for preparticipation screening strategies[END_REF][START_REF] Pelliccia | Outcomes in athletes with marked ECG repolarization abnormalities[END_REF] Preparticipation screening and emergency response planning, for example, have been associated with a significant decrease in the incidence and mortality of such events in young athletes. [START_REF] Corrado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF][START_REF] Maron | Comparison of U.S. and Italian experiences with sudden cardiac deaths in young competitive athletes and implications for preparticipation screening strategies[END_REF][START_REF] Drezner | Effectiveness of emergency response planning for sudden cardiac arrest in United States high schools with automated external defibrillators[END_REF][START_REF] Drezner | Survival trends in the United States following exercise-related sudden cardiac arrest in the youth: 2000 -2006[END_REF] Nevertheless, the relatively low absolute number of such events occurring annually among young competitive athletes (estimated at 50 to 75 per year in the United States) appears to have influenced authorities in other countries, [START_REF] Maron | Sudden deaths in young competitive athletes: analysis of 1866 deaths in the United States, 1980 -2006[END_REF] such as France (Projet d'amendement relatif a ` l'installationde de ´fibrillateurs dans les installations sportives de grande taille: Mission d'evaluation des politiques publiques-Re ´publique Franc ¸aise, Mai 2009), where comprehensive preventive strategies (eg, widespread access to automatic external defibrillators) have not been instituted in public sports arenas because of a perceived lack of cost-effectiveness.

The extent to which data on sports-related SD based on young competitive athletes are generalizable to the wider community remains unknown. We hypothesized that sportsrelated SD would be more prevalent in the general population and thus designed a prospective nationwide observational study over a 5-year period to examine this issue. Our aims were to determine the overall sports-related SD burden in the community, to characterize the circumstances and features of sports-related SD, and to identify the independent factors that influence outcome.

Methods

Setting, Definitions, and Survey Methods

This prospective study was performed in France between April 2005 and April 2010. We included subjects 10 to 75 years old because the context of physical activity can be difficult to assess in the very young, and those older than 75 years only rarely participate in sports. The project was commissioned in 2004 by the Ministe `re de la Sante ´ et des Sports and was performed by the Institut National de la Sante ´ et de la Recherche Me ´dicale (INSERM). We obtained close collaboration from the French system of mobile intensive care units (Service d'Aide Me ´dicale Urgente [SAMU]). All of the 96 administrative districts throughout France were invited to participate, and 60 of these districts eventually participated (see Appendix for participating centers).

A death was considered to be sudden if it occurred within 1 hour of symptom onset. [START_REF] Corrado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF][START_REF] Maron | Sudden deaths in young competitive athletes: analysis of 1866 deaths in the United States, 1980 -2006[END_REF][START_REF] Fishman | Sudden cardiac death prediction and prevention: report from a National Heart, Lung, and Blood Institute and Heart Rhythm Society workshop[END_REF] Sports-related SD was defined if death occurred during sport or within 1 hour of cessation of sports activity. In most cases, this resulted from an unexpected collapse during physical exertion or immediately after, associated with a previously uneventful clinical course. Sports-related SDs related to trauma (for example, runner or cyclist hit by a car) were excluded, other than for cases of commotio cordis. A "young competitive athlete" was defined as any person 10 to 35 years old who participated in an organized sports program (team or individual sport) that required regular competition and training according to the National Registry from the Ministe `re de la Sante ´ et des Sports (individuals participating in college-sponsored intramural sports were not considered young competitive athletes). 16 Sports-related SDs that occurred in competitive athletes outside the context of physical activity were not included. Sports participants who survived cardiac arrest after defibrillation and/or cardiopulmonary resuscitation (CPR) were considered to have experienced sports-related SD for the purpose of the present analysis.

To maximize case detection, 2 complementary independent methods were instituted that have already been used by other groups for young competitive athletes. [START_REF] Maron | Sudden deaths in young competitive athletes: analysis of 1866 deaths in the United States, 1980 -2006[END_REF] We tested the efficacy of the following case-detection methodology in France by performing a preliminary 6-month pilot study over 5 districts in 2004. First, a prospective case reporting system was instituted via the mobile intensive care services (SAMU); the French SAMU system is a nationwide emergency medical service accessed via a well-known nationwide free call number. Thus, SAMU is systematically called in cases of cardiac arrest or sudden collapse. Emergency health technicians in SAMU assess these subjects and start resuscitation where required, on site. For each case of sports-related SD during the study period, a specific detailed report form was completed by SAMU and then transmitted to the data collection center (INSERM U970, Paris, France) within the following 3 months. In parallel, a media search program screened for cases from continuous Web-based monitoring via Google Reader using the Really Simple Syndication system for local, regional, and national newspapers (n=275) with the key words "death," "cardiac arrest," and "dizziness." Manual checking of these reports was performed to identify verifiable sports-related SDs in 10-to 75-yearolds, and these cases were followed up if we had not already been notified of them via SAMU; complementary information was then obtained from written accounts or telephone interviews with the local emergency team of the SAMU.

A systematic review of all sports-related SD notifications was undertaken by an independent events committee every 6 months. Three experts in sports-related SD adjudicated cases to exclude SDs unrelated to sports, nonfatal sports-related cardiovascular events, and deaths that did not meet the SD definition, and finally to identify incomplete files that required further information.

Collected Data

Data were collected and registered according to revised Utstein guidelines, with an additional specific focus related to sports activity information. [START_REF] Cummins | Recommended guidelines for uniform reporting of data from out-of-hospital cardiac arrest: the Utstein style: a statement for health professionals from a task force of the American Heart Association, the European Resuscitation Council, the Heart and Stroke Foundation of Canada, and the Australian Resuscitation Council[END_REF][START_REF] Jacobs | Cardiac arrest and cardiopulmonary resuscitation outcome reports: update and simplification of the Utstein templates for resuscitation registries: a statement for healthcare professionals from a task force of the International Liaison Committee on Resuscitation (American Heart Association, European Resuscitation Council, Australian Resuscitation Council, New Zealand Resuscitation Council, Heart and Stroke Foundation of Canada, Interamerican Heart Foundation, Resuscitation Councils of Southern Africa)[END_REF] Case report forms used by the emergency team of the SAMU included (1) date of arrest, patient identifier, and district of each event; (2) clinical and demographic information (age or date of birth, sex, any personal history of known heart disease, and any cardiovascular symptoms during the 4 preceding weeks); (3) data regarding the setting (recreational or competitive) and level of exercise at the time of the sports-related SD, as assessed by the ambulance officers using a prospectively defined scale of 1 to 8 metabolic equivalents (MET), classified as light (<4 MET), moderate (4 to 6 MET), or vigorous (>6 MET), [START_REF] Whang | Physical exertion, exercise, and sudden cardiac death in women[END_REF][START_REF] Mittleman | Determinants of Myocardial Infarction Onset Study Investigators. Triggering of acute myocardial infarction by heavy physical exertion: protection against triggering by regular exertion[END_REF] as well as the sports activity being undertaken at the time of the event and the regular training regimen of each subject; (4) circumstances of collapse, location type (sports facility or other area), presence of witnesses, and bystander CPR and defibrillation attempt; (5) detailed information regarding the resuscitation (time of collapse, SAMU call receipt, time SAMU vehicle arrived, CPR initiation, time of first rhythm analysis, and time of achieved return of spontaneous circulation), the initial cardiac rhythm recorded, and the number and timing of delivered defibrillator shocks; and (6) survival status to hospital admission.

Complementary data on survival status and causes of death were managed directly by INSERM U970 in collaboration with hospitals, and in some cases after contact with the patient's regular physician. Status of the patients at hospital discharge was available in the great majority of admitted cases (97.6%). Cause of death could only be obtained in cases in which an autopsy was performed or clinical features at the time of death permitted a confident diagnosis for those who died on site. In France, the performance of an autopsy is at the discretion of the physician certifying the death and is only mandated when the death is regarded as potentially "suspicious." For survivors, neurological status at discharge was evaluated with the Cerebral Performance Categories score. [START_REF]The Brain Resuscitation Clinical Trial II Study Group. A randomized clinical trial of calcium entry blocker administration to comatose survivors of cardiac arrest: design, methods, and patient characteristics[END_REF] Importantly, no specific recommendations were specified in the study design regarding emergency care modalities or documentation of causes of death, because the objective of the investigators was to offer a "real-life" descriptive picture of sports-related SD features in France.

Statistical Methods

This report was prepared in compliance with the STROBE (Strengthening the Reporting of Observational Studies in Epidemiology) checklist for observational studies. [START_REF] Vandenbroucke | The making of STROBE[END_REF] The primary objective of the study was to provide an estimate of the incidence of sports-related SD in the general population of subjects 10 to 75 years of age. Secondary objectives were to describe population characteristics and identify variables associated with survival. The overall incidence of sports-related SD in France was estimated by extrapolation of the incidence of sports-related SD in the 60 participating administrative districts (which represent approximately 70% [33 948 400] of the French population 10 to 75 years of age) to the overall 96 French districts. We calculated the incidence in each participating district by dividing the average number of sportsrelated SDs reported per year by the locally registered 10-to 75year-old population. [START_REF]Population Française[END_REF] Because there is no evidence to support a triggering effect of light exertion for SD, the primary result regarding calculation of incidence only included cases that occurred during moderate and vigorous exertion (n=780). Moreover, because our estimated incidence of sports-related SD in the general population is almost certainly an underestimate, in addition to providing the global incidence with its 95% confidence interval (CI), 2 scenarios were then calculated based on the 80th and 20th percentiles of the highest reported incidences of sports-related SD across different districts in France.

The specific incidence of sports-related SD among young competitive athletes was calculated according to data from the Ministe `re de la Sante ´ et des Sports. 16 The participating number of 10-to 35year-old athletes over the study period was 1 015 293, and this figure was considered stable over time. 16 Because of the constant high level of media coverage of sports-related SDs in young sports participants in France (that is, such reports were considered likely to be comprehensive), we provided only 1 incidence rate calculation for this population.

Two additional sensitivity analyses were performed: First, on cases that occurred during vigorous exertion among the entire population (n=322), and second, among all degrees of exertion among the 10-to 60-year-old age group (n=667). The characteristics of sports-related SDs are described as mean±SD, proportions, or median and interquartile ranges, as appropriate. Qualitative variables were compared with the x 2 test. All covariates that reached a significance level of P<0.15 for survival were then included in an initial multivariable regression model. Odds ratios (ORs) and their 95% CIs were calculated. All tests were 2-tailed, and P<0.05 was considered to indicate statistical significance. All data were analyzed at INSERM, Unit 970, Cardiovascular Epidemiology and Sudden Death, Paris, using STATA software version 11.0.

The authors had full access to the data and take full responsibility for its integrity. All authors have read and agree to the manuscript as written.

Results

Incidence Rates of Sports-Related Sudden Death

After 5 years of observation, representing 169 742 000 agespecific person-years, a total of 820 sports-related SDs were recorded (Figures 1 and2); of these, 51% were first reported by SAMU and 49% by press reports. The proportion of cases reported first by SAMU varied between participating districts (from 32% to 100%). The overall estimate of sports-related SD was 4.6 (95% CI 3.9 to 5.3) per million population per year.

Additional sensitivity analyses were performed to address suspected underreporting of sports-related SD in each district; considering the 80th and 20th highest percentiles of reporting districts, the incidence of sports-related SD throughout France was calculated to be 5.4 and 16.7 cases per million per year, respectively (Table 1). Considering the 60-to 75-yearold population separately did not alter the nature of our findings, because the calculated incidence when we excluded the cases in the older age group was 15.9 (95% CI 13.2 to 18.6) per million per year (20th percentile). Among the 820 reported sports-related SDs, only 50 cases (6%) occurred in young competitive athletes, which yielded an incidence in this group of approximately 10 cases per million per year (Figure 2; Table 1). Compared with young (10 to 35 years old) noncompetitive sports participants, young competitive athletes had a higher risk of sports-related SD (relative risk 4.5, 95% CI 2.3 to 8.7). There were 777 SDs in males and 43 SDs in females (male/female ratio of 20:1), equivalent to incidence rates (20th percentile) of 32.8 and 1.9 million persons per year, respectively (relative risk 17.3, 95% CI 5.9 to 50.0).

Patient Characteristics and Sports-Related Sudden Death Settings

The mean age of the sports-related SD subjects was 46±15 years (range 11 to 75 years; Table 2). Ninety-four subjects (11.7%) had a known history of cardiovascular disease or presented with more than 1 classic cardiovascular risk factor for coronary heart disease. The majority (86.5%) of sportsrelated SD subjects performed regular sports and/or training. Symptoms, mainly chest discomfort, were reported in the days preceding the sports-related SD in 104 patients (12.7%). These patients were more likely to have sports-related SD due to ischemic heart disease than other participants (92.3% versus 70.7%, P=0.005). Data are n (%), unless otherwise specified. *Data were missing for history of known heart disease (for 8 subjects), cardiovascular risk factors (11), intensity of exercise (8), presence of witness (4), and monitored rhythm (12). Percentages were calculated on the basis of the total number of known events. †Others denote operated congenital heart disease (2), history of Kawasaki disease (1), or Marfan syndrome (1).

‡Cardiovascular risk factor included known/treated diabetes mellitus, dyslipidemia, systemic hypertension, obesity, current smoker, and family history of premature coronary heart disease.

Sports-related SD actually occurred during sports activity in 92%, immediately after (within 30 minutes) in 7.4%, and only rarely more than 30 minutes after cessation of sports activity. Most sports-related SDs were observed during cycling (30.6%), running (21.3%), or soccer (13.0%), the sport activities most practiced in France (Figure 3). Sports work in a sheltered environment. The highest survival rates (=50%) were in the Burgundy (Co ˆte-d'Or) and Nord-Pas-de- Calais (Nord) districts, where bystanders initiated CPR in more than 90% of cases.

Univariable analyses for factors associated with survival to hospital discharge are shown in Table 3. According to multiple logistic regression analysis, the following factors were independently associated with survival to hospital discharge: Bystander CPR (OR 3.73, 95% CI 2.19 to 6.39, P<0.0001), time from collapse to start of CPR (OR 1.32, 95% CI 1.08 to 1.61, P=0.006), and the initial use of cardiac defibrillation (OR 3.71, 95% CI 2.07 to 6.64, P<0.0001). Additional sensitivity analyses among cases of sports-related SD associated with vigorous exertion and among the 10-to 60-year-old population did not demonstrate significant differences regarding predictors for survival compared with the overall population.

Cardiovascular Abnormalities Associated With Sports-Related Sudden Deaths

The proportion of cases with a clear diagnosis remained low (24.7%), particularly among subjects who died on site (5.1%). Cause-of-death analysis among young competitive athletes and the general population is presented in Figure 4. Although 64% of sports-related SDs among young competitive athletes had the cause elucidated, this was only available for 22% of SDs among the general population.

Among the 203 cases in which it was possible to ascertain the cause of death with clinical certainty or from anatomopathological investigations, causes were cardiac in origin in all but 4 cases (98.0%). Acute coronary syndrome was the principal cause of cardiac arrest during sports (152 subjects, 74.9%); other cardiovascular causes of death or pathological findings included hypertrophic cardiomyopathy (7 subjects), possible hypertrophic cardiomyopathy (6), congenital cardiac abnormality (5, including 2 cases of aberrant coronary artery), dilated cardiomyopathy (5), myocarditis (5, including 1 case of sarcoidosis), arrhythmogenic right ventricular dysplasia (3), early repolarization syndrome (2), Wolff-Parkinsonwhite syndrome (1), familial long-QT syndrome (1), mitral valve prolapse (2), and commotio cordis (1). Noncardiac causes were found in 4 cases, ascribed to epilepsy, ruptured cerebral artery, stroke, and ruptured ascending aorta, respectively. The remaining 9 cases were ascribed to idiopathic ventricular fibrillation because no cardiac abnormality was observed. Among the 820 patients, drugs or anabolic steroids were reported to have been implicated in death in 2 cases.

Discussion

In this prospective study of 820 cases of sports-related SDs and resuscitated cardiac arrests in a general population, we have identified a higher burden of this event than previously suspected from comprehensive surveys of sports-related SD in young competitive athletes. We found that the majority of such events were witnessed (93%), although bystander CPR was initiated in fewer than one third of the cases. Given the often predictable setting of sports-related SD and that prompt interventions were significantly associated with improved survival, these data have implications for health services planning. In particular, education about CPR for sports participants and availability of public access defibrillation in sports facilities have the potential to achieve substantial improvements in outcomes.

Previously published data on the incidence of sportsrelated SDs observed in competitive young athletes have documented an incidence of 6 to 9 per million such competitors per year. [START_REF] Corrado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF][START_REF] Maron | Sudden deaths in young competitive athletes: analysis of 1866 deaths in the United States, 1980 -2006[END_REF] The present data among young competitive athletes are very consistent with these estimates. The largest previous European study of sports-related SDs, from Veneto, Italy, included 55 cases of sports-related SDs in young competitive athletes documented over a 25-year period. [START_REF] Corrado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF][START_REF] Corrado | Does sports activity enhance the risk of sudden death in adolescents and young adults?[END_REF] Maron et al 8 reported 1866 cases of SD in young competitive athletes over a 27-year period in the United States, 50% of which were traumatic, of noncardiac origin, or did not occur during sporting activity. Taken together, these data suggest that approximately 50 to 75 SDs in young competitive athletes would occur annually in the United States, and approximately 10 to 15 such events would occur in France each year. The focus on this group of healthy young athletes is highly understandable given the substantial social and emotional impact of sudden and unexpected deaths in this group. [START_REF] Maron | Sudden death in young athletes[END_REF] However, data showing relatively low absolute numbers of events in young athletes per year have influenced health authorities in many countries regarding the institution of widespread early response measures. In France in May 2009, for example, draft legislation proposing the installation of automatic external defibrillators in sporting arenas was eventually rejected after evaluation by an economic expert com-mittee that concluded that this would not be cost-effective based on the 10 to 15 expected such cases yearly.

Data concerning the risk of sports-related SD among recreational sports participants, who accounted for more than 90% of cases in the present study, are relatively sparse. These data come from a limited number of retrospective studies that involved small numbers of cases and most often were in regard to only 1 sports activity. 9,10,26 -32 To date, only 1 study attempted to evaluate sports-related acute cardiovascular events in the general population, with no specific focus on SD. [START_REF] Chevalier | Sports-related acute cardiovascular events in a general population: a French prospective study[END_REF] Other available data from autopsy series contribute information on likely causes of SD; however, these are associated with bias and do not allow calculation of the population burden of sports-related SD. 34 -38 Finally, to the best of our knowledge, sports-related SD has never been evaluated in the general population, including athletes and recreational sports participants across a wide age range. The present study revealed a much higher prevalence of sportsrelated SD in the general population than previously suspected (estimated at approximately 4250 cases in the United States or 800 cases in France each year in the 10-to 75year-old age group). The present data demonstrated that although risk for sports-related SD remains higher in young competitive athletes than in 10-to 35-year-old noncompetitive sports participants, the absolute risk appears to be higher in the general population. In the present series, although the survival rate to hospital discharge after sports-related SD (approximately 16%) was substantially higher than the published rates of survival after SD in general, at approximately 7% to 8%, [START_REF] Sasson | Predictors of survival from out-of-hospital cardiac arrest: a systematic review and meta-analysis[END_REF] there may still be substantial room for improvement. For example, 90% of sports-related SDs in the present series were witnessed, but bystander CPR was commenced in only one third of these cases. The data from 2 French districts where resuscitation rates of more than 90% were associated with survival rates of approximately 50% support calls for more extensive population-based CPR education. Because the population is relatively young and such settings offer optimal chances for survival, these data should prompt the development of preventive measures based on public education. Moreover, these results also raise the question of systematic screening, especially for ischemic heart disease, among male recreational sports participants with cardiovascular risk factors.

We note the striking male predominance in sports-related SD in the present series. Based on sports participation rates in France, this is not simply the reflection of a greater male participation but likely the consequence of a higher risk profile in men. The reasons for this sex difference is unclear, but it could be related to the known higher prevalence of coronary disease in young and middle-aged men than in women. Differences in the mode of physical activity and training between sexes may also be relevant.

We acknowledge that the present study has a number of potential limitations. First, we observed important differences in incidences among districts. One possible explanation is that this may reflect real differences in incidences due to subject characteristics or sports activities. However, the ratio of emergency medical services cases to total cases varied markedly among districts, which leads us to favor a different explanation, ie, that the variability of reported incidences across different districts reflects a variable degree of underreporting. Our estimated incidence of sports-related SD in the general population is also almost certainly an underestimate. In addition to underreporting, this is also due in part to the consideration of the entire 10-to 75-year-old population as being potentially at risk of sports-related SD rather than estimating the proportion who were actually participating in sports (this is because there are no reliable data to estimate actual recreational sports participation). This has been the rationale for the use of sensitivity analyses to provide a range of incidence of community-based sports-related SD. Second, autopsy rates are very low in France, where they are performed at the discretion of the certifying physician rather than mandated by law. The present results among young competitive athletes, including the rate of cases without precise diagnosis, are consistent with the study of the largest cohort to date performed by Maron and colleagues in the United States. [START_REF] Maron | Sudden deaths in young competitive athletes: analysis of 1866 deaths in the United States, 1980 -2006[END_REF] On the other hand, we offered a different view from the general population with a majority of cases considered ischemic cardiovascular events but that ultimately remained without a precise diagnosis. Further efforts for better identification of cause of death, including postmortem molecular analysis, are crucial in this context, in which early diagnosis of familial cardiomyopathy or channelopathy may be vital for relatives. [START_REF] Chugh | Determinants of prolonged QT interval and their contribution to sudden death risk in coronary artery disease: the Oregon Sudden Unexpected Death Study[END_REF] In summary, we have documented a substantially higher incidence of sports-related SD in the general population than suspected previously from studies focused on competitive young athletes. Depending on the degree of underreporting present in our sample, we estimate from 5 to 17 cases of sports-related SD occur per million individuals residing in France per year. Most such events occur in previously healthy men 35 to 65 years old, happen in a public area, and are witnessed by bystanders. Given the often predictable setting of sports-related SD and that prompt interventions including bystander CPR and defibrillation were significantly associated with improved survival, these data have implications for health services planning.

Figure 2 .

 2 Figure 2. Distribution by age of sports-related sudden deaths (SDs) in the overall population (blue) and among young competitive athletes (red).

Figure 3 .

 3 Figure 3. Sports engaged in at the time of sudden death (SD) in 820 sports participants. Shaded proportions of the bars represent the female ratio.

of Signs of Circulation and/or Considered for Resuscitation N=820

	Absence
	Resuscitation Not Attempted	Resuscitation Attempted
	N=15	All cases, N=805
		Any defibrillation attempt, N=494
		Chest compressions, N=805
		Assisted ventilation, N=523
	CPR Before EMS Arrival, N=252
		Outcome
		Any ROSC
		Yes, N=494
		No, N=296
		Unknown, N=15
		Survived Event, N=253
		Discharged Alive, N=128
		Neurologic Outcome at Discharge
		CPC 1 or 2, N= 103
		CPC 3 or 4, N=23
		Unknown, N=8

Table 1 . Estimated Incidence Rates of Sports-Related Sudden Death in the General Population, Amongst Young Competitive Athletes and Young Noncompetitive Sport Participants

 1

			No. of	Incidence Rates, per Million	Estimated Absolute
		Population	Districts	per Year (95% CI)	per Year in France
	General population (10 -75 y)
	Overall	33 948 400	60	4.6 (3.9-5.3)	226
	Male	16 195 232	60	9.2 (7.7-10.7)	216
	Female	17 753 168	60	0.4 (0.1-0.7)	10
	20th percentile	7 239 500	12	16.7 (13.7-19.7)	814
	Male	3 474 960	12	32.8 (26.8-38.8)	772
	Female	3 764 540	12	1.9 (0.5-3.2)	42
	80th percentile	27 950 100	48	5.4 (4.5-6.3)	263
	Male	13 116 500	48	10.6 (8.8-12.4)	250
	Female	14 833 600	48	0.8 (0.4-1.3)	13
	Young competitive athletes	1 015 293	60	9.8 (3.7-16.0)	15
	Young noncompetitive participants	12 358 498	60	2.2 (1.4-3.0)	39
	CI indicates confidence interval.

Table 2 . Characteristics of Patients and Selected Features of 820 Sports-Related Sudden Deaths Included in the Study ValueTable 2 .

 22 Continued

	Value
	Delay, median (IQR), min

Bystander Cardiopulmonary Resuscitation, Advanced Life Support, and Predictors of Survival

  -related SDs during team sports activities occurred at a younger age than during individual sports (33±11 versus 51±13 years,

	Week		P<0.0001). Half of the sports-related SDs (51.9%) occurred
	Weekend	336 (41.0)	in public sports facilities (such as a gymnasium or a stadium),
	Time of day		and 99.8% of these were witnessed. Bystanders in sports
	Morning (6 AM to 12 PM)	287 (35.0)	facilities were slightly more likely to initiate CPR than were
	Afternoon (12 PM to 6 PM)	359 (43.8)	bystanders outside these settings, but this difference was not
	Evening (6 PM to 12 AM)	174 (21.2)	statistically significant (35.5% versus 30.8%, P=0.15).
	Sport setting Sport activities Team sports Individual sports	273 (33.3) 547 (66.7)	Among cases outside these settings, 83.4% were witnessed. There were more sports-related SDs on Sunday (P<0.0001, asymptotic test for binomial proportions), but no significant circadian variation was observed.
	Location of occurrence
	Sport facilities	426 (52.0)
	Others	394 (48.0)
	Intensity of exercise* Light Moderate	32 (3.9) 458 (56.4)	Bystander CPR was initiated in 252 cases (30.7%). Median time from collapse to CPR initiation was 4.8 minutes (inter-quartile range 1.2 to 8.5 minutes). The initial cardiac rhythm was ventricular fibrillation or pulseless fast ventricular
	Vigorous	322 (39.7)	tachycardia in 377 cases (46.1%), with a mean number of
	Management of sports-related SD		shocks in this group of 5.3±3.8 (range 1 to 19).
	Sudden death witnessed*	756 (92.6)	A total of 253 of the 820 subjects (30.8%; 95% CI 27.7%
	Bystander CPR	252 (30.7)	to 34.0%) survived to hospital admission, and among them,
	First monitored rhythm*		vital status at time of discharge was known for 247 (97.6%).
	Ventricular fibrillation or tachycardia	377 (46.7)	One hundred twenty-eight of these subjects survived to
	Asystole	343 (42.5)
	Pulseless electrical activity	87 (10.8)
	Sinus rhythm	1 (0.1)

hospital discharge, giving an overall survival rate to hospital discharge of 15.7% (95% CI 13.2% to 18.2%). The majority of these (103 of 126 for whom neurological assessment was Shock by external defibrillator established) presented with a Cerebral Performance Categories 494 (61.1) score of 1 or 2, good enough to allow normal life or part-time (Continued)

Table 3 . Univariate Analysis for Survival to Hospital Discharge

 3

	Predictors of Survival: Variables	OR	95% CI	P
	Women vs men	1.59	0.76 -3.32	0.21
	Age, y
	10-30	1.00		<0.0001
	31-50	0.60	0.36-0.99
	51-75	0.26	0.15-0.44
	Known CVD vs others	0.56	0.28-1.12	0.10
	Seasons
	Winter	1.00		0.18
	Spring	1.61	0.92-2.84
	Summer	1.13	0.62-2.04
	Autumn	1.64	0.89-3.00
	Weekend	1.60	1.10-2.32	0.01
	Team sports	0.98	0.66-1.45	0.92
	Intensity of exercise
	Light	1.00		0.002
	Moderate	1.70	0.40-7.51
	Vigorous	3.36	0.78-14.5
	Sport facilities vs others	2.83	1.88-4.25	<0.0001
	Competitive vs recreational setting	1.32	0.83-2.12	0.24
	Young competitive athlete	1.62	0.82-3.18	0.16
	Presence of witness	6.73	1.62-27.8	0.009
	Bystander CPR	9.83	6.42-15.04	<0.0001
	Collapse to start of CPR delay, min
	0-2
	3-4
	5-10
	11-20
	Initial shockable rhythm	9.60	5.88-15.70	<0.0001
	OR indicates odds ratio; CI, confidence interval; CVD, cardiovascular disease;
	and CPR, cardiopulmonary resuscitation.

Acknowledgments

We would like to acknowledge the contributions of all participating units of SAMU of France, as well as Dr Pe ´ne ´lope Troude for her technical assistance.

Sources of Funding

This study was supported in part by INSERM (Institut National de la Sante ´ et de la Recherche Me ´dicale) and IRMES (Institut de Recherche BioMe ´dicale et d'E ´ pide ´miologie du Sport).

Appendix Participating Centers

The investigators and coordinators participating in the study were as follows (

CLINICAL PERSPECTIVE

To date, the majority of data regarding sports-related sudden death (SD) have focused on young competitive athletes, and few data concerning sports-related SD in the general population have been available. In this 5-year observational study of SD occurring during sports in competitive and recreational participants 10 to 75 years of age, we have found that although the risk for sports-related SD remains higher in young competitive athletes than in young noncompetitive sports participants, the absolute risk is otherwise higher in the general population. The present study revealed approximately 800 cases in France each year compared with the 15 annual cases among young competitive athletes. Case subjects were relatively young (46±15 years), predominantly males (95%), and without any prior history of heart disease. Half of the observed cases occurred in sports facilities, and almost 90% of events were witnessed; however, the rate of bystander cardiopulmonary resuscitation was low, initiated in fewer than one third of cases. Given the often predictable setting of sports-related SD and that prompt interventions including bystander CPR and defibrillation were significantly associated with improved survival in the present study, these data have implications for health services planning.


Maissa Chagmani
email: chagmani@univ-tln.fr

Bernard Xerri
email: xerri@univ-tln.fr

Bruno Borloz
email: borloz@univ-tln.fr

Claude Jauffret
email: jauffret@univ-tln.fr

The Constrained Stochastic Matched Filter Subspace Tracking

Keywords: tracking, signal-to-noise ratio, detection, subspace projection algorithm, subspace. I

This paper introduces a new fast algorithm

named CSMFST which estimates the p-dimensional optimal subspace, i.e. where the signal-to-noise ratio is maximized in the case of n-dimensional nonstationary signals.

We assume that we treat both signal and noise which are characterized by their samples. This algorithm is an SP-type algorithm and uses the same principles as the Yet Another Subspace Tracking (YAST) algorithm when estimating the covariance matrices. At each step, it estimates a matrix which spans the optimal subspace.

INTRODUCTION

The main goal of the CSMFST algorithm that we present in this paper is to estimate the principal and unique p-dimensional subspace in which the signal-to-noise ratio is maximized, from known (n x n) correlation matrices computed from the signal and noise realizations at each iteration. This leads us to resolve an equation with a linear combination of the signal matrix, the noise matrix and the signal-to-noise ratio that depends on the weighting matrix 𝑾(𝑡).

In literature, we can find algorithms which have applications in blind channel estimation, adaptive filtering and spectral analysis [START_REF] Veen | Subspace-based signal analysis using singular value decomposition[END_REF]. They are classified according to their complexity: high for 𝑂(𝑛 2 𝑝) and 𝑂(𝑛 2 ) such as Rank revealing QR [START_REF] Bischof | On updating signal subspaces[END_REF], medium for 𝑂(𝑛𝑝 2 ), and finally methods with low complexity for 𝑂(𝑛𝑝) such as the PAST algorithm presented in [START_REF] Yang | Projection approximation subspace tracking[END_REF], its orthogonal version OPAST [START_REF] Abed-Meraim | Fast orthonormal PAST algorithm[END_REF] FAPI [START_REF] Badeau | Fast approximated power iteration subspace tracking[END_REF] and FDPM [START_REF] Doukopoulos | Fast and Stable Subspace Tracking[END_REF]. Their common goal is to estimate and track the p-dimensional principal subspace generated by the p eigenvectors associated with the largest eigenvalues of the data correlation matrix within a ndimensional subspace for fixed p<n. However, the problem treated by the CSMFST algorithm is different and can be resolved by implementing a subspace tracker which could be able to estimate the subspace weighting matrix even with a timevarying signal-to-noise ratio and subspace. In order to implement the CSMFST algorithm, we will rely on one of the SP-type algorithms [START_REF] Davila | Efficient, high performance, subspace tracking for time-domain data[END_REF] which is the YAST algorithm first introduced in [START_REF] Badeau | Yet Another Subspace Tracker[END_REF] and [START_REF] Badeau | YAST Algorithm for Minor Subspace Tracking[END_REF] then presented in [START_REF] Badeau | Fast and stable YAST algorithm for principal and minor subspace tracking[END_REF] since it outperforms many subspace trackers and guarantees the orthonormality of the subspace weighting matrix at each iteration with a low complexity. Indeed, a performance comparison in stationary and nonstationary cases between the different tracking algorithms shows that the YAST algorithm has the best steady-state error and convergence rate among all fast subspace trackers as presented in [START_REF] Arjomandi-Lari | Generalized YAST algorithm for signal subspace tracking[END_REF]. In this paper, we will only present the principle of the CSMFST algorithm without detailing the calculations and doing all the demonstrations, however this will be done in the near future. The paper is organized as follow. In section 2, the principle of the YAST algorithm is presented. Then, we introduce the approach used to implement the CSMFST algorithm in section 3. Numerical applications of our new algorithm are illustrated in section 4. Finally, the main conclusions of this paper are summarized in the last section.

II. PRINCIPLE OF THE YAST ALGORITHM

Let {𝒙(𝑡) 𝑡𝜖ℤ } be a sequence of n-dimensional data vectors. Denoting the time-varying correlation matrix associated to this data vector sequence received until the instant t, 𝑪 𝒙𝒙 (𝑡) , the goal is to track the principal subspace 𝑬 𝒑 (𝑡) of 𝑪 𝒙𝒙 (𝑡) which is spanned by the p eigenvectors associated with the largest eigenvalues of the matrix 𝑪 𝒙𝒙 (𝑡). For each time step t, receiving 𝒙(t) requires the updating of 𝑪 𝒙𝒙 (𝑡). This is usually done recursively according to: 𝑪 𝒙𝒙 (𝑡) = 𝛽 * 𝑪 𝒙𝒙 (t -1) + 𝒙(t) * 𝒙(𝑡) 𝐻 (1

)
where 0 < 𝛽 < 1 is the forgetting factor. 𝛽 is chosen in order to allow the YAST algorithm to track 𝑬 𝒑 (𝑡) even in case we are dealing with nonstationary signals.

The (n x p) orthonormal matrix 𝑾(𝑡) selected by the YAST algorithm spans 𝑬 𝒑 (𝑡) and also maximizes the criterion:

𝜁(𝚷(𝒕)) = 𝑡𝑟𝑎𝑐𝑒( 𝑾(𝑡) 𝐻 𝑪 𝒙𝒙 (𝑡) 𝑾(𝑡)) = 𝑡𝑟𝑎𝑐𝑒( 𝑪 𝒙𝒙 (𝑡)𝚷(𝒕) ) (2)
where 𝚷(𝒕) is the orthogonal projector on 𝑬 𝒑 (𝑡).

𝚷(𝒕)=𝑾(𝑡)𝑾(𝑡) 𝐻 , (3)
Unfortunately, implementing this optimization over all orthonormal matrices is computationally demanding, and does not lead to a simple recursion between 𝑾(𝑡) and 𝑾(𝑡 -1). In order to reduce the complexity of the algorithm, the search of the updated weighting matrix 𝑾(𝑡) is limited in the range space of 𝑾(𝑡 -1) plus one additional search direction given by the additional data vector 𝒙(𝑡)

i.e. in a (p + 1)-dimensional subspace.

Let 𝚷(𝑡) be the orthogonal projector on the augmented subspace and the n x (p + 1) orthonormal matrix 𝐖(𝑡) such that

𝚷(𝑡) = 𝑾(𝑡)𝑾(𝑡) 𝐻

(4) where

𝐖(𝑡) = [𝐖(t -1) 𝐮(t)]

(5) and

𝒆(𝑡) = 𝒙(𝑡) -𝑾(𝑡 -1)𝑾(𝑡 -1) 𝐻 𝒙(𝑡). (6)
𝐮(t) is the unit-norm variant of 𝒆(𝑡).

Then any p-dimensional subspace of span(𝚷(𝑡)) can be written in the form

𝚷(𝒕) = 𝚷(𝑡) -𝒗(𝑡)𝒗(𝑡) 𝐻 (7)
If we denote by 𝒗(𝑡) the unitary vector which belongs to the range space of 𝚷(𝑡) then

𝒗(𝑡) = 𝐖(𝑡)𝝓(𝑡), (8)
where 𝝓(𝑡) is a (p+1)-dimensional unitary vector, Substituting ( 4) and ( 7) into equation ( 1), are led to a new optimization problem with the criterion:

𝜁(𝚷(𝒕)) = 𝑡𝑟𝑎𝑐𝑒(𝑪 𝑦𝑦 (𝑡) -𝝓(𝑡) 𝐻 𝑪 𝑦𝑦 (𝑡)𝝓(𝑡)) (9)
where 𝑪 𝑦𝑦 (𝑡) is the (p + 1) x (p + 1) matrix given by:

𝑪 𝑦𝑦 (𝑡) = 𝐖(𝑡) 𝐻 𝑪 𝒙𝒙 (𝑡) 𝐖(𝑡) (10)
It is well known that 𝜁 is maximized when 𝝓(𝑡) is the minor eigenvector of the matrix 𝑪 𝑦𝑦 (𝑡). At each iteration, the YAST receives a new data vector 𝒙(𝑡), consequently, it updates the previous weighting matrix 𝑾(𝑡 -1). Thus, computing the subspace weighting matrix 𝑾(𝑡), can be decomposed into three steps:

 Compute an orthonormal basis 𝐖(𝑡) of the augmented subspace.



Construct the matrix 𝑪 𝑦𝑦 (𝑡) defined in equation [START_REF] Badeau | Fast and stable YAST algorithm for principal and minor subspace tracking[END_REF].  Find the minor eigenvector 𝛟(𝑡) of 𝑪 𝑦𝑦 (𝑡) and 𝑾(𝑡) of the range space of the projector 𝚷(𝒕)defined in equation [START_REF] Davila | Efficient, high performance, subspace tracking for time-domain data[END_REF].

III. PRINCIPLE OF THE CSMFST ALGORITHM

In this section, we introduce our new subspace tracking algorithm, the CSMFST algorithm. The YAST approach is applied when estimating the signal and noise matrices. In addition, our new subspace tracker attempts to properly optimize the signal-to-noise ratio 𝜌 and generate the optimal p-dimensional subspace spanner 𝑾(𝑡) at each iteration. This ratio is defined as:

𝜌(𝑾(𝑡)) = 𝑡𝑟𝑎𝑐𝑒(𝑾(𝑡) 𝐻 𝑨(𝒕) 𝑾(𝑡) ) 𝑡𝑟𝑎𝑐𝑒(𝑾(𝑡) 𝐻 𝑩(𝒕) 𝑾(𝑡) ) = 𝜌(𝑡) (11)
where A(t) and B(t) are respectively estimated (n x n) covariance matrices directly computed from signal and noise realizations at each iteration.

To estimate 𝑾(𝑡) which generates the p-dimensional optimal subspace, in which 𝜌(𝑡) is maximized [START_REF] Borloz | Subspace SNR Maximization: The Constrained Stochastic Matched Filter[END_REF], we introduce the (n x n) following matrix: 𝑨(𝑡) -𝜌(𝑡)𝑩(𝑡) [START_REF] Borloz | Subspace SNR Maximization: The Constrained Stochastic Matched Filter[END_REF] Therefore, the main goal of the CSMFST algorithm is to estimate the p-dimensional subspace associated with the matrix defined in (1), however, we note that in this case, the signal-to-noise ratio 𝝆(𝑡) is unknown and must be computed at each iteration. Besides, receiving new signal and noise realizations at each time step, requires the reestimating of the correlation matrices A(t) and B(t) in addition to the weighting matrix 𝑾(𝑡). In order to do this, we implemented an algorithm that computes 𝑾(𝑡) related to the matrix presented in [START_REF] Arjomandi-Lari | Generalized YAST algorithm for signal subspace tracking[END_REF], it's named optSNR [START_REF] Borloz | Subspace SNR Maximization: The Constrained Stochastic Matched Filter[END_REF][START_REF] Cavassilas | A matched filter extension. Application in short signals detection degraded by noise[END_REF][START_REF] Borloz | Estimation, détection, classification par maximisation du rapport signal-à-bruit : le filtre adapté stochastique sous contrainte[END_REF].

IV. IMPLEMENTATION OF THE CSMFST ALGORITHM

In this part, we introduce the CSMF subspace tracking algorithm that permit a fast convergence of 𝑬 𝒑 (𝑡) to the optimal subspace 𝑬 𝒑 * (𝑡).

This algorithm uses the same technique as the YAST algorithm to estimate the signal and noise correlation matrices.

Below, an implementation of the CSMFST algorithm is presented. It can be decomposed into three steps:

A. Computation of 𝑾(𝑡)

By applying the same principle to our optimization problem, we compute the (n x p) weighting matrix denoted by 𝑾(𝑡), we introduce signal and noise data samples both received at a step time t then we compute two different (n x p) matrices which characterizes respectively the complement of the orthogonal projection of the signal data vector denoted by 𝒙 𝑨 (t) then the noise data vector denoted 𝒙 𝑩 (t) onto the subspace spanned by 𝑾(𝑡 -1) 𝒆𝒆 𝑨 (t)= 𝒙 𝑨 -𝑾(𝑡 -1)𝑾(𝑡 -1) 𝐻 𝒙 𝑨 (t) (13) and 𝒆𝒆 𝑩 (t) = 𝒙 𝑩 -𝑾(𝑡 -1)𝑾(𝑡 -1) 𝐻 𝒙 𝑩 (t)

Considering the signal error 𝒆𝒆 𝑨 and the noise error 𝒆𝒆 𝑩 , we obtain a (n x n) matrix 𝑴(𝑡) defined as 𝑴(𝑡) = 𝒆𝒆 𝑨 (𝑡)𝒆𝒆 𝑨 (𝑡) 𝐻 -𝜌(𝑡 -1) * 𝒆𝒆 𝑩 (𝑡)𝒆𝒆 𝑩 (𝑡) 𝐻 (15)

In the next step, the augmented weighting matrix 𝑾(t) is obtained by adding the new search direction which refers to the major eigenvector 𝒖(𝑡) of 𝑴(𝑡) and 𝑾(t) is calculated from (5).

B. Computation of 𝑪 𝒚𝒚 (𝑡)

We denote by 𝚷(t) the orthogonal projector on the augmented subspace 𝑾(t) as seen in ( 4).

Any p-dimensional subspace of span((𝚷(t)) can be represented by an orthogonal projector 𝚷(t) defined in [START_REF] Davila | Efficient, high performance, subspace tracking for time-domain data[END_REF]. In order to compute the (n x p) weighting matrix 𝑾(𝑡), we select the major p-dimensional subspace of the orthogonal projector Π(t).

The signal-to-noise ratio 𝜌(𝑡) can be computed using equation [START_REF] Arjomandi-Lari | Generalized YAST algorithm for signal subspace tracking[END_REF].

V. SOME NUMERICAL APPLICATIONS

In all the simulations we choose 𝑛 = 8 and 𝑝 = 3.

The projection error power is the distance between the subspaces spanned by 𝑊 1 and 𝑊 2 [START_REF] Golub | Matrix Computations[END_REF] defined by

𝑑(𝑊 1 , 𝑊 2 ) = ‖𝑊 1 𝑊 1 𝐻 -𝑊 2 𝑊 2 𝐻 ‖ 𝐹

A. Performance of the CSMFST algorithm in stationary case

In this part, at each t, we compare the subspace returned by the CSMFST algorithm in a stationary case 𝑬 𝒑 (𝑡) with the one returned by the OptSNR algorithm 𝑬 𝒑 * (𝑡) then with the empirical subspace  𝒑 which is directly computed from all signal and noise samples. Since we are in a stationary case, the forgetting factor 𝛽 is chosen equal to 1.

In this first simulation all the eigenvectors of the signal covariance matrix 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) are considered to be fixed and three of the corresponding eigenvalues 𝜆 𝑖 are chosen larger than 1, however we consider a white noise obtained by fixing a scalar noise matrix 𝑪 𝒙 𝑩 𝒙 𝑩 (𝑡) with all eigenvalues 𝜇 𝑖 = 1.

Respectively, Fig. 1a and b represent the projection error powers i.e. the distance between 𝑬 𝒑 (𝑡) and 𝑬 𝒑 * (𝑡) and between 𝑬 𝒑 (𝑡) and

 𝒑 .
It shows that the CSMFST algorithm presents a good rate of convergence as the resulting subspace returned by the algorithm 𝑬 𝒑 (𝑡) is close to 𝑬 𝒑 * (𝑡) and also  𝒑 .

Fig. 1c shows the difference between theoretical signal-to-noise ratio and the one computed by the CSMFST algorithm at each step, we can notice that they are quite similar Since we are in a stationary case with a white noise, we can also compare the result obtained by the YAST on the signal samples and the CSMFST algorithm.

B. Performance of the CSMFST algorithm in nonstationary case

In this second part, we test the algorithm in a nonstationary environment with a forgetting factor 𝛽 equal to 0,999.

In this third scenario we keep the same eigenvectors of 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) and the same white noise corresponding to the scalar matrix 𝑪 𝒙 𝑩 𝒙 𝑩 (𝑡) than the first scenario. However we create four different stationary zones in which we change the principal subspace by switching two eigenvectors of 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡). In each zone we switch an eigenvector of the principal subspace with another eigenvector chosen outside the principal subspace.

The signal-to-noise-ratio is constant. Fig. 4a represents the distance between 𝑬 𝒑 (𝑡) and 𝑬 𝒑 * (𝑡); we can notice that the CSMFST algorithm presents a good convergence since the resulting subspace is quite near to 𝑬 𝒑 * (𝑡) (we have exactly the same result when we compare it to empirical subspace). The same figure shows that we have some peaks obtained exactly at the beginning of each zone due to the subspace change.

In Fig. 4b we show the difference between theoretical and estimated signal-to-noise-ratio.

The Fourth scenario is the following: we create four zones in which we change the principal subspace by modifying the eigenvalues and the eigenvectors associated of 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) in each zone.

The signal-to-noise-ratio varies from a zone to another. The result can be seen on Fig. 5, we can observe that the SNR 𝜌(𝑡) is well estimated with a fast convergence rate. The two last tests show that the CSMFST algorithm presents a good rate of convergence even in a nonstationary environment.

VI. CONCLUSION

In this paper, we present a new fast tracking algorithm which estimates at each step the p-dimensional subspace where the signalto-noise ratio is maximal.

The main advantage of the CSMF algorithm is that it can be used not necessary in white noise environments while all the SP-type algorithms consider white noise. The CSMFST algorithm estimates the covariance matrices as well as the optimal subspace at each time step using the signal and noise realizations and presents a good rate of convergence in stationary and nonstationary environments. In a non-white noise environment, it can be compared only with the optSNR algorithm which needs heavy calculations. In a white noise environment, it can be compared with classical SP-algorithms such as YAST.

Fig. 2

 2 Fig.1cshows the difference between theoretical signal-to-noise ratio and the one computed by the CSMFST algorithm at each step, we can notice that they are quite similar Since we are in a stationary case with a white noise, we can also compare the result obtained by the YAST on the signal samples and the CSMFST algorithm. Fig. 2 presents the projection error power between the two algorithms. In a second scenario, the eigenvectors 𝒖 𝑖 of 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) and 𝑪 𝒙 𝑩 𝒙 𝑩 (𝑡) are the same. The corresponding eigenvalues are chosen as following : 𝜆 𝑖 = [20,18,16,1,1,1,1,1] for the signal and 𝜇 𝑖 = [2,3,4,1,2,3,4,5] for the noise. It can be shown that the subspace returned by the algorithm 𝐸 𝑝 (𝑡) = 𝑠𝑝𝑎𝑛(𝑢 1 , 𝑢 2 , 𝑢 4 ) with 𝜌 = 6.5, even though a fast and wrong analysis would say 𝐸 𝑝 (𝑡) = 𝑠𝑝𝑎𝑛(𝑢 1 , 𝑢 2 , 𝑢 3 ) where 𝜌 = 6. Convincing results are given in fig. 3.

Fig. 1 .

 1 Fig. 1. Performance of the CSMFST algorithm in the stationary case in term of projection error power: the distance between 𝑬 𝒑 (𝑡) and 𝑬 𝒑 * (𝑡) (a) and between 𝑬 𝒑 (𝑡) and  𝒑 (b) and SNR(c).

Fig. 2 .

 2 Fig. 2. Comparison in stationary case between the YAST and the CSMFST in presence of white noise.

Fig. 3 .

 3 Fig. 3. Performance of the CSMFST algorithm in the stationary case.

Fig. 4 .

 4 Fig. 4. Performance of the CSMFST algorithm in the nonstationary case in term of projection error (a) and SNR (b).

Fig. 5 .

 5 Fig. 5. Performance of the CSMFST algorithm in the nonstationary case in term of projection error (a) and SNR (b).

  𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) and 𝑪 𝒙 𝑩 𝒙 𝑩 (𝑡) are respectively the signal and noise covariance matrices which must be updated by the CSMFST algorithm at each time step t as: 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) = 𝜷 𝑪 𝒙 𝑨 𝒙 𝑨 (t -1) + 𝒙 𝑨 (t) 𝒙 𝑨 (t) 𝐻 (18) 𝑪 𝒙 𝑩 𝒙 𝑩 (𝑡) = 𝜷 𝑪 𝒙 𝑩 𝒙 𝑩 (t -1) + 𝒙 𝑩 (t) 𝒙 𝑩(t)

	The unitary vector 𝒗(𝑡) verifies (8).
	𝐂 𝐲𝐲 (t) is such that
	𝐂 𝐲𝐲 (t) = 𝑾(t) 𝐻 𝑪 𝒙𝒙 (𝑡)𝑾(𝑡)	(16)
	where
	𝑪 𝒙𝒙 (𝑡) = 𝑪 𝒙 𝑨 𝒙 𝑨 (𝑡) -𝜌(𝑡 -1) 𝑪 𝒙 𝑩 𝒙 𝑩 (𝑡)	(17)

𝐻 (19)

C. Update of 𝑾(𝑡) and 𝜌(𝑡)


Yoann Poher
email: yoann.poher@imbe.fr

Philippe Ponel

Frédéric Médail

Valérie Andrieu-Ponel

Frédéric Guiter

Long-term dynamics of a remarkable Corsican wetland (Bagliettu peatland, 206 m a.s.l.)

RESULTS

->Zone-1: from 240 to 200 cm (ca. before 3700 cal. BP) Scarcity of insect and seed remains in the assemblages may be due to a rapid deposition of silty gyttja sediments in a side-arm of the Ascu river.

->Zone-2: from 200 to 180 cm (ca. 3700 to 3650 cal. BP) Rheophilous/ripicolous and limnophilous/paludicolous taxa present in the same assemblages indicate the proximity of a stream but also a lentic environment. Saproxylophagous (lobed-leaved tree dependent) and open-land insects suggest that an open, marshy environment was established in an disconnected side-arm close to the Ascu river, with scattered trees in the riverine vegetation.

->Zone-3: from 180 to 80 cm (ca. 3650 to 2900 cal. BP) Rheophilous taxa and Trichoptera testify to the proximity of the river and possibly ood events. From ca. 3300 cal. BP, their decline suggests that the active Ascu river was embanked. Seeds of Alnus, Rubus, Vitis and saproxylic insects attest to the rapid colonisation of the Ascu margins by the riparian forest.

->Zone-4: from 75 to 0 cm (ca. 1700 cal. BP to the present) Signi cant amounts of open-land, coprophilous insects and herbaceous seeds indicate that pastoralism was intensi ed in the Ascu valley. The expansion of Erica terminalis recorded by seeds since ca. 800 cal. years could be a consequence of the vegetation opening. This event marks an unprecedented change in the Bagliettu peatland history.

This study allows us to propose a scenario for the palaeoenvironmental history of a unique wetland of Corsica: -a side-arm of the Ascu hydrographic system probably evolved from an abandoned side-arm to marshes and swampy riparian forest while the active river was embanked -human activities induced a return to herbaceous wet habitats Sediments from Bagliettu o er a rare occasion to:

1). reconstruct the dynamics of a unique low-elevation peatland during the last 3700 cal. BP, 2). caracterise past anthropic activities and their impacts on the ecosystems.

  Insect fossils represent a valuable tool to reconstruct past hydrological changes and impacts of human activities in fragile ecosystems such as Bagliettu. of the Ascu valley, Bagliettu (206 m a.s.l.) and Valdu (240 m a.s.l.) peatlands present remarkable habitats of patrimonial interest for Corsica: vegetation and insects from rivulets which comprise rare and endemic species are concentrated in a small area (35 ha).


Nabil Amirach
email: amirach@univ-tln.fr

Bernard Xerri

Bruno Borloz

Claude Jauffret

A New Approach for Event Detection and Feature Extraction for NILM

Keywords: Non-intrusive load monitoring (NILM), event detection, feature extraction, estimation. I

The method proposed in this paper aims both to detect the events occurred on the overall current measurement and to estimate the features of the electric load underlying these events. This method is within the scope of the non-intrusive approach consisting in extracting information from the electric consumption only measured at the electrical switchboard.

INTRODUCTION

Since last years, in order to save energy, the need to monitor the power consumption of residential and industrial processes has became an important problem.

The purpose of the smart grid is to have an accurate and reliable knowledge of a given electrical network in order to optimize the electricity consumption through the analysis of the overall current. A main issue is to decompose the total electrical consumption of a given grid in order to provide a detailed analysis of electrical consumption of each device present in the electrical network.

One can find two types of approaches: the intrusive ones which use one sensor for each appliance, and the non-intrusive load monitoring (NILM) one which aims at estimating the load consumption from a unique overall current and voltage measurement. This last problem is wildly addressed in the literature since the last 30 years [START_REF] Hart | Residential energy monitoring and computerized surveillance via utility power flows[END_REF][START_REF] Hart | Nonintrusive Appliance Load Monitoring[END_REF][START_REF] Norford | Nonintrusive Electrical Load Monitoring[END_REF][START_REF] Leeb | Transient Event Detection in Spectral Envelope Estimates for Noninstrusive Load Monitoring[END_REF][START_REF] Cole | Algorithm for non-intrusive identification of residential appliances[END_REF][START_REF] Laughman | Power Signature Analysis[END_REF][START_REF] Leeb | Transient event detection in spectral envelop estimates for nonintrusive load monitoring[END_REF][START_REF] Guedri | Time-frequency characterisation for electric load monitoring[END_REF][START_REF] Guedri | Caractérisation aveugle de la courbe de charge électrique: Détection, classification et estimation des usages dans les secteursz résidentiel et tertiaire[END_REF][START_REF] Hala | Méthode d'identification et de classification de la consommation d'énergie par usages en vue de l'intégration dans un compteur d'énergie électrique[END_REF].

NILM was first introduced by G.W. Hart [START_REF] Hart | Residential energy monitoring and computerized surveillance via utility power flows[END_REF] [START_REF] Hart | Nonintrusive Appliance Load Monitoring[END_REF] where only the active power was considered. An appliance turning on or off is detected when the difference in the power consumption exceeds a predefined threshold. Then Hart proposed an improvement by using reactive power and harmonics in order to classify two loads with a same active power consumption.

Other solutions use the current signal instead of active and reactive power for event detection and feature extraction. In [START_REF] Hala | Méthode d'identification et de classification de la consommation d'énergie par usages en vue de l'intégration dans un compteur d'énergie électrique[END_REF], the matrix pencil method estimates the parameters of a current modeled as a sum of exponentially damped sinusoids characterizing the loads.

Most approaches proposed in the literature are divided into four stages. Foremost the event detection and the feature extraction stage are processed. Then a classification stage uses the information provided by the preceding modules, in order to associate each event with the appliance which caused it. Finally, a last stage estimates the power consumption of each appliance.

In this study, we focus on the event detection and the feature extraction stages and the paper is organized as follows:

• in section II, we propose a new model of current matched to accurate transition detection,

• in section III, we develop a parameters estimation method for our new model,

• in section IV, we present some application examples.

II. PROBLEM STATEMENT

The only available information is the overall current and voltage measurement (respectively ݔ ሺݐሻ and ݕ ሺݐሻ). One has chosen to modelize the voltage in steady state by:

ݕሺݐሻ = ܣ ݂ߨ2ሺݏܿ ݐ + ߮ ሻ ே ୀଵ
where ܰ is the number of frequencies ݂ with 0 ≤ ݂ < The events occur on the current signal; we will try to detect and extract the features from ݔ ሺݐሻ and ݕ ሺݐሻ. Considering Kirchhoff's law, the overall current is composed by the sum of the currents consumed by each system in use which contains information about frequencies. The model of the current consumed by a linear load in steady state is similar to [START_REF] Hart | Residential energy monitoring and computerized surveillance via utility power flows[END_REF].

Unlike to the matrix pencil method which estimates the frequencies ݂ and their number ܰ [START_REF] Hua | Matrix Pencil Method for Estimating Parameters of Exponentially Damped/Undamped Sinusoids in Noise[END_REF] [START_REF] Sarkar | Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials[END_REF], they are here supposed to be known.

In previous studies, the current was modelized by a sum of damped sinusoids. However, the current measured on a grid is made of a sequence of events (transitions) due to the switching on and off of devices. Figure 1 shows the different scenarios that may be encountered during processing (different cases of observing window):

1) This case represents the current on initial steady state, before that another load turns on.

2) Here, there is a "positive" transition in the observation window. Before the transition is the current consumed

by the already running loads, and after, there is the sum of the previous current and the current consumed by the load which has switched on.

3) In this and the fifth cases, there is no transition. We observe the current consumed by the overall system in steady state.

4) Now we observe a "negative" transition. A load is turned off.

If the model previously defined is used for the cases 2) and 4), it will estimate the parameters of an "average" signal of the signals before and after the transition. The model above is not matched to these scenarios: it must take into account the existence of a transition in the observation window.

Figure 2 shows the case of an observation window of duration ܶcontaining a transition at time ܶ ଵ (e.g. case 2)). ܶ ଶ is the duration of the signal added by the load which turned on. This transition is in fact related to a system that turns on or turns off. From this window, we must detect the transition instant and characterize the load which is the source of this transition. When a transition occurs, the model (eq (1)) must be rewritten as two sums of cosine: it is simplified as a single sum of cosine when there is no transition. The first sum represents the overall system before the transition, and the second one the system which was turned on.

In the observing window, the signal is modelized as: Comparing to Figure 2, we can see that the model takes into account all possible scenarios. It corresponds to the cases of an observing window with both a transition ሺ0 < ܶ ଶ < ܶሻ and without a transition (ܶ ଶ = 0 or ܶ ଶ = ܶ).

ݔሺݐሻ = ܣ ଵ, ݂ߨ2൫ݏܿ ݐ + ߮ ଵ, ൯ ே ୀଵ . ⨅ ் ሺݐሻ + ܣ ଶ, ݂ߨ2൫ݏܿ ݐ + ߮ ଶ, ൯ ே ୀଵ . ⨅ ் మ ሺݐ െ ܶ ଵ ሻ
The theoretical development above is expressed in the continuous time domain. However we work with discrete signals. So let us transpose these expressions in the discrete time domain.

We define ܰ the number of samples in the observing window, ܰ ଵ the sample number where the transition begins and ܰ ଶ = ܰ െ ܰ ଵ + 1 the number of samples of the added signal.

The duration of the observing window and others are defined by (with ݐ∆ the sampling period):

ܶ = ሺܰ െ 1ሻ. ݐ∆ , ܶ ଵ = ሺܰ ଵ െ 1ሻ. ,ݐ∆ ܶ ଶ = ሺܰ ଶ െ 1ሻ. ݐ∆
The unknown parameters are A ଵ,୫ , φ ଵ,୫ , A ଶ,୫ , φ ଶ,୫ and N ଶ .

III. EVENT DETECTION AND FEATURE EXTRACTION

Let us develop a method to estimate these parameters. And characterize the load in the frequency domain. Using the discrete Fourier transform, the model is more towable to estimate ܰ ଶ . Thus, one obtains the given frequency model:

ܺሺ݂ሻ = ቊ ܣ ଵ, 2 ݁ ఝ భ, . ݊݅ݏሺߨሺ݂ െ ݂ ሻܶሻ ݊݅ݏሺߨሺ݂ െ ݂ ሻ∆ݐ ሻ . ݁ ିగሺି ሻ் ே ୀଵ + ܣ ଵ, 2 ݁ ିఝ భ, . ݊݅ݏሺߨሺ݂ + ݂ ሻܶሻ ݊݅ݏሺߨሺ݂ + ݂ ሻ∆ݐ ሻ ݁ ିగሺା ሻ் + ܣ ଶ, 2 ݁ ఝ మ, . ݊݅ݏሺߨሺ݂ െ ݂ ሻܶ ଶ ሻ ݊݅ݏሺߨሺ݂ െ ݂ ሻ∆ݐ ሻ . ݁ ିଶగሺି ሻቀ ் మ ଶ ା் భ ቁ + ܣ ଶ, 2 ݁ ିఝ మ, . ݊݅ݏሺߨሺ݂ + ݂ ሻܶ ଶ ሻ ݊݅ݏሺߨሺ݂ + ݂ ሻ∆ݐ ሻ . ݁ ିଶగሺା ሻቀ ் మ ଶ ା் భ ቁ ቋ
During the steady state preceding the transition, one can obtain an estimation of the vectors A ଵ and φ ଵ (using Least Squares method). That's why these vectors will be considered to be known. And we know that A ଶ = 0 and φ ଶ = 0. Knowing A ଵ and φ ෝ ଵ , the current measured before the transition can be removed by calculating the error between the measured signal in the observing window, and the signal estimate constructed with A ଵ and φ ෝ ଵ . The residue obtained, denoted X ෩ ଶ ሺfሻ is in fact the current consumed by the system which turns on. Now, using X ෩ ଶ ሺfሻ, we can characterize this system by estimating the parameters of its model denoted X ଶ ሺfሻ. Therefore, using

࣠ሺN ଶ , fሻ = sinሺN ଶ . πf∆tሻ sinሺπf∆t ሻ . e ି୨ሺଶିଵି మ ሻ∆୲ (2)
one has

X ଶ ሺfሻ = A ଶ,୫ 2 e ୨ మ,ౣ . ࣠ሺN ଶ , f െ f ୫ ሻ ୫ୀଵ + A ଶ,୫ 2 e ି୨ మ,ౣ . ࣠ሺN ଶ , f + f ୫ ሻ
One estimates N ଶ , A ଶ and φ ଶ from X ଶ ሺfሻ using a least squares (LS) criterion:

CሺN ଶ , A ଶ , φ ଶ ሻ = หܺ ෨ ଶ ሺ݂ ሻ െ ܺ ଶ ሺ݂ ሻห ଶ ୩ୀଵ where f ୩ = ሺk െ 1ሻ∆f, ∆f = ౩ = ଵ ே.∆௧ .
The estimate in the least squares sense is the vector ൫ܰ ଶ , ܣ መ ଶ , ߮ ො ଶ ൯ that minimizes ܥሺܰ ଶ , ܣ ଶ , ߮ ଶ ሻ. We note that:

ܺ ଶ ሺ݂ ሻ = 1 2 . ۏ ێ ێ ێ ێ ۍ ࣠ሺܰ ଶ , ݂ െ ݂ ଵ ሻ ࣠ሺܰ ଶ , ݂ + ݂ ଵ ሻ ⋮ ࣠ ቀܰ ଶ , ݂ െ ݂ ே ቁ ࣠ ቀܰ ଶ , ݂ + ݂ ே ቁ ے ۑ ۑ ۑ ۑ ې ் . ۏ ێ ێ ێ ێ ێ ۍ ܣ ଶ,ଵ ݁ ఝ మ,భ ܣ ଶ,ଵ ݁ ିఝ మ,భ ⋮ ܣ ଶ,ே ݁ ఝ మ,ಿ ܣ ଶ,ே ݁ ିఝ మ,ಿ ے ۑ ۑ ۑ ۑ ۑ ې = ܤ ሺܰ ଶ ሻ. ߠ
The minimization of CሺN ଶ , A ଶ , φ ଶ ሻ is performed using the two following stages.

Stage 1: set N ଶ , and compute the vector θ ሺN ଶ ሻ that minimizes the following criterion

ܥ ே మ ሺߠሻ = หܺ ෨ ଶ ሺ݂ ሻ െ ܤ ሺܰ ଶ ሻ. ߠห ଶ ே

ୀଵ

This minimization is performed by:

ߠ ሺܰ ଶ ሻ = ൭ ܤ ு ሺܰ ଶ ሻܤ ሺܰ ଶ ሻ ൱ ିଵ ܤ ு ሺܰ ଶ ሻܺ ෨ ଶ ሺ݂ ሻ
Note: Here . ு represents the conjugate transpose of a vector or a matrix.

Stage 2: search ܰ ଶ which minimizes the following criterion

ܥሺܰ ଶ ሻ = หܺ ෨ ଶ ሺ݂ ሻ െ ܤ ሺܰ ଶ ሻ. ߠ ሺܰ ଶ ሻห ଶ ே ୀଵ
Then, we select the estimate θ corresponding to this value of N ଶ . Finally, calculating the modulus and argument of θ provides the estimated amplitudes and phases (A ଶ and φ ෝ ଶ ).

The estimation is also performed on, the amplitudes and phases of the voltage in order to calculate the admittance (or impedance) associated with each frequency. The voltage is assumed without transition. But, it may have some variations when an event occurs on the current. However, we do not have to estimate the transition instant. This allows to enhance the information vector. Unlike to most of the methods found in the literature, it does not only work on the fundamental frequency. At each transition, the features extracted are:

• the instant and duration of the transitions • the frequencies present in the current consumed by loads that turn on,

• the admittances associated with each of these frequencies.

These features are then processed by the classification algorithm where is compared the information of each transition. If two transitions have the same frequencies, and if, for each frequency, the modulus of the admittance is the same and the phases are shifted by 180°, then these transitions correspond to a load which has been turned on and off.

IV. RESULTS

In this section, one presents an application on simulated signals and another one on real signals.

A voltage ݕ is created containing following harmonics J = ሼ1, 3, 5ሽ, so:

yሺt ୩ ሻ = U ଵ,୨ cos൫2πf ୨ t ୩ + ψ ଵ,୨ ൯ ୨∈ and f = ሾ50Hz, 150Hz, 250Hzሿ ܷ ଵ = ሾ50, 1, 0.5ሿ, ψ ଵ = ሾ90, 60, 45ሿ
Then three currents are simulated and noise free, each representing the current consumed by a load. These currents compose the overall current :ݔ

xሺt ୩ ሻ = x ଵ ሺt ୩ ሻ + x ଶ ሺt ୬ ሻ + x ଷ ሺt ୫ ሻ
where: Four transitions underlied by two loads are simulated (consuming x ଶ and x ଷ ). The first one turns on at 1.018s and turns off at 3.1605s. The second one turns on at 2.006s and turns off at 4.506s. Each simulated admittance modulus and phase are calculated as follow:

x ୧ ሺt ୩ ሻ = I ୧,୨ cos൫2πf ୨ t ୩ + φ ୧,
• the modulus is the ratio of ܫ and ܷ

• the argument is the difference between ߮ and ߮ ୧ and are given here:

M ଶ = ሾ0.84, 3, 2ሿ, P ଶ = ሾ0, െ24, െ135ሿ M ଷ = ሾ0.24, 1, 1ሿ, P ଷ = ሾെ60, െ60, 45ሿ

Figure 3 show the result of the detection algorithm. It perfectly detects these transitions. From table I which contain numerical results, we note that the first and third transitions have the same amplitudes. In addition estimates of the third transition phases are shifted by 180° from the first transition. It is the same for the second and fourth transitions. The extracted information allows us to say that the first and third transitions are caused by the same load (turning on and off). And the two other are caused by a second load.

Table II contains the result using our algorithm on real measurements (realized at the University of Toulon). These are the voltage and the current measured across an appliance given as resistor.

The extracted information corroborates the results obtained on simulated signals. Indeed we can see that the modulus is the same for both the first and second transition. And the phases are shifted by 180°. So these transitions are due to the same load switching on and off. From table II, the phase difference between the current and the voltage is almost equal to 0° for all frequencies, except for 150Hz. The admittance modulus of this load is approximately equal to 0.24 except for 150Hz. Then, it appears that the detected load is a resistor with impedance about 4Ω and with an unexpected behaviour around 150Hz perhaps due to a non-linearity.

V. CONCLUSION

The method proposed in this paper allows to detect each transition and extract its date and duration, and then to estimate the frequencies parameters (amplitudes and phases). Moreover, other information is the number of frequencies (often harmonics) which is different depending on the appliance. Thus the estimated features can be used to improve the classification of the appliances detected in the network.

Studies have been made on measurements disturbed with additive random noise. Or when there exists frequencies which are not taken into account. And it appears that the results provided by our algorithm are still good.

  phase of the ݉ݐℎ frequency component are ܣ ≥ 0 and ߮ ∈ ሾ0,2ߨሻ. Let's define the vectors ݂ = ቂ݂ ଵ ⋯ ݂ ே ቃ ் , ܣ = ቂܣ ଵ ⋯ ܣ ே ቃ ் and ߮ = ቂ߮ ଵ ⋯ ߮ ே ቃ ் , where ሾ. ሿ ் denotes the transposition operator.

  with ܣ = ቂܣ ,ଵ ⋯ ܣ ,ே ቃ ் and ߮ = ቂ߮ ,ଵ ⋯ ߮ ,ே ቃ ் : ݅ = 1 means before the transition, and ݅ = 2 represent the load which causes the transition. ⨅ ் మ ሺݐ െ ܶ ଵ ሻ denotes the rectangular window function of length ܶ ଶ starting in ܶ ଵ (ܶ = ܶ ଵ + ܶ ଶ ).

Figure 1 .Figure 2 .

 12 Figure 1. Example of voltage and current (cases 1) to 5))

  = 2kHz with ܦ ୧ ሺ1ሻ < ݇ < ܦ ୧ ሺ2ሻ D ଵ = ሾ1; 11000ሿ, D ଶ = ሾ2036; 6321ሿ, D ଷ = ሾ4012; 9012ሿ I ଵ = ሾ30, 2, 0.25ሿ, φ ଵ = ሾ30, 0, 90ሿ I ଶ = ሾ42, 3, 1ሿ, φ ଶ = ሾ90, 36, െ90ሿ I ଷ = ሾ12, 1, 0.5ሿ, φ ଷ = ሾ30, 0, 90ሿ

Figure 3 .

 3 Figure 3. Current signal and transition instant detected

TABLE I

 I

	.	FEATURES EXTRACTION RESULT
	Harmonic frequencies (Hz)		Estimated transition instant: Estimated admittances	ૢ
	Modulus	0.84	0.24	0.84	0.24
	50
	Phase (°)	0	-60	-180	120
	Modulus	3	1	3	1
	150
	Phase (°)	-24	-60	156	120
	Modulus	2	1	2	1
	250
	Phase (°)	-135	45	45	-135
	TABLE II.	RESULTS ON REAL SIGNALS
	Harmonic frequencies (Hz)		Estimated transition instant: ૡ ૡ Estimated admittances
	Modulus		0.246	0.239
	50
	Phase (°)		-0.253	179.752
	Modulus		0.382	0.386
	150
	Phase (°)		47.9	-132.821
	Modulus		0.249	0.224
	350

Phase (°)

ACKNOWLEDGMENT

The authors would like to thank the Provence Alpes Côte d'Azur (PACA) region and the company Qualisteo for their financial support.


Bruno Borloz
email: borloz@univ-tln.fr

Bernard Xerri
email: xerri@univ-tln.fr

SUBSPACE SIGNAL-TO-NOISE RATIO MAXIMIZATION: THE CONSTRAINED STOCHASTIC MATCHED FILTER

This paper deals with signal-to-noise ratio (SNR) maximization in a subspace. This criterion can be applied to estimation or detection of a stochastic signals embedded in additive noise. Both signal and noise are considered to be realizations of random functions whose statistics are assumed known up to second-order. The proposed method leads to a filter we call "constrained stochastic matched filter" (CSMF), optimal in the sense that it maximizes the SNR in a subspace whose dimension is chosen a priori. It is an extension of the stochastic matched filter (SMF), itself an extension of the matched filter.

INTRODUCTION

The study presented in this paper is concerned with detection or estimation of stochastic signals embedded in noise. The available information is the covariance matrices of signal and noise.

When key quantities like the probability density function of the signals studied are not available, it is legitimate to try to use methods based on energetic criterion. Hence the output SNR maximization, i.e. after a filtering operation, seems a natural way to attack problems of detection or estimation. It requires the knowledge of second-order statistics of both signal and noise.

In this scope, several approaches can be considered, two main of which are the matched filter [START_REF] Van | Trees, Detection, Estimation and Modulation Theory[END_REF][2] [START_REF] Kailath | Detection of Stochastic Processes[END_REF] and derived approaches [START_REF] Verdu | Signal Selection for Robust Matched Filtering[END_REF][5] [START_REF] Proakis | Digital Communications[END_REF][7] [START_REF] Cavassilas | Extension de la notion de filtre adapté. Contribution à la détection de signaux courts en présence de termes perturbateurs[END_REF] and classification techniques as, for example, the Fisher linear discriminant analysis applied to the particular case of a two-classes problem [START_REF] Duda | Stork Pattern Classification[END_REF] [START_REF] Fukunaga | introuduction to Statistical Pattern Recognition[END_REF]. The method proposed is derived from the first approach in the sense that it shares the same philosophical framework: a SNR maximization.

For example, it is well known that the linear filter that has maximum output SNR at some instant of time when input is a deterministic signal embedded in an additive corrupting random noise is the matched filter [START_REF] Kailath | Detection of Stochastic Processes[END_REF]. When noise is Gaussian, the output of this filter provides a sufficient statistic for any Likelihood Ratio Test for detection of presence of the input signal. A complete specification of signal and second-order statistics of noise is necessary to derive the corresponding matched filter.

Derived methods have been proposed to face this problem when the signal to detect is no longer perfectly known, due to channel nonlinearities, timing jitter, nonstationarities, modeling uncertainties, etc. For example, the minimax robust matched-filter [START_REF] Verdu | Signal Selection for Robust Matched Filtering[END_REF][5] optimizes the worst-case performance over an uncertainty class. The problem of detecting a transmitted signal when one of M known signals is transmitted (e.g. in digital communications [START_REF] Proakis | Digital Communications[END_REF] [START_REF] Eldar | Orthogonal matched filter detection[END_REF]) leads, when the additive noise is white and gaussian, to an optimal detector consisting in a bank of matched filters followed by a detector which chooses as the detected signal the one of which the output is maximal.

The stochastic matched filter (SMF) was introduced to take into consideration the problem of detecting a stochastic signal whose second-order statistics are known [START_REF] Cavassilas | Extension de la notion de filtre adapté. Contribution à la détection de signaux courts en présence de termes perturbateurs[END_REF]. It extends the matched filter approach because it gives an optimal filter in the sense that it makes maximum the output SNR expressed in the form of Rayleigh's quotient. This problem is a generalized eigenvalues one with suitable matrices, say the covariance matrices of signal and noise. Filtering consists of a projection onto an optimal subspace with direction of the eigenvector associated to the largest eigenvalue, which indeed can be interpreted as the output SNR.

A natural extension is to take into account more eigenvectors deduced from this method: this could be called Extended SMF (ESMF). Actually, it has been shown [START_REF] Cavassilas | Extension de la notion de filtre adapté. Contribution à la détection de signaux courts en présence de termes perturbateurs[END_REF] that other eigenvectors (associated to the largest eigenvalues) can statistically contribute to improve detection.

The method proposed in this paper is a natural extension of the SMF method, naturally inferred from the remark above concerning a projection onto a subspace of dimension two or higher; it aim is to make maximum the output SNR in an aptly chosen subspace. This is why we call this optimal filter 'Constrained Stochastic Matched Filter' (CSMF). It takes place in the class of the subspace projection methods whose interest is increasing [START_REF] Lindgren | Subspace selection techniques for classification problems Thesis[END_REF][12] [START_REF] Brandts | A comparison of subspace methods for Sylvester equations[END_REF].

NOTATIONS AND CLASSICAL METHODS

Throughout this paper each signal is discrete and then represented by a vector s of E N . Moreover, we will suppose that s is zero mean.

Notations and basic developments

A N -dimensional random vector s can be expressed as follows:

s = N i=1 α i v i = Vα
where the α i are random variables, the

{v i } linearly inde- pendent unit vectors, V = [v 1 ...v N ] and α = [α 1 ...α N ] .
As we are interested in calculating the power of signals in a p-dimension subspace E p , quantity that must not depend on the basis used to described this subspace, we can without loss of generality assume that the {v i } are orthonormal.

Covariance matrix A = E ss and the power P s , after a projection onto the subspace spanned by the vectors

V p = [v 1 , v 2 , ...v p ]
are related by:

P s = tr V p AV p = p i=1 v i Av i .
(

Similar calculations can be performed for the noise vector n with covariance matrix B.

The matched filter

If s is a known deterministic signal, the objective is to find a linear filter (a vector) h so as to maximize the output SNR defined by: s h

2 h Bh = h ss h h Bh .
It is well established that the optimal filter is a correlation filter called the "matched filter", h = B -1 s and that the output SNR is s B -1 s.

The stochastic matched filter

This approach was developed [START_REF] Cavassilas | Extension de la notion de filtre adapté. Contribution à la détection de signaux courts en présence de termes perturbateurs[END_REF] to take into consideration the case where s is a stochastic signal uncorrelated with noise in lieu of the deterministic one. The output SNR can be written like a Rayleigh quotient:

h Γ s h h Bh
where h is the filter, i.e. a N -dimensional vector. This is a stationary-value problem which can be resolved by solving the so-called generalized eigenvalue problem:

Γ s h = λBh.
The maximal SNR ρ max is obtained for the filter h = u 1 , say the eigenvector associated with the largest eigenvalue λ 1 of B -1 A; then ρ max = λ 1 : this filtering comes down to projecting the signal onto a one-dimensional subspace, which was the initial objective. Calculations lead to the eigenvectors of B -1 A, one of them being optimal. Using all the eigenvectors of B -1 A, we perform a simultaneous diagonalization of A and B.

Made from intuitive arguments, experiments show that the use of more than one eigenvector could improve the detection of the signal of interest. But even if this basis maximizes the SNR in a one-dimensional subspace, there is no reason that it would do it also in a p-dimensional subspace. This is why the approach of the constrained stochastic matched filter has been developed.

THE CONSTRAINED STOCHASTIC MATCHED FILTER

There is no reason why the eigenvectors of B -1 A should span a subspace of dimension p > 1 where the SNR is maximum. We propose here to define and find such a subspace whose dimension is set a priori.

SNR in a p-dimensional subspace

From (1) it comes that the output SNR after projection onto E p spanned by p orthonormal vectors {v i } is:

ρ = tr V p AV p tr V p BV p = p i=1 v i Av i p i=1 v i Bv i . ( 2
)
Our objective is to maximize this ratio. That means, p beeing chosen a priori, find the p-dimensional subspace E p in which this value is maximum: this optimal subspace will be noted E * p . Without loss of generality, we can state that trA = trB = 1. That means that the initial SNR trA trB is 1 and then ρ appears to be a gain on the SNR.

Properties of the optimal subspace E *

p p is given and the unknowns of the problem are the SNR ρ defined by (2) and the v i 's which must be calculated to maximize ρ. The constraints can be expressed by v i v j = δ ij . This optimization problem with constraints is usually solved with a Lagrange multipliers method. Hence let's define the function:

L (v i , ω ij ) = ρ + p i=1 p j=1 ω ij v i v j -δ ij or L (V p , Ω) = tr V p AV p tr V p BV p + tr Ω V p V p -I p ,
where Ω ≡ [ω ij ] is a p × p symmetric matrix. This value is maximum when ∂L ∂V p = 0. Calculation leads to:

(A -ρB) V p = V p Ω 0 (3)
where Ω 0 is a p × p real symmetric matrix. There exist a p × p real orthogonal matrix Φ and a p × p real diagonal matrix such as ∆ such that Ω 0 = Φ∆Φ . Then equation (3) becomes:

(A -ρB) V p Φ = V p Φ∆. ( 4
)
As Φ is invertible, V p and V p Φ span the same subspace. Then E * p is spanned by p vectors v i among the N satisfying the eigenvalue equation ( 4).

(A -ρB) is a symmetric matrix depending on ρ, diagonalizable, of unitary eigenvectors matrix T = [t 1 ...t N ]:

(A -ρB) t i = µ i t i ∀ i = 1, ...N.
As ρ is unknown, this is not a common eigenvalue problem. Let's choose a subset I of {1, ...N } satisfying card(I) = p; then

i∈I t i At i i∈I t i Bt i -ρ = i∈I µ i i∈I t i Bt i . ( 5
)
In the subspace E * p the left expression is null. Hence the p optimal vectors {t i } {i∈I} which span E * p are those for which:

i∈I µ i = 0.
The t i , µ i and ρ are unknown. In fact, the solution is directly calculable only for the particular cases p = 1 and p = N -1. An algorithm to reach the solution is proposed in section 3.3.

Remarks 1.

If p = 1, the eigenvalue µ 1 to take into account is null.

Hence, (A -ρB) t 1 = 0 i.e. At 1 = ρBt 1 . ρ is the largest eigenvalue of B -1 A and t 1 its associated eigenvector. Naturally we find in this case the SMF.

2.

From simple examples, it is possible to infer two significant conclusions. First, the optimal subspace E * p is not necessarily spanned by the eigenvectors of B -1 A, and even though this is the case, the eigenvectors are not necessarily associated with the largest eigenvalues. Secondly, it is no use thinking of a recursive formulation on p to find E * p .

Relations on the eigenvectors and eigenvalues

A study of the behavior of the µ i w.r.t. ρ shows that µ i (0

) = µ A i > 0 (eigenvalue of A) and for ρ → +∞ µ i (ρ) → -ρµ B i → -∞. In fact, ∀i, ∂µ i ∂ρ = -t i Bt i < 0 (6)
the curves µ i (ρ) are decreasing with a linear slope at infinity.

Algorithm to find E

* p 3.3.1. Presentation
Let's take an initial value of ρ (a reasonable value is the largest eigenvalue of B -1 A) denoted ρ 0 ; we obtain the matrix A -ρ 0 B. One can calculate the N eigenvectors of it, t i (ρ 0 ), and choose among them the p ones for which

ρ 1 = i∈I0 t i (ρ 0 )At i (ρ 0 ) i∈I 0 t i (ρ 0 )Bt i (ρ 0 ) (7)
is maximum. These p vectors span a subspace E (0) p . At last, one can calculate A -ρ 1 B, I 1 and the new subspace E [START_REF] Van | Trees, Detection, Estimation and Modulation Theory[END_REF] p and iterate the process until ∆ρ n = ρ n+1 -ρ n < ε. It remains to prove that this algorithm converges to the good solution ρ max .

Study of convergence

Noting µ i,n = µ i (ρ n ), t i,n = t i (ρ n ) and for the optimal values µ * i = µ i (ρ max ), let's demonstrate that the algorithm proposed converges when starting near from the solution. We know that for ρ max , there exists a subset I * ⊂ {1, ...N } such that card(I) = p and i∈I * µ * i = 0. At step n, from ( 5) and ( 7), [START_REF] Proakis | Digital Communications[END_REF], the variation of ρ is:

∆ρ n ρ n+1 -ρ n = i∈I n µ i,n i∈In t i,n Bt i,n = i∈I n µ i,n - i∈I n ∂µi,n ∂ρ . Note that if i∈I n µ i,n = i∈I µ * i = 0, then ∆ρ n = 0. ρ max is
an attractive fixed-point of the algorithm.

If we note f (ρ) = i∈I µ i , this equation can be written

∆ρ n = ρ n+1 -ρ n = - f (ρ n ) f (ρ n ) .
This algorithm is nothing else than the Newton-Raphson method applied to f twice differentiable, which owns a zero (ρ max ) with f (ρ max ) = 0. It is known that such an algorithm converges with a quadratic speed.

simulation

We present here a simulation. Each realization is made up of twenty one samples. We build ROC curves, for the two methods, with 410 realizations. Signal and noise are zero mean. Covariance matrices are estimated with 100 realizations. The CSMF is as good as ESMF or better.

CONCLUSION

The method proposed in this paper takes place in the set of methods of decomposition of signals on adapted basis and of subspace-based methods.

When trying to detect stochastic signals with known covariance matrix but with no a priori knowledge on their probability density function, people usually try to project on the signal subspace (SVD,...). It is possible to take into account the structure (covariance) of the embedding noise: classically the SMF is used in such a point of view and in this case, a projection onto a one-dimension subspace is made. In theory this method can be proved to be equivalent to a two stages method: a whitening stage of noise followed by an SVD. Obviously, the other filters calculated with the SMF method can bring information for detection (or filtering, reconstruction, ...) and it is tempting to use several filters to improve the results. Such an approach has been already adopted: people naturally use the filters corresponding to the largest eigenvalues. This position comes down to project onto a subspace of dimension greater than 1. However, there is no guarantee that the signal-to-noise ratio is maximum in the subspace spanned by this vectors.

In this paper, we calculate a set of optimal filters in the sense that they maximize the signal-to-noise ratio in a subspace of dimension chosen a priori. This subspace is not necessarily those spanned by the vectors calculated by the SMF.

Prospects of applications of the CSMF can easily be imagined in image processing or stochastic transient signals detection or estimation (acoustic signals), and a extension to the classification problem can be envisaged.

Figure 1 :

 1 Figure 1: COR curves for CSMF and ESMF : SNR -14 dB