Nicola Peter Todorov

whether they are published or not. The documents may come

Review by Nicola P. Todorov, University of French Guiana.

In 1903, the great historian Herbert A. L. Fisher described illiterate rustics in a Hessian village puzzled by the new Westphalian administration's bureaucratic demands, concluding that "The old-fashioned folk looked on all these busy doings with great disdain." [START_REF] Herbert | Studies in Napoleonic Statesmanship[END_REF] Such descriptions might have been one source of inspiration for the title Sam A. Mustafa gave to his book Napoleon's Paper Kingdom: The Life and Death of Westphalia, 1807-1813. The other might have been the huge number of archival files that, seemingly, impressed him when visiting several, mostly German, archives. "Empires have risen and fallen with less of a paper trail," he contends, "than that left behind by the Kingdom of Westphalia" (p. 314). Indeed, the short-lived states, like this satellite kingdom modeled on Napoleonic France, implemented in the territories conquered by Napoleon at the beginning of the nineteenth century in many parts of Europe, left behind them an archival record that appears a bit exotic in present-day non-French archives.

In his historiographical preface, Mustafa argues that scholars writing under the German Empire at the end of the nineteenth century had agreed that the Westphalian regime's supposed collapse matched the expectations of Prussian "liberators" (p. 286). The presumed revision of this old nationalist view from the 1970s onwards, initiated in Western Germany, was due to the desire to strengthen the Franco-German friendship, downplaying the oppression Napoleonic rule had represented for the people. Suspecting postwar historians to have "overcorrected," Mustafa attempts to challenge this view, stating that "Old stories … are not necessarily wrong only because they are old" (p. xix).

The author admits having written an old-fashioned narrative history aimed at being lively and readable, dealing with individuals without necessarily categorizing them. His purpose, he claims, was to contextualize these individual actors and to examine to what extent they understood what was happening around them, while placing Westphalia into the larger context of the Napoleonic Wars. But he also aimed at writing a comprehensive history of the satellite kingdom, given that the abundant recent literature on the topic is either thematically or regionally restricted in scope. To the end, Sam A. Mustafa has divided his book into thirteen chapters that cleverly alternate between chronological and thematic sequences.

The first chapter is devoted to an account of the occupation of provinces later welded together into the Kingdom of Westphalia and to descriptions of life during this period of the war between France and the Fourth Coalition (from October 1806 until July 1807). Chapter two, entitled "A Kingdom on Paper," deals with the kingdom's organization, giving a brief description of its institutions and administrative divisions, the rational character of which the author attempts to refute. Taking up the idea of a state designed on paper and modeled on foreign country's administration without any consideration to local resources, disseminated by the older historiography, he concludes that Westphalia was simply over-governed, its administrative staff being too replete given the size of the country and the number of its inhabitants. Mustafa analyses the motivations of those who served the new state, concluding that most were mainly driven by the desire to keep their jobs. He underlines the abundant contemporary literature about the new state and its legislation, showing the new state's need to legitimize itself. As many historians before him, he tackles Napoleon's domanial policy as one reason for Westphalia's financial difficulties. Napoleon required the new state to cede a substantial part of all crownland laying in Westphalia to give it as fiefs to his officers. Chapter three describes Napoleon's brother Jerome as the least ambitious of all his siblings, who rightly received his lasting reputation as a womanizer living an extravagant lifestyle at his court. But, as Mustapha believes, it was less the court's expensive splendor and Jerome's lack of interest in governing that discredited his regime than the burden of Napoleon's excessive demands of troops and money that prevented the Westphalian state from winning popular support and loyalty.

The following thematic chapters are devoted to economic life and taxation, the army and military conscription, and, finally, the new legislation and the police. The author repeats several commonplaces created by the older literature. Heavy taxes rested upon the people, economic life was gloomy because of the continental blockade and this was, he claims, a continent-wide problem. Only a few people understood the new laws, and there were not enough judges who could read French and hence be able to understand the Civil Code. Of course, families hated conscription and compulsory military service, as they did elsewhere. Westphalia's army was thus weakened by significant rates of desertion. In chapter seven, Mustafa describes the conspiracies that sought to overthrow the kingdom in 1809, just before and during Napoleon's war against Austria, and he contends that they showed the new polity's fragility. Chapters eight, nine, and ten deal with reforms in religious policy, the emergence of a new Westphalian identity, and attitudes towards the new regime. The final sequence is devoted to the alleged collapse of Westphalia and the restoration of the old order. All that remained of Westphalia was paper.

There is no doubt that the author argues that the old nineteenth-century story written by German historians is finally what matches best a scholarly (and evidence-based) reconstruction of Westphalia's history. According to the author, recent German and French scholarship has not only overlooked the miseries under Napoleon's rule, but also exaggerated the modernization of society and of the so-called model state. It is debatable whether historians have sanitized the unpleasant aspects of Napoleonic domination for political reasons. They simply opened their research to new fields that had been previously neglected. One might recall that the German historian Helmut Berding has insisted that the alleged failure of Napoleon's reform policy occurred because the military and financial pressures exerted by the emperor compromised all efforts to win the people's support. [START_REF] Berding | Les dotations impériales dans le Royaume de Westphalie[END_REF] Berding did not question the existence of what Mustafa calls a "plunder state" disseminated by the old nationalist historiography (pp. 73-74). However, more recent research has offered more nuanced views of this issue. [START_REF]Konig Lustik!? Jeŕome Bonaparte und der Modellstaat Konigreich Westphalen[END_REF] Challenging recent views on Westphalia's life would require exploring large amounts of primary sources and analyzing them according to scholarly standards of the twenty-first century. It would involve a serious discussion of recent research, especially given that the author's project was ambitious enough to deal with Westphalia in its entirety. Does Mustafa match these requirements?

Those who are a bit familiar with the satellite kingdom's history will be astonished by the incredible number of factual errors in Mustafa's book. Kassel, the capital, was not Westphalia's largest city, in contrast to Mustafa's repeated assertions. Despite growing from 18,450 people in 1807 to 22,803 in 1812, mainly due to the prosperity induced by its function as capital of a much larger state than the former Hessian electorate and not only because of immigration from France, Kassel remained only the third most populous city in the kingdom after Braunschweig (with a population of 31,714 in 1807) and Magdeburg (about 30,000). [START_REF] Bosse | Esquisse de la statistique générale et particulière du royaume de Westphalie[END_REF] It is quite clear that the French garrison of Magdeburg was never "as large as the city's population" (p. 209), as the author claims, and the number of soldiers stationed there rarely exceeded 7000-8000. Historians used to recognize the famous French jurist's Siméon sincere aim to deliver the peasants from feudal obligations. He kept the portfolios of the Ministries of the Interior and Justice up to the end of 1808 and then left the Ministry of the Interior to Wolffradt. However, according to Mustafa, Wolffradt was the Minister of Justice and Simeon that of Interior (p. 42, p. 131) throughout the whole reign of Jerome. According to a statement ascribed to Napoleon, Westphalians were not citizens but only subjects. Therefore, the word citizen was systematically avoided in official texts, as the author asserts repeatedly. However, these claims can be easily refuted by reading the text of the Westphalian Constitution, which uses both terms interchangeably.

Mustafa has not really understood Westphalia's administrative hierarchy. Confusing the canton mayors--a specific Westphalian rank created at the end of 1808--with the sub-prefects, he asserts that each district was governed by a certain number of sub-prefects (up to eight in certain cases). Like in France, every district was administered by one sub-prefect. The great number of communes administered by non-professional mayors, who often came from the peasantry and accepted relatively easily administrative tasks, shows the broad and interested involvement of the people in public local administration, rather than over-government.

Mustafa's description of the demesne policy, beginning with a wrong definition of what a domain was, confuses the uninformed reader. In the 1970s, scholars claimed that by depriving the Westphalian government of a substantial part of its crownland, Napoleon favored maintaining feudalism in these parts of the kingdom. However, he had declared that Westphalia should be a model of modernity for the other German states of the Rhenish Confederation, among other things, by abolishing serfdom and privileges. The owners of these domains, composed of the manors whose lord was the prince, were the Landgrave of Hesse, the Duke of Braunschweig, and the King of Prussia. For all reductions of these estates' revenues by Westphalian anti-seignorial legislation, the Westphalian government had to compensate the recipients. Repeatedly, Mustafa describes these estates as private or aristocratic property confiscated by Napoleon (p. 46)! According to Mustafa, the aristocrats who feared confiscations of their lands, sent delegations to the emperor in 1807, since Napoleon's administrators were about to survey all these aristocratic properties (p. 29). Napoleonic administrators did indeed describe all the provinces before organizing the new kingdom, but they surveyed the domanial land of the former princes, never confiscating aristocratic and commoners' properties, considered as private property by the Civil Code. Private property was normally confiscated only in cases of civil servants and military officers who passed to the enemy (or betrayed the Westphalian state). Such confiscations were relatively rare. What the aristocratic delegates sent to Napoleon in 1807 feared was the complete abolition of the peasants' seignorial obligations, as this had been done in France in 1794. The princely estates were run by farmers who were mostly agrarian pre-capitalist entrepreneurs and who saw their contracts confirmed; they continued to exploit these estates on behalf of the new French remote owners and were not expelled, as the author contends (p. 29). However, they had to pay the ground tax levied on the ground's revenue.

To show how the new system puzzled many Germans, Mustafa quotes a German author who published a detailed survey of Westphalian territories and who "got a bit confused over the difference between a department and a district" (p. 43). The cited author did not write on the new territorial division at all, but simply presented the statistics of the provinces integrated into the new kingdom. [START_REF] Bosse | Esquisse de la statistique générale[END_REF] These are only a few examples showing that the meticulous reader will have some difficulty finding confirmation of what the author claims in the works cited in the endnotes. Moreover, Mustafa rounds up figures provided by primary sources very roughly to make it look like Westphalia's balance sheet worse than it actually was. For example, Mustafa says that even the last Westphalian Minister of Finance, Malchus, conceded "that Westphalia's tax burden at least 50% higher than that of the predecessor states, even without counting the forced bonds, special war taxes and French contributions" (p. 101). An analysis of the former minister's post-Napoleonic justifications reveals that Malchus acknowledged an increase of less than 44 percent and finally only around 30 percent, once the taxes imposed on the former privileged aristocrats were deducted and the old pre-Westphalian taxes suppressed in 1811 were taken into account. [START_REF] Malchus | Ueber die Verwaltung der Finanzen des Königreichs Westphalen[END_REF] This increase did not affect all social classes equally.

The numerous scans of archival files uploaded by the Hessian Archives allow us to verify

Mustafa's high estimates for desertion.[7] He contends that the different police organs caught between 150 and 200 deserters per month or between 1800 and 2400 a year. The cited source file contains sixteen monthly accounts of gratifications paid to those who had caught draft dodgers or deserters for the districts of the Werra department. When counting the deserters and not mixing up them with the draft dodgers, as Mustafa clearly does, there were 29 deserters in the whole Werra department caught in 1811. If we assume that the evidence is incomplete, which is not certain at all, because gratifications could be paid irregularly, an extrapolation of the average numbers would lead to an estimate of 520 to 590 deserters caught in the whole kingdom in 1811, or 1,7% of the army's size. This would figure with the data published by the regime's press, which Mustafa attempts to question. But desertion varied regionally. The two lists available for the Fulda department, both of which are not accounts of gratifications, contain more arrested deserters: 28 in April 1811 and 14 in June. But since deserters were likely to desert when their units stood near to their homes--three quarters of the caught deserters came from the Fulda department that garrisoned half the Westphalian troops--these desertion rates are hardly generalizable. That only two lists are available might be due to incomplete preservation, or it might reflect periodical men-hunting practices known in other satellite states. Draft dodging like desertion were forms of resistance to conscription in Napoleonic France and elsewhere. But normally, historians relate the numbers of draft dodgers to the number of young men in the agegroups liable to military service. Compared to other Napoleonic armies, there was relatively little desertion and draft-dodging in Westphalia, which is not astonishing given the highly militarized societies in which people had lived before. The troops standing in the predecessor states amounted to at least 42,000 soldiers, a number the Westphalian army never reached, and the country was able to supply the French troops standing in the kingdom. The Westphalian army's losses have been exaggerated.

Neither the predicted "broadly-based rebellion" in the case of a war against Russia nor the collapse of Westphalia expected by Prussian strategists happened (p. 238). Westphalia was simply occupied by the regular forces of the Sixth Coalition victorious at Leipzig. Even before, deprived of the main part of its troops and engaged at the Saxon theater of operation alongside Napoleon's Grande Armée, the kingdom was unable to prevent the enemy's systematic incursions aimed at disorganizing its administrations. After these short incursions, Westphalian authority was immediately restored, and the alleged collapse of Westphalia expresses nothing more than the wishful and teleological thinking of nationalist historians. During the nearly four weeks between the "dissolution" of Westphalia on October 1 st , 1813 by the Russians (p. 286), who briefly occupied Kassel, and the week after the Battle of Leipzig, 16 th to 19 th of October 1813, and even beyond, the Westphalian administration continued to work as normally as possible in a context of military threat. The idea of widespread popular reprisals against Westphalian officials after the withdrawal of Napoleonic troops is illustrated by some cases that are largely exaggerated by the older historiography, and it is not based on carefully studied evidence. In the former duchy of Braunschweig, these incidents were mostly no more than charivaris, and they affected a little bit more than 6 percent of the villages and towns. [START_REF] Hoffmann | Die gerichtliche Verfolgung von Selbstjustiz und Exzessen bei der Auflösung des Königreichs Westphalen im Gebiet des Herzogtums Braunschweig-Wolfenbüttel[END_REF] Mustafa is obliged to acknowledge that most Westphalian mayors remained in place. In North-western Westphalia, officials and inhabitants hesitated to side openly Napoleon's enemies as long the scope of his defeat at Leipzig was not clear. [START_REF] Sunderbrinck | Revolutionäre Neuordnung auf Zeit. Gelebte Verfassungskultur im Königreich Westphalen: Das Beispiel Minden-Ravensberg 1807-1813[END_REF] In contrast to his declared purpose, Mustafa categorizes people. For instance, he stresses rightly the aristocrats' hostility to the regime but does not apply his categorization to approach their writings critically. But above all, and without the slightest discussion, Mustafa uses national categories the relevance of which for the early nineteenth century has been largely questioned by recent scholarship. [START_REF]Das Vaterland der Feinde. Studien zum nationalen Feindbegriff und Selbstverständnis in Deutschland und Frankreich 1792-1918[END_REF] What did it mean for the popular classes to be a German at that time? Beyond these weaknesses Mustafa's work reveals some methodological limitations characteristic of many other histories of Napoleon's satellites. Many historians specializing in these polities study exclusively the archival records of these short-lived administrations, leaving aside the huge mass of archival files left by the previous bureaucracies. When Prussia imposed her administration on the Polish provinces occupied after the last partitions (1795), the Poles perceived the new bureaucratic rules as petty. Their reactions resembled those of the Hessians, whose disdain for the Westphalian bureaucracy is described at the beginning of this review: "Most Poles reacted badly to a civil service that overwhelmed them with an avalanche of rules and regulations. Many of these created an impression of absurd pettiness." [START_REF] Czubaty | A Napoleonic Outpost in Central Europe[END_REF] But it was this kind of administration that Napoleon's men found in many parts of Westphalia. Bureaucratic tasks imposed on villagers were nothing new. And if Sam A. Mustafa had studied seriously these pre-Westphalian files, he would have found lists of Prussian deserters and draft dodgers whose properties were confiscated. Even if this is not always an easy task, measuring the rejection of Prussian and Westphalian military service in a comparative perspective would have allowed him to conclude that the Prussian military system was so much hated that in 1813 and 1814 Prussia's occupational authorities felt obliged to justify the lacking will to enroll in the Landwehr by the fear of corporal punishments. The same criticism applies to the author's impression of a strange hybridization of the German language with French words during the Westphalian period. The German language had much more French words than today, and they were largely imported in the seventeenth and eighteenth centuries, and not under Napoleon's rule. Some acquaintance with pre-Napoleonic German texts and familiarity with language history would have allowed the author to avoid such hasty conclusions.

It matters little that the author does not mention some works published in 2015 and 2016. [START_REF]for example: Sunderbrinck, Revolutionäre Neuordnung auf Zeit and the essays[END_REF] He had decided to avoid seriously engaging with most of the recent scholarly literature, instead using largely memoirs written in the nineteenth century as evidence to depict public opinion in Westphalia. Above all, he has systematically selected only those facts in both the old and new scholarly literature that support his theory of Westphalia's complete failure. It is easy to complicate and even to refute many of Mustafa's geographically and socially unnuanced assertions about the impact of the continental blockade, the effects of quartering troops, and many other aspects with information taken from the same scholarly literature he cites in his own bibliography. For example, in Willy Kohl's scholarly work, courageously published in the unfavorable context of Nazi Germany, he would have found that in 1817 peasants confessed "guilelessly" to a Prussian minister making a tour throughout the reoccupied province of Magdeburg that "they would like to have such a constitution back" (the Westphalian one). [START_REF] Kohl | Die Verwaltung der östlichen Departements des Königreichs Westphalen[END_REF] The numerous factual errors, the shallow reading of sources, the miscalculations obviously aimed at supporting his negative view of Westphalia, the very partial selection of facts, and the many unreliable references to the sources and the secondary literature disqualify Mustafa's book as scholarly work. Unfortunately, the English-speaking reader will still be advised to refer to the old works of Herbert Fisher and Owen Connelly, that certainly do not take account of recent research, but provide a much more well-balanced judgement on Napoleonic Westphalia and certainly contain fewer factual errors than Sam Mustafa's polemical writing. [START_REF] Connelly | Napoleon's Satellite Kingdoms. Managing Conquered Peoples[END_REF] NOTES

H

  -France Review Vol. 19 (March 2019), No. 37 Sam A. Mustafa, Napoleon's Paper Kingdom: The Life and Death of the Kingdom of Westphalia, 1807-1813. London and New York: Rowman & Littlefield Publishers, 2017. xxi+341 pp. Maps, figures, notes, bibliography, and index. $99.00 U.S. (hb). ISBN 978-1-5381-0829-1; $94.00 U.S. (eb). ISBN 978-1-5381-0831-4.


domain: sde spi stat

text:
Rachid Ouaret
email: rachid.ouaret@toulouse-inp.fr

Anda Ionescu

O Ramalho

Non-negative Matrix Factorization for the analysis of particle number concentrations: characterization of the temporal variability of sources in indoor workplace

Keywords: Non-negative Matrix Factorization (NMF), Temporal source apportionment, Particle Number (PN) concentration, Open-plan Office

come

Introduction

Nowadays, it is becoming increasingly evident that indoor environments play a critical role to understand and assess total human exposure to air pollution. Indoor air quality became a matter of particular interest for the following three main reasons: (i) people spend about 85% of their time indoors, (ii) indoor pollutant concentrations can be significantly higher from those outdoors and (iii) many potentially health-hazardous pollutants are emitted from indoor sources [40]. The assessment of the indoor source variability is now necessary for the design and implementation of effective control strategies. Airborne Particulate Matter (PM) is one of the major types of contaminants in indoor air due to their ubiquitous occurrence and toxicity [42,[START_REF] Christopher M Long | A pilot investigation of the relative toxicity of indoor and outdoor fine particles: in vitro effects of endotoxin and other particulate properties[END_REF]. Particulate matter is a mixture of many different chemical species and the assessment of human exposure to PM requires some knowledge about the sources, and the determination of the time variability of the PM time series.

PM represent a complex mixture of organic and inorganic species which vary in size, composition, and origin. Regarding particle size, PM is usually classified according their aerodynamic diameter such as coarse (2.5 µm-10 µm), fine (<2.5 µm), and ultra-fine particles (< 0.1 µm) [START_REF] Albright | Algorithms, initializations, and convergence for the nonnegative matrix factorization[END_REF]. Since different particles originate from different sources and their concentration in a given place is influenced by different factors, scientific knowledge about the occurrence, strength, and temporal variability of the sources is required to reduce the indoor exposure impact. Several studies focused the extent to which human exposure to outdoor PM occurs indoors [64,[START_REF] Nezis | Particulate matter and health effects in offices-a review[END_REF]. From outdoors to indoors, PM enters by infiltrating through cracks and gaps in the building envelope, via natural or mechanical ventilation [11,[START_REF] Isaxon | Contribution of indoor-generated particles to residential exposure[END_REF]. Consequently, the temporal patterns of indoor PM concentrations are resulting from both indoor and outdoor sources variability which constantly varies over time. In addition, the PM temporal patterns are influenced by many factors: (i) penetration and ventilation efficiency, (ii) indoor occupancy and occupants' activity, and (iii) building's volume and interaction with surfaces.

Important indoor sources of PM include, but not limited to, household chores (combustion, candles, and cooking, ...) [START_REF] Abt | Relative contribution of outdoor and indoor particle sources to indoor concentrations[END_REF][START_REF] Isaxon | Contribution of indoor-generated particles to residential exposure[END_REF] and different human activities [START_REF] Ferro | Source strengths for indoor human activities that resuspend particulate matter[END_REF]. In indoor offices, particles also can be generated from some equipment, such as copier and printers machines, computers, and other electronic devices [START_REF] Luoma | Characterization of particulate emissions from occupant activities in offices[END_REF][START_REF] Cacho | Air pollutants in office environments and emissions from electronic equipment: A review[END_REF][START_REF] Salonen | Human exposure to ozone in school and office indoor environments[END_REF]. Furthermore, many studies claim that particle re-suspension from walking is an important indoor source of PM [START_REF] Qian | Walking-induced particle resuspension in indoor environments[END_REF]57]. The source emissions and sinks are generally variable and depend on many environmental factors, such as temperature, humidity, and air exchange rates. Particle deposition and resuspension within the microenvironments are mechanisms that can extremely change the time variability of the indoor PM concentration. Overall, as detailed in [40], three main factors determine indoor pollutant variations in occupied spaces: (i) properties of pollutants, (ii) occupants' behavior and (iii) building characteristics. In the case of the testing chambers or experimental automated houses, these parameters are usually well defined [68,[START_REF] Bruce A Tichenor | Evaluating sources of indoor air pollution[END_REF].

To the best of our knowledge there are few studies in the literature focusing on the impact of the occupancy and windows state in offices for a long period with a short time step.

The originality of the study presented in this paper is due to the following features: it is based on a long-term monitoring campaign in real conditions; parameters are monitored with a short time step; it contributes to indoor source identification taking into account the occupancy and the window opening state.

The objective of this work is to get a better understanding of the temporal variability of PM's and their determinants, i.e. the influencing factors, as well as their relative contribution to PM indoor concentrations. The measurement campaign was carried out in an occupied open-plan office.

By retracing pollutants to their origins, emission sources variability can be characterized. The source identification and the assessment of their relative contribution to total indoor exposures can provide valuable information for reducing their emission with the aim to protect human health.

The practice of deriving information about pollution sources and the amount they emit is called source apportionment [START_REF] Belis | European guide on air pollution source apportionment with receptor models[END_REF].

In principle, there are two basic approaches used in the environmental field to perform a source apportionment analysis: source-oriented models (dispersion model) and receptor-oriented models.

Dispersion models require the knowledge of emission rates and dispersion factors together with local topography and meteorology for the estimation of source impacts [START_REF] John | Receptor oriented methods of air particulate source apportionment[END_REF][START_REF] Ronald | Review of receptor model fundamentals[END_REF]. It is a direct modeling (from sources to receptors). By contrast, receptor modeling, which is an inverse one (from receptor to sources), is based on the mass conservation and it can be used to identify and apportion sources of contaminants in the air [23]. The main idea in the latter is to solve a mass balance equation using multivariate factor analysis. Thus, the receptor models are used to estimate the contribution of different sources to ambient PM concentrations based on PM measurements and subsequent chemical analysis [START_REF] Pant | Receptor modelling studies of airborne particulate matter in the United Kingdom and India[END_REF]. Exogenous variables can be used for interpretation.

Indoor environments are modeled sometimes by means of experimental chambers, where climatic parameters are controlled. In this case, physical models can be developed easier than for the real environments. Indeed, in the case of the real buildings the development of such models is more complicated for the following reasons: difficulty in measuring source emissions [58]; unknown or changing airflow patterns [START_REF] Liang | Indoor formaldehyde in real buildings: Emission source identification, overall emission rate estimation, concentration increase and decay patterns[END_REF]10]; difficulty to model room configuration and lack of information about occupancy and its impact on the indoor environment.

The advantage of receptor models in occupied spaces is that this type of models doesn't require all the information mentioned previously, as input. They are based on the decomposition of measured concentration signals without predetermining the pollutant transformation, transportation, and sink processes [START_REF] Viana | Source apportionment of particulate matter in europe: a review of methods and results[END_REF].

In recent years, there have been tremendous advances in the development of source apportionment techniques based on statistical analysis, particularly the use of Factor Analysis (FA) for temporally varying constituent concentrations [START_REF] Belis | European guide on air pollution source apportionment with receptor models[END_REF][START_REF] Ronald | Review of receptor model fundamentals[END_REF]23,67]. All these techniques are based on the factorization of the initial matrix (database of concentrations in our case) in two matrices: matrix of source profiles and matrix of source contributions, under the constraint of the RMSE minimization and by imposing different constraints, such as: decorrelation (Principal Component Analysis), statistical independence (Independent Component Analysis), non-negativity (Non-negative Matrix Factorization and Positive Matrix Factorization).

Positive Matrix Factorization (PMF), developed by Paatero et al. 1994 [47] is widely used for source apportionment in environmental applications such as atmospheric pollution. This method imposes positivity constraints in the factor computational process by calculating dominant positive factors based on measurements. The factorization is achieved without detailed prior knowledge of source profiles or chemical fingerprints. In contrast to many other linear factorizations such as (PCA) and Independent Component Analysis (ICA), Non-negative Matrix Factorization (NMF) [START_REF] Daniel | Learning the parts of objects by non-negative matrix factorization[END_REF][START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF], as well as PMF, makes positive latent structure explicit.

The Principal Component Analysis (PCA) method is one of the most used tools in factorial methods.

In indoor environment, we can find hybridization with other linear methods (regressions, ...) to quantify the factors [START_REF] Mohd Yasreen Ali | Quantitative source apportionment and human toxicity of indoor trace metals at university buildings[END_REF].

In the same vein, we have used NMF and simulated several factorizations, which are obtained by a group of initialization-computation algorithms. For this reason, the different extensions on NMF comprising the PMF method which is considered as a special case

Materials and Methods

Materials

PN concentration monitoring

Particle Number concentrations (PN) were acquired continuously using an optical particle counter, a GRIMM 1.108 Dust Monitor (Grimm Technologies, Inc., Douglasville, GA, USA). This monitor counts the number of particles with a diameter within the range of 0.3 to 20 µm, and more in detail in 15 different size channels limited by: 0.3, 0.4, 0.5, 0.65, 0.8, 1.0, 1.6, 2.0, 3.0, 4.0, 5.0, 7.5, 10, 15 and 20 µm diameters (Figure 1 It should be noted that motion detection greatly underestimates the actual occupancy of the office space. The situation where the occupant is static is considered to be a vacancy state. This would reflect a high vacancy rate over the measurement period (≈ 94%).

The open-plan environment

The concentration of indoor particles is highly variable and indoor-specific. The measurements were performed in a building (Scientific and Technical Center for Building (CSTB), cf. The indoor materials were carpets on the floor, painted walls, and ceiling tiles. The furniture comprises typical L-shaped desks melamine-faced particleboard and aluminium closets. A laser multi-function copier was in use in the office plan. No specific major sources of particles, such as combustion, were identified in the office.

Methods

Related work

The number of existing receptor models applied to the environmental field is relatively large and it includes methods such as Principal Component Analysis [START_REF] George | A quantitative assessment of source contributions to inhalable particulate matter pollution in metropolitan boston[END_REF]20], Positive Matrix Factorization (PMF) [START_REF] Paatero | Matrix factorization methods for analysing diffusion battery data[END_REF][START_REF] Paatero | Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values[END_REF], Independent Component Analysis (ICA) [4] and recently Non-negative Matrix Factorization (NMF) in outdoor [17,59,27,[START_REF] Kfoury | Pm 2.5 source apportionment in a french urban coastal site under steelworks emission influences using constrained non-negative matrix factorization receptor model[END_REF][START_REF] Liu | Development and assessment of a receptor source apportionment model based on four nonnegative matrix factorization algorithms[END_REF] and indoor [START_REF] Rösch | Relationship between sources and patterns of vocs in indoor air[END_REF][START_REF] Ouaret | Analysis of the temporal variability of indoor particulate matter concentrations using Blind Source Separation methods: a comparative study[END_REF][START_REF] Ouaret | Particulate matter variability sources in an open-plan office: comparison of two monitoring campaigns[END_REF][START_REF] Ouaret | Indoor air pollutant sources using blind source separation methods[END_REF] environment.

Compared to the PCA and ICA, the PMF and NMF methods have the advantage of more realistic non-negative constraint on factor profiles and contributions. Following this criterion, this study used non-negativity for indoor source apportionment and particularly employed NMF that offers a wide range of algorithms and extensions compared to PMF. To take into consideration both constraints simultaneously, in the previous works [START_REF] Limem | Nonnegative matrix factorization using weighted beta divergence and equality constraints for industrial source apportionment[END_REF]34] a new parametrization was proposed by incorporating a new unconstrained matrix [START_REF] Chreiky | Split gradient method for informed non-negative matrix factorization[END_REF]. The update rules in [START_REF] Chreiky | Split gradient method for informed non-negative matrix factorization[END_REF] are based on the framework of the Split Gradient Method of Lantéri et al. [START_REF] Lantéri | Split gradient method for nonnegative matrix factorization[END_REF]. To identify distinctive PM 10 patterns across Europe on the airborne data obtained from 1097 monitoring stations for 3 years, Žibert et.al [START_REF] Žibert | Particulate matter (pm10) patterns in europe: An exploratory data analysis using non-negative matrix factorization[END_REF] used NMF and the autocorrelation function (ACF) to enhance the interpretation of factors features. In the indoor air quality (IAQ) field, the positive (non-negative) matrix factorization is not such popular as for outdoor air quality. We find only one study assessing the use of NMF in IAQ, which is conducted by Rösch et al. in 2014 [START_REF] Rösch | Relationship between sources and patterns of vocs in indoor air[END_REF]. Also in the aircraft cabins context, PMF coupled with information related to VOC sources have been applied to identify the major VOC sources [65]. The study pointed out the importance of service and humans as major source (29%) of the total VOC emissions. Overall, there is not sufficient research to provide meaningful information on indoor source time variability.

As the IAQ continues to face the realities of climate change [40], our understanding of temporal patterns of exposure is crucial and this research gap requires further investigation. Therefore, this study aims to address the elements of the research gap in indoor source identification. This paper is an extended version of the works presented in [START_REF] Ouaret | Particulate matter variability sources in an open-plan office: comparison of two monitoring campaigns[END_REF], [START_REF] Ouaret | Analysis of the temporal variability of indoor particulate matter concentrations using Blind Source Separation methods: a comparative study[END_REF] as well as in [START_REF] Ouaret | Indoor air pollutant sources using blind source separation methods[END_REF].

Non-negative Matrix Factorization

Non-negative matrix factorization (NMF) is a multivariate data analysis technique which is aimed to estimate physically meaningful latent components from non-negative data. Mathematically, the factorization is carried out by a linear superposition of non-negative basis components and nonnegative weights.

NMF was initially introduced by Paatero's works [START_REF] Paatero | Matrix factorization methods for analysing diffusion battery data[END_REF][START_REF] Paatero | Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values[END_REF], which refer to the problem as Positive Matrix Factorization (PMF) that corresponds to the mass-balance model. The NMF has gained popularity by the works of Lee and Seung who presented multiplicative update algorithms for computing the NMF to optimize a cost function based on either a Euclidean distance measure or a generalized Kullback-Leibler divergence [START_REF] Daniel | Unsupervised learning by convex and conic coding[END_REF][START_REF] Daniel | Learning the parts of objects by non-negative matrix factorization[END_REF][START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF]. A multitude of NMF variants and generalizations is summarised in Cichocki's concise lecture note [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation[END_REF].

In this paper, a matrix is denoted with an uppercase bold letter, e.g., X, its elements with the corresponding lowercase letter, e.g., x it , and a column vector in lowercase boldface, e.g., x i .

Given an input non-negative raw data matrix X = [x 1 , . . . , x T ] ∈ R I×T + and a positive integer reduced rank J, (J ≤ min (I, T )), the non-negative matrix factorization problem consists in finding two non-negative matrices W = [w 1 , . . . , w J ] ∈ R I×J + and an encoding matrix

H = h jt ∈ R J×T + that approximate X, i.e. X ≈ W H = x t ≈ J j=1 w j h jt . ( 1
)
Depending on the application field, these factors W and H are interpreted differently. In environmental source apportionment, H plays the role of mixing matrix (weights) which represents the source contributions, while W expresses temporal factor profiles (time series, see Appendix C).

Thus, if a weight h jt in a column of H is high, then the corresponding basis vector w j is very important in approximating x t . Geometrically the basis vectors generate a simplicial cone and the columns of the matrix W are basis vectors spanning a subspace in J ≤ I. Once estimated, the H and W matrices will be presented in sections 3.2.1 and 3.2.2, respectively. Indeed, the H matrix will allow us to analyze the contribution of the factors and the W matrix to visualize the diurnal profiles of the factors (time series).

The time series of the sources and their weights are calculated iteratively by minimizing a suitable measure f for the divergence between W and H:

arg min W ,H≥0 f (W , H) = arg min W ,H≥0 [D (X W H) + R (W , H)] , (2)
where

D : R I×J + × R J×T + → R + is a
D F (X W H) = 1 2 X -W H 2 F = 1 2 I i T t (X it -(W H) it ) 2 . ( 3
)
The Frobenius similarity measure is a special case of the so-called β -divergence [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation[END_REF]. In this study we will consider only Kullback-Leibler divergence in case of Brunet algorithm [START_REF] Brunet | Metagenes and molecular pattern discovery using matrix factorization[END_REF] as follows:

D KL (X W H) = I i T t X it ln X it (W H) it -X it + (W H) it . (4)

Algorithms for solving the NMF problem

Many numerical algorithms have been developed to solve the NMF problem expressed in the equation (2). They can be divided into three general classes: (i) Alternating Least Squares (ALS) algorithms, (ii) multiplicative update algorithms, and (iii) gradient descent algorithms [START_REF] Michael W Berry | Algorithms and applications for approximate nonnegative matrix factorization[END_REF][START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation[END_REF]. The ALS algorithm computes the optimal solution of the unconstrained least squares problem, then it optimizes alternatively over one of the two factors W or H while keeping the other fixed. This subproblem is then reduced to one-factor convex.

By minimizing two criteria the squared Euclidean distance (or equivalently the squared Frobenius norm D F ) and the generalized Kullback-Leibler divergence D KL , Lee and Seung [START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF] proposed the multiplicative update algorithm to solve the equation [START_REF] Abt | Relative contribution of outdoor and indoor particle sources to indoor concentrations[END_REF]. Simple multiplicative update formulas based on D F are given by

w ij ← w ij XH ij XHH ij + ε , ( 5
)
h jt ← h jt W X jt W W H jt + ε . ( 6
)
For the implementation purpose, a small positive constant ε is added to the denominator in each update rule to avoid division by zero. Lee and Seung claimed that the above algorithm converges to a local minimum [START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF], which was later shown to be incorrect (see for example [14,[START_REF] Michael W Berry | Algorithms and applications for approximate nonnegative matrix factorization[END_REF][START_REF] Lin | On the convergence of multiplicative update algorithms for nonnegative matrix factorization[END_REF]): the above algorithm 5, 6 can only keep the non-increasing property of the objective.

Based on the Kullback-Leibler divergence, Brunet et al. [START_REF] Brunet | Metagenes and molecular pattern discovery using matrix factorization[END_REF] used modified versions of Lee and Seung's (2001) [START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF] simple multiplicative updates to avoid numerical underflow. At each step, W and H are updated by using the coupled divergence equations:

W ia ← W ia µ HaµXiµ (W H) iµ ν H aν , ( 7
)
H aµ ← H aµ i W iaX iµ (W H) iµ j W ja . ( 8
)
The stopping criterion for Lee and for Burnet optimization is the variance of the connectivity matrix.

Initialization of NMF

Most NMF objectives are not convex and they are sensitive to the initialization of matrices W and We tested three different kinds of initialization techniques for the two different NMF algorithms described above. In the first seeding method, a random starting point has been used, where the entries of W and H are drawn from a uniform distribution, within the same range as the X matrix's entries. That is the entries of each factor are drawn from a uniform distribution over [0, max {x}],

where x is the column vector of X. We used a maximum of 100 runs for each algorithm to achieve stability. The second initialization method that we tested consists in using the results of the Independent Component Analysis (ICA) (FastICA algorithm [START_REF] Hyvärinen | Independent component analysis: algorithms and applications[END_REF]) where only the positive part is used to initialize the factors. The last tested initialization algorithm is the Non-Negative Double Singular Value Decomposition (NNSVD) [START_REF] Boutsidis | Svd based initialization: A head start for nonnegative matrix factorization[END_REF].

Determining the number of components

One of the critical parameters in NMF is the number of components J to select for the factorization in equation 1. An appropriate decision on the value of J is critical in practice, but it is usually chosen such that J min (I, T ) in which case W H represents a compressed form of the initial matrix X. The factorization rank parameter can be estimated by computing the Residual Sum of Squares (RSS) or by the explained variance (EVAR) between a target matrix X and its estimate X:

RSS = i t X it -X it 2 (9) EVAR = 1 - RSS i,t X it (10)
The "optimal" rank is chosen using the graph of the EVAR (or RSS); it corresponds to the first point where the graph shows an inflection point, as Hutchins et al.

[24] did with the algorithm of Lee et al. [START_REF] Daniel | Learning the parts of objects by non-negative matrix factorization[END_REF].

Results and discussion

In the following, a preliminary analysis of the original data is conducted (section 3.1). Then, the choice of the number of factors and their profiles are discussed in the section 3.2. Finally, for interpretation purposes, theses factors are analyzed and related to different other parameters such as occupancy and windows state.

In this paper, the terms factor, component and patterns are equivalent. The results and analyses were conducted in R [54] using the NMF R package [19] and figures were produced using the ggplot2 package [START_REF] Wickham | ggplot2: Elegant Graphics for Data Analysis[END_REF].

Since particles in the fine range (<2.5 µm) dominate the total particle number, the obtained NMF patterns emphasize this range. As the range variation of the number of particles can be very different from one bin to another, we considered very carefully the standardization need of the raw data matrix as input for NMF. This standardization procedure produces negative values that should be avoided due to the non-negativity constraint. For this reason we have made a translation of the standardized values by adding a constant equal to 2 particles.cm -3 to each one. This constant shift does not impact the factorization results.

Data description and preliminary analysis

During the sampling campaign, the indoor temperature values varied from 15°C to 40°C and relative humidity ranged between 18 % and 65%. higher than the median ones. As the data distribution was positively skewed, the median (P 50 ) was preferred for interpretation purposes instead of the arithmetic mean.

For most of the particle sizes, the median values of PN are very similar regardless the windows state, while the 10 th and 90 th percentile values are different. For instance, 90 % of PN 0.35 concentrations are less than 64 #.cm -3 in the case of closed windows state and 42 #.cm -3 when at least one window is opened. These observations are related to the total number of minutes in each opening state. On average, windows remain closed around 85% of the total time throughout the study period (24/7), week-end and holidays included.

Indoor particles can originate from outdoor sources and also from various indoor sources. Thus, it is possible that the levels of indoor particles exceed outdoor ones. The Figure 2 shows the diurnal variability of the median value of indoor PN concentrations according to occupancy and windows states for the following bins: PN 0.35 , PN 0.9 , PN 2.5 and PN 17.5 . The diurnal pattern for PN 0.35 shows a convex shape when all the windows are closed and no occupancy detected. The I/O ratio of the corresponding bin is always less than 0.6 for closed windows and it ranges between 0.6 and 1 in the case of opened windows. The variation profile can be explained by the fact that the outdoor particles penetrate inside by mechanical ventilation and infiltration through the gaps in the building, given that the windows are closed. These fine particles come mostly from outdoor sources, such as traffic which is higher during the rush hours. The diurnal profile of the PN of size 0.9 µm when the windows are opened is characterized by a significant peak between 8:00 and 9:30 a.m. (when usually work starts) and then it decreases until 7:00 p.m. On the other hand, the same variation is observed when a movement of occupants is detected, even when the windows are closed. For PN 2.5 , there is a significant difference between the case when the windows are opened or closed, regardless the occupancy. The values are higher for PN 2.5 when windows are opened showing the importance of outdoor sources for these size bins. By contrast, for the coarse particles (PN 17.5 ), the occupancy variable discriminates the profiles revealing that these particles are generated indoors (occupants' activities, such as walking).

To summarize, opening the windows results in similar trends in PN for both occupation and nonoccupation conditions (except for PN 17.5 due to less contribution of coarse particles in the air).

Mostly, PN concentrations with occupancy are higher than PN concentrations without occupancy.

However, real contribution of occupation is marked when the windows are closed.

The particulate number concentration decrease in the indoor environment occurs mainly by two mechanisms: ventilation and deposition. In general, ventilation could play a positive role in the loss of particles from indoor air, but sometimes it may cause entering the outdoor pollutants via the supplied air coming indoors.

NMF results

As presented in the subsection 2.2.4 and 2.2.5, the method should be initialized and an optimal number of components should be determined.

Values in the range J = 2, . . . , 10 were tested based on 20 000 randomly sampled 1-minute dataset using three different initialization algorithms: ICA, NNSVD and Random for the two optimization methods proposed by Brunet [START_REF] Brunet | Metagenes and molecular pattern discovery using matrix factorization[END_REF] and Lee [START_REF] Daniel | Learning the parts of objects by non-negative matrix factorization[END_REF][START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF]. q q q q q q q q q q q q q q q q q q Brunet Lee

ICA NNDSVD Random

Figure 3: Explained Variance (EVAR) variations of Brunet [START_REF] Brunet | Metagenes and molecular pattern discovery using matrix factorization[END_REF] or Lee [START_REF] Daniel | Learning the parts of objects by non-negative matrix factorization[END_REF][START_REF] Daniel | Algorithms for non-negative matrix factorization[END_REF] algorithms according to different initializations.

Figure 3 shows the evolution of the explained variance according to initialization algorithms (ICA, NNSVD, and Random) according to the rank number. Unexpectedly, for each initialization method, NMF generated extremely resembling results for Lee's algorithm. A similar observation is observed for the first 5 rank factors in Brunet's algorithm. Note that random initialization may make the experiments unrepeatable because of their local minima property. Several studies have revealed that methods with non-random seeding demonstrate their superiority either in the fast convergence in prophase or the structure preservation [START_REF] Zheng | Initialization enhancer for non-negative matrix factorization[END_REF][START_REF] Boutsidis | Svd based initialization: A head start for nonnegative matrix factorization[END_REF]. This is yet another reason for not choosing random initialization in our experiments.

The interpretability of the factors can be a selection criterion for the rank choice. The use of expert insight in the case of indoor air quality was taken into account. It can be observed that using the interpretability criterion of the value of J by increasing or decreasing by 1, either the new NMF-factor shows a mixture of parts of existing factors or contains completely new environmental patterns. To represent as many different patterns as possible and taking into account the explained variance variation, we found for the considered dataset that J should be about 4-6. To draw comparisons between different results, the best physical interpretation and factorization error at the same time were obtained for J = 5. With this rank corresponding thus to 5 factors, the explained variance corresponds to 96% and 97.5% for Brunet's and Lee's algorithms, respectively.

Factors contributions

The time series of the factors obtained by NMF are provided in the supplementary material of this article (cf. Figure C.1). The relative contribution of factors to each particle size is shown in Figure 4. The relative contributions of the five components are distributive i.e. all the fractions can be found in at least one factor.

For the fine size fractions (0.3 µm, 0.45 µm, 0.57 µm), the contribution of the component 2 obtained by NMF is nearly 75 % of the total contributions. By contrast, this component contributes the least for larger sizes of particles, so it is specific to the fine size.

The fourth component is related to coarse fraction, which represents 80 % of all the other contri-butions. For fractions between 1.3 µm and 4.5 µm, the components 3 and 5 share over 50 % of all contributions. The first component is involved in about 20% for particles smaller than 2.5 µm and can reach 80% for particles with a size of 12.5 µm.

To facilitate the interpretation of each component, we compute the correlations between the original data (the monitored time series corresponding to each fraction) and each NMF component.

The interpretation of the NMF components is based on finding which PN time series are the most strongly correlated with each component. Figure 5 shows the correlation values between the components time series and fraction variables time series. The color variations express the strength and direction of the correlation relationship.

q q q q q q q q q q q q q qq Note that all the significant correlations, i.e. greater than 0.6, are positive. The first NMF component is strongly correlated with particles ranging from 6.25 µm to 12.5 µm, the correlation coefficients vary form +0.6 and +0.82. The contribution of this component to the variability of the of particles of sizes between 3.5 µm and 4.5 µm is not negligible (∼ 40%) although their correlations are not very strong (+0.45, +0.51).

The 2 nd factor is associated with particle size ranges below 0.8 µm with a strong positive correlation coefficient (> 0.8). Its contribution is higher than 63% for particles of sizes 0.575 µm, 0.45 µm and 0.35 µm.

Factor 3 contributes (up to 50%) to sizes between 0.9 µm and 2.5 µm, with a correlation varying between +0.58 and +0.95. We notice that the diurnal profiles of the F3 and PN 0.9-1.8 are very similar (cf. Figure 2); both of them are characterized by a significant peak between 8:00 a.m. and 9:30 a.m. followed by a decreases until 7:00 p.m.

Note that for particles of size 2.5 µm, two factors contribute simultaneously to its variation: F3

(correlation = 0.58) and F5 (correlation =0.78). The same observation can be made for the particles of size 6.25 µm: two factors contribute to its variation (F1 and F5). Coarse particles (>17.5 µm) are associated with the 4 th factor with a correlation of +0.73. The latter does not contribute to any of the other particle ranges. In Figure 3, there is a peak of contribution of about 80% for particle sizes 17.5 µm and > 20 µm. Particles in the intermediate size range (between 2.5 and 6.25 µm) are associated to F5 with correlations ranging from +0.74 to +0.85. The contribution of F5 to the variability of particles of 3.5 µm size, is close to 50%.

Time variation of the factors

Having examined the overall contributions of NMF components, we now take a closer look to interpretate them. This section sheds light on the nature of the hidden components obtained using NMF. To do so, we combine three sources of information: To facilitate the interpretation, diurnal variations in CO 2 concentrations have been added in the same Figure . The CO 2 variations are used here as a fingerprint of occupant presence in order to allow identification of similar factors with this type of variation.

qqq qqq qq q q q q q q q qq q q q qqqq q qqqqqq q q q q q q q q qq q q q qq q q Occup q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q qq CLOSE OPEN Occup q qq q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q CLOSE OPEN 0 Occup qq q q q q q q q qq q qqq q q q q q qqqq q q q q qq q q q q q q q q q q q q q q q q q q CLOSE OPEN Occup q q q qq q q q q q q q q q qqq qq q q qqq q q q q qq q q q q q q q q q q q q q q qq q q The diurnal profile of the first factor is very similar to the diurnal variations of indoor CO 2 concentrations (cf. Figure 6), suggesting that this component is related to the presence of the occupants, who are the main source of CO 2 . When all the windows are closed, the occupancy curve (red dots) is separated (p-value < 0.05) from the non-occupancy one.

When at least one window is open, the values of component 1 are higher. It is as if another source coming from outside is added or the ventilation changes the transport mechanisms of large particles (resuspension). Nevertheless, several studies report that the mechanism of resuspension is especially related to particles of sizes less than 10 µm [START_REF] Qian | Estimating the resuspension rate and residence time of indoor particles[END_REF][START_REF] Qian | Resuspension of dust particles in a chamber and associated environmental factors[END_REF]. As previously mentioned (Figure 5), the peak contribution of the first component is associated with particles of sizes between 8.75and 12.5 µm.

The diurnal profile of the second component illustrates a decreasing effect during the period from 6 a.m. to 7 p.m.. For F2, there is no significant difference for the "occupancy" variable, the two curves overlap. As shown in previous sections, this component is related to fine particles (less than 0.75 µm). Components F3 and F5 could be attributed to the influence of outdoor sources on the indoor environment. The seasonal appearance (of component 3) is mainly due to diurnal variations of outdoor sources. This finding provides additional insight into the sources of particles ranging in size from 1.8 µm to 6.25 µm. Thus, by combining them with the information provided in Table 2, we notice the important influence of outdoor conditions on indoor concentration levels. At this stage, one of the major points of interest has been the outdoor source identification i.e. when outdoor concentrations and characteristics are the main contributing factors. Overall, outdoor sources have been mainly associated with fine particles in accumulation mode (0.1-1 µm), probably because these particles can persist in the air since they are too small for inertial deposition and too large for diffusion removal processes. These particles are capable of entering in the buildings and remaining airborne for longer periods.

The forth component (F4) is characterized mainly by the coarse particles (> 17.5 µm). Figure 6 shows a strong variability in the diurnal profile during the occupation period. The profile is very random because it is mainly associated with the activities of the occupants. These activities, and in particular walking, are responsible of resuspension of the coarse particles. We remind that the office is equipped with a carpet covering the entire floor. The statistical tests cannot be used because the probability density is bimodal for this profile. We notice that when at least one window is open, the profile is further modified. Statistical tests confirm this observation (p-value <5%).

Limitations and future directions

Firstly, a generic NMF, which is a method to solve a linear system, was employed and the underlying model was used as a set of linear Chemical Mass Balance (CMB) to estimate individual source components from their mixtures. However, the indoor mixing phenomena are better represented by non-linear relationships, such as infiltration, sink, inertial deposition, and diffusive removal processes. A research question that remains is how these latter processes can be considered in a more general model. In the same vein, future work would include the hybridization of physical models with factorial source separation methods.

Secondly, although NMF has a realistic non-negative constraint on factor profiles and contributions, it still does not cover all the indoor environment characteristics. Thus, many parameters as occupant behavior and windows opening, which have "random impacts" over the primary determinants of the decay rate, have to be incorporated in the NMF optimization problems. Similarly, the including time-activity patterns could improve the interpretability of the NMF results.

One critical question that remains to be answered about the NMF method for indoor air quality is: how to integrate inherent indoor specificity and constraints in the NMF formulation?

Recently the NMF field expanded to multidimensional data arrays, called Non-negative Tensor Factorization (NTF) [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation[END_REF], which could offer valuable new insights on IAQ modeling issues. Future work also involves validating the performance of the NMF model using data over a much longer period and above all including more information about the different activities such as walking, cleaning, printing, ....

From the instrumentation point of view, it is clear that the motion detection has been largely underestimated by the measurement method. It might be more appropriate to introduce the "number of occupants" as a parameter in the data processing as well as in the modeling step.

Conclusions

Evidence continues to mount that indoor particles concentrations are one of the major determinants for individual exposure. That is why it is necessary to characterize the source time variability and to assesses the different source impacts on the total personal exposure. Attempts to estimate the source of indoor particles concentrations are complicated because indoor air quality is both building-specific and occupant-specific. Many researchers are using experimental chambers; however quantifying continuous ventilation rates and penetration factors for an occupied building can be tedious, time-consuming and expensive. While most receptor-oriented methods use the pollutant compositions (chemical fingerprint) to apportion the contribution of individual sources, our work focuses on the time variability source characterization, namely "temporal fingerprint". This study outlines the basic temporal characterization and source apportionment using Non-negative Matrix Factorization (NMF).

This study has shown that continuous measurements of indoor particle number concentrations with additional information of occupancy and windows opening are useful for the source apportionment models. The factorization has been successfully employed to study particle number time series fluctuations in an occupied open-plan office. Five distribution profiles were resolved using different initialization and optimization algorithms. Thanks to NMF and correlation analysis, the impact of occupancy and windows states associated with outdoor traffic were identified. The outdoor sources are captured by two components (very likely the traffic impact) with major number modes at 1.3 µm.

Besides, a potential association with primary outdoor pollutants (NO x ) has been captured and a diurnal pattern similar to traffic can be associated. A common pattern between F1 and PN 0.9-1.8

has been identified, such as diurnal profile which is characterized by a significant peak between 8:00 a.m. and 9:30 a.m, decreasing until 7:00 p.m.

Taking into consideration the results presented above, throughout the six-month measurements of indoor and outdoor PN concentrations, it can be concluded the followings:

• This study demonstrates the importance of recording real-time concentrations over a longer duration (i.e., several months with a short time step). The exploitation of such data has the potential to extract and capture the different patterns of temporal variability and their major determinants.

• These results contribute significantly to the very small data set available in the literature on the time variability source characterization.

• The NMF technique -and its variants-shows considerable promise for further application to the indoor environment and the possibility to identify other sources and their contributions.

• Continuous monitoring of climatic parameters as well as active instrumentation of the building (windows, occupancy) are necessary for the evaluation of the total exposure to indoor pollutants.

To sum up, the results of this study entail that the degree of human's exposure to different sources varies with many parameters (building and occupants specific). These sources could be captured using NMF and its variants.

  [59]. The NMF factors were analyzed and connected with their influencing parameters, such as opening windows and occupancy rate of the open-plan office. In other words, we identify patterns that shared by several sources. To date, few research studies have been conducted to identify temporal patterns of different sources. Most of them focus on the chemical fingerprint. The contribution of this study lies in the characterization of the time variability of the most frequent source-patterns in an occupied open office. The interpretation of NMF factors and time variability patterns were supported by correlation analysis as well as by temporal variation visualization.

Figure A. 1 in

 1 supplementary material) located in a suburban area, at 30 km East of Paris. The measurement campaign was conducted in an open-plan office space with a total area of 132 m 2 and a volume of 364 m 3 . Figure1shows the plan of the open space located at the 2 nd floor. A virtual tour of the open space office is given in the supplementary material of this article.

Figure 1 :

 1 Figure 1: Plan of the office space (132 m 2 , 364 m 3 ). The configuration of the tables varies with the number of occupants. An example of motion detection is materialized by two pink half spheres.

  NMF and its generalizations have been used for different purposes such as dimensionality reduction, feature extraction, clustering, blind source separation (BSS), and classification. In this paper, NMF was used only for BSS purposes.In the environmental field, NMF is a new method of a wide range of receptor modeling. It is used to analyze the series of chemical concentration measurements and to find underlying explanatory sources[59,[START_REF] Kfoury | Pm 2.5 source apportionment in a french urban coastal site under steelworks emission influences using constrained non-negative matrix factorization receptor model[END_REF]. Some research papers proposed an extension of the standard NMF form to incorporate some physical proprieties.[START_REF] Limem | Nonnegative matrix factorization using weighted beta divergence and equality constraints for industrial source apportionment[END_REF] proposed an informed NMF with a specific parametrization which involves constraints about some known components of the factorization[START_REF] Limem | Nonnegative matrix factorization using weighted beta divergence and equality constraints for industrial source apportionment[END_REF]. Plouvin et al. (2014) extended the latter work by adding some information provided by a physical dispersion model [50]. Limem et al. (2014) introduced the use of basic equality constraints and have derived theoretical expressions of constrained Weighted NMF (WNMF) to characterize industrial source apportionment of PM 10 [34].

H.

  A good initialization can then sidestep some of the convergence problems, especially for a large input dataset. If the starting values for the algorithm are chosen randomly, every run of the NMF algorithm may find a different local minimum of the objective function. Therefore a reasonable initialization of matrix W and H is necessary and helps the physical interpretation of the obtained patterns [59, 50]. Particular emphasis has to be placed on the initialization of NMF because of its local convergence. Several approaches have then been proposed to choose the appropriate NMF initialization (see [2] for a review). Thiem et al. [59] suggested the use of Non-Negative Double Singular Value Decomposition (NNDSVD, developed in [7]) for PM source apportionment and did not recommend random initialization. Meanwhile, Hutchins et al. [24] claim that performing 30-50 runs for random initialization is considered sufficient to get a robust estimate of the factorization.

Figure 2 :

 2 Figure 2: Diurnal variability of the median value of indoor PN concentrations for the following size ranges: 0.35 µm, 0.9 µm 2.5 µm and 17.5 µm. PN 0.35 = PN [0.30-0.40] , PN 0.9 = PN [8-10] , PN 2.5 = PN [2-3] and PN 17.5 = PN [15.5-19.5] .

Figure 4 :

 4 Figure 4: Relative contributions of the factors to temporal variability of the 15 fractions.

Figure 5 :

 5 Figure 5: Pearson correlation coefficient between NMF factors (H matrix) and the PN concentrations time series of each fraction.

  figures are presented for each component corresponding to the window states (closed or open) and in each figure, two curves (red dot and green triangle) are associated with the occupancy status.

Figure 6 :

 6 Figure 6: Diurnal variation of the five NMF factors and CO 2 concentrations according to windows state and occupancy.

  It turns out that the parameter "opening windows" perfectly discriminates the component 5 (p-value <0.05), but the importance of occupancy is not conclusive. When at least one window is open, the component 5 levels are high. The diurnal profiles of the two components 1 and 5 have comparable patterns, for the case of open windows. As a matter of fact, the relative contribution of F1 and F5 are important when the open-plan is occupied (7 a.m.-7 p.m.).For component 3, a peak at 9 a.m. is observed in the case of open windows and the diurnal profile gives a sinusoidal appearance. On the other hand, N O x emissions vary according to the time of day: they are very high in the morning and late afternoon (the rush hours). A hypothesis can be put forward suggesting that the peak of component 3 is of external origin, related to some sources such as road traffic.

Table 1 :

 1 depends on the size of the particle. Indeed, the CV values are: 113% for PN 0.35 , 80% for PN 0.9 , 92% for PN 2.5 , 250% for PN 8.75 and 750% for PN 17.5 . Descriptive statistics of the selected indoor PN size bins (#.cm -3 for number per cm 3 ) according to occupancy (Occup. or Non Occup.) and windows opening state (Open or Close). The percentile P corresponds to percentage of the total values are the same as or below the measurement concentrations: the 25 th percentile (P 50 ), the 50 th percentile corresponds to the median and the 75th percentile (P75). n represents the number of samples (minutes).

	Descriptive statistics for the five size bins concentrations according to occupancy and windows
	opening state are shown in Table 1. The mean values show very large variations; mean values are

The highest humidity values were recorded during May and June. Fifteen size bins were collected, but only five representative fractions are described in this paper: PN 0.35 = PN [0.30-0.40] , PN 0.9 = PN [8-10] , PN 2.5 = PN [2-3] , PN 8.75 = PN [6.25-12.25] and PN 17.5 = PN [15.5-19.5] . The indoor PN 0.35 varied from 1 to 263 #.cm -3 (#.cm -3 for number per cm 3 ) with a median of 16 #.cm -3 and 90 % (percentile P 90 ) of the values are less than 18 #.cm -3 . There is no significant difference in the monthly levels excepting for March. The range variations for PN 0.9 , PN 8.75 and PN 17.5 are 2611, 119 and 26 #.L -1 , respectively. The variability expressed by means of the coefficient of variation (CV = Standard Deviation/M ean)

  Table 2 shows the Indoor/Outdoor (I/O) ratio for the selected five size bins levels in different configurations. Diurnal time variability of the ratio (I/O) is provided in the supplementary material of this paper (cf. Figure B.1).

	Median I/O ratios are less than 1 for smaller range (≤ 2.5 µm) regardless the windows state or
	indoor occupancy.
	It is worth noting that median I/O ratios of PN during the non-occupancy are very similar to those
	observed for closed windows. In the absence of known indoor sources, the reported median I/O
	ratios have been ranged from 0.37 to 0.53 in the case of opening windows.

Table 2 :

 2 Descriptive statistics of I/O PN ratios for the selected size ranges according to occupancy (Occup. and Non Occup.) and opening windows state (Open and Close). n represents the number of samples (minutes) and P the percentile values.

	I/O ratio	Group	n	Mean(sd)	min.	max.	P25	P50	P75	P90
	PN0.35	Close Open	104393 496	0.49(0.2) 0.73(0.21)	0.008 0.347	4.318 0.363 0.47 0.59 0.71 1.476 0.601 0.7 0.88 1
	PN0.9	Close Open	104393 496	0.57(0.25) 0.76(0.18)	0.002 0.36	7.436 0.425 0.53 0.65 0.81 1.378 0.643 0.76 0.87 1
	PN2.5	Close Open	104393 496	0.59(4.57) 0.62(0.33)	0.0003 0.1	611 2.212 0.376 0.57 0.83 1.07 0.243 0.37 0.57 0.91
	PN8.75	Close Open	104393 496	2.29(6.22) 6(11.33)	0.0004 0.012	311 71	1 0.524	1 1	1 3.72	1.91 21
	PN17.5	Close Open	104393 496	1.17(1.9) 2.43(5.2)	0.002 0.032	201 51	1 1	1 1	1 1	1 11
	PN0.35	Non Occup. Occup.	99902 4987	0.49(0.2) 0.51(0.26)	0.043 0.008	4.318 0.365 0.47 0.56 0.71 4.143 0.342 0.47 0.63 0.8
	PN0.9	Non Occup. Occup.	99902 4987	0.57(0.25) 0.62(0.27)	0.002 0.075	7.436 0.424 0.53 0.65 3.849 0.448 0.58 0.73	0.8 0.9
	PN2.5	Non Occup. Occup.	99902 4987	0.6(4.6) 0.9(4.25)	0.0003 0.008	611 291	0.239 0.36 0.56 0.398 0.59 0.2	0.8 1.4
	PN8.75	Non Occup. Occup.	99902 4987	2.1(5.67) 6.6(12.7)	0.0004 0.001	301 311	1 0.524	1 1	1 11	1 21
	PN17.5	Non Occup Occup	99902 4987	1.15(1.7) 2(3.8)	0.002 0.009	201 71	1 1	1 1	1 1	1 1

Table 3

 3 shows statistical tests performed to determine whether the differences between the average values of all the NMF factors are significant in two situations: (i) when all windows are closed vs. when at least one window is open, and (ii) in the case of occupancy vs. non-occupancy. For a p-value <0.05, the null hypothesis "H 0 : No difference in mean" is rejected. For example, for the first factor, we can say that there is no significant difference when all the windows are closed or at least one window is open (p-value >0.05, gray cells in the table3). Whereas occupancy is an important parameter for component 1 (F1): the variability of the two profiles (occupancy vs.

	non-occupancy) are different (p-value <0.05).

Table 3 :

 3 Significant differences between diurnal profiles to discriminate the role of windows and occupancy parameters using statistical tests.

	p -value	Windows t -test Wilcoxon	Occupancy t -test Wilcoxon
	F1	0.36	0.98	4.69E-06	2.54E-06
	F2	0.0003301 0.0006599 0.0005568	0.0009668
	F3	0.0001015 3.69E-05	0.00245	0.004618
	F4	6.77E-08	1.97E-08	0.1147	0.02706242
	F5	0E-10	0E-10	0.02706242	0.002432

Acknowledgments

This work received financial support from the French research program on air quality (PRIME-QUAL) through the Grant No 12-MRES-PRIMEQUAL-4-CVS-09.