Allison Marie Loconto
email: allison-marie.loconto@inrae.fr

Nadine Arnold
email: n.a.arnold@vu.nl

Governing Value(s) and Organizing through Standards

Keywords: Governance, metrics, organization, standards, valuation

Standards are strongly intertwined with values in economic contexts, which goes far beyond economic value. Standards' diversity is expressed in local spaces where standards are made, put into action, circulate and commensurate. In these mutually linked and globally distributed spaces, we can analyze the ways in which standards and value(s) encounter each other and what consequences this brings for individuals, organizations, communities, and societies. Examining different settings of food production and organization brings new insights into the sociological explanations and understandings of how standards are guided by values and create value(s). These insights highlight new tensions between global and local social dynamics and offer two ways forward for the sociology of standards. First, is the importance of intermediation between the values of the standards and the governance effects of these same standards. Second, is the relevance of valuelessness, where the prioritization of some values devalues others or loses them completely.

Introduction

While standards have been acknowledged as a fundamental form of organizing (Brunsson and Jacobsson, 2000), their interrelation with value(s) is empirically underexamined and undertheorized. The relationship between standards and values needs our attention, because we know that standards infuse value chains with valueladen qualities [START_REF] Ponte | Quality standards, conventions and the governance of global value chains[END_REF], shape businesses and markets globally [START_REF] Djelic | Globalization and Business Regulation[END_REF] and are part of today's moral economy [START_REF] Busch | The Moral Economy of Grades and Standards[END_REF]. Standards are thus strongly intertwined with values in economic contexts (e.g., production, trade, sales, consumption) -and this goes far beyond economic value [START_REF] Beckert | The Worth of Goods: Valuation and Pricing in the Economy[END_REF].

In situations where nature is the basis of economic activity, like agriculture and food systems, standards have a long history of use (Busch, 2011). Here, due to ecological crises (e.g., pollution, loss of species, climate change), standards are mobilized to account for abstract values, such as biodiversity, health, or carbonneutrality. At the same time, standards can refer to values that seem more social, such as equality or fairness, since labor conditions in the production and processing of resources can be problematic. All these various values often culminate under the guiding principle of sustainable development (United Nations, 2021). Sustainability thus brings together many, possibly competing values [START_REF] Constance | Contested Sustainability Discourses in the Agrifood System[END_REF][START_REF] Ratner | Sustainability" as a Dialogue of Values: Challenges to the Sociology of Development[END_REF], that are interpreted and practiced differently depending on the actors involved [START_REF] Adloff | Futures of sustainability as modernization, transformation, and control: A conceptual framework[END_REF]. We can thus assume a locally contingent "patterning of values" [START_REF] Rokeach | Understanding Human Values[END_REF]. We see this patterning of values in the articles that are published as part of this special issue, as a group, the articles address the following questions: What value(s) do standards make visible and which are made invisible? Conversely, how do values shape the development, implementation, and diffusion of particular standards? How are value(s) mobilized as an or ganizing principle of production, agriculture, extraction, or conservation?

The purpose of this introduction is to situate a set of articles that help to better understand the role that standards play in these value-oriented processesand with what societal consequencesby examining different settings of food production and organization around the world.

Standard Value(s)

Value(s) is a topic in sociology that is reemerging since [START_REF] Parsons | The structure of social action: a study in social theory with special reference to a group of recent European writers[END_REF][START_REF] Parsons | The structure of social action: a study in social theory with special reference to a group of recent European writers[END_REF]) superficial demarcation that value was the concern of economics, while values were the domain of sociology [START_REF] Granovetter | The Sociology of economic life[END_REF]; also known as Parson's Pact, see [START_REF] Stark | The sense of dissonance : accounts of worth in economic life[END_REF]. Economic sociology has reclaimed the notion of value over the past 15 years with the emergence of valuation studies [START_REF] Doganova | Valuation Studies and the Critique of Valuation[END_REF]. These studies bring together different strands of sociological theory such as conventions theory, actor-network theory, science and technology studies (STS) and pragmatic sociologyall of which are employed by the articles in this special issue in order to understand the values that are encoded within a range of international standards.

Values are conceptions of the desirable that are learned and guide action. Standards, on the other hand, are formalized rules and material measures (Busch, 2011) As socio-technical devices that appear to be rational and based on expertise [START_REF] Ransom | Science and Standards[END_REF], standards are designed to diffuse worldwide [START_REF] Strang | Institutional Conditions for Diffusion[END_REF]. Their voluntary nature makes it possible that state borders are no obstacle for standards, so that they regulate transnationally [START_REF] Bartley | Institutional Emergence in an Era of Globalization: The Rise of Transnational Private Regulation of Labor and Environmental Conditions[END_REF][START_REF] Djelic | Transnational governance: Institutional dynamics of regulation[END_REF]. However, it would be naïve to believe that standards therefore lead to global homogenization and uniformity [START_REF] Timmermans | A world of standards but not a standard world: Toward a sociology of standards and standardization[END_REF]. Their diversity is expressed in local spaces where standards are made, put into action, circulate and commensurate [START_REF] Chavinskaia | Keep Diversity--Make Standards! Spaces of Standardization for Diversity Analyzed through Cattle Breeding Industry[END_REF][START_REF] Loconto | Standardization as spaces of diversity[END_REF].

In these mutually linked and globally distributed spaces, we can analyze the ways in which standards and value(s) encounter each other and what consequences this brings for individuals, organizations, communities, and society. The articles that compose this special issue bring specific attention to what is pushed into the background, becomes invisible and is not taken into account when standards meet value(s) [START_REF] Lampland | Standards and their stories: How quantifying, classifying, and formalizing practices shape everyday life[END_REF]. As an ensemble of empirical cases and theoretical discussions, the articles that we review here offer new insights into the sociological explanations and understandings of how standards are guided by values and create value(s), which are increasingly demanded in a world charact erized by crises in nature.

Standards making

In the spaces of standards making, it becomes clear that the setting of standards is a fundamentally social, messy act [START_REF] Lampland | Standards and their stories: How quantifying, classifying, and formalizing practices shape everyday life[END_REF]. This raises not only the important question of who are the rule-makers and what power do they have [START_REF] Boström | NGO Power in Global Social and Environmental Standard-Setting[END_REF][START_REF] Cheyns | Multi-stakeholder Initiatives for Sustainable Agriculture: Limits of the "Inclusiveness" Paradigm[END_REF][START_REF] Renard | Fair trade: quality, market and conventions[END_REF], but also which values do (not) guide the process. While some argue for actively considering values as orientations, as for example envisaged in the case of value sensitive design [START_REF] Friedman | Value Sensitive Design: Shaping Technology with Moral Imagination[END_REF], [START_REF] Thévenot | Postscript to the Special Issue: Governing Life by Standards A View from Engagements[END_REF] reminds us that s tandards inevitably imply evaluations at the expense of some others. That is also why , we should take seriously the recent call for increased engagement with detachment, which studies how we lose touch with some value(s) [START_REF] Goulet | L'innovation par retrait. Contribution à une sociologie du détachement[END_REF][START_REF] Brembeck | Letting go: Economies of detachment[END_REF]. [START_REF] Renard | Values and the making of standards in "sustainable" coffee networks. The case of 4C and Nestlé in México[END_REF] explores the making of standards for sustainable coffee in Mexico. She explores how Nestlé, the largest buyer of certified coffee in the world, has participated in the making of a private standard for coffee quality (the 4C standard).

Here, the values that are encoded in the standard focus on a tripartite understanding of sustainability: economic, environmental and social. The logic that underpins the design of the standard is a supply chain approach, whereby the most v aluable benefit from applying the standard will be supply chain security. [START_REF] Renard | Values and the making of standards in "sustainable" coffee networks. The case of 4C and Nestlé in México[END_REF] argues that the consequences of designing the standard in such a way is that the reality of practice is not in line with the values set out in the standard. Her argumen t is that this gap between the vision and the practice is not due to farmers not knowing how to comply with the standard, but rather there is an obligatory decoupling between the values and the way that Nestlé manages the standard as part of its supply cha in logic (see [START_REF] Busch | Performing the economy, performing science: from neoclassical to supply chain models in the agrifood sector[END_REF]. Farmers are not actually paid fair prices for their products because despite the claims to autonomy, they are reliant upon selling only t o Nestlé's intermediaries, Nestlé in turn sets the prices that create value for the company. The seedlings and suggested environmental practices end up reducing biodiversity, increasing the risk of disease and increase deforestation because of a narrow framing of environmental sustainability that allows for monocropping. [START_REF] Blancaneaux | Seasonality as value(s) in organic farming: On the conflict on heating greenhouses in France[END_REF] studies the process of making the value of a "season" within organic standards for tomato production in France. He explores how the modern, industrial food system has devaluedsymbolically and economicallythe seasonal quality of food through the use of technologies and international trade among different agro-ecological zones. While local food movements have begun to promote the value of seasonality in their attempts to shorten supply chains. Blancaneux explores these tensions within organic farming around the practice of heated greenhouse grown tomatoes, which are simultaneously local and de -seasonalized. He illustrates how different organizational forms in and of the market create spaces of diversity where the season is either valued or devalued depending on the form of organization proposed by value chain actors. Nonetheless, both relocalization processes result in differentiated standardization of the value of " seasonal" tomatoes, but in the end neither necessarily remove inequalities in global value chains.

Enacting standards

When standards are put into action, they construct value(s). Given the abstractness of standards and the fact that they can organize at a distance, their implementation requires investments from the rule-adopters. They translate the standards into local practices [START_REF] Arnold | Serving Magically Perfect Fruit Globally: Local nesting in translating multiple standards[END_REF], while reorganizing their activities [START_REF] Bingen | Agricultural standards: The shape of the global food and fiber system[END_REF][START_REF] Henson | Understanding the complexities of private standards in global agri-food chains as they impact developing countries[END_REF]. Thus, we can ask to what extent standards can be part of an "active society" [START_REF] Etzioni | The active society a theory of societal and political processes[END_REF] where values are realized without reifying humanity? This question is pressing because the implementation of standards is usually enforced with forms of surveillance, such as audits and certifications (Arnold, 2022;[START_REF] Gustafsson | How Standards Rule the World: The Construction of a Global Control Regime[END_REF][START_REF] Loconto | Standards, Techno-economic Networks, and Playing Fields: Performing the Global Market Economy[END_REF], and more recently digital means of control [START_REF] Herlin | Animal Welfare Implications of Digital Tools for Monitoring and Management of Cattle and Sheep on Pasture[END_REF]. The impact of the different forms of surveillance is often unclear, but we know, for example, that audits can erode democratic, professional values locally [START_REF] Shore | Audit Culture Revisited: Rankings, Ratings, and the Reassembling of Society[END_REF][START_REF] Sloan | I'll See: How Surveillance Undermines Privacy by Eroding Trust[END_REF].

Sánchez-García et al., (2022) and [START_REF] Moreno | Health as an emerging value in the construction of quality of berries in Huelva (Spain)[END_REF] both examine the application of the GlobalG.A.P. standard in different agricultural production systems in Spain. These two articles demonstrate how the GlobaG.A.P standard, which was originally developed as a stepwise approach to encourage farmers to adopt good agricultural practices, does this by prioritizing the values of modernity. In this standard, the "good" practices are those that make rational use of synthetic inputs and are focused mainly on ensuring that the food that is produced is not toxic for human consumption. In the case of the Mar Minor (small sea) near Murcia, once this standard is put into action in fresh fruit and vegetable production, the value of profitability is prioritized over a nature that is uncontrollable and unpredictable.

The consequence is first ecological collapse as the unique inland lagoon in the Mediterranean becomes home to a massive phytoplankton bloom feeding on the nitrates and phosphates that have leached from the inland intensive farming operations, and second a socio-cultural collapse as the tourism industry suffers. The nature that is valued through a bureaucratic process of managing human and nonhuman agents renders it abstract and detached from the territorial reality. [START_REF] Moreno | Health as an emerging value in the construction of quality of berries in Huelva (Spain)[END_REF] look at the same standard in action in strawberry production near the Atlantic coast of Southern Spain. The criteria established by Gl obal G.A.P. standard focuses almost exclusively on consumers' health while neglecting the health costs and inequalities experienced by female agricultural workers. In this way, healthiness is an emergent value of Huelva berries. However, the people who benefit from this value are not the women who are principally responsible for creating it, instead, they are valued as potential contaminating agents. In this case, we see a completely different value of "seasonal" emerge from industrial agriculture where a risk framing that values food safety for consumers prevents one class of workers from being equitably healthy.

Raynolds (2022) picks up the concern of worker welfare by examining the Fairtrade International standard in action in Ecuadorian flower plantatio ns. The values of 'trade fairness' and 'worker empowerment' are examined, but these values are not reproduced in action. Fairtrade has only marginally reduced floral buyer power due to retailer resistance, low-bar certification competition, and programmatic regulatory gaps. In other words, producers are not seeing an increased economic value from the sale of the flowers. In terms of the empowerment of workers, most actions are focused on individual empowerment through training and strengthening of workers' committees. This individualisation of the collective value of empowerment has, nonetheless, brought about better working conditions for Fairtrade certified plantation workers when compared to other plantation workers in Ecuador. The challenge is that this empowerment is conditional upon the plantation management's decision to continue to value the Fairtrade certification, which will only occur if it remains an attractive reputational and profitable marketing tool for reaching distant consumers. [START_REF] Schermer | Private standards for animal welfare in Austrian dairy husbandry: Consequences for farmers in mountain regions[END_REF] examines the welfare of animals in dairy farming in Austria. Here, consumers value dairy products where the cows are pastured and there are good human-animal relations in small farms. What Schermer documents is a slow, insidious push from retailer standards to remove the traditional approach of tethering in the winter and pasturing in the summer in favor of free stalls where 'sufficient space' is prioritized as the value for cow welfare. This value is remunerated in terms of extra cents per liter by the only two retail chains that purchase products from all across the country. The unintended consequence of such a valuing is widespread change in terms of farm size, structural change and land use, and a change in the cultural landscape whereby the summer tourism suffers from a lack of cows in pastures. These structural shifts are embodied in the shift from cow farmers to dairy farmers who are focused on valuing economies of scale, rather than the welfare of the animals and the humans who are forced to leav e profitable markets.

All four of these articles demonstrate how international standards struggle to build sustainable relationships between agriculture designed for distant consumers and the nature (human and non-human) of the territory. In other words, when standards meet values, this does not necessarily lead to a win -win situation in everyday life, but creates new problems, challenges and misfires.

Circulating standards

The interactions of standards with rule-makers, rule-intermediaries and ruleadopters [START_REF] Abbott | Theorizing Regulatory Intermediaries: The RIT Model[END_REF] are often the basis for further circulation of standards into other spaces [START_REF] Loconto | Standardization as spaces of diversity[END_REF]. For example, the intrusion of transnational standards into specific locales provoke the making of new, local standards [START_REF] Foley | The territorialization of transnational sustainability governance: Production, power and globalization in Iceland's fisheries[END_REF][START_REF] Schouten | The emergence of Southern standards in agricultural value chains: A new trend in sustainability governance[END_REF]. Or competitive relationships form between the multiple standards, which then lead to harmonization endeavors [START_REF] Loconto | Politics of Private Regulation: ISEAL and the Shaping of Transnational Sustainability Governance[END_REF] or to the formation of new markets [START_REF] Fouilleux | Dans les coulisses des labels: Régulation tripartite et marchés imbriqués[END_REF][START_REF] Reinecke | The emergence of a standards market: Multiplicity of sustainability standards in the global coffee industry[END_REF]. Moreover, standards also circulate in official spaces of the State when interacting with public regulations [START_REF] Bartley | Transnational governance as the layering of rules: Intersections of public and private standards[END_REF]. The extent to which values exert influence on these processes and how values are co-constituted through standards' circulation, is still understudied.

Feur (2022) and Sekine (2022) examine the circulation of international standards that have been designed specifically to value cultural and environmental heritage of territories. Feur takes on geographical indications (GI) as an international standard that values a European logic of protecting intellectual property and 'traditional' practices, particularly the food processing techniques of well-defined geographic regions. By tracing the history of how GIs were introduced into international intellectual property regimes, then into Cambodia and Japan, we learn of a neocolonial approach to governing trade in traditional products that are desired by consumers in the former colonial powers and where technocracy emerges as a federating value for registering GIs around the world. Feur argues that the standardized and piloted approach to the spread of GIs has resulted in a crowding out of useful pre-existing mechanisms of heritage protection that existed in East Asian legal systems. A second effect has been the displacement of exchanges that were embedded in local, moral economies to international trade regimes that value individual GI products that are successfully sold in Europe. A lack of reflexivity in the approach alienates the GIs from the cultural values they are supposed to protect.

Sekine ( 2022) looks instead at a more recently developed standard called the Globally Important Agricultural Heritage Systems (GIAHS). Created by the FAO in 2002, these standards value traditional agro-ecosystems and associated biodiversity, outstanding landscapes and cultural heritage. Sekine explores how the circulation of this standard into Japan via the intermediation of the Ministry of Agriculture, Forestry and Fisheries (MAFF) facilitates of process of neo-liberalisation of the standards through its interpretation into a new national level standard. Because Japanese consumers do not recognize, and thus don't value, the GIAHS standard, the ministry has authorised the use of other standards such as GIs, Slow Food Presidia and other territorial labels in order to help promote the sale of GIAHS products. The consequence of this approach, we learn, is the tendency of large farmers to dominate the system. This dominance emerges as there is a culture of inclusiveness in rural communities where they prefer to offer a diversity of economic opportunities in order to preserve the community connections. This approach has resulted in a shift in production from small-scale polycultures to graindominated monocultures that could have two effects: a reduction in the biodiversity that was meant to be protected and the unaffordability of the food that is produced in the GIAHS for local consumers. The valuing of economy over community poses challenges to the use of standards for rural development.

The consequences of governing value(s) and organizing by standards

The articles included in this special issue demonstrate that nature is 'valued' and codified into hierarchies of value within all these spaces of standardization with consequences for the use of standards to organize production, sectors, industries, value chains, built environments and nature. Specifically, values can be extended, specified and explicated as well as destroyed or ignored through the use of standards. Put differently, values are not given, but are done [START_REF] Kjellberg | Valuation Studies? Our Collective Two Cents[END_REF][START_REF] Muniesa | A flank movement in the understanding of valuation[END_REF].

Examining the ways in which standards are deployed to construct and back up values promises insights into the dynamic linkage of the local to the global and vice versa [START_REF] Congretel | Found again in translation? Standardizing the authenticity of guaraná among the Sateré-Mawé people (Brazilian Amazon)[END_REF]. By studying the intersection of value(s) and standards, the SI addresses tensions between global and local social dynamics in an innovative way. Standards are a phenomenon that is a driver and expression of a globalized world, although their consequences are primarily experienced in locally, situated spaces. Neither on a global nor on a local level has the interplay of standards with values received any dedicated attention thus far, particularly not in eco nomic, political or organizational sociology.

The articles in this special issue offer two interesting ways forward in a sociology of standards. First, a number of the articles discussed the importance of intermediation between the values of the standards and the governance effects of these same standards. Recent efforts have been made to understand regulatory intermediation through standards [START_REF] Abbott | Theorizing Regulatory Intermediaries: The RIT Model[END_REF], but the articles in this special issue suggest that intermediation is not necessarily occurring through the delegation of authority, but rather through network reconfigurations as standards move from one space to the next. STS-inspired approaches to intermediationthat take into account the human and non-human intermediaries -could be usefully theorized in order to improve our understanding of governing by standards. The second refers to an emerging issue in valuation studies, which is the notion valuelessness ( Arnold, 2022;Greeson et al., 2020). Through the invisibilities explored in this special issue, we see that in the prioritization of some values, others are devalued or lost completely. We see this in the unintended consequences like deforestation, inflation of food prices and marginalization of seasonal female workers. Future research could usefully take up this concept in order to better understand how different forms of organization and valuing might change what loses value and how other values might fill that void.

Author biographies

Allison Marie Loconto, PhD and HDR in sociology, is Research Professor at the French Research Institute for Agriculture, Food and the Environment (INRAE) and Deputy Director of the Interdisciplinary Laboratory for Science, Innovation and Society at Gustave Eiffel University. Her research focuses on the governance of transitions to sustainable food systems, specifically on the metrics, models, standards and systems of certification that are part of emerging institutional innovations.

Nadine Arnold, PhD in sociology, is Assistant Professor of Organisational Theory at the Vrije Universiteit Amsterdam. Her research interests include organisation and institutional theory, economic sociology, and sociology of standards. Empirically, she examines organization and valuation in the context of sustainability, paying particular attention to the role of standards and waste.

Résumé

Les normes sont fortement imbriquées avec les valeurs dans les contextes économiques, ce qui va bien au-delà de la valeur économique. La diversité des normes s'exprime dans les espaces locaux où les normes sont élaborées, mises en pratique, mises en circulation et adaptées. Dans ces espaces mutuellement liés et distribués à l'échelle mondiale, on peut analyser comment les normes et les valeurs se rencontrent et les conséquences que cela entraîne pour les individus, les organisations, les communautés et les sociétés. L'examen de différents contextes de production et d'organisation alimentaires apporte de nouveaux éclairages sur les explications et interprétations sociologiques relatives à la manière dont les normes sont guidées par les valeurs et créent de la valeur/des valeurs. Ces perspectives mettent en lumière de nouvelles tensions entre les dynamiques sociales mondiales et locales et offrent deux pistes de réflexion pour la sociologie des normes. La première concerne l'importance de l'intermédiation entre les valeurs des normes et les effets de gouvernance de ces mêmes normes. La seconde se rapporte à la pertinence de l'absence de valeur, lorsque la priorité accordée à certaines valeurs en devalue d'autres ou les élimine complètement.

Mots-clés valuation, gouvernance, métriques, normes, organisation

Resumen

Los estándares están fuertemente entrelazados con los valores en los contextos económicos, lo que va mucho más allá del valor económico. La diversidad de los estándares se expresa en los espacios locales donde se elaboran, ponen en práctica, circulan y se adecúan estos estándares. En estos espacios mutuamente vinculados y distribuidos globalmente, podemos analizar las formas en que los estándares y los valores se encuentran entre sí y qué consecuencias tiene esto para los individuos, las organizaciones, las comunidades y las sociedades. A través del análisis de diferentes escenarios de producción y organización de alimentos se aportan nuevas ideas sobre las explicaciones e interpretaciones sociológicas de cómo los estándares están guiados por valores y crean valor(es). Estas ideas ponen de manifiesto nuevas tensiones entre las dinámicas sociales globales y locales y ofrecen dos caminos para avanzar para la sociología de los estándares. El primero sería la importancia de la intermediación entre los valores de los estándares y los efectos en la gobernanza de estos estándares. En segundo lugar, estaría la relevancia de la falta de valor, en la que la priorización de unos valores devalúa otros o los elimina por completo. Palabras clave estándares, gobernanza, métricas, organización, valoracion


Antoine Baudiquez

Éric Lantz

Enrico Rubiola

Franc ¸ois Vernotte

The Statistics of the Cross-Spectrum and the Spectrum Average: Generalization to Multiple Instruments

Keywords: Bayesian statistics, Monte Carlo simulation, confidence interval, cross-spectrum, spectrum average, Karhunen-Loève transform, QR decomposition, characteristic function, probability density function

This article addresses the measurement of the power spectrum of red noise processes at the lowest frequencies, where the minimum acquisition time is so long that it is impossible to average on a sequence of data record. Therefore, averaging is possible only on simultaneous observation of multiple instruments. This is the case of radio astronomy, which we take as the paradigm, but examples may be found in other fields such as climatology and geodesy. We compare the Bayesian confidence interval of the red-noise parameter using two estimators, the spectrum average and the cross-spectrum. While the spectrum average is widely used, the cross-spectrum using multiple instruments is rather uncommon. With two instruments, the cross-spectrum estimator leads to the Variance-Gamma distribution. A generalization to q devices is provided, with the example of the observation of millisecond pulsars with 5 radio telescopes.

I. INTRODUCTION

T he term red noise refers to a variety of processes sharing the property that the power spectral density (PSD) grows at low frequency as 1/f 2 (Brownian noise) or 1/f α , with α > 2. We are interested in the estimation of the PSD of such random signals out of the background noise of the instrument in the specific case of very slow phenomena, which take too long acquisition time for the average on a sequence of data sets to be viable. Therefore, averaging out the background is possible only by exploiting simultaneous measurements of the same signal taken with multiple instruments, under the obvious hypothesis that they are independent. The frequency stability of the millisecond pulsars is the example we have in mind. Such rapidly rotating neutron stars, emitting highly stable periodic pulses out of the magnetic poles, rival the best atomic clocks [START_REF] Rawley | Millisecond pulsar psr 1937+21: A highly stable clock[END_REF]- [START_REF] Manchester | Millisecond pulsars, their evolution and applications[END_REF]. Among other fields, slow phenomena are found in climatology [START_REF] Hansen | Paleoclimate implications for human-made climate change[END_REF] and geodesy, the latter nowadays measured with Very Large Baseline Interferometry [START_REF] Nothnagel | Elements of geodetic and astrometric very long baseline interferometry[END_REF].

A. Baudiquez With the purposes stated in mind, we compare the efficiency of the spectrum average (s.a) and with the cross-spectrum (cs) measuring the signal with q instruments simultaneously. The s.a estimator is the average of the q observed spectra S i , weighted with the background noise σ 2 N,i of the i-th instrument. The c-s method is the average of the all combinatorial choices of the cross-spectrum S j,i , i = j. The s.a is the classical estimator used in these cases [START_REF] Tiuri | Radio-telescope receivers[END_REF], while the c-s is rather uncommon. Data are analyzed with the Bayesian statistics, also known as the inverse problem, which consists of estimating the most probable value of the signal (the slowest spectral components) from the experimental outcomes and their statistical properties. We take the 95% upper limit as the efficiency criterion. Accordingly, the most efficient estimator is the one that provides the most stringent upper limit with the same data set. Our previous article [START_REF] Baudiquez | Cross-spectrum measurement statistics: Uncertainties and detection limit[END_REF] shows that the Variance-Gamma (V Γ) distribution is the exact solution for the probability density function (PDF) of the cross-spectrum in the case of two instruments. We generalize the result to the case of the cross spectrum of q instruments, each with its own background noise σ 2

Ni , assessing the confidence interval on the signal level σ 2 R . Of course, the PDF is no longer a V Γ, and can only be calculated numerically. The case of equally noisy instruments is simpler, and at first sight similar to that of q = 2, but it has no analytical solution. We run a simulation with up to five instruments, inspired to the LEAP experiment [START_REF] Bassa | Leap: the large european array for pulsars[END_REF]. Such experiment gathers the five largest European radio telescopes (RTs) in order to increase the sensitivity of high-precision pulsar timing. Interestingly, Pulsar Timing Arrays seem a promising option to explore the low-frequency gravity waves crossing our Galaxy [START_REF] Verbiest | Pulsar timing array experiments[END_REF], [START_REF] Goncharov | On the evidence for a common-spectrum process in the search for the nanohertz gravitational-wave background with the parkes pulsar timing array[END_REF]. The simulation shows that the s.a is by a small amount more efficient than the c-s, chiefly when the background exceeds the signal. Indeed, this depends on the numerical values. In the end, the use of both estimators may be a wise choice.

II. STATEMENT OF THE PROBLEM A. Spectral Measurement

Let us consider a red noise r(t) which is measured by q independent instruments as shown in Fig. 1. We assume that each instrument adds a white noise n i (t) to the measurement and that all these white noises are uncorrelated. The output of each channel is then Each RT adds a white noise to the output x(t) whose Fourier transform is X(f ). Then the estimate S is computed. where the subscript i corresponds to the i-th instrument, ↔ stands for the Fourier transform and inverse Fourier transform pair, lower case is time domain, upper case is frequency domain, and the variables t and f are implied. Let us remind that the Fourier transform of a white noise is a white noise, at least for sampled signals. Indeed even if continuous pure white noise have an infinite power, the Fourier transform for discrete simulation can be defined. A realistic white noise corresponds to a Markov process of the first order, more details about colored noise are given in [START_REF] Kasdin | Discrete simulation of colored noise and stochastic processes and 1/f power law noise generation[END_REF]. On the other hand, a red noise can be described as a filtered white noise. Its spectrum is then the product of a white spectrum by a deterministic function; so the random part of a red noise is uncorrelated for each frequency bin. Consequently in term of random variable, working in the frequency domain gives a precious advantage because the Fourier components (frequency bins) are statistically independent unlike the time data.

x i = r + n i ↔ X i = R + N i (1
f 0 log[|X i (f)| 2 ] log(f) Periodogram PSD
In the following we focus solely on one frequency bin, thanks to energy equipartition it follows,

V [N i ] = 2V [ [N i ]] = 2V [ [N i ]] = σ 2 N,i V [R] = 2V [ [R]] = 2V [ [R]] = σ 2 R ( 2
)
where

V[•], [•], [•]
respectively denote the variance, the real and imaginary part of the quantity within the brackets.

B. Periodogram and Power Spectral Density

First, let us recall some basics of frequency analysis. Using a data record of duration T sampled at a suitable frequency, the periodogram is

P x (f ) = 2 T |X(f )| 2 , f > 0 (3)
where the factor "2" is needed for energy conservation after deleting the negative frequencies. The expectation of the periodogram is the Power Spectral Density (PSD),

S x (f ) = E 2 T |X(f )| 2 , f > 0. (4)
Figure 2 shows the periodogram and the PSD. We estimate the PSD as the average periodogram, with the ultimate goal of expecting the red noise parameters of r out of the measurement noise n. Of course r is the same for all instruments, while the n i are specific to the i-th instrument and its environnement. The total duration of the experiment is the major problem, as the lowest frequency of interest sets T . In turn, a long T goes with a small number p of averages because the total duration of the experiment is pT . In this paper we focus on the slowest red noise phenomena, up to years, for which we have to set p = 1. In other words, the phenomena of interest are so slow that we cannot average on multiple acquisitions.

C. Estimators

We are now focusing on one bin of the periodogram of a single simultaneous measurement with q instruments, e.g. f 0 as represented on Fig. 2. Let us emphasize on the term periodogram which designates a unique realization of the red noise since all instruments observe this red noise realization at the same time. Nevertheless, taking into account the uncorrelated white noises coming from the instruments, we have to deal with the PSD S. One bin of S represents the power in a given bandwidth, i.e. the 2-nd central moment, or variance. Hereinafter, we work on a generic bin, thus S(f ) at that frequency is replaced with σ 2 . Because the N i are all different, it is appropriate to use a weighted average, where the weights α i are to be found for the optimum detection of R. We denote the estimates with a "hat", then

µ = q i α i X i q i α i , ( 5
)
where q is the number of instruments. The variance of the estimate µ is

V [ µ] = q i α 2 i σ 2 N,i + σ 2 R [ q i α i ] 2 . ( 6
)
An optimal choice is obtained by solving,

∂V [ µ] ∂α i = 0 (7)
which leads to the solution,

α i = 1 σ 2 N,i . (8)
Therefore the inverse-variance weighted average, described in [START_REF] Hartung | Statistical Meta-Analysis with Applications[END_REF] with applications examples, has the least variance among all weighted averages. Then Eq. ( 6) becomes

σ 2 µ = V [ µ] = q i 1 σ 2 N,i -1 . (9)
Let us define now the two estimators of interest: the spectrum average weighted by the noise variance σ 2 N,i and the crossspectrum,

S sa = σ 2 µ q i X i σ 2 N,i 2 + σ 2 µ q i X i σ 2 N,i 2 S cs = X i • Xj m with i = j. (10)
Moreover σ 2 µ corresponds to the noise weight normalization factor defined in Eq. ( 9). Finally • stands for the m average over the different combinations of instruments with m = q 2 and • stands for the complex conjugate of the quantity which is below. Furthermore we have omitted in Eq. ( 10) the measurement time factor 2 T which is necessary to have the dimension of a power per frequency for a better readability thereafter. In addition, only the random part has a direct influence on the probability density function. Denoting E[•] the mathematical expectation of the quantity within the brackets,

     E S sa = σ 2 R + σ 2 µ E S cs = σ 2 R ( 11
)
which means that the spectrum average estimator is biased.

Usually one removes the bias to have the s.a estimate average over realizations which tends towards the sought signal level σ 2 R . This gives a clear advantage to the c-s estimator. However, we will see that the computation of the confidence interval over the signal level σ 2 R requires an estimation of this bias σ 2 µ whatever the chosen estimator, s.a or c-s. Therefore we want to estimate the PSD and we assume it follows a 1/f α power law, then we only have to estimate a level and exponent of the first frequency bins. We now compare the estimator defined in Eq. ( 10) by determining their variance. We can demonstrate provided that ∀i,

σ 2 N,i = σ 2 N (see Annexe A), V S cs ≈      V S sa if σ 2 R σ 2 N q q-1 V S sa if σ 2 R σ 2 N . (12)
This is confirmed by Fig. 3 which exhibits the variance of the estimates of both estimators applied to a signal composed of a mixture of uncorrelated white noise of level 1 arbitrary unit (a.u.) and a common f -4 noise of level 4096 a.u. for 2 instruments. Therefore the variance decreases in f -8 and Fig. 3 compares these variances to the square of the PDF. At f = 4 a.u., the signal PSD is 16 times higher than the white level and therefore its square is 256 times higher. In this case, the variances of both estimators coincide. On the other hand, for frequencies higher than 16 a.u., the signal PSD is less than 16 times lower than the white level (256 for their squares) and the variance of the c-s estimates is 2 times higher than the variance of the s.a estimates. This seems to indicate a better efficiency of the s.a estimator. Indeed the spectrum average estimator is a sufficient estimator which means of minimal variance. However what about the pdf of the estimates knowing the parameter σ 2 R for a given frequency?

III. PROBABILITY DENSITY FUNCTION

A. Spectrum Average Method

The spectrum average estimator leads to the following χ 2 distribution with 2 degrees of freedom resulting from the real and imaginary part of the spectrum,

p( S sa |σ 2 R ) = e -Ssa 2σ 2 2σ 2 (13)
where,

σ 2 = 1 2 σ 2 µ + σ 2 R . ( 14
)
where σ 2 µ is the weighted noise level according to Eq. ( 9) and σ 2 R the signal level of interest.

B. Karhunen-Loève Transform

The KLT method, denoting to the Karhunen-Loève transform, has been developed in [START_REF] Lantz | KLTS: A rigorous method to compute the confidence intervals for the three-cornered hat and for Groslambert covariance[END_REF]. It uses the statistics of the data themselves instead of the statistics of the estimates. This method has the advantage to combine linearly independent Gaussian estimates. Furthermore it also forms a sufficient statistics like the s.a method. It is based on determining the covariance matrix M associated to the real or imaginary part of the measurement X i obtained by the q instruments,

     M ii = 1 2 σ 2 N,i + σ 2 R M ij = 1 2 σ 2 R with i = j (15)
where the extra factors 1/2 come from Eq (2). This covariance matrix has to be diagonalized and we denote the eigenvalues λ i . Their associated normalized eigenvectors are V i and the pdf is then given by

p( S KLT |σ 2 R ) = q i=1 1 (2πλ i ) ν/2 e - ν j=1 w 2 ij 2λ i (16)
where j highlights the real and imaginary part obtained through the Fourier transform therefore ν = 2. Let us remind that X corresponds to the matrix containing the set of Fourier transform of the measurements at the output of each instrument. The numerator of the exponential argument is then

w = X • V ( 17
)
where V are the eigenvectors obtained from the diagonalized covariance matrix.

C. Cross-spectrum

The cross-spectrum estimator leads to the variance-gamma (VΓ) distribution for 2 instruments as described in Section III in [START_REF] Baudiquez | Cross-spectrum measurement statistics: Uncertainties and detection limit[END_REF] but for more than 2 instruments it is no longer the case. Having no exact solution known nowadays, we give an approximation of it. The process is the same until the establishment of the χ 2 linear combination. First we perform an orthonormalization by using the Householder transformation to define a basis of unit vectors that are orthogonal to each other. We define W the matrix where each column contains the standard deviation of the spectrum according to Eq. ( 1) as

W = 1 √ 2        σ N,1 0 . . . . . . 0 0 σ N,2 0 . . . 0 0 0 σ N,3 . . . 0 . . . . . . . . . . . . σ N,q σ R σ R σ R . . . σ R        . ( 18
)
All the measurement noises are independent, as assumed, whereas the signal is common. Then W is projected onto the orthogonal basis and we compute the eigenvalues λ j of the resulting components. This leads to a linear combination of χ 2 distribution as follows,

S cs = q j λ j χ 2 k ( 19
)
where k is the number of degrees of freedom corresponding to each eigenvalue, e.g. equal to 2 for the real and imaginary part without degeneration. We respectively used the DGEQRF and DSYEV LAPACK subroutine to perform the orthonormalization and compute the eigenvalues. In the special case of 2 instruments we obtain the subtraction of two χ 2 random variables with the same number of degrees of freedom. The characteristic function of the χ 2 k distribution is defined as

φ j (t) = (1 -2iλ j t) -k/2 ( 20
)
where i is the imaginary unit and we apply a variable change of -t for the negative eigenvalues. The χ 2 distributions according to Eq. ( 19) being independent, the characteristic function of the c-s becomes

φ(t) = q j φ j (t). (21)
It leads to the moment generating function of the VΓ distribution for 2 instruments but it is no longer the case for more instruments. When all the instruments have the same level of intrinsic noise σ 2 n , the diagonalization of the matrix W defined by Eq. (18) leads to two eigenvalues. One is unique and the second one has a degeneration of q -1 with q the number of instruments. Consequently, it leads to the difference of two χ 2 random variables with different degrees of freedom. However even if it looks like the case with 2 instruments, the difference in the degrees of freedom of the χ 2 distributions has no analytical solution. Therefore the probability density function of the c-s for any noise level is defined as

p( S cs |σ 2 R ) = 1 2π R e -it Scs φ(t)dt. ( 22
)
We perform the integration by using the Simpson method only on the positive side because the real part of this function is even whereas the imaginary part is odd. Figure 4 shows that the theoretical probability density function fits very well the histogram obtained by 10 7 Monte Carlo simulations for 5 instruments. The variance of each white noise is the same σ 2 N = 10 a.u. whereas the signal level is σ 2 R = 6 a.u.

D. Bayesian inference 1) A posteriori distribution:

We seek to determine a confidence interval on σ 2 R , but Eq. ( 13), ( 16) and ( 22) define the pdf of a set of measurement X given the sought parameter σ 2 R . So we have to solve the inverse problem which means to determine the pdf of σ 2 R given a set of measurement X called the posterior distribution. The Bayes theorem leads to the following relation,

p(σ 2 R |X) ∝ p(X|σ 2 R ) • π(σ 2 R ) ∞ 0 p(σ 2 R |X)dσ 2 R = 1 (23)
where π(σ 2 R ) is the prior, i.e. the pdf before any measurement. One of the main issue of Bayesian analysis concerns the choice of this prior.

2) Choice of the prior: In order to be as general as possible, we will assume a total ignorance of the signal level. In such a case, it is generally considered that any order of magnitude has the same probability which suggests a constant prior in a logarithmic scale, i.e π σ 2 R = 1/σ 2 R . However, our perfect knowledge of the noise level induces an implicit scale factor. In other words, since we did not remove the "bias" σ 2 µ in Eq. ( 11), the s.a estimator is shifted by σ 2 µ . In a very similar case [START_REF] Mchugh | Pulsar timing and the upper limits on a gravitational wave background : a Bayesian approach[END_REF], we decided that the true parameter should be the sum of both levels θ = σ 2 µ + σ 2 R . Moreover according to Eq. ( 9) higher noise will have lower weight and in our case, since the mathematical expectation of the s.a estimator is σ 2 µ + σ 2 R , it comes naturally that the true parameter should be:

θ = σ 2 µ + σ 2 R . (24)
From these considerations, we will choose

π(θ) = 1/θ = 1 σ 2 µ +σ 2 R
and then, our prior for the s.a estimator will be

π(σ 2 R ) ∝ 1 σ 2 µ + σ 2 R . (25)
In order to be fair in the trial of c-s against s.a, the same prior will be used for both estimators.

In the following we will compare the different methods, starting with the spectrum average and KLT in Sec. IV.

IV. SPECTRUM AVERAGE AND KLT COMPARISON

A. A particular case: all the instruments have the same variance

Let us define ∀i, σ 2 N,i = σ 2 N , i.e. all the q instruments have the same noise level. At a first step we determine the s.a pdf, in this case according to Eq. ( 9) and ( 2), the variance defined by Eq. ( 14) leads to the following expression,

σ 2 = 1 2 σ 2 N q + σ 2 R . (26)
From Eq. ( 10), the estimate S sa now becomes

S sa = σ 2 µ q i X i σ 2 N,i 2 + σ 2 µ q i X i σ 2 N,i 2 = 1 q 2   q i X i 2 + q i X i 2   (27)
According to Eq. ( 13), the s.a pdf is given by

p( S sa |σ 2 R ) = e - 1 q 2 { [ q i X i ] 2 + [ q i X i ] 2 } σ 2 N q +σ 2 R σ 2 N q + σ 2 R . (28)
In a second step let us define the KLT pdf. The eigenvalues of the covariance matrix resulting from Eq. ( 15) are given by

λ 1 = 1 2 σ 2 N + qσ 2 R λ i = 1 2 σ 2 N with i ∈ {2, ..., q} (29)
The first and highest eigenvalue being the only one to depend of σ 2 R , we solely define its associated eigenvector

V 1 = J q,1 √ q (30)
where J q,1 is the all-ones column vector. Then the numerator in the exponential in Eq. ( 16) is

ν j ŵ2 1,j = ν j [X j • V 1 ] 2 = 1 q ν j [X j • J q,1 ] 2 = 1 q ν j [ q i X ij ] 2 = 1 q [ q i X i ] 2 + [ q i X i ] 2 . ( 31
)
The KLT pdf defined by Eq. ( 16) is given by

p( S KLT |σ 2 R ) = C e - 1 q { [ q i X i ] 2 + [ q i X i ] 2 } σ 2 N +qσ 2 R π (σ 2 N + qσ 2 R ) ( 32
)
where C is the Gaussian remaining product with a variance depending only on the measurement noise level. However what we want to characterize is not the estimates but the parameter σ 2 R . According to Eq. ( 23), the pdf of the true parameter σ 2 R is proportional to the prior π(σ 2 R ) multiplied respectively by Eq. ( 28) and (32) for the s.a and KLT estimates. The Bayes theorem leads then to

p(σ 2 R | S sa ) ∝ π(σ 2 R ) e - 1 q { [ q i X i ] 2 + [ q i X i ] 2 } σ 2 N +qσ 2 R σ 2 N + qσ 2 R ( 33
)
and

p(σ 2 R | S KLT ) ∝ π(σ 2 R ) e - 1 q { [ q i X i ] 2 + [ q i X i ] 2 } σ 2 N +qσ 2 R σ 2 N + qσ 2 R . ( 34
)
Multiplying respectively Eq. ( 33) and (34) by a factor 1/q and π does not change the pdf since it is normalized. It is exactly the same for Eq. (34) where C does not depend on σ 2 R and vanish through the normalization. Therefore both expressions are exactly the same. It should also be noted that the noise level σ 2 N is necessary in both cases and the bias does not influence the sought parameter density whereas it does regarding the estimates. This implies a very interesting consequence: both pdf for the s.a and KLT leads to the exact same confidence interval for the same noise level.

B. General case

In this part any number of instruments and different noise level for each of them can be considered. In Section IV-A, we showed analytically that both methods lead to the same pdf of the signal level knowing the estimates in the event that all noise levels are the same. However when each noise level is different Eq. ( 29) giving the relation between the eigenvalues and the signal becomes much more complicated without degeneration. In this case, let us consider a number of instruments solely up to 5, refering as instance to the number of radio telescopes (RTs) part of the LEAP project. Then we make several empirical comparisons by computing the upper limit at 95% for the spectrum average and KLT methods. It should be noticed that the 5% lower bound has no interest since we are more particularly interested in the case where the signal is weaker than the noise level. This bound then greatly depends on the prior and is very close to zero. Table I gives the average over 1 000 realizations of the 95% upper bound for 2 to 5 RTs. The signal and noise levels are respectively σ 2 R = 1 a.u. and σ 2 N,i = i a.u. where i is the i-th RT. Then the 2-nd and 3-rd RT are respectively 2 and 3 times more noisy than the first one and so forth. First, these comparisons show as expected that the 95% bounds obtained by both estimators as in Sec. IV-A for the same noise variance, are exactly the same. Second, the mean and median are decreasing as the number of RTs increases. As a consequence adding measuring instruments or RTs always add information about the signal level or in the worst case is useless but never worsen it. On the other side the upper bound maximum values obtained depend strongly on the stochastic behavior of the measurements. Finally, it should be noticed that both methods require the noise level knowledge for the expression of the probability density function. The spectrum average method being the fastest way to compute the confidence interval is then to be privileged. Therefore we will only compare the spectrum average method with the cross-spectrum in the next section.

V. 95% UPPER LIMIT: SPECTRUM AVERAGE VS

CROSS-SPECTRUM

We have set the direct problem, i.e. the statistics of the s.a or c-s knowing the signal level and noise level (which is assumed to be known), respectively in Sections III-A and III-C. Now we tackle the inverse problem from the direct problem, i.e. the statistics of the signal level knowing the s.a or c-s estimate. The Bayes theorem enables us to establish this link as described in section III-D. The posterior distribution of the s.a and c-s are given by

p(σ 2 R | S sa ) ∝ 1 (σ 2 µ +σ 2 R ) 2 e -Ssa σ 2 µ +σ 2 R (35) and p(σ 2 R | S cs ) ∝ 1 2π(σ 2 µ +σ 2 R ) R e -it Scs φ(t)dt (36)
where σ 2 µ is the noise variance weighting according to Eq. ( 9). Let us describe our simulation algorithm in order to assess the 95% upper limit. First simulation (S 1 to S 3 ): simulate a set of real data from q instruments, assuming the red noise level is known (as well as, of course, the measurement noise levels). S 1 : Assign the number of RTs, the noise variance of each one and the sought true signal level. S 2 : Generate a set of spectral measurement according to Eq.

(1) S 3 : Compute the s.a and c-s estimates, as stated in Eq. [START_REF] Verbiest | Pulsar timing array experiments[END_REF], which are now fixed as parameters. Second simulation: we no longer modify the data (these are acquired measurement results) and we look for a confidence interval on the red noise, assuming the level of the measurement noise is known. S 4 : Define any basis and perform an orthogonalization and normalization of it by using the DGEQRF subroutine from LAPACK S 5 : Establish, from Eq. (18), one W matrix for each signal level varying from 0 to an upper limit for which Eq. ( 35) and (36) are close enough to zero according to the required precision. S 6 : Peform S 7 to S 11 for each σ 2 R value. S 7 : Project the W matrix onto the orthogonal basis. S 8 : Compute the c-s denoted Z from the result of S 6 . S 9 : Determine the eigenvalues of Z by using the DSYEV subroutine from LAPACK which has now the form of Eq. ( 19). S 10 : Define the product of each characteristic function defined by Eq. (20). S 11 : Compute the posterior distribution respectively of the s.a and c-s estimates according to Eq. ( 35) and (36). For the c-s, we perform a numerical integration of one signal value by using the Simpson method. S 12 : Normalize the s.a and c-s posterior pdf. S 13 : Determine the cumulative distribution function (cdf) by integrating the s.a and c-s posterior pdf and find the 95% upper limit corresponding onto the cdf value associated to the signal level. The loops for the different values of the signal are computed in parallel in order to save computing time. Let us give an example of such a process. We set the number of RTs to 5 and the variances of the signal and noise are respectively σ 2 R = 6 a.u., σ 2 n = 10 a.u. Then we produce 2 sets of random measurement with these parameters, shown in Table II. The first measurement set gives respectively S sa,1 = 14.886 a.u. for the c-s. Furthermore the second set gives us 167.1 for the s.a and 164.8 for the c-s. These results show that either the c-s or the s.a can be the most efficient even with the same parameters, then it only depends on the measurement set. However, the difference between the 95% upper limit for both methods is relatively low.

Let us now compare the s.a and c-s 95% upper limit over 100 simulations as shown in Table III for the sought signal level set to 6 a.u. and a noise level equal to 10 a.u. for each RT. The 95% upper limit is given respectively for, from the top of the Table to the bottom, the spectrum average, the crossspectrum and the ratio of the 95% bound of s.a over c-s. The mean and median are decreasing when the number of RTs is increasing. However for 4 RTs the results are much more lower but it is just an artefact of "luck". Indeed the maximum value is 1.4 times lower than for 5 RTs and the standard deviation (std) is also very much more lower. The sample size can have a significant effect on the values obtained but is necessary to have a good precision with a reasonable computation time. However, the minimum value of the 95% bound obtained for both methods permits to override this randomness. Indeed when the cross-spectrum estimate is negative or the spectrum average estimate tends towards zero it leads to the smallest 95% bound. Whereas the maximum 95% bound obtainable for a reasonable amount of simulations can "wriggle" a lot as the tail of the posterior pdf is very long especially with higher noise level than signal level which is of interest. The minimum value shows as expected an improvement with the increase in number of RTs. It seems that the s.a method gives the most stringent confidence interval. Figure 5 shows the histogram of the 95% limit with 5 RTs for 10 000 realizations, σ 2 R = 6 a.u. and σ 2 N = 10 a.u. Both histograms exhibit a similar distribution which extend up to high values. However the first bin corresponding to the lowest 95% bound shows a high number of realizations for the c-s method. This can be explained by a negative estimate for the cross-spectrum which may corresponds to a spectrum average estimate having a not so small value and so a higher 95% bound. Figure 6 shows the comparison of the 95% upper limit for the s.a and c-s methods for a window of hundred data among the same set of realizations. The 6 620-th realization framed by a blue rectangle highlights the fact that the c-s can sometimes be much more stringent than the s.a method. TABLE III: 95% upper limit statistics for the s.a (top), c-s (middle) and the ratio of the s.a by the c-s over 100 simulations where σ 2 R = 6 a.u. and σ 2 N = 10 a.u. Each rows respectively from the left to the right corresponds to the number of RTs, the mean, median, standard deviation, minimun and maximum value of the 95% upper bound. However in most of the other realizations we notice that the 95% limit is almost the same. Figure 7 depict the 95% upper bound median among 1 000 simulations with 5 RTs, for the s.a over c-s ratio depending on the signal-to-noise level ratio (with σ 2 N = 1 a.u.). When σ 2 R σ 2 N then the s.a seems to be the most stringent most of the time. However when the signal level becomes higher than the noise level, both the s.a and the c-s methods give in median the same 95% limit. Considering all these observations it is wiser to compute both estimators and use the most restrictive one. Even if most of the time both estimators give a very close upper bound, sometimes the gap is clearly significant.

VI. CONCLUSION

First, we demonstrated that the spectrum average variance is q/(q -1) lower than the cross-spectrum variance. Second, in order to assess the confidence interval of the signal level we defined its probability density function knowing the s.a and c-s estimates but also the noise of each instruments (radio telescopes). In addition a method directly using the statistics of the measurement (KLT) has also been compared. It turns out that the KLT and the s.a methods lead to the exact same pdf of the signal level σ 2 R knowing the estimates, so the precision is the same. Furthermore whereas the exact density of the cross-spectrum leads to the VΓ distribution for 2 instruments. it is no longer the case for more instruments. We proposed a generalized method which implies a numerical integration of the characteristic function product. This method works very well according to the Monte Carlo simulations.

Finally the efficiency of both estimators, the spectrum average versus the cross-spectrum, is highlighted through the comparison of the 95% Bayesian upper limit. We found a slight advantage for the spectrum average estimator when the noise level is higher than the signal level. However we showed that sometimes the c-s gives the most stringent confidence interval but above all a little more often than the s.a for the lowest upper limit. Nevertheless it is the s.a method which gives us the minimum 95% limit reachable. To conclude it is wiser to compute both estimates and use the most stringent.

Fig. 1 :

 1 Fig.1: Array of q instruments measuring the signal r(t). Each RT adds a white noise to the output x(t) whose Fourier transform is X(f ). Then the estimate S is computed.

Fig. 2 :

 2 Fig. 2: Periodogram of x (white noise plus red noise). The PSD is the expectation of the periodogram.

Fig. 3 :

 3 Fig. 3: Variance of the estimate with the signal variance which is of the form (σ 2 R /f α ) 2 , where σ 2 R = 4096 a.u. is the signal level and α = 4 the red noise exponent. The noise model is a white noise of level σ 2 N = 1 a.u. with 2 instruments.

Fig. 4 :

 4 Fig. 4: Comparison of the empirical (red boxes) and theoretical (green line) pdf of the c-s for 5 instruments where the variances are σ 2 R = 6 a.u. and σ 2 N = 10 a.u.

Fig. 5 :

 5 Fig. 5: Histogram comparison of 95% upper bound between the c-s and s.a for 10 000 realizations. The parameters are set for 5 RTs, σ 2 R = 6 a.u. and σ 2 N = 10 a.u.

Fig. 6 :Fig. 7 :

 67 Fig.6: Set of 100 realizations for 5 RTs of 95% bounds for cross-spectrum (green +) and spectrum average (red ×) where σ 2 R = 6 a.u. and σ 2 N = 10 a.u.

  and E. Rubiola are with FEMTO-ST, Department of Time and Frequency, UMR 6174, Université Bourgogne Franche-Comté, France.

Antoine's ORCID is 0000-0002-7007-5273 E. Lantz is with FEMTO-ST, Département d'Optique P.M. Duffieux, UMR 6174 CNRS, Université Bourgogne Franche-Comté, France. E. Rubiola is also with the Division of Quantum Metrology and Nanotechnology, Istituto Nazionale di Ricerca Metrologica (INRiM), 10135 Turin, Italy. Enrico's ORCID is 0000-0002-5364-1835 F. Vernotte is with FEMTO-ST, Department of Time and Frequency, Observatory THETA, UMR 6174 CNRS, Université Bourgogne Franche-Comté, France. Franc ¸ois's ORCID is 0000-0002-1645-5873

TABLE I :

 I Upper limit average of the parameter σ 2 R taking into account 2 to 5 RTs. These data were obtained from a set of 1 000 simulated spectra. The signal and noise level used for the computation are σ 2 R = 1 and σ 2 N,i = i where i is the index of the RT.

		Spectrum average / KLT 95% upper limit
	RTs number	Mean	Median	Std	Min	Max
	2	17.44	12.88	3.10	6.30	115.32
	3	16.32	11.78	2.39	5.16	91.78
	4	15.66	11.10	2.95	4.54	108.82
	5	14.84	10.67	2.28	4.14	86.99

TABLE II :

 II Measurement set for the outputs of each RT (5 in total) where σ 2 R = 6 a.u. and σ 2 N = 10 a.u. = 13.226 a.u. for the s.a and c-s estimates whereas the second one gives S sa,2 = 20.730 a.u. and S cs,2 = 18.564 a.u. It leads for the first set to the 95% upper limit on the signal σ 2 R following value, 125.8 for the s.a and 127.3

		measurement set 1	measurement set 2
		Real part	Imaginary part	Real part	Imaginary part
	X 1	-3.8947	-1.7994	-0.1494	8.9456
	X 2	-5.0950	-3.9125	-0.5275	4.4659
	X 3	-2.5133	-5.5431	0.2176	5.7742
	X 4	0.6433	-1.9566	1.6044	3.2146
	X 5	-0.2294	-2.5738	-0.5284	0.3563
	and S cs,1

ACKNOWLEDGEMENT

This work was partially funded by the ANR Programmes d'Investissement d'Avenir (PIA) Oscillateur IMP (Project 11-EQPX-0033) and FIRST-TF (Project 10-LABX-0048).

APPENDIX

Glossary of symbols

q Number of instruments r(t)

Common signal measured by q RTs (red noise) R(f ) Fourier transform of r(t) S r (f)

Power spectral density of r(t) n i (t) Intrinsic white noise of the i-th RT N i (f ) Fourier transform of n i (t) S n,i (f) Power spectral density of n i (t) x i (t)

x i (t) = r(t) + n i (t), received at the output of the i-th RT X i (f ) Fourier transform of x i (t) S x,i (f) Power spectral density of x i (t) estimate as in S. Here we consider three estimators, S sa Spectrum average

R

Variance of R in a bandwidth, i.e. the power in one bin of S(f ). It takes three different flavors: s.a, KLT or c-s σ 2

N,i

Same as above, with the noise of the i-th RT σ 2 µ Noise weight factor, inverse of the sum of the inverse of σ 2

N,i

Variance of the estimators S sa and S cs 1) Measurements: Let us define q instruments measurements X 1 , X 2 , . . . and X q as X j = N j + iN j + R + iR where N j , N j are independent Gaussian centered random variables of variance σ 2 N /2 and S, S are independent Gaussian centered random variables of variance σ 2 R /2. 2) Estimators: The estimator S cs is defined by Eq. 10 as

On the other hand, S sa is defined by Eq. 10 as

3) Statistics reminder: If A and B are 2 independent random variables of zero expectation

according to Eq. (a) from [START_REF] Barnett | The variance of the product of two independent variables and its application to an investigation based on sample data[END_REF] where V[•] stands for the variance of the quantity within the brackets. Moreover according to the Isserlis' theorem [START_REF] Isserlis | On a Formula for the Product-Moment Coefficient of any Order of a Normal Frequency Distribution in any Number of Variables[END_REF],

(41) If A, B, C, D are 4 independent Gaussian centered random variables, this can be derived to the following particular cases (Isserlis' theorem [START_REF] Isserlis | On a Formula for the Product-Moment Coefficient of any Order of a Normal Frequency Distribution in any Number of Variables[END_REF]):

since the only mathematical expectation which is not null,

4) Variance of S cs : From (37), it comes

Then,

where all covariance terms are null thanks to Isserlis'theorem.

From the properties (39) and (40), it comes

Therefore,

5) Variance of S sa : From (38), it comes

where all covariance terms are null thanks to Isserlis'theorem.

From the properties (39) and (40), it comes

(43) 6) Variance ratios: Let us compare the cross-spectrum and spectrum average estimates variances for limit signal to noise ratio values.

Consequently,