Patrick Flandrin

Pierre Borgnat

Revisiting and testing stationarity

come

Introduction

The concept of stationarity is ubiquitous in the signal and image processing literature, but its actual theoretical definition fails to be operational when confronted to practice. Strictly speaking, stationarity is a stochastic concept and-even when restricted to second order-its definition relies on an invariance with respect to any shift (in time or space), no matter how large this shift may be. This obviously contrasts with common practice which often accepts to loosely extend the definition to deterministic periodicities and to implicitly restrict its application to some finite observation range. As a result, practical "stationarity" turns out to be a relative concept, the very same physical object having the ability of appearing as stationary or not, depending on the observation scale. A schematic example of this situation is given in Fig. 1, in which zooming in on a scene makes successively appear, in an intertwined way, "stationary" and "nonstationary" features. 1

A general framework

In order to cope with the aforementioned issues, a methodology has recently been proposed [START_REF] Xiao | Testing stationarity with time-frequency surrogates[END_REF][START_REF] Xiao | Testing stationarity with surrogates -A one-class SVM approach[END_REF], that basically relies on two ingredients: (i) selecting an appropriate representation space in which local features can be compared to global ones so as to assess variability; (ii) giving this assessment a statistical significance by comparing the actual observation with stationarized data sharing the same global structure.

A time-frequency/space-scale approach

Considered from either a stochastic or a deterministic point of view, stationarity of a time series is usually meant for an invariance of spectral properties over time. This naturally makes of the time-frequency plane [START_REF] Flandrin | Time-Frequency/Time-Scale Analysis[END_REF] a natural representation space, with stationarity defined as an equivalence between local spectra and the global spectrum obtained by marginalization (see, e.g., [START_REF] Martin | Measuring the degree of non-stationarity by using the Wigner-Ville spectrum[END_REF][START_REF] Martin | Detection of changes of signal structure by using the Wigner-Ville spectrum[END_REF] for early attempts in this direction). Whereas the initial setting was developed for Figure 1. Zooming in on a given scene makes appear structures that, depending on the corresponding observation scale, can be interpreted as "stationary" or "nonstationary". In this example, observation at a large scale (top left) evidences large distinct regions that make the overall scene "nonstationary". At some smaller scale (top right), the scene is dominated by roughly periodic structures that turn it into a "stationary" one. Zooming in further (bottom left) turns back to "nonstationarity" whereas an observation at an even smaller scale (bottom right) reveals again a "stationary" structure attached to an homogeneous texture.

1D time series only, with spectra considered the usual way as a function of frequency (via the Fourier transform), it is clear that the proposed picture extends trivially to scale-based spectra (via the wavelet transform) as well as to 2D data in space. In any case, the rationale is to test for the variablity of differences (in a sense to be precised further) between local and global features over a given observation domain.

Stationarization via surrogates

From a practical point of view, local frequency/scale features will always exhibit fluctuations, and the question is to give them some statistical significance. For this to make sense, there is a need for some reference for the null hypothesis of stationarity. Since, in general, such a reference is not available, it is proposed to create it in a data-driven way by synthesizing a set of surrogate data that all share with the actual one the same global structure while being stationarized. For a given marginal spectrum, "nonstationary" processes differ from "stationary" ones by some organized structure in time or space. Recognizing that such an organization is encoded in the spectrum phase, stationarized surrogates can then be easily constructed by randomizing uniformly the phase of the actual data spectrum while keeping its magnitude unchanged (this is in fact a new way of using the technique of surrogate data that has been primarily introduced in the physics literature and mostly used for testing nonlinearity [START_REF] Theiler | Testing for nonlinearity in time series: the method of surrogate data[END_REF]).

Tests

Within the above general framework, the principle of any surrogates-based test is therefore very simple, since it essentially amounts to (i) creating a suitable set of stationarized surrogates from the observed data under test;

(ii) elaborating from this set a statistical characterization of what the distribution of natural fluctuations is supposed to be in the corresponding stationary situation; (iii) computing the actual fluctuation for the observed data and deciding whether it is an outlier or not with respect to this distribution.

The 1D case

Two variations have been proposed in the 1D case, both based on (multitaper) spectrograms.

In the first one [START_REF] Xiao | Testing stationarity with time-frequency surrogates[END_REF], local and global frequency spectra are compared via a suitable "distance" combining the Kullback-Leibler divergence and the log-spectral deviation [START_REF] Basseville | Distance measures for signal processing and pattern recognition[END_REF].

The mean-square deviation in time (over the considered interval) of this dissimilarity measure has been shown to follow a Gamma distribution. For a given confidence level, this modelling allows for the determination of an objective threshold on the basis of the two Gamma parameters, as estimated (e.g., in a maximum likelihood sense) from a limited number of surrogates (typically, about 50) [START_REF] Xiao | Testing stationarity with time-frequency surrogates[END_REF]. (Matlab routines are available at http://perso.ens-lyon.fr/patrick.flandrin/stat test.html.) A second variation [START_REF] Xiao | Testing stationarity with surrogates -A one-class SVM approach[END_REF] is aimed at by-passing the pre-requisite of the null hypothesis distribution modelling by directly considering the family of surrogates as a learning set. This viewpoint makes possible the use of (kernel-based) machine learning methods, and in particular of one-class support vector machines [START_REF] Schölkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF]. For the sake of learning, different time-frequency features can be envisioned but, as previously, nonstationarity is assessed by the fact that the actual data feature vector lies outside the domain of the empirical distribution derived from the training.

Some 2D extensions

Going from 1D to 2D can be made in a straightforward manner, replacing mutatis mutandis time by space and spectrograms by scalograms (i.e., squared magnitude of wavelet transforms). An example-in the spirit of the distance-based approach-is given in Figs. 2 and3: it makes use of an undecimated dyadic wavelet transform, the overall test being constructed as the ℓ 1 -norm of a distance map computed pointwise. In order to compensate for the estimation variability, the wavelet transform is dyadically weighted in scale and, for both the distance map and its fluctuations, the comparison "local vs. global" is achieved in a directional way, ending up in fact with 3 different test measures attached to the horizontal, vertical and "diagonal" features extracted by the (tensor) 2D wavelet transform.

One further example is given in Fig. 4, dealing with a 2D fractional Brownian motion (fBm) with Hurst exponent H = 0.3. From a strictly theoretical point of view [START_REF] Embrechts | Self-Similar Processes[END_REF], such a self-similar process is classically referred to as nonstationary whereas, within the framework considered here, it appears as "stationary". This result is however in accordance with the principles underlying the approach proposed here, since identifying "stationarity" with a statistical equivalence between local and global properties tends naturally to label as stationary those fractal processes (such as fBm) for which "the part is identical to the whole". Moreover, the distance measure aimed at quantifying the difference local vs. global is based on detail wavelet coefficients only, getting rid of the companion approximation coefficients that are known to carry the nonstationary part of processes with stationary increments [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF]. Revisiting stationarity this way appears therefore as operational, adding a quantitative characterization to a meaningful interpretation.

Conclusion

A general methodology has been proposed for testing stationarity in an operational way, i.e., in a relative sense that explicitly includes the observation scale, and with a statistical significance stemming from the construction of an adequate set of surrogate data. Only the principle has been outlined, and the efficiency of the approach has been supported by simple, schematic examples. Whereas more details can be found in [START_REF] Xiao | Testing stationarity with time-frequency surrogates[END_REF][START_REF] Xiao | Testing stationarity with surrogates -A one-class SVM approach[END_REF], further studies are necessary to thoroughly evaluate performance and to make comparisons with related approaches.

Figure 2 .

 2 Figure 2. A "nonstationary" scene and one of its stationarized surrogates are displayed in the left column, while the middle column represents the corresponding (horizontal) distance maps between local and global scalogram spectra, the bottom one being an average based the use of 50 surrogates. The "nonstationary" nature of the scene at the considered scale is assessed in the far right diagram by the 3 test values (in horizontal (H), vertical (V) and "diagonal" (D) directions) for the actual scene (red dots) that appear as outliers when compared to the distributions boxplots of the corresponding surrogates test values.

Figure 3 .Figure 4 .

 34 Figure 3. Same as in Fig. 2 when zooming in on a "stationary" region. At such a refined scale, stationarity is assessed by the test values for the actual scene (red dots) that lie within the corresponding distributions of the surrogates test values.

The images used in Figs. 1 to 3 of this paper have been downloaded from http://maps.google.com/.

Acknowledgments

Part of this work is supported by ANR-07-BLAN-0191-01 StaRAC and conducted in collaboration with Pierre-Olivier Amblard (GIPSA-lab, Grenoble), Cédric Richard (UTT, Troyes) and Jun Xiao (ENS Lyon and ECNU, Shanghai).


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text:
G Ollivier

S Bellon

& Penvern

Thematic and citation structure dynamics of Organic Food & Farming research

Keywords: OF&F, scientometrics, document co-citation analysis, specialty detection

This paper analyses the Organic Food & Farming (OF&F) scientific domain dynamic throught a "progressive document co-citation analysis" based on peer-reviewed papers from Web of Science. The dataset of OF&F domain displayed an exponential growth and a thematic diversification pattern. Both dominant and marginal clusters in association with their main cited articles were identified. This study enables to pinpoint major themes addressed or emerging. It can feed further research work and projects, namely with the definition of information system and research policy.

Introduction: need for research and research needs in OF&F

The organic sector is increasingly considered by stakeholders as a tangible alternative to address current challenges faced by agriculture. This leads to a continuous expansion of both the knowledge base and the demand of synthesis in OF&F. Several authors suggested topics for research, whether as general themes [START_REF] Watson | Research in organic production systemspast, present and future[END_REF][START_REF] Rahmann | Good, but not good enough? Research and development needs in Organic Farming[END_REF] or specific fields [START_REF] Kristiansen | Organic Agriculture: A global perspective[END_REF]. Others addressed the specificity of organic research approaches [START_REF] Lockeretz | Organic farming research, today and tomorrow[END_REF][START_REF] Watson | Research in organic production systemspast, present and future[END_REF][START_REF] Drinkwater | Ecological Knowledge:Foundation for Sustainable Organic Agriculture[END_REF]. However, such reviews do not analyse evolutions in topics within the OF&F literature. This assessment is important as transnational projects and journals dedicated to OF&F develop. We propose, using scientometrics, to identify dynamic of specialties and associated research topics in OF&F domain.

Material and methods

The progressive document co-citation analysis (PDCA), developped inside the CiteSpace Software [START_REF] Chen | The Structure and Dynamics of Co-Citation Clusters: A Multiple-Perspective Co-Citation Analysis[END_REF], allows to find the intellectual basis of specialties developed within the OF&F domain by mapping its internal citation structure. A co-citation link occurs when two documents are cited in the same citing documents. PDCA identifies high-density patterns (clusters) inside the co-citation network. Theses clusters gather the documents often cited together (intellectual basis) upon which authors build their specialties. The analysis is progressive since the dataset is divided into n-years slices, thus useful to interprete the clusters evolutions. As citation clusters are hierarchically organised, we iteratively tested various threshold parameters to produce different co-citation networks in order to detect a common robust structure emerging from the different resulting clustering. To better synthetise information, we gathered structurally and topically coherent clusters thanks to the CiteSpace automatic cluster labeling which detects specific cluster terms from citing documents [START_REF] Chen | The Structure and Dynamics of Co-Citation Clusters: A Multiple-Perspective Co-Citation Analysis[END_REF].

The Web of Science was used because it includes the cited references for each publication. The challenge was to build a query representing all the aspects of OF&F research (production modes, products, social dimensions) without inducing noises. Indeed, some terms related to OF&F are also used in other domains. After a first query, we extracted terms using "organic" to test their specificity in the OF&F domain.

Comment citer ce document : Ollivier, G., Bellon, S., Penvern, S. (2011). Thematic and citation structure dynamics of Organic Food & Farming research. In: Organic is life. Knowledge for tomorrow. Proceedings (p.

321-325). Presented at 3. ISOFAR Scientific Conference at the 17. IFOAM Organic World Congress, Gyeonggi Paldang, KOR (2011-09-28 -2011-10-01). Bonn, DEU : ISOFAR -International Society

We finally built a complex query, composed of 120 specific terms representing the domain and resulting in a dataset of 4401 journal articles from 1975 to 2009.

Results

Figure 1: Evolution of significative subject areas 2

The dataset displayed a clear exponential growth associated with a thematic diversification pattern (Figure 1). The OF&F domain was firstly developed inside Agricultural sciences. Every subject area (SA) increased in the 90's except for Sociology whose growth remained relatively low. Lately, three SA have dominated the domain. Agriculture (Multidisciplinary) and Agronomy prevailed jointly until 2001 and afterwards; they exhibit a rather similar growth.

The third SA, Food Science & Technology, remained marginal until 1999 before being the second most important one after 2002. While Ecology was also a major SA, it did not face such a development and increased constantly after the late 90's. In the meantime, secondary SA emerged with significant growth during the past years: Dairy and Animal Sciences, Economics, Applied Chemistry and Nutrition & Dietetics. Comment citer ce document : Ollivier, G., Bellon, S., Penvern, S. ( 2011). Thematic and citation structure dynamics of Organic Food & Farming research. In: Organic is life. Knowledge for tomorrow. Proceedings (p.

321-325). Presented at 3. ISOFAR Scientific Conference at the 17. IFOAM Organic World Congress, Gyeonggi Paldang, KOR (2011-09-28 -2011-10-01). Bonn, DEU : ISOFAR -International Society

The first major co-citation cluster identified concerns performances in the 80's turn, with studies tackling the energetic and environmental crisis of agriculture. This introduced the multiple farming performances assessment in the domain. The domain "diversification" occured since the turn of the 90's in correlation with its global growth A specialty related to Soil historically dominated while new specialties emerged. They are related to Biodiversity, Food content, Consumers, and Social aspects of OF&F development.

The main speciality (Soil), composed of many citation sub-clusters (almost 30% of detected clusters and included references) also contained the most cited references of the domain, with various focus of Soil. The most structured approach, first developed, concerned soil biological activity. It essentially referred to the Journal Soil Biology & Biochemistry, which introduced methodologies to measure soil biological features. Related smaller and peripheric clusters focussed on specific biological compartments. A second and more recent approach focussed on nutrient cycling, particularly on the role of nutrient processes in soil fertility. Some clusters combined the previous two approaches with a holistic view of soil. This approach is supported by comparative studies in agronomy or soil ecology, centered on the effects of organic/conventional systems or practices (fertilisation, tillage) on different soil features.

The Biodiversity specialty is the most distinctive of the domain due to the weakness of its interconnexion to other clusters. Emerging at the 90's turn, it concerns studies and reviews on farming systems or practices comparisons to assess the effects of intensification on diverse elements of biodiversity (landscape, bird, invertebrates, weed communities). Surprisingly, no citation cluster appears with Biodiversity as a source of services for farming.

Another well-structured specialty concerns organic food content in comparison with conventional food. Two linked clusters, without clear delineation, emerged in the beginning of the 2000's. The specialty focussed on chemical characterisation and to a lesser extent on nutritional and sensory quality concerns. Few studies appear in citing articles on the effects of organic food on human health, but not as a specialty per se, ie without a structured intellectual basis. The specialty is weakly linked to clusters focussing on farming system while some papers refer to the effect of specific production practices on food quality.

The connexion between production and products issues is made throught references related to consumer and sociological approaches of OF&F dynamics. The first specialty is based on two main clusters. The first one, occuring as a dense cluster around the 90's turn concerns factors determining consumer purchase decisions. Corresponding studies reveal the importance of Nutrition and Health determinants that are now emerging issues in the "food content" specialty. The second cluster largely extends this approach with a slight evolution in the type of references used. Whereas the first one is mainly anchored in Food Science & Technology, the second one includes more Economics and Sociology corresponding to the integration of commodity chains, linking production and consumption with individual-focused studies.

This partly explains the connexions between the last specialty and the one concerning social sciences approaches of OF&F development dynamics. Indeed, the clusters of this specialty, emerging at the turn of the 90's, share a common intellectual basis specific to their discipline whereas they mix different aspects of OF&F from producers to consumers. The main cluster concerns issues about OF&F sector development from the farmer level, with adoption/conversion/abandonment determinants, to the institutional level, with policy issues and certification devices, also addressing Comment citer ce document : Ollivier, G., Bellon, S., Penvern, S. ( 2011). Thematic and citation structure dynamics of Organic Food & Farming research. In: Organic is life. Knowledge for tomorrow. Proceedings (p.

321-325). Presented at 3. ISOFAR Scientific Conference at the 17. IFOAM Organic World Congress, Gyeonggi Paldang, KOR (2011-09-28 -2011-10-01). Bonn, DEU : ISOFAR -International Society collective actions and geographical diffusion. These aspects, emerging in the beginning of 2000's, are gathered under the conventionalisation thesis in OF&F development. A second cluster is focused on OF&F as an element of power/politics transformation in food system/chain/network through the localism issue.

In addition, some marginal clusters, ie with less structured intellectual basis, occured in the DCA tested. Some expected topics concerning production practices were marginal or absent. Clusters concerning dairy production may arise in some networks but are still in a marginal position. When present, this topic concerned animal health in organic production. We also expected to find specialties about crop protection and weed control but these topics are not based upon a highly cited reference pool.

Discussion and conclusion

The OF&F domain is very dynamic as shown with the continuous emergence of specialties. This outlines the gradual recognition and incorporation of OF&F into the larger agricultural scientific and policy communities where it is progressively viewed both as one credible variant of "alternative" agriculture and as a new research field. The analysis highlights the recurrence of comparative approaches of performances and effects. Researchers rarely design new production systems or techniques and they often tend to value OF&F in confrontation with a conventional model, still considered as the reference. This questions the autonomy of OF&F research. This growth suggests early stages of development. As well as OF&F sector, research development could suffer forms of conventionalisation. Even if the multidisciplinarity and holistic approaches are fundamental in the emergence and the identity of the domain, recent specialisations induce some issues about the maintenance of OF&F as a separate domain. Yet organic research proved to be rarely different from those used in conventional agriculture [START_REF] Lockeretz | Organic farming research, today and tomorrow[END_REF][START_REF] Watson | Research in organic production systemspast, present and future[END_REF].

We highlighted the institutionalisation but also some lacks in topics we expected. There is a methodological issue about detecting this kind of emerging or lightly structured topics, because statistical tools tend to detect patterns with high frequency. We could deepen our study by using lower thresholds or other paramaters in DCA to detect more clusters and crossing with other types of analysis to detect marginal topics [START_REF] Chen | The Structure and Dynamics of Co-Citation Clusters: A Multiple-Perspective Co-Citation Analysis[END_REF].

Table 1 : main features of specialties detected Specialty year 1st Author title Journal # cited Farming

 1

	Science	36

performances 1973 Pimentel Food production and the energy crisis Science 11 1978 Berardi Organic and conventional wheat production: examination of energy and economics Agroecosystems 13 1981 Lockeretz Organic farming in the Corn Belt

UR 0767 Ecodéveloppement, INRA E-Mail : gollivier@avignon.inra.fr

Each journal in Web of Science is associated with 1 or more SA. For more details see: http://science.thomsonreuters.com/mjl/scope/scope_sci/