Frédéric M Vanwindekens
email: frederic.vanwindekens@uclouvain.be

Didier Stilmant
email: stilmant@cra.wallonie.be

Philippe V Baret
email: philippe.baret@uclouvain.be

The Relevance of Fuzzy Cognitive Mapping Approaches for Assessing Adaptive Capacity and Resilience in Social-Ecological Systems

Keywords: Agriculture, Social-Ecological Systems, Systems of Practices, Fuzzy Cognitive Mapping, Resilience Assessment

Social-Ecological Systems (SES) are complex due to uncertainty related to their nature and their functions. In these systems, decision-making processes and practices of managers are often valueladen and subjective, dominated by their world-views and their own knowledge. People's knowledge are central in building their adaptive capacity but are seldom taken into account by traditional decision-making approaches in modelling SES management. In this paper, we introduce a Fuzzy Cognitive Mapping approach to study the dynamic behaviour of managers' systems of practices. As a case study, we aim to assess farmers' forage management under different climatic scenarios. Results show that summer drought have varying consequences according to farmers' systems of practices. Fuzzy Cognitive Mapping approaches are particularly relevant in studying systems of practices in SES. Their utilisation is promising for the evaluation of adaptive capacity and resilience in SES at local scale (exploitation, community) and regional scale (ecological areas, country).

Introduction

The management of Social-Ecological Systems (SES, [START_REF] Folke | Resilience thinking: Integrating resilience, adaptability and transformability[END_REF]) is complex due to the intricacy of their components, to the uncertainty related to their nature and to the various societal, institutional, physical, ecological, economical processes involved in their functions [START_REF] Ascough | Future research challenges for incorporation of uncertainty in environmental and ecological decision-making[END_REF]. Managers' strategies are largely influenced by their perceptions of the ecological, economical and social environments of SES [START_REF] Ascough | Future research challenges for incorporation of uncertainty in environmental and ecological decision-making[END_REF]. These influences have been particularly pointed out and studied in the agricultural context [START_REF] Darré | Le sens des pratiques. Conceptions d'agriculteurs et modèles d'agronomes[END_REF][START_REF] Edwards-Jones | Modelling farmer decision-making: concepts, progress and challenges[END_REF][START_REF] Ingram | Revealing different understandings of soil held by scientists and farmers in the context of soil protection and management[END_REF]. In order to help farmers in managing their farm, Decision Support Systems (DSS) have been developed by 'management scientists' [START_REF] Mccown | Changing systems for supporting farmers' decisions: problems, paradigms, and prospects[END_REF]. But unexpectedly, farmers pay little attention to these DSS [START_REF] Ascough | Future research challenges for incorporation of uncertainty in environmental and ecological decision-making[END_REF][START_REF] Edwards-Jones | Modelling farmer decision-making: concepts, progress and challenges[END_REF][START_REF] Mccown | Changing systems for supporting farmers' decisions: problems, paradigms, and prospects[END_REF][START_REF] Özesmi | Ecological models based on people's knowledge: a multistep fuzzy cognitive mapping approach[END_REF].

Recent scientific approaches have been developed to cope with incorporation of human, social and institutional aspects in SES models by explicitly accommodating relations between the natural and human environment [START_REF] Ascough | Future research challenges for incorporation of uncertainty in environmental and ecological decision-making[END_REF]. Fuzzy cognitive maps (FCM) are particularly relevant tools in modelling SES based on people explicit knowledge [START_REF] Özesmi | Ecological models based on people's knowledge: a multistep fuzzy cognitive mapping approach[END_REF] as they can be considered as a model of a belief system [START_REF] Kok | The potential of fuzzy cognitive maps for semi-quantitative scenario development, with an example from brazil[END_REF] constituted by concepts, the key drivers of the system, and edges, causal relationships between concepts. They have been developed by Bart Kosko in 1986 [START_REF] Kosko | Fuzzy cognitive maps[END_REF] in introducing the notion of 'fuzziness' and 'fuzzy weigth' to relationships of Robert Axelrod's cognitive maps (CM) [START_REF] Axelrod | Institute of public policy studies[END_REF].

In the agricultural context (see [START_REF] Vanwindekens | Development of a broadened cognitive mapping approach for analysing systems of practices in social-ecological systems[END_REF]), CM and FCM have been successfully applied for (i) analysing people knowledge, beliefs [START_REF] Popper | Knowledge and beliefs regarding agricultural pesticides in rural guatemala[END_REF] and decision-making on farm [START_REF] Isaac | Assessing local knowledge use in agroforestry management with cognitive maps[END_REF], (ii) studying adoption of agri-environment measures [START_REF] Ortolani | Analysis of farmers' concepts of environmental management measures: An application of cognitive maps and cluster analysis in pursuit of modelling agents' behaviour[END_REF], (iii) modelling farmers perceptions of how their ecosystem works [START_REF] Fairweather | Farmer models of socio-ecologic systems: Application of causal mapping across multiple locations[END_REF] and of the sustainability of farms [START_REF] Fairweather | Can farmers map their farm system? causal mapping and the sustainability of sheep/beef farms in new zealand[END_REF], and (iv) predicting yield production [START_REF] Papageorgiou | Fuzzy cognitive map based approach for predicting yield in cotton crop production as a basis for decision support system in precision agriculture application[END_REF][START_REF] Papageorgiou | Yield prediction in apples using fuzzy cognitive map learning approach[END_REF]. In order to study farmers' systems of practices (SOP) based on their own conceptions, we developed an approach for building cognitive maps by coding people's open-ended interviews. This approach was named CMASOP for 'Cognitive Mapping Approach for Analysing Actors' Systems of Practices' in SES. In a previous paper, we presented the core principles of CMASOP [START_REF] Vanwindekens | Development of a broadened cognitive mapping approach for analysing systems of practices in social-ecological systems[END_REF]. In this first publication, we applied CMASOP to the general description of farmers' SOP for managing grasslands in two Belgian agroecological areas [START_REF] Vanwindekens | Development of a broadened cognitive mapping approach for analysing systems of practices in social-ecological systems[END_REF]. In a second step, we developed complementary applications of CMASOP : a comparative one and a typological one. For comparing SOP between groups of managers defined a priori, we coupled CMASOP and descriptive statistical methods. For classifying SOP in a posteriori typological groups, we coupled CMASOP, clustering methods and statistical analysis. Results of the comparative and typological applications of CMASOP are being submitted [START_REF] Vanwindekens | Using a cognitive mapping approach for studying diversity of farmers' systems of practices in agricultural systems[END_REF].

In the present paper, we coupled CMASOP and auto-associative neural networks methods [START_REF] Özesmi | Ecological models based on people's knowledge: a multistep fuzzy cognitive mapping approach[END_REF] for carrying out inferences about farmers' adaptations to climatic uncertainties. The objectives of this development is to model the dynamic behaviour of managers' SOP for assessing their adaptive capacity, and indirectly, the resilience of their SES. The aim of this paper is to present this new development of CMASOP and to demonstrate the relevance of using FCM approaches for assessing adaptive capacity of managers and resilience of their exploitation in social-ecological systems. The management of the second cut in grassland based livestock farming systems of southern Belgium is used as a case-study.

Resilience is defined by Folke et al. as the "capacity of a system to absorb disturbance and reorganize [. . . ] " [START_REF] Folke | Resilience thinking: Integrating resilience, adaptability and transformability[END_REF]. Adaptive capacity is defined as the "capacity of actors in a system to influence resilience" [START_REF] Folke | Resilience thinking: Integrating resilience, adaptability and transformability[END_REF]. Different studies have used FCM for scenario analysis in SES [START_REF] Özesmi | Ecological models based on people's knowledge: a multistep fuzzy cognitive mapping approach[END_REF][START_REF] Kok | The potential of fuzzy cognitive maps for semi-quantitative scenario development, with an example from brazil[END_REF][START_REF] Soler | Using fuzzy cognitive maps to describe current system dynamics and develop land cover scenarios: a case study in the brazilian amazon[END_REF][START_REF] Wise | Qualitative modelling of fishermen's behaviour in a pelagic fishery[END_REF]. They rely on the possibility to compare the steady state calculation under various conditions : (i) current situation, (ii) evolution of some environmental variables (prices, rainfall) or (iii) implementation of different policy options (laws, tax). These concepts are closed to the concept of 'vulnerability' that has been analysed using FCM by [START_REF] Murungweni | Application of fuzzy cognitive mapping in livelihood vulnerability analysis[END_REF] in the study of livelihood [START_REF] Murungweni | Application of fuzzy cognitive mapping in livelihood vulnerability analysis[END_REF].

Materials and Methods

We studied farmers' Systems of practices (SOP) in Ardenne and Famenne, two grassland based agroecological areas in southern Belgium. We collected qualitative and quantitative data during forty-nine open-ended interviews on management of farms systems (structure, technical orientation, world views) and subsystems (forage, herd, grazing). We developed a cognitive mapping based approach for analysing systems of practices (CMASOP3 [START_REF] Vanwindekens | Development of a broadened cognitive mapping approach for analysing systems of practices in social-ecological systems[END_REF][START_REF] Vanwindekens | Using a cognitive mapping approach for studying diversity of farmers' systems of practices in agricultural systems[END_REF]). We applied it for studying grass forage management in our surveyed area.

The core principles of CMASOP consists in coding open-ended interviews of managers in order to create individual cognitive maps (ICM). These ICM can then be used to build a social cognitive map (SCM). As open-ended interviews focus on managers practices in social-ecological systems (SES), the ICM and SCM are considered as inductive models of SOP based on people conceptions [START_REF] Vanwindekens | Development of a broadened cognitive mapping approach for analysing systems of practices in social-ecological systems[END_REF]. The SCM is inter alia constituted by thirteen highly related concepts classed in seven core hubs (First, Second and Third cuts, Silo, Bale wrap, Hay and Cattle movement) and six peripheral hubs (Plot utilization, Plot-Farm distance, Forage quality, Forage quantity, Cutting date and Weather). A quote-retrieving module has been implemented in order to permanently relate each relationships to managers' quotations.

We developed applications for using CMASOP in comparative and typological ways [START_REF] Vanwindekens | Using a cognitive mapping approach for studying diversity of farmers' systems of practices in agricultural systems[END_REF]. Differences in SOP between groups of managers can be highlighted in coupling CMASOP and descriptive statistical methods. Typology of systems of practices can be processed by coupling CMASOP, clustering methods and statistical analysis. These developments have been applied to our case study, grass forage management in farming systems. The clustering of SOP in these systems highlighted two contrasted groups of farmers based on the management of their second grass cut(figure 1). The first group of farmers (A, n=24, figure 1(a)) are more prone for silaging [START_REF]R: A Language and Environment for Statistical Computing[END_REF] than they are for bale wrapping [START_REF] Fairweather | Farmer models of socio-ecologic systems: Application of causal mapping across multiple locations[END_REF] or haying [START_REF] Fairweather | Farmer models of socio-ecologic systems: Application of causal mapping across multiple locations[END_REF]. Conversely, the second group of farmers (B, n=25, figure 1(b)) are more prone for bale wrapping [START_REF] Özesmi | Ecological models based on people's knowledge: a multistep fuzzy cognitive mapping approach[END_REF] and haying [START_REF] Kosko | Fuzzy cognitive maps[END_REF] than they are for silaging [START_REF] Ascough | Future research challenges for incorporation of uncertainty in environmental and ecological decision-making[END_REF]. As a result of previous works, the drought has been quoted by farmers as a typical risk in grassland management in Famenne. The potential drought mainly occurs during the summer and have damageable consequences on grass growth and, in parallel, on milk production and animal performances in general. In order to cope with drought, farmers' adaptations are contrasted : grazed area increase or supplementation in grazing plots.

The two SCM (figures 1) have been taken as patterns to build two synthetic FCM (figures 2) for studying systems resilience and farmers' adaptive capacity linked with summer drought. These synthetic FCM show the grassland plots allocated for harvesting (Silo, Bale wrap, Hay) or for grazing (Grazed area). The weights of relationships between Second cut and these four concepts are proportional to their weights in the SCM of the two clusters (0.8, 0.2, 0.2 and 0.4 respectively in A and 0, 0.6, 0.4 and 0.4 in B). We considered that the products harvested on cutting plots constitute the Stock of forage (0.33 for Hay, Bale wrap and Silo in A ; 0.5 for Hay and Bale wrap in B). In case of Summer drought, two adaptations are simulated : the increase of Grazed area (0.9 for A, 0.1 for B) or the Supplementation of forage in grazed plots (0.1 in A, 0.9 in B). The increase of Grazed area involve a decrease of the harvested area (-0.5 for Silo in A and -0.25 for Bale wrap and Hay in B). Two self-reinforcing relationships have been added for the driver concepts Second cut and Summer drought.

Farmers are more prone to distribute forage conditioned in Bale wrap or in Hay, available in individual elements (bales), than to open a whole Silo done with the harvest of the first cut. Therefore, farmers of the cluster B have the possibility to supplement herd in grazing plots because their stock are mostly constituted by Bale wrap and Hay. For the simulations, we supposed that these farmers choose to cope with Summer drought in supplementing. Conversely, farmers of cluster A has only few Bale wrap and Hay in their Stock. For simulations, we supposed then that these farmers choose the other adaptation, in increasing of Grazed area. For the same reason, we supposed that the reduction of Grazed area for these farmers (A) only affect the most important conditioning, Silo.

The simulations have been processed using the auto-associative neural networks method described by Özesmi and Özesmi [START_REF] Özesmi | Ecological models based on people's knowledge: a multistep fuzzy cognitive mapping approach[END_REF] in order to calculate the activation degrees of each concepts at all time steps till convergence. Activation degrees are semi-quantitative values of concepts that can only be interpreted relative to each other [START_REF] Kok | The potential of fuzzy cognitive maps for semi-quantitative scenario development, with an example from brazil[END_REF]. The two scenarios have been implemented in forcing the activation degree of Summer Drought to 0 ('No Summer Drought') or 1 ('Summer Drought').

Results

Figure 3 shows the comparisons of the scenarios 'No Summer Drought' and 'Summer Drought' for the two Systems of Practices (SOP) assessed. The evolution of the two driver concepts ('Second cut' and 'Summer drought') is logically not shown nor analysed.

For SOP A cluster based on silaging, (i) the activation degree of Grazed area strongly increase from 0.380 (No Summer Drought) to 0.862 (Summer Drought), (ii) the activation degree of Supplementation slightly increase from 0.000 to 0.100, (iii) the activation degrees of Silo and of Stock decrease from 0.544 and 0.300 respectively to 0.353 and 0.194 respectively (table 1).

For SOP B cluster based on bale wrapping and haying, (i) the activation degree of Supplementation strongly increase from 0.000 to 0.716 while (ii) the activation degree of Stock strongly decrease from 0.246 to -0.118, (iii) the acti- vation degree of Grazed area increase from 0.380 to 0.462 and (iv) the activation degrees of Bale wrap and Hay slightly decrease from 0.466 and 0.296 respectively to 0.450 and 0.277 respectively (table 1). The increases of Grazed area for SOP A cluster and of Supplementation for SOP B cluster are adaptations of each groups of farmers in case of Summer drought (figure 4). In the FCM of SOP A cluster, a direct consequence of increasing of Grazed area is the decrease of Silo and a subsequent decrease of Stock that is limited. In the case of SOP B cluster, as a direct consequence of the increase of Supplementation, FCM shows a decrease of the activation degree of Stock more important than for SOP A cluster (figure 4).

Discussion and conclusion

This article presents an original method for assessing managers' adaptive capacity under uncertain environmental conditions. This method gain in coherence and relevance through its integration in CMASOP, a complete Cognitive Mapping approach [START_REF] Vanwindekens | Development of a broadened cognitive mapping approach for analysing systems of practices in social-ecological systems[END_REF][START_REF] Vanwindekens | Using a cognitive mapping approach for studying diversity of farmers' systems of practices in agricultural systems[END_REF]. The method is grounded on various kind of qualitative data collected during open-ended interviews. Its descriptive application allows to inductively model SOP of individual and groups of managers based on their own conceptions of their system. Its comparative and typological applications allow to objectively compare and cluster SOP. Finally, CMASOP allows to construct FCM of managers' SOP. In computing the steady states of FCM of various SOP under various environmental conditions, it is relevant in assessing adaptive capacity of managers in complex SES.

This paper present a first application of FCM for studying SOP in SES. The SOP we model are basic in terms of concepts and relations. As a consequence, the dynamic behaviours of the FCMs are elementary. Nevertheless, results confirm influences of farming practices on the whole functioning of the production system. They confirm also that various systems of practices have various effects on the system.

Beyond these results that could appear as relatively evident, results shows the possibility to model a wide variety of SOP in a simple way in order to assess them under various scenarios. The easy way of building model and simulating scenarios is a major advantage of the method presented. We illustrate it in processing further simulations in order to test two other SOP : C, based on a mixed sources of harvested forage (0.33 silo, 0.33 bale wrap and 0.33 hay) and D, whose stock is only constituted by purchased feed. Results are shown in table 1. Technically, our tool is easy-to-use for researchers or even for farmers : the relations and their weights are entered in a spreadsheet that is subsequently processed by an R program [START_REF]R: A Language and Environment for Statistical Computing[END_REF]. Comparisons of two SOP in case of Summer drought showed differences in terms of Stock between farmers' FCM. These differences could also have significant consequences on concepts not modelled in the present study : feeding of herd during the winter (stocks are smaller), cows selling (stocks are insufficient) or feed purchase (for restoring stocks), treasury (due to purchasing) and, finally, resilience of the whole farms. Although these concepts are beyond the scope of our model, this reasoning illustrates how resilience of farms and adaptive capacity of farmers could be assessed using FCM approaches.

Most of previous studies of managers' practices in SES were conducted in a qualitative way by social scientists or modelled in reductionist DSS. The twofold nature of FCM (qualitative and quantitative) bring another advantage to our method. It allows building SOP model based on people's knowledge and processing simulations.

Further works could include developments of more elaborated FCM including various indicators of economic fields (e.g. production, profit), ecological sciences (e.g. environmental footprint) or social sciences and psychology (e.g. personal fulfilment, happiness). It has not escaped our notice that the broadening of the map could represent an opportunity of measuring resilience of social-ecological systems at local or regional scales.

As asserted by van Vliet et al. (2010, [START_REF] Van Vliet | Linking stakeholders and modellers in scenario studies: The use of fuzzy cognitive maps as a communication and learning tool[END_REF]), FCM could be a relevant communication and learning tools between managers and scientists. It would be very interesting to carry out qualitative surveys in order to discuss the results of FCM simulations and adaptive capacity assessment with managers of SES. The results of these surveys could also constitute relevant data for an inductive and qualitative evaluation of resilience and adaptive capacity.

Fig. 1 .Fig. 2 .

 12 Fig.1. The 49 farmers' Systems of Practices (SOP) have been classified in two groups using the clustering application of CMASOP[START_REF] Vanwindekens | Using a cognitive mapping approach for studying diversity of farmers' systems of practices in agricultural systems[END_REF]. Farmers of the first cluster (A) are specialized and have SOP based on silaging. Farmers of the second cluster (B) are more diversified and have SOP based on bale wrapping and haying. The social maps of two clusters of SOP have been used to build and calibrate two synthetic FCM used in the present study (figure2)

Fig. 3 .

 3 Fig. 3. Evolution of activations degrees of concepts in the four Fuzzy Cognitive Maps. Comparisons of the scenarios 'No Summer Drought' and 'Summer Drought' for the two Systems of Practices (SOP) assessed : A (SOP based on silaging) and B (SOP based on bale wrapping and haying). X-axis: iteration step, Y -axis: activation degree

Fig. 4 .

 4 Fig. 4. Evolution of activations degrees of major concepts in the four Fuzzy Cognitive Maps. Comparisons of the adaptations of the two Systems of Practices (SOP) to cope with 'Summer Drought'. X-axis: iteration step, Y -axis: activation degree

Table 1 .

 1 Values of activations degrees of major concepts at steady states. Eight simulations are compared. Four Systems of Practices (SOP) : A, based on silageing ; B, based on haying ; C, based on mixed sources and D, feed purchasing. Two climatic scenarios : Normal (No Summer Drought) and Drought (Summer Drougth)

		SOP-A	SOP-B	SOP-C	SOP-D
		Normal Drought Normal Drought Normal Drought Normal Drought
	Silo	0.544	0.353			0.259	0.205
	Bale wrap	0.197	0.197	0.466	0.450	0.259	0.205
	Hay	0.197	0.197	0.296	0.277	0.259	0.205
	Supplementation 0.000	0.100	0.000	0.716	0.000	0.462	0.000	0.462
	Grazed area	0.380	0.862	0.380	0.462	0.380	0.716	0.762	0.762
	Stock	0.300	0.194	0.364	0.005	0.251 -0.028	0.000 -0.432

CMASOP was developed in R[START_REF]R: A Language and Environment for Statistical Computing[END_REF]. Figures1 and 2were done using Rgraphviz[START_REF] Gentry | Rgraphviz: Provides plotting capabilities for R graph objects[END_REF]. Figures3 and

were made using ggplot2[START_REF] Wickham | ggplot2: elegant graphics for data analysis[END_REF]

Acknowledgements. The Walloon Agricultural Research Centre provided financial support for this work through the MIMOSA project. We would also like to thank two anonymous reviewers for their helpful comments.


Haider Raza
email: raza-h@email.ulster.ac.uk

Girijesh Prasad
email: g.prasad@ulster.ac.uk

Yuhua Li
email: y.li@ulster.ac.uk

EWMA based Two-Stage Dataset Shift-Detection in Non-Stationary Environments

Keywords: Non-stationary, Dataset shift, EWMA, Online Shift-detection

Dataset shift is a major challenge in the non-stationary environments wherein the input data distribution may change over time. In a time-series data, detecting the dataset shift point, where the distribution changes its properties is of utmost interest. Dataset shift exists in a broad range of real-world systems. In such systems, there is a need for continuous monitoring of the process behavior and tracking the state of the shift so as to decide about initiating adaptive corrections in a timely manner. This paper presents a novel method to detect the shift-point based on a two-stage structure involving Exponentially Weighted Moving Average (EWMA) chart and Kolmogorov-Smirnov test, which substantially reduces type-I error rate. The algorithm is suitable to be run in real-time. Its performance is evaluated through experiments using synthetic and realworld datasets. Results show effectiveness of the proposed approach in terms of decreased type-I error and tolerable increase in detection time delay.

Introduction

In the research community of statistics and machine learning, detecting abrupt and gradual changes in time-series data is called shift-point detection [START_REF] Liu | Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation[END_REF]. Based on the delay of the detection, shift-point detection methods can be classified into two categories: retrospective detection and real-time detection. The retrospective shift-point detection tends to give more accurate and robust detection; however, it requires longer reaction periods so it may not be suitable for the real-time applications where initiating adaptation closest to the shift-point is of paramount importance. Also, in the realtime systems, each observation coming in the data stream may be processed only once and then discarded; hence retrospective shift-point detection may not be possible to implement. The real-time detection is also called a single pass method and requires an immediate reaction to meet the deadline. One important application of such a method is in pattern classification based on streaming data, performed in several key areas such as electroencephalography (EEG) based brain-computer interface, robot navigation, remote sensing and spam-filtering. Classifying the data stream in non-stationary environments requires the developments of a method which should be computationally efficient and able to detect the shift-point in the underlying distribution of the data stream. The key difference between the conventional classification and streaming classification is that, in a conventional classification problem, the input data distribution is assumed not to vary with time. In a streaming case, the input data distribution may change over time during the testing/operating phase [START_REF] Shimodaira | Improving Predictive Inference Under Covariate Shift by Weighting the Log-Likelihood Function[END_REF]. Based on the detected shift, a classifier needs to account for the dataset shift. In non-stationary environments there are several types of dataset shift, a brief review of the dataset shift and the types of dataset shift are presented in the next section. Some pioneering works [3][4] have demonstrated good shift-point detection performance by monitoring the moving control charts and comparing the probability distributions of the time-series samples over past and present intervals. These methods follow different strategies such as CUSUM (Cumulative SUM) [START_REF] Basseville | Detection of Abrupt Changes: Theory and Application[END_REF], Computational-Intelligence CUSUM (CI-SUSUM) [START_REF] Alippi | Just-in-Time Adaptive Classifier--Part I: Detecting Nonstationary Changes[END_REF], Intersection of Confidence Interval rule (ICI) for change-point detection [START_REF] Alippi | A Just-In-Time Adaptive Classification System Based on the Intersection of Confidence Intervals Rule[END_REF]. However, most of these methods rely on predesigned parametric models such as underlying probability distribution and autoregressive models for tracking some specific statistics such as the mean, variance, and spectrum. However, to overcome these weaknesses, recently some researchers have proposed a different strategy, which estimates the ratio of two probability densities directly without density estimation [START_REF] Sugiyama | Density Ratio Estimation in Machine Learning[END_REF]. Thus, the aforementioned methods are not robust against different types of the shift-detection tests because of the delay and the high rate of type-I error (i.e., false-positive). On account of the need for identifying models, this may significantly limit their range of applications in fast data streaming problems.

In this paper we present an approach consisting of a Two-Stage Dataset Shift-detection based on an Exponentially Weighted Moving Average chart (TSSD-EWMA). This TSSD-EWMA in an extension of our work on Dataset Shift-Detection based on EWMA (SD-EWMA) [START_REF] Raza | Dataset Shift Detection in Non-Stationary Environments using EWMA Charts[END_REF], which suffers from the high false-positive rate. In TSSD-EWMA, the stage-I consists of a shift-detection test that activates the stage-II when a shift is alarmed at the stage-I. At stage-II, the suspected shift from stage-I is validated and is either confirmed as a shift or declared as a false alarm. The structure of test is given in Figure 1. It is demonstrated to outperform other approaches in terms of non-stationarity detection with a tolerable time delay and decreased false-positive alarms. So, this scheme can be deployed along with any classifier such as k-nearest neighbor or support vector machine (SVM) in an adaptive online learning framework.

This paper proceeds as follows: Section 2 presents a background of dataset shift and EWMA control chart. Section 3 details the shift-detection algorithm and the structure of two-stage test. Section 4 presents the datasets used in the experiment. Finally, Section 5 presents the results and discussion. The term dataset shift [START_REF] Moreno-Torres | A Unifying View on Dataset Shift in Classification[END_REF] was first used in the workshop of neural information processing systems (NIPS, 2006). Assume a classification problem is described by a set of features or inputs , a target variable , the joint distribution , the prior probability and conditional probability , respectively. The dataset shift is then defined as "cases where the joint distribution of inputs and outputs differs between training and test stages, i.e., when " [START_REF] Sugiyama | Dataset Shift in Machine Learning[END_REF]. Dataset shift was previously defined by various authors and they gave different names to the same concept such as, concept shift, changes of classification, changing environment, contrast mining, fracture point, and fracture between data. There are three types of dataset shift that usually occur, (i) covariate shift, (ii) prior probability shift and (iii) concept shift; we will briefly describe these shifts in following subsections.

Covariate Shift:

It is defined as a case, where ( and ( [START_REF] Moreno-Torres | A Unifying View on Dataset Shift in Classification[END_REF] (e.g., the input distribution often changes in the session-tosession transfer of an EEG based brain-computer interface). Prior Probability Shift: It is defined as a case, where and [START_REF] Moreno-Torres | A Unifying View on Dataset Shift in Classification[END_REF] (e.g., the survivorship in a stock market where there is bias in obtaining the samples). Concept Shift: It is defined as a case, where [START_REF] Moreno-Torres | A Unifying View on Dataset Shift in Classification[END_REF] (e.g., fraud detection).

The shifts discussed above are the most commonly present in the real-world problems, there are other shifts also that could happen in theory, but we are not discussing those because they appear rarely.

EWMA control chart

An exponentially weighted moving average (EWMA) control chart [START_REF] Roberts | Control chart tests Based on Geometric Moving Averages[END_REF] is from the family of control charts in the statistical process control (SPC) theory. EWMA is an efficient statistical method in detecting the small shifts in the time-series data. The EWMA control chart outperforms the other control charts because it combines current and historical data in such a way that small changes in the time-series can be detected more easily and quickly. The exponentially weighted moving average model is defined as

where is the smoothing constant ( and z is a EWMA statistics. Moreover, the EWMA charts are used for both uncorrelated and auto-correlated data. We are only considering the auto-correlated data in our study and simulation.

EWMA for auto-correlated data:

If data contain a sequence of auto-correlated observations, , then the EWMA statistics in (1) can be used to provide 1-step-ahead prediction model of auto-correlated data. We have assumed that the process observations can be defined as the equation ( 2) below, which is a first-order integrated moving average (ARIMA) model. This equation describes non-stationary behavior, wherein the variable shifts as if there is no fixed value of the process mean.

where is a sequence of independent and identically distributed (i.i.d) random signal with zero mean and constant variance. It can be easily shown that the EWMA with is the optimal 1-step-ahead prediction for this process [START_REF] Dougla | Introduction to Statistical Quality Control[END_REF]. That is, if ̂ is the forecast of the observation in the period made at the end of the period , then, ̂

, where equation ( 1) is the EWMA. The 1-stepahead error are calculated as [START_REF] Ye | Computer Intrusion Detection Through EWMA for Autocorrelated and Uncorrelated Data[END_REF]:

̂
where ̂ is the forecast made at the end of period . Assume, that the 1-step-ahead prediction errors are normally distributed. It is given in [START_REF] Dougla | Introduction to Statistical Quality Control[END_REF], that it is possible to combine information about the statistical control and process dynamics on a single control chart. Then, the control limits of the chart on these errors satisfy the following probability statements by substituting the right hand side of equation (3) as given below.

[ ] [ ̂ ̂ ]

where is the standard deviation of the errors, L is the control limit multiplier and (1-α) is a confidence interval. The standard deviation of 1-step-ahead error can be estimated in various ways such as mean absolute deviation or directly calculate the smoothed variance [START_REF] Dougla | Introduction to Statistical Quality Control[END_REF]. If the EWMA is a suitable 1-step-ahead predictor, then one could use as the center line for the period with UCL (Upper Control Limit) and LCL (Lower Control Limit).

) Whenever, the moves out of and , the process is said to be out of control. This method is also known as a moving center-line EWMA control chart [START_REF] Dougla | Introduction to Statistical Quality Control[END_REF]. The EWMA control chart is robust to the non-normality assumption if properly designed for the and gamma distributions [START_REF] Connie | Robustness of the EWMA Control Chart to Non-Normality[END_REF]. Following the above assumption, we have designed a two-stage algorithm for the shift detection in the process observation of auto-correlated data, which is discussed in the next section. So, the two-stage SD-EWMA (TSSD-EWMA) test can be employed when there is concern about the normality assumption.

Methodology

A control chart is the graphical representation of the sample statistics. Commonly it is represented by three lines plotted along horizontal axis. The center line and two control lines (control limits) are plotted on a control chart, which correspond to target value and acceptable deviation from either side of the target value respectively, where is the control limit multiplier and is the standard deviation.

The proposed two-stage shift-detection based on EWMA (TSSD-EWMA) test works at two-stages. In the stage-I, the method employs a control chart to detect the dataset shift in the data stream. The stage-I works in an online mode, which continuously process the upcoming data from the data stream. The Stage-II uses a statistical hypothesis test to validate the shift detected by the stage-I. The stage-II operates in retrospective mode and starts validation once the shift is detected by the stage-I.

Stage-I

At the stage-I, the test works in two different phases. The first phase is training phase and the second phase is an operation or testing phase. In the first phase, the parameters are calculated to decide the null hypothesis that there is no shift in the data. To calculate the parameters, first obtain the sequence of observations and calculate the mean. Use the mean as the initial value and obtain the EWMA statistics by equation [START_REF] Liu | Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation[END_REF]. The sum of the squared 1-step-ahead prediction error divided by the length of the training dataset is used as an initial value of for the testing data. The test has been performed on the several values of , chosen as suggested in [START_REF] Dougla | Introduction to Statistical Quality Control[END_REF], [START_REF] Connie | Robustness of the EWMA Control Chart to Non-Normality[END_REF] and the choice of is discussed later in the result and discussion section.

In the testing phase, for each observation use equation ( 1) to obtain the EWMA statistics and follow the steps given in algorithm TSSD-EWMA in Table 1. Next, check if each observation falls within the control its [ ], otherwise the shift is detected and alarm is raised at the stage-I. Furthermore, the shift detected by the stage-I is passed to the stage-II for validation.

Stage-II

In stage-II, the shift detected by the stage-I needs to be validated. This phase works in retrospective mode and it executes only when a shift is detected at stage-I. In particular, to validate the shift detected by the stage-I, the available information need to be partitioned into two-disjoint subsequences and then the statistical hypothesis test is applied. In literature, the statistical hypothesis tests are well established. The twosample Kolmogorov-Smirnov test [START_REF] Snedecor | Statistical Methods, Eight[END_REF] is used to validate stationarity in the subsequences because of its non-parametric nature. The two-sample Kolmogorov-Smirnov test returns a test decision for the null hypothesis that the data in the subsequences are stationary with equal means and equal but unknown variances. The Kolmogorov-Smirnov statistics is briefly described as follows:

where is the supremum and and are the empirical cumulative distribution function on the first and second samples respectively. The and are the lengths of the two samples given as (( ) ) and where is the number of observations in the window of data used for the test. The null hypothesis is rejected at level α and (H=1) is returned if, √ where is the critical value and can be found in, e.g., [START_REF]Table of Critical Values for The Two-Sample Test[END_REF].

Table 1. Algorithm-TSSD-EWMA

An important point to note here is that we have assumed that nonstationarity occurs due to changes in the input distribution only. So, it is said to be covariate shift-detection in non-stationary time-series by the TSSD-EWMA test. In Input: A process x(i), generates independent and identically distributed observations over time (i).

Output: Shift-detection points

Stage-I Training Phase

1. Initialize training data x(i) for i=1:n, where n is the number of observations in training data 2. Calculate the mean of x(i) and set as z(0) 3. Compute the z-statistics for each observation x(i) z(i)=lambda*x(i)+(1-lambda)*z(i-1) 4. Compute the 1-step-ahead prediction error: err(i)=x(i)-z (i-1) 5. Estimate the variance of error for the testing phase 6. Set lambda by minimizing squared prediction error Testing Phase 1. For each data point x(i+1) in the operation/testing phase 2. Compute z(i)=lambda*x(i)+(1-lambda)*z(i-1) 3. Compute err(i)= x(i)-z(i-1) 4. Estimate the variance sigma_hat_err(i)^2=phi*err(i)^2 + (1phi)* sigma_hat_err(i-1)^2 5. Compute UCL(i+1) and LCL(i+1) 6. IF(LCL(i+1)< x(i+1)< UCL(i+1))

THEN (Continue processing) ELSE (Go to Stage-II)

Stage-II 1. For each x(i+1) 2. Wait for m observations after the time i, organise the sequential observations around time i into two partitions, one containing x((i-(m-1)):i), another x((i+1):(i+m)).

Execute the hypothesis test on the partitioned data IF(H==1) THEN (test rejects the null hypothesis): Alarm is raised ELSE(The detection received by stage-I is a false-positive)

Table 1, the designed algorithm TSSD-EWMA is presented in the form of pseudo code.

Dataset and Feature Analysis

To validate the effectiveness of the suggested TSSD-EWMA test, a series of experimental evaluations have been performed on three synthetic datasets and one realworld dataset. The datasets are described as follows.

Synthetic Data

Dataset 1-Jumping Mean (D1):

The dataset used here is same as the toy dataset given in [START_REF] Liu | Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation[END_REF] for detecting change point in a time-series data. The dataset is defined as in which 5000 samples are generated where is a noise with mean and standard deviation 1.5. The initial values are set as . A change point is inserted at every 100 time steps by setting the noise mean at time as { where is a natural number such that .

Dataset 2-Scaling Variance (D2):

The dataset used here is the same as the toy dataset given in [START_REF] Liu | Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation[END_REF], for detecting change-point in time-series data. The dataset is defined as the auto-regressive model, but the change point is inserted at every 100 time steps by setting the noise standard deviation at time as

{
where is a natural number such that

Dataset 3-Positive-Auto-correlated (D3):

The dataset is consisting of 2000 datapoints, the non stationarity occurs in the middle of the data stream, shifting from to , where denotes the normal distribution with mean and standard deviation respectively.

Real-world Dataset

The real-world data used here are from BCI competition-III (IV-b) dataset [START_REF] Klaus | BCI Competition III: Data set IVb[END_REF]. This dataset contains 2 classes, 118 EEG channels (0.05-200Hz), 1000Hz sampling rate which is down-sampled to 100Hz, 210 training trials, and 420 test trials. We have selected a single channel (C3) and performed the band-pass filtering on the mu ( ) band [START_REF] Raza | Dataset Shift Detection in Non-Stationary Environments using EWMA Charts[END_REF][START_REF] Moreno-Torres | A Unifying View on Dataset Shift in Classification[END_REF][START_REF] Sugiyama | Dataset Shift in Machine Learning[END_REF][START_REF] Roberts | Control chart tests Based on Geometric Moving Averages[END_REF][START_REF] Dougla | Introduction to Statistical Quality Control[END_REF] to obtain bandpower features. This real-world dataset is a good example to validate if the proposed TSSD-EWMA is able to detect the types of shifts in the data generating process with decreased false-positive. The results and discussion of the experiments are given in the next section.

Results

On each dataset, the proposed TSSD-EWMA technique is evaluated by the following metrics to measure the performance of the tests: False-positive (FP): it counts the times a test detects a shift in the sequence when it is not there i.e., (false alarm); True-positive (TP): it counts the times a test detects a shift in the sequence when it is there i.e., (hit alarm); True-negative (TN): it counts the times a test does not detect the shift when it is not there; False-negative (FN): it counts the times a test does not detect a shift in the sequence when it is there i.e., (miss); Not applicable (NA): it denotes not applicable situation, where the dataset is not suitable to be executed on the test; Average delay (AD): it measures the average delay in shift-detection, i.e., sum of delay in each shift-detection is divided by the number of shifts detected; Accuracy (ACC): it measures the accuracy of the results i.e., It is important to note that the stage-I of the TSSD-EWMA is based on the current observation of the data to detect the shift. So, it is coming without the delay and this is the advantage of this approach that the stage-I of the test always operate in online mode. Once, the shift is detected by the stage-I, the test moves into the retrospective mode to validate the suspected shift in the process. Figure 2 represents the stage-I of the TSSD-EWMA based shift-detection test results from the Dataset 1. The solid line is the observation plotted on the chart and the two dotted lines are the ULC and LCL, whenever the solid line crosses the dotted line (control limits), it is the shiftpoint detected. The tests have been performed on several values of as suggested in [START_REF] Dougla | Introduction to Statistical Quality Control[END_REF], [START_REF] Connie | Robustness of the EWMA Control Chart to Non-Normality[END_REF].

According to Table 2, for the case of D1 with , the TSSD-EWMA gives an optimal result with all the shift-detections and no false-positive, however with a delay of 10 samples, whereas, SD-EWMA detects four false-positive with no delay. The delay in TSSD-EWMA is resulting from the stage-II of the test. For the

9

D2, the TSSD-EWMA is not applicable because of the nature of the dataset. For D3 in the TSSD-EWMA, when it detects all the shifts with no false-positive and a low delay of 10 samples, whereas, the SD-EWMA test suffers twelve false-positives and no delay. To assess the performance and compare the results of the TSSD-EWMA, we have chosen other shift-detection methods such as the SD-EWMA [START_REF] Raza | Dataset Shift Detection in Non-Stationary Environments using EWMA Charts[END_REF] and the ICI-CDT [START_REF] Alippi | Change Detection Tests Using the ICI Rule[END_REF] because these are state-of-the-art non-parametric sequential shift-point detection tests. Table 3 compares the results of ICI-CDT and SD-EWMA with TSSD-EWMA. The rate of false-positive is much reduced for D1 and D3 datasets. However, in the case of D2 the test is not applicable because of the nature of dataset. The delay in the shift-detection for the TSSD-EWMA is less than that for the ICI-CDT method. To perform the shift-detection test, on the real-world dataset we have made use of bandpower features from the EEG data of session one from the aforementioned BCI competition-III (IV-b) and assumed that it is in stationary state. It contains 70 trials and the parameters are calculated in training phase to be used in the testing/operational phase. Next, the test is applied to the sessions two and three and the results are given in Table 4, wherein TSSD-EWMA is compared with the SD-EWMA. For evaluation purposes, we have performed the test on a fixed number of trials from each session and monitored the points where the shifts are detected. For the value of λ=0.01, and the trials 21-45 and 46-70, one false-positive is reduced by the TSSD-EWMA in each case. With increased value of , the number of shifts increased, so the possibility of getting the number of false-positive also increased. Hence, by executing the TSSD-EWMA, the number of false alarms decreased. In case of the EEG signals, no comparison can be done because the results of actual shift points are not provided in [START_REF] Klaus | BCI Competition III: Data set IVb[END_REF].

Discussion and Conclusion

According to Table 3, the TSSD-EWMA provides better performance in terms of shift-detection with reduced number of false-positives compared to other state-ofthe-art methods such as the SD-EWMA [START_REF] Raza | Dataset Shift Detection in Non-Stationary Environments using EWMA Charts[END_REF] and ICI-CDT [START_REF] Alippi | Change Detection Tests Using the ICI Rule[END_REF], and it also has much smaller time delay compared to ICI-CDT. The TSSD-EWMA test outperforms over other methods in terms of low false-positive rate for all datasets because of its twostage structure. The choice of smoothing constant is an important issue in using EWMA control charts. The TSSD-EWMA results show that the detection of shift in the time-series data depends upon the value of the . The value of can be obtained by minimizing the least mean square prediction error on the datasets. However, selecting the value of by minimizing the least mean square prediction error does not always lead to a good choice. Moreover, if the value of the smoothing constant is large the test becomes more sensitive to spurious small shifts and it contains more number of false-positives. Hence, there is a trade-off between the smoothing constant and shift detection.

For the real-world dataset, we have assumed that the data from BCI session 1 is in stationary state and investigated for an optimal value of the smoothing constant considering the session 1 data as the training dataset. We have tested several values of in the range of (0.01-0.1). As the value of lambda increases, the number of detected shifts increases. Thus the smaller value of is a better choice for shiftdetection in EEG based BCI, as it avoids shift-detections resulting from noise or spurious changes through much more intense smoothing of the EEG signal. Moreover, for correlated data, the smaller values of produce smaller prediction errors thereby resulting in smaller estimated standard error. If is too small, the performance of the test results in less false-positive rate but it tends toward getting more false-negatives, because chance of missing the shift is increased by much smoothing of the EEG signal. In summary, the experimental results demonstrate that TSSD-EWMA based test works well for shift-detection in the non-stationary environments.

This paper presented a method of TSSD-EWMA for detecting the shift in data stream based on the two-stage detection architecture. The advantage of using this method over other methods is primarily in terms of having a reduced number of falsepositive detections. Our method is focused on auto-correlated data, which contain non-stationarities. Experimental analysis shows that the performance of the approach is good in a range of non-stationary situations. This work is planned to be extended further by employing it into pattern recognition problems involving multivariate data and an appropriate classifier.

Fig. 1 .

 1 Fig. 1. The structure of the proposed two-stage shift-detection test

Fig. 2 .

 2 Fig. 2. Shift-detection based on TSSD-EWMA: Dataset 1 (jumping mean): (a) the shift point is detected at every 100 th point. (b) Zoomed view of figure a: shift is detected at 401 st sample.

Table 2 .

 2 TSSD-EWMA shift-detection

					SD-EWMA			TSSD-EWMA
		Total	Lambda	#	#	#		#	#	#
		shifts	( )	TP	FP	FN	ACC	TP	FP	FN	AD	ACC
	D1	9	S 0.20 h 0.30 i f 0.40	8 8 All	1 3 4	1 1 0	99.8 99.6 99.6	8 8 All	0 0 0	1 1 0	10 10 10	99.9 99.9 100
			t 0.60	All	11	1	99.8	NA	NA	NA	NA	NA
	D2	5	s 0.70	All	6	0	99.4	NA	NA	NA	NA	NA
			0.80	4	8	1	99.1	NA	NA	NA	NA	NA
			0.40	All	13	0	99.1	All	0	0	10	100
	D3	1	0.50	All	12	0	99.2	All	0	0	10	100
			0.60	All	15	0	99.0	All	0	0	10	100

TABLE 3 :

 3 Comparison on different shift-detection tests

			ICI-CDT			SD-EWMA			TSSD-EWMA
		#	#	#		#	#	#		#	#	#
		TP	FP	FN	AD	TP	FP	FN	AD	TP	FP	FN	AD
	D1	6	0	3	35	9	4	0	0	9	0	0	10
	D2	1	0	4	60	5	6	0	0	NA	NA	NA	NA
	D3	1	0	0	80	1	12	0	0	1	0	0	10

TABLE 4 :

 4 TSSD-EWMA shift-detection in BCI data

	Lambda
	( )

Number of Trials # Shift-points SD-EWMA # Shift-points TSSD-EWMA Session 2 Session 3 Session 2 Session 3

		1-20	0	0	0	0
	0.01	21-45	3	1	2	0
		46-70	4	2	3	1
		1-20	10	4	6	3
	0.05	21-45	9	6	6	4
		46-70	7	5	6	4
		1-20	16	9	8	6
	0.10	21-45	22	21	15	15
		46-70	25	17	17	13


Georgios M Milis
email: milis.georgios@ucy.ac.cy

Christos G Panayiotou
email: christosp@ucy.ac.cy

Marios M Polycarpou

Exploring semantic mediation techniques in feedback control architectures

Keywords: Control system architecture, interoperability, scalability, semantic knowledge models, plug&play of components

Modern control systems implementations, especially in largescale systems, assume the interoperation of different types of sensors, actuators, controllers and software algorithms, being physical or cyber. In most cases, the scalability and interoperability of the control system are compromised by its design, which is based on a fixed configuration of specific components with certain knowledge of their specific characteristics. This work presents an innovative feedback control architecture framework, in which classical and modern feedback control techniques can be combined with domain knowledge (thematic, location and time) in order to enable the online plugging of components in a feedback control system and the subsequent reconfiguration and adaptation of the system.

Introduction

Nowadays, systems are designed and built not as monolithic entities but as collection of smaller physical and cyber components that, many times, can be considered as separate systems themselves with their own dynamics and objectives. This system of systems paradigm ( [START_REF] Samad | Systems of Systems[END_REF]) necessitates the easy interaction and interoperability of the components that comprise a larger system. Components are expected to take informed decisions and act intelligently towards meeting (or balancing) the system's objectives. This description is valid also for modern control systems, where different types of components, being physical or cyber, interoperate in a larger control system implementation. However, in most cases, the design of feedback control systems is based on a fixed configuration of specific components, with certain knowledge of their specific characteristics. This causes lack of scalability and interoperability for the control system, thus considerably limiting its potential lifetime. There are cases where faulty sensors need to be replaced or additional sensors need to be installed (e.g. due to recent availability of this type of components or due to upgrading to new technology), and this should not require redesign of the overall feedback control system since such action would be impractical and costly.

The detection and identification of non-modelled events in linear and nonlinear systems is currently addressed by the fault diagnosis research area. The authors in [START_REF] Gertler | Fault detection and diagnosis in engineering systems[END_REF] and [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] provide a thorough overview on the classical algorithms that identify deviations from the normal behaviour of a system, attributed to faults or other external events. The adaptive and fault-tolerant control research areas address the design of intelligent control type algorithms, that aim to facilitate the flexibility of the control system with respect to on-line adaptation, and accommodation of faults, system uncertainties and/or time variations. Approaches to designing fault tolerant and reconfigurable control systems are presented in [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF]. In [START_REF] Isermann | Fault-diagnosis systems: an introduction from fault detection to fault tolerance[END_REF] the author also addresses the issue of fault-tolerant components, while the authors in [START_REF] Farrell | Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches[END_REF] provide methodologies for designing adaptive approximation-based control systems. Recent efforts in plug&play control ( [START_REF] Stoustrup | Plug & Play Control: Control Technology Towards New Challenges[END_REF] and more recently in [START_REF] Bendtsen | Plug-and-Play ControlModifying Control Systems Online[END_REF]), propose methodologies for the online identification of newly introduced dynamics when new components are plugged in a closedloop system and the subsequent online adaptation of the feedback laws. Also, the authors of the present paper have recently presented initial results of their work [START_REF] Milis | Towards a Semantically Enhanced Control Architecture[END_REF] on the exploitation of ontology-based semantic mediation techniques in feedback control systems.

The main contribution of our work is the design of an innovative feedback control architecture framework, in which classical and modern feedback control techniques can be combined with domain knowledge (thematic, location and time) in order to enable the online plugging of components in feedback control systems and their subsequent reconfiguration and adaptation. The control system becomes able to make use of and enrich thematic, location and time related structured knowledge about the environment in which it operates.

The rest of the paper is organised as follows: Section 2 formulates the problem, to facilitate the presentation of the solution. Then, section 3 presents the proposed architecture and framework, followed by section 4 where a case-study scenario is given. Finally, section 5 shows a simulation with results and section 6 concludes the report.

Problem Formulation

Consider a closed-loop system with sensors measuring plant outputs, actuators acting on controlled inputs following instruction by a control law that considers an error trajectory. The actual plant states are estimated by an observer (e.g. a Luenberger observer [START_REF] Luenberger | Observers for multivariable systems. Automatic Control[END_REF]), to compensate for the case when some of them are missing, or for redundancy and noise cancellation.

Consider the following cases:

1 A deployed sensor fails and is replaced by a new one having different (and not compliant with the closed-loop system implementation) characteristics. 2 Sensor(s) enter the plant, at different locations and at different times. These sensors measure physical quantities that are already considered as states in the closed-loop system design.

3 Sensor(s) enter the plant as above, but this time some or all of them measure quantities not already taken into consideration for the initial design of the closed-loop system.

When changes happen in the components' synthesis of the closed-loop system, as explained in Section 2, the altered measurement vectors carry new sensing capabilities that can be potentially exploited using different models of the same plant. Therefore, at discrete time steps, the closed-loop system may assume a different model, with different types and/or dimensions of variables and parameters respectively. Without loss of generality, we assume this system is described by the state-space model in [START_REF] Antoniou | Web ontology language: Owl[END_REF]. The top-pointer I = 1, 2, ..., is utilised to distinguish among different models.

ẋ(I) = A (I) x (I) + B (I) u (I) a + G (I) d (I) y (I) a = C (I) x (I) + D (I) u (I) a + H (I) d (I) + ν (I) (1)
where (avoiding the pointer I for simplicity): x ∈ R n is the vector of system states, u a ∈ R m is the vector of controlled inputs, d ∈ R q is the vector of uncontrolled inputs, y a ∈ R p is the vector of outputs (measurements), ν ∈ R p is the vector of measurement noise and A, B, G, C, D, H are the parameter matrices of proper dimensions and content.

The output part in [START_REF] Antoniou | Web ontology language: Owl[END_REF], can be written as follows. Note that the signal is split into two parts to facilitate the analysis. Moreover, an extra top-pointer is used to indicate the signals that are changing between cases.

y (0) a = y (0) a1 y (0) a2 = C (0) 1 C (0) 2 x (0) + D (0) 1 D (0) 2 u (0) a + H (0) 1 H (0) 2 g (0) + ν (0) 1 ν (0) 2 (2)
Then, the three cases identified above, lead to the equations:

y (1) a = y (0) a1 y (1) a2 = C (0) 1 C (1) 2 x (0) + D (0) 1 D (1) 2 u (0) a + H (0) 1 H (1) 2
g (0) + ν (0) 1 ν (1) 2
(3)

y (2) a =    y (0) a1 y (0) a2 y (0) a2    =    C (0) 1 C (0) 2 C (0) 2    x (0) +    D (0) 1 D (0) 2 D (0) 2    u (0) a +    H (0) 1 H (0) 2 H (0) 2    g (0) +    ν (0) 1 ν (0) 2 ν (0) 2    (4)
y (3) a =    y (0) a1 y (0) a2 y (3) a2    =    C (0) 1 x (0) + D (0) 1 u (0) a + H (0) 1 g (0) + ν (0) 1 C (0) 2 x (0) + D (0) 2 u (0) a + H (0) 2 g (0) + ν (0) 2 C (3) 2 x (3) + D (3) 2 u
(3)

a + H (3) 2 g (3) + ν (3)    (5)
Equation 3 shows that a part of the sensing signals have been modified, comparing to specifications, resulting in a modified output vector. Equation 4shows that the output vector has been modified not only in terms of content but also in terms of dimension, while still measuring same quantities. Finally, [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] shows that the output vector has been modified in terms of dimension and the newly introduced part measures different quantities. All described cases need to be properly accommodated in the closed-loop system by utilising available new knowledge and tools.

Proposed Architecture and Framework

In an earlier work, [START_REF] Milis | Towards a Semantically Enhanced Control Architecture[END_REF], the authors presented a basic introduction of the semantic interoperability concepts and the ontologies as a tool to implement knowledge models. Such models have been also used in domestic robotics (DOGont, [START_REF][END_REF]) to face the interoperation issues by implementing structured representations of domain knowledge. In this work, we adopt knowledge models in combination with control engineering mathematical representations. Efforts to represent the mathematical models in ontological knowledge models can be found in [START_REF] Suresh | OntoMODEL: Ontological Mathematical Modeling Knowledge Management[END_REF] and [START_REF] Lange | Ontologies and languages for representing mathematical knowledge on the semantic web[END_REF]. The proposed architecture for the closed-loop system is depicted in fig. 1. As illustrated, the set-up comprises the: i) plant with its parameters and selfdynamics, ii) the physical control system components, like sensors (producing y) and actuators of different types (producing u a ), iii) a tools' base, which stores the implementations of software functions such as observer design implementations producing the state estimation x, functions performing transformations among measurement units, etc., iv) the humans (e.g. Control Systems Engineer, Knowledge Models Engineer), v) a communication infrastructure (the orangedashed line shows the BACnet/IP protocol stack communication [START_REF]ASHRAE: BACnet Website[END_REF], whereas the blue-dashed line shows communication through any other protocol), vi) a semantic mediation module, M , which is responsible for the scalability of the control system and vii) a Knowledge Model, implemented as OWL ontology(ies) [START_REF] Antoniou | Web ontology language: Owl[END_REF].

A critical component introduced here is M , which has a multi-fold scope as it implements the physical interaction interface among all components. The semantic mediation module strongly relies on the knowledge model to analyse each time's situation and take reasonable and optimal decisions for the operation of the system. It is therefore, of utmost important for the knowledge model to be well designed and defined based on the "closed-world" assumption [START_REF] Mazzocchi | Closed World vs. Open World: the First Semantic Web Battle[END_REF]. We want the knowledge model to support control systems, that might comprise also safety-critical deployments, so the decisions taken should be based on explicit knowledge such as to avoid instability.

The knowledge model

The knowledge model comprises the agreement between all interacting physical and cyber components, about the interpretation of their environment.

This model is implemented as a set of objects' symbols, a set of classes/types for these objects and a set of properties of objects that also implement relations/mappings among them, that is, A = { T H , C L , P R }. For the purpose of this work, we define specific objects, types of objects and properties. In order to keep it simple, we developed our own mini knowledge model. In future practical implementations, this model can be replaced by more complete efforts from the literature, such as combinations of the knowledge models in [START_REF] Holger | The Semantic Sensor Network Ontology : A Generic Language to Describe Sensor Assets[END_REF] to describe the environment and interactions of components, and the ones in [START_REF] Suresh | OntoMODEL: Ontological Mathematical Modeling Knowledge Management[END_REF] and [START_REF] Lange | Ontologies and languages for representing mathematical knowledge on the semantic web[END_REF] to describe the knowledge in mathematical representations.

The set of objects is defined as:

T H = { o i | i = 1, 2, ..., }
, where o i is the reference to an object's literal (e.g. the physical property "temperature") or to the real implementation of the object (e.g. "Sensor1" meaning the device with that identification).

The following classes of objects have been defined: C L = {P lant, M odel, State, ControlledInput, U ncontrolledInput, Output, P lantLocation, P hysicalP roperty, M easurementU nit, Sensor, Actuator, F unction} where: P lant is the set of plants served by the knowledge model, M odel is the set of system models (e.g. a state-space model of the system), State is the set of system states, ControlledInput is the set of controlled inputs, U ncontrolledInput is the set of uncontrolled inputs (disturbances to the plant), Output is the set of measurable outputs of plant, P lantLocation is the set of identified locations in the plant, P hysicalP roperty is the set of defined physical properties (e.g. temperature, energy), M easurementU nit is the set of units for the defined physical properties, Sensor is the set of sensing devices deployed in the plant, Actuator is the set of actuating devices deployed in the plant, F unction is the set of functions/mappings defined to represent the mathematical relations among variables.

Then, relations are defined, to represent the properties of objects. A relation is a mapping of the form: relationN ame : C L(i) × C L(j) → { , ⊥}. These may define whether an object belongs to a specific class, the relation between a plant and a model, the relation between a model and a state of the plant, the relation between an input/output of the plant and a physical property, the physical property that is measured in a specific unit, the location where a sensing/actuating device is located in, etc.

The controller and observer implementations

Upon a shift to a different model of the plant, as a result of the inference step, the implementations of the controller and the state-observer change. The new implementations, are either given and retrieved from the knowledge base or they are calculated online. We assume the actuators are driven by a simple proportional controller, while the system states are estimated (mostly for compensation of missing measurements) with a simple Luenberger full-state observer [START_REF] Luenberger | Observers for multivariable systems. Automatic Control[END_REF], as shown in [START_REF][END_REF].

u = Ke + u 0 e = x d - x ˆx = Ax + Bu + G d + LW (y -C x) (6)
where K ∈ R m×n is the control gain matrix, e ∈ R n is the error signal (difference between desired and estimated state values), u 0 ∈ R m is the control bias that is used to cancel system disturbances, model uncertainties and retain the system at desired operation, x d ∈ R n is the desired system states' vector, x ∈ R n is the estimated system states' vector, d ∈ R q is the estimated uncontrolled inputs' vector, if such option exists, L ∈ R n×n is the observer gain, implemented such as the pair (A, W C) is stable and W ∈ R n×p is a weight matrix that represents the trust on each of the p measurements.

Case-Study Scenario

We assume an apartment with three rooms as shown in figure 2a. The apartment is equipped with a central heating installation, however, for budget reasons there is only one heating radiator in the bedroom, accompanied by one temperature sensor in the same room that measures in degrees Celsius. The equipment is used to regulate the temperature of the apartment at desired value. The design also assumes an uncontrolled input to the plant produced by the ambient temperature and modelled by a slightly open window (for simplicity we consider zero transfer of heat through the walls).

The case of replacing a sensor with another one of not compatible specifications, is described by (3) and has been specifically addressed in [START_REF] Milis | Towards a Semantically Enhanced Control Architecture[END_REF]. Here we consider the owner of the apartment buying a smart phone, which is equipped with temperature sensor. This mobile sensor is entering and leaving the apartment at different occasions during a day, therefore, at discrete sampling times the mediation component is retrieving more than one measurements. This case is described by [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF]. The knowledge base helps the mediation component to retrieve the measurements and also build the weight matrix W , while the observer continues to producing estimates of the state and the mediation component now feeds the control law with the fused sensors' measurements. This allows benefiting from the availability and accuracy of redundant information.

Later in time, the apartment is bought by an ICT company and is turned into an open-plan, as shown in fig. 2b. Soon after, they notice that people working in Room 3, do not feel comfortable and wear heavy clothes. So, they install temperature sensors in the other two rooms as well. In parallel, a control engineer is asked to design higher-order models of the apartment heating system. For simplicity we consider a manual design of the models, while an alternative would be for an adaptive algorithm like the one in [START_REF] Bendtsen | Plug-and-Play ControlModifying Control Systems Online[END_REF] to be used in closed-loop operation. The closed-loop system now fully incorporates the sensing information available (increases the order of the model) and it is now able to maintain better temperature conditions across all rooms (of course with the limited capacity of the single actuator). This case is described by [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF].

Simulations and Results

The devices are implemented as virtual BACnet/IP-enabled devices, using the BACnet4J API [START_REF]Serotonin-Software: BACnet I/P for Java[END_REF]. Their semantic descriptions (e.g. for a sensor, the measurement unit, its location in the plant, etc.) are created and stored in the knowledge base. Next, three plant models (1st-, 2nd-and 3rd-order) are created and stored in the knowledge base. We use the Newton's law of cooling, Q = cA(x i -x j ), i = j, with Q the heat transfer in J/sec, c the heat transfer coefficient, A the area of the surface through which the heat flows and x i , x j the temperatures in the two sides of the surface, to derive linear state-space models of the closed-loop system in the form of [START_REF][END_REF]. For each of the models, the ambient temperature is acting as an uncontrolled input. In addition, further scenariorelated parameters are defined, like the steps of the electric radiator output, a model for the outside temperature, the simulation time (50 hours), the desired temperatures of rooms (25 0 Celcius) and the initial temperatures of the rooms.

Initially, sensor1 and radiator1 are installed in Room1 (bedroom). At time 08:00, sensor1 breaks and is replaced by sensor4. At time 10:00 mobile sensor5 enters Room1. At time 22:00, sensor2 is installed in Room2 and sensor3 is installed in Room3. Finally, at time 36:00, sensor6 and sensor7 enter Room3 and both leave at time 39:00. The simulation runs in 1-minute steps. During each step, the mediation component reads and stores the sensors' measurements together with their time-stamp. The processing of the measurements and the calculation of the control input is performed at 5-minute intervals. At each such step, the mediation component retrieves information about the current measurements. These are discarded if they were taken more than 2.5 minutes earlier. Moreover, in case a value is in a different unit than the one required by the current control law, the mediation component runs an inference rule [START_REF]W3C: SWRL: A Semantic Web Rule Language Combining OWL and RuleML[END_REF] and retrieves the literal name of the function to invoke (from those in the Tools base) in order to perform the required transformation. The rule says: "Find the name of the function that takes as input a real value of the given sensor's measurement unit and produces a real value in the desired measurement unit". If no such function is returned, the measurement is discarded. In the implementation of the rule, the given and the desired measurement units are denoted as o 1 , o 2 ∈ M easurementU nit and any symbols starting with "?" denote a variable that can take as value an object from the knowledge base of the class accepted as argument by the specific relation. The rule is written as:

z 1 ∈ Z = F unction(?x) ∧ hasDomain(?x, o 1 ) ∧ hasRange(?x, o 2 ) -→ Inf erredInd(?x)
At that moment, in case there was any change in the sensors that comprise the closed-loop system, the mediation component retrieves the best available plant model to use for the operation of the controller, given the locations and the measurement properties/units. To this end, several inference rules are executed in the knowledge base. The first one is the:

z 2 ∈ Z = Output(?x) ∧ [associatedW ithLocation(?x, o 3 ) ∨ [associatedW ithLocation(?x, ?y) ∧ isP artOf (o 3 , ?y)]] ∧ isP hysicalP roperty(?x, o 4 ) -→ Inf erredInd(?x)
where o 3 ∈ Location and o 4 ∈ P hysicalP roperty are the given measurement location and the measured physical property of the sensor, respectively. The above rule means: "Find all available outputs of models, that are either associated directly with the given location or are associated with a different location which is, however, defined as part of the given location, and that are associated with the given physical property (e.g. temperature)". At the end, a decision algorithm is invoked which finds the model that is of the highest order, while still controllable and observable under current situation, and returns its constant parameter matrices as defined in ( 1) and [START_REF][END_REF]. At this moment, the dimensions of all vectors and all parameters of the model to be used, are considered known. Given the new model, the mediation component invokes functions to calculate the observer's and the controller's gain matrices, L (I) and K (I) respectively. This is performed with simple pole placement for observability (pair A, W C) and stability (pair A, B). The new state vector estimation is based each time on the model and the designed observer. The observed state values are used by the controller to compute the next control input value. It is noted that the control input retains the previous value until a new one is produced.

The result is that the closed-loop system is able to transparently integrate any new component and use the new information to operate smoothly despite the events introduced during operation. No downtime or manual re-configuration are required.

Conclusions

We have presented a new architecture that can be adopted in the design of feedback control systems, in order to take advantage of the scalability characteristics offered by the combination of the classical control capabilities with a cyber infrastructure and semantic interoperability protocols and interfaces.

The scope of the work was not to advance the control algorithms as such. The current industrial practice suggests using standard controllers (e.g. PID) and applying the interoperability of components at higher application levels. We believe that much more advance intelligent control algorithms, already developed in the literature, can enormously impact the industrial applications if there is a framework for their deployment in large feedback control systems.

There is still lot of work to be done, before we can claim achieving the objectives of this work. Our immediate next steps will be the thorough investigation of the closed-loop system stability within the proposed architecture, as well as, the implementation of a demonstration setup that will pilot test the applicability in real-life scenarios.

Fig. 1 .

 1 Fig. 1. Block diagram of proposed feedback-control architecture. Details about the content of this figure are given in section 3

Fig. 2 .

 2 Fig. 2. a. The apartment plant. Q1 represents the heating input produced by the electric radiator, Ti, i ∈ 1, 2, 3, a represent the temperature of the three rooms and the ambient respectively, qij, with i, j ∈ 1, 2, 3, a representing the flow of heat among the rooms and the ambient, b. The office plant with open doors

Acknowledgments. This work is partially funded by the European Research Council (ERC) under the project "Fault-Adaptive Monitoring and Control of Complex Distributed Dynamical Systems".