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Spatial statistics — A watery business
Spatial Statistics 1 (2012) 121–132
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Spatial Statistics journal homepage: www.elsevier.com/locate/spasta
Spatial statistics — A watery business E. Marian Scott ∗ , J. Campbell Gemmell 1 School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, UK
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Article history: Received 22 December 2011 Accepted 22 March 2012 Available online 29 March 2012 Keywords: Environmental policy Statistical models Regulation and evidence
abstract Spatial statistics as a sub-discipline has a long tradition, but modern environmental science is offering new challenges. In this short commentary paper, we consider the specific challenges posed by environmental policy, regulation and management for the freshwater environment, focussing on two specific pieces of European legislation, namely the Water Framework and the Floods Directives. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Environmental statistics typically but not always have a spatial context, so spatial statistics become important in the process of exploring and discovering the relationships, drivers and pressures and responses of the environment. Especially striking are the environmental (and spatial) challenges in dealing with the water environment, such as monitoring and modelling water quality or in evaluating flood risks. The environment is a natural landscape on which we can try to drape/fold our statistical model outputs, and thinking in this way recognises the holistic view of the complex interconnections which operate at different spatial and temporal scales. The current reality of what is possible means that we are only part of the way towards this ideal view—what we have in reality is (sometimes) a poor surrogate for those complex inter-connections, limited by our inability to measure what we are interested in with the spatial and temporal support that reflects the scales at which the environmental processes operate. New sensor system developments may open up opportunities in the near future, offering spatially distributed, high frequency observations (with their own statistical challenges Kirchner et al., 2004).
∗ Correspondence to: School of Maths and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK. Tel.: +44 141 330 5125; fax: +44 141 330 4814. E-mail addresses: [email protected] (E.M. Scott), [email protected] (J.C. Gemmell). 1 Environment Protection Authority, South Australia, Adelaide 5001, Australia (formerly of the Scottish Environment Protection Agency). 2211-6753/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spasta.2012.03.004
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Many of the common tools used in a spatial analysis have developed in different disciplines—and this is reflected in some of the emphases and languages used. GIS (geographic information systems), traditionally was very commonly used in Geography and Earth Sciences departments but not so widely used by statisticians, while geostatistics was developed in engineering, while at the same time, statisticians were developing their own framework and language for spatial models. However, now there is much more commonality, convergence and cross-disciplinary working, driven in part by the urge to visualise and communicate outside our silos, and to tackle challenges such as those in climate change research (where tools that can handle massive datasets are needed). We make progress, often driven by practical challenges, and this journal offers, along with others, a forum for our communities to reflect on that progress and drive it forward. So in the light of challenges—scientific, political and societal, we have chosen to focus this commentary on the freshwater environment and to consider some of the spatial (and statistical) challenges set by the current European regulatory framework. 1.1. Why water? The IPCC (2008) in its 2008 technical paper on Climate Change and Water wrote ‘‘observational records and climate projections provide abundant evidence that freshwater resources are vulnerable and have the potential to be strongly impacted by climate change with wide ranging consequences on human societies and ecosystems’’. A simple summary would be that both quantity and quality of water are of global, regional and local importance. The first step in improving our understanding of potential impacts and consequences to water quality and quantity is to consider the current state of the water environment (a snapshot in time) and the historical evidence of past state and its trajectory (over space and time), in the light of the regulatory framework and any environmental measures. Within the European Union, there are a number of regulatory frameworks dealing with the aquatic environment, of which the Water Framework Directive (WFD, 2000) which brought together a large suite of earlier directives and policy elements, transposed in Scotland as the Water Environment and Water Services Act (Scottish Government, 2003), to protect, improve and promote sustainable use of Scotland’s water environment, is perhaps the most significant. The other that will be briefly considered in this paper is the Floods Directive (FD, 2009), subsequently the Flood Risk Management (Scotland) Act (Scottish Government, 2009) which requires a national assessment of flood risk by the end of 2011, and flood risk and hazard maps by 2013. The WFD has created a six year cycle of policy implementation and reporting which will be captured in the synchronous and systematic reporting of water quality in River Basin Plan Reports, beginning from the baselines reported in 2009/10 in 2015. In terms of reporting on policy effectiveness and ultimately on the state of the freshwater environment, there are three basic questions to consider: (a) What is happening? (b) Why is it happening? (c) Are the changes significant? We consider the elements of spatial statistics used in answering these three questions for each of two regulatory regimes (WFD and FD) for Scotland and consider the role of statistical modelling in delivering improved understanding and modelling changes in the hydrological cycle at a scale relevant to decision making. 2. The policy framework Two EU directives are briefly introduced below, namely WFD and FD to illustrate the policy language. The objectives of the WFD include:
• To prevent deterioration in the status of surface water bodies. • To protect, enhance, and restore all bodies of surface water. • The establishment of systems for managing water environments, underpinned by extensive environmental monitoring and scientific investigation.
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A key component of the WFD is River Basin Management Planning, setting out the environmental pressures and what can be done to address them—in terms of regions and catchments and the control of all impacts—physical, polluting and otherwise—on the water environment with the aim of achieving ‘‘good’’ ecological status for water bodies by 2015. Status is determined on the basis of ecology not solely by chemical composition. Fig. 1 shows a map of the overall status of surface waters in Scotland for 2009, and underlines the spatial context of the WFD. The Floods Directive requires Member States to undertake (i) a preliminary flood risk assessment, (ii) develop flood hazard and flood risk maps and (iii) produce flood risk management plans for zones at risk of flooding. An important change is the introduction of flood risk management plans. These plans consider all forms of flood risk from rivers, groundwater and coastal areas, as well as floods in towns and cities. They include both flood risk issues now and in the future. 2.1. Flood hazard maps and flood risk maps (article 6) Flood hazards and risks will be mapped for the river basins and sub-basins with significant potential risk of flooding for three scenarios:
• Floods with a low probability or extreme event scenarios • Floods with a medium probability (likely return period >100 years) • Floods with high probability, where appropriate. The maps may show information related to flood extent, depths and velocity of water and the potential adverse consequences. Again the spatial context is key and indeed the development of an online web tool allows citizens to discover the flood risk for their area. So, returning to the more conventional statistical components when confronting a spatial problem, the first question to be addressed would concern the framework for sampling and monitoring. Monitoring networks (ideally long term and stable) provide the key evidence base for change, yet (IPCC, 2008) commented that ‘‘observational data and data access are pre-requisites for adaptive management, yet many observational networks are shrinking’’. Today’s data network for Scottish monitoring purposes is considerably reduced from that reported and interpreted by Marsden and Mackay (2001) although arguably broader in scope as well as more focussed in application. 2.2. Sampling and monitoring Some directives offer general advice concerning the essential sampling and monitoring. The directives frequently define in a generic way the sampling that is required, e.g. (WFD, 2000, Monitoring for the water framework directive 2000/60/EC): ‘‘Member states must ensure that enough individual water bodies of each water type are monitored and determine how many stations are required to determine the ecological and chemical status of the water body’’. Within Scotland, the regulatory agency responsible for implementing these directives is the Scottish Environment Protection Agency (SEPA), which operates a risk based sampling and monitoring programme classifying monitoring sites in three ways: surveillance, operational and investigative. A surveillance network is geographically distributed, designed to assess long-term changes in natural conditions and long-term changes due to widespread anthropogenic activity. Operational monitoring is driven by risk assessments based on pressure information and located in areas of known risk. Investigative monitoring is a more variable network responsive to unplanned events and emerging risks. In the coming years, SEPA faces the challenge of delivering an informed report on the state of the Scottish water environment with reduced monitoring effort. Fig. 2 shows the operational network that currently SEPA operates. There are many papers about design of sampling and monitoring, but there is one further added but very interesting twist in this context. Currently classification under WFD is based on identifying a single representative loch (or lake) of a group, and groups are based on typology (altitude, alkalinity, depth).
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Fig. 1. Overall classification status for Scotland (SEPA, 2010).
It is then of interest to look at the monitoring data from lochs in each group to see whether or not there is similarity within the group. This may sound trivial, but when we try to match the sampling points in time over space, we encounter a problem, since the matching is limited—not all samples are collected on the same day or even the same week, so instead another approach is needed. Fig. 3 shows the pattern of observations over time for a set of lochs, their group being identified by colour.
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Fig. 2. Operational and surveillance loch monitoring (SEPA, 2010).
A functional data analysis approach considers the time series of data collected for each determinand at each lake as observations of a continuous function collected at a finite series of time points. The first step in the analysis is to estimate a smooth function of the observed data, so that a curve becomes a data point. Functional clustering can then be applied, where a functional data object (a curve) is first
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Fig. 3. Temporal sampling patterns for the groups and individual lochs within groups.
estimated for each individual (loch) using a set of basis splines, and then individuals are grouped by applying a clustering method to the basis coefficients that define these smooth functions. Functional data analysis (Ramsay and Silverman, 1997) is where a smooth function is fitted to the observed data at each site. This enables the time series to be considered and compared as curves over time. The time series of data is thought of as observations of a continuous function collected at a finite series of time points. Regarding the data in this way makes it easier to see if there are common longterm patterns in the data across sites. Current work at the University of Glasgow (Haggarty et al., in preparation) involves grouping lakes based on alkalinity, chlorophyll and phosphorus over time using functional clustering models (James and Sugar, 2003; Pastres et al., 2010). This enables lakes to be grouped not only on the basis of mean levels, but also on the basis of any common patterns over time. This is part of an investigation as to whether lakes with common trends are clustered spatially. 3. Spatial models for freshwater systems—nitrates Traditionally, we can develop spatio-temporal models within hydrological catchments to investigate nutrient trends and seasonal patterns in observed concentrations, including drivers and pressures such as environmental factors, agricultural practice and population. Non-parametric smooth models (Bowman and Azzalini, 1997; Wood, 2006) such as additive models are extremely useful, since they offer flexibility in terms of the patterns. Regression type models can be developed to examine the temporal and spatial trends and seasonality within each catchment, to incorporate catchment covariates and to incorporate space/time interactions and an appropriate covariance structure. Regression models are used in which the explanatory variables are incorporated as smooth functions instead of linear relationships. The benefit of such an approach is the flexibility to model smooth trends in space and time along with flexible non-restricted seasonal patterns, improving knowledge of detailed changes across space and time. Data can be incorporated for a number of monitoring locations to estimate the trends and seasonal patterns that appear, on average, across the area of the monitoring locations. The approach makes fewer assumptions about the nature of trends and seasonal patterns within the data compared to a parametric approach, and the methods are more robust to the presence of outliers in the data. However, it can become computationally intensive for data that is high dimensional (Bowman et al., 2009). These techniques are data driven and additional smooth or linear covariates for environmental factors can be incorporated. To investigate temporal and spatial trends and seasonality, a model such as: y = µ + s(East, North) + s(year.day) + s(day) + ε could be useful, and is a generalisation of geographical regressions.
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Such a model can also be extended to incorporate trends including covariates: y = µ + s(East, North) + s(year.month) + s(month) + s(covariates) + ε. Finally, space–time interactions can also be fit: y = µ + s(East, North) + s(year.day) + s(day) + s(day) : s(year.day)
+ s(East, North) : s(year.day) + ε. The smooth for day within year, or the seasonal pattern is circular (Wood, 2006), and appropriate assumptions concerning ε can be incorporated to reflect spatial and temporal correlations (GonzalesManteiga and Crujeiras, 2012). However, in river networks there are some additional and slightly unusual modelling challenges. 3.1. River networks The WFD (and Nitrates directive) requires an investigation of the pattern of nutrient concentrations in surface waters since nitrogen and phosphorus can lead to eutrophication. This needs us to model the spatio-temporal distribution of nutrient concentrations in surface waters within and between catchments and to recognise that spatial patterns of change may be important For a river catchment, we could anticipate that there would be several monitoring locations along the main channel and tributaries, in the example below for the River Tweed there are more than 80 monitoring locations. This sounds like a very conventional spatial problem, but there are two issues. First, we might wish to consider the matter of preferential sampling (Diggle et al., 2010), the sampling locations are unlikely to be randomly distributed and secondly, there is the natural structure of the river to consider, points close in Euclidean space may be unconnected, so that interpolation over the entire network is possible, but needs a spatial mode. In particular, a river distance model (where river distance is defined as the shortest distance between two locations, along the river network) is useful. One other important issue concerns the connectedness of locations. River network modelling in a spatial sense requires some thought to be given to the distance metric. Euclidean distance is commonly used for spatial problems and has been used in the analysis of problems on a river network, Cressie et al. (2006), but this ignores the almost tree like structure of the network, so another possibility would be to use ‘‘stream distance’’, defined to be ‘‘the shortest distance between two locations where distance is only computed along the stream network’’ (Ver Hoef et al., 2006). This distance measure does not however address the flow connectedness in that it may not be applicable to two stations that are not ‘‘flow connected’’ in the river network. Two stations are described as being flow connected if the water at either location flows into the water at the other location. Readily available from GIS software, Ver Hoef et al. (2006) suggest that it may not be appropriate to use these alternative distance measures in statistical models developed for Euclidean Distance. Ver Hoef et al. (2006) demonstrate that standard autocovariance models (with the exception of the exponential model) are not valid when using the stream in place of Euclidean distance. In order to avoid such issues, Ver Hoef and Peterson (2010) define two types of model, termed ‘tailup’ and ‘tail-down’, to model the spatial dependence across river networks using stream distance. The names ‘tail-up’ and ‘tail-down’ refer to the direction in which the tail of the moving average process moves (either upstream or downstream). The use of variance component models incorporating stream distance based covariance structures such as the tail-up and tail-down models was first suggested in Cressie et al. (2006). Further work is ongoing developing such models. However, this problem can be considered in a different way, which is quite common in ecology, by asking ‘‘What are the common features (if any) when we look at multiple sites over time?’’ This is a spatial problem. Identifying common signals, patterns and trends in environmental time series data is often referred to as ‘coherency’, which has commonly been investigated through describing and quantifying the association between two or more time series. The possible dependence between the series is not limited to simultaneous values but may include leading, lagged and smoothed relationships. The main aim of measuring coherency, is to achieve a final outcome, which includes ‘the clustering of time series into similar-groups’ (Potamias and Moustakis, 2001).
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Fig. 4. River Tweed network and map of predicted nitrate levels (O’Donnell, 2011).
Fig. 5a. Time series plot of TOC at 4 sites.
The example used here is part of an investigation of total organic carbon (TOC) in Scottish rivers and lochs. Changes in Scotland’s climate will affect water quality in a number of ways, including a possible increase in organic carbon. Increases in water based organic carbon have already been documented at a number of UK sites (Evans et al., 2005; Worrall and Burt, 2007) although there may be decreasing trends at a few sites (Worrall et al., 2004). Despite the high carbon content of Scotland’s soils, information is limited on trends in organic carbon in surface water in Scotland (SEPA, 2009), so that using data provided by SEPA an investigation was carried out. A small subset of the data for 4 sites in the River Dee in Aberdeenshire (Reid, 2011) in shown in Fig. 5a as well as the map showing the results from an additive model fit incorporating river distance in Fig. 5b. Whether we consider the common trends and patterns in the time series as in Fig. 5a, or build a spatio-temporal model as in Fig. 5b, the key challenge is the identification of common trends and
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Fig. 5b. Additive model results fit to the River Dee network (Reid, 2011).
perhaps also phenological change, and to identify regions which behave similarly. Classically this has been done in the ecological literature in a pairwise fashion using tools such as cross-wavelet coherency and phase and more recently dynamic factor analysis (Zuur et al., 2003a) an alternative. Multivariate approaches such as dynamic factor analysis, min/max autocorrelation factor analysis (Alonso et al., 2011; Zuur et al., 2003b; Lopes et al., 2008; Strickland et al., 2009) and dynamic principal components analysis (Salvador et al., 2003) enable a study of many sites simultaneously to establish how the time series of response at each spatial location evolve throughout the year and over the time period. Such approaches enable the assessment of common trends for multiple time series, which may lead to a spatial interpretation. 4. The Floods Directive (FD) Understanding temporal patterns in river flows and their relationship to flooding is critical to flood planning and risk management. Delivering effective and efficient flood risk management in future requires new approaches: in Scotland, the Flood Risk Management Act (FD, 2009) was passed with the aim of introducing ‘‘a more sustainable and modern approach to flood risk management’’ (Scottish Government, 2009). To do so, new and improved estimates of flood risk which take into account the impact of climate change and possible spatial heterogeneity are needed. The Flood Risk Management (Scotland) Act 2009 was enacted on June 16, 2009 introducing new approaches to flood risk management. Of specific interest is the assessment of flood risk, and the spatial aspects of extremes in river flow. River flow records have formed the basis of many flood risk estimates, their modelling based on classical statistical models, that have assumed stationarity. However, under climate change and climate change scenarios, there is an expectation that the flow series may no longer be stationary and therefore statistical models that do not make this assumption are required. In Scotland, these changes may be apparent in west to east differences in terms of rainfall and river flow. Wavelet analyses are being applied to individual river flow series to identify the local behaviour of potentially non stationary series, resulting in a time–frequency representation of the data (Percival and Walden, 2006). The resulting representations of variability can then be explored in terms of climatic drivers (such as the North Atlantic Oscillation and Atlantic Meridional Oscillation), and to explore spatial differences (Franco-Villoria et al., 2011).
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Fig. 6. Maps of the scale, location and shape parameters for GEV distribution fit to each river flow time series.
However, the potentially more interesting statistical question concerns the spatial pattern in extreme flows which is one of considerable scientific interest in the wider context of climate change. Generalisation and extension from the univariate extreme value theory is complex, with recent developments exploring max-stable stochastic processes (Schlather, 2002), and building spatial hierarchical models (Sang and Gelfand, 2009). As an illustration, thirty five rivers of different catchment sizes across Scotland were selected on the basis of data quality and quantity, and the monthly maxima were calculated for the time period January 1985 to December 2005. In a preliminary analysis, at each site, the monthly overall mean was removed to de-seasonalise the series and then, a GEV distribution was fitted separately to each of 35 series and a geostatistical analysis of the parameter estimates carried out. Approaches like these will allow us to address, within the context of the legislation, what the flood risks are (using the historical data), to evaluate complex change in time, and to explore the drivers of change. 5. Conclusions This paper has brought together policy and regulatory issues for the aquatic environment, and used some specific illustrations to show the statistical challenges that arise from the spatial and spatio-temporal contexts that the questions are posed within. The ability to answer some of the deceptively simple questions posed in the introduction and to consider the evidence base for the effectiveness of environmental policy, requires us to have the ability to identify often complex spatial and spatio-temporal trends. To deliver the information (contextual and interpretational) value of routine monitoring data, more innovative spatial analysis is needed. Many environmental issues are set within a spatio-temporal framework. There are challenges presented by the data, often collected for one purpose (e.g. to demonstrate compliance) and then used for another, the sampling strategies used and their spatial support and temporal frequency. There are technical challenges in the statistical modelling, defining spatio-temporal covariance structures, handling massive datasets, dealing with non-stationary series. There are challenges in uncertainty evaluation and visualisation. The environment is multivariate, so we need holistic assessment tools (everything is related somehow and at some scale). Modelling the water environment can present a variety of specific statistical challenges, when related to policy targets as exemplified by the Water Framework Directive and Floods Directive. Sophisticated statistical models for spatial and spatio-temporal trends can give
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added value to routine monitoring data, provide better descriptions of complex change behaviour and begin to tease out climate change driven effects in environmental quality. Acknowledgments My Ph.D. and M.Sc. students who helped me with figures: Ruth Haggarty (Fig. 3), David O’Donnell (Fig. 4), Stephen Reid (Fig. 5) and Maria Franco-Villoria (Fig. 6). References Alonso, A.M., Garcia-Martos, C., Rodriguez, J., Sanchez, M.J., 2011. Seasonal dynamic factor analysis and bootstrap inference: application to electricity market forecasting. Technometrics 53 (2), 137–151. Bowman, A., Azzalini, A., 1997. Applied Smoothing Techniques for Data Analyis: The Kernel Approach with S-Plus Illustrations. Oxford University Press. Bowman, A.W., Giannitrapani, M., Scott, E.M., 2009. Spatiotemporal smoothing and sulphur dioxide trends over Europe. Journal of Applied Statistics 58 (5), 737–753. Cressie, N., Frey, J., Harch, B., Smith, M., 2006. Spatial prediction on a river network. Journal of Agricultural, Biological, and Environmental Statistics 11 (2), 127–150. Diggle, P., Menezes, R., Su, T., 2010. Geostatistical inference under preferential sampling. Journal of the Royal Statistical Society: Series C (Applied Statistics) 59 (2), 191–232. Evans, C.D., Monteith, D.T., Cooper, D.M., 2005. Long-term increases in surface water dissolved organic carbon: observations, possible causes and environmental impacts. Environmental Pollution 137 (1), 55–71. Floods Directive Scotland, 2009. Scottish Government. Downloaded October 2011. http://www.scotland.gov.uk/Publications/2010/05/14113652/2. Franco-Villoria, M., Scott, E.M., Hoey, T., Fischbacher-Smith, D., 2011. Conditional probability of flood risk in Scotland. In: IWSM Proceedings, Valencia July 2011, pp. 228–234. Gonzales-Manteiga, W., Crujeiras, R.M., 2012. Recent advances in spatio-temporal stochastic modelling. Environmetrics 23 (1), 1–2. Haggarty, R., Miller, C., Scott, E.M., 2011. Functional clustering of water quality data in Scotland, Water Research (in preparation). IPCC, 2008. Technical paper on climate change and water. http://www.ipcc.ch/pdf/technical-papers/climate-change-wateren.pdf. James, G.M., Sugar, C.A., 2003. Clustering for sparsely sampled functional data. Journal of the American Statistical Association 98 (462), 397–408. Kirchner, J., Feng, X., Neal, C., Robson, A., 2004. The fine structure of water-quality dynamics: the (high-frequency) wave of the future. Hydrological Processes 18, 1353–1359. Lopes, H.F., Salazar, E., Gamerman, D., 2008. Spatial dynamic factor analysis. Bayesian Analysis 3 (4), 759–792. Marsden, M.W., Mackay, D.W., 2001. Water quality in Scotland: the view of the regulator. The Science of the Total Environment 265, 369–386. O’Donnell, D., 2011. Spatial prediction and spatio-temporal modelling on river networks. Ph.D. Thesis. University of Glasgow. Pastres, R., Pastore, A., Tonellato, S.F., 2010. Looking for similar patterns among monitoring stations, venice lagoon application. Environmetrics 22, 712–724. Percival, D.B., Walden, A.T., 2006. Wavelet Methods for Time Series Analysis. In: Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press. Potamias, G., Moustakis, V.S., 2001. Mining coherence in time series data. In: Proceedings of the Ninth International Conference on Human–Computer Interaction, pp. 928–932. Ramsay, J., Silverman, B.W., 1997. Functional Data Analysis, first ed. In: Springer Series in Statistics, Springer. Reid, S., 2011. Trends of organic carbon in Scottish rivers and lochs. M.Sc. Thesis. University of Glasgow. Salvador, M., Gallizo, J.L., Gargallo, P., 2003. A dynamic principal components analysis based on multivariate matrix normal dynamic linear models. Journal of Forecasting 22, 457–478. Sang, H., Gelfand, A.E., 2009. Hierarchical modeling for extreme values observed over space and time. Environmental and Ecological Statistics 16, 407–426. Schlather, M., 2002. Models for stationary max-stable random fields. Extremes 5 (1), 33–44. Scottish Government, 2003. Water Environment and Water Services (Scotland) Act 2003 http://www.hmso.gov.uk/legislation/ scotland/acts2003/20030003.htm. Scottish Government, 2009. Flood Risk Management (Scotland) Act 2009 http://www.legislation.gov.uk/asp/2009/6/contents. SEPA, 2009. Trends in organic carbon in Scottish rivers and lochs, Downloaded October 2011. www.sepa.org.uk/climate_change/ idoc.ashx?docid=b44060fe-5acf-4383-9efb-80b941405c66\&version=-1-09Dec2009-downloadedOctober2011. SEPA, 2010. Water monitoring and classification, downloaded October 2010. http://www.sepa.org.uk/water/monitoring_and_classification/scottish_monitoring_strategy.aspx. Strickland, C.M., Simpson, D.P., Turner, I.W., Denham, R., Mengersen, K.L., 2009. Fast Bayesian analysis of spatial dynamic factor models for large space time data sets. Ver Hoef, J.M., Peterson, E., 2010. A moving average approach for spatial statistical models of stream networks. Journal of the American Statistical Association. Ver Hoef, J.M., Peterson, E., Theobald, D., 2006. Spatial statistical models that use flow and stream distance. Environmental and Ecological Statistics 13 (4), 449–464. Water Framework Directive, 2000. Directive 2000/60/EC of the European Parliament and of the Council establishing a framework for the community action in the field of water policy. Downloaded April 2010 from EC/Environment web site.
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Wood, S.N., 2006. Generalized Additive Models. An Introduction with R. Chapman and Hall, CRC. Worrall, F., Burt, T.P., 2007. Trends in DOC concentration in Great Britain. Journal of Hydrology 346, 81–92. Worrall, F., Harriman, R., Evans, C.D., Watts, C.D., Adamson, J., Neal, C., et al., 2004. Trends in dissolved organic carbon in UK rivers and lakes. Biogeochemistry 70 (3), 369–402. Zuur, A.F., Fryer, R.J., Joliffe, I.T., Dekker, R., Beukema, J.J., 2003a. Estimating common trends in multivariate time series using dynamic factor analysis. Environmetrics 14, 665–685. Zuur, A.F., Tuck, I.D., Bailey, N., 2003b. Dynamic factor analysis to estimate common trends in fisheries time series. Canadian Journal of Fisheries and Aquatic Sciences 60, 542–552.
Contents lists available at SciVerse ScienceDirect
Spatial Statistics journal homepage: www.elsevier.com/locate/spasta
Spatial statistics — A watery business E. Marian Scott ∗ , J. Campbell Gemmell 1 School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, UK
article
info
Article history: Received 22 December 2011 Accepted 22 March 2012 Available online 29 March 2012 Keywords: Environmental policy Statistical models Regulation and evidence
abstract Spatial statistics as a sub-discipline has a long tradition, but modern environmental science is offering new challenges. In this short commentary paper, we consider the specific challenges posed by environmental policy, regulation and management for the freshwater environment, focussing on two specific pieces of European legislation, namely the Water Framework and the Floods Directives. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Environmental statistics typically but not always have a spatial context, so spatial statistics become important in the process of exploring and discovering the relationships, drivers and pressures and responses of the environment. Especially striking are the environmental (and spatial) challenges in dealing with the water environment, such as monitoring and modelling water quality or in evaluating flood risks. The environment is a natural landscape on which we can try to drape/fold our statistical model outputs, and thinking in this way recognises the holistic view of the complex interconnections which operate at different spatial and temporal scales. The current reality of what is possible means that we are only part of the way towards this ideal view—what we have in reality is (sometimes) a poor surrogate for those complex inter-connections, limited by our inability to measure what we are interested in with the spatial and temporal support that reflects the scales at which the environmental processes operate. New sensor system developments may open up opportunities in the near future, offering spatially distributed, high frequency observations (with their own statistical challenges Kirchner et al., 2004).
∗ Correspondence to: School of Maths and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK. Tel.: +44 141 330 5125; fax: +44 141 330 4814. E-mail addresses: [email protected] (E.M. Scott), [email protected] (J.C. Gemmell). 1 Environment Protection Authority, South Australia, Adelaide 5001, Australia (formerly of the Scottish Environment Protection Agency). 2211-6753/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spasta.2012.03.004
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Many of the common tools used in a spatial analysis have developed in different disciplines—and this is reflected in some of the emphases and languages used. GIS (geographic information systems), traditionally was very commonly used in Geography and Earth Sciences departments but not so widely used by statisticians, while geostatistics was developed in engineering, while at the same time, statisticians were developing their own framework and language for spatial models. However, now there is much more commonality, convergence and cross-disciplinary working, driven in part by the urge to visualise and communicate outside our silos, and to tackle challenges such as those in climate change research (where tools that can handle massive datasets are needed). We make progress, often driven by practical challenges, and this journal offers, along with others, a forum for our communities to reflect on that progress and drive it forward. So in the light of challenges—scientific, political and societal, we have chosen to focus this commentary on the freshwater environment and to consider some of the spatial (and statistical) challenges set by the current European regulatory framework. 1.1. Why water? The IPCC (2008) in its 2008 technical paper on Climate Change and Water wrote ‘‘observational records and climate projections provide abundant evidence that freshwater resources are vulnerable and have the potential to be strongly impacted by climate change with wide ranging consequences on human societies and ecosystems’’. A simple summary would be that both quantity and quality of water are of global, regional and local importance. The first step in improving our understanding of potential impacts and consequences to water quality and quantity is to consider the current state of the water environment (a snapshot in time) and the historical evidence of past state and its trajectory (over space and time), in the light of the regulatory framework and any environmental measures. Within the European Union, there are a number of regulatory frameworks dealing with the aquatic environment, of which the Water Framework Directive (WFD, 2000) which brought together a large suite of earlier directives and policy elements, transposed in Scotland as the Water Environment and Water Services Act (Scottish Government, 2003), to protect, improve and promote sustainable use of Scotland’s water environment, is perhaps the most significant. The other that will be briefly considered in this paper is the Floods Directive (FD, 2009), subsequently the Flood Risk Management (Scotland) Act (Scottish Government, 2009) which requires a national assessment of flood risk by the end of 2011, and flood risk and hazard maps by 2013. The WFD has created a six year cycle of policy implementation and reporting which will be captured in the synchronous and systematic reporting of water quality in River Basin Plan Reports, beginning from the baselines reported in 2009/10 in 2015. In terms of reporting on policy effectiveness and ultimately on the state of the freshwater environment, there are three basic questions to consider: (a) What is happening? (b) Why is it happening? (c) Are the changes significant? We consider the elements of spatial statistics used in answering these three questions for each of two regulatory regimes (WFD and FD) for Scotland and consider the role of statistical modelling in delivering improved understanding and modelling changes in the hydrological cycle at a scale relevant to decision making. 2. The policy framework Two EU directives are briefly introduced below, namely WFD and FD to illustrate the policy language. The objectives of the WFD include:
• To prevent deterioration in the status of surface water bodies. • To protect, enhance, and restore all bodies of surface water. • The establishment of systems for managing water environments, underpinned by extensive environmental monitoring and scientific investigation.
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A key component of the WFD is River Basin Management Planning, setting out the environmental pressures and what can be done to address them—in terms of regions and catchments and the control of all impacts—physical, polluting and otherwise—on the water environment with the aim of achieving ‘‘good’’ ecological status for water bodies by 2015. Status is determined on the basis of ecology not solely by chemical composition. Fig. 1 shows a map of the overall status of surface waters in Scotland for 2009, and underlines the spatial context of the WFD. The Floods Directive requires Member States to undertake (i) a preliminary flood risk assessment, (ii) develop flood hazard and flood risk maps and (iii) produce flood risk management plans for zones at risk of flooding. An important change is the introduction of flood risk management plans. These plans consider all forms of flood risk from rivers, groundwater and coastal areas, as well as floods in towns and cities. They include both flood risk issues now and in the future. 2.1. Flood hazard maps and flood risk maps (article 6) Flood hazards and risks will be mapped for the river basins and sub-basins with significant potential risk of flooding for three scenarios:
• Floods with a low probability or extreme event scenarios • Floods with a medium probability (likely return period >100 years) • Floods with high probability, where appropriate. The maps may show information related to flood extent, depths and velocity of water and the potential adverse consequences. Again the spatial context is key and indeed the development of an online web tool allows citizens to discover the flood risk for their area. So, returning to the more conventional statistical components when confronting a spatial problem, the first question to be addressed would concern the framework for sampling and monitoring. Monitoring networks (ideally long term and stable) provide the key evidence base for change, yet (IPCC, 2008) commented that ‘‘observational data and data access are pre-requisites for adaptive management, yet many observational networks are shrinking’’. Today’s data network for Scottish monitoring purposes is considerably reduced from that reported and interpreted by Marsden and Mackay (2001) although arguably broader in scope as well as more focussed in application. 2.2. Sampling and monitoring Some directives offer general advice concerning the essential sampling and monitoring. The directives frequently define in a generic way the sampling that is required, e.g. (WFD, 2000, Monitoring for the water framework directive 2000/60/EC): ‘‘Member states must ensure that enough individual water bodies of each water type are monitored and determine how many stations are required to determine the ecological and chemical status of the water body’’. Within Scotland, the regulatory agency responsible for implementing these directives is the Scottish Environment Protection Agency (SEPA), which operates a risk based sampling and monitoring programme classifying monitoring sites in three ways: surveillance, operational and investigative. A surveillance network is geographically distributed, designed to assess long-term changes in natural conditions and long-term changes due to widespread anthropogenic activity. Operational monitoring is driven by risk assessments based on pressure information and located in areas of known risk. Investigative monitoring is a more variable network responsive to unplanned events and emerging risks. In the coming years, SEPA faces the challenge of delivering an informed report on the state of the Scottish water environment with reduced monitoring effort. Fig. 2 shows the operational network that currently SEPA operates. There are many papers about design of sampling and monitoring, but there is one further added but very interesting twist in this context. Currently classification under WFD is based on identifying a single representative loch (or lake) of a group, and groups are based on typology (altitude, alkalinity, depth).
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Fig. 1. Overall classification status for Scotland (SEPA, 2010).
It is then of interest to look at the monitoring data from lochs in each group to see whether or not there is similarity within the group. This may sound trivial, but when we try to match the sampling points in time over space, we encounter a problem, since the matching is limited—not all samples are collected on the same day or even the same week, so instead another approach is needed. Fig. 3 shows the pattern of observations over time for a set of lochs, their group being identified by colour.
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Fig. 2. Operational and surveillance loch monitoring (SEPA, 2010).
A functional data analysis approach considers the time series of data collected for each determinand at each lake as observations of a continuous function collected at a finite series of time points. The first step in the analysis is to estimate a smooth function of the observed data, so that a curve becomes a data point. Functional clustering can then be applied, where a functional data object (a curve) is first
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Fig. 3. Temporal sampling patterns for the groups and individual lochs within groups.
estimated for each individual (loch) using a set of basis splines, and then individuals are grouped by applying a clustering method to the basis coefficients that define these smooth functions. Functional data analysis (Ramsay and Silverman, 1997) is where a smooth function is fitted to the observed data at each site. This enables the time series to be considered and compared as curves over time. The time series of data is thought of as observations of a continuous function collected at a finite series of time points. Regarding the data in this way makes it easier to see if there are common longterm patterns in the data across sites. Current work at the University of Glasgow (Haggarty et al., in preparation) involves grouping lakes based on alkalinity, chlorophyll and phosphorus over time using functional clustering models (James and Sugar, 2003; Pastres et al., 2010). This enables lakes to be grouped not only on the basis of mean levels, but also on the basis of any common patterns over time. This is part of an investigation as to whether lakes with common trends are clustered spatially. 3. Spatial models for freshwater systems—nitrates Traditionally, we can develop spatio-temporal models within hydrological catchments to investigate nutrient trends and seasonal patterns in observed concentrations, including drivers and pressures such as environmental factors, agricultural practice and population. Non-parametric smooth models (Bowman and Azzalini, 1997; Wood, 2006) such as additive models are extremely useful, since they offer flexibility in terms of the patterns. Regression type models can be developed to examine the temporal and spatial trends and seasonality within each catchment, to incorporate catchment covariates and to incorporate space/time interactions and an appropriate covariance structure. Regression models are used in which the explanatory variables are incorporated as smooth functions instead of linear relationships. The benefit of such an approach is the flexibility to model smooth trends in space and time along with flexible non-restricted seasonal patterns, improving knowledge of detailed changes across space and time. Data can be incorporated for a number of monitoring locations to estimate the trends and seasonal patterns that appear, on average, across the area of the monitoring locations. The approach makes fewer assumptions about the nature of trends and seasonal patterns within the data compared to a parametric approach, and the methods are more robust to the presence of outliers in the data. However, it can become computationally intensive for data that is high dimensional (Bowman et al., 2009). These techniques are data driven and additional smooth or linear covariates for environmental factors can be incorporated. To investigate temporal and spatial trends and seasonality, a model such as: y = µ + s(East, North) + s(year.day) + s(day) + ε could be useful, and is a generalisation of geographical regressions.
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Such a model can also be extended to incorporate trends including covariates: y = µ + s(East, North) + s(year.month) + s(month) + s(covariates) + ε. Finally, space–time interactions can also be fit: y = µ + s(East, North) + s(year.day) + s(day) + s(day) : s(year.day)
+ s(East, North) : s(year.day) + ε. The smooth for day within year, or the seasonal pattern is circular (Wood, 2006), and appropriate assumptions concerning ε can be incorporated to reflect spatial and temporal correlations (GonzalesManteiga and Crujeiras, 2012). However, in river networks there are some additional and slightly unusual modelling challenges. 3.1. River networks The WFD (and Nitrates directive) requires an investigation of the pattern of nutrient concentrations in surface waters since nitrogen and phosphorus can lead to eutrophication. This needs us to model the spatio-temporal distribution of nutrient concentrations in surface waters within and between catchments and to recognise that spatial patterns of change may be important For a river catchment, we could anticipate that there would be several monitoring locations along the main channel and tributaries, in the example below for the River Tweed there are more than 80 monitoring locations. This sounds like a very conventional spatial problem, but there are two issues. First, we might wish to consider the matter of preferential sampling (Diggle et al., 2010), the sampling locations are unlikely to be randomly distributed and secondly, there is the natural structure of the river to consider, points close in Euclidean space may be unconnected, so that interpolation over the entire network is possible, but needs a spatial mode. In particular, a river distance model (where river distance is defined as the shortest distance between two locations, along the river network) is useful. One other important issue concerns the connectedness of locations. River network modelling in a spatial sense requires some thought to be given to the distance metric. Euclidean distance is commonly used for spatial problems and has been used in the analysis of problems on a river network, Cressie et al. (2006), but this ignores the almost tree like structure of the network, so another possibility would be to use ‘‘stream distance’’, defined to be ‘‘the shortest distance between two locations where distance is only computed along the stream network’’ (Ver Hoef et al., 2006). This distance measure does not however address the flow connectedness in that it may not be applicable to two stations that are not ‘‘flow connected’’ in the river network. Two stations are described as being flow connected if the water at either location flows into the water at the other location. Readily available from GIS software, Ver Hoef et al. (2006) suggest that it may not be appropriate to use these alternative distance measures in statistical models developed for Euclidean Distance. Ver Hoef et al. (2006) demonstrate that standard autocovariance models (with the exception of the exponential model) are not valid when using the stream in place of Euclidean distance. In order to avoid such issues, Ver Hoef and Peterson (2010) define two types of model, termed ‘tailup’ and ‘tail-down’, to model the spatial dependence across river networks using stream distance. The names ‘tail-up’ and ‘tail-down’ refer to the direction in which the tail of the moving average process moves (either upstream or downstream). The use of variance component models incorporating stream distance based covariance structures such as the tail-up and tail-down models was first suggested in Cressie et al. (2006). Further work is ongoing developing such models. However, this problem can be considered in a different way, which is quite common in ecology, by asking ‘‘What are the common features (if any) when we look at multiple sites over time?’’ This is a spatial problem. Identifying common signals, patterns and trends in environmental time series data is often referred to as ‘coherency’, which has commonly been investigated through describing and quantifying the association between two or more time series. The possible dependence between the series is not limited to simultaneous values but may include leading, lagged and smoothed relationships. The main aim of measuring coherency, is to achieve a final outcome, which includes ‘the clustering of time series into similar-groups’ (Potamias and Moustakis, 2001).
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Fig. 4. River Tweed network and map of predicted nitrate levels (O’Donnell, 2011).
Fig. 5a. Time series plot of TOC at 4 sites.
The example used here is part of an investigation of total organic carbon (TOC) in Scottish rivers and lochs. Changes in Scotland’s climate will affect water quality in a number of ways, including a possible increase in organic carbon. Increases in water based organic carbon have already been documented at a number of UK sites (Evans et al., 2005; Worrall and Burt, 2007) although there may be decreasing trends at a few sites (Worrall et al., 2004). Despite the high carbon content of Scotland’s soils, information is limited on trends in organic carbon in surface water in Scotland (SEPA, 2009), so that using data provided by SEPA an investigation was carried out. A small subset of the data for 4 sites in the River Dee in Aberdeenshire (Reid, 2011) in shown in Fig. 5a as well as the map showing the results from an additive model fit incorporating river distance in Fig. 5b. Whether we consider the common trends and patterns in the time series as in Fig. 5a, or build a spatio-temporal model as in Fig. 5b, the key challenge is the identification of common trends and
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Fig. 5b. Additive model results fit to the River Dee network (Reid, 2011).
perhaps also phenological change, and to identify regions which behave similarly. Classically this has been done in the ecological literature in a pairwise fashion using tools such as cross-wavelet coherency and phase and more recently dynamic factor analysis (Zuur et al., 2003a) an alternative. Multivariate approaches such as dynamic factor analysis, min/max autocorrelation factor analysis (Alonso et al., 2011; Zuur et al., 2003b; Lopes et al., 2008; Strickland et al., 2009) and dynamic principal components analysis (Salvador et al., 2003) enable a study of many sites simultaneously to establish how the time series of response at each spatial location evolve throughout the year and over the time period. Such approaches enable the assessment of common trends for multiple time series, which may lead to a spatial interpretation. 4. The Floods Directive (FD) Understanding temporal patterns in river flows and their relationship to flooding is critical to flood planning and risk management. Delivering effective and efficient flood risk management in future requires new approaches: in Scotland, the Flood Risk Management Act (FD, 2009) was passed with the aim of introducing ‘‘a more sustainable and modern approach to flood risk management’’ (Scottish Government, 2009). To do so, new and improved estimates of flood risk which take into account the impact of climate change and possible spatial heterogeneity are needed. The Flood Risk Management (Scotland) Act 2009 was enacted on June 16, 2009 introducing new approaches to flood risk management. Of specific interest is the assessment of flood risk, and the spatial aspects of extremes in river flow. River flow records have formed the basis of many flood risk estimates, their modelling based on classical statistical models, that have assumed stationarity. However, under climate change and climate change scenarios, there is an expectation that the flow series may no longer be stationary and therefore statistical models that do not make this assumption are required. In Scotland, these changes may be apparent in west to east differences in terms of rainfall and river flow. Wavelet analyses are being applied to individual river flow series to identify the local behaviour of potentially non stationary series, resulting in a time–frequency representation of the data (Percival and Walden, 2006). The resulting representations of variability can then be explored in terms of climatic drivers (such as the North Atlantic Oscillation and Atlantic Meridional Oscillation), and to explore spatial differences (Franco-Villoria et al., 2011).
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Fig. 6. Maps of the scale, location and shape parameters for GEV distribution fit to each river flow time series.
However, the potentially more interesting statistical question concerns the spatial pattern in extreme flows which is one of considerable scientific interest in the wider context of climate change. Generalisation and extension from the univariate extreme value theory is complex, with recent developments exploring max-stable stochastic processes (Schlather, 2002), and building spatial hierarchical models (Sang and Gelfand, 2009). As an illustration, thirty five rivers of different catchment sizes across Scotland were selected on the basis of data quality and quantity, and the monthly maxima were calculated for the time period January 1985 to December 2005. In a preliminary analysis, at each site, the monthly overall mean was removed to de-seasonalise the series and then, a GEV distribution was fitted separately to each of 35 series and a geostatistical analysis of the parameter estimates carried out. Approaches like these will allow us to address, within the context of the legislation, what the flood risks are (using the historical data), to evaluate complex change in time, and to explore the drivers of change. 5. Conclusions This paper has brought together policy and regulatory issues for the aquatic environment, and used some specific illustrations to show the statistical challenges that arise from the spatial and spatio-temporal contexts that the questions are posed within. The ability to answer some of the deceptively simple questions posed in the introduction and to consider the evidence base for the effectiveness of environmental policy, requires us to have the ability to identify often complex spatial and spatio-temporal trends. To deliver the information (contextual and interpretational) value of routine monitoring data, more innovative spatial analysis is needed. Many environmental issues are set within a spatio-temporal framework. There are challenges presented by the data, often collected for one purpose (e.g. to demonstrate compliance) and then used for another, the sampling strategies used and their spatial support and temporal frequency. There are technical challenges in the statistical modelling, defining spatio-temporal covariance structures, handling massive datasets, dealing with non-stationary series. There are challenges in uncertainty evaluation and visualisation. The environment is multivariate, so we need holistic assessment tools (everything is related somehow and at some scale). Modelling the water environment can present a variety of specific statistical challenges, when related to policy targets as exemplified by the Water Framework Directive and Floods Directive. Sophisticated statistical models for spatial and spatio-temporal trends can give
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added value to routine monitoring data, provide better descriptions of complex change behaviour and begin to tease out climate change driven effects in environmental quality. Acknowledgments My Ph.D. and M.Sc. students who helped me with figures: Ruth Haggarty (Fig. 3), David O’Donnell (Fig. 4), Stephen Reid (Fig. 5) and Maria Franco-Villoria (Fig. 6). References Alonso, A.M., Garcia-Martos, C., Rodriguez, J., Sanchez, M.J., 2011. Seasonal dynamic factor analysis and bootstrap inference: application to electricity market forecasting. Technometrics 53 (2), 137–151. Bowman, A., Azzalini, A., 1997. Applied Smoothing Techniques for Data Analyis: The Kernel Approach with S-Plus Illustrations. Oxford University Press. Bowman, A.W., Giannitrapani, M., Scott, E.M., 2009. Spatiotemporal smoothing and sulphur dioxide trends over Europe. Journal of Applied Statistics 58 (5), 737–753. Cressie, N., Frey, J., Harch, B., Smith, M., 2006. Spatial prediction on a river network. Journal of Agricultural, Biological, and Environmental Statistics 11 (2), 127–150. Diggle, P., Menezes, R., Su, T., 2010. Geostatistical inference under preferential sampling. Journal of the Royal Statistical Society: Series C (Applied Statistics) 59 (2), 191–232. Evans, C.D., Monteith, D.T., Cooper, D.M., 2005. Long-term increases in surface water dissolved organic carbon: observations, possible causes and environmental impacts. Environmental Pollution 137 (1), 55–71. Floods Directive Scotland, 2009. Scottish Government. Downloaded October 2011. http://www.scotland.gov.uk/Publications/2010/05/14113652/2. Franco-Villoria, M., Scott, E.M., Hoey, T., Fischbacher-Smith, D., 2011. Conditional probability of flood risk in Scotland. In: IWSM Proceedings, Valencia July 2011, pp. 228–234. Gonzales-Manteiga, W., Crujeiras, R.M., 2012. Recent advances in spatio-temporal stochastic modelling. Environmetrics 23 (1), 1–2. Haggarty, R., Miller, C., Scott, E.M., 2011. Functional clustering of water quality data in Scotland, Water Research (in preparation). IPCC, 2008. Technical paper on climate change and water. http://www.ipcc.ch/pdf/technical-papers/climate-change-wateren.pdf. James, G.M., Sugar, C.A., 2003. Clustering for sparsely sampled functional data. Journal of the American Statistical Association 98 (462), 397–408. Kirchner, J., Feng, X., Neal, C., Robson, A., 2004. The fine structure of water-quality dynamics: the (high-frequency) wave of the future. Hydrological Processes 18, 1353–1359. Lopes, H.F., Salazar, E., Gamerman, D., 2008. Spatial dynamic factor analysis. Bayesian Analysis 3 (4), 759–792. Marsden, M.W., Mackay, D.W., 2001. Water quality in Scotland: the view of the regulator. The Science of the Total Environment 265, 369–386. O’Donnell, D., 2011. Spatial prediction and spatio-temporal modelling on river networks. Ph.D. Thesis. University of Glasgow. Pastres, R., Pastore, A., Tonellato, S.F., 2010. Looking for similar patterns among monitoring stations, venice lagoon application. Environmetrics 22, 712–724. Percival, D.B., Walden, A.T., 2006. Wavelet Methods for Time Series Analysis. In: Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press. Potamias, G., Moustakis, V.S., 2001. Mining coherence in time series data. In: Proceedings of the Ninth International Conference on Human–Computer Interaction, pp. 928–932. Ramsay, J., Silverman, B.W., 1997. Functional Data Analysis, first ed. In: Springer Series in Statistics, Springer. Reid, S., 2011. Trends of organic carbon in Scottish rivers and lochs. M.Sc. Thesis. University of Glasgow. Salvador, M., Gallizo, J.L., Gargallo, P., 2003. A dynamic principal components analysis based on multivariate matrix normal dynamic linear models. Journal of Forecasting 22, 457–478. Sang, H., Gelfand, A.E., 2009. Hierarchical modeling for extreme values observed over space and time. Environmental and Ecological Statistics 16, 407–426. Schlather, M., 2002. Models for stationary max-stable random fields. Extremes 5 (1), 33–44. Scottish Government, 2003. Water Environment and Water Services (Scotland) Act 2003 http://www.hmso.gov.uk/legislation/ scotland/acts2003/20030003.htm. Scottish Government, 2009. Flood Risk Management (Scotland) Act 2009 http://www.legislation.gov.uk/asp/2009/6/contents. SEPA, 2009. Trends in organic carbon in Scottish rivers and lochs, Downloaded October 2011. www.sepa.org.uk/climate_change/ idoc.ashx?docid=b44060fe-5acf-4383-9efb-80b941405c66\&version=-1-09Dec2009-downloadedOctober2011. SEPA, 2010. Water monitoring and classification, downloaded October 2010. http://www.sepa.org.uk/water/monitoring_and_classification/scottish_monitoring_strategy.aspx. Strickland, C.M., Simpson, D.P., Turner, I.W., Denham, R., Mengersen, K.L., 2009. Fast Bayesian analysis of spatial dynamic factor models for large space time data sets. Ver Hoef, J.M., Peterson, E., 2010. A moving average approach for spatial statistical models of stream networks. Journal of the American Statistical Association. Ver Hoef, J.M., Peterson, E., Theobald, D., 2006. Spatial statistical models that use flow and stream distance. Environmental and Ecological Statistics 13 (4), 449–464. Water Framework Directive, 2000. Directive 2000/60/EC of the European Parliament and of the Council establishing a framework for the community action in the field of water policy. Downloaded April 2010 from EC/Environment web site.
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Wood, S.N., 2006. Generalized Additive Models. An Introduction with R. Chapman and Hall, CRC. Worrall, F., Burt, T.P., 2007. Trends in DOC concentration in Great Britain. Journal of Hydrology 346, 81–92. Worrall, F., Harriman, R., Evans, C.D., Watts, C.D., Adamson, J., Neal, C., et al., 2004. Trends in dissolved organic carbon in UK rivers and lakes. Biogeochemistry 70 (3), 369–402. Zuur, A.F., Fryer, R.J., Joliffe, I.T., Dekker, R., Beukema, J.J., 2003a. Estimating common trends in multivariate time series using dynamic factor analysis. Environmetrics 14, 665–685. Zuur, A.F., Tuck, I.D., Bailey, N., 2003b. Dynamic factor analysis to estimate common trends in fisheries time series. Canadian Journal of Fisheries and Aquatic Sciences 60, 542–552.