Quantifying flow–ecology relationships with functional ...

Quantifying flow–ecology relationships with functional linear models

Ben Stewart-Koster1,2, Julian D. Olden1 and Keith B. Gido3

1School of Aquatic and Fishery Sciences, University of Washington, Box 355020, Seattle, Washington 98195, USA2Present address: Australian Rivers Institute, Griffith University, 170 Kessels Road, Nathan, Queensland 4111, [email protected] of Biology, Kansas State University, Ackert Hall, Manhattan, Kansas 66506, USA

Received 25 October 2012; accepted 16 August 2013; open for discussion until 1 October 2014

Editor Z.W. Kundzewicz; Guest editor M. Acreman

Citation Stewart-Koster, B., Olden, J.D., and Gido, K.B., 2014. Quantifying flow–ecology relationships with functional linear models.Hydrological Sciences Journal, 59 (3–4), 1–16.

Abstract Hydrologic metrics have been used widely to quantify flow-ecology relationships; however, there areseveral challenges associated with their use, including the selection from a large number of available metrics andthe limitation that metrics are a synthetic measure of a multi-dimensional flow regime. Using two case studies offish species density and community composition, we illustrate the use of functional linear models to provide newinsights into flow–ecology relationships and predict the expected impact of environmental flow scenarios,without relying on hydrologic metrics. The models identified statistically significant relationships to river flowover the 12 months prior to sampling (r2 range 36–67%) and an environmental flow scenario that may enhancenative species’ densities while controlling a non-native species. Hydrologic metrics continue to play an importantrole in ecohydrology and environmental flow management; however, functional linear models provide anapproach that overcomes some of the limitations associated with their use.

Key words hydrologic metrics; environmental flows; ecohydrology; flow variation; river regulation; functional data analysis;dams

Quantification des relations débit–écologie par des modèles linéaires fonctionnelsRésumé Les mesures hydrologiques ont été largement utilisées pour quantifier les relations débit–écologie,cependant, il y a plusieurs difficultés associées à leur utilisation, comme le choix parmi un grand nombre devariables disponibles et la limitation due au fait que ces variables sont des mesures synthétiques d’un régimed’écoulement multi-dimensionnel. En nous basant sur deux études de la densité des espèces de poissons et de lacomposition des communautés, nous avons illustré l’utilisation de modèles linéaires fonctionnels pour donner denouvelles perspectives aux relations débit–écologie et pour prévoir l’impact de scénarios de débits environne-mentaux, sans s’appuyer sur les variables hydrologiques. Les modèles ont identifié des relations statistiquementsignificatives au débit de la rivière au cours des 12 mois précédant l’échantillonnage (r2 entre 36 et 67%) et unscénario de débit environnemental susceptible d’améliorer les densités des espèces indigènes tout en contrôlantune espèce non indigène. Les mesures hydrologiques continuent de jouer un rôle important dans l’écohydrologieet la gestion des débits de l’environnement, mais les modèles linéaires fonctionnels fournissent cependant uneapproche permettant de surmonter certaines des limitations liées à leur utilisation.

Mots clefs mesures hydrologiques ; débits environnementaux ; écohydrologie ; variation de débit, réglementation de la rivière,analyse fonctionnelle des données ; barrages

INTRODUCTION

Spatial and temporal variation in natural water flows isfundamental to the long-term sustainability and produc-tivity of riverine ecosystems and their riparian areas(Poff et al. 1997). It is well recognized that river flow

is the “master” variable that governs lotic ecosystems,driving strong ecological consequences at local to regio-nal scales, and at time intervals ranging from days (eco-logical effects) to millennia (evolutionary effects)(Naiman et al. 2008). Past research has generated a

Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 2014http://dx.doi.org/10.1080/02626667.2013.860231

Special issue: Hydrological Science for Environmental Flows

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wealth of knowledge aimed at establishing relationshipsbetween biotic assemblages and characteristics of theflow regime (e.g. Kennard et al. 2007, Stewart-Kosteret al. 2011, Gido et al. 2013), particularly to supportenvironmental flow efforts in human-altered systems(Arthington et al. 2006, Poff and Zimmerman 2010,Poff et al. 2010). Researchers often approach such ques-tions by describing key components of the hydrograph(i.e. magnitude, frequency, timing, duration and rate ofchange of flows) using hydrologic metrics, or indicesbased on a given time series of discharge data (Oldenand Poff 2003, Mathews and Richter 2007, Gao et al.2009, Kennard et al. 2010).

The long history of hydrologic metrics usage inboth hydrology and ecology (Olden et al. 2012) hasresulted in greater ecological understanding and moreinformed management of water resources (Richter et al.2006). Pioneering work in this area focused on singlemetrics, or a small number that described typical hydro-logic conditions, variability around averages, predict-ability of flow events or statistical characteristics offlow events such as skewness (Clausen and Biggs2000, Olden and Poff 2003). With increasing interestand research on flow–ecology relationships came theproliferation of hydrologic metrics to simultaneouslydescribe multiple characteristics of the flow regime(e.g. Poff and Ward 1989, Richter et al. 1996,Puckridge et al. 1998). There are now hundreds ofdifferent metrics (Olden and Poff 2003), many ofwhich are associated with high statistical uncertainty(see Kennard et al. (2010) for multiple sources of uncer-tainty), from which researchers and managers mustchoose a conceptually and ecologically tractable subset.

Although the history of hydrologic metrics hasyielded many important advances in river ecology andmanagement, there continue to be several practicalchallenges associated with their use. The first is thetask of selecting a subset of hydrologic metrics from apotentially bewildering array of possible choices. Thiscan be aided by a formal redundancy analysis; how-ever, it is generally understood that ecological knowl-edge should also be incorporated in such decisions(Olden and Poff 2003). A second task is to determineenvironmental flow standards based on relationshipsbetween ecological response variables and potentiallydisparate hydrologic metrics (Arthington et al. 2006).With so many metrics available, there is often littlecoordination among studies, making synthesis andgeneralizations of flow–ecology relationships verydifficult (Poff and Zimmerman 2010).

A synthetic summary of a complex hydrographprovided by hydrologic metrics is beneficial in many

circumstances, but this also creates a practical chal-lenge. Hydrologic metrics are descriptors of a hydro-graph, or some facet of it, rather than depicting theflow itself. The implication of this is that limitedinformation is available about the mechanistic linkbetween the response variable and the actual dis-charge reflected by the hydrograph; rather, it is arelationship between the response variable and a spe-cific characteristic of the discharge. Related to this isthe extreme example, where two very differenthydrographs may have the same or similar valuesfor a set of hydrologic metrics over a given interval(e.g. Fig. 1). Although this is a conveniently chosenexample, it does help to illustrate that a single valueof a hydrologic metric could be derived from analmost infinite number of differently shaped hydro-graphs. In such cases, using hydrologic metrics with-out judicious selection may lead to difficulties inestablishing flow–ecology relationships, and couldcompromise the success of environmental flow man-agement. Consequently, it is worthwhile exploringalternative approaches to relying solely on hydrologicmetrics for river research and management.

Recent statistical advances can provide a moreholistic characterization of the flow regime in anattempt to develop meaningful hydrologic descrip-tors. Spectral analysis techniques, such as Fourierand wavelet analysis, offer quantitative approachesthat incorporate a more continuous hydrologic

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Fig. 1 Comparison of observed discharge and four keysummary statistics over a 12-month period for two differ-ent systems in the USA. The two very different hydro-graphs have very similar hydrologic metrics over theinterval shown.

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perspective to quantify meaningful descriptors of theflow regime (see Steel and Lange (2007) and Saboand Post (2008) for examples). These methods aim todecompose the mathematical functions that underliethe observed time series for use in subsequent statis-tical analyses (Priestley 1991). Building on theseanalytical approaches is the field of functional dataanalysis, which has provided a framework to incor-porate such spectral decompositions of functionaldata into statistical models (Ramsay and Silverman2006). Functional data can have many forms, buttypically consist of continuous, or effectively contin-uous observations of a process over space or timethat are derived from mathematical functions(Ramsay and Silverman 2002), such as temperatureor river discharge (e.g. Ainsworth et al. 2011).Quantifying and understanding the functional dataseries may be the goal of the analysis, or they maybe used as predictor variables in a functional linearmodel (Ramsay and Silverman 2002, Müller andStadtmüller 2005). In such models, the actual timeseries, in this case the hydrograph, becomes the pre-dictor variable in a linear model as opposed to a setof hydrologic metrics that aim to describe it. Thesefunctional methodologies may provide a valuabletool to overcome some, but not necessarily all, ofthe challenges associated with the use of hydrologicmetrics in ecohydrology. In addition, they may beuseful in identifying flow–ecology relationships forhitherto unstudied species.

In this study we demonstrate the use of func-tional linear models to quantify flow–ecology rela-tionships in riverine ecosystems. We use two casestudies to illustrate the flexibility of the statisticalapproach; the first consists of data characterizingspatial variation in the frequency of species occur-rence (summarized as life-history strategies) for asingle year from 32 streams across North America,and the second comprises time series data on fishdensities in a single river for 18 years. These casestudies were selected to represent the most commondata types being analysed in the literature to establishflow–ecology relationships and inform the science ofenvironmental flows.

METHODS

Functional linear models

Functional linear models can have several forms,including the case where the predictor variable is atime series and the response variable is a scalar

(Ramsay and Silverman 2002). In the context ofecohydrology, such a model would relate theobserved value of a response variable at a singlepoint in time (or space) to the hydrograph over aspecified time interval. That is, each observationconsists of a time series of flow and a single responsevariable. Thus, hydrologic metrics are replaced aspredictor variables by the time series of flow fromwhich they would be calculated. Using the timeseries of flow in this manner requires a functionaltransformation of the hydrograph, such as a Fourieror B-spline transformation. From this, the integral ofthe curve can be estimated and then used as thefunctional predictor variable (Ramsay andSilverman 2006). Thus, it is seen that the predictorvariable in the statistical model is the actual hydro-graph. The mathematical form of the model is that ofan integral equation:

yi ¼ β0 þZ

xi tð Þβ1 tð Þdt þ "i (1)

where yi is the ith value of the response variable, β0 isthe intercept term of the regression, xi(t) is the valueof the hydrograph for the ith observation at time t, β1(t)is the regression coefficient describing the relation-ship between the response variable yi and the predic-tor variable xi at time t, and εi is the residual of the ithobservation in the model. This is the general form ofthe functional linear model; however, the residualsmay be assumed to follow a normal distribution orcertain exponential distributions (e.g. Müller andStadtmüller 2005). In this form of a functional linearmodel, the regression coefficient is itself a mathema-tical function that describes the relationship betweenthe response variable and the predictor variable ateach time t. In the context of flow–ecology, thismodel quantifies the relationship between theobserved values of the response variable (which arescalars) and each day of the time series of flow. Thiseffectively quantifies the importance of the timing,duration and magnitude of key flow events.Additional scalar predictor variables, i.e. those usedin standard regression modelling, such as habitatdescriptors, can also be included in addition to thefunctional predictor variable (Ramsay and Silverman2006). An advantage of functional linear models isthat by using basis functions in the modelling pro-cess, the regression coefficient can be estimated usingthe standard linear regression equation (Ramsay andSilverman 2006). A brief explanation follows below.

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A key step in fitting a functional linear model isto define the functional predictor variable, in this casethe hydrograph, in an appropriate form for the model.This step essentially converts the observed flow datainto a mathematical function that describes the timeseries, which can then be used in the regression. Thefunction itself is not interpreted; it is only required toconvert the observed time series into a form that canbe used as a predictor variable in the functional linearmodel. This can be done by transforming the hydro-graph into either a Fourier series or a B-spline series(Ramsay and Silverman 2006). Fourier series are thedecomposition of a periodic series into a sum of sineand cosine waves and are typically used to quantifythe frequencies of periodic patterns in time seriessuch as daily temperatures (Sabo and Post 2008).Although seasonality due to rainfall and snowmeltclearly exists in long-term hydrologic data, periodicpatterns are likely to be absent across relatively shortintervals (i.e. one year or less). In such cases, a B-spline curve can be used to define the hydrograph asa functional predictor variable (e.g. Ainsworth et al.2011). B-spline curves, which are a series of joinedpolynomial functions, offer great flexibility in quan-tifying the functional form of time series data(Ramsay and Silverman 2006). Converting the timeseries to a set of basis functions creates a linearcombination of basis functions with a set of its owncoefficients:

xi tð Þ ¼XKx

k¼1

cikψkðtÞ (2)

This, in the matrix format, is xi(t) = Cψ(t), where C isthe N × Kx coefficient matrix for the vector of K basisfunctions, ψ.

The next step in model development is definingthe form of the functional regression coefficient,β1(t), the shape of which is estimated in the modelfitting process (Ramsay and Silverman 2006). Similarto the predictor variable, this can take the form of aFourier series or a B-spline curve, thereby producinga linear combination of basis functions:

β tð Þ ¼XKβ

k¼1

bkθkðtÞ (3)

In the matrix form, this is β = θ′b, where θ is a vectorof basis functions and b is the vector of Kβ

coefficients.

Having defined the functional predictor variableand the functional regression coefficient, the modelcan now be defined as

yi ¼Z

Cψ tð Þθ sð Þ0bdt (4)

In the matrix form, this is yi = CJψθb, where Jψθ is aKx × Kβ matrix defined by

Zψ tð Þθ0 sð Þdt (5)

The matrix form of the model can be further simpli-fied so that the Kβ +1 vector consists of an interceptβ0 and all k b coefficients = ζ , and the intercept andcoefficient matrix associated with the basis functionsof the functional predictor = Z. Then the model canbe written as y ¼ Zζ , which can be solved using thestandard linear regression equation:

ζ ¼ ðZ 0ZÞ1Z 0y (6)

An alternative to using basis functions would be to fita standard multiple regression model using eachobserved daily flow value separately for the entiretime period of interest, thereby using t predictorvariables. However, this would very likely result inan overfit model with little predictive value, and withserious problems of multicollinearity. A functionallinear model provides an efficient alternative to sum-marize the time series into a smaller number of basisfunctions; however, the potential for overfitting stillexists with functional linear models. Therefore, thechallenge is to use a functional regression coefficientthat is sufficiently complex to capture relationshipsbetween the response variable and the time series ofthe predictor, and yet is simple enough to interpretand not result in an overfit model (Ramsay et al.2010, Ainsworth et al. 2011). As is common inmost statistical models, this amounts to determiningthe trade-off between detecting signal from noise.One way of achieving this is to start with a largenumber of basis functions and use a smoothing para-meter to penalize the “roughness,” or amount ofcurvature of the regression coefficient (Ramsay andSilverman 2006). Such penalty terms are common ininformation theoretic approaches to model selection,such as AIC and BIC. The smoothing parameter isused as a coefficient for the “roughness” of the pre-dictor function in an estimate of penalized sums ofsquares (PENSSE):


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PENSSEλ α0; βð Þ ¼X

yi α0 Z

xt tð Þβ tð Þdt 2

þ λZ

Lβ tð Þ½ 2dt(7)

whereRLβ tð Þ½ 2dt is a linear differential operator

used to define the roughness of the functional regres-sion coefficient and λ is the smoothing parameter thatpenalizes it (Ramsay et al. 2010). Minimizing thePENSSE provides a functional regression coefficientfor a given value of λ, with degrees of freedom equalto the trace of the hat matrix from the regression. Thevalue of λ can be selected via cross-validation tests ofmultiple possible values providing an automaticselection of the smoothed regression coefficient forthe best-fitting model (Ramsay and Silverman 2006).This results in a functional regression coefficient thatminimizes the penalized sum of squares, and isdirectly interpretable as the relationship between thepredictor variable over the time series and theresponse variable.

The shape of the fitted functional regression coef-ficient can be illustrated graphically as a curve overthe time interval of interest. In the context of flow–ecology relationships, the response variable shows apositive association with higher flows at times of theinterval that the curve is above the zero line.Conversely, at times of the interval that the curve isbelow the zero line, the response variable shows anegative association with higher flows. A significantrelationship to the predictor variable at any given timeis evident when the 95% confidence intervals aroundthe functional regression coefficient do not overlapzero (Ainswoth et al. 2011). Approximate point-wiseconfidence intervals are estimated as two times thestandard error of the regression coefficient at time t(Ramsay et al. 2010). Although it is currently notavailable in freely downloadable software, it is alsopossible to derive joint confidence regions to enablesimultaneous inference, thereby better accounting formultiple testing (Ainswoth et al. 2011).

Permutation tests are used to test the significanceof the functional linear model, because it is difficult toderive a theoretical null distribution (Ramsay et al.2010). The process follows a standard permutationtesting approach, where an F-statistic for the initialmodel fit is compared against a null distributionobtained by calculating the same statistic for a largenumber of model fits from random permutations of theresponse variable (Manly 2007). In the case of

functional linear models, the F-statistic is a comparisonof the residual variance and the predicted variance foreach permuted model fit (Ramsay et al. 2010).Comparing the original F-statistic with the permutednull distribution provides an estimate of a p-value forthe statistical significance of the model fit.

For a full explanation of functional data analysis,including functional linear models, we refer thereader to Ramsay and Silverman (2006) andRamsay et al. (2010). Below we demonstrate theuse of functional linear models with two case studieswhere the predictor variable is a time series of flowfor each observation and the response variable is asingle value, or scalar, for each observation.

Case study #1: flow determinants of fish life-history composition in the midwestern UnitedStates

The first case study comprised a data set of fishspecies occurrences from one-time community sur-veys at 32 free-flowing streams with minimal humanimpact in the midwestern region of the USA (Mimsand Olden 2012). All streams are located within fivecontiguous freshwater ecoregions (Abell et al. 2008):Upper Missouri, Middle Missouri, English-WinnipegLakes, Upper Mississippi and Laurentian GreatLakes (Fig. 2). These streams are arrayed across arange of elevations and experience broadly similarcontinental climates (Abell et al. 2008). The studystreams are predominantly perennial with peak flowsfrom March to May each year (spring) and low flowsoccurring from August to October (late summer). Inall cases, sampling occurred during summer months,defined for our purposes as 1 July. The data set usedhere is a subset of that used in Mims and Olden(2012) and is fully described therein.

At each sampling site, species were classified asopportunistic, periodic or equilibrium according tothe life-history model of Winemiller and Rose(1992). These classifications were determined basedon each species’ relative investment in reproductiveparameters of fecundity, onset and duration of repro-ductive period and parental care (following Mimsand Olden 2012). Periodic strategists are those thatare large-bodied, long-lived and late-maturing spe-cies; opportunistic strategists are small-bodied,short-lived, early-maturing species; and equilibriumstrategists are medium-bodied species with moderateage to maturation and generally provide parental care(Winemiller and Rose 1992). Each of these life-his-tory strategies has different hydrologic preferences:


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periodic strategists favouring highly seasonal andperiodically suitable hydrology, opportunistic strate-gists favour highly variable and unpredictable hydro-logic environments and equilibrium strategists tend tofavour habitats with relatively stable hydrology(Olden et al. 2006, Tedesco et al. 2008, Olden andKennard 2010). At each site, we calculated the rela-tive proportion of species in each life-history strategyaccording to species presence/absence.

Equilibrium species are known to prefer stableflows; however, flow stability was confounded withflow magnitude across the sampling reaches in thiscase study. Lower elevation reaches with more stabledischarge also had generally higher flows. To preventthis confounding from influencing the model fit, westandardized daily discharge by drainage area andrelated the relative proportion of each life-historystrategy to daily runoff (discharge/drainage area).

Case study #2: flow determinants of fish speciesdensities in the San Juan River

The second case study consists of a single sample offish abundance collected each year over 18 yearsfrom secondary channels in a single 138-km-long

reach of the San Juan River in New Mexico andUtah from 1993 to 2010 (see Gido and Propst2012). Being a semi-arid river, the San Juan Riverhydrology was historically highly seasonal and alsohighly variable, both inter- and intra-annually, withpeak flows occurring during spring snowmelt. Thesampling reach is in a segment of the river that ispartly regulated by the Navajo Dam, which is mana-ged to mimic the natural flow regime where possible(Propst and Gido 2004). To this end, there is approxi-mately 370 000 ML of water available per year forenvironmental flows (Holden 1999). This allocationhas been used predominantly to deliver flows thataugment the spring flows to improve conditions forendangered species downstream (Holden 1999). Insome years, such as 2008 (Fig. 3(b)), releases maycontinue into early summer (Bliesner et al. 2009).Despite the mimicry of the natural flow regime bydam operators, the biggest change since impound-ment has been the reduction in hydrologic variability,as well as reduced mean spring flows (Propst andGido 2004).

For this case study, we focused on three specieswith known relationships to temporal variation in dis-charge: the native species, flannelmouth sucker

= sampling site, case

San Juan River, case study 0 2040 80 km

Fig. 2 Map of the USA showing the location of sampling sites for the two case studies. Case study #1 comprises a databaseof fish species occurrences from one-time standardized surveys at 32 free-flowing streams across the midwestern USA.Case study #2 consists of a single sample of the density of three different species collected each year over 18 years fromsecondary channels in a single 138-km-long reach of the San Juan River in New Mexico and Utah from 1993 to 2010.


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Catostomus latipinnis and speckled dace Rhinichthysosculus, and the non-native species red shinerCyprinella lutrensis. The flannelmouth sucker is con-sidered a periodic species (Olden et al. 2006), withspawning in the San Juan River beginning as early asMarch and concluding in July (Gido and Propst 1999),but potentially continuing until October in other rivers(Minckley and Marsh 2009). It has shown positiveassociations with increased flows in spring (Gido andPropst 2012), whereas high flows in summer have beenassociated with improved fish condition (Paukert andRogers 2004). Speckled dace is a riffle-dwelling speciesthat generally prefers shallow, higher velocity habitats(Minckley and Marsh 2009). It is considered an equili-brium species (Olden et al. 2006), with spawningoccurring in spring, with a possible second spawningevent in the late summer (John 1963). Speckled dacehas shown positive associations with high flows inspring (Gido and Propst 2012). The non-native redshiner is a hardy species that is very tolerant to highlyvariable flows (Moore and Thorp 2008) and is

recognized as an opportunistic species (Zeug andWinemiller 2007). It has a long potential spawningseason that can run from April to September(Farringer et al. 1979); however, in the San JuanRiver, spawning is often delayed during years withhigh spring discharge (Franssen et al. 2007).Moreover, extended low-flow events during the latterpart of the spawning season are positively associatedwith increased densities (Propst and Gido 2004, Gidoand Propst 2012).

The data collection for this case study occurredbetween 15 September and 15 October each year, andstandardized sampling methods were used to ensureeach habitat type (e.g. riffle, run or pool) within thereach was sampled according to its proportional area(Propst and Gido 2004). Species abundance was repre-sented by densities and based on the number of indivi-duals caught within the area of habitat sampled (count/m-2). Standardizing by habitat area sampled facilitatedinter-annual comparisons for this analysis.Subsequently, we related the log-transformed singleobserved density for each species in the autumn ofeach year to the log-transformed daily discharge forthe full year prior to sampling. Transforming the dis-charge data was required to reduce the skewness in thepredictor variable and to facilitate parameter estimation.See Propst andGido (2004) andGido and Propst (2012)for more details regarding the sampling methods andresults from earlier analyses using hydrologic metrics.

Having developed the functional linear modelsfor the three species, we then illustrate the utility ofthe approach for managing flow regimes by predict-ing species responses to hypothetical hydrographs.We derived five hypothetical hydrographs by “releas-ing” the water available for environmental flows atdifferent times of the year. To do this we subtractedapproximately 370 000 ML from the spring flowrelease in the 2009 water year and added this waterto the hydrograph at different times of the year,resulting in five hypothetical environmental flowscenarios (Fig. 4). These scenarios consisted of sin-gle-flow releases for 30, 60 or 90 days, which wereachieved by “releasing” 12 335, 6167 or 4070 MLper day, respectively. This equates to mean daily flowrates of 143, 71 and 48 m-3 s-1, respectively. Wesubsequently applied the fitted functional linear mod-els to predict the density of each species expectedunder these hypothetical hydrographs.

Each of these hypothetical hydrographs used todemonstrate the predictive capacity of the approachfalls within the bounds of the observed hydrographsduring the 18 years of data used to develop the models.

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Functional predictorObserved discharge (2008)

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Fig. 3 Observed discharge in the San Juan River (logscale) for two different years (thick dashed line) with thefunctional predictor variable used for each year overlaid(thin dotted line).


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Model parameterization

We developed the functional linear models with nor-mally distributed errors in the R statistical environ-ment (R Development Core Team 2012) using thecontributed package, fda (Ramsay et al. 2011). Agraphical residual analysis for each model showedno systematic pattern in the residuals (results notshown). For both case studies, we followedAinsworth et al. (2011) by using B-splines withknots placed every 5 days to define the functionalpredictor variable for each observation. Two func-tional predictor variables are shown in Fig. 3 toillustrate the output from the mathematical functionthat is used as the predictor variable. We used thesame basis to define the functional regression coeffi-cient. We tested 29 different values for the smoothingparameter, λ (101, 101.5, 102, 102.5… 1014.5, 1015),and identified the best fitting, smoothed regressioncoefficient via generalized cross-validation (Ramsayet al. 2010). Ainsworth et al. (2011) provide a usefuldiscussion on the selection of smoothing parameterswith respect to generalized cross-validation.

The process of estimating predicted values underthe model given the environmental flow scenarios

follows the same approach as an ordinary linearmodel, where the new predictor variables are appliedto the coefficients in the regression equation. All thatis required is to convert the hypothetical hydrographsinto functional predictor variables using the samebasis (in this case, a B-spline curve). This is easilyachieved using the fda package in R (Ramsay et al.2011).

As noted above, functional linear models areflexible enough to accommodate functional and sca-lar predictor variables in the same regression. Thismay be particularly useful to capture relationshipsbetween fish and hydrologic metrics describing flowvariability, or counts of certain types of flow days,which would not otherwise be well quantified by afunctional predictor. To demonstrate this, we also fita hybrid model for red shiner in Case study # 2, usinga functional predictor for one part of the year and twohydrologic metrics for other times of the year. Giventhe relationships between this species and flow,described in the literature and outlined above, weused spring flow (1 March–31 May) as a functionalpredictor variable, again using a B-spline with knotsplaced every 5 days. Based on previous research(Propst and Gido 2004, Gido and Propst 2012), weused the number of days with flow <14 m-3 s-1

between 1 June and 31 August (LFS) and the numberof days with flows above the 75th percentile for theperiod 1 October–31 December (HFA) as hydrologicmetrics in the model. When using additional scalarpredictors in this way, the interpretation of the regres-sion coefficients is similar to that of any multipleregression model: each coefficient quantifies theeffect of each predictor variable having controlledfor all others in the model (Ramsay and Silverman2006). Statistical significance of the regression coef-ficient(s) across the time interval of each model wasassessed using point-wise confidence intervals(Ramsay et al. 2010); however, using the more strin-gent joint confident regions that account for multipletesting may produce slightly different results(Ainsworth et al. 2011).

RESULTS

Case study #1

As predicted by ecological theory, the magnitude anddirection of the relationship between runoff and theproportion of each life-history strategy varied amongthe life-history strategies. The models explained aconsiderable amount of variation for each strategy

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No e-flow

Fig. 4 Five hypothetical environmental flow scenariosusing the approximately 370 000 ML of water availablefor such use in the Navajo Dam. In each scenario the entirevolume of water is delivered as part of a single-flowrelease over 30, 60 or 90 days, which equates to releasesat mean daily flow rates of 143, 71 and 48 m-3 s-1,respectively.


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(range: 36–62%), where permutation testing sug-gested each model was at least marginally significant(p-values shown on Fig. 5(a)–(c)). The functionalregression coefficients for periodic strategists(p = 0.03, 5.8 degrees of freedom) and equilibriumstrategists (p = 0.07, 5.8 degrees of freedom) showedvery similar relationships to runoff in both magnitudeand direction. This is despite these two groups show-ing little correlation in their relative richness across

sites (r = 0.18). The two coefficients indicate signifi-cant negative associations with higher runoff inMarch and April and a positive association withhigher runoff in May and June. A key differencebetween the regression coefficients of these twogroups was the marginally significant positive rela-tionship to runoff for equilibrium strategists inDecember–January, during which periodic strategistsshowed no relationship. By contrast, the functionalcoefficient for opportunistic strategists (p < 0.001,6.6 degrees of freedom) was almost contrary to thatof equilibrium strategists (Fig. 5(b)), with a signifi-cant positive association with runoff found in Marchand April, but a negative association in May andJune. The positive association between equilibriumstrategies and runoff in July and August is mirroredby corresponding negative associations for opportu-nistic strategists.

Case study #2

The functional linear models identified significantrelationships to flow at different times of the yearfor each species in the San Juan River (Fig. 6(a)–(c)).The models for speckled dace (p = 0.03, 3.8 degreesof freedom) and flannelmouth sucker (p = 0.13, 4.2degrees of freedom) both explained approximately40–45% of the temporal variation in the species’densities, although the model for flannelmouthsucker was not a significant fit. The model for redshiner was also not a significant fit and onlyexplained 14% of the variation. Speckled dace andflannelmouth sucker showed negative relationships tohigher flows in October and November, althoughonly the relationship for speckled dace was signifi-cant at this time (Fig. 6(a) and (b)). The regressioncoefficients for these two species diverged inFebruary and March, when flannelmouth suckershowed a positive relationship to flows. By contrast,the model for speckled dace showed a more gradualchange to positive associations with flows, whichpeaked in May and June. Flows in late summerwere not significantly related to the density of eitherspecies.

The hybrid model for red shiner, whichincluded a three-month functional predictor andtwo hydrologic metrics, was a considerably betterfit than the first model that used an entire year’sflow as the functional predictor variable (p < 0.001,r2 = 0.67; Fig. 7). Interestingly, by including thetwo hydrologic metrics along with the shorter inter-val functional coefficient, the model identified a

Jul Aug

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Fig. 5 Functional regression coefficients for the proportionof the three life-history strategies—periodic, opportunisticand equilibrium—in streams across the midwestern USA.The functional coefficients quantify the relationshipbetween the richness of each life-history strategy and riverflow on each day of the year prior to sampling. Streamswith higher runoff at times of the year when the coefficientis above zero tend to have higher proportions of eachstrategy during the summer sampling period. In contrast,streams with higher runoff when the coefficient is belowzero tend to have lower proportions of each strategy duringthe summer sampling period. The solid line represents theparameter estimate, with dotted lines being the 95% con-fidence interval. The p values on each plot indicate thesignificance of that model based on the permutation testsdescribed in the “Methods” section.


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significant positive relationship to higher flows dur-ing April–May (Fig. 7(a)), which was not evident inthe first model (Fig. 6(c)). In addition, the modelidentified a positive association with higher numbersof low-flow days in summer (LFS), and a negativeassociation with higher numbers of high-flow daysin the late autumn period of October–December(HFA) (Fig. 7(b)).

As expected, the results from the environmentalflow scenarios were consistent with the shapes of thefitted regression coefficients. Predicted density forthe two native species was generally higher than therespective 18-year average when the timing of thehypothetical release coincided with a positive rela-tionship to flow for that species. For example, flan-nelmouth sucker showed the highest predicteddensity when the environmental flow was “released”in early spring for 60 days (Table 1), the time whenthis species showed a significant positive relationshipto higher observed flows (Fig. 6(b)). Equally, thedensity of all species was much lower than averagewhen the flow was released in late autumn (Table 1).Speckled dace and flannelmouth sucker both showeda negative relationship to higher flows at this time(Fig. 6(a) and (b)); however, red shiner showed anegative relationship to the number of days abovethe 75th percentile of flow (HFA, Fig. 7). The 60-dayearly spring flow showed the greatest capacity toenhance densities of the two native species, whilesomewhat depressing the densities of the non-nativered shiner.

Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep



(a) Speckled dace

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Fig. 6 Functional regression coefficients for the density ofspeckled dace, flannelmouth sucker and red shiner in theSan Juan River. See Fig. 5 caption for an explanation ofthe functional coefficient.

Mar Apr May LFS HFA

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Fig. 7 (a) Functional and (b) scalar regression coefficients for the second model fitted for red shiner. This model provided amuch better fit to the data than the first model for red shiner (Fig. 5(c)). As with the functional coefficient, the scalarregression coefficients are shown with their approximate 95% confidence intervals, neither of which cross zero, indicating asignificant relationship. The two scalar predictors are number of days in summer <14 m-3 s-1 (LFS) and the number of daysover the 75th percentile of flow in autumn (HFA).


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DISCUSSION

In the present study, we outlined an alternativeapproach to fitting flow–ecology relationships madeavailable by advances in the area of functional dataanalysis. We sought to apply this approach to facil-itate an accurate description of the hydrograph over agiven time interval for use in statistical models. Themodels provided a good fit to the data in each casestudy and identified periods of each water year inwhich there was a significant relationship to riverflow. The regression coefficients in our models quan-tify relationships between the response variables andactual river flow, and there is limited capacity toquantify relationships to flow variability with a func-tional predictor variable. However, the models canprovide managers with a specific understanding ofthe timing, magnitude and duration of ecologicallyimportant flow events to replicate when setting envir-onmental flow standards. In addition, functionalmodels may provide a valuable tool for ecologistsseeking to explore the flow–ecology relationships ofpoorly understood species.

Flow–ecology relationships: the case studies

Functional linear models provided new insight intoflow–ecology relationships in both case studies.

Previous analyses of relationships between fish life-history strategies and hydrology have combined datafrom broad geographic regions, and subsequentlyrelied on hydrologic descriptors of variability andpredictability for analysis (e.g. Olden and Kennard2010, Mims and Olden 2012). These studies sup-ported the expectation that periodic strategists areassociated with streams with highly predictable, sea-sonal flows, opportunistic strategists are associatedwith streams with highly variable flows and equili-brium strategists show variable responses to theseflow characteristics. In the present study, werestricted the analysis to a specific region and useddaily runoff (flow divided by catchment area) as thepredictor variable to quantify associations with thetiming, magnitude and duration of annual dischargeevents.

Periodic strategists demonstrate strong relianceon seasonally predictable changes, such as high-flowpulses at specific times of the year (Winemiller andRose 1992). This was reflected in the regressioncoefficient for these species, as the significant rela-tionship to elevated runoff was confined to therelatively predictable spring/summer period.Opportunistic strategists showed significant relation-ships to runoff at different times of the year, includ-ing a positive association with higher flows in spring.This is consistent with the ecological expectation thatrapidly maturing opportunistic species can leveragethe quickly changing flows associated with springthaw and rain events (Winemiller and Rose 1992).However, the significant negative association withrunoff from May to September highlights that pro-tracted disturbance events in the form of high flowsacross the entire spawning season have a negativeimpact on the reproductive success of these species(Cooke et al. 2005, Roberts et al. 2006). Similar toopportunistic strategists, equilibrium strategists showsignificant associations with runoff at several timesof the year, including a positive association withrunoff in early winter. Such a relationship may reflectthe gain from extended access to near-shore andfloodplain habitat, and associated warmer tempera-tures in the lead up to overwintering that is beneficialfor species with high parental investment. For exam-ple, overwinter survival of juvenile smallmouth bassis enhanced by warmer temperatures and higher win-ter flows provided by groundwater-fed springs(Peterson and Rabeni 1996, Brewer 2013).

In the second case study, functional linear mod-els were successful in identifying flow–ecology rela-tionships over time in the San Juan River. The model

Table 1 Predicted densities of each species (count/m-2)under five different environmental flow scenarios (shownin Fig. 4) to release the approximately 370 000 ML avail-able for this purpose in the San Juan River (Holden 1999).Each flow scenario is defined by the timing and length ofthe environmental flows with 30-, 60- and 90-day flowscomprising mean daily flow rates of 143, 71 and48 m-3 s-1, respectively. The mean and standard deviation(SD) of fish density over the 18 years of sampling areincluded for comparison.

Hydrograph Predicted densities (fish/m-2)

Speckled dace Flannelmouthsucker

Red shiner

No e-flow 0.183 0.034 0.133Early spring60 days

0.256 0.055 0.137

Late spring30 days

0.235 0.036 0.480

Late autumn30 days

0.067 0.009 0.002

Summer90 days

0.263 0.026 0.11

Mean density/m-2 (SD)

0.311 (0.243) 0.039 (0.04) 2.745 (5. 672)


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for flannelmouth sucker revealed a positive associa-tion between early spring flows and higher densities,likely reflecting their use of backwater habitats forspawning that are accessible following elevatedspring discharge (Bliesner et al. 1999). Thus, higherflows in spring would potentially result in a corre-sponding higher density of young-of-year suckers inthe early autumn when the data were collected.Although not a significant model fit according togeneralized cross-validation, the explanatory powerof this model (r2 = 0.43) is comparable to that of amultiple regression model fit with the same data butusing hydrologic metrics (r2 = 0.44; Gido and Propst2012). Similarly, for speckled dace, a riffle-dwellingspecies that spawns in spring and summer, higherflows throughout this period were positively asso-ciated with higher densities likely related to spawningsuccess (Bliesner et al. 1999, Gido and Propst 2012).The explanatory power of this model (r2 = 0.42) isconsiderably higher than that of the multiple regres-sion model using hydrologic metrics (r2 = 0.29; Gidoand Propst 2012). The first model for red shiner,using only a functional predictor variable for theentire year, failed to identify a significant relationshipto river flow. This is perhaps not surprising given redshiner is a habitat generalist that is tolerant to a widevariety of hydrologic conditions (Minckley andMarsh 2009).

The second and better-fitting model for red shi-ner was more complicated and thereby able to quan-tify relationships to very specific characteristics ofthe hydrograph as well as actual discharge. Themodel quantified the likely negative impact of highernumbers of high-flow days in autumn, as well as thepositive effect of spring flows and higher numbers oflow-flow days in summer. These relationships areprobably related to the spawning strategy of thisspecies and the impact of the timing of flow eventson spawning success. High flows in spring, whichhave a positive association with red shiner densityaccording to the functional regression, provide accessto previously inaccessible riffle habitat that is usedfor spawning at this time of year (Gido and Propst2012, Gido et al. 2013). The positive relationship tohigher numbers of low-flow days in summer is likelyrelated to the warmer temperatures of such conditionsthat enable the rapid growth of larvae, as well asmore frequent reproductive bouts for this repeatspawner (Moore and Thorp 2008, Gido and Propst2012). Finally, the negative association with high-flow days in autumn may be due to the cooling effectof such late season events, which also deposit fine

sediment over spawning habitat, thereby reducingspawning and recruitment success (Gido and Propst2012). The hydrologic metrics, which were found tobe statistically significant in earlier work (Gido andPropst 2012), were chosen because they quantified afacet of the flow regime that the functional predictorcould not account for, i.e. the frequency of a type offlow day, as opposed to the timing, magnitude orduration of a flow event. In this instance, the expla-natory power of the model (r2 = 0.67) is almostdouble that of the model fit given by Gido andPropst (2012) (r2 = 0.35); it was also able to identifynew flow–ecology relationships for this species thatwere not previously detected with this data set.

Use and interpretation of functional linear models

Functional linear models offer a novel approach toquantifying flow–ecology relationships that is con-ceptually appealing and complements the existingapproach of using hydrologic metrics. Our modelswere successful in quantifying the relationshipsbetween fish species densities and community com-position, and the timing, magnitude and duration ofhigher flows. In using a mathematical decompositionof the actual discharge, a functional approachremoves the uncertainty that can be associated withcalculating hydrologic metrics from flow records ofdifferent lengths (e.g. Kennard et al. 2010). However,the functional predictor variable and, therefore, theregression coefficient do not readily quantify theinfluence of variability of flows, although it is possi-ble to quantify relationships to the rate of change offlow by considering the first derivative of the func-tional predictor variable (see Ramsay and Silverman2006). We illustrated the flexibility of the approach inthe red shiner model using hydrologic metrics thatdescribed a different facet of the flow regime from afunctional predictor, for summer and autumn flows intandem with a functional predictor for spring flows.This also demonstrates the obvious fact that a purelyfunctional approach will not always yield betterresults; a statistical model can only detect relation-ships that exist in the data. Nonetheless, using scalarpredictor variables along with a functional predictorvariable provides an avenue to incorporate hydrolo-gic metrics that describe flow variability, an impor-tant hydrological driver of diversity (Poff and Allan1995, Puckridge et al. 1998, Biggs et al. 2005). Itwould also be plausible to include scalar habitatdescriptors of water quality (i.e. temperature) andhabitat quality (i.e. substrate composition) in future


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analyses. Decisions about variable selection shouldideally involve some expert knowledge in tandemwith information theoretic indices (such as cross-validation or AIC) to ensure models have biologicalrelevance. Similar to multiple regression models, thecoefficients in a model combining scalar and func-tional predictors quantify the effect of each predictorvariable having controlled for the others in the model(Ramsay et al. 2006).

There are at least two clear applications where afunctional approach may be useful for river ecologyand management. The first is to identify importantflow–ecology relationships in relatively unstudiedspecies, where hypotheses about specific hydrologicmetrics may not exist. The functional linear modelcan be used to identify time periods when higherflows have a significant relationship to such a spe-cies. If required, this information can be used toguide further analyses using hydrologic metrics cal-culated over specific intervals. The second applica-tion would be to use functional linear models todevelop and test environmental flow recommenda-tions. The predictive analysis in the second casestudy demonstrated how it is possible to test a setof environmental flow scenarios directly by applyinga fitted functional model to each. Our results showedconcordance between the fitted regression coeffi-cients and predicted outcomes under the environmen-tal flow scenarios. Predicted densities for eachspecies were above and below the 18-year averagewhen flows were released at times of the year thatcoincided with positive and negative relationships tohigher flows for each species, respectively. However,it is important to note that the lack of a statisticallysignificant regression model for flannelmouth suckersuggests that predictions for this species should beinterpreted with caution. Nonetheless, the results ofan analysis such as this can be used to determine thepreferred environmental flow scenario given the spe-cifics of the management objectives. We identified aflow prescription (i.e. flow regime) that was pre-dicted to result in a lower than average density ofthe invasive species, red shiner, and a greater thanaverage density of the two native species, flannel-mouth sucker and speckled dace. When the responsesof multiple species are important, a formal decisionanalysis may still be required to decide which sce-nario would be preferable given the predicted out-comes of each species. However, the use offunctional models provides a methodology to linkthe likely response of organisms of interest to poten-tial environmental flows.

In addition to the functional linear models pre-sented here, there are other functional approachesthat may also be of use in prescribing environmentalflows, including models for functional response vari-ables, the use of wavelets, Fourier series and harmo-nic analyses. An ecohydrological model for afunctional response variable is one where the predic-tor variable may be river discharge and the responsevariable would be another time series such as habitatvolume defined by water velocity and depth (Ramsayand Silverman 2006). Such a model could be used tomake real-time predictions of available habitat (e.g.Stewart et al. 2005). Wavelets are a statistical toolthat can be used to quantify multi-scaled variabilityin time series data, such as river flow or streamtemperature, that may arise due to daily, seasonal orannual periodicities (Steel and Lange 2007). Fourieranalysis has also been used to decompose the fre-quency of periodic fluctuations within the hydro-graph and quantify metrics of flow variability,autocorrelation and the frequency, magnitude andtiming of key flow events (Sabo and Post 2008).These latter two approaches essentially provide alter-native methods to generating ecologically relevanthydrologic metrics based on a functional transforma-tion of the hydrograph. Each of these approaches hasits own strengths and weaknesses for quantifyingdifferent elements of the flow regime, and the choiceamong them will be naturally informed by the goal ofthe analysis (Table 2).

Although functional approaches and, in particu-lar, functional linear models have the conceptualappeal of using the actual hydrograph as a predictor,there are obviously challenges associated with theiruse. A familiar challenge for researchers using hydro-logic metrics is to select a subset that describes somefacet(s) of the flow regime that is hypothesized toinfluence the response variable. There are statisticaland ecological approaches for this (e.g. Olden andPoff 2003, Jowett and Biggs 2009, Poff et al. 2010)and important guidelines to minimize metric uncer-tainty (Kennard et al. 2010); however, the selectionof metrics will ultimately influence the results of anyecohydrological analysis. The challenges in fitting afunctional model revolve around the nature of thepredictor variable and the subsequent regressioncoefficient(s). The selection of the temporal windowfor the functional predictor variable has the potentialto influence the results of the analysis (e.g.Ainsworth et al. 2011). As with standard regressionanalysis, the regression coefficients may change withthe inclusion of additional predictor variables.


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Moreover, the type of functional transformation usedwill influence the analysis. For example, a Fouriertransformation may be more appropriate to capturethe seasonality of a multi-year predictor variable thanthe B-spline approach we used in this study. Thesemay be new challenges for researchers and managersusing functional linear models; however, they can beovercome by invoking ecological intuition.

We have shown that functional linear modelshave great promise in quantifying relationshipsbetween ecological response variables and river dis-charge. Functional data analysis is an area of activeresearch in the statistical literature (e.g. Ainsworthet al. 2011), and there are ongoing developmentsincluding generalized functional linear models thatleverage different error distributions (Müller andStadtmüller 2005) and Bayesian functional regression(Crainiceanu and Goldsmith 2010). Developmentssuch as these will enable ecologists and river man-agers to use different types of response variables,such as species occurrences, as well as incorporateissues of scale and different types of predictor vari-ables, such as microhabitat variables, into these mod-els. We encourage ecologists and river managers toinvestigate further this suite of statistical tools for usein ecohydrology and when developing standards forenvironmental flows.

Acknowledgements We thank Meryl Mims for use-ful discussions and assistance with the data for thefirst case study. We also thank Giles Hooker forhelpful discussions on functional data analysis. DrAngus Webb, Mike Acreman and one anonymous

reviewer also provided helpful comments, whichimproved the manuscript.

Funding JDO was supported by the H. MasonKeeler Endowed Professorship and the Departmentof Defense—Strategic Environmental Research andDevelopment Program [RC-1724].

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Key facets of the flow regimeMagnitude High High High HighDuration High High High ModerateFrequency High Low High HighTiming High High High HighRate of change High High Low LowVariability High Unsuitable High High


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