Tools for quantifying changes in ecosystem service delivery through time CWES Seminary Series York January 2009

Time series data... the questions Acknowledgement: Zuur, Ieno & Smith (2007) ‘Analysing Ecological Data’, Springer publishing much more readable – and more applicable to ‘ecological’-sized datasets – than standard econometric tomes e.g. Green’s ‘Econometric Analysis’ Time series ? any variable measured repeatedly over time generates ‘time series’ data total fish catch or CPUE, number of breeding pairs of oyster catchers, number of children in primary school, average farm income..... The questions ? what is going on.... is there a trend ? are explanatory variables responsible for the trend ? are different time series data linked or interacting ? are there any sudden changes ? (of direction or slope ?) are there cyclic patterns ? can we predict future trends and/or future values ?

Time series data... the problems Serial correlation in errors produces incorrect standard errors and therefore incorrect t-values, p- values and F-statistics in linear regression, and related problems in PCA and redundancy analysis. Appropriate tools are required to answer the ‘interesting questions’ whilst avoiding the pitfalls of inappropriate statistical inference typically from small data sets The questions ? what is going on.... is there a trend ? are explanatory variables available (or ‘responsible’) ? are separate time series linked or interacting ? are there any sudden changes ? (of direction or slope ?) are there cyclic patterns ? can we predict future trends and/or future values ?

Investigative tools Initial data exploration Correlations Appropriate time series regressions Tools for ‘trends’ Identifying sudden changes

CPUE : Nephrops 11 areas (Eiríksson 1999) Scanned from Zuur, Ieno & Smith (2007) Chp 16

Auto-correlation – investigative tool Reports similarity between data points in the same time series displaced by a certain number of time steps (k) Pearson’s sample autocorrelation coefficient Statistical significance of result adjusted for time displacement being investigated relative to length of the full time series

Auto-correlation: single site Scanned from Zuur, Ieno & Smith (2007) Chp 16

Auto-correlation – basic findings Oscillating positive / negative autocorrelation as time lag increases suggests cycling Seasonal cycles: +/- switching can be predicted Unknown frequency: +/- patterns help identify the periodicity Long term trends: declining autocorrelation with time indicates, becoming negative for longer time lags, indicates long term downward trend (upward trend vice versa) Box-Pierce and Ljung-Box portmanteau tests look at auto-correlations across a number of different time lags and provide a more convincing test of temporal association between data points

Cross-correlation – investigative tool Reports similarity between data points in the time series from different measurement sites displaced by a certain number of time steps (k) Time series being cross correlated can report the same data or different types of data (CPUE at two different locations, or CPUE at location 1 cross correlated with water temperature at location 2) Test statistic again derived from a variant of Pearson’s correlation Statistical significance bands can again be established (Diggle 1990)

Cross-correlation: CPUE at two site Scanned from Zuur, Ieno & Smith (2007) Chp 16

Cross-correlation – basic findings Oscillating positive / negative cross-correlation as time lag increases suggests seasonal or periodic cycling between sites Long term trends: similar interpretations to auto- correlation results Interesting to know at which time lag cross-correlation is at its maximum for any pair of sites Patterns in peak cross-correlations may be made more evident by multi-dimensional scaling methods (Ask Alain Zuur !)

Cross-correlation: Mean SST and NAO Scanned from Zuur, Ieno & Smith (2007) Chp 16

Deseasonalised SST:NAO Scanned from Zuur, Ieno & Smith (2007) Chp 16

Deseasonalised SST:NAO Cross-correlations Scanned from Zuur, Ieno & Smith (2007) Chp 16

Multivariate methods Can show strong associations clearly

Abundance indices for Scottish ducks Scanned from Zuur, Ieno & Smith (2007) Chp 16 Data from Musgrove et al (Wetland Bird Survey) Which abundance series are related ? Try PCA on the time series

Scottish ducks Abundance: PCA biplot Scanned from Zuur, Ieno & Smith (2007) Chp 16 Data from Musgrove et al (Wetland Bird Survey)

Generalised Least Squares for Time series Standard linear regression assumes that data points, and therefore errors around the regression estimates, are independent of one another Time series data are usually auto-correlated and therefore not independent – BIG problem, leading to over-inflated t-statistics and (heavily) increased risk of a ‘false positive’ effect Generalised least squares – allows for covariance structure in the errors surrounding the estimated regression, typically by assuming that covariance between observations is present, but decreases as the time lag between observations increases. Can provide excellent ‘explanation’ of behaviour by identifying significant driving factors

AR, ARIMA and ARIMAX Use ‘lagged’ values of the dependent variable to predict the future path of the dependent variable –Notice ‘predict’ here, not ‘explain’ [not usually anyway] AR, ARIMA, ARIMAX can be terrific tools for prediction BUT These models require stationary time series (time series data which do not contain a trend and data for which the variation is approximately the same across the whole timespan) Stationarity can usually be manufactured by ‘differencing’. Differencing removes trends, so (stating the obvious) trends cannot be detected by these models Statistical validity of these models rests on asymptotic normality – requires 25 – 30 observations in the time series BIG problem for many eco-service datasets

Tools to identify trends: 1 Repeated LOESS Repeated LOESS smoothing – main time series Scanned from Zuur, Ieno & Smith (2007)

Tools to identify trends: 1b Repeated LOESS smoothing – second smoother on resids from first LOESS smoothers Scanned from Zuur, Ieno & Smith (2007)

Tools to identify common trends: 2 MAFA MAFA – min/max auto-correlation factor analysis Identifies underlying trends in multiple time series Weighting factors associated with each time series adjusted so that the first principal component Z 1 (termed the first MAFA trend) has maximum auto-correlation with time lag 1. This represents the strongest trend, or underlying pattern in the dataset. The second MAFA identifies the second most important pattern, and so on. Scanned from Zuur, Ieno & Smith (2007)

Tools to identify trends: 2 MAFA Scanned from Zuur, Ieno & Smith (2007)

Tools to identify trends: 2 MAFA Scanned from Zuur, Ieno & Smith (2007)

Tools to identify trends: 2 MAFA Scanned from Zuur, Ieno & Smith (2007)

Tools to identify trends: 2 MAFA Scanned from Zuur, Ieno & Smith (2007)

Tools to identify common trends: 3 DFA DFA – dynamic factor analysis identifies common trends, effects of explanatory variables and interactions in multivariate time series data sets..... Scanned from Zuur, Ieno & Smith (2007) Tools to identify sudden changes: Chronological clustering..... check out Zuur, Ieno & Smith !

FINISH

Bio-economic modelling Stages 1.identify key ecological and economic relationships underlying the ‘problem’ 2.express these relationships in ‘models’ (equations !) 3.parameterise these models for the study site(s) 4.combine ecological and economic relationships to produce an integrated bio-economic model of the system 5.use the bio-economic model to investigate possible ‘solutions’ to the ‘problem’ 6.identify sensitivity of proposed solutions to variation in the ecological and economic parameters within the models 7.develop robust policies for system management