Tools for quantifying changes in ecosystem service delivery through time

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Tools for quantifying changes in ecosystem
service delivery through time

• CWES Seminary Series
• York
• January 2009

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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. Greens 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
  ?

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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
  ?

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Investigative tools

• Initial data exploration
• Correlations
• Appropriate time series regressions
• Tools for trends
• Identifying sudden changes

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CPUE Nephrops 11 areas (Eiríksson 1999)
Scanned from Zuur, Ieno Smith (2007) Chp 16
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Auto-correlation investigative tool

• Reports similarity between data points in the
  same time series displaced by a certain number of
  time steps (k)
• Pearsons sample autocorrelation coefficient
• Statistical significance of result adjusted for
  time displacement being investigated relative to
  length of the full time series

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

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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
  Pearsons correlation
• Statistical significance bands can again be
  established (Diggle 1990)

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

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Cross-correlation Mean SST and NAO
Scanned from Zuur, Ieno Smith (2007) Chp 16
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Deseasonalised SSTNAO
Scanned from Zuur, Ieno Smith (2007) Chp 16
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Deseasonalised SSTNAO Cross-correlations
Scanned from Zuur, Ieno Smith (2007) Chp 16
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Multivariate methods

• Can show strong associations clearly

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Abundance indices for Scottish ducks
Which abundance series are related ? Try PCA on
the time series
Data from Musgrove et al. 2002 (Wetland Bird
Survey)
Scanned from Zuur, Ieno Smith (2007) Chp 16
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Scottish ducks Abundance PCA biplot
Data from Musgrove et al. 2002 (Wetland Bird
Survey)
Scanned from Zuur, Ieno Smith (2007) Chp 16
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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

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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

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Tools to identify trends 1 Repeated LOESS

• Repeated LOESS smoothing main time series

Scanned from Zuur, Ieno Smith (2007)
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Tools to identify trends 1b

• Repeated LOESS smoothing second smoother on
  resids from first LOESS smoothers

Scanned from Zuur, Ieno Smith (2007)
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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 Z1 (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)
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Tools to identify trends 2 MAFA
Scanned from Zuur, Ieno Smith (2007)
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Tools to identify trends 2 MAFA
Scanned from Zuur, Ieno Smith (2007)
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Tools to identify trends 2 MAFA
Scanned from Zuur, Ieno Smith (2007)
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Tools to identify trends 2 MAFA
Scanned from Zuur, Ieno Smith (2007)
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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 .....

Tools to identify sudden changes Chronological
clustering .....

• check out Zuur, Ieno Smith !

Scanned from Zuur, Ieno Smith (2007)
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FINISH
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Bio-economic modelling

• Stages
• identify key ecological and economic
  relationships underlying the problem
• express these relationships in models
  (equations !)
• parameterise these models for the study site(s)
• combine ecological and economic relationships to
  produce an integrated bio-economic model of the
  system
• use the bio-economic model to investigate
  possible solutions to the problem
• identify sensitivity of proposed solutions to
  variation in the ecological and economic
  parameters within the models
• develop robust policies for system management