introduction to numerical methods in community ecology

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introduction to numerical methods in community
ecology

• 01.20.20

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species environmental data
classification define groups
Indicator Species Analysis MRPP Discriminant
Analysis
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introduction to ordination

• Ordination - positioning of species or samples in
  order along an abstracted gradient

A
D
B
C
E
F
LATENT GRADIENT
why would you want to do this??
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why ordination?

• reduced dimensionality/data reduction
• (so that reduced space similarity reflects full
  dimensional/ecological similarity)
• ecological data often do not meet assumptions of
  traditional statistics.
• explore community responses to environmental
  gradients

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environmental gradient response

• who lives with whom and why?
• how are environmental gradients associated with
  ordinal arrangement of sites?

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confusing vocabulary? eigenanalysis

• techniques resulting in a linear reduction in
  dimensionality
• aka singular value decomposition
• PCA, CA, DCA, CCA
• eigenvalue strength of an axis
• eigenvector set of sample scores

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confusing vocabulary? Three Cs

• Canonical - refers to simultaneous analysis of
  two or more related data matrices
• Correspondence repeating a weighted averaging
  of site scores to yield species scores and vice
  versa
• Correlation test of relationship between two
  matrices

McCune and Grace 2002
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confusing vocabulary? site, sample, and species
scores

• Sitesample scores - refers to the coordinate
  along an ordination axis specifying the location
  of a site/sample
• Species scores refers to the coordinate along
  an ordination axis specifying the location of a
  species

Mike Palmer 2004
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history of ordination and methods

• 1957 (Bray and Curtis) - Polar Ordination (BC)
• 1954 (Goodall) Principal Components analysis
  (PCA)
• 1964 (Kruskal) Nonmetric Multidimensional
  Scaling (NMS, NMDS)
• 1973 (Hill) Correspondence Analysis (CA)
• 1980 (Hill and Gauch) the very popular
  Detrended Correspondence Analysis (DCA)
• 1986 (ter Braak) Canonical Correspondence
  Analysis (CCA)
• 2000s revival of NMS probably current method
  of choice for peer-review

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direct vs. indirect

• choice depends on data collected/available
• Direct requires environmental data matrix for
  ordination (samples environmental variables)
• Indirect ordination independent of
  environmental variables (samples only)

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direct gradient analysis

• requires full set of environmental variables
• mapping of species into a measured environmental
  space.
• knowledge of relevant and important variables
  (somewhat subjective?)
• weighted averaging/CCA? (CCA slightly different
  bird because it does not assume relative
  importance of env. vars)

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indirect gradient analysis

• no environmental data required, but can be
  overlaid after the fact.
• Gradients assumed from species data
• no assumptions regarding the specific
  species?environment responses in initial
  ordination

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indirect vs. direct
does your dataset include relevant and dependable
weights of species and environmental vars?
no
yes
indirect gradient analysis current methods of
choice
direct gradient analysis (often best to confirm
results with indirect method)
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weighted averaging (Whittaker 1967)

• direct gradient analysis
• a set of previously assigned species weights is
  used to calculate scores for the sites. The
  calculation is a weighted averaging for species
  present in the sample unit.

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weighted averaging (Whittaker 1967)

• Algorithm
• Where
• vi ordination score for site i
• aij abundance of species j at site i
• wj weight of species j

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weighted averaging (Whittaker 1967)

• Assumptions knowledge of species weights
• Advantages - simple, easy to use, understand,
  communicate good for regulators and
  nonscientists
• Disadvantages - focuses on a single gradient,
  potential for species optimum to occur outside of
  sampled range

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Bray Curtis - Sorenson(Bray Curtis 1957)

• indirect gradient analysis method
• polar ordination - two end point samples are
  used as the poles of the natural gradient

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Bray Curtis - Sorenson(Bray Curtis 1957)

• Algorithm (method of scaling)
• end points (poles) selected
• distance matrix applied to position sites along
  axes with respect to poles

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Bray Curtis - Sorenson(Bray Curtis 1957)

• Assumptions end points are true gradient ends?
• Advantages
• - no assumption of linearity among species
• - flexible distance measure
• Disadvantages
• - dependent on endpoint selection
• - sites ordered according to relation to
  endpoints

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Bray Curtis - Sorenson(Bray Curtis 1957)
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principal components analysis (PCA) (Pearson
1901 Goodall 1954)

• indirect gradient analysis
• performed to reduce data to a smaller set of
  synthetic variables
• based on principal that strongest covariation
  among variables emerges in the first few axes, or
  components
• ideal technique for data with approximately
  linear relationships among variables (infrequent
  with ecological data)

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principal components analysis (PCA) (Pearson
1901 Goodall 1954)

• Algorithm
• Calculate variance/covariance matrix (cross
  products among columns) related to the
  correlation matrix
• Calculate eigenvalues
• Determine eigenvectors
• Find scores for each site (original data matrix
  matrix of eigenvectors)
• Calculate loading matrix (eigenvector matrix
  sqrt of eigenvalue matrix) gives sensitivities
  to changes in principal components

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principal components analysis (PCA) (Pearson
1901 Goodall 1954)

• Assumptions
• - component variables are normally distributed
  and linearly related
• - component variables are uncorrelated
  (infrequently true for ecological data)
• - linear, monotonic response of species to
  environmental gradients

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principal components analysis (PCA) (Pearson
1901 Goodall 1954)

• Disadvantages
• - linear response model
• - often difficult to interpret
• - even moderately heterogeneous data sets will
  be severely distorted by PCA
• - horseshoe effect
• - implicit use of Euclidean distance
• - strongly affected by outliers

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linear responses?
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Horseshoe effect of principal components analysis
(PCA)

• The second axis is curved and twisted relative
  to the first and does not represent a true
  secondary gradient occurs with very long
  gradients

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principal components analysis (PCA) (Pearson
1901 Goodall 1954)
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reciprocal averaging (RA)/ correspondence
analysis (CA)(Hill 1973)

• similar to PCA, but implicitly uses chi-square
  distance measure
• double weighting of species (rows) and stands
  (columns) in CA distinguishes from PCA

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reciprocal averaging (RA)/ correspondence
analysis (CA)

• Algorithm (eigenanalysis problem)
• Arbitrary position of sites along axes
• Weighted average of species is taken using
  abundance values as weights (species scores)
• Calculate site scores by weighted averaging of
  species score (step 2 used to locate site on
  environmental gradient)
• Center and standardize site scores
• Return to step 2 until convergence

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reciprocal averaging (RA)/ correspondence
analysis (CA) (Hill 1973)

• Assumptions
• Homogeneous distribution of sites along gradients
• Equal tolerances of species to environment
• Assumptions regarding species optima (homogeneity
  and independence of abundance)
• Advantages - very effective for detecting major
  environmental gradients

McCune and Grace 2002
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reciprocal averaging (RA)/ correspondence
analysis (CA) (Hill 1973)

• Disadvantages
• arch effect on second and higher axes
• end compression of axes
• Chi-square distance measure exaggerates samples
  with numerous rare species
• Best applied when beta diversity is low

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arch effect of correspondence analysis (CA)
(Hill 1973)

• Second axis is an arched function of the first
  axis, due to the unimodal distribution of the
  species along gradients

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correspondence analysis (CA) (Hill 1973)
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detrended correspondence analysis (DCA)(Hill
Gauch 1980)

• Indirect gradient analysis
• Similar to CA but with two corrections
• One of the more popular methods, but subject to
  recent criticism

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detrended correspondence analysis (DCA)(Hill
Gauch 1980)

• Solve eigenanalysis problem (like CA)
• Detrend axes
• Rescale axes

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detrended correspondence analysis (DCA)(Hill
Gauch 1980)

• rescaling corrects problems of axis end
  compression in order to have species turn over
  at uniform rate along gradient is this
  appropriate?
• accomplished by Hills method - the axes are
  scaled so that species have unit dispersion along
  axes (beta diversitygradient length constant)

McCune and Grace 2002
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Detrending by DCA(Hill Gauch 1980)

• arch effect removed by detrending data
• division of first axis into segments
• for each segment on axis 2, setting average score
  to 0.

McCune and Grace 2002
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detrended correspondence analysis (DCA)(Hill
Gauch 1980)
McCune and Grace 2002
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detrended correspondence analysis (DCA)(Hill
Gauch 1980)

• Assumptions
• unimodal species response
• similar maxima and dispersions of species
• Advantages
• popular, well-known
• Disadvantages
• criticized for being a fix of a fix
• number of segments affects results (demonstrated
  in lab)
• implicitly use of chi-square distance measure

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detrended correspondence analysis (DCA)(Hill
Gauch 1980)
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canonical correspondence analysis (CCA) (ter
Braak 1986)

• direct gradient analysis example of constrained
  ordination
• relevant environmental variables are known, but
  not the relative degree of influence
• axes are constrained to be linear functions of
  measured environmental variables

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canonical correspondence analysis (CCA) (ter
Braak 1986)

• 3 scores reported
• Species scores
• Sample scores as weighted averages of species
  (WA)
• Sample scores as Linear Combinations (LC) of
  environmental variables

• which to interpret?? Good question!
• Palmer (1993) recommended using the LC-scores
  because they are best fitted to the environmental
  variables and are logically most consistent with
  premise of CCA.
• McCune (1997) argued that WA scores are more
  robust because LC scores may be more sensitive to
  noise in the environmental data.

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canonical correspondence analysis (CCA) (ter
Braak 1986)

• Same as CA, but before species scores are
  re-calculated, sample weighted average (WA)
  scores are further weighted by Linear Combination
  (LC) scores from environmental variables

Figure from Dean Urban 2004
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canonical correspondence analysis (CCA) (ter
Braak 1986)

• Assumptions
• unimodal species response
• influential environmental variables are recorded
• Advantages
• ignores community structure that is unrelated to
  environmental variables
• Disadvantages
• Lots of outputs difficult to know what is
  meaningful
• Unimodal species response

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canonical correspondence analysis (CCA) (ter
Braak 1986)
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non-metric multidimensional scaling (NMS)
(Kruskal 1964)

• Scaling method (similar to Bray Curtis)
• iterative optimization based on rank order of
  dissimilarities

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non-metric multidimensional scaling (NMS)
(Kruskal 1964)

• Algorithm
• calculate distance matrix
• assign random starting positions
• tweak positions to minimize the ranks of the
  differences in the original, fully dimensional
  distance matrix and the reduced dimension matrix
• Remember the point is to find the best
  positions of your sites on a to-be-determined
  number of axes to duplicate the original distance
  matrix as closely as possible

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non-metric multidimensional scaling (NMS)
(Kruskal 1964)

• Stress departure
• from monotonicity
• between the original
• distances and the
• reduced, ordinated
• distances.

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non-metric multidimensional scaling (NMS)
(Kruskal 1964)

• Assumptions?
• Advantages
• any distance measure (sort of.well demonstrate
  this in lab)
• no assumption of linearity between variables
• ranked distances good for reducing stress
• Disadvantages
• computationally intensive
• local vs. global minima
• solution dependent on the number of axes selected

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non-metric multidimensional scaling (NMS)
(Kruskal 1964)
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choosing the right method

• as demonstrated, each method has advantages and
  disadvantages
• consider method assumptions can you be
  comfortable with making these statements about
  your data?
• run several different methodsdo the results
  generally correspond?

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choosing the right method

• run several different methodsdo the results
  generally correspond?
• compare a direct and an indirect gradient method
  Wentworths theory

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brief look at outputs

• more comprehensive discussion during lab
• lots of outputs and differences between methods

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another look at outputsbiplots

• closeness of samples (sites) on ordination plots
  indicate similarity
• vectors illustrate relationships of environmental
  variables to species the relative length of the
  vector indicates the relative strength (and
  direction)

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some references

• Dean Urban (2004) - ENV358 lecture notes
• http//www.nicholas.duke.edu/landscape/classes/env
  358
• McCune and Grace (2002) Analysis of Ecological
  Communities
• http//home.centurytel.net/mjm/
• Mike Palmer (OK State) The Ordination Website
• http//www.okstate.edu/artsci/botany/ordinate