Constrained ordination

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Constrained ordination

• Regression is the key to understanding

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Studying community means studying individual
species and comparing them
MoistureManure (nutrients)
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Linear Regression the model
Y b0 b1X e
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Linear Regression the quality

• Total sum of squares (TSS)
• Model sum of squares (MSS)
• Residual sum of squares (RSS)

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Multiple responses,multiple predictors
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The best predictors ever principal components
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Comparing two regressions with first two PCA axes
PCA l1 l2 0.51
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PCA ordination diagram
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What to do with measured environmental factors? -
I
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What to do with measured environmental factors? -
II

• Predicting species values using PCA1 and PCA2
  yik b1k PCA1i b2k PCA2i e
• Constraining scores definition PCA-gt RDA
  RDA1i c11Moisturei c12Manurei
• Similarly RDA2i c21Moisturei
  c22Manurei
• Consequently yik b1kc11Moistureib1kc12Ma
  nurei b2kc21Moistureib2kc22Manurei

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The boiled-down predictors constrained axes
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Definition of constrained ordination axes
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Comparing regressions, PCA axes, and RDA axes
RDA l1 l2 0.37
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RDA alternative interpretation
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When linearity is not a good idea

• Weighted regression on proportional data leads to
  weighted averaging approach yik ? (yik/
  yk)/(yi/ y) case weights are yi , variable
  weights are yk
• Roughly
• resulting gradients are best predictors for
  unimodal response model
• species scores represent optima

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Species scores vs. optima