Fitted vectors. Ordination and environment. Alternatives to vectors. Example: River bryophytes

Transcription

1 Vegetation Analysis Ordination and environment Slide 1 Vegetation Analysis Ordination and environment Slide Ordination and environment We take granted that vegetation is controlled by environment, so 1. Two sites close to each other in ordination have similar vegetation, and. If two sites have similar vegetation, they have similar environment; moreover. Two sites far away from each other in ordination have dissimilar vegetation, and perhaps. If two sites have different vegetation, they have different environment Fitted vectors Direction of fitted vector shows the gradient, length shows its importance. For every arrow, there is an equally long arrow into opposite direction: Decreasing direction of the gradient. Implies a linear model: roject sample plots onto the vector for expected value. Class values as weighted averages. Dimension! Fe ph 1 Mo N S ! Zn Dimension 1 Ca Mg Humdepth Mn 8 Vegetation Analysis Ordination and environment Slide 1 ternatives to vectors Vegetation Analysis Ordination and environment Slide 1 Example: River bryophytes Fitted vectors natural in constrained ordination, since these have linear constraints. Distant sites are different, but may be different in various ways: Environmental variables may have a non-linear relation to ordination. mds artsize r = 0. mds Currvel r = 0. mds slope r = 0.1 mds Depth r = 0.1 Dimension!1.0! Linear r = 0. Ca Dimension!1.0! Observed values Ca Dimension!1.0! GAM Ca mds!1.!1.0! width r = 0. mds!1.!1.0! Instab r = 0.0 mds!1.!1.0! ph r = 0.88 mds!1.!1.0! Conduc r = 0.! Dimension 1! Dimension 1! Dimension 1!1.!1.0! !1.!1.0! !1.!1.0! !1.!1.0!

2 Vegetation Analysis Ordination and environment Slide 1 Vegetation Analysis Ordination and environment Slide 1 Lessons from environmental interpretation Constrained vs. unconstrained aims Environmental variables need not be parallel to ordination axes. Axes cannot be taken as gradients, but gradients are oblique to axes: You cannot tear off an axis from an ordination. Never calculate a correlation between an axis and an environmental variable. Environmental variables need not be linearly correlated with the ordination, but locations in ordination can be exceptional. Unconstrained ordination tries to display the variation in data. Constrained ordination tries to display only the variation that can be explained with constraining variables. You can observe only things that you have measured. V V1 Vegetation Analysis Ordination and environment Slide Vegetation Analysis Ordination and environment Slide 1 The constraining toolbox Constrained Correspondence Analysis (CCA) Linear tools based on CA framework: Discriminant analysis, Canonical Correlations Redundancy Analysis (RDA). Only RDA useful in community ecology if linear model is adequate. Unimodal tools based on CA framework: Constrained or Canonical Correspondence Analysis (CCA). Absolutely the most important constrained ordination in ecology: The only one dealt with in these lectures. Ordinary Correspondence analysis gives: 1. Site scores which may be regarded as describing the gradients.. Species scores which may be taken as location of species optima in the space spanned by site scores. Constrained or Canonical Correspondence Analysis gives in addition:. Environmental scores which define the gradient space. And optimizes the interpretability of results.

3 Vegetation Analysis Ordination and environment Slide 0 Cla.ste CCA: gorithm 1. Fit weighted linear regression to all species individually using all constraints as explonatory variables.. Analyse fitted values using CA! le.sch! Vegetation Analysis Ordination and environment Slide 1 Arbitrary Site Scores Species scores as weighted averages of Site Scores Site scores as weigtehd averages of Species Scores Orthogonalize Site Scores Any Change in Scores? CCA: ternating regression algorithm CA DCA CCA Arbitrary Site Scores Species Scores as weighted averages of Site Scores Site Scores as weighted averages of Species Scores Detrend Site Scores Any Change in Scores? Arbitrary LC Scores Species Scores as weighted averages of LC Scores WA Scores as weighted averages of Species Scores LC Site Scores as predicted values of linear regression Any Change in Scores? Two kind of site scores: 1. LC scores are predicted values of multiple regression with constraining variables = the constraints.. WA Scores are weighted averages of species scores. Vegetation Analysis Ordination and environment Slide Vegetation Analysis Ordination and environment Slide Those numbers... Eigenvalues exactly like in CA. CCA eigenvalue should be lower than in CA or constraining may have been useless. Eigenvalue has nothing to do with variance, so there is neither variance explained. Species Environment correlation: Multiple correlation from constraining regression: Usually high even with poor models. ointwise goodness of fit can be expressed either as residual distance from the ordination space or as proportion of projection from thet total Chi-squared distance exactly like in CA. Like CA plot, but now a triplot: Vectors for linear constraints. Classes as weighted averages. Most use LC scores: These are the constraints. opular to scale species relative to eigenvalues, but keep sites unscaled. Sites do not display the configuration, but their projections onto environmental vectors are the estimated values. CCA plot Bar.lyc Bet.pub 1 ti.cil 1 1 Cla.bot Cal.vul Led.pal ol.com Ich.eri Dic.fus Vac.uli Dic.pol Dip.mon Cla.ama Cla.def Cla.arb Cla.cri 0 Emp.nig Cla.fim Cet.isl Ste.sp 1 Cla.unc Cla.gra Cla.ran Vac.myr Vac.vit Cla.coc Cla.cor ol.jun in.syl oh.nut Cla.chl Cet.eri ol.pil el.aph Cla.sp Des.flele.sch Nep.arc Cla.ste 1 Cla.cerCet.niv Dic.sp Cla.phy 1 Hyl.spl 8!

4 Vegetation Analysis Ordination and environment Slide Class constraints Vegetation Analysis Ordination and environment Slide redicted values of constraints Class variables usually as dummy variates: Make m 1 indicator variables out of m levels Indicator scoring: 1 if site belongs the the class, 0 otherwise One dummy less than levels, because all are redundant Ordered factors may be better expressed with polynomial constraints! X1 X X X X X X1 X 1 X X X X X X1 X1 Moisture.Q Moisture.C X1 Moisture.L X8 X1 X0 X1 roject a site point onto environmental arrow: rediction Exact with two constraints: Multidimensional space warped! Vegetation Analysis Ordination and environment Slide LC or WA Scores? Vegetation Analysis Ordination and environment Slide WA and LC scores with class constraints Mike almer: Use LC scores, because they give the best fit with the environment, and WA scores are a step from CCA towards CA. Bruce McCune: LC scores are excellent, if you have no error in constraining variables. Even with small error, LC scores become miserable, but WA scores are good even in noisy data. Error!free Constraints Noisy Constraints LC Scores LC Scores x.t, Normal error sd=0. Error!free Constraints Noisy Constraints WA Scores WA Scores 1. Make class centroids as distinct as possible.. Make clouds about centroids as compact as possible. Success λ. LC scores are the class centroids: The expected locations. If high λ, WA scores are close to LC scores. With several class variables, or together with continuous variables, the simple structure becomess blurred. X1 X1 LC Scores X X1 X1 X X8 X X1 X X1 X0 X X X1 X X1 X X X X!! 0 WA Scores X1 X1 X1 X0 X X X X X X X8 X X1X X X X1X!! 0 Dune meadows, Landuse constraints

5 Vegetation Analysis Ordination and environment Slide 8 LC scores are the constraints Vegetation Analysis Ordination and environment Slide Number of constraints and the plot LC scores are the scaled and weighted constraints and shuffling the order of sites of the community data does not change the configuration Shuffled sites, LC scores Cla.phy 1 Hyl.spl 1 Cla.ste Ca S Cla.chl Mg Zn 1 Des.fle Vac.myr le.sch ol.com Cet.isl Cla.sp Humdepth Mn Dic.sp ol.jun oh.nut in.syl Nep.arc el.aph Dic.pol Cla.unc Cla.cor Emp.nig Vac.vit Led.pal Cla.cri ph Cla.def ti.cil Cla.fim Cet.eri Cla.cer 1 Cla.coc Bet.pub Bar.lyc Cla.bot Cla.gra Mo Fe 0 Dic.fus ol.pil Cla.ran 1 Cet.niv Cla.arb 1 N Dip.mon Cal.vul Cla.ama Vac.uli Ste.sp Ich.eri Cet.niv Cla.cer 1 Cla.phy 1 Cla.ste el.aph Nep.arc Cla.spCet.eri Cla.chl oh.nut ol.jun ol.pil le.sch Cla.cor Vac.vit Dip.mon Des.fle in.syl Cla.gra Cla.unc Emp.nig Dic.pol Cla.ranCla.ama Cla.coc Cet.isl Cla.def 0 Vac.myr Cla.fim Vac.uliCla.arb Ste.sp Cla.cri ol.com 1 Ich.eri Cal.vul Led.pal ti.cil Cla.bot Dic.fus 1 Bet.pub Bar.lyc 1 Dic.sp 1 Hyl.spl 8 1! Vegetation Analysis Ordination and environment Slide Vegetation Analysis Ordination and environment Slide Number of constraints and curvature Curvature cured because forced to linear constraints. High number of constraints = no constraint. Absolute limit: Number of constraints = min(s, N) 1, but release from the constraints can begin much earlier. Reduce environmental variables so that only the important remain: Heuristic value better than statistics. Reduces multicollinearity as well. Eigenvalue Unconstrained 1 Constraints Constraints 1 0 Axis DECORANA in Disguise Constrained Correspondence Analysis replaced Decorana as the canonical method and indeed, it is Decorana in disguise Detrending: Based on fitted values from linear regression Rescaling: Linear combinations of environmental variables scaled similarly Downweighting: Rare species fit poorly

6 Vegetation Analysis Ordination and environment Slide 1 olynomial Constraints: A Bad Idea Vegetation Analysis Ordination and environment Slide Constrained horseshoe Unconstrained CA produces curves, because species have non-linear responses to gradients Constrained CA straightens up curves, because it forces linear species responses olynomial constraints produce quadratic fitted values: Ordination will be quadratic. y (sd units)! Truth 0 1 x (sd units) CCA, Linear Constraints y x CA!1! !!! 0 1 CA! CA1 CCA, olynomial Constraints x^ x y^ y!.0! Curve is removed in CCA because the solution is forced to linear constraints If contraints have a quadratic relation to each other, a curve may re-appear X olynomial constraints and interactions are X1 generally a bad idea! X1 X X X X 1 X X X1 X X X X1 Moisture.Q Moisture.C X Moisture.L X8 X1 X0 X1 X1 Vegetation Analysis Ordination and environment Slide 1 Levels of environmental intervention Vegetation Analysis Ordination and environment Slide 1 Significance of constraints Background variables (covariates) Environemntal variables (constraints) Environmental variates (correlates) constrain constrain correlate (residual) (residual) partial CCA CCA CA DCA artial CCA removes the effect of background variables before proper (C)CA: random or nuisance variables. Residual ordinations may be analysed at all level: artitioning of variation. Constraints are linear: If levels of environmental variables are not orthogonal, this may result in negative components of variation. Information of lower levels mixed with upper. CCA maximizes eigenvalue with constraints. ermutation tests can be used to assess significance: ermute lines of environmental data. Repeat CCA with permuted data. If observed λ higher than (most) permutations, regarded as significant. Many constraints = much opportunity for optimizing: Significance usually lower. Eigenvalue ermutation test with constraints 0 permutations 1 Dimension

7 Vegetation Analysis Ordination and environment Slide 1 ermutation statistic Without constraints, sum of all eigenvalues is a natural choice Testing of first eigenvalue has an unclear meaning In partial models, use pseudo-f : ( p ) ( q ) F p,q = λ (c) i /p / λ (r) i /q i with constrained (c) and residual (r) eigenvalues λ and respective number of axes p and q. Not at all distributed like real F, but used in permutation tests i Vegetation Analysis Ordination and environment Slide 1 Number of permutations Too few permutations: Cannot detect significant response when it is close to a critical limit Too many permutations waste time Sequential testing: ermute so many times that assessed significance is certainly outside the grey zone where it could be either significant or nonsignificant Number of ermutations Vegetation Analysis Ordination and environment Slide What is permuted? Vegetation Analysis Ordination and environment Slide 1 Selecting constraining variables No conditioning variables: Community data or constraints can be permuted In partial models ith conditioning variables: Community data cannot be permuted, because it is dependent on conditions Small number of variables means stricter constraints, reduced curvature, improve interpretation, increases signficance: Try to end up with one or two constraints for each independent factor. Significance tests and visual inspection help in selecting environmental variables. Constraints cannot be permuted, because they correlate with conditions Residuals are exchangeable if they are independent and identically distributed... Automated selection dangerous: Small changes in data set can change the whole selection history, and omission of a variable does not mean it is unimportant. Final selection must be made with heuristic criteria. Reduced model permutes residuals after conditions, Full model residuals after conditions and constraints The purpose of computation is insight, not numbers

8 Vegetation Analysis Ordination and environment Slide Vegetation Analysis Ordination and environment Slide Automatic stepping is dangerous Automatic model selection may give different results depending on stepping direction, scope or small changes in the data set! Forward ! 8 8 Forward w/interactions : 1 0 : Humdepth Backward ! S 1 Components of Variation Take a partial model CCA(Y X Z) The explained Inertia can be decomposed into two components: 1. Explained by X in a simple model CCA(Y X). The residual effect of X after removing the variation caused by the conditioning variable Z After conditioning by Z, the eigenvalue of X decreases by the amount of shared component of variation Vegetation Analysis Ordination and environment Slide 1 Negative Components of Variation Constraints: Ca ph Red line Ca lightblue levels ph CCA(Y X Z) is equal to CCA(Res(CCA(Y Z)) X Z) If variables are better predictors together than in isolation: λ XZ > λ X λ Z Constraints allow the reappearance of the curve λ Ca = 0.1 λ Ca ph = 0.