Unconstrained Ordination

Unconstrained Ordination Sites Species A Species B Species C Species D Species E 1 0 (1) 5 (1) 1 (1) 10 (4) 10 (4) 2 2 (3) 8 (3) 4 (3) 12 (6) 20 (6) 3 8 (6) 20 (6) 10 (6) 1 (2) 3 (2) 4 4 (5) 11 (5) 8 (5) 11 (5) 14 (5) 5 1 (2) 6 (2) 2 (2) 2 (3) 6 (3) 6 3 (4) 10 (4) 6 (4) 0 (1) 0 (1) 1 Unconstrained Ordination Sites Species A Species B Species C Species D Species E 1 0 (1) 5 (1) 1 (1) 10 (4) 10 (4) 2 2 (3) 8 (3) 4 (3) 12 (6) 20 (6) 3 8 (6) 20 (6) 10 (6) 1 (2) 3 (2) 4 4 (5) 11 (5) 8 (5) 11 (5) 14 (5) 5 1 (2) 6 (2) 2 (2) 2 (3) 6 (3) 6 3 (4) 10 (4) 6 (4) 0 (1) 0 (1) -ABC 1 5 2 +DE 4 3 +ABC -DE 6 2

Important Characteristics of Unconstrained Ordination Techniques P A family of techniques with similar goals. P Organize sampling entities (e.g., species, sites, observations, etc.) along continuous ecological gradients. P Assess relationships within a single set of variables; no attempt is made to define the relationship between a set of independent variables and one or more dependent variables. 3 Important Characteristics of Unconstrained Ordination Techniques P Extract dominant, underlying gradients of variation (e.g., principal components) among sampling units from a set of multivariate observations; emphasizes variation among samples rather than similarity (as in cluster analysis). P Reduce the dimensionality of a multivariate data set by condensing a large number of original variables into a smaller set of new composite dimensions (e.g., principal components) with a minimum loss of information. 4

Important Characteristics of Unconstrained Ordination Techniques P Summarize data redundancy by placing similar entities in proximity in ordination space and producing a parsimonious understanding of the data in terms of a few dominant gradients of variation. P Define new composite dimensions (e.g., principal components) as weighted, linear combinations of the original variables. P Eliminate noise from a multivariate data set by recovering patterns in the first few composite dimensions (e.g., principal components) and deferring noise to subsequent axes. 5 Principal Components Analysis Data Matrix Canopy Snag Canopy Obs Cover Density Height X3 1 80 1.2 35 2 75 0.5 32 3 72 0.8 28........ N 25 0.6 15 Centroid PC1 X 1 PC1 =.8x 1 -.4x 2 +.1x 3 PC2 = -.1x 1 -.1x 2 +.9x 3 X 2 PC2 6

PCA: The Data Set P Single set of variables; no distinction between independent and dependent variables. P Continuous, categorical, or count variables (preferably all continuous); mixed data sets unknown but probably not appropriate. P Every sample entity must be measured on the same set of variables. P Ideally there should be more samples (rows) than number of variables (columns) [i.e., data matrix should be of full rank]. 7 PCA: The Data Set P Common 2-way ecological data: < Sites-by-Environmental Parameters < Species-by-Niche Parameters < Species-by-Behavioral Characteristics < Samples-by-Species < Specimens-by-Characterisitcs Variables Sample x 1 x 2 x 3... x p 1 x 11 x 12 x 13... x 1p 2 x 21 x 22 x 23... x 2p 3 x 31 x 32 x 33... x 3p................ n x n1 x n2 x n3... x np 8

PCA: The Data Set 1 1S0 20 1 5 0 0 55 4 2 0 0 16 25 0 0 0 55 0 5 0 0 55 5 2 1S1 35 1 5 0 0 90 0 2 0 0 20 0 10 0 0 60 0 0 0 0 60 0 3 1S2 20 10 25 0 0 25 3 6 0 0 45 0 0 3 1 21 0 43 0 0 21 43 4 1S3 10 1 40 1 0 25 4 3 0 0 41 0 0 5 0 7 0 72 15 15 7 99........................................................................ 48 6S7 30 3 30 0 0 40 2 2 0 0 45 5 5 1 0 21 0 52 3 0 21 55 9 PCA: Assumptions P Descriptive use of PCA requires "no" assumptions! P Inferential use of PCA requires assumptions! 1. Multivariate Normality PCA assumes that the underlying structure of the data is multivariate normal (i.e., hyperellipsoidal with normally varying density around the centroid). Such a distribution exists when each variable has a normal distribution about fixed values on all others. 10

PCA: Assumptions P Multivariate Normality 11 PCA: Assumptions Multivariate Normality --Consequences: P Invalid significance tests. P Lose strict independence (i.e., orthogonality) among principal components. P Later principal components (i.e., those associated with smaller eigenvalues) will often resemble the earlier components, but will have smaller principal component loadings. 12

PCA: Assumptions Multivariate Normality Univariate Diagnostics: P Conduct univariate tests of normality for each variable. P Visually inspect distribution plots (e.g., histogram, box plot, and normal quantile-quantile plot) for each variable. P < "Univariate" normality does not equal "multivariate" normality. < Often used to determine whether the variables should be transformed prior to the PCA. < Usually assumed that univariate normality is a good step towards multivariate normality. 13 PCA: Assumptions Multivariate Normality Univariate Diagnostics: 14

PCA: Assumptions Multivariate normailty multivariate diagnostics: P Conduct a multivariate test of normality (e.g., E- statistic). P Visually inspect distribution plots (e.g., histogram, box plot, normal quantilequantile plot) for each principal component (PC). 15 PCA: Assumptions Multivariate Normality Solutions: P Collect a larger sample; although even an infinitely large sample will not normalize an inherently nonnormal distribution. P Ignore the problem and do not make inferences. P Use a nonparametric ordination technique like NMDS 16

PCA: Assumptions 2. Independent Random Sample (and effects of outliers) PCA assumes that random samples of observation vectors have been drawn independently from a P-dimensional multivariate normal population; that is, that sample points represent an independent, random sample of the multidimensional space. Transect From Urban 17 PCA: Assumptions Independent Random Sample (and outliers) Consequences: P Invalid significance tests. P Outliers and point clusters exert undue pull on the direction of the component axes and therefore strongly affect the ecological efficacy of the ordination. 18

PCA: Assumptions Outliers Univariate Diagnostics: P Standardize the data and inspect for entities with any value more than, e.g., 2.5 standard deviations from the mean on any variable. Obs var1 var2 var3 1 1 5 8 2 6 4 2 3 2 3 0 4 8 23 7 5 5 7 3 6 7 9 5 7 4 12 8 8 5 13 6 9 25 9 26 10 9 3 4 x i s x Obs stndvar1 stndvar2 stndvar3 1-0.921-0.620 0.153 2-0.178-0.784-0.680 3-0.773-0.947-0.958 4 0.119 2.319 0.014 5-0.327-0.294-0.541 6-0.030 0.033-0.264 7-0.476 0.522 0.153 8-0.327 0.686-0.125 9 2.645 0.033 2.652 10 0.267-0.947-0.403 19 PCA: Assumptions Outliers Univariate Diagnostics: P Construct univariate stem-and-leaf, box, and normal probability plots for each variable and check for suspected outliers. 20

Outliers Multivariate Diagnostics: P Examine deviations of the sample average (Euclidean) distances to other samples. PCA: Assumptions Standard deviation scores >3 Extreme observations 21 Outliers Multivariate Diagnostics: P Examine each sample s Mahalanobis distance to the group of remaining samples. PCA: Assumptions D 2 22

Outliers Multivariate Diagnostics: P Construct histograms, box plots, and normal quantile-quantile plots of the principal component scores for each principal component and check for suspected outliers. PCA: Assumptions 23 Outliers Multivariate Diagnostics: P Construct scatter plots of principal components and check for suspect points. PCA: Assumptions 24

PCA: Assumptions Independent Random Sample (and outliers) Solutions: P Intelligent sampling plan (large, representative sample). P Use stratified random sampling when appropriate. P Eliminate outliers. P Ignore the problem and do not make inferences. 25 PCA: Assumptions 3. Linearity PCA assumes that variables change linearly along underlying gradients and that there exists linear relationships among variables such that the variables can be combined in a linear fashion to create principal components. X3 Centroid PC1 X 1 PC1 =.8x 1 -.4x 2 +.1x 3 PC2 = -.1x 1 -.1x 2 +.9x 3 X 2 PC2 26

Linearity Consequences: P Failure to identify and interpret the gradient. PCA: Assumptions Sampling over A-C A B C Environmental Gradient 27 Linearity Consequences: P Failure to identify and interpret the gradient. PCA: Assumptions 28

Linearity Diagnostics: (A) Scatter plots of variables. PCA: Assumptions 29 Linearity Diagnostics: (B) Scatter plots of principal component (PC) scores. PCA: Assumptions 30

Linearity Diagnostics: (C) Scatter plots of variables vs. principal component (PC) scores. PCA: Assumptions 31 Linearity Solutions: PCA: Assumptions P Sample a shorter range of the environmental gradient. P Use alternative ordination methods, such as Detrended Correspondence Analysis, Detrended Principal Components Analysis, or Nonmetric Multidimensional Scaling. P Interpret results cautiously. 32

General Rules: PCA: Sample Size Considerations P More samples (rows) than variables (columns). P Enough samples should be taken to adequately describe each distinctive community. P Enough samples should be taken to ensure that the covariance structure of the population is estimated accurately and precisely by the sample data set (i.e., enough to insure stable parameter estimates). Rule of Thumb: N $3@P 33 PCA: Sample Size Considerations Sample Size Solutions: P Eliminate unimportant variables. P Sample sequentially until the mean and variance of the parameter estimates (e.g., the eigenvalues and eigenvectors) stabilize. P Examine the stability of the results using a resampling procedure. P Interpret results cautiously; don't extrapolate findings. 34

PCA: Deriving the Principal Components Correlation vs. Covariance Matrices: Raw Data Matrix OBS CCov Snag 1 80 1.2 2 35 3.3 3 5 2.1 CHgt 35 20 5 Variance ij j i i1 n x x n 2 σ (chgt) = 1/3[(35-20) 2 + (20-20) 2 + (5-20) 2 ] = 150 s jk Covariance n i1 xij x j xik xk Cov (ccov-snag) = 1/3[(80-40)(1.2-2.2)+ (35-40)(3.3-2.2)+ (5-40)(2.1-2.2)] = -14 n CCov Snag CHgt Covariance Matrix CCov 950.000-14.000 375.000 Snag 0.740-4.500 CHgt 150.000 Diagonals = variances Off-diagonals = covariances 35 PCA: Deriving the Principal Components Correlation vs. Covariance Matrices: Raw Data Matrix OBS CCov Snag 1 80 1.2 2 35 3.3 3 5 2.1 CHgt 35 20 5 Correlation ij ik ij ik 2 2 n x x x x 2 n x x n x x ij ij ik ik 2 CCov Snag CHgt Correlation Matrix CCov 1.000-0.528 0.993 Snag Cor (ccov-snag) = 3[(80)(1.2) + (35)(3.3) + (5)(2.1)] -[(120)(6.6)] = -0.528 {[3(80 2 +35 2 +5 2 ) -(120 2 )] [3(1.2 2 +3.3 2 +2.1 2 ) -(6.6 2 )] }1/2 1.000-0.427 Cor (ccov-ccov) = 3[(80)(80) + (35)(35) + (5)(5)] -[(120)(120)] = 1.000 {[3(80 2 +35 2 +5 2 ) -(120 2 )] [3(80 2 +35 2 +5 2 ) -(120 2 )] }1/2 CHgt 1.000 Diagonals = internal association Off-diagonals = correlations 36

PCA: Deriving the Principal Components Correlation vs. Covariance Matrices: Raw Data Matrix Standardized Data Matrix OBS 1 2 3 CCov 80 35 5 Snag 1.2 3.3 2.1 CHgt 35 20 5 x ij x s j OBS 1 2 3 CCov 1.298-0.162-1.136 Snag -1.162 1.279-0.116 CHgt 1.225 0.000-1.225 Correlation Matrix Covariance Matrix CCov Snag CHgt CCov Snag CHgt CCov Snag 1.000-0.528 1.000 = CCov Snag 1.000-0.528 1.000 CHgt 0.993-0.427 1.000 CHgt 0.993-0.427 1.000 37 PCA: Deriving the Principal Components Correlation vs. Covariance Matrices: Correlation Matrix Covariance Matrix CCov Snag CHgt CCov Snag CHgt CCov 1.000 CCov 950.000 Snag -0.528 1.000 Snag -14.000 0.740 CHgt 0.993-0.427 1.000 CHgt 375.000-4.500 150.000 P Correlation matrix treats all variables as equally important (i.e., gives equal weight to all variables). P Raw covariance matrix gives more weight to variables with larger variances (i.e., gives weights to variables proportionate to their variance). 38

PCA: Deriving the Principal Components Correlation vs. Covariance Matrices: P Note that the solutions obtained from the correlation and raw covariance matrices will be different. P Correlation matrix almost always preferred, and is always more appropriate if the scale or unit of measurement differs among variables. P Correlation matrix indicates how parsimoniously the PCA will be able to summarize the data. 39 PCA: Deriving the Principal Components Eigenvalues: Characteristic Equation: R I 0 P An NxP data set has P eigenvalues. Where: R = correlation or covariance matrix λ = vector of eigenvalue solutions I = identity matrix P Eigenvalues = variances of the corresponding PC s. P λ 1 > λ 2 > λ 3 >... > λ p P Cor Approach: Σλ i = P = trace of correlation matrix P Cov Approach: Σλ i = Σσ i = trace of covariance matrix 40

PCA: Deriving the Principal Components Eigenvalues: 41 PCA: Deriving the Principal Components Eigenvectors: Characteristic Equation: R I v 0 i Where: λ i = eigenvalue corresponding to the i th PC v i = eigenvector associated with the i th PC P Eigenvectors equal the coefficients (weights) of the variables in the linear equations that define the principal components. P Cor Approach: v i proportional to structure coefficients - loadings (s i ). P Cov Approach: v i "not" proportional to s i. i 42

PCA: Deriving the Principal Components Eigenvectors: PC1 =.8x 1 -.4x 2 +.1x 3 PC2 = -.1x 1 -.1x 2 +.9x 3... X3 PC1 Centroid X 1 X 2 PC2 43 PCA: Deriving the Principal Components Eigenvectors: Correlation Approach Covariance Approach 44

Sample Scores: PCA: Deriving the Principal Components z a x a x... a x * * * ij i1 j1 i2 j2 ip jp Z ij = a ik = x * jk = score for i th PC and j th sample eigenvector coefficient for i th PC and k th variable standardized value for j th sample and k th variable P Scores represent the values of the new uncorrelated variables (components) that can serve as the input data for subsequent analysis by other statistical procedures. 45 PCA: Assessing the Importance of the PCs P How important (significant) is each component? P How "many" components to retain and interpret? 1. Latent Root Criterion: Retain components with eigenvalues >1 (correlation approach only), because components with eigenvalues <1 represent less variance than is accounted for by a single variable. P To determine "maximum" number of components to retain. P Most reliable when the number of variables is between 20 and 50. Too few when P < 20; too many when P > 50. 46

PCA: Assessing the Importance of the PCs 1. Latent Root Criterion: Keep 8 principal components. 47 PCA: Assessing the Importance of the PCs 2. Scree Plot Criterion: The point at which the scree plot curve first begins to straighten out is considered to indicate the maximum number of components to retain. Keep 4 principal components? 48

PCA: Assessing the Importance of the PCs 3. Broken Stick Criterion: The point at which the scree plot curve crosses the broken stick model distribution is considered to indicate the maximum number of components to retain. * i p 1 ki k Broken stick: Keep 2 or 3 principal components 1 1 1 1... 1 369. 1 2 3 22 2 1 1 1 2 3... 22 269. 49 PCA: Assessing the Importance of the PCs 4. Relative Percent Variance Criterion: Compare the relative magnitudes of the eigenvalues to see how much of the total sample variation in the data set is accounted for by each principal component. i p i1 i i 50

PCA: Assessing the Importance of the PCs 4. Relative Percent Variance Criterion: #Measures how much of the total sample variance is accounted for by each principal component. #Cumulative percent variance of all eigenvalues equals 100%. #Used to evaluate the "importance" of each principal component. #Used to determine how many principal components to retain. #Used to evaluate the effectiveness of the ordination as a whole in parsimoniously summarizing the data structure. #Influenced by the number of variables in the data set (decreases as P increases). #Influenced by the number of samples (decreases as N increases). #Should only be used in conjuction with other measures. 51 PCA: Assessing the Importance of the PCs 5. Significance Tests: A. Parametric Tests: Rarely employed because of the assumptions involved (e.g., multivariate normal, independent random sample). B. Nonparametric Tests Based on Resampling Procedures: Jackknife/Bootstrap/Randomization Procedures conceptually simple, computer-intensive, nonparametric procedures, involving resampling the original data, for determining the variability of statistics with unknown or poorly known distributions. 52

PCA: Assessing the Importance of the PCs 5. Significance Tests: Remember, statiscal significance does not always mean ecological significance. P Component may not describe enough variance to meet your ecological needs. P Component may not have a meaningful ecological interpretation as judged by the principal component loadings. P Ultimately, the utility of each principal component must be grounded on ecological criteria. 53 PCA: Assessing the Importance of the PCs Bootstrap Procedure: The premise is that, through resampling of the original data, confidence intervals may be constructed based on the repeated recalculation of the statistic under investigation. [ n x p ] PCA Φ i(all) Bootstrap sample [ n x p ] [ n x p ] PCA Φ * i(1) Repeat M times [ n x p ] PCA 54 Φ * i(m) Bootstrap estimates

PCA: Assessing the Importance of the PCs Bootstrap Procedure: Bootstrap estimate * i j M 1 M * i( j) * SE( ) i M ji * * i(( j) i M 1 2 Bootstrap error ratio * i * SE( ) i t-statistic with N 1 df 55 PCA: Assessing the Importance of the PCs Bootstrap Procedure: Test Hypothesis: H o : Φ i(observed) = Φ i(no real data structure) H A : Φ i(observed) > Φ i(no real data structure) Translated distribution Translation α- level Distribution of bootstrap estimates 0 Φ i(ho) Φ i(observed) 56

PCA: Assessing the Importance of the PCs Randomization Procedure: The premise is that, through resampling of the original data, we can generate the actual distribution of the statistic under the null hypothesis, and test the observed sample against this distribution. [ n x p ] PCA Φ i(all) Randomize w/i Columns Random permutation [ n x p ] PCA Φ * i(1) Repeat M times [ n x p ] PCA 57 Φ * i(m) Permutation estimates PCA: Assessing the Importance of the PCs Randomization Procedure: Test Hypothesis: H o : Φ i(observed) = Φ i(no real data structure) H A : Φ i(observed) > Φ i(no real data structure) Distribution of permutation estimates Direct and intuitive interpretation of the type 1 error rate! α- level 0 Φ i(ho) Φ i(observed) 58

PCA: Assessing the Importance of the PCs Randomization Procedure: 59 PCA: Interpreting the Principal Components 1. Principal Component Structure (also loadings ): s v ( ) ij i j i S ij = v i(j) = λ i = 60 correlation between the i th PC and the j th variable eigenvector element of the j th variable in the i th PC derived from correlation matrix i th eigenvalue (i th PC) #Bivariate product-moment correlations between the principal components and original variables. #The squared loadings indicate the percent of the variable's variance accounted for by that component. #Note that the structure is different depending on whether the correlation matrix or raw covariance matrix is used in the eigenanalysis.

PCA: Interpreting the Principal Components 1. Principal Component Structure (also loadings ): s v ( ) ij i j i s 11. 16944 * 5309. 61 PCA: Interpreting the Principal Components Significance of Structure Correlations: P s ij > ±0.30 significant, when N > 50 P s ij > ±0.26 significant, when N = 100 P s ij > ±0.18 significant, when N = 200 P s ij > ±0.15 significant, when N = 300 # The disadvantage of these rules is that the number of variables being analyzed and the specific component being examined are not considered. < As you move from the 1 st to later components, the acceptable level for considering a loading significant should increase. < As N or P increases, the acceptable level for considering a loading significant should decrease. 62

PCA: Interpreting the Principal Components Interpreting Structure Correlations: P The larger the sample size, the smaller the loading to be considered significant. P The larger the number of variables being analyzed, the smaller the loading to be considered significant. P The larger the number of components, the larger the size of the loading on later factors to be considered significant for interpretation. P Note, that significant correlation coefficients may not necessarily represent ecologically important variables. 63 PCA: Interpreting the Principal Components Interpreting Structure Correlations: P Highlight the highest significant loading for each variable (red) P Highlight other significant loadings (blue) PC1 Gradient Conifer Cedar/Fir Huckle Fern/Grape Salal/Plum Currant Hardwood Alder/Salmon 64

PCA: Interpreting the Principal Components Interpreting Structure Correlations: P Highlight the highest significant loading for each variable (red) P Highlight other significant loadings (blue) PC2 Gradient Vine Fir Fern Hazel Plum/Salal Hemlock Cedar Forb 65 PCA: Interpreting the Principal Components 2. Final Communalities: Communality of a variable is equal to the squared multiple correlation for predicting the variable from the principal components; that is, the proportion of a variable's variance that is accounted for by the principal components. c j P s 2 ij i1 P Prior communalities = 1.0 P Final communality estimates = the squared multiple correlations for predicting the variables from the "retained" principal components 66

PCA: Interpreting the Principal Components 2. Final Communalities: c j P s 2 ij i1 P Final communality estimates indicate how well the original variables are accounted for by the "retained" principal components. P Final communality estimates increase from zero to one as the number of retained principal components increases from zero to P. 67 PCA: Interpreting the Principal Components 3. Principal Component Scores and Biplots: P Scatter plots of scores graphically illustrate the relationships among entities. P Axes can be scaled in a variety of ways (don t sweat over it). P Scatter plots of scores can be useful in evaluating model assumptions (e.g., linearity, outliers). 68

PCA: Interpreting the Principal Components Enhanced Ordination Plots: 3d plots Samples Variables 69 PCA: Interpreting the Principal Components Enhanced Ordination Plots: Overlays Sample scores scaled in relation to magnitude of variable 70

PCA: Interpreting the Principal Components Enhanced Ordination Plots: Overlays Scatter plot envelopes using quantile or robust spline regression P Scatter plot envelopes are useful for assessing the shape of the response function (i.e., linear, unimodal) and, thus, determining model appropriateness. 71 PCA: Interpreting the Principal Components Enhanced Ordination Plots: Displaying groups Ordihulls Ordispider 72

PCA: Interpreting the Principal Components Enhanced Ordination Plots: Displaying groups Ordiellipse Ordiarrows 8 7 5 6 4 1 2 3 73 PCA: Interpreting the Principal Components Enhanced Ordination Plots: Example of worm tracks Marten habitat trajectories under simulated management regimes Start Habitat extent 74

PCA: Interpreting the Principal Components Enhanced Ordination Plots: Fitting other variables Permutation Tests: Vector fitting: 75 PCA: Interpreting the Principal Components Enhanced Ordination Plots: Fitting other variables Permutation Tests: Factor fitting: 76

PCA: Interpreting the Principal Components Enhanced Ordination Plots: Fitting other variables GAM results: GAM surface fitting: 77 PCA: Rotating the Principal Components Purpose: To improve component interpretation by redistributing the variance from earlier components to later ones to achieve a simpler, theoretically more meaningful, principal component structure; that is, by increasing loadings of important variables and decreasing loadings of unimportant variables. P Orthogonal rotation: Axes maintained at 90 o < varimax rotation < quartimax rotation < equimax rotation P Oblique rotation: Axes "not" maintained at 90 o 78

PCA: Rotating the Principal Components Orthogonal Rotation +1.0 PC2 V 1 Oblique Rotation +1.0 PC2 V 1 V 2 V 2 Loadings -1.0 0 +1.0 V 4 V 3 PC1 Loadings -1.0 0 +1.0 V 4 V 3 PC1 V 5 V 5-1.0-1.0 79 PCA: Rotating the Principal Components Orthogonal Rotations: P Varimax...column rotation to simplify structure within component and improve component interpretation (increase high loadings and decrease low loadings). P Quartimax...row rotation for simplifying interpretation of variables in terms of wellunderstood components (variables load high on fewer components). 80

PCA: Rotating the Principal Components Use and Limitations: P Effective when sample is not multivariate normal. P Rotations always reduce the eigenvalue (variance) of the first component. P Rotations always maintain the cumulative percent variance (or total variance accounted for by the retained components). P Only true test of the usefulness of component rotation is whether the component interpretation is better. 81 Limitations: Principal Components Analysis P PCA can produce severely distorted data sets with long gradients other techniques perform better in most cases under these conditions. P PCA assumes an underlying multivariate normal distribution which is unlikely in most ecological data sets. P PCA assumes linear response model, e.g., that species respond linearly to underlying environmental gradients unrealistic in many (but not all) ecological data sets, especially for long gradients. 82

Principal Components Analysis Review: P Eigenvalues... P Eigenvectors... P Structure coefficients (loadings)... P Principal component scores... P Final communalities... Variances of PC s. Variable weights in PC linear combinations. Correlations between original variables and PC s. Location of samples on PC s. % of variance in original variables explained by retained PCs. 83