Multivariate Ordination Analyses: Principal Component Analysis. Dilys Vela

Transcription

1 Multivariate Ordination Analyses: Principal Component Analysis Dilys Vela Tatiana Boza

2 Multivariate Analyses A multivariate data set includes more than one variable ibl recorded dd from a number of replicate sampling or experimental units, sometimes referred to as objects.

3 If these objects are organisms, the variables might be morphological or physiological measurements If the objects are ecological sampling units, the variables might be physicochemical measurements or species abundances

4 What ordinations analyses are? Ordination is arranging items along a scale (axis) or multiples li l axes. The proposed of ordination i is summarized graphically complex relationships, extracting one or few dominant patterns from an infinite number of possible patterns. The placement of variables along an axis it is possible because the ordination it is base on the variables correlation.

5 What ordination analyses help us to see? Select the most important variables from multiple variables imagined or hypothesized. Reveal unforeseen patterns and suggest unforeseen processes.

6 What type of question can we answer with ordination analysis? In ecology, to seek and describe pattern of process. In community ecology, to describe the strongest patterns in species composition. I i i d dfi i In systematics, to recognize and to define species boundaries.

7 Multivariate Analysis Ordination Analysis Clasification (or Clustering Analysis) Direct Gradient Analysis Indirect Gradient Analysis Linear Regression (Few Species) Detrended CA (DCA) Corresponden ce Analysis (CA) (Many Species) Canonical CA (CCA) Redundancy Analysis (RDA) Distant Values Raw Data available Pi Principal i Non metric ti Coordinate Dimensional Analysis Analysis (PCoA (NMDS) Principal Components Analysis (PCA) Non metric Dimensional Analysis (NMDS) Detrended CA (DCA) Canonical CA (CCA)

8 Principal Components Analysis Principal component analysis (PCA) is a statistical technique that has been specifically developed to address data reduction. In general terms, the major aim of PCA is to reduce the complexity of the interrelationships among a potentially large number of observed variables to a relatively small number of linear combinations of them, which hare referred to as principal components. Principal components analysis finds a set of orthogonal standardized linear combinations which together explain all of the variation in the original data.

9 What are the assumptions of PCA? Assumes relationships among variables. cloud of points in p dimensional space has linear dimensions that can be effectively summarized by the principal axes. If the structure in the data is NONLINEAR (the cloud of points twists and curves its way through hp dimensional space), the principal axes will not be an efficient and informative summary of the data.

10 Considerations before to run a PCA Normal Distributions Data Outliers Transformationsf i Standardization Data Matrix

11 Normal Distributions When using PCA data normality is not essential. However, these methods are based on the correlation or covariance matrix, which is strongly affected by non normally distributed data and thepresence of outliers.

12 Data outliers Extreme values as well as outliers can have a severe influence on PCA, since they are based on the correlation or covariance matrix (Pison et al., 2003). Outliers should thus be removed prior to the statistical analysis, or statistical methods able to handle outliers should be employed, and the influence of extreme values needs to be reduced (e.g., via a suitable transformation).

13 Transformations Transformations, which change the scale of measurement of the data, in relation to meeting the normality assumption of parametric analyses and the homogeneity of variance assumption of most of these analyses. Transformations are particularly important for multivariate procedures based on eigenanalysis (e.g. principal components analysis) because covariances and correlations measure linear relationships between variables. Transformations that improve linearity will increase the Transformations that improve linearity will increase the efficiency with which the eigenanalysis extracts the eigenvectors.

14 Standardization The first stage in rotating the data cloud is to standardize the data by subtracting the mean and dividing by the standard deviation. It may be argued that we should not divide by the standard deviation. By standardizing, we are giving all species the same variation, i.e. a standard deviation of 1.

15 Data Matrix We actually can have it both ways: A PCA without dividing by the standard deviation is an analysis of the covariance matrix. A PCA in which you do indeed divide by the standard deviation is an analysis of the correlation matrix. When using species/variables measured in different units, you must use a correlation matrix.

16 Look at Descriptors Homogeneous nature? All Same Kind? Same Units? Same Order of Magnitude Heterogenous nature? Different kind? Different Units? Different order of Magnitude? S matrix R matrix (Covariance) (Correlation)

17 Advantages Disadvantages Correlation The results of There are considerable differences in the Matrix analyses for different sets of random variables standard deviations, caused mainly by differences in scale. None of the correlations is particularly large in are more directly comparable. absolute value. PCs has moderate sized coefficients for several of the variables. PCs give coefficients for standardized variables and are therefore less easy to interpret directly. Covariance Matrix PCs for the covariance matrix are each dominated by a single variable. The variances and total variance are more meaningful indices for measuring variability in data sets that are symmetric. The sensitivity of the PCs to the units of measurement used for each element of the variables. If there are large differences between the variances of the elements of the variables, then those variables whose variances are largest will tend to dominate the first few PCs.

18 Eigenvalues & Eigenvectors The eigenvectors are the loadings of the principal components spanning the new PCA coordinate system. The amount of variability contained in each principal component is expressed by the eigenvalues which are simply the variances of the scores.

19 PCA searches for the direction in the multivariate space that contains the maximum variability. This is the direction of the first principal component (PC1). The second principal p component (PC2) has to be orthogonal (perpendicular) to PC1andwill contain the maximum amount of the remaining data variability. Subsequent principal i components are found by the same principle.

20 Biplots A biplot is a visualization tool to present results of PCA. The PCA biplot is called the scaling process. The loadings(arrows) represent the elements. The lengths of the arrows in the plot are directly proportional to the variability included in the two components (PC1 and PC2) displayed, and the angle between any two arrows is a measure of the correlation between those variables.

21 Misconceptions PCA cannot cope with missing values (but neither can most other statistical methods). It does not require normality. It is not a hypothesis test. There are no clear distinctions between response variables and explanatory variables.

22 When should PCA be used? In community ecology, PCA is useful for summarizing variables whose relationships are approximately linear or at least monotonic. e.g. A PCA of many soil properties might be used to extract a few components that summarize main dimensions of soil variation PCA is generally NOT useful for ordinating community data. Why? Because relationships among species are highly nonlinear.

23 Community trends along environmenal gradients appear as horseshoes in PCA ordinations. None of the PC axes effectively summarizes the trend in species composition along the gradient. 2 Axis Beta Diversity 2R - Covariance Axis 1

24 The Horseshoe Effect Curvature of the gradient and the degree of infolding of the extremes increase with beta diversity. PCA ordinations are not useful summaries of community data except when beta diversity is very low Using correlation generally does better than covariance. This is because standardization by species improves the correlation between Euclidean distance and environmental distance.

25 What if there s more than one underlying ecological lgradient? When two or more underlying gradients with high beta diversity a horseshoe is usually not detectable. Interpretation problems are more severe.

26 Data Set

27 Morphological and anatomical variation of Calophyllum L. (Calophyllaceae) in South America. D. Vela

28 Kielmeyeroideae Calophylleae Calophyllum Neotatea Marila Mahurea Clusiella Kielmeyera Caraipa Haploclathra Poeciloneuron Mesua Kayea Mammea Kayea Caraipa Endodesmieae Endoodesmia Lebrunia Stevens, 2006 Calophyllum

29 Wurdarck & Davis (2009)

30 Distribution of Calophyllaceae species species Stevens,

31

32 Vein Resin canal

33

34

35 Calophyllum brasiliense calophyllum inophyllum/

36 There is infraspecific variation in tepal number between individuals of the same species, and between flowers from the same inflorescence. Stevens (1974,1980)

37 Calophyllum brasiliense Calophyllum lanigerum Calophyllum pisiferum

38 1. Mi Main objective 1.A To distinguish species limits of Calophyllum in South America. 2. Specific objectives 2.A To analyze morphological and anatomical variation. iti

39 Data collection for morphological observations Herbarium and personal collections. Collection sort: qualitative characteristics (Systematic Association Committee for descriptive Biological Terminology (cited by Stearn 2006). Measurement. Ruler and a digital caliper. E ldt ti Excel data matrix. Specimen collections in rows and variables in columns.

40 Leaf characters Flower characters Fruit characters External Fruit length mm Petiole length mm (PTL) Pedicel length mm (PDL) (FrLEx) Leaf length cm (LL) Perianth width mm (PW ) External Fruit width mm (FrWEx) Leaf flength at widest part cm (LWWP) Perianth length mm (PRL) Internal Fruit length mm (FrLIn) Leaf width cm (LW) Anther length mm (AL) Internal Fruit width mm (FrWIn) Apex length mm (PL) Anther width mm (AW) Stigma remained mm (StygR) Midrib width at abaxial side mm (MW) Stamen length mm (STL) Basal discoloration mm (BsDis) Vein angle degree (VA) Filament length mm (FL) Stone mm (Stn) Venation density (VD) Style length mm (STYL) Corky mm (CRK) Gynoecium length mm (GL) Ovary length mm (OL) Stigma width mm (SL)

41 REFERENCES Claude, Julien Morphometrics with R. Springer. Gotelli, Nicholas J., and Aaron M. Ellison A primer of ecological statistics. Sinauer Associates Publishers. Jolliffe, I. T Principal component analysis. Springer. Legendre, Pierre, and Louis Legendre Numerical ecology. Elsevier. Q i G ldp d Mi h l J K h 2002 E i l Quinn, Gerald Peter, and Michael J. Keough Experimental design and data analysis for biologists. Cambridge University Press.