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Ordination

Ordination. Ordination contains a number of techniques to classify data according to predefined standards . . The simplest ordination technique is cluster analysis . An easy but powerful technique is principal component analysis (PCA). Factor analysis.

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Ordination

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  1. Ordination Ordinationcontains a number of techniques to classify data according to predefinedstandards. Thesimplestordinationtechniqueisclusteranalysis. An easy but powerfultechniqueisprincipal componentanalysis(PCA).

  2. Factoranalysis Isitpossible to group thevariablesaccording to theirvalues for thecountries? Thetaskis to findcoefficients of correlationetweentheoriginalvariables and theexctractedfactorsfromtheanalysis of thecoefficiencts of correlationbetweentheoriginalvariables. Factor 3 Factor 2 Factor 1 Correlations Elev GDP/C GDP T (July) Mean T T (Jan) Diff T

  3. Because the f values are also Z-transformed we have Eigenvalue

  4. How to computethefactorloadings? Thedotproduct of orthonormalmatricesgivesthe unity matrix Fundamental theorem of factor analysis

  5. Cases n F1 F2 Factors F 1 f11 f12 2 f21 f22 Z-trans-formed Factor values b 3 f31 f32 4 f41 f42 5 f51 f52 6 f61 f62 Factors are new variables. They have factor values (independent of loadings) for each case. These factors can now be used in further analysis, for instance in regression analysis.  

  6. We arelooking for a newx,y system werethe data areclosest to thelongestaxis. PCA infactrotatestheoriginal data set to find a solutionwherethe data areclosest to theaxes. PCA leavesthenumber of axesunchanged. Only a few of theserotatedaxescan be interpretedfromthedistances to theoriginalaxes. Principal axesareeigenvectors. X’1 Y’1 Y1 PCA is an eigenvectormethod X1 We interpretthenewaxis on thebasis of theirdistance (measured by theirangle) to theoriginalaxes. Thenewaxesarethe principal axes (eigenvectors) of thedispersionmatrixobtainedfromraw data.

  7. Theprogramsdifferinthedirection of eigenvectors. Thisdoes not changetheresults but mightposeproblemswiththeinterpretation of factorsaccording to theoriginalvariables.

  8. Pincipalcoordinateanalysis PCoAusesdifferentmetrics to generatethedispersionmatrix

  9. Using PCA orPCoA to group cases v A factor might be interpreted if more than two variables have loadings higher than 0.7. A factor might be interpreted if more than four variables have loadings higher than 0.6. A factor might be interpreted if more than 10 variables have loadings higher than 0.4.

  10. Correspondenceanalysis (reciprocalaveraging, seriation, contingencytableanalysis) Correspondenceanalysisordinatesrows and columns of matricessimultaneouslyaccordingtheir principal axes. Itusesthec2-distances instead of correlationscoefficientsorEuclideandistances. Contingencytable c distances

  11. Joint plot We takethetransposedraw data matrix and calculateeigenvectorsinthe same way Correspondenceanalyisisrow and columnordination.

  12. Theplotsaresimilar but differnumerically and inorientation. Theorientation problem comesagainfromthewayEcxelcalculateseigenvalues. Row and columneigenvectorsdifferinscale. For a joint plot thevectorshave to be rescaled.

  13. Reciprocalaveraging Sortingaccording to row/columneigenvaluesrearrangesthematrixin a waywherethelargestvaluesare near thematrix diagonal.

  14. Seriationusingreciprocalaveraging Weighedmean =(B85*B$97+C85*C$97+D85*D$97+E85*E$97)/$F85 =los() =(H85-H$94)/H$95 Z-transformedweighedmeans Repeatuntilscoresbecomestable

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