Workshop 15: Q-mode MVA Murray Logan 06 Aug 2016
R-mode analyses • preserve euclidean (PCA/RDA) or χ 2 (CA/CCA) distances • based on either correlation or covariance of variables ◦ data restricted by assumptions
Q-mode analyses • based on object similarity • unrestricted choice of similarity/distance matrix ◦ filter data to comply • resulting scores not independent
Dissimilarity Sp1 Sp2 Sp3 Sp4 Site1 2 0 0 5 Site2 13 7 10 5 Site3 9 5 55 93 Site4 10 6 76 81 Site5 0 2 6 0
15.329710 105.546198 107.596468 Site2 8.306624 Site5 24.228083 Site4 107.944430 100.707497 98.939375 Site3 104.129727 16.431677 Site4 Site3 Site2 Site1 > library (vegan) Distance measures • Euclidean distance √∑ d jk = ( y ji − y ki ) 2 > vegdist (Y,method="euclidean")
Site5 1.0000000 0.6279070 0.9058824 0.9116022 Site1 Site4 0.9222222 0.7019231 0.1044776 Site3 0.9171598 0.7055838 Site2 0.6666667 Site4 Site3 Site2 > library (vegan) Distance measures • Bray-Curtis distance ∑ | y ji − y ki | d jk = ∑ y ji + y ki = 1 − 2 ∑ min ( y ji , y ki ) ∑ y ji + y ki > vegdist (Y,method="bray")
Site5 1.4142136 0.7918120 0.9028295 0.8159425 > library (vegan) Site4 0.7657483 0.5608590 0.1092925 Site3 0.6836053 0.5999300 Site2 0.8423744 Site4 Site3 Site2 Site1 Distance measures • Hellinger distance ∑ (√ y ji √ y ki √ ) 2 d jk = − ∑ ∑ y j y k > dist ( decostand (Y,method="hellinger"))
Multidimensional scaling • re-project objects (sites) in reduced dimensional space • must nominate the number of dimensions up-front • optimized patterns for the nominated dimensions
. . . . . . . . . . Q-mode analyses Site 6 Site 7 Site 5 Site 4 Site 8 Site 3 Site 2 Site 9 Site 10 Site 1 Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10 Site1 Site1 5 0 0 65 5 0 0 0 0 0 Site2 Site2 0 0 0 25 39 0 6 23 0 0 Site3 Site3 0 0 0 6 42 0 6 31 0 0 Site4 Site4 0 0 0 0 0 0 0 40 0 14 Site5 Site5 0 0 6 0 0 0 0 34 18 12 Site6 Site6 0 29 12 0 0 0 0 0 22 0 Site7 Site7 0 0 21 0 0 5 0 0 20 0 Site8 Site8 0 0 0 0 13 0 6 37 0 0 Site9 Site9 0 0 0 60 47 0 4 0 0 0 Site10 Site10 0 0 0 72 34 0 0 0 0 0
Site3 0.6428571 1.0000000 1.0000000 1.0000000 1.0000000 0.5862069 0.4128440 Site7 1.0000000 1.0000000 1.0000000 1.0000000 0.6390977 Site6 1.0000000 0.7177914 0.6000000 0.2580645 Site5 1.0000000 0.6870748 0.5539568 Site4 0.8625000 0.1685393 Site10 0.2265193 0.4070352 0.5811518 1.0000000 1.0000000 1.0000000 1.0000000 0.8395062 0.1336406 Site2 Site9 0.3010753 0.3333333 0.4693878 1.0000000 1.0000000 1.0000000 1.0000000 0.7964072 Site9 Site8 Site7 Site6 Site5 Site4 Site3 Site2 Site1 > data.dist > data.dist <- vegdist (data[,-1], "bray") Site8 0.9236641 0.4362416 0.2907801 0.3272727 0.4603175 1.0000000 1.0000000 Multidimensional scaling 1. generate distance matrix Data Distances
Multidimensional scaling 2. choose # dimensions (k=2)
Multidimensional scaling 3. random configuration Site8 ● Site10 ● Site5 ● Dimension (axis) 2 Site7 ● Site3 ● Site6 ● Site9 Site4 ● ● Site1 Site2 ● ● Dimension (axis) 1
Multidimensional scaling 4. measure Kruskals stress Site8 ● Data Distances Site10 ● Site5 ● Dimension (axis) 2 Ordination Site7 ● Site3 Distances ● Iterations = 0 Site6 ● Stress = 0.368 ● ● ● ● ● ● Ordination distance Site4 ● Site9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Site1 ● ● ● ● ● ● Site2 ● ● ● ● ● ● Dimension (axis) 1 Data distance
Multidimensional scaling 5. iterate - gradient descent Site8 ● Site7 ● Data Distances Site3 ● Dimension (axis) 2 Site5 ● Ordination Distances Site10 ● Site6 ● Iterations = 1 Stress = 0.329 ● ● Site1 ● ● ● Ordination distance ● ● ● ● ● ● ● ● ● ● ● Site9 ● ● ● ● ● ● ● ● ● Site2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Site4 ● ● ● ● ● ● ● Dimension (axis) 1 Data distance
Multidimensional scaling 6. continual to iterate Site8 ● Site4 ● Site7 ● Site5 ● Data Distances Site3 ● Dimension (axis) 2 Ordination Distances Site2 ● Iterations = 10 Stress = 0.155 ● ● ● Site6 ● Site10 ● Site9 ● Ordination distance ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Site1 ● ● ● ● ● ● ● ● ● Dimension (axis) 1 Data distance
Multidimensional scaling 7. continual (stopping criteria) Site4 ● Site8 ● Data Site5 ● Distances Dimension (axis) 2 Ordination Site3 ● Distances Site2 ● Iterations = 34 Site7 ● Stress = 0.055 ● ● ● ● ● ● ● ● ● ● ● ● ● ● Site6 Ordination distance ● ● ● ● ● ● ● ● Site9 ● Site10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Site1 ● ● ● ● ● ● ● ● Dimension (axis) 1 Data distance
Multidimensional scaling a e r i r i t g c p i n t o p S • convergence tolerance (stress below a threshold) • the stress ratio (improvement in stress) • maximum iterations Ideally stress 0.2
Multidimensional scaling o n t i u r a f i g c o n a l F i n • axes have no real meaning • operate together to create ordination space • orientation of points is arbitrary
Multidimensional scaling i o n r a t i g u o n t g c t i n t a r S • completely random • based on eigen-analysis • repeated random starts
Multidimensional scaling o n a t i r o t e s s t c r u P r o Site4 Site4 ● ● Site5 ● Site7 ● Site8 ● Site5 ● Site6 Site8 ● ● Site3 ● Site3 Site2 ● ● Site7 ● Site2 ● Site6 ● Site9 ● Site10 ● Site9 ● Site10 ● Site1 Site1 ● ●
Multidimensional scaling n t i o o t a s r s t e c r u r o P Site4 Site4 Site4 Site4 Site4 Site4 Site4 ● ● ● ● ● ● ● Site5 ● Site7 ● Site8 Site8 Site8 Site8 ● ● ● ● Site5 Site5 Site5 Site5 ● ● ● ● Site8 Site8 ● ● Site5 Site5 ● ● Site6 ● Site8 ● Site3 Site3 Site3 Site3 ● ● ● ● Site3 Site3 ● ● Site2 Site2 ● Site3 ● ● Site2 Site2 Site2 Site2 ● ● ● ● Site7 Site7 Site7 Site7 ● ● ● ● Site2 Site7 Site7 ● ● ● Site6 Site6 Site6 Site6 ● ● ● ● Site6 Site6 ● ● Site9 Site9 Site9 Site9 ● ● Site10 Site10 ● ● Site10 Site10 ● ● ● ● Site9 Site9 ● Site10 ● Site10 ● ● Site9 ● Site10 ● Site1 Site1 Site1 Site1 Site1 Site1 Site1 ● ● ● ● ● ● ● Root mean square error (rmse) • rmse 0.01, and no one 0.005 stopping criteria
Multidimensional scaling metaMDS() • transform and scale • generates dissimilarity • PCoA for starting config • up to 20 random starts • procrustes used to determine final config • final scores scaled ◦ PCA-like axes rotations
max resid 0.1897497 Run 15 stress 0.002519856 Run 10 stress 9.756922e-05 ... procrustes: rmse 0.09787159 max resid 0.2172152 Run 11 stress 9.903768e-05 ... procrustes: rmse 0.1377402 max resid 0.2070494 Run 12 stress 0.07877608 Run 13 stress 9.67808e-05 ... procrustes: rmse 0.105466 max resid 0.2688797 Run 14 stress 0.00122647 Run 16 stress 9.401245e-05 ... procrustes: rmse 0.03799946 ... procrustes: rmse 0.01935187 max resid 0.04030324 Run 17 stress 9.207175e-05 ... New best solution ... procrustes: rmse 0.1056994 max resid 0.2401655 Run 18 stress 0.00139044 Run 19 stress 9.781088e-05 ... procrustes: rmse 0.1089896 max resid 0.25471 Run 20 stress 9.529673e-05 ... procrustes: rmse 0.09327002 max resid 0.06399347 Run 9 stress 9.636878e-05 > library (vegan) max resid 0.2286374 Square root transformation Wisconsin double standardization Run 0 stress 9.575935e-05 Run 1 stress 9.419141e-05 ... New best solution ... procrustes: rmse 0.02551179 max resid 0.04216078 Run 2 stress 0.0005652599 ... procrustes: rmse 0.1885613 max resid 0.2962192 Run 3 stress 0.0003846328 ... procrustes: rmse 0.128145 max resid 0.2569595 Run 4 stress 0.1071568 Run 5 stress 0.08642 Run 6 stress 0.0002599177 ... procrustes: rmse 0.1766213 max resid 0.2933088 Run 7 stress 9.39752e-05 ... New best solution ... procrustes: rmse 0.1050724 max resid 0.2432038 Run 8 stress 9.692363e-05 ... procrustes: rmse 0.101578 Multidimensional scaling > data.nmds <- metaMDS (data[,-1])
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