Nonmetric Multidimensional Scaling NMDS

1. Nonmetric Multidimensional Scaling(NMDS)

3. The model of NMDS NMDS works in a space with a specified number (small) of dimensions (e.g., 2 or 3) The objects on interest (usually SUs in ecological applications) are points in this ordination space The data on which NMDS operates are in the dissimilarity matrix among all pairs of objects (e.g., Bray-Curtis dissimilarities computed from community data).

4. The model of NMDS NMDS seeks an ordination in which the distances between all pairs of SUs are, as far as possible, in rank-order agreement with their dissimilarities in species composition.

6. Model of NMDS Let Dij be the dissimilarity between SUs i and j, computed with any suitable measure (e.g. Bray-Curtis) let ?ij be the Euclidean distance between SUs i and j in the ordination space.

7. Model of NMDS The objective is to produce an ordination such that:Dij < Dkl ? ?ij ? ?kl for all i, j, k, l if any given pair of SUs have a dissimilarity less than some other pair, then the first pair should be no further apart in the ordination than the second pair a scatter plot of ordination distances, ?, against dissimilarities, D, is known as a Shepard diagram.

8. Shepard Diagram

9. Model of NMDS The degree to which distances agree in rank-order with dissimilarities can be determined by fitting a monotone regression of the ordination distances ? onto the dissimilarities D A monotone regression line looks like an ascending staircase: it uses only the ranks of the dissimilarities The fitted values, , represent hypothetical distances that would be in perfect rank-order with the dissimilarities.

10. Shepard Diagram with Monotone Regression

11. Badness-of-fit: �Stress� The badness-of-fit of the regression is measured by Kruskal�s stress, computed as:

12. Contributions to Stress (Badness-of-fit)

13. Model of NMDS Stress decreases as the rank-order agreement between distances and dissimilarities improves The aim is therefore to find the ordination with the lowest possible stress There is no algebraic solution to find the best ordination: it must be sought by an iterative search or trial-and-error optimization process.

14. Basic Algorithm for NMDS Compute dissimilarities, D, among the n SUs using a suitable choice of data standardization and dissimilarity measure Specify the number of ordination dimensions to be used Generate an initial ordination of the SUs (starting configuration) with this number of axes This can be totally random or an ordination of the SUs by some other method might be used.

15. Basic Algorithm for NMDS Calculate the distances, ?, between each pair of SUs in the current ordination Perform a monotone regression of the distances, ?, on dissimilarities, D Calculate the stress Move each SU point slightly, in a manner deemed likely to decrease the stress Repeat steps 4 � 7 until the stress either approaches zero or stops decreasing (each cycle is called an iteration).

16. Basic Algorithm for NMDS Any suitable optimization method can be used at step 7 to decide how to move each point Stress can be considered a function with many independent variables: the coordinates of each SU on each axis The aim is to find the coordinates that will minimize this function This is a difficult problem to solve, especially when n is large.

17. Local Optima There is no guarantee that the ordination with the lowest possible stress (global optimum) will be found from any given initial ordination The search may arrive at a local optimum, where no small change in any coordinates will make stress decrease, even though a solution with lower stress does exist.

18. Local Optima

19. Local Optima run the entire ordination from several different starting configurations (typically at least 100) if the algorithm converges to the same minimum stress solution from several different random starts, one can be confident the global optimum has been found.

20. Worked Example of NMDS Densities (km-1) of 7 large mammal species in 9 areas of Rweonzori National Park, Uganda.

21. Worked Example of NMDS Bray-Curtis dissimilarity matrix among the 9 areas. Really only need the lower triangle, without the zero diagonals.

22. Initial Ordination (Random)

23. Initial Shepard Diagram

24. Stress stress of initial (random) ordination is 0.4183 This is high, reflecting the poor rank-order agreement of distances with dissimilarities at this stage. each SU point is now moved slightly

25. How the ordination evolved

26. How stress changed

27. Shepard Diagram � Iteration 5

28. Shepard Diagram � Iteration 10

29. Shepard Diagram � Iteration 15

30. Shepard Diagram � Iteration 20

31. Shepard Diagram � Iteration 30

32. How the ordination evolved

33. The Journey of Area 3

34. How stress changed

35. Final NMDS Ordination

36. Dimensionality There is no way of knowing in advance how many dimensions you need for a given data set To determine how many dimensions are needed, NMDS must be run in a range of dimensionalities The vast majority of community data sets can be adequately summarized with 2 or 3 NMDS axes; rarely 4 or 5 may be needed.

37. Dimensionality no simple relationship between axes in NMDS solutions for different numbers of dimensions e.g. axes 1 and 2 of a 3-D NMDS are not the same as axes 1 and 2 of the 2-D solution always possible to achieve a lower stress with an increase in dimensionality

38. Scree Plots A line plot of minimum stress (Y axis) against number of dimensions is called a scree plot It can be used as a guide in deciding on the number of dimensions required A sharp break in slope of the curve, beyond which further reductions in stress are small, suggests dimensionality.

39. Scree Plot

40. Scree Plots only a rough guide sharpness of the �break� in slope depends on the �signal to noise ratio� of the data the scree plot should be used to estimate the minimum number of axes required If scree plot suggests k axes, also save and examine the k+1 dimensional solution.

41. Scree Plot

42. How low is low? Kruskal and later authors suggest guidelines for interpreting the stress values in NMDS NMDS ordinations with stresses up to 0.20 can be ecologically interpretable and useful.

43. Recommended Strategy for Choosing Number of Dimensions use scree plot as a guide to minimum number of dimensions needed if scree plot suggests k, save k-dimensional and (k+1)-dimensional solutions interpret k-dimensional ordination patterns of community change correlations with environmental or other explanatory variables then see if the extra axis of the (k+1)-D ordination adds interpretable information.

44. �Unstable� Solutions & PCORD Trials with PCORD suggest that its algorithm for NMDS is much more prone to being trapped at local optima than other programs (e.g. SAS, DECODA) The best way to avoid �unstable� solutions is not to use PCORD for NMDS.

45. Evaluation of NMDS Performance Minchin (1987) compared of NMDS with other ordination methods (PCA, DCA) using simulated data Model properties varied included: Shape of response curves (symmetric, skewed, monotonic) Beta diversity of gradients Sampling pattern in �gradient space� Amount of random variation (�noise�).

46. Simulated Gradient Space(Target �Ideal� Ordination Result)

51. Overall Conclusions NMDS recovers gradient structure better and more reliably than CA, DCA, and PCA (for community data) NMDS is more robust than the eigenanalysis methods to variations in model properties Other studies (e.g. Kenkel & Orl�ci 1986) confirm NMDS superior to DCA Why is NMDS so successful?

52. NMDS model is compatible with ecological models Dissimilarity measures like Bray-Curtis have the highest rank correlation with environmental distance NMDS assumes a rank-order relationship between dissimilarity and the distances between SUs in the ordination NMDS using Bray-Curtis is therefore successful in producing ordinations in which distances approximate environmental distances.

53. Treatment of Tied Dissimilarities Kruskal noted that there are two ways to treat tied dissimilarities Primary approach: pairs of SUs with equal dissimilarities need not be equally distant in the ordination Secondary approach: pairs of SUs with equal dissimilarities should be equally distant in the ordination.

54. Treatment of Tied Dissimilarities For community data, it is essential to use the primary approach, especially for SU pairs with no species in common (dissimilarity = 1.0) Such pairs of SUs generally have different environmental distances Primary tie treatment allows NMDS to correctly estimate their distances Use of secondary approach forces NMDS to represent gradients of high beta diversity as horseshoes.