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Nonmetric Multidimensional Scaling (NMDS)PowerPoint Presentation

Nonmetric Multidimensional Scaling (NMDS)

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Nonmetric Multidimensional Scaling (NMDS). Nonmetric Multidimensional Scaling (NMDS). Developed by Shepard (1962) and Kruskal (1964) for psychological data First applied in ecology by Anderson (1971)

Nonmetric Multidimensional Scaling (NMDS)

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- Developed by Shepard (1962) and Kruskal (1964) for psychological data
- First applied in ecology by Anderson (1971)
- Based on a fundamentally different approach than the eigenanalysis methods PCA, CA (and DCA)
- Axes of NMDS are not rotated axes of a high-dimensional “species space”.

- 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).

- 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.

SU 4

SU 2

SU 1

Axis 2

SU 3

SU 5

Axis 1

- 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.

- 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.

Distance,

Dissimilarity, D

- 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.

Distance,

Distance,

Dissimilarity, D

Dissimilarity, D

- The badness-of-fit of the regression is measured by Kruskal’s stress, computed as:

Residual sum ofsquares of monotone

regression

Sum ofsquares of distances

Distance,

Dissimilarity, D

- 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.

- 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.

- 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).

- 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.

- 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.

Stress

Local Optimum

Global Optimum

- 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.

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

- Bray-Curtis dissimilarity matrix among the 9 areas.
- Really only need the lower triangle, without the zero diagonals.

- 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

Axis 2

Axis 1

Stress

Iteration Number

Axis 2

Axis 1

Start

Final

Axis 2

10

5

20

Axis 1

Stress

Once stress starts to level out, most SUs don’t change much in their position

Iteration Number

Stress = 0.0139

- 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.

- 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

- 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.

- 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.

- 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.

- 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.

- 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.

- 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”).

Gradient 2

Gradient 1

NMDS=0.13

DCA=0.09

CA=0.10

NMDS=0.08

DCA=0.19

CA=0.16

NMDS=0.08

DCA=0.19

CA=0.29

DCA=0.34

NMDS=0.09

CA=0.29

- 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 & Orlóci 1986) confirm NMDS superior to DCA
- Why is NMDS so successful?

- 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.

- 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.

- 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.