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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)

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Nonmetric multidimensional scaling nmds2 l.jpg
Nonmetric Multidimensional Scaling(NMDS)

  • 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”.

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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).

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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.

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SU 4

SU 2

SU 1

Axis 2

SU 3

SU 5

Axis 1

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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.

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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.

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Shepard Diagram

Distance, 

Dissimilarity, D

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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.

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Shepard Diagram with Monotone Regression

Distance, 

Distance, 

Dissimilarity, D

Dissimilarity, D

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Badness-of-fit: “Stress”

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

Residual sum ofsquares of monotone


Sum ofsquares of distances

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Contributions to Stress (Badness-of-fit)

Distance, 

Dissimilarity, D

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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.

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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.

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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).

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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.

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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.

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Local Optima


Local Optimum

Global Optimum

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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.

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Worked Example of NMDS

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

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Worked Example of NMDS

  • Bray-Curtis dissimilarity matrix among the 9 areas.

  • Really only need the lower triangle, without the zero diagonals.

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  • 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

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How stress changed


Iteration Number

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The Journey of Area 3



Axis 2




Axis 1

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How stress changed


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

Iteration Number

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Final NMDS Ordination

Stress = 0.0139

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  • 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.

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  • 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

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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.

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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.

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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.

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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.

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“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.

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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”).

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Simulated Gradient Space(Target “Ideal” Ordination Result)

Gradient 2

Gradient 1

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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 & Orlóci 1986) confirm NMDS superior to DCA

  • Why is NMDS so successful?

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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.

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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.

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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.