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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
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”.
the model of nmds
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).
the model of nmds4
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.
slide5

SU 4

SU 2

SU 1

Axis 2

SU 3

SU 5

Axis 1

model of nmds
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.
model of nmds7
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.
shepard diagram
Shepard Diagram

Distance, 

Dissimilarity, D

model of nmds9
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.
shepard diagram with monotone regression
Shepard Diagram with Monotone Regression

Distance, 

Distance, 

Dissimilarity, D

Dissimilarity, D

badness of fit stress
Badness-of-fit: “Stress”
  • 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

contributions to stress badness of fit
Contributions to Stress (Badness-of-fit)

Distance, 

Dissimilarity, D

model of nmds13
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.
basic algorithm for nmds
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.
basic algorithm for nmds15
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).
basic algorithm for nmds16
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.
local optima
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.
local optima18
Local Optima

Stress

Local Optimum

Global Optimum

local optima19
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.
worked example of nmds
Worked Example of NMDS
  • Densities (km-1) of 7 large mammal species in 9 areas of Rweonzori National Park, Uganda.
worked example of nmds21
Worked Example of NMDS
  • Bray-Curtis dissimilarity matrix among the 9 areas.
  • Really only need the lower triangle, without the zero diagonals.
stress
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
how stress changed
How stress changed

Stress

Iteration Number

the journey of area 3
The Journey of Area 3

Start

Final

Axis 2

10

5

20

Axis 1

how stress changed34
How stress changed

Stress

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

Iteration Number

final nmds ordination
Final NMDS Ordination

Stress = 0.0139

dimensionality
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.
dimensionality37
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
scree plots
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.
scree plots40
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.
how low is low
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.
recommended strategy for choosing number of dimensions
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.
unstable solutions pcord
“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.
evaluation of nmds performance
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”).
slide47

NMDS=0.13

DCA=0.09

CA=0.10

slide48

NMDS=0.08

DCA=0.19

CA=0.16

slide49

NMDS=0.08

DCA=0.19

CA=0.29

slide50

DCA=0.34

NMDS=0.09

CA=0.29

overall conclusions
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?
nmds model is compatible with ecological models
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.
treatment of tied dissimilarities
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.
treatment of tied dissimilarities54
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.
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