Discriminant Analysis

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CHAPTER 26 Discriminant Analysis
Tables, Figures, and Equations
From McCune, B. J. B. Grace. 2002. Analysis
of Ecological Communities. MjM Software Design,
Gleneden Beach, Oregon http//www.pcord.com
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Purposes 1. Summarizing the differences between
groups (often used as a follow-up to clustering,
to help describe the groups) "descriptive
discriminant analysis." With community data, you
could use indicator species analysis as a
nonparametric alternative.
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Purposes 1. Summarizing the differences between
groups (often used as a follow-up to clustering,
to help describe the groups) "descriptive
discriminant analysis." With community data, you
could use indicator species analysis as a
nonparametric alternative. 2. Multivariate
testing of whether or not two or more groups
differ significantly from each other. For
ecological community data this is better done
with MRPP, thus avoiding the assumptions listed
below.
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Purposes 1. Summarizing the differences between
groups (often used as a follow-up to clustering,
to help describe the groups) "descriptive
discriminant analysis." With community data, you
could use indicator species analysis as a
nonparametric alternative. 2. Multivariate
testing of whether or not two or more groups
differ significantly from each other. For
ecological community data this is better done
with MRPP, thus avoiding the assumptions listed
below. 3. Determining the dimensionality of group
differences.
5
Purposes 1. Summarizing the differences between
groups (often used as a follow-up to clustering,
to help describe the groups) "descriptive
discriminant analysis." With community data, you
could use indicator species analysis as a
nonparametric alternative. 2. Multivariate
testing of whether or not two or more groups
differ significantly from each other. For
ecological community data this is better done
with MRPP, thus avoiding the assumptions listed
below. 3. Determining the dimensionality of group
differences. 4. Checking for misclassified items.
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Purposes (cont.) 5. Predicting group membership
or classifying new cases ("predictive
discriminant analysis").
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Purposes (cont.) 5. Predicting group membership
or classifying new cases ("predictive
discriminant analysis"). 6. Comparing occupied
vs. unoccupied habitat to determine the habitat
characteristics that allow or prevent a species'
existence. DA has been widely used for this
purpose in wildlife studies and rare plant
studies.
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Assumptions 1. Homogeneous within-group
variances 2. Multivariate normality within
groups. 3. Linearity among all pairs of
variables. 4. Prior probabilities.
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How it works The "direct" procedure is described
below. 1. Calculate variance/covariance matrix
for each group.
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How it works The "direct" procedure is described
below. 1. Calculate variance/covariance matrix
for each group. 2. Calculate pooled
variance/covariance matrix (Sp) from the above
matrices.
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How it works The "direct" procedure is described
below. 1. Calculate variance/covariance matrix
for each group. 2. Calculate pooled
variance/covariance matrix (Sp) from the above
matrices. 3. Calculate between group variance
(Sg) for each variable.
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4. Maximize the F-ratio
where the y is an the eigenvector associated with
a particular discriminant function. We seek y
to maximize F.
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Maximize this ratio by finding the partial
derivatives with a characteristic equation
The number of roots is g-1, where g is number of
groups. In other words, the number of functions
(axes) derived is one less than the number of
groups. The eigenvalues thus express the
percent of variance among groups explained by
those axes.
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6. Solve for each eigenvector y (also known as
the "canonical variates" or "discriminant
functions").
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7. Locate points (sample units) on each axis.
X scores (coordinates) for n rows (sample
units) on m dimensions, where m g-1. A
original data matrix of n rows by p columns Y
matrix of m eigenvectors with loadings for p
variables. Each eigenvector is known as a
discriminant function.
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These unstandardized discriminant functions Y can
be used as (linear) prediction equations,
assigning scores to unclassified items.
Standardized discriminant function coefficients
standardize to unit variance. The absolute value
of these coefficients indicate the relative
importance of the individual variables in
contributing to the discriminant function.
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• 8. Classification phase.
• Derive a classification equation for each group,
  one term in the equation for each variable, plus
  a constant.
• Insert data values for a given SU to calculate a
  classification score for each group for that SU.
  
• The SU is assigned to the group in which it had
  the highest score.
• The coefficients in the equation are derived
  from
• p ? p within-group variance-covariance matrix
  (Sp) and
• p ? 1 vector of the means for each variable in
  group k, Mk.
• First, calculate W by dividing each term of Sp by
  the within-group degrees of freedom. Then

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8. Classification phase, cont. The
coefficients in the equation are derived from p
? p within-group variance-covariance matrix (Sp)
p ? 1 vector of the means for each variable in
group k, Mk. First, calculate W by dividing
each term of Sp by the within-group degrees of
freedom. Then
The constant is derived as The constant and
the coefficients in Ck define a linear equation
of the usual form, one equation for each group k.
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• Summary statistics
• Wilk's lambda (l). Wilk's l is the error sum of
  squares divided by the sum of the effect sum of
  squares and the error sum of squares. Thus, it
  is the variance among the objects not explained
  by the discriminant functions. It ranges from
  zero (perfect separation of groups) to one (no
  separation of groups).
• Statistical significance of lambda is tested with
  a chi-square approximation.
• Chi-square (derived from Wilks lambda).
• Variance explained.

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Figure 26.1. Comparison of DA and PCA. Groups
are tighter in DA than in PCA because DA
maximizes group separation while PCA maximizes
the representation of variance among individual
points. Groups were superimposed on an
ordination of pine species in ecological trait
space (after McCune 1988). Pinus resinosa was
not assigned to a group, so it does not appear in
the DA ordination.
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Table 26.1. Predictions of goshawk nesting sites
from DA compared to actual results, in one case
using equal prior probabilities, in the other
case using prior probabilities based on the
occupancy rate of landscape cells. The first
value of 0.83 means that 83 of the sites that
were predicted by DA to be nesting sites actually
were nesting sites.
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Table 26.1. Predictions of goshawk nesting sites
from DA compared to actual results, in one case
using equal prior probabilities, in the other
case using prior probabilities based on the
occupancy rate of landscape cells. The first
value of 0.83 means that 83 of the sites that
were predicted by DA to be nesting sites actually
were nesting sites.
0.5
0.07
0.5
0.93
priors
priors
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EQUAL priors No. non-nests predicted nests
p(predicted nest but not nest) ? number of
non-nests 0.17 ? 93 15.8 No.
nests predicted non-nests p(predicted not nest
but nest) ? number of nests 0.17 ?
7 1.2 Total number of errors 15.8
1.2 17
False positives
False negatives
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UNEQUAL priors No. non-nests predicted nests
p(predicted nest but not nest) ? number of
non-nests 0.02 ? 93 1.9 No.
nests predicted non-nests p(predicted not nest
but nest) ? number of nests 0.52 ?
7 3.6 Total number of errors 1.9
3.6 5.5
False positives
False negatives