This vignette deals with biplot for separating classes. Topics
discussed are
1. What is a CVA biplot
Consider a data matrix \(\mathbf{X}:n
\times p\) containing data on \(n\) objects and \(p\) variables. In addition, a vector \(\mathbf{g}:n \times 1\) contains
information on class membership of each observation. Let \(G\) indicate the total number of classes.
CVA is closely related to linear discriminant anlaysis, in that the
\(p\) variables are transformed to
\(p\) new variables, called canonical
variates, such that the classes are optimally separated in the canonical
space. By optimally separated, we mean maximising the between class
variance, relative to the within class variance. This can be formulated
as follows:
Let \(\mathbf{G}:n \times G\) be an
indicator matrix with \(g_{ij} = 0\)
unless observation \(i\) belongs to
class \(j\) and then \(g_{ij} = 1\). The matrix \(\mathbf{G'G}\) is a diagonal matrix
containing the number of observations per class on the diagonal. We can
form the matrix of class means \(\bar{\mathbf{X}}:G \times p =
(\mathbf{G'G})^{-1} \mathbf{G'X}\). With the usual
analysis of variance the total variance can be decomposed into a between
class variance and within class variance:
\[
\mathbf{T} = \mathbf{B} + \mathbf{W}
\]
\[
\mathbf{X'X} = \mathbf{\bar{\mathbf{X}}'C \bar{\mathbf{X}}} +
\mathbf{X' [I - G(G'G)^{-1}C(G'G)^{-1}G'] X}
\]
The default choice for the centring matrix \(\mathbf{C = G'G}\) leads to the
simplification
\[
\mathbf{X'X} = \mathbf{\bar{\mathbf{X}}'G'G
\bar{\mathbf{X}}} + \mathbf{X' [I - G(G'G)^{-1}G'] X}.
\]
Other options are \(\mathbf{C = I}\)
and \(\mathbf{C} = (\mathbf{I}_G -
\frac{1}{G}\mathbf{11'})\). To find the canonical variates we
want to maximise the ratio
\[
\frac{\mathbf{m'Bm}}{\mathbf{m'Wm}}
\]
subject to \(\mathbf{m'Wm} =
1\). It can be shown that this leads to the following equivalent
eigen equations:
\[
\mathbf{W}^{-1}\mathbf{BM} = \mathbf{M \Lambda}
\]
\[
\mathbf{BM} = \mathbf{WM \Lambda}
\]
\[
(\mathbf{W}^{-\frac{1}{2}} \mathbf{B} \mathbf{W}^{-\frac{1}{2}})
\mathbf{M} = (\mathbf{W}^{-\frac{1}{2}} \mathbf{M}) \mathbf{\Lambda}
\]
with \(\mathbf{M'BM}=
\mathbf{\Lambda}\) and \(\mathbf{M'WM}= \mathbf{I}\).
Since the matrix \(\mathbf{W}^{-\frac{1}{2}} \mathbf{B}
\mathbf{W}^{-\frac{1}{2}}\) is symmetric and positive
semi-definite the eigenvalues in the matrix \(\mathbf{\Lambda}\) are positive and
ordered. The rank of \(\mathbf{B} = min(p,
G-1)\) so that only the first \(rank(\mathbf{B})\) eigenvalues are
non-zero. We form the canonical variates with the transformation
\[
\bar{\mathbf{Y}} = \bar{\mathbf{X}}\mathbf{M}.
\]
To construct a 2D biplot, we plot the first two canonical variates
\(\bar{\mathbf{Z}} =
\bar{\mathbf{X}}\mathbf{MJ}_2\) where \(\mathbf{J}_2' = \begin{bmatrix} \mathbf{I}_2
& \mathbf{0} \end{bmatrix}\). We add the individual sample
points with the same transformation
\[
\mathbf{Z} = \mathbf{X}\mathbf{MJ}_2.
\] Interpolation of a new sample \(\mathbf{x}^*:p \times 1\) follows as \({\mathbf{z}^*}':2 \times 1
={\mathbf{x}^*}' \mathbf{MJ}_2\). Using the inverse
transformation \(\mathbf{x}' =
\mathbf{y}'\mathbf{M}^{-1}\), all the points that will
predict \(\mu\) for variable \(j\) will have the form
\[
\mu = \mathbf{y}'\mathbf{M}^{-1} \mathbf{e}_j
\]
where \(\mathbf{e}_j\) is a vector
of zeros with a one in position \(j\).
All the points in the 2D biplot that predict the value \(\mu\) will have
\[
\mu = \begin{bmatrix} z_1 & z_2 & 0 & \dots &
0\end{bmatrix}\mathbf{M}^{-1} \mathbf{e}_j
\]
defining the prediction line as
\[
\mu = \mathbf{z}_{\mu}' \mathbf{J}_2 \mathbf{M}^{-1} \mathbf{e}_j.
\]
Writing \(\mathbf{h}_{(j)} = \mathbf{J}_2
\mathbf{M}^{-1} \mathbf{e}_j\) the construction of biplot axes is
similar to the discussion in the biplotEZ vignette for PCA biplots. The
direction of the axis is given by \(\mathbf{h}_{(j)}\). To find the
intersection of the prediction line with \(\mathbf{h}_{(j)}\) we note that \[
\mathbf{z}'_{(\mu)}\mathbf{h}_{(j)} = \| \mathbf{z}_{(\mu)} \|^2 \|
\mathbf{h}_{(j)} \|^2 cos(\mathbf{z}_{(\mu)},\mathbf{h}_{(j)}) =
\| \mathbf{p} \|^2 \| \mathbf{h}_{(j)} \|^2
\] where \(\mathbf{p}\) is the
length of the orthogonal projection of \(\mathbf{z}_{(\mu)}\) on \(\mathbf{h}_{(j)}\).
Since \(\mathbf{p}\) is along \(\mathbf{h}_{(j)}\) we can write \(\mathbf{p} = c\mathbf{h}_{(j)}\) and all
points on the prediction line \(\mu =
\mathbf{z}'_{\mu}\mathbf{h}_{(j)}\) project on the same point
\(c_{\mu}\mathbf{h}_{(j)}\). We solve
for \(c_{\mu}\) from \[
\mu = \mathbf{z}'_{\mu}\mathbf{h}_{(j)}=\| \mathbf{p} \|^2 \|
\mathbf{h}_{(j)} \|^2 =
\| c_{\mu}\mathbf{h}_{(j)} \|^2 \| \mathbf{h}_{(j)} \|^2
\]
\[
c_{\mu} = \frac{\mu}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}.
\] If we select ‘nice’ scale markers \(\tau_{1}, \tau_{2}, \cdots \tau_{k}\) for
variable \(j\), then \(\tau_{h}-\bar{x}_j = \mu_{h}\) and
positions of these scale markers on \(\mathbf{h}_{(j)}\) are given by \(p_{\mu_{1}}, p_{\mu_{2}}, \cdots
p_{\mu_{k}}\) with \[
p_{\mu_h} = c_{\mu_h}\mathbf{h}_{(j)}
= \frac{\mu_h}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}\mathbf{h}_{(j)}
\]
\[
= \frac{\mu_h}{\mathbf{e}_{(j)}' \mathbf{M'}^{-1} \mathbf{J}
\mathbf{M}^{-1} \mathbf{e}_{(j)}}\ \mathbf{J}_2 \mathbf{M}^{-1}
\mathbf{e}_{(j)}
\]
with \[
\mathbf{J} = \begin{bmatrix}
\mathbf{I}_2 & \mathbf{0}\\
\mathbf{0} & \mathbf{0}
\end{bmatrix}.
\]
3. The function means()
This function controls the aesthetics of the class means in the
biplot. The function accepts as first argument an object of class
biplot where the aesthetics should be applied. Let us first
construct a CVA biplot of the state.x77 data with samples
optimally separated according to state.division.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
legend.type(means = TRUE) |> plot()
Instead of adding a legend, we can choose to label the class means.
Furthermore, the colour of each class mean defaults to the colour of the
samples. We wish to select a different colour and plotting character for
the class means.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
means(label = TRUE, col = "olivedrab", pch = 15) |> plot()
If we choose to only show the class means for the central states, the
argument which is used either indicating the number(s) in
the sequence of levels (which = 4:7), or as shown below,
the levels themselves:
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
means (which = c("West North Central", "West South Central", "East South Central",
"East North Central"), label = TRUE) |>
plot()
The size of the labels is controlled with label.cex
which can be specified either as a single value (for all class means) or
a vector indicating size values for each individual sample. The colour
of the labels defaults to the colour(s) of the class means. However,
individual label colours can be spesified with label.col,
similar to label.cex as either a single value of a vector
of length equal to the number of classes.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
means (col = "olivedrab", pch = 15, cex = 1.5,
label = TRUE, label.col = c("blue","green","gold","cyan","magenta",
"black","red","grey","purple")) |>
plot()
We can also make use of the functionality of the ggrepel
package to place the labels.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
samples (label = "ggrepel", label.cex=0.65) |>
means (label = "ggrepel", label.cex=0.8) |> plot()
4. The functions fit.measures() and
summary()
There is a number of fit measures that are specific to CVA biplots.
The measures are computed with the function fit.measures()
and the results are displayed by the function
summary().
Canonical variate analysis can be considered as a transformation of
the original variables to the canonical space followed by constructing a
PCA biplot of canonical variables. The matrix of class means \(\bar{\mathbf{X}} = (\mathbf{G'G})^{-1}
\mathbf{G'X}\) is transformed to \(\mathbf{\bar{X}L}\) where \(\mathbf{L}\) is a non-singular matrix such
that \(\mathbf{LL'=W}^{-1}\).
Pricipal component analysis finds the orthogonal matrix \(\mathbf{V}\) such that
\[
\mathbf{(L'\bar{X}'C\bar{X}L)V=V \Lambda}
\]
where \(\mathbf{M = LV}\) as defined
in section 1. The predicted values for the class means is given by
\[
\mathbf{\hat{\bar{X}}} = \mathbf{\bar{X}MJ}\mathbf{M}^{-1}.
\]
Overall quality of fit
Based on the two-step process described above, there are two measures
of quality of fit. The quality of the approximation of the canonical
variables \(\mathbf{\bar{X}L}\) in the
\(2\)-dimensional display is given
by
\[
Quality (canonical \: variables) = \frac{tr(\mathbf{\Lambda
J})}{tr(\mathbf{\Lambda)}}
\] and the quality of the approximation of the original variables
\(\mathbf{\bar{X}}\) in the 2D CVA
biplot is given by
\[
Quality (original \: variables) = \frac{tr(\mathbf{\Lambda
J})}{tr(\mathbf{\Lambda)}}
\]
Adequacy of representation of variables
The adequacy with which each of the variables is represented in the
biplot is given by the elementwise ratios
\[
Adequacy = \frac{diag(\mathbf{MJM'})}{diag(\mathbf{MM'})}.
\]
Predictivity
Between class predictivity
The axis and class mean predictivities are defined in terms of the
weighted class means.
Axis predictivity
The elementwise ratios for the predictivity of each of the axes are
given by
\[
axis \: predictivity =
\frac{diag(\mathbf{\hat{\bar{X}}}'\mathbf{C\hat{\bar{X}}})}{diag(\mathbf{\bar{X}}'\mathbf{C\bar{X}})}.
\]
Class predictivity
Similarly for each of the class means the elementwise ratio is
computed from
\[
class \: predictivity =
\frac{diag(\mathbf{C}^{\frac{1}{2}}\mathbf{\hat{\bar{X}}}'\mathbf{W^{-1}}\mathbf{\hat{\bar{X}}}\mathbf{C}^{\frac{1}{2}})}{diag(\mathbf{C}^{\frac{1}{2}}\mathbf{\bar{X}}'\mathbf{W^{-1}}\mathbf{\bar{X}}\mathbf{C}^{\frac{1}{2}})}.
\]
Within class predictivity
We define the matrix of samples as deviations from their class means
as
\[
(\mathbf{I-H})\mathbf{X}=(\mathbf{I}_n-\mathbf{G}(\mathbf{G'G})^{-1}\mathbf{G}')\mathbf{X}
\]
where \(\mathbf{H} =
\mathbf{G}(\mathbf{G'G})^{-1}\mathbf{G}'\).
Within class axis predictivity
The within class axis predictivity is computed as the elementwise
ratios
\[
within \: class \: axis \: predictivity =
\frac{diag(\mathbf{\hat{X}}'(\mathbf{I-H)\hat{X}})}{diag(\mathbf{X}'(\mathbf{I-H)X})}.
\]
Within class sample predictivity
Unlike PCA biplots, sample predictivity for CVA biplots are computed
for the observations expressed as deviations from their class means. The
elementwise ratios is obtained from
\[
within \: class \: axis \: predictivity =
\frac{diag(\mathbf{(I-H)\hat{X}}\mathbf{W}^{-1}\mathbf{\hat{X}'(I-H)})}{diag(\mathbf{(I-H)X}\mathbf{W}^{-1}\mathbf{X'(I-H)})}.
\] To display the fit measures, we create a biplot
object with the measures added by the function
fit.measures() and call summary().
obj <- biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
fit.measures() |>
plot()
summary (obj)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> 9 classes: New England Middle Atlantic South Atlantic East South Central West South Central East North Central West North Central Mountain Pacific
#>
#> Quality of fit of canonical variables = 70.7%
#> Quality of fit of original variables = 70.5%
#> Adequacy of variables:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.41716176 0.15621549 0.16136381 0.09759664 0.19426796 0.55332679 0.50497634
#> Area
#> 0.40661470
#> Axis predictivity:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.1859124 0.4019427 0.8195756 0.6925389 0.7685373 0.9506355 0.7819324
#> Area
#> 0.8458143
#> Class predictivity:
#> New England Middle Atlantic South Atlantic East South Central
#> 0.7922047 0.6570417 0.8191791 0.8777759
#> West South Central East North Central West North Central Mountain
#> 0.7416085 0.6370315 0.3265978 0.6825966
#> Pacific
#> 0.6700194
#> Within class axis predictivity:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.04212318 0.09357501 0.25675620 0.19900223 0.29474972 0.75215233 0.31027358
#> Area
#> 0.12741853
#> Within class sample predictivity:
#> Alabama Alaska Arizona Arkansas California
#> 0.722548912 0.163442379 0.333341120 0.268976273 0.229139828
#> Colorado Connecticut Delaware Florida Georgia
#> 0.264963758 0.082284385 0.593415987 0.461070888 0.636531435
#> Hawaii Idaho Illinois Indiana Iowa
#> 0.015640188 0.113711473 0.338612599 0.389208196 0.507060148
#> Kansas Kentucky Louisiana Maine Maryland
#> 0.784831952 0.314119027 0.078465054 0.008388471 0.306141816
#> Massachusetts Michigan Minnesota Mississippi Missouri
#> 0.076563044 0.218470793 0.645446212 0.046129058 0.710971640
#> Montana Nebraska Nevada New Hampshire New Jersey
#> 0.086279776 0.810374638 0.090490164 0.298187909 0.003496353
#> New Mexico New York North Carolina North Dakota Ohio
#> 0.007134343 0.024268121 0.422776032 0.446240464 0.277262145
#> Oklahoma Oregon Pennsylvania Rhode Island South Carolina
#> 0.450104680 0.108636860 0.033945796 0.415029328 0.261568299
#> South Dakota Tennessee Texas Utah Vermont
#> 0.134881180 0.247921823 0.110537439 0.500454605 0.159941068
#> Virginia Washington West Virginia Wisconsin Wyoming
#> 0.310439564 0.030877305 0.066303623 0.295499472 0.474458397
The call to biplot(), CVA() and
fit.measures() is required to (a) create an object of class
biplot, (b) extend the object to class CVA and
(c) compute the fit measures. The call to the function
plot() is optional. It is further possible to select which
fit measures to display in the summary() function where all
measures default to TRUE.
obj <- biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.region) |>
fit.measures()
summary (obj, adequacy = FALSE, within.class.axis.predictivity = FALSE,
within.class.sample.predictivity = FALSE)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> 4 classes: Northeast South North Central West
#>
#> Quality of fit of canonical variables = 91.9%
#> Quality of fit of original variables = 95.3%
#> Axis predictivity:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.9873763 0.9848608 0.8757913 0.9050208 0.9955088 0.9970346 0.9558192
#> Area
#> 0.9344651
#> Class predictivity:
#> Northeast South North Central West
#> 0.8031465 0.9985089 0.6449906 0.9988469
