biplotEZ

The package provides users with an EZ-to-use way of constructing multi-dimensional scatterplots of their data. The simplest form of a biplot is the principal component analysis (PCA) biplot which will be used for illustration in this vignette.

1. What is a PCA biplot

Consider a data matrix $$\mathbf{X}^{*}:n \times p$$ containing data on $$n$$ objects and $$p$$ variables. To produce a 2D biplot, we need to optimally approximate $$\mathbf{X} = (\mathbf{I}_n-\frac{1}{n}\mathbf{11}')\mathbf{X}^{*}$$ (typically of rank $$p$$ with $$p<n$$) with a rank $$2$$ matrix. In terms of the least squares error, we want to

$min \| \hat{\mathbf{X}}-\mathbf{X} \|^2$ where $$rank(\hat{\mathbf{X}})=2$$. It was shown by Eckart and Young (1936) that if the singular value decomposition of $$\mathbf{X} = \mathbf{UDV'}$$ then $\hat{\mathbf{X}} = \mathbf{UJDJV'}$ with $\mathbf{J} = \begin{bmatrix} \mathbf{I}_2 & \mathbf{0}\\ \mathbf{0} & \mathbf{0} \end{bmatrix}$ essentially selecting only the first two columns of $$\mathbf{U}$$, the diagonal matrix of the first (largest) two singular values and the first two rows of $$\mathbf{V}'$$. Define $\mathbf{J}_2 = \begin{bmatrix} \mathbf{I}_2\\ \mathbf{0} \end{bmatrix}$ then $$\mathbf{J}_2\mathbf{J}_2' = \mathbf{J}$$ and we can write $$\hat{\mathbf{X}} = (\mathbf{UDJ}_2)(\mathbf{VJ}_2)'$$.

Gabriel (1971) shows that any rank $$2$$ matrix can be written as $$$\hat{\mathbf{X}} = \mathbf{G} \mathbf{H}' \tag{1}$$$ where $$\mathbf{G}:n \times 2$$ and $$\mathbf{H}:p \times 2$$.The $$n$$ rows of $$\mathbf{G}$$ provide the $$n$$ pairs of 2D coordinates representing the rows of $$\hat{\mathbf{X}}$$ and the $$p$$ rows of $$\mathbf{H}$$ provide the $$p$$ pairs of 2D coordinates representing the columns of $$\hat{\mathbf{X}}$$. Since $$\hat{\mathbf{X}} = (\mathbf{UDJ}_2)(\mathbf{VJ}_2)'$$, by setting $$\mathbf{G}=\mathbf{UDJ}_2$$ and $$\mathbf{H}=\mathbf{VJ}_2$$ we obtains the best least squares approximation of $$\mathbf{X}$$. Gabriel (1971) further shows that the approximation of distances between the rows are optimal, while the approximation of correlations by the cosines between the angles of the rows of $$\mathbf{H}$$ is sub-optimal.

The rows of $$\mathbf{G}$$ is plotted as points, representing the samples. The rows of $$\mathbf{H}$$ provide the directions of the axes for the variables. Since we have $x^{*}_{ij}-\bar{x}_j = x_{ij} \approx \hat{x}_{ij} = \mathbf{g}_{(i)}'\mathbf{h}_{(j)}$ all the values that will predict $$\mu$$ for variable $$j$$ is of the form $\mu = \mathbf{g}'_{\mu}\mathbf{h}_{(j)}$ which defines a straight line orthogonal to $$\mathbf{h}_{(j)}$$ in the biplot space (see the dotted red line in Figure 1(a)). To find the intersection of this prediction line with $$\mathbf{h}_{(j)}$$ we note that $\mathbf{g}'_{(i)}\mathbf{h}_{(j)} = \| \mathbf{g}_{(i)} \| \| \mathbf{h}_{(j)} \| cos(\mathbf{g}_{(i)},\mathbf{h}_{(j)}) = \| \mathbf{p} \| \| \mathbf{h}_{(j)} \|$ where $$\mathbf{p}$$ is the length of the orthogonal projection of $$\mathbf{g}_{(i)}$$ on $$\mathbf{h}_{(j)}$$. This is illustrated in Figure 1(b) with triangle ABC: $$cos(\theta) = \frac{AC}{AB}$$ or $$AC = AB cos(\theta)$$ The length of $$AC$$, written as $$\| \mathbf{p} \|$$ is equal to the cosine times the length of $$AB$$, i.e. $$cos(\mathbf{g}_{(i)},\mathbf{h}_{(j)}) \| \mathbf{g}_{(i)} \|$$.

Since $$\mathbf{p}$$ is along $$\mathbf{h}_{(j)}$$ we can write $$\mathbf{p} = c\mathbf{h}_{(j)}$$ and all points on the prediction line $$\mu = \mathbf{g}'_{\mu}\mathbf{h}_{(j)}$$ project on the same point $$c_{\mu}\mathbf{h}_{(j)}$$. We solve for $$c_{\mu}$$ from $\mu = \mathbf{g}'_{\mu}\mathbf{h}_{(j)}=\| \mathbf{p} \| \| \mathbf{h}_{(j)} \| = \| c_{\mu}\mathbf{h}_{(j)} \| \| \mathbf{h}_{(j)} \|$

$c_{\mu} = \frac{\mu}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}.$ 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 $p_{\mu_h} = c_{\mu_h}\mathbf{h}_{(j)} = \frac{\mu_h}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}\mathbf{h}_{(j)} \tag{2}$ To obtain a PCA biplot of the $$48\times 4$$ rock data in R we call

biplot(rock, scale = TRUE) |> PCA() |> plot()

2. The function biplot()

The function biplot() takes a data set (usually) and outputs an object of class biplot.

state.data <- data.frame (state.region, state.x77)
biplot(state.data)
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.

Apart from specifying a data set, we can specify a single variable for classification purposes.

biplot(state.x77, classes=state.region)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> 4 classes: Northeast South North Central West

If we want to use the variable state.region for formatting, say colour coding the samples according to region, we instead specify grouping.aes to indicate it pertains to the aesthetics, rather than data structure. We can include or exclude the aestethics variable from the data set.

biplot(state.x77, group.aes=state.region)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.

Next, we look at centring and scaling of the numeric data matrix. As we saw in section 1 above, PCA is computed from the centred data matrix. For most methods, centring is either required or has no effect on the methodology, therefore the default is center = TRUE. Since centring is usually assumed, you will get a warning message, should you explicitly choose to set center = FALSE. The default for scaled is FALSE, but often when variables are in different units of measurement, it is advisable to divide each variable by its standard deviation which is accomplished by setting scale = TRUE’.

biplot(state.data)                 # centred, but no scaling
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.
biplot(state.data, scale = TRUE)   # centered and scaled
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.
biplot(state.data, center = FALSE) # no centring (usually not recommended) or scaling
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.

The final optional argument to the function is specifying a title for your plot. We notice in the output above, that centring and / or scaling has no effect on the print method. It does however have an effect on the components of the object of class biplot in the output.

out <- biplot(state.data)                 # centred, but no scaling
out$center #> [1] TRUE out$scaled
#> [1] FALSE
out$means #> Population Income Illiteracy Life.Exp Murder HS.Grad Frost #> 4246.4200 4435.8000 1.1700 70.8786 7.3780 53.1080 104.4600 #> Area #> 70735.8800 out$sd
#> [1] 1 1 1 1 1 1 1 1
out <- biplot(state.data, scale = TRUE)   # centered and scaled
out$center #> [1] TRUE out$scaled
#> [1] TRUE
out$means #> Population Income Illiteracy Life.Exp Murder HS.Grad Frost #> 4246.4200 4435.8000 1.1700 70.8786 7.3780 53.1080 104.4600 #> Area #> 70735.8800 out$sd
#>   Population       Income   Illiteracy     Life.Exp       Murder      HS.Grad
#> 4.464491e+03 6.144699e+02 6.095331e-01 1.342394e+00 3.691540e+00 8.076998e+00
#>        Frost         Area
#> 5.198085e+01 8.532730e+04
out <- biplot(state.data, center = FALSE) # no centring (usually not recommended) or scaling
out$center #> [1] FALSE out$scaled
#> [1] FALSE
out$means #> [1] 0 0 0 0 0 0 0 0 out$sd
#> [1] 1 1 1 1 1 1 1 1

Note that the components means and sd only contain the sample means and sample sds when either/or center and scaled is TRUE. For values of FALSE, these components contain zeros for the means and/or ones for the sd to ensure back transformation will not have any affect.

2.1 Using biplot() with princomp() or prcomp()

Should the user wish to construct a PCA biplot after performing principal component analysis via the built in functions in the stats package, the output from either of these functions can be piped into the biplot function, where the piping implies that the argument data now takes the value of an object of class prcomp or princomp.

princomp(state.x77) |> biplot()
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
out <- prcomp(state.x77, scale.=TRUE) |> biplot()
rbind (head(out$raw.X,3),tail(out$raw.X,3))
#>               Population Income Illiteracy Life Exp Murder HS Grad Frost   Area
#> Alabama             3615   3624        2.1    69.05   15.1    41.3    20  50708
#> Alaska               365   6315        1.5    69.31   11.3    66.7   152 566432
#> Arizona             2212   4530        1.8    70.55    7.8    58.1    15 113417
#> West Virginia       1799   3617        1.4    69.48    6.7    41.6   100  24070
#> Wisconsin           4589   4468        0.7    72.48    3.0    54.5   149  54464
#> Wyoming              376   4566        0.6    70.29    6.9    62.9   173  97203
rbind (head(out$X,3),tail(out$X,3))
#>                Population      Income Illiteracy   Life Exp     Murder
#> Alabama       -0.14143156 -1.32113867   1.525758 -1.3621937  2.0918101
#> Alaska        -0.86939802  3.05824562   0.541398 -1.1685098  1.0624293
#> Arizona       -0.45568908  0.15330286   1.033578 -0.2447866  0.1143154
#> West Virginia -0.54819682 -1.33253061   0.377338 -1.0418703 -0.1836632
#> Wisconsin      0.07673438  0.05240289  -0.771082  1.1929438 -1.1859550
#> Wyoming       -0.86693413  0.21188994  -0.935142 -0.4384705 -0.1294853
#> Alabama       -1.4619293 -1.62482920 -0.2347183
#> Arizona        0.6180514 -1.72101848  0.5002047
#> West Virginia -1.4247868 -0.08580083 -0.5469045
#> Wisconsin      0.1723413  0.85685405 -0.1906996
#> Wyoming        1.2123316  1.31856256  0.3101835
out$center #> [1] TRUE out$scaled
#> [1] TRUE
out$means #> Population Income Illiteracy Life Exp Murder HS Grad Frost #> 4246.4200 4435.8000 1.1700 70.8786 7.3780 53.1080 104.4600 #> Area #> 70735.8800 out$sd
#>   Population       Income   Illiteracy     Life Exp       Murder      HS Grad
#> 4.464491e+03 6.144699e+02 6.095331e-01 1.342394e+00 3.691540e+00 8.076998e+00
#>        Frost         Area
#> 5.198085e+01 8.532730e+04

3. The functions PCA(), plot() and legend.type()

The first argument to the function PCA() is an object of class biplot, i.e. the output of the biplot() function. By default we c onstruct a 2D biplot (argument dim.biplot = 2) of the first two principal components (argument e.vects = 1:2). The group.aes argument, if not specified in the function biplot(), allows a grouping argument for the sample aesthetics. A PCA biplot of the state.x77 data with colouring according to state.region is obtained as follows:

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.region) |> plot()

The output of PCA() is an object of class PCA which inherits from the class biplot. Four additional components are present in the PCA object. The matrix Z contains the coordinates of the sample points, while the matrix Vr contains the “coordinates” for the variables. In the notation of equation (1), Z=$$\mathbf{G}:n \times 2$$ and Vr=$$\mathbf{H}:p \times 2$$. The component Xhat is the matrix $$\hat{\mathbf{X}}$$ on the left hand side of equation (1). The final component ax.one.unit contains as rows the expression in equation (2) with $$\mu_h=1$$, in other words, one unit in the positive direction of the biplot axis.

By piping the PCA class object (inheriting from class biplot) to the generic plot() function, the plot.biplot() function constructs the biplot on the graphical device. To add a legend to the biplot, we call

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.region) |>
legend.type(samples = TRUE) |> plot()

It was mentioned in section 1 that the default choice $$\mathbf{G}=\mathbf{UDJ}_2$$ and $$\mathbf{H}=\mathbf{VJ}_2$$ provides an exact representation of the distances between the rows of $$\mathbf{\hat{X}}$$ which is an optimal approximation in the least squares sense of the distances between the rows of $$\mathbf{X}$$ (samples). Alternatively, the correlations between the variables (columns of $$\mathbf{X}$$) can be optimally approximated by the cosines of the angles between the axes, leaving the approximation of the distances between the samples to be suboptimal. In this case $$\mathbf{G}=\mathbf{UJ}_2$$ and $$\mathbf{H}=\mathbf{VDJ}_2$$ and this biplot is obtained by setting the argument correlation.biplot = TRUE.

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.region, correlation.biplot = TRUE) |>
legend.type(samples = TRUE) |> plot()

4. The function samples()

This function controls the aesthetics of the sample points 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 PCA biplot of the state.x77 data with samples coloured according to state.division.

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
legend.type(samples = TRUE) |> plot()

Since the legend interferes with the sample points, we choose to place the legend on a new page, by setting new = TRUE in the legend.type function. Furthermore, we wish to select colours, other than the defaults, for the divisions.

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "#000000", "blue")) |>
legend.type(samples = TRUE, new = TRUE) |> plot()

Furthermore we want to use a different plotting character for the central regions.

levels (state.division)
#> [1] "New England"        "Middle Atlantic"    "South Atlantic"
#> [4] "East South Central" "West South Central" "East North Central"
#> [7] "West North Central" "Mountain"           "Pacific"

We want to use pch = 15 for the first three and final two divisions and pch = 1 for the remaining four divisions.

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "black", "blue"),
pch = c(15, 15, 15, 1, 1, 1, 1, 15, 15)) |>
legend.type(samples = TRUE, new = TRUE) |> plot()

To increase the size of the plotting characters of the eastern states, we add the following:

biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "black", "blue"),
pch = c(15, 15, 15, 1, 1, 1, 1, 15, 15),
cex = c(rep(1.5,4), c(1,1.5,1,1.5))) |>
legend.type(samples = TRUE, new = TRUE)  |> plot()

If we choose to only show the samples 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) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "black", "blue"),
which = c("West North Central", "West South Central", "East South Central",
"East North Central")) |>
legend.type(samples = TRUE, new = TRUE)  |> plot()

Note that since four regions are selected, the colour (and other aesthetics) is applied to these regions in the order they are specified in which. To add the sample names, the label argument is set to TRUE. For large sample sizes, this is not recommended, as overplotting will render the plot unusable. The size of the labels is controlled with label.cex which can be specified either as a single value (for all samples) or a vector indicating size values for each individual sample. The colour of the labels defaults to the colour(s) of the samples. 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 samples.

biplot(state.x77, scaled = TRUE) |> PCA() |>
samples (label = TRUE) |> plot()

We can use the arguments label.cex, label.side and label.offset to make the plot more legible with a little effort.

rownames(state.x77)[match(c("Pennsylvania", "New Jersey", "Massachusetts",
"Minnesota"), rownames(state.x77))] <- c("PA", "NJ", "MA", "MN")
"North Carolina", "Colorado", "Washington", "Illinois", "Michigan", "Arizon",
"Florida", "Ohio", "NJ", "Kansas"), rownames(state.x77))
right.side <- match(c("South Carolina", "Kentucky", "Rhode Island", "New Hampshire", "Virginia",
"Missouri", "Delaware", "Hawaii", "Oregon", "PA", "Nebraska", "Montana",
"Maryland", "Indiana", "Idaho"), rownames(state.x77))
left.side <- match(c("Wyoming", "Iowa", "MN", "Connecticut"), rownames(state.x77))
label.offset <- rep(0.3, nrow(state.x77))
label.offset[match(c("Colorado", "Kansas", "Idaho"), rownames(state.x77))] <- c(0.8, 0.5, 0.8)
label.side <- rep("bottom", nrow(state.x77))
label.side[above] <- "top"
label.side[right.side] <- "right"
label.side[left.side] <- "left"
biplot (state.x77, scaled=TRUE) |> PCA() |>
samples (label=TRUE, label.cex=0.6, label.side=label.side, label.offset=label.offset) |>
plot()

We can also make use of the functionality of the ggrepel package to place the labels.

biplot(state.x77, scaled = TRUE) |> PCA() |>
samples (label = "ggrepel", label.cex=0.65) |> plot()
#> Warning: ggrepel: 1 unlabeled data points (too many overlaps). Consider increasing max.overlaps
#> ggrepel: 1 unlabeled data points (too many overlaps). Consider increasing max.overlaps

If the data plotted in the biplot is a multivariate time series, it can make sense to connect the data points in order. Let us consider the four quarters of the UKgas data set as four variables and we represent the years as sample points in a PCA biplot.

gas.data <- matrix (UKgas, ncol=4, byrow=T)
colnames(gas.data) <- paste("Q", 1:4, sep="")
rownames(gas.data) <- 60:86
even.labels <- rep(c(TRUE, FALSE), 14)
biplot(gas.data, scaled = TRUE) |> PCA() |>
samples (connected = TRUE, connect.col="red", label = even.labels, label.cex=0.6) |>
plot()

4. The function means()

The function means() allow changing the aesthetics for group means specified by group.aes, when the argument show.group.means = TRUE in the call to the function PCA(). The functionality of means() mirrors that of samples() and is discussed in detail in the vignette Class separation where class means are more prominent than in PCA biplots.

5. The function axes()

Similar to the samples() function, this function allows for changing the aestethics of the biplot axes. The first argument to axes() is an object of class biplot. The X.names argument is typically not specified by the user, but is required for the function to allow specifying which axes to display in the which argument, by either speficying the column numbers
or the column names. The arguments col, lwd and lty pertains to the axes themselves and can be specified either as a scaler value (to be recycled) or a vector with length equal to that of which.

To construct a PCA biplot of the rock data, displaying only the axes for peri and shape with different colours for the two axes, different line widths and line type 2, we need to following code:

biplot(rock, scaled = TRUE) |> PCA() |>
axes(which = c("shape","peri"),
col=c("lightskyblue","slategrey"),
lwd = c(1,2), lty=2) |>
plot()

The following four arguments deal with the axis labels. The argument label.dir is based on the graphics parameter las and allows for labels to be either orthogonal to the axis direction (Orthog), horisontal (Hor) or parallel to the plot Paral. The argument label.line fulfills the role of the line argument in mtext() to determine on which margin line (how far from the plot) the label is placed while label.col and label.cex is self-explanatory and defaults to the axis colour and size 0.75. Note in for the illustration the in the code below the colour vector has only three components, so that recycling is applied.

biplot(rock, scaled = TRUE) |> PCA() |>
axes(col=c("lightskyblue","slategrey","blue"),
label.dir="Hor", label.line=c(0,0.5,1,1.5)) |>
plot()

The function pretty() finds ‘nice’ tick marks where the value specified in the argument ticks determine the desired number of tick marks, although the observed number could be different. The other tick.* arguments are similar to their naming counterparts in par() or text(). Since the tick labels are important to follow the direction of increasing values of the axes, setting tick.label = FALSE does not remove the tick marks completely, but limits the labels to the smallest and largest value visible in the plot. If the user would like to specify alternative names for the axes, this can be done in the argument ax.names.

biplot(rock, scaled = TRUE) |> PCA() |>
axes(label.dir="Paral",
ticks = c(3, 5, 5, 10), tick.label=c(F, F, T, T),
ax.names = c("area", "perimeter", "shape",
"permeability in milli-Darcies")) |>
plot()

6. The functions fit.measures() and summary()

The print method provides a short summary of the biplot object.

  obj <- biplot(airquality)
#> Warning in biplot(airquality): 42 rows deleted due to missing values
obj
#> Object of class biplot, based on 111 samples and 6 variables.
#> 6 numeric variables.
#>
#> The following 42 sample-rows where removed due to missing values
#>  5 6 10 11 25 26 27 32 33 34 35 36 37 39 42 43 45 46 52 53 54 55 56 57 58 59 60 61 65 72 75 83 84 96 97 98 102 103 107 115 119 150

The output from summary() will be very similar.

  summary(obj)
#> Object of class biplot, based on 111 samples and 6 variables.
#> 6 numeric variables.
#>
#> The following 42 sample-rows where removed due to missing values
#>  5 6 10 11 25 26 27 32 33 34 35 36 37 39 42 43 45 46 52 53 54 55 56 57 58 59 60 61 65 72 75 83 84 96 97 98 102 103 107 115 119 150

Additional information about the biplot object is added by the fit.measures() function.

Quality of approximation

$\mathbf{X} = \mathbf{\hat{X}} + \mathbf{X-\hat{X}}$ which decomposes $$\mathbf{X}$$ into a fitted part

$\mathbf{\hat{X}} = \mathbf{UJDJV'} = \mathbf{UDJ}_2(\mathbf{VJ}_2)' = \mathbf{UDV'VJ}_2(\mathbf{VJ}_2)' = \mathbf{XVJV'}$

and the residual part $$\mathbf{X-\hat{X}}$$. The lack of fit is quantified by the quantity we are minimising

$\| \hat{\mathbf{X}}-\mathbf{X} \|^2$ where we have the orthogonal decomposition

$\|\mathbf{X}\|^2 = \|\hat{\mathbf{X}}\|^2 + \|\hat{\mathbf{X}}-\mathbf{X} \|^2.$ The overall quality of fit is therefore defined as

$quality = \frac{\|\hat{\mathbf{X}}\|^2}{\|\mathbf{X}\|^2} = \frac{tr(\mathbf{XX}')}{tr(\mathbf{\hat{X}\hat{X}'})} = \frac{tr(\mathbf{X'X})}{tr(\mathbf{\hat{X}'\hat{X}})} = \frac{tr(\mathbf{VD}^2\mathbf{V'})}{tr(\mathbf{VD}^2\mathbf{JV'})}.$ In biplotEZ the overall quality is displayed as a percentage:

$quality =\frac{d_1^2+d_2^2}{d_1^2+\dots+d_p^2}100\%.$

Adequacy of representation of the variables

Researchers who construct the PCA biplot representing the columns with arrows (vectors) often fit the biplot with a unit circle. The rationale being that perfect representation of a variable will have unit length and the length of each arrow vs the distance to the unit circle represent the adequacy with which the variable is represented.

By fitting the biplot with calibrated axes, it is much easier to read off values for the variables, but the adequacy values can still be computed from

$\frac{diag(\mathbf{V}_r\mathbf{V}_r')}{diag(\mathbf{VV}')}= diag(\mathbf{V}_r\mathbf{V}_r')$

due to the orthogonality of the matrix $$\mathbf{V}:p \times p$$.

Predictivities

The predictivity provides a measure of who well the original values are recovered from the biplot. An element that is well represented will have a predictivity close to one, indicating that the sample or variable values from prediction is close to the observed values. If an element is poorly represented, the predicted values will be very different from the original values and the predictivity value will be close to zero.

Axis predictivity

The predictivity for each of the $$p$$ variables is computed as the elementwise ratios

$axis \: predictivity = \frac{diag(\mathbf{\hat{X}'\hat{X}})}{diag(\mathbf{X'X})}$

Sample predictivity

The predictivity for each of the $$n$$ samples is computed as the elementwise ratios

$sample \: predictivity = \frac{diag(\mathbf{\hat{X}\hat{X}'})}{diag(\mathbf{XX'})}$ By calling the function fit.measures() these quantities are computed for the specific biplot object. The values are displayed with the summary() function.

obj <- biplot(state.x77, scale = TRUE) |> PCA() |>
fit.measures() |> plot()

summary (obj)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#>
#> Quality of fit = 65.4%
#> Population     Income Illiteracy   Life Exp     Murder    HS Grad      Frost
#>  0.1848016  0.3586383  0.2215201  0.1760908  0.2915819  0.2696184  0.1513317
#>       Area
#>  0.3464170
#> Axis predictivity:
#> Population     Income Illiteracy   Life Exp     Murder    HS Grad      Frost
#>  0.3330216  0.7609185  0.7917091  0.6206172  0.8640485  0.7947530  0.4982299
#>       Area
#>  0.5675169
#> Sample predictivity:
#>        Alabama         Alaska        Arizona       Arkansas     California
#>     0.95126856     0.61373919     0.26327256     0.86308539     0.57062754
#>       Colorado    Connecticut       Delaware        Florida        Georgia
#>     0.83358779     0.59003002     0.18284712     0.49725356     0.94461052
#>         Hawaii          Idaho       Illinois        Indiana           Iowa
#>     0.01984127     0.70337480     0.33405270     0.30082350     0.96367113
#>         Kansas       Kentucky      Louisiana          Maine       Maryland
#>     0.86554676     0.87758262     0.93717163     0.66553856     0.06362508
#>             MA       Michigan             MN    Mississippi       Missouri
#>     0.47386267     0.26050188     0.89207404     0.93073099     0.11321791
#>     0.44603781     0.93570441     0.22393876     0.87499561     0.15979033
#>     New Mexico       New York North Carolina   North Dakota           Ohio
#>     0.29304145     0.40609063     0.93004841     0.69011551     0.08810179
#>       Oklahoma         Oregon             PA   Rhode Island South Carolina
#>     0.37520943     0.36273523     0.02176080     0.58625617     0.93187284
#>   South Dakota      Tennessee          Texas           Utah        Vermont
#>     0.83804787     0.96006357     0.73748654     0.66209083     0.80365601
#>       Virginia     Washington  West Virginia      Wisconsin        Wyoming
#>     0.58564755     0.33877314     0.85231725     0.82519206     0.42499724

If is not necessary to call the plot() function to obtain the fit measures, but one of the biplot methods, such as PCA() is required, since the measures differ depending on which type of biplot is constructed. To suppress the output of some fit measures, for instance if the interest is in the axis predictivity and there are many samples which result in a very long output, these can be set in the call to summary(). By default all measures are set to TRUE.

obj <- biplot(state.x77, scale = TRUE) |> PCA() |>
fit.measures()
summary (obj, adequacy = FALSE, sample.predictivity = FALSE)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#>
#> Quality of fit = 65.4%
#> Axis predictivity:
#> Population     Income Illiteracy   Life Exp     Murder    HS Grad      Frost
#>  0.3330216  0.7609185  0.7917091  0.6206172  0.8640485  0.7947530  0.4982299
#>       Area
#>  0.5675169

The axis predictivities and sample predictivities can be represented in the biplot in two ways: setting either axis.predictivity and / or sample.predictivity to TRUE, applies shading for axes and shrinking for samples according to the predictivity values.

biplot(state.x77, scale = TRUE) |> PCA(group.aes = state.region) |>
samples (which = "South", pch = 15, label = T, label.cex=0.5) |>
axes (col = "black") |>
fit.measures() |> plot (sample.predictivity = TRUE,
axis.predictivity = TRUE) 

Comparing the plot with the summary output it is clear that the variables Population and Frost are not very well represented and it can be expected that predictions on these variables will be less accurate. Furthermore, the samples located close to the origin are not as well represented as those located towards the bottom right. This is typically the case where samples nearly orthogonal to the PCA plane are projected close to the origin and due to their orthogonality, very poorly represented.

7. The function alpha.bags()

An $$\alpha$$-bag encloses the $$\alpha100\%$$ inner data points in a cloud of points. It is based on the concept of halfspace location depth as defined by Tukey (1975). Rousseeuw, Ruts, and Tukey (1999) generalised a boxplot to a two-dimensional bagplot where the box is replaced by a bag containing the inner $$50\%$$ of the observations. Gower, Lubbe, and Roux (2011) replaces the $$50\%$$-bag contour by a general $$\alpha100\%$$ contour referred to as an $$\alpha$$-bag.

When the number of samples in the biplot is larger, it becomes difficult to isolate individual observations. Often, when a grouping variable is present, the interest is not so much in the indivdiual samples, but rather in the location and spread of the groups. In the plot below, we enclose each century’s number of sunspots by a $$95\%$$-bag where the months are used as 12 different variables for each year (sample point). Note that the legend displays the $$\alpha$$-bags while samples = FALSE is left at the default. Both can be displayed, but since the $$\alpha$$-bags’ colour defaults to the colour of the sample points, both are not necessary here.

sunspots <- matrix (sunspot.month[1:(264*12)], ncol = 12, byrow = TRUE)
years <- 1749:2012
rownames(sunspots) <- years
colnames(sunspots) <- c("Jan", "Feb", "Mar", "Apr", "May", "Jun",
"Jul", "Aug", "Sep", "Oct", "Nov", "Dec")
century <-paste(floor((years-1)/100)+1, ifelse (floor((years-1)/100)+1<21, "th","st"), sep = "-")
biplot(sunspots, group.aes=century) |> PCA() |>
axes (label.dir = "Hor", label.line = c(0.8, rep(0,10), 0.8)) |>
alpha.bags () |>
legend.type(bags = TRUE)  |> plot()
#> Computing 0.95 -bag for 18-th
#> Computing 0.95 -bag for 19-th
#> Computing 0.95 -bag for 20-th
#> Computing 0.95 -bag for 21-st

By default one $$95\%$$-bag is constructed for each group. In general, the alpha.bags() function accepts an object of class biplot as first argument. The next argument alpha can be specified as a single value, or to construct a series of $$\alpha$$-bags for a group, alpha can be a vector argument. The argument which specifies the groups to be fitted with $$\alpha$$-bags.

biplot(sunspots, group.aes=century) |> PCA() |>
axes (label.dir = "Hor", label.line = c(0.8, rep(0,10), 0.8)) |>
alpha.bags (alpha = c(0.9, 0.95, 0.99), which = c(1,4)) |>
legend.type(bags = TRUE)  |> plot()
#> Computing 0.9 -bag for 18-th
#> Computing 0.9 -bag for 21-st
#> Computing 0.95 -bag for 18-th
#> Computing 0.95 -bag for 21-st
#> Computing 0.99 -bag for 18-th
#> Computing 0.99 -bag for 21-st

In the biplot above, the colours were recycled for each alpha value. To specify differential colours, we can use the col argument and similarly the lty and or lwd arguments. Since we are mostly interested in the location and overlap of the clouds of points we can remove the indivdiual samples by setting samples (which = NULL). The default colours will still be used for the $$\alpha$$-bags and we chose to specify different line types for different $$\alpha$$ values.

#biplot(sunspots, group.aes=century) |> PCA() |>
#        axes (label.dir = "Hor", label.line = c(0.8, rep(0,10), 0.8)) |>
#        samples (which = NULL) |>
#        alpha.bags (alpha = c(0.9, 0.95, 0.99), lty = c(1,3,5)) |>
#        legend.type(bags = TRUE, new = TRUE)  |> plot()

For a completely custom combination of $$\alpha$$-bags, we do not rely on any recycling and specify each of the arguments alpha, which, col, lty, lwd as a vector. Since the calculate of halfspace location depth is very computationally intensive, a random sample of size 2500 is chosen for each group to construct the $$\alpha$$-bag. This sample size can be changed with the argument max. Setting trace = FALSE will suppress the message “Computing $$\alpha$$” -bag for groupX.”

biplot(sunspots, group.aes=century) |> PCA() |>
axes (label.dir = "Hor", label.line = c(0.8, rep(0,10), 0.8)) |>
samples (which = NULL) |>
alpha.bags (alpha = c(   0.9,   0.95,   0.99,             0.5,         0.6,     0.7),
which = c(     1,      1,      2,               3,           3,      3),
col   = c("brown", "red", "gold",  "deepskyblue2", "steelblue3","blue"),
lty   = c(     1,      2,     10,               2,           2,      2),
lwd   = c(     1,      1,      3,               1,           2,      1)) |>
legend.type(bags = TRUE) |> plot()
#> Computing 0.9 -bag for 18-th
#> Computing 0.95 -bag for 18-th
#> Computing 0.99 -bag for 19-th
#> Computing 0.5 -bag for 20-th
#> Computing 0.6 -bag for 20-th
#> Computing 0.7 -bag for 20-th

8. The function ellipses()

If we observe a random sample from a $$p$$-variate normal distribution with $$\bar{\mathbf{x}}$$ and $$\mathbf{S}$$ the usual unbaised estimates of the mean vector and covariance matrix, then

$(\mathbf{x} - \bar{\mathbf{x}})' \mathbf{S}^{-1} (\mathbf{x} - \bar{\mathbf{x}}) = \kappa^2$

traces an ellipsiod in $$p$$ dimensions. For $$p=2$$, chosing $$\kappa = {(\chi^{2}_{2,1-\alpha})}^{\frac{1}{2}}$$ where $$\chi^{2}_{2,1-\alpha}$$ denotes the $$(1-\alpha)100$$-th percentage point of the $$\chi^2_2$$ distribution results in an ellipse covering approximately $$100\alpha\%$$ of the configuraion of two-dimensional points. With default arguments df = 2 and alpha = 0.95, the value of $$\kappa$$ is $$2.447747$$ and the ellipse function constructs an ellipse that would enclose approximately $$95\%$$ of the observations from a bivariate normal distribution. The argument kappa can be specified directly, and will take precedence over the specification of alpha. The other arguments of the ellipses() function operates identically to the corresponding arguments of the function alpha.bags(). Using $$\alpha$$-bags, rather than ellipses is recommended in general, since the construction of the ellipses are based on the underlying assumption of a random sample observed from a normal distribution.

biplot(sunspots, group.aes=century) |> PCA() |>
axes (label.dir = "Hor", label.line = c(0.8, rep(0,10), 0.8)) |>
samples (which = NULL) |>
ellipses (alpha = c(0.9, 0.95), lty = c(1,3,5)) |>
legend.type(ellipses = TRUE)  |> plot()
#> Computing 2.15 -ellipse for 18-th
#> Computing 2.15 -ellipse for 19-th
#> Computing 2.15 -ellipse for 20-th
#> Computing 2.15 -ellipse for 21-st
#> Computing 2.45 -ellipse for 18-th
#> Computing 2.45 -ellipse for 19-th
#> Computing 2.45 -ellipse for 20-th
#> Computing 2.45 -ellipse for 21-st


biplot(sunspots, group.aes=century) |> PCA() |>
axes (label.dir = "Hor", label.line = c(0.8, rep(0,10), 0.8)) |>
samples (which = NULL) |>
ellipses (kappa = 1:2, lty = c(1,3,5)) |>
legend.type(ellipses = TRUE) |> plot()
#> Computing 1 -ellipse for 18-th
#> Computing 1 -ellipse for 19-th
#> Computing 1 -ellipse for 20-th
#> Computing 1 -ellipse for 21-st
#> Computing 2 -ellipse for 18-th
#> Computing 2 -ellipse for 19-th
#> Computing 2 -ellipse for 20-th
#> Computing 2 -ellipse for 21-st

9. The functions interpolate() and newsamples()

The process of interpolation is described by Gower and Hand (1996) as the process of finding the coordinates of a $$p$$-dimensional sample in the lower dimensional biplot space. For PCA we showed in section 1 that the sample points are represented by $$\mathbf{G}=\mathbf{UDJ}_2$$ which can be written as $$\mathbf{G}=\mathbf{UDV'VJ}_2=\mathbf{XVJ}_2$$. Finding the position of a new sample $$\mathbf{x}^*:p \times 1$$ make use of the same transformation so that the 2D coordinates is given by $${\mathbf{z}^*}':2 \times 1 ={\mathbf{x}^*}' \mathbf{VJ}_2$$.

Adding samples to the plot is facilitated by the function interpolate(). Note that the samples to be interpolated did not contribute to the construction of the biplot. This is the reason why Greenacre (2017) term these supplementary points.

The function interpolate() accepts two arguments, the first an object of class biplot and the second a matrix or data frame containing the samples to be interpolated. The second argument, newdata, needs to have a similar structure to the data set sent to biplot(). If biplot() received a data frame, newdata can be either another data frame or a matrix containing the subset of numerical variables.

Suppose we construct a PCA biplot of the first $$40$$ samples in the data set rock and then $$8$$ new samples is to be interpolated the call will be:

biplot(rock[1:40,], scale = TRUE) |> PCA() |>
interpolate (rock[41:48,]) |> plot()

The function newsamples() operates similar to samples, allowing changes to the aestethics of the interpolated new samples. There is no argument which for newsamples() since it is assumed that samples are interpolated to be represented in the biplot. All the other arguments are vectors of length similar to the number of samples in newdata. To change the colour of the interpolated samples and add labels, the following call will be used:

biplot(rock[1:40,], scale = TRUE) |> PCA() |>
interpolate (rock[41:48,]) |>
newsamples (label = TRUE, label.side = "top", col = rainbow(10)) |> plot()`

References

Eckart, C., and G. Young. 1936. “The Approximation of One Matrix by Another of Lower Rank.” Psychometrika, 211–18.
Gabriel, K. R. 1971. “The Biplot Graphic Display of Matrices with Application to Principal Component Analysis.” Biometrika, 453–67.
Gower, J. C., and D. J. Hand. 1996. Biplots. Chapman & Hall.
Gower, J. C., S. Lubbe, and N. J. le Roux. 2011. Understanding Biplots. Wiley.
Greenacre, M. J. 2017. Correspondence Analysis in Practice. CRC press.
Rousseeuw, P. J, I Ruts, and J. W. Tukey. 1999. “The Bagplot: A Bivaraite Boxplot.” American Statistician, 382–87.
Tukey, J. W. 1975. “Mathematics and the Picturing of Data.” Proceedings of the International Congress of Mathematicians, 523–31.