4. lpda function
Palmdates: Chemical variables
First we apply the method with the first two variables:
fibre and sorbitol to show the performance of
the package. The application of the method with two variables allows the
visualization of the hyperplane in two dimensions, in this case, a
straight line.
group = as.factor( c(rep("Spanish",11), rep("Other",10)) )
model1 = lpda(data = palmdates$conc[,1:2], group = group )
model1
#> Call:
#> lpda(data = palmdates$conc[, 1:2], group = group)
#> Coefficients:
#> [1] -1.116131 -3.002923 -6.563646
names(model1)
#> [1] "coef" "data" "group" "scale" "pca" "call"One of the outputs of lpda is a matrix with the
coefficients of the hyperplane: \(a'x=b\) for each pair-wise comparison.
In this example, -1.116 and -3.003 the coefficients of
fibre and sorbitol respectively and -6.564 the
constant \(b\).
In cases with 2 variables, as this simple example, we can plot the line on the points, that represents the samples, with the following code:
plot(palmdates$conc[,1:2], col = as.numeric(group)+1, pch = 20,
main = "Model 1. Palmdates: fibre & sorbitol")
abline(model1$coef[3]/model1$coef[2], -model1$coef[1]/model1$coef[2], cex = 2)
legend("bottomright", c("Other","Spanish"),col = c(2,3), pch = 20, cex=0.8)
Predicted group with the model is obtained with predict
function. Confusion matrix can be obtained with summary. We
observe that all the samples are well classified.
predict(model1)
#>
#> elche1 elche2 elche3 elche4
#> "Spanish" "Spanish" "Spanish" "Spanish"
#> elche5 elche6 elche7 elche8
#> "Spanish" "Spanish" "Spanish" "Spanish"
#> elche9 elche10 elche11 salomon.israel
#> "Spanish" "Spanish" "Spanish" "Other"
#> hayani.israel medjool.israel barhi.israel deglet.noor.tunez
#> "Other" "Other" "Other" "Other"
#> tunez.alligh argelia.deglet.noor iran arabia.saudi.perny
#> "Other" "Other" "Other" "Other"
#> sud.africa.medjool
#> "Other"
summary(model1)
#> Call:
#> lpda(data = palmdates$conc[, 1:2], group = group)
#>
#> CONFUSION MATRIX
#> Real
#> Predicted Other Spanish
#> Other 10 0
#> Spanish 0 11By considering the five concentration variables, all samples are well classified as well. As a 5-dimensional plot can not be showed, we offer the possibility of seeing a plot that shows the situation of samples with respect to \(H\). Y-axis represents order in which they appear in the data matrix and X-axis distances of each sample to \(H\).
model2 = lpda(data = palmdates$conc, group = group )
plot(model2, main="Model 2. Palmdates: all conc variables")
Another option is the application of lpda to the first
PCs scores. When data is highly correlated, two components are usually
preferred. In such case choosing as plot arguments
PCscores = TRUE we will visualize the first two PCs,
indicating in the axis the proportion of explained variance by each PC.
Moreover if there are two groups, as in this example, the optimal
hyperplane is also showed.
model3 = lpda(data = palmdates$conc, group = group, pca = TRUE, Variability = 0.7)
plot(model3, PCscores = TRUE, main = "Model 3. Palmdates: PCA on conc variables")
Palmdates: Spectra variables
When having data sets with more variables than individuals PCA is not
directly applicable. In lpda we have implemented the
possibility of dealing with such problem, working with the equivalences
between the PCA of the data matrix and the transposed matrix.
In palmdates$spectra there are 2050 very correlated
measurements as it is showed in the following figure:
Due to the high correlation the application of lpda to
all the spectra variables and to the first 2 PCs (model4) give the same
solution: 0 prediction errors.
model4 = lpda(data = palmdates$spectra, group = group, pca = TRUE, Variability = 0.9)
plot(model4, PCscores = TRUE, main = "Model 4. Palmdates: PCA on Spectra variables")
Following example shows how predict the group of a test set that does not participate in the model. In this set we include two samples of each group.
test = c(10,11,12,13)
model5 = lpda(data = palmdates$spectra[-test,], group = group[-test], pca = TRUE,
Variability = 0.9)
predict(model5, datatest=palmdates$spectra[test,])
#>
#> elche10 elche11 salomon.israel hayani.israel
#> "Spanish" "Spanish" "Other" "Other"
summary(model5, datatest=palmdates$spectra[test,], grouptest = group[test])
#> Call:
#> lpda(data = palmdates$spectra[-test, ], group = group[-test],
#> pca = TRUE, Variability = 0.9)
#>
#> CONFUSION MATRIX
#> Real
#> Predicted Other Spanish
#> Other 2 0
#> Spanish 0 2iris data
To show an example with more than two groups we use the famous
(Fisher’s or Anderson’s) iris data set that is available in
base R. The iris data set gives the
measurements in centimeters of the variables sepal length and width and
petal length and width, respectively, for 50 flowers from each of 3
species of iris. The species are Iris setosa,
versicolor, and virginica.
Results with the four variables give 2 classification errors. In this
case the model computes 3 hyperplanes for each one of the 3 pairwise
comparisons. Now plot function gives 3 plots.
model.iris = lpda(iris[,-5], iris[,5])
summary(model.iris)
#> Call:
#> lpda(data = iris[, -5], group = iris[, 5])
#>
#> CONFUSION MATRIX
#> Real
#> Predicted setosa versicolor virginica
#> setosa 50 0 0
#> versicolor 0 49 1
#> virginica 0 1 49
The application of lpda with 3 PCs gives the same
classification error. Function plot with
Pcscores = TRUE gives the scores in the first two PCs, but
in this case without the hyperplanes.
model.iris2 = lpda(iris[,-5], iris[,5], pca=TRUE, PC=3)
summary(model.iris2)
#> Call:
#> lpda(data = iris[, -5], group = iris[, 5], pca = TRUE, PC = 3)
#>
#> CONFUSION MATRIX
#> Real
#> Predicted setosa versicolor virginica
#> setosa 50 0 0
#> versicolor 0 49 1
#> virginica 0 1 49
plot(model.iris2, PCscores= TRUE)
