Differences between linear and canonical discriminant analyses (LDA and CDA)

Question

I'm using R to try and compare the results of variable chemical compositions, following on from an article I've read. In it, the authors used CDA to do something very similar to what I want to do, but I've been told by another researcher (without much of an explanation) that LDA would be better suited. I could go into the specifics of why supervised learning is the avenue chosen, etc. but I won't post that unless someone asks.

After doing some background reading (which hasn't really cleared up the difference between the two), I figured I'd try to explore this myself and compare the results. The primary difference between my data and that in this article is that instead of just using the compositions, I've created 3 new variables (S-, F- and V-) for the CDA that are functions of the original compositional data (see code below).

However, when I run the two analyses I get EXACTLY the same results - identical plots. This doesn't seem possible, but I can't find an error in my coding.

My two questions are:

1. Is it possible for LDA and CDA to return the exact same result?

2. What are the practical differences between LDA and CDA?

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Data:

library(MASS)
library(candisc)
library(ggplot2)

al2o3<-runif(20,5,10)
sio2<-runif(20,10,30)
feo<-runif(20,40,60)
country<-c(rep("England",6), rep("Scotland",6), rep("Wales",4), rep("France",4))
df<-data.frame(country,al2o3,sio2,feo)

LDA:

lda <- lda(country ~ feo+sio2+al2o3, data=df)
plda <- predict(object = lda, newdata = df)
dataset = data.frame(country = df[,"country"], lda = plda$x)
ggplot(dataset) + geom_point(aes(lda.LD1, lda.LD2, colour = country))

CDA:

fvalue<-(df$also3/df$sio2)
svalue<-((2.39*df$feo)/(df$al2o3+df$sio2))
vvalue<-(df$sio2/df$feo)

mod <- lm(cbind(feo,sio2,al2o3) ~ country, data=df)
can2 <- candiscList(mod)
mod2 <- lm(cbind(fvalue,svalue,vvalue) ~ country, data=df)
can3 <- candiscList(mod2)
ggplot(can2$country$scores, aes(x=Can1,y=Can2)) + geom_point(aes(color=country))

Answer 1

These are two names for the same thing.

Linear discriminant analysis (LDA) is called a lot of different names. I have seen

• canonical discriminant analysis
• canonical linear discriminant analysis
• descriptive discriminant analysis (see What is "Descriptive Discriminant Analysis"?)
• Fisher's discriminant analysis

and possibly some others. I suspect different names might be used in different applied fields. In machine learning, "linear discriminant analysis" is by far the most standard term and "LDA" is a standard abbreviation.

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The reason for the term "canonical" is probably that LDA can be understood as a special case of canonical correlation analysis (CCA). Specifically, the "dimensionality reduction part" of LDA is equivalent to doing CCA between the data matrix $\mathbf X$ and the group indicator matrix $\mathbf G$. The indicator matrix $\mathbf G$ is a matrix with $n$ rows and $k$ columns with $G_{ij}=1$ if $i$-th data point belongs to class $j$ and zero otherwise. [Footnote: this $\mathbf G$ should not be centered.]

This fact is not at all obvious and has a proof, which this margin is too narrow to contain.