Introduction

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Canonical Correlation Analysis (CCA) explores relationships between two sets of variables.

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Unlike simple correlation or multiple regression, CCA examines how two groups of variables relate to each other simultaneously.

Essentially, CCA finds linear combinations of variables from each set that maximise the correlation between them. These linear combinations, called “canonical variates”, represent the strongest possible relationship between the two variable sets.

Example

Imagine you have two big boxes of children’s toys.

One box has cars, trucks, and planes, and the other box has dolls, action figures, and stuffed animals.

For some reason, you want to find out if there’s a connection between the toys in the first box and the toys in the second box. You notice that every time you pick a fast car from the first box, you also pick a superhero from the second box. That’s a pattern…

Canonical correlation is like a game where you match up groups of toys from each box to see which ones go together the best.

First, you find the strongest match, like the car and superhero combo. Then, you look for the next-best match, but with a rule: the new pair has to be totally different from the first one, like a slow truck with a cuddly teddy bear. This way, each pair of toys shows a different kind of connection between the boxes.

So, canonical correlation helps us find these "best pairs" between two groups of things, showing us hidden patterns we might not have noticed just by looking at the toys one by one.

The process of canonical correlation

Step 1: Two sets of variables

• We start with two different datasets (but they are related).
• Each dataset contains three variables.
  • Set 1 (X variables): Three physiological measures
    • $X_1$ = Blood Pressure
    • $X_2$ = Heart Rate
    • $X_3$ = Oxygen Saturation
  • Set 2 (Y variables): Three psychological scores
    • $Y_1$ = Stress Score
    • $Y_2$ = Anxiety Score
    • $Y_3$ = Depression Score

Step 2: Goal of CCA

• We want to find relationships between the two sets of variables.
• But instead of looking at correlations one by one (e.g., $X_1$ vs $Y_1$), CCA creates weighted combinations of variables from both sets that maximise correlation.