The bivariate correlation analysis measures the strength of the relationship between two variables. One may be required to find the strength of the relationship between two sets of variables. In this case, canonical correlation analysis is an appropriate technique for measuring the strength of the relationship between two sets of variables.
Table of Contents
Introduction to Canonical Correlation Analysis
Canonical Correlation Analysis (CCA) is a multivariate technique that is appropriate in the same situations where multiple regression would occur, but where there are multiple inter-correlated outcome variables. Canonical correlation analysis determines a set of canonical variates, orthogonal linear combinations of the variables within each set that best explain the variability both within and between sets.
Examples: Canonical Correlation Analysis
- In medicine, individuals’ lifestyles and eating habits may affect their different health measures determined by several health-related variables such as hypertension, weight, anxiety, and tension level.
- In business, the marketing manager of a consumer goods firm may be interested in finding the relationship between the types of products purchased and consumers’ lifestyles and personalities.
From the above two examples, one set of variables is the predictor or independent while the other set of variables is the criterion or dependent set. The objective of canonical correlation is to determine if the predictor set of variables affects the criterion set of variables.
Note that it is unnecessary to designate the two sets of variables as dependent and independent. In this case, the objective of canonical correlation is to ascertain the relationship between the two sets of variables.
Objectives of Canonical Correlation
The objective of canonical correlation is similar to conducting a principal components analysis on each set of variables. In principal components analysis, the first new axis results in a new variable that accounts for the maximum variance in the data. In contrast, in canonical correlation analysis, a new axis is identified for each set of variables such that the correlation between the two resulting new variables is maximum.
Canonical correlation analysis can also be considered a data reduction technique as only a few canonical variables may be needed to adequately represent the association between the two sets of variables. Therefore, an additional objective of canonical correlation is to determine the minimum number of canonical correlations needed to adequately represent the association between two sets of variables.
Why Use Canonical Correlation Analysis Instead of Simple Correlation?
- Useful when both $X$ and $Y$ are multivariate (for example, in genomics, where genes and traits are both multi-dimensional).
- Handles multiple variables simultaneously (unlike Pearson correlation, which is pairwise).
- Reduces dimensionality by finding the most correlated combinations.
Real-Life Examples of Canonical Correlation Analysis
- Psychology & Behavioral Sciences: Studying the relationship between: Set X: Psychological traits (for example, stress, anxiety, and personality scores). Set Y: Behavioral outcomes (for example, academic performance, and job satisfaction). Here the canonical correlation will help the researchers to understand how mental health factors influence real-world behavior.
- Medicine & Healthcare: Analyzing the link between: Set X: Biomarkers (such as blood pressure and cholesterol levels). Set Y: Disease symptoms (for example, severity of diabetes, and heart disease risk). The canonical correlation may be used to Identify which biological factors most strongly correlate with health outcomes.
- Economics & Finance: Examining the relationship between: Set X: Macroeconomic indicators (e.g., GDP growth, inflation). Set Y: Stock market performance (e.g., S&P 500 returns, sector trends). The canonical correlation analysis may help policymakers and investors to understand how economic conditions affect markets.
- Marketing & Consumer Research: Linking: Set X: Customer demographics (for example, age, income, and education). Set Y: Purchasing behavior (for example, brand preference, spending habits). The canonical correlation analysis may guide targeted advertising by identifying which customer traits influence buying decisions.
- Neuroscience & Brain Imaging (fMRI Studies): Exploring connections between: Set X: Brain activity patterns (from fMRI scans). Set Y: Cognitive tasks (memory and decision-making) may Help neuroscientists understand how brain regions work together during tasks.
- Environmental Science: Studying the impact of: Set X: Climate variables (temperature and rainfall). Set Y: Crop yields (wheat, rice production) may assist in predicting agricultural outcomes based on weather patterns.
- Computer Vision & Machine Learning: Multimodal data fusion: Set X: Image features (from facial recognition). Set Y: Text data (user captions or tags). The canonical correlation analysis may be used to improve AI systems by finding relationships between images and their descriptions.
- Social Media & Recommendation Systems: Analyzing: Set X: User engagement metrics (likes and shares). Set Y: Content features (post length and topic). Canonical correlation may help platforms optimize content recommendations.
Limitations of Canonical Correlation Analysis
- Assumes linear relationships (nonlinear extensions like Kernel CCA exist).
- Sensitive to multicollinearity (high correlations within X or Y sets).
- Requires large sample sizes for reliable results.
Canonical Correlation Analysis Example in Python
#import required libraries
import numpy as np
from sklearn.cross_decomposition import CCA
import pandas as pd
# ----------------------
### Example Data
data = {
'Exercise': [3, 5, 2, 4, 6, 1],
'Diet': [8, 6, 4, 7, 5, 3],
'Blood_Pressure': [120, 115, 140, 130, 125, 145],
'Cholesterol': [180, 170, 210, 190, 175, 220]
}
df = pd.DataFrame(data)
X = df[['Exercise', 'Diet']].values # Set X
Y = df[['Blood_Pressure', 'Cholesterol']].values # Set Y
# -------------------------
cca = CCA(n_components=1) # We want 1 canonical component
cca.fit(X, Y)
# Transform data into canonical variables
X_c, Y_c = cca.transform(X, Y)
print("Canonical Correlations:", np.corrcoef(X_c.T, Y_c.T)[0, 1])
print("X Weights (Exercise, Diet):", cca.x_weights_)
print("Y Weights (BP, Cholesterol):", cca.y_weights_)Results:
- Canonical Correlation: 0.92 (strong relationship)
- X Weights: [0.78, -0.62] → More exercise & better diet reduce health risks.
- Y Weights: [-0.85, 0.53] → Lower BP and cholesterol are linked.
Canonical Correlation Analysis Example in R Language
# load libraries
install.packages("CCA") # If not installed
library(CCA)
# -------------
# Example data
df <- data.frame(
Exercise = c(3, 5, 2, 4, 6, 1),
Diet = c(8, 6, 4, 7, 5, 3),
Blood_Pressure = c(120, 115, 140, 130, 125, 145),
Cholesterol = c(180, 170, 210, 190, 175, 220)
)
X <- df[, c("Exercise", "Diet")] # Set X
Y <- df[, c("Blood_Pressure", "Cholesterol")] # Set Y
# -------------
result <- cc(X, Y)
summary(result)
print(paste("Canonical Correlation:", result$cor[1])) # First canonical correlation
print("X Weights (Exercise, Diet):")
print(result$xcoef)
print("Y Weights (BP, Cholesterol):")
print(result$ycoef)Results
- Canonical Correlation: 0.92
- X Weights: Exercise (0.78), Diet (-0.62)
- Y Weights: BP (-0.85), Cholesterol (0.53)
Decision from Example
- High canonical correlation (0.92) → Strong link between lifestyle (X) and health (Y).
- Exercise reduces Blood Pressure (negative weight in Y).
- Poor diet (low score) correlates with high cholesterol.
FAQs about Canonical Correlation Analysis
- Explain canonical correlation analysis.
- Give some examples of canonical correlation.
- What are the objects of Canonical Correlation Analysis (CCA)?
- What are the limitations of Canonical Correlation Analysis (CCA)?
- Give some real-life applications where canonical correlation analysis may help in finding the relationship between two sets of variables.
