Correlation and covariance in MATLAB - Mini Project: Build & Apply

Correlation and covariance

📖 Scenario: You are analyzing the relationship between two sets of data collected from a small experiment. You want to understand how the two variables move together and how strong their relationship is.

🎯 Goal: Build a MATLAB script that calculates the covariance and correlation between two data vectors.

📋 What You'll Learn

Create two vectors with exact numeric values

Create a variable to hold the number of data points

Calculate covariance using MATLAB's built-in function

Calculate correlation using MATLAB's built-in function

Display the covariance and correlation values

💡 Why This Matters

🌍 Real World

Covariance and correlation help understand relationships between variables in science, finance, and engineering.

💼 Career

Data analysts and engineers use these concepts to find patterns and make predictions.

1

Create data vectors

Create two vectors called dataX and dataY with these exact values: dataX = [2, 4, 6, 8, 10] and dataY = [1, 3, 5, 7, 9].

MATLAB

% Your code here

Hint

Use square brackets to create vectors in MATLAB.

2

Set number of data points

Create a variable called n that stores the number of elements in dataX using the length function.

MATLAB

dataX = [2, 4, 6, 8, 10];
dataY = [1, 3, 5, 7, 9];
% Your code here

Hint

Use length(dataX) to find how many elements are in dataX.

3

Calculate covariance and correlation

Calculate the covariance matrix of dataX and dataY using cov([dataX; dataY]') and store it in covMatrix. Then calculate the correlation matrix using corrcoef(dataX, dataY) and store it in corrMatrix.

MATLAB

dataX = [2, 4, 6, 8, 10];
dataY = [1, 3, 5, 7, 9];
n = length(dataX);
% Your code here

Hint

Use MATLAB's cov and corrcoef functions to calculate covariance and correlation matrices.

4

Display covariance and correlation

Use disp to print the text 'Covariance matrix:' and then display covMatrix. Then print 'Correlation matrix:' and display corrMatrix.

MATLAB

dataX = [2, 4, 6, 8, 10];
dataY = [1, 3, 5, 7, 9];
n = length(dataX);
covMatrix = cov([dataX; dataY]');
corrMatrix = corrcoef(dataX, dataY);
% Your code here

Hint

Use disp twice to show the labels and matrices.

Practice

1.

What does the MATLAB function cov compute?

easy

A. Standard deviation of a variable

B. Covariance matrix showing how two variables vary together

C. Mean value of a dataset

D. Correlation coefficient between two variables

Solution

1. Step 1: Understand the purpose of cov

   The cov function calculates the covariance matrix, which shows how two variables change together.

2. Step 2: Differentiate covariance from correlation

   Correlation measures strength and direction, but cov specifically returns covariance values.

3. Final Answer:

   Covariance matrix showing how two variables vary together -> Option B

4. Quick Check:

   cov = covariance matrix [OK]

Hint: Remember: cov = covariance matrix, corrcoef = correlation [OK]

Common Mistakes:

• Confusing covariance with correlation
• Thinking cov returns a single value instead of a matrix
• Mixing up cov with mean or std functions

2.

Which of the following is the correct syntax to calculate the correlation matrix of variables X and Y in MATLAB?

?

easy

A. R = cov(X, Y);

B. R = correlation(X, Y);

C. R = corrcoef(X, Y);

D. R = corr(X, Y);

Solution

1. Step 1: Identify MATLAB function for correlation

   The correct MATLAB function to compute correlation coefficients is corrcoef.

2. Step 2: Check syntax correctness

   The syntax R = corrcoef(X, Y); correctly computes the correlation matrix between X and Y.

3. Final Answer:

   R = corrcoef(X, Y); -> Option C

4. Quick Check:

   corrcoef syntax = R = corrcoef(X, Y) [OK]

Hint: Use corrcoef for correlation, not cov [OK]

Common Mistakes:

• Using cov instead of corrcoef for correlation
• Using non-existent functions like correlation or corr
• Missing parentheses or semicolon

3.

What is the output of the following MATLAB code?

X = [1 2 3];
Y = [4 5 6];
R = corrcoef(X, Y);
disp(R);

medium

A. [[1 0]; [0 1]]

B. [[1 0.5]; [0.5 1]]

C. [[1 -1]; [-1 1]]

D. [[1 1]; [1 1]]

Solution

1. Step 1: Understand the data vectors

   Vectors X and Y are perfectly linearly related: Y = X + 3.

2. Step 2: Calculate correlation coefficient

   Since they have a perfect positive linear relationship, correlation coefficient is 1.

3. Final Answer:

   [[1 1]; [1 1]] -> Option D

4. Quick Check:

   Perfect positive correlation = 1 [OK]

Hint: Perfect linear relation means correlation = 1 [OK]

Common Mistakes:

• Assuming correlation less than 1 for linear relation
• Confusing covariance with correlation values
• Misreading matrix output format

4.

Identify the error in this MATLAB code snippet:

X = [1 2 3];
Y = [4 5 6];
R = corrcoef(X Y);
disp(R);

medium

A. Missing comma between X and Y in corrcoef function

B. Vectors X and Y must be column vectors

C. corrcoef cannot take two inputs

D. disp cannot display matrices

Solution

1. Step 1: Check function call syntax

   The function call corrcoef(X Y) is missing a comma between X and Y.

2. Step 2: Confirm correct syntax

   The correct syntax is corrcoef(X, Y) with a comma separating inputs.

3. Final Answer:

   Missing comma between X and Y in corrcoef function -> Option A

4. Quick Check:

   Function arguments need commas [OK]

Hint: Separate function arguments with commas [OK]

Common Mistakes:

• Omitting commas between function arguments
• Thinking vectors must be columns for corrcoef
• Believing disp cannot show matrices

5.

You have two datasets A = [1 2 3 4] and B = [2 4 6 8]. You want to check if B is exactly twice A using correlation and covariance. Which MATLAB code snippet correctly confirms this?

hard

A. covAB = cov(A, B); corrAB = corrcoef(A, B); disp(covAB(2,1)); disp(corrAB(2,1));

B. covAB = cov(A, B); corrAB = corrcoef(A, B); disp(covAB); disp(corrAB);

C. covAB = cov(A, B); corrAB = corrcoef(A, B); disp(covAB(1,2)); disp(corrAB(1,2));

D. covAB = cov(A, B); corrAB = corrcoef(A, B); disp(covAB(1,1)); disp(corrAB(1,1));

Solution

1. Step 1: Calculate covariance and correlation matrices

   Using cov(A, B) and corrcoef(A, B) returns 2x2 matrices with variances/covariances and correlations.

2. Step 2: Extract covariance and correlation between A and B

   The off-diagonal elements (1,2) or (2,1) give covariance and correlation between A and B.

3. Step 3: Confirm correct display of these values

   covAB = cov(A, B); corrAB = corrcoef(A, B); disp(covAB(2,1)); disp(corrAB(2,1)); correctly displays covAB(2,1) and corrAB(2,1), confirming the relationship.

4. Final Answer:

   covAB = cov(A, B); corrAB = corrcoef(A, B); disp(covAB(2,1)); disp(corrAB(2,1)); -> Option A

5. Quick Check:

   Use off-diagonal elements for covariance and correlation [OK]

Hint: Off-diagonal matrix elements show variable relationships [OK]

Common Mistakes:

• Displaying entire matrix instead of specific covariance/correlation
• Using diagonal elements which are variances
• Confusing matrix indices