Correlation and covariance in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Calculate covariance matrix output

What is the output of the following MATLAB code snippet that calculates the covariance matrix of two variables?

MATLAB

X = [1 2 3 4 5; 2 4 6 8 10];
covMatrix = cov(X');
disp(covMatrix);

A [2.5 2.5; 5 5]

B [2.5 5; 5 10]

C [2.5 5; 2.5 5]

D [2.5 10; 5 20]

💡 Hint

Remember that cov(X') computes covariance between rows of X, so transpose is important.

The covariance matrix shows how two variables vary together. Here, the first variable is [1 2 3 4 5] and the second is [2 4 6 8 10]. The covariance matrix is symmetric and the off-diagonal elements represent covariance between the two variables.

❓ Predict Output

intermediate

Correlation coefficient calculation

What is the output of this MATLAB code that calculates the correlation coefficient between two vectors?

MATLAB

A = [1 2 3 4 5];
B = [2 4 6 8 10];
r = corrcoef(A, B);
disp(r(1,2));

A 0.5

B -1

C 0

D 1

💡 Hint

Check if B is a perfect linear multiple of A.

Since B is exactly 2 times A, the correlation coefficient is 1, indicating perfect positive linear correlation.

🔧 Debug

advanced

Identify the error in covariance calculation

What error does this MATLAB code produce when trying to calculate covariance?

MATLAB

X = [1 2 3 4 5; 2 4 6 8 10];
covMatrix = cov(X);
disp(covMatrix);

A Outputs a 5x5 covariance matrix

B No error, outputs a 2x2 covariance matrix

C Error: Undefined function 'cov' for input arguments of type 'double'

D Error: Inputs must be vectors or matrices with observations in rows

💡 Hint

Check how cov treats rows and columns as observations or variables.

In MATLAB, cov treats rows as observations and columns as variables. Since X has 2 rows and 5 columns, cov(X) treats 2 observations of 5 variables, resulting in a 5x5 covariance matrix.

🧠 Conceptual

advanced

Interpret correlation matrix output

Given the MATLAB code below, what does the output matrix represent?

MATLAB

data = [1 2 3; 4 5 6; 7 8 9];
corrMatrix = corrcoef(data);
disp(corrMatrix);

A Correlation coefficients between columns of data

B Correlation coefficients between rows of data

C Covariance matrix of rows of data

D Covariance matrix of columns of data

💡 Hint

corrcoef by default computes correlation between columns.

corrcoef computes the correlation coefficients between columns of the input matrix. Here, data has 3 columns, so the output is a 3x3 matrix showing correlations between these columns.

❓ Predict Output

expert

Covariance matrix with missing data

What is the output of this MATLAB code that calculates covariance with missing data (NaN)?

MATLAB

X = [1 NaN 3 4 5; 2 4 NaN 8 10];
covMatrix = cov(X', 'partialrows');
disp(covMatrix);

A [NaN NaN; NaN NaN]

B [2.5 5; 5 10]

C [2.9167 5.8333; 5.8333 14.3333]

D Error: Invalid input argument 'partialrows'

💡 Hint

Check how cov handles NaN values with 'partialrows' option.

The 'partialrows' option tells cov to use all available pairs of data ignoring NaNs. This results in a covariance matrix computed from partial data, giving the shown values.

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