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Calculate Correlation Matrix

Write a Python function to calculate the correlation matrix for a given dataset. The function should take in a 2D numpy array X and an optional 2D numpy array Y. If Y is not provided, the function should calculate the correlation matrix of X with itself. It should return the correlation matrix as a 2D numpy array.

Example

Example:
    X = np.array([[1, 2],
                  [3, 4],
                  [5, 6]])
    output = calculate_correlation_matrix(X)
    print(output)
    # Output:
    # [[1. 1.]
    #  [1. 1.]]
    
    Reasoning:
    The function calculates the correlation matrix for the dataset X. In this example, the correlation between the two features is 1, indicating a perfect linear relationship.

Understanding Correlation Matrix

A correlation matrix is a table showing the correlation coefficients between variables. Each cell in the table shows the correlation between two variables. The value is between -1 and 1, indicating the strength and direction of the linear relationship between the variables.

The correlation coefficient between two variables \(X\) and \(Y\) is given by:

\[ \text{corr}(X, Y) = \frac{\text{cov}(X, Y)}{\sigma_X \sigma_Y} \]

Where:

  • \(\text{cov}(X, Y)\) is the covariance between \(X\) and \(Y\).
  • \(\sigma_X\) and \(\sigma_Y\) are the standard deviations of \(X\) and \(Y\), respectively.

In this problem, you will write a function to calculate the correlation matrix for a given dataset. The function will take in a 2D numpy array \(X\) and an optional 2D numpy array \(Y\). If \(Y\) is not provided, the function will calculate the correlation matrix of \(X\) with itself.

import numpy as np

def calculate_correlation_matrix(X, Y=None):
    # Helper function to calculate standard deviation
    def calculate_std_dev(A):
        return np.sqrt(np.mean((A - A.mean(0))**2, axis=0))
    
    if Y is None:
        Y = X
    n_samples = np.shape(X)[0]
    # Calculate the covariance matrix
    covariance = (1 / n_samples) * (X - X.mean(0)).T.dot(Y - Y.mean(0))
    # Calculate the standard deviations
    std_dev_X = np.expand_dims(calculate_std_dev(X), 1)
    std_dev_y = np.expand_dims(calculate_std_dev(Y), 1)
    # Calculate the correlation matrix
    correlation_matrix = np.divide(covariance, std_dev_X.dot(std_dev_y.T))

    return np.array(correlation_matrix, dtype=float)

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