Covariance (14 of 17) Covariance Matrix "Normalized" - Correlation Coefficient By Michel van Biezen

Summary:

1. Introduction to Normalizing Covariance Matrix:

   • The lecturer introduces the concept of normalizing a covariance matrix derived from sample data sets in variables x, y, and z.
   • They explain the intention to find correlation coefficients for the variances themselves, rather than just for the covariance.

2. Variance Calculation:

   • Variance for each variable (x, y, z) is computed.
   • Variance of x is 1.67, variance of y is 2, and variance of z is 0.9167.

3. Correlation Coefficient Calculation:

   • The lecturer finds the correlation coefficients for each variable's variance.
   • When comparing each variable to itself, correlation coefficients are 1, as expected.
   • Correlation coefficients for off-diagonal elements are calculated using covariance divided by the square root of the product of variances.

4. Calculation of Correlation Coefficients:

   • The correlation coefficient between x and y is found to be negative 0.547, indicating a weak negative correlation.
   • The correlation coefficient between x and z is negative 0.404, showing a relatively weaker negative correlation.
   • For y and z, the correlation coefficient is 0.246, suggesting a very weak negative correlation, almost random.

5. Interpretation of Results:

   • The lecturer explains that negative correlation coefficients indicate one variable increasing while the other decreases.
   • The strength of correlation varies, with x and y showing a slightly stronger negative correlation compared to x and z.
   • The correlation between y and z is almost negligible, suggesting a near-random relationship between the two variables.

6. Summary of Normalized Covariance Matrix:

   • The lecturer concludes by presenting the normalized covariance matrix.
   • The diagonals contain ones, representing perfect correlation of each variable with itself.
   • Off-diagonal elements are filled with correlation coefficients, indicating the strength and direction of the relationship between variables.

7. Final Remarks:

   • The lecturer emphasizes the importance of understanding correlation coefficients in analyzing relationships between variables.
   • They highlight the significance of the normalized covariance matrix in visualizing and interpreting correlations in data sets.

In essence, the lecture demonstrates the process of normalizing a covariance matrix and deriving correlation coefficients to better understand the relationships between different variables in a data set.