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Covariance (11 of 17) Covariance Matrix with 3 Data Sets (Part 2) By Michel van Biezen

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Covariance (11 of 17) Covariance Matrix with 3 Data Sets (Part 2) By Michel van Biezen


Summary:


  1. Introduction to Covariance Calculation:

    • The lecturer introduces the topic of finding covariances between different data sets to complete the covariance matrix.

  2. Recap of Previous Video:

    • They recap that in the previous video, variances of three data sets (x, y, and z) were determined, which now serve as the diagonal elements of the covariance matrix.

  3. Definition of Covariance:

    • Covariance between data sets x and y is explained as the sum of products of differences between each element and the average of each dataset.
    • The formula for covariance between x and y is applied to calculate the covariance value, which turns out to be 12.

  4. Calculation of Covariances:

    • Similar calculations are performed to find the covariance between x and z and between y and z.
    • Covariance between x and z yields a value of -8, while covariance between y and z results in -12.

  5. Interpretation of Covariance:

    • Negative covariance indicates one dataset increasing while the other decreases.
    • The lecturer explains that while covariance size indicates correlation strength, it can be difficult to interpret solely based on magnitude.

  6. Completion of Covariance Matrix:

    • Using the calculated covariances, the lecturer fills in the covariance matrix, ensuring symmetry across the diagonal.
    • Covariance values are placed accordingly: 12 for x-y and y-x, -8 for x-z and z-x, and -12 for y-z and z-y.

  7. Overview of Covariance Matrix:

    • The completed covariance matrix is described as a 3x3 matrix.
    • Diagonal elements represent variances, while off-diagonal elements represent covariances.
    • The lecturer notes the symmetry of covariances across the diagonal.

  8. Conclusion:

    • The lecturer concludes by affirming the completion of the covariance matrix for three data sets and hints at calculating correlation coefficients in the next video.

This summary provides a comprehensive overview of the key points discussed in the video regarding the calculation and interpretation of covariances between different data sets to form a covariance matrix.