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Covariance (10 of 17) Covariance Matrix with 3 Data Sets (Part 1) By Michel van Biezen

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Covariance (10 of 17) Covariance Matrix with 3 Data Sets (Part 1) By Michel van Biezen


Summary:


  1. Introduction to Covariance Matrix for Three Data Sets:

    • The lecturer introduces the concept of a covariance matrix for three data sets (x, y, and z) instead of two.

  2. Calculation of Averages:

    • They explain that the diagonal elements of the covariance matrix represent the variances of each data set.
    • To calculate variances, the averages of each data set are needed. The averages of x, y, and z are computed.

  3. Calculation of Variances:

    • The equations for calculating variances for each data set are presented.
    • Each element is squared, summed, and divided by the number of elements to obtain the variances.
    • Variances for data sets x, y, and z are calculated as 8, 18, and 8 respectively.

  4. Structure of the Covariance Matrix:

    • The lecturer explains the structure of the covariance matrix, which is a 3x3 matrix.
    • Variances are placed along the diagonal, and covariances are placed in the off-diagonal elements.

  5. Explanation of Covariances:

    • Covariances represent the relationship between different data sets.
    • There are six off-diagonal elements, but only three unique covariances need to be calculated.
    • Covariances between x-y, x-z, and y-z are calculated, with duplicates omitted due to symmetry.

  6. Prediction of Covariance Relationships:

    • The lecturer predicts the expected signs of covariances based on the trend of data sets.
    • Positive covariance is expected between increasing data sets (x and y), while negative covariance is expected between increasing and decreasing data sets (x and z, y and z).

  7. Upcoming Calculation:

    • The lecturer mentions that in the next video, they will calculate all the covariances to complete the matrix.
    • They highlight the significance of understanding how covariances relate to each other and to the data sets.

  8. Conclusion:

    • The audience is encouraged to stay tuned for the next video to witness the completion of the covariance matrix and the verification of covariance relationships as predicted.

This summary provides a comprehensive breakdown of the key points discussed in the video regarding the structure and calculation of a covariance matrix for three data sets.