Data Science in Theory and Practice. Maria Cristina Mariani

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      3.6.2 Linear Combinations of Sample Variance and Covariance

      The sample variance of z Subscript i Baseline equals bold a Superscript upper T Baseline bold x Subscript i Baseline comma i equals 1 comma 2 comma ellipsis comma n can be found as the sample variance of z 1 comma z 2 comma ellipsis comma z Subscript n Baseline or directly from bold a and bold upper S, where bold upper S is the sample covariance matrix of x 1 comma x 2 comma ellipsis comma x Subscript n Baseline:

      If we define another linear combination u equals bold b Superscript upper T Baseline bold upper X equals b 1 x 1 plus b 2 x 2 plus midline-horizontal-ellipsis plus b Subscript p Baseline x Subscript p Baseline, where b Superscript upper T Baseline equals left-parenthesis b 1 comma b 2 comma ellipsis comma b Subscript p Baseline right-parenthesis is a vector of constants different from bold a Superscript upper T, then the sample covariance of z equals bold a Superscript bold upper T Baseline bold upper X and u equals bold b Superscript bold upper T Baseline bold upper X is given by

      where n is the number of measurements.

      Please refer to Johnson and Wichern (2014) for the proof of (3.14).

      3.6.3 Linear Combinations of Sample Correlation

      The sample correlation between z equals bold a Superscript upper T Baseline bold upper X and u equals bold b Superscript upper T Baseline bold upper X is obtained as follows:

      We note that the sample results in Section 3.6 have population counterparts. We briefly state them below:

      The population mean of z equals bold a Superscript upper T Baseline bold upper X is defined as follows:

upper E left-parenthesis z right-parenthesis equals upper E left-parenthesis bold a Superscript upper T Baseline bold upper X right-parenthesis equals bold a Superscript upper T Baseline upper E left-parenthesis bold upper X right-parenthesis equals bold a Superscript upper T Baseline mu comma

      where mu denotes the population mean vector. The population variance of z is defined as follows:

sigma Subscript z Superscript 2 Baseline equals var left-parenthesis bold a Superscript upper T Baseline bold upper X right-parenthesis equals bold a Superscript upper T Baseline cov left-parenthesis bold upper X right-parenthesis bold a equals bold a Superscript upper T Baseline sigma-summation bold a comma

      where sigma-summation denotes the population covariance matrix which is defined in (3.5) as

bold sigma-summation equals cov left-parenthesis bold upper X right-parenthesis equals Start 4 By 4 Matrix 1st Row 1st Column sigma Subscript 1 comma 1 Baseline 2nd Column sigma Subscript 1 comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column sigma Subscript 1 comma p Baseline 2nd Row 1st Column sigma Subscript 2 comma 1 Baseline 2nd Column sigma Subscript 2 comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column sigma Subscript 2 comma p Baseline 3rd Row 1st Column vertical-ellipsis 
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