Data Science in Theory and Practice. Maria Cristina Mariani

can be found either by averaging the values or as a linear combination of , the sample mean vector of .

(3.12)

      3.6.2 Linear Combinations of Sample Variance and Covariance

      The sample variance of can be found as the sample variance of or directly from and , where is the sample covariance matrix of :

(3.13)

We recall from Section 2.3 that variance is always nonnegative. Thus, we have , and therefore, , for every .

      If we define another linear combination , where is a vector of constants different from , then the sample covariance of and is given by

(3.14)

      where 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 and is obtained as follows:

(3.15)

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

      The population mean of is defined as follows:

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

      where denotes the population covariance