Читать книгу Data Science in Theory and Practice - Maria Cristina Mariani - Страница 35

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 matrix which is defined in (3.5) as

Let , where is a vector of constants different from . The population covariance of and is defined as

where denotes the population covariance matrix which is defined in (3.5).

Finally, the population correlation of and is defined as

where denotes the population covariance matrix which is defined in (3.5).

Remarks 3.1 If is a scalar matrix and are random vectors, then represents several linear combinations. The population mean vector and covariance matrix are given by

where denotes the population mean vector and denotes the population covariance matrix which is defined in (3.5).

The proof of Remark 3.1 is left as an exercise. Please see Problem 2.

Please refer to Johnson and Wichern (2014), Rencher (2002), and Axler (2002) and references therein for more details of multivariate analysis.