Data Science in Theory and Practice. Maria Cristina Mariani

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.

3.7 Problems

      1 The following are four measurements on the variables and :91251600143587Use the above information to answer the following questions:Find the sample mean vector, .Find the sample covariance matrix, .Find the sample correlation matrix, .

      2 For random vectors and , scalar matrices and , and scalar vectors and , prove the following:..., where denotes the population covariance matrix which is defined in (3.5)...

      3 Consider the five pairs of measurements :3426855.54410CalculateThe sample means and .The sample variances and .The sample covariances

      4 Consider the data matrix:Calculate the matrix of deviations (residuals), . Is this matrix of full rank?Note: A matrix is of full rank if all rows and columns are linearly independent. A square matrix is full rank if and only if its determinant is nonzero.

      5 Calculate the sample covariance matrix using the data matrix in Problem 4.

      6 Consider the data matrix:Obtain the mean corrected data matrix, sample covariance matrix and verify that the columns are linearly dependent.

      7 If for , show that , where is a constant.

      8 If for , show that , where is a constant.

      9 The data in Table 3.1 (Elston and Grizzle 1962) consist of measurements , and of the ramus bone at four different ages on each of 20 boys.Find .Find .Find .Table 3.1 Ramus Bone Length at Four Ages for 20 Boys.AgeIndividual147.848.849.049.7246.447.347.748.4346.346.847.848.5445.145.346.147.2547.648.548.949.3652.553.253.353.7751.253.054.354.5849.850.050.352.7948.150.852.354.41045.047.047.348.31151.251.451.651.91248.549.253.055.51352.152.853.755.01448.248.949.349.81549.650.451.251.81650.751.752.753.31747.247.748.449.51853.354.655.155.31946.247.548.148.42046.347.651.351.8

      10 For the data in Table 3.1, define and .Find , , , and using (3.12) and (3.13).Find and using (3.14) and (3.15).