Читать книгу Data Science in Theory and Practice - Maria Cristina Mariani - Страница 36

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).