(14) 5. Consider a data set to daily log returns of 8 stocks in the American market in 2010-2011 (504 trading days); the stocks are: LO (Lorillard Tobacco), MO (Altria Tobacco), PM (Philip Morris Tobacco), RAI (Reynolds Tobacco), DPS (Dr Pepper Snapple Beverage), KO (Coca-Cola), MNST (Monster Beverage). PEP (Pepsico). Principal component analysis was applied to both the sample covariance matrix and sample correlation matrix. A summary of some results are below. stock LO MO PM RAI DPS KO MNST PEP sample mean 0.0009 0.0010 0.0011 0.0011 0.0007 0.0005 0.0017 0.0003 sample SD 0.0149 0.0100 0.0125 0.0119 0.0158 0.0104 0.0202 0.0101 row 1 -0.0272 0.0006 -0.0088 0.0014 -0.0116 -0.0122 0.0041 0.0120 sample covariance matrix | sample correlation matrix Importance of components: Compl Comp2 Comp3 Comp4 I Comp1 Comp2 Comp3 Comp4 SD 0.028 0.016 0.013 0.010 2. 102 0.913 0.832 0.802 PropofVar 0.515 0.175 0.117 0.071 0. 552 0. 104 0.087 0.080 CumProp 0.515 0.690 0.807 0.878 0. 552 0.657 0.743 0.824 Loadings: sample covariance matrix I sample correlation matrix Compl Comp2 Comp3 Comp4 Comp1 Comp2 Comp3 Comp4 LO -0.363 -0.389 0.383 0.689 -0.331 -0.504 -0.401 MO -0.274 -0.173 -0. 139 -0.395 -0.216 PM -0.337 -0.206 0.187 -0.347 | -0.381 -0.194 -0.102 0.163 RAI -0.335 -0.218 0.145 -0. 122 -0. 400 -0.250 DPS -0.362 -0.174 -0.879 0.223 -0. 285 0.469 0.600 -0.544 KO -0.278 -0. 102 -0. 402 | -0.384 0.133 0. 455 MNST -0.546 0.823 0. 126 | -0.271 0.536 -0.749 -0.257 PET -0.246 -0.105 -0.378 | -0.356 0.271 0.255 0.483 (a) How many components are needed to explain 80% of the variation in the data using the sample covariance matrix? (b) Interpret the first two components from the coefficients of the loadings for both sets of output. (c) The first observation in the data set is given in the above table along with the sample means and SDs of the 8 variables. For observation 1, what is the value of compl for the left-hand side (in the new coordinate system centred at the vector of sample means). Write down an expression without doing the calculation