Proportion of total sample variance

Question 1 You are given a sample:
x1 : 3 1 2 0 −1 −2 −3
x2 : 2 4 6 0 −6 −2 −4.
(a) Give the formula for the first and second sample principal components ˆy1 and ˆy2;
(b) Determine the proportion of total sample variance due to the first sample principal component;
(c) Compare the contributions of the two variates to the determination of the first sample principal component based on loadings;
(d) Compare the contributions of the two variates to the determination of the first sample principal component based on sample correlations;
(e) Redo (a)-(d) on the standardized dataset.
Question 2 If the first principal component of X1 and X2 is
Y1 =
√
2
2
X1 −
√
2
2
X2,
is it possible that Corr(X1, X2) > 0? Explain.
Question 3 Problem 8.12 on Page 474.
Question 4 Consider two samples with equal sizes n1 = n2:
~x11, . . . , ~x1n1
and
~x21, . . . , ~x2n2
with summary statistics
~x1 =

6
0

, S1 =

4 2
2 5
,
~x2 =

0
2

, S2 =

4 2
2 3
.
For a new observation ~x0 =

x01
x02
, consider the following classifiers:

1. Classifier 1: Fisher’s rule if only x01 is observed;
2. Classifier 2: Fisher’s rule if only x02 is observed;
3. Classifier 3: Fisher’s rule based on ~x0.
   Does there exist a ~x0, such that Classifier 1 and Classifier 2 give consensus assignment, while Classifier 3
   gives a different assignment?
   1
   Question 5 Redo Question 4 with the following summary statistics:
   ~x1 =
   
   6
   2
   
   , S1 =
   
   4 2
   2 5
   ,
   ~x2 =
   
   0
   0
   
   , S2 =
   
   4 2
   2 3
   .
   Question 6 For the dataset on Table 1.6 (Page 42), construct Fisher’s rule. Moreover, calculate the
   apparent error rate as well as the expected actual error rate by Lachenbruch’s holdout.