1. From the calculated correlation matrix for all four variables given below, we can see that the first highest (in absolute value) correlation coefficient is between the variables ozone and temperature with the value 0.6985414, and the second highest is between ozone and wind with the absolute value 0.6129508.
ozone radiation temperature wind
ozone 1.0000000 0.3483417 0.6985414 -0.6129508
radiation 0.3483417 1.0000000 0.2940876 -0.1273656
temperature 0.6985414 0.2940876 1.0000000 -0.4971459
wind -0.6129508 -0.1273656 -0.4971459 1.0000000


2. The standard deviations of the scores are 1.5363, 0.9458, 0.6898, 0.5191. The calculated loadings of the principal components are given in the following table:
PC1 PC2 PC3 PC4
ozone 0.5890271 -0.0630412 0.1137638 0.7975780
radiation 0.3168987 0.8985548 -0.2773707 -0.1234503
temperature 0.5527125 -0.0612848 0.6585842 -0.5069713
wind -0.4971228 0.4299643 0.6902102 0.3026705


3. The sum of squares of the loadings of each principal component are all equal to 1 as shown here:


4. The two largest loadings in absolute value of the first principle component are for the variables ozone and temperature with the values 0.5890271 and 0.5527125 respectively. This corresponds with the correlation analysis of the first question because with that we had found that the variables ozone and temperature had the highest correlation coefficient.


5. The largest loading in absolute value of the second principle component is for the variable radiation with the value of 0.8985548.


6. We can see in the Importance table output below, the first principal component explains about 59% of the variance, and the fourth principal component explains about 6.7% of the variance.


7. No, it is not quite enough to use the first two principal components if we want to explain at least 90% of the total variance because we can easily see from the summary output table above in problem 6 that the cumulative proportion of the first two principal components only explains about 81.4% of the variance.


8. From the figure below, we see that the first principal component captures the variables ozone and temperature, so the first principal component is a linear combination of ozone and temperature. Additionally, the variables ozone and temperature appear to be highly correlated because their lines in the biplot seem to overlap (they appear to be almost the same line). Then, we see that the second principal component captures the variables radiation and wind. Also, the variables radiation and wind seem to be more or less independent from the variables ozone and temperature.