Total Variation in Data - Overall Importance of the Principal Components

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#1
As the system was not accepting spaces I gave comma instead of space.

Table 6.1 indicates the overall importance of the principal components. It
displays for each component ph the standard deviation, given by
√λh, the proportion of variance explained by each component, equal to λh/j λj, and the cumulative proportion explained by the first components up to ph included.The analysis shows that the first two components explain 84% of the total variation in the data while the first 5 explain 94%.

Table 6.1 Overall importance of principal components in the mtcars dataset
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Comp.1,,,,,,,,,Comp.2,,,,,,Comp.3,,,,,Comp.4
Standard deviation,,,,,,2.5706809,,,,1.6280258,,,0.7919578,,0.5192277
Proportion of variance,,0.6007637,,,,0.2409516,,,0.0570179,,0.0245088
Cumulative proportion,,0.6007637,,,,0.8417153,,,0.8987332,,0.9232420

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Comp.5,,,,,,,,,Comp.6,,,,,,Comp.7,,,,,,Comp.8
Standard deviation,,,,,,0.4727061,,,0.4599957,,,,0.3677798,,,0.3505730
Proportion of variance,,0.0203137,,,0.0192360,,,,0.0122965,,,0.0111728
Cumulative proportion,,0.9435558,,,0.9627918,,,0.9750883,,,0.9862612

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Comp.9,,,,,,,,,Comp.10,,,,Comp.11
Standard deviation,,,,,,,0.2775727,,,,0.2281127,,,0.1484735
Proportion of variance,,,0.0070042,,,,0.0047304,,,0.0020040
Cumulative proportion,,,0.9932654,,,,,,0.9979959,,,,1.0000000



Can someone please the formula for total variation in data, how did we get the values 84% and 94%

Table 6.2 shows the values of the coefficients whj = uhj ; equivalently, it
indicates the eigenvectors of the covariance matrix V. The first component,
which alone explains 60% of the variance, is negatively correlated with the
attributes {cyl, disp,wt, carb}, whose meaning is explained in Appendix B,
while it is positively correlated with all the other attributes.

Table 6.2 Principal component coefficients for the mtcars dataset
,,,,,,,,,,Comp.1,,,,,,,Comp.2,,,,Comp.3,,,,Comp.4,,,Comp.5,,,,Comp.6
mpg,,,,,0.363,,,,,,−0.226,,,,,,−0.103,,,,,−0.109,,,,,0.368,,,,,,,0.754
cyl,,,,,−0.374,,,,,,−0.175,,,,,,,,0.169,,,,,,,0.231,,,−0.846
disp,, −0.368,,,,,,,,,0.257,,,,,,−0.394,,,,,−0.336,,,,,,0.214,,,,,,,0.198
hp,,,,,,−0.33,,,,,,,,−0.249,,,,,,,,0.14,,,,,,,−0.54,,,,,,,0.222,,,,,−0.576
drat,,,,,,0.294,,,,,,,−0.275,,,,,,,0.161,,,,,,,,0.855,,,,,0.244,,,,,,−0.101
wt,,,,,,−0.346,,,,,,,,,0.143,,,,,,,0.342,,,,,,,0.246,,,,,−0.465,,,,,, 0.359
qsec,,,,,,0.2,,,,,,,,,,,,0.463,,,,,,,0.403,,,,,,,0.165,,,,−0.33,,,,,,,,0.232
vs,,,,,,,,,0.307,,,,,,,,,0.232,,,,,,,0.429,,,,,,,−0.215,,−0.6,,,,,,,,,0.194
am,,,,,,,0.235,,,,,,,,−0.429,,,,,,−0.206,,,,,,−0.571,,−0.587,,,,,−0.178
gear,,,,,,0.207,,,,,,,,−0.462,,,,,,,,0.29,,,,,,,−0.265,,−0.244,,,,,,,0.605
carb,,,,−0.214,,,,,,,,−0.414,,,,,,,0.529,,,,,,−0.127,,,,0.361,,,,,,,,0.184

,,,,,,,,,,,Comp.7,,,,,Comp.8,,,,,Comp.9,,,,,Comp.10,,,Comp.11
mpg,,,,,0.236,,,,,,,,,0.139,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0.125
cyl,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0.141
disp,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,−0.661
hp,,,,,,,0.248,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0.256
drat
wt,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0.567
qsec,,,,,−0.528,,,,, −0.271,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,−0.181
vs,,,,,,,,−0.266,,,,,,,,0.359,,,,, −0.159
am
gear,,,,,−0.336,,,,,,−0.214
carb,,,,,−0.175,,,,,,,,,,0.396,,,,,,,,0.171,,,,,,,,,,,,,,,,,,,,−0.32

Can anyone please explain how the corelation is calculated like how did we get the value 60%

Thanks in advance
 
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#2
This is pretty advanced statistics for this forum. I recommend asking this question on Talkstats.com. Talkstats is a statistics discussion forum, and I have seen PCA and Factor Analysis discussed periodically.
 
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