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Due to the large number of variables in the dataset, it is hard to comprehend all of
the relationships between the variables using a scatter plot or correlation matrix. Using a
data reduction technique such as principal components analysis (PCA) reduces the
dimensionality of the dataset whilst retaining as much of the variability in the data as
possible. The first few principal components retain most of the variation in the original
variables, and, to make interpretation simpler, they can be used to describe the
relationships between the original variables and similarities between observations.
a mathematical technique that reduces dimensionality by creating a new set of variables
called principal components. The first principal component is a linear combination of
the original variables and explains as much variation as possible in the original data.
Each subsequent component explains as much of the remaining variation as possible under
the condition that it is uncorrelated with the previous components.
The variances table shows the amount of variance in the original data explained
by each principal component (also called the eigenvalues). Because the data was
standardized, a principal component with a variance of 1 indicates that the
component accounts for variation equivalent to one of the original variables. Also,
the sum of all the variances equals the number of original variables.
There are many ad-hoc rules regarding the number of components to retain to
adequately describe the data. According to the table, the first two principal
components account for nearly 70% of the variance in the original 12 variables,
whilst the first three components account for nearly 80%.
table shows the linear combinations that make each principal component, and the
color map shows the structure of the components. Absolute values near zero indicate
that a variable contributes little to the component, whereas larger absolute values
indicate variables that contribute more to the component. The sign of the
coefficients is irrelevant and may even differ when the analysis is performed on
There is not necessarily a simple interpretable structure to the principal
components because they are created to maximize the amount of variance whilst
remaining uncorrelated with the other components. By trying to interpret the
coefficients in the table, we can see that the first component is an average of many
different variables; the second component represents mainly crime, wellness, and, to
a lesser extent, schools and housing quality; and the third component – although it
still has some reasonable sized contribution from other variables – represents
mainly green space.