First Principal Component
Forrest Young's Notes
Copyright © 1999 by Forrest W. Young.
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The First Principal Component
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The first principal component is the linear
function of the set of variables which fits the variables as well
as possible in a least squares sense. The linear combination has
certain properties:
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The first principal component line is as close as possible, in a specific
average
least squares sense, to all of the points.
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The first principal component line identifies the "central tendency"
of the set of variables, just as the mean identifies the "central tendency"
of a single variable.
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The first principal component line provides a simplified description ---
a model --- of the set of variables.
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The first principal component line gives us a way to summarize the
set of variables by a single linear combination.
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Equation for the first Principal Component
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The n variables, denoted
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Y1, Y2,
... Yn
are described by the following linear equation, where X1
is
the vector of scores on the first principal component, and b1
is the coefficient of the first principal component:
Y1, Y2,
... Yn = a + b1X1
+
r
where r is the "residual" information in
the Y's not fit by the component's linear combination.
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