WEBVTT
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If our objective is to keep the
mean squared estimation
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error small, then the best
possible estimator is the
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conditional expectation.
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But sometimes the conditional
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expectation is hard to calculate.
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Maybe we're missing the
details of the various
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probability distributions.
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Or maybe we have the
distributions that we need but
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the formulas are complicated.
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After all, the conditional
expectation can be a
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complicated non-linear function
of the observations.
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For this reason, we may want to
consider an estimator that
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has a simpler structure, an
estimator that is a linear
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function of the data.
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And then, within this class of
estimators, find the one that
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results in the smallest possible
mean squared error.
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In this lecture we will
formulate this linear least
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squares estimation problem
and then solve it.
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We will see that the solution
is given by a simple formula
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that involves only the means,
variances, and covariances of
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the random variables involved.
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Because of the simplicity of the
method, linear estimators
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are used quite often, especially
in systems where
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estimates need to be computed
quickly in real time as
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observations are obtained.
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We will look into some of the
mathematical properties of the
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linear least mean squares
estimator and the associated
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mean squared error, revisit an
example from the previous
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lecture, and finally close with
some comments on the ways
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that this estimator
can be used.