We return to the topic of classification, and we assume an input
(feature) space
$\mathcal{X}$ and a binary output (label) space
$\mathcal{Y}=\{0,1\}$ . Recall that the Bayes classifier (which minimizes the
probability of misclassification) is defined by
One way to construct a classifier using the training data
${\{{X}_{i},{Y}_{i}\}}_{\phantom{\rule{4pt}{0ex}}i=1}^{n}$ is to estimate
$\eta \left(x\right)$ and then plug-it
into the form of the Bayes classifier. That is obtain an estimate,
Therefore, in this sense plug-in methods are solving a more complicated
problem than necessary. However, plug-in methods can perform well,as demonstrated by the next result.
Theorem
Plug-in classifier
Let
$\tilde{\eta}$ be an approximation to
$\eta $ , and consider the plug-in
rule
and the second inequality is simply a result of the fact that
${\mathbf{1}}_{\{{f}^{*}\left(x\right)\ne f\left(x\right)\}}$ is either 0 or 1.
The theorem shows us that a good estimate of
$\eta $ can produce a good
plug-in classification rule. By “good" estimate, we mean an estimator
$\tilde{\eta}$ that is close to
$\eta $ in expected
${L}_{1}\text{-norm}$ .
The histogram classifier
Let's assume that the (input) features are randomly distributed over
theunit hypercube
$\mathcal{X}={[0,1]}^{d}$ (note that by scaling and
shifting any set of bounded features we can satisfy this assumption),and assume that the (output) labels are binary, i.e.,
$\mathcal{Y}=\{0,1\}$ . A histogram classifier is based on a partition the hypercube
${[0,1]}^{d}$ into
$M$ smaller cubes of equal size.
Partition of hypercube in 2 dimensions
Consider the unit square
${[0,1]}^{2}$ and partition it into
$M$ subsquares of equal area (assuming
$M$ is a squared integer). Let
the subsquares be denoted by
$\left\{{Q}_{i}\right\},\phantom{\rule{4pt}{0ex}}i=1,...,M$ .
Define the following
piecewise-constant estimator of
$\eta \left(x\right)$ :
Like our previous denoising examples, we expect that the bias of
${\widehat{\eta}}_{n}$ will decrease as
$M$ increases, but the variance will
increase as
$M$ increases.
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Rafiq
Rafiq
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Mahi
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How this robot is carried to required site of body cell.?
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Rafiq
Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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