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.
Questions & Answers
where we get a research paper on Nano chemistry....?
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?