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
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?