where
$L>0$ is a constant. A function satisfying condition
[link] is said to be Lipschitz on
$[0,1]$ . Notice
that such a function must be continuous, but it is not necessarilydifferentiable. An example of such a function is depicted in
[link] (a).
where
$\stackrel{i.i.d.}{\sim}$ means
independently and identically
distributed .
[link] (a) illustrates this
setup.
In many applications we can sample
$\mathcal{X}=[0,1]$ as we like, and not
necessarily at random. For example we can take
$n$ samples
uniformly on [0,1]
We will proceed with this setup (as in
[link] (b)) in the rest of the lecture.
Our goal is to find
${\widehat{f}}_{n}$ such that
$E[\parallel {f}^{*}-{\widehat{f}}_{n}{\parallel}^{2}]\to 0,\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to 0$ (here
$\parallel \xb7\parallel $ is the usual
${L}_{2}$ -norm; i.e.,
$\parallel {f}^{*}-{\widehat{f}}_{n}{\parallel}^{2}={\int}_{0}^{1}{|{f}^{*}\left(t\right)-{\widehat{f}}_{n}\left(t\right)|}^{2}dt$ ).
Let
$0<{m}_{1}\le {m}_{2}\le {m}_{3}\le \cdots $ be a sequence of integers
satisfying
${m}_{n}\to \infty $ as
$n\to \infty $ , and
${k}_{n}{m}_{n}=n$ for some integer
${k}_{n}>0$ . That is, for each value of
$n$ there is an associated integer value
${m}_{n}$ . Define the Sieve
${\mathcal{F}}_{1},\phantom{\rule{0.277778em}{0ex}}{\mathcal{F}}_{2},\phantom{\rule{0.277778em}{0ex}}{\mathcal{F}}_{3},\phantom{\rule{0.277778em}{0ex}}...,$
The above implies that
$\parallel {f}^{*}-{f}_{n}{\parallel}^{2}\to 0\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty ,$ since
$m={m}_{n}\to \infty \phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty $ . In words, with
$n$ sufficiently large we can
approximate
${f}^{*}$ to arbitrary accuracy using models in
${\mathcal{F}}_{n}$ (even if the functions we are using to approximate
${f}^{*}$ are not
Lipschitz!).
For any
$f\in {\mathcal{F}}_{n},$$f={\sum}_{j=1}^{m}{c}_{j}\phantom{\rule{0.166667em}{0ex}}{\mathbf{1}}_{\{t\in {I}_{j,m}\}},$ we have
Note that
$E\left[{\widehat{c}}_{j}\right]={c}_{j}^{*}$ and therefore
$E\left[{\widehat{f}}_{n}\left(t\right)\right]={f}_{n}\left(t\right)$ . Lets analyze now the expected risk of
${\widehat{f}}_{n}$ :
where the final step follows from the fact that
$E\left[{\widehat{f}}_{n}\left(t\right)\right]={f}_{n}\left(t\right)$ . A couple of important remarks pertaining
the right-hand-side of equation
[link] : The
first term,
$\parallel {f}^{*}-{f}_{n}{\parallel}^{2}$ , corresponds to the approximation
error, and indicates how well can we approximate the function
${f}^{*}$ with a function from
${\mathcal{F}}_{n}$ . Clearly, the larger the class
${\mathcal{F}}_{n}$ is, the smallest we can make this term. This term is
precisely the squared bias of the estimator
${\widehat{f}}_{n}$ . The second
term,
$E[\parallel {f}_{n}-{\widehat{f}}_{n}{\parallel}^{2}]$ , is the estimation error, the
variance of our estimator. We will see that the estimation erroris small if the class of possible estimators
${\mathcal{F}}_{n}$ is also small.
The behavior of the first term in
[link] was
already studied. Consider the other term:
What is the best choice of
$m$ ? If
$m$ is small then the approximation
error (
i.e.,$O(1/{m}^{2})$ ) is going to be large, but the estimation error
(
i.e.,$O(m/n)$ ) is going to be small, and vice-versa. This two
conflicting goals provide a tradeoff that directs our choice of
$m$ (as a function of
$n$ ). In
[link] we depict
this tradeoff. In
[link] (a) we considered a
large
${m}_{n}$ value, and we see that the approximation of
${f}^{*}$ by a
function in the class
${\mathcal{F}}_{n}$ can be very accurate (that is, our
estimate will have a small bias), but when we use the measured dataour estimate looks very bad (high variance). On the other hand, as
illustrated in
[link] (b), using a very small
${m}_{n}$ allows our estimator to get very close to the best approximating
function in the class
${\mathcal{F}}_{n}$ , so we have a low variance estimator, but
the bias of our estimator (
i.e., the difference between
${f}_{n}$ and
${f}^{*}$ )
is quite considerable.
We need to balance the two terms in the right-hand-side of
[link] in order to maximize the rate of decay (with
$n$ ) of the expected risk. This implies that
$\frac{1}{{m}^{2}}=\frac{m}{n}$ therefore
${m}_{n}={n}^{1/3}$ and the Mean
Squared Error (MSE) is
produces a
$\mathcal{F}$ -consistent
estimator for
${f}^{*}=E\left[Y\right|X+x]\in \mathcal{F}$ .
It is interesting to note that the rate of decay of the MSE we obtainwith this strategy cannot be further improved by using more
sophisticated estimation techniques (that is,
${n}^{-2/3}$ is the
minimax MSE rate for this problem). Also, rather surprisingly, we
are considering classes of models
${\mathcal{F}}_{n}$ that are actually not
Lipschitz, therefore our estimator of
${f}^{*}$ is not a Lipschitz
function, unlike
${f}^{*}$ itself.
Questions & Answers
find the 15th term of the geometric sequince whose first is 18 and last term of 387
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.