# 4.17 Non-parametric model evaluation

 Page 1 / 1

If very uncertain about model accuracy, assuming a form for the nominal density may be questionable orquantifying the degree of uncertainty may be unreasonable. In these cases, any formula for the underlying probability densities may be unjustified, but the model evaluationproblem remains. For example, we may want to determine a signal presence or absence in an array output (non-zero mean vs. zeromean) without much knowledge of the contaminating noise. If minimal assumptions can be made about the probability densities, non-parametric model evaluation can be used ( Gibson and Melsa ). In this theoretical framework, no formula for the conditional densities isrequired; instead, we use worst-case densities which conform to the weak problem specification. Because few assumptions about theprobability models are used, non-parametric decision rules are robust: they are insensitive to modeling assumptions because sofew are used. The "robust" test of the previous section are so-named because they explicitly encapsulate model imprecision. Ineither case, one should expect greater performance (smaller error probabilities) in non-parametric decision rules than possible froma "robust" one.

Two hypothesized models are to be tested; ${}_{0}$ is intended to describe the situation "the observed data have zero mean" and the other "a non-zero mean is present."We make the usual assumption that the $L$ observed data values are statistically independent. The only assumption we will make about the probabilistic descriptions underlying these models is that the median of the observationsis zero in the first instance and non-zero in the second. The median of a random variable is the "half-way" value: the probability that the random variable is less than themedian is one-half as is the probability that it is greater. The median and mean of a random variable are not necessarily equal;for the special case of a symmetric probability density they are. In any case, the non-parametric models will be stated interms of the probability that an observation is greater than zero. ${}_{0}:({r}_{l}\ge 0)=\frac{1}{2}$ ${}_{1}:({r}_{l}\ge 0)> \frac{1}{2}$ The first model is equivalent to a zero-median model for the data; the second implies that the median is greater thanzero. Note that the form of the two underlying probability densities need not be the same to correspond to the two models;they can differ in more general ways than in their means.

To solve this model evaluation problem, we seek (as do all robust techniques) the worst-case density, the densitysatisfying the conditions for one model that is maximally difficult to distinguish from a given density under theother. Several interesting problems arise in this approach. First of all, we seek a non-parametric answer: thesolution must not depend on unstated parameters (we should not have to specify how large the non-zero mean might be). Secondly,the model evaluation rule must not depend on the form for the given density. These seemingly impossible properties are easilysatisfied. To find the worst-case density, first define ${p}^{+}({r}_{l}, {}_{1}, {r}_{l})$ to be the probability density of the ${l}^{\mathrm{th}}$ observation assuming that ${}_{1}$ is true and that the observation was non-negative. A similar definition for negative values isneeded. ${p}^{+}({r}_{l}, {}_{1}, {r}_{l})=p({r}_{l}, {}_{1}, {r}_{l}\ge 0, {r}_{l})$ ${p}^{-}({r}_{l}, {}_{1}, {r}_{l})=p({r}_{l}, {}_{1}, {r}_{l}< 0, {r}_{l})$ In terms of these quantities, the conditional density of an observation under ${}_{1}$ is given by $p({r}_{l}, {}_{1}, {r}_{l})=({}_{1}, {r}_{l}\ge 0){p}^{+}({r}_{l}, {}_{1}, {r}_{l})-1{p}^{-}({r}_{l}, {}_{1}, {r}_{l})$ The worst-case density under ${}_{0}$ would have exactly the same functional form as this one for positive and negative valueswhile having a zero median.

Don't forget that the worst-case density in model evaluation agrees with thegiven one over as large a range as possible.
As depicted in , a density meeting these requirements is $p({r}_{l}, {}_{1}, {r}_{l})=\frac{{p}^{+}({r}_{l}, {}_{1}, {r}_{l})+{p}^{-}({r}_{l}, {}_{1}, {r}_{l})}{2}$

The likelihood ratio for a single observation would be $2({}_{1}, {r}_{l}\ge 0)$ for non-negative values and $2(1-({}_{1}, {r}_{l}\ge 0))$ for negative values. While the likelihood ratio depends on $({}_{1}, {r}_{l}\ge 0)$ , which is not specified in out non-parametric model, the sufficient statistic will not depend on it! To see this, note that the likelihood ratio varies only withthe sign of the observation. Hence, the optimal decision rule amounts to counting how many of the observations are positive;this count can be succinctly expressed with the unit-step function $u()$ as $\sum_{l=0}^{L-1} u({r}_{l})$ .

We define the unit-step function as $u(x)=\begin{cases}1 & \text{if x> 0}\\ 0 & \text{if x< 0}\end{cases}$ , with the value at the origin undefined. We presume that the densities have no mass at the origin under eithermodel. Although appearing unusual, $\sum u({r}_{l})$ does indeed yield the number of positively values observations.
Thus, the likelihood ratio for the $L$ statistically independent observations is written $2^{L}({}_{1}, {r}_{l}\ge 0)^{\sum u({r}_{l})}(1-({}_{1}, {r}_{l}\ge 0))^{(L-\sum u({r}_{l}))}\underset{{}_{0}}{\overset{{}_{1}}{}}$ Making the usual simplifications, the unknown probability $({}_{1}, {r}_{l}\ge 0)$ can be maneuvered to the right side and merged with the threshold. The optimal non-parametric decision rule thuscompares the sufficient statistic - the count of positive-valued observations - with a threshold determined by the Neyman-Pearsoncriterion.
$\sum_{l=0}^{L-1} u({r}_{l})\underset{{}_{0}}{\overset{{}_{1}}{}}$
This decision rule is called the sign test as it depends only on the signs of the observed data. The sign test isuniformly most powerful and robust.

To find the threshold  , we can use the Central Limit Theorem to approximate theprobability distribution of the sum by a Gaussian. Under ${}_{0}$ , the expected value of $u({r}_{l})$ is $\frac{1}{2}$ and the variance is $\frac{1}{4}$ . To the degree that the Central Limit Theorem reflects the false-alarm probability (see this problem ), ${P}_{F}$ is approximately given by ${P}_{F}=Q(\frac{-\frac{L}{2}}{\sqrt{\frac{L}{4}}})$ and the threshold is found to be $=\frac{\sqrt{L}}{2}Q({P}_{F})^{(-1)}+\frac{L}{2}$ As it makes no sense for the threshold to be greater than $L$ (how many positively values observations can there be?), the specified false-alarmprobability must satisfy ${P}_{F}\ge Q(\sqrt{L})$ . This restriction means that increasing stringent requirements on the false-alarm probability can only be met ifwe have sufficient data.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!