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"Robust" is a technical word that implies insensitivity to modeling assumptions. As we have seen, some algorithms arerobust while others are not. The intent of robust signal processing is to derive algorithms that are explicitly insensitive to the underlying signal and/or noise models. The way in which modelingincertainties are described is typified by the approach we shall use in the following discussion of robust model evaluation.

We assume that two nominal models of the generation of the statistically independent observations areknown; the "actual" conditional probability density that describes the data under the assumptions of each model is notknown exactly, but is "close" to the nominal. Letting p be the actual probability density for each observation and p o the nominal, we say that ( Huber; 1981 ) p x 1 p o x p d x where p d is the unknown disturbance density and is the uncertainty variable ( 0 1 ). The uncertainty variable specifies how accurate the nominal model is through to be: the smaller , the smaller the contribution of the disturbance. It is assumed that some valuefor can be rationally assigned. The disturbance density is entirely unknown and isassumed to be any value probability density function. The expression given above is normalized so that p has unit density ranging about it. An example of densities described this way are shown in .

The nominal density, a Gaussian, is shown as a dashed line along with example densities derived from it having anuncertainty of 10% ( 0.1 ). The left plot illustrates a symmetric contamination and the right an asymmetric one.

The robust model evaluation problem is formally stated as 0 : p r 0 r l 0 L 1 1 p o r l 0 r l p d r l 0 r l 1 : p r 1 r l 0 L 1 1 p o r l 1 r l p d r l 1 r l The nominal densities under each model correspond to the conditional densities that we have been using until now. Thedisturbance densities are intended to model imprecision of both descriptions; hence, they are assumed to be different in thecontext of each model. Note that the measure of imprecision is assumed to be the same under either model.

To solve this problem, we take what is known as a minimax approach : find the worst-case combinations of a priori densities (max), then minimize the consequences of this situation (mini)according to some criterion. In this way, bad situations are handles as well as can be expected while the more tolerable onesare (hopefully) processed well also. The "mini" phase of the minimax solution corresponds to the likelihood ratio for manycriteria. Thus, the "max" phase amounts to finding the worst-case probability distributions for the likelihood ratiotest as described in the previous section: find the disturbance densities that can result in a constant value for the ratio overlarge domains of functions. When the two nominal distributions scaled by 1 can be brought together so that they are equal for some disturbance, then the likelihood ratio will be constant inthat domain. Of most interest here is the case where the models differ only in the value of the mean, as shown in . "Bringing the distributions together" means, in this case, scaling the distribution for 0 by 1 while adding the constant to the scaled distribution for 1 . One can shown in general that if the ratio of the nominal densities is monotonic, this procedure finds theworst-case distribution ( Huber; 1965 ). The distributions overlap for small and for large values of the data with no overlap in a central region. As weshall see, the size of this central region depends greatly on the choice of . The tails of the worst-case distributions under each model are equal; conceptually, we consider that theworst-case densities have exponential tails in the model evaluation problem.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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Daniel
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Cesar
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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