# 0.1 Detection performance criteria  (Page 2/3)

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From a grander viewpoint, these expressions represent an achievable lower bound on performance (as assessed by the probability oferror). Furthermore, this probability will be non-zero if the conditional densities overlap over some range of values of $r$ , such as occurred in the previous example. Within regions of overlap, the observed values are ambiguous: eithermodel is consistent with the observations. Our "optimum" decision rule operates in such regions by selecting that modelwhich is most likely (has the highest probability) of generating the measured data.

## Neyman-pearson criterion

Situations occur frequently where assigning or measuring the a priori probabilities ${\pi }_{i}$ is unreasonable. For example, just what is the a priori probability of a supernova occurring in any particular region of the sky? We clearlyneed a model evaluation procedure that can function without a priori probabilities. This kind of test results when the so-called Neyman-Pearson criterion is used to derive the decision rule.

Using nomenclature from radar, where model ${ℳ}_{1}$ represents the presence of a target and ${ℳ}_{0}$ its absence, the various types of correct and incorrect decisions have the following names.

In statistics, a false-alarm is known as a type I error and a miss a type II error .
• ## Detection probability

we say it's there when it is; ${P}_{D}=({ℳ}_{1}\text{true}, \text{say}{ℳ}_{1})$
• ## False-alarm probability

we say it's there when it's not; ${P}_{F}=({ℳ}_{0}\text{true}, \text{say}{ℳ}_{1})$
• ## Miss probability

we say it's not there when it is; ${P}_{M}=({ℳ}_{1}\text{true}, \text{say}{ℳ}_{0})$
The remaining probability $({ℳ}_{0}\text{true}, \text{say}{ℳ}_{0})$ has historically been left nameless and equals $1-{P}_{F}$ . We should also note that the detection and miss probabilities are related by ${P}_{M}=1-{P}_{D}$ . As these are conditional probabilities, they do not depend on the a priori probabilities. Furthermore, the two probabilities ${P}_{F}$ and ${P}_{D}$ characterize the errors when any decision rule is used.

These two probabilities are related to each other in an interesting way. Expressing these quantities in terms of thedecision regions and the likelihood functions, we have ${P}_{F}=\int p(R, {ℳ}_{0}, r)\,d r$ ${P}_{D}=\int p(R, {ℳ}_{1}, r)\,d r$ As the region ${Z}_{1}$ shrinks, both of these probabilities tend toward zero; as ${Z}_{1}$ expands to engulf the entire range of observation values, they both tend toward unity. This rather directrelationship between ${P}_{D}$ and ${P}_{F}$ does not mean that they equal each other; in most cases, as ${Z}_{1}$ expands, ${P}_{D}$ increases more rapidly than ${P}_{F}$ (we had better be right more often than we are wrong!). However, the "ultimate" situation where a rule isalways right and never wrong ( ${P}_{D}=1$ , ${P}_{F}=0$ ) cannot occur when the conditional distributions overlap. Thus, to increase the detection probability we mustalso allow the false-alarm probability to increase. This behavior represents the fundamental tradeoff in detection theory .

One can attempt to impose a performance criterion that depends only on these probabilities with the consequent decision rulenot depending on the a priori probabilities. The Neyman-Pearson criterion assumes that the false-alarm probability is constrained to be less than orequal to a specified value $\alpha$ while we maximize the detection probability ${P}_{D}$ . $\forall {P}_{F}, {P}_{F}\le \alpha \colon \max\{{Z}_{1} , {P}_{D}\}$ A subtlety of the solution we are about to obtain is that the underlying probability distribution functions may not becontinuous, with the consequence that ${P}_{F}$ can never equal the constraining value $\alpha$ . Furthermore, a (unlikely) possibility is that the optimum value for the false-alarm probability is somewhat lessthan the criterion value. Assume, therefore, that we rephrase the optimization problem by requiring that the false-alarmprobability equal a value $\alpha ^\prime$ that is the largest possible value less than or equal to $\alpha$ .

#### Questions & Answers

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
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
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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