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Tests and decision regions

Consider the general hypothesis testing problem where we have N d -dimensional observations x 1 , , x N and M hypotheses. If the data are real-valued, for example, then a hypothesistest is a mapping : d N 1 M For every possible realization of the input, the test outputs a hypothesis. The test partitions the input space into a disjoint collection R 1 , , R M , where R k ( x 1 , , x N ) | x 1 x N k The sets R k are called decision regions . The boundary between two decision regions is a decision boundary . depicts these concepts when d 2 , N 1 , and M 3 .

Simple versus composite hypotheses

If the distribution of the data under a certain hypothesis is fully known, we call it a simple hypothesis. All of the hypotheses in the examples above are simple. In many cases, however, we onlyknow the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know thevariance of the noise. In this case, a hypothesis is said to be composite .

Consider the problem of detecting the signal s n 2 f 0 n k n n 1 N where k is an unknown delay parameter. Then H 0 : x w H 1 : x s w is a binary test of a simple hypothesis ( H 0 ) versus a composite alternative. Here we are assuming w n 0 2 , with 2 known.

Often a test involving a composite hypothesis has the form H 0 : 0 H 1 : 0 where 0 is fixed. Such problems are called two-sided because the composite alternative "lies on both sides of H 0 ." When is a scalar, the test H 0 : 0 H 1 : 0 is called one-sided . Here, both hypotheses are composite.

Suppose a coin turns up heads with probability p . We want to assess whether the coin is fair( p 1 2 ). We toss the coin N times and record x 1 , , x N ( x n 1 means heads and x n 0 means tails). Then H 0 : p 1 2 H 1 : p 1 2 is a binary test of a simple hypothesis ( H 0 ) versus a composite alternative. This is also a two-sided test.

Errors and probabilities

In binary hypothesis testing, assuming at least one of the two models does indeed correspond to reality, thereare four possible scenarios:

  • Case 1

    H 0 is true, and we declare H 0 to be true
  • Case 2

    H 0 is true, but we declare H 1 to be true
  • Case 3

    H 1 is true, and we declare H 1 to be true
  • Case 4

    H 1 is true, but we declare H 0 to be true
In cases 2 and 4, errors occur. The names given to these errors depend on the area of application. In statistics, theyare called type I and type II errors respectively, while in signal processing they are known as a false alarm or a miss .

Consider the general binary hypothesis testing problem H 0 : x f x , 0 H 1 : x f x , 1 If H 0 is simple, that is, 0 0 , then the size (denoted ), also called the false-alarm probability ( P F ), is defined to be P F 0 declare H 1 When 0 is composite, we define P F sup 0 declare H 1 For 1 , the power (denoted ), or detection probability ( P D ), is defined to be P D declare H 1 The probability of a type II error, also called the miss probability , is P M 1 P D If H 1 is composite, then is viewed as a function of .

Criteria in hypothesis testing

The design of a hypothesis test/detector often involves constructing the solution to an optimizationproblem. The optimality criteria used fall into two classes: Bayesian and frequent.

Representing the former approach is the Bayes Risk Criterion . Representing the latter is the Neyman-Pearson Criterion . These two approaches are developed at length in separate modules.

Statistics versus engineering lingo

The following table, adapted from Kay, p.65 , summarizes the different terminology for hypothesis testing from statistics and signal processing:

Statistics Signal Processing
Hypothesis Test Detector
Null Hypothesis Noise Only Hypothesis
Alternate Hypothesis Signal + Noise Hypothesis
Critical Region Signal Present Decision Region
Type I Error False Alarm
Type II Error Miss
Size of Test ( ) Probability of False Alarm ( P F )
Power of Test ( ) Probability of Detection ( P D )

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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