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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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I think
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nanocopper obvius
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Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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
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Adin Reply
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what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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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