Hypothesis testing

 Page 1 / 2

Suppose you measure a collection of scalars ${x}_{1},,{x}_{N}$ . You believe the data is distributed in one of two ways. Your first model, call it ${H}_{0}$ , postulates the data to be governed by the density ${f}_{0}(x)$ (some fixed density). Your second model, ${H}_{1}$ , postulates a different density ${f}_{1}(x)$ . These models, termed hypotheses , are denoted as follows: ${H}_{0}:({x}_{n}, {f}_{0}(x)),n=1N$ ${H}_{1}:({x}_{n}, {f}_{1}(x)),n=1N$ A hypothesis test is a rule that, given a measurement $x$ , makes a decision as to which hypothesis best "explains" the data.

Suppose you are confident that your data is normally distributed with variance 1, but you are uncertain aboutthe sign of the mean. You might postulate ${H}_{0}:({x}_{n}, (-1, 1))$ ${H}_{1}:({x}_{n}, (1, 1))$ These densities are depicted in .

Assuming each hypothesis is a priori equally likely, an intuitively appealing hypothesis test is to compute the sample mean $\langle x\rangle =\frac{1}{N}\sum_{n=1}^{N} {x}_{n}$ , and choose ${H}_{0}$ if $\langle x\rangle \le 0$ , and ${H}_{1}$ if $\langle x\rangle > 0$ . As we will see later, this test is in fact optimal under certain assumptions.

Generalizations and nomenclature

The concepts introduced above can be extended inseveral ways. In what follows we provide more rigorous definitions, describe different kinds of hypothesis testing, andintroduce terminology.

Data

In the most general setup, the observation is a collection ${x}_{1},,{x}_{N}$ of random vectors. A common assumption, which facilitates analysis, is that the data are independent and identicallydistributed (IID). The random vectors may be continuous, discrete, or in some cases mixed. It is generally assumedthat all of the data is available at once, although for some applications, such as Sequential Hypothesis Testing , the data is a never ending stream.

Binary versus m-ary tests

When there are two competing hypotheses, we refer to a binary hypothesis test. When the number of hypotheses is $M\ge 2$ , we refer to an M-ary hypothesis test. Clearly, binary is a special case of $M$ -ary, but binary tests are accorded a special status for certain reasons. These includetheir simplicity, their prevalence in applications, and theoretical results that do not carry over to the $M$ -ary case.

Phase-shift keying

Suppose we wish to transmit a binary string of length $r$ over a noisy communication channel. We assign each of the $M=2^{r}$ possible bit sequences to a signal ${s}^{k}$ , $k=\{1, , M\}$ where ${s}_{n}^{k}=\cos (2\pi {f}_{0}n+\frac{2\pi (k-1)}{M})$ This symboling scheme is known as phase-shift keying (PSK). After transmitting a signal across the noisy channel, the receiver faces an $M$ -ary hypothesis testing problem: ${H}_{0}:x={s}^{1}+w$  ${H}_{M}:x={s}^{M}+w$ where $(w, (0, ^{2}I))$ .

In many binary hypothesis tests, one hypothesis represents the absence of a ceratinfeature. In such cases, the hypothesis is usually labelled ${H}_{0}$ and called the null hypothesis. The other hypothesis is labelled ${H}_{1}$ and called the alternative hypothesis.

Waveform detection

Consider the problem of detecting a known signal $s=\left(\begin{array}{c}{s}_{1}\\ \\ {s}_{N}\end{array}\right)$ in additive white Gaussian noise (AWGN). This scenario is common in sonar and radar systems. Denotingthe data as $x=\left(\begin{array}{c}{x}_{1}\\ \\ {x}_{N}\end{array}\right)$ , our hypothesis testing problem is ${H}_{0}:x=w$ ${H}_{1}:x=s+w$ where $(w, (0, ^{2}I))$ . ${H}_{0}$ is the null hypothesis, corresponding to the absence of a signal.

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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
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
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
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!