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T ( y ) = y a ( η ) = - log ( 1 - Φ ) = log ( 1 + e η ) b ( y ) = 1

This shows that the Bernoulli distribution can be written in the form of Equation  [link] , using an appropriate choice of T , a and b .

Let's now move on to consider the Gaussian distribution. Recall that, when deriving linear regression, the value of σ 2 had no effect on our final choice of θ and h θ ( x ) . Thus, we can choose an arbitrary value for σ 2 without changing anything. To simplify the derivation below, let'sset σ 2 = 1 . If we leave σ 2 as a variable, the Gaussian distribution can also be shown to be in the exponential family, where η R 2 is now a 2-dimension vector that depends on both μ and σ . For the purposes of GLMs, however, the σ 2 parameter can also be treated by considering a more general definition of the exponential family: p ( y ; η , τ ) = b ( a , τ ) exp ( ( η T T ( y ) - a ( η ) ) / c ( τ ) ) . Here, τ is called the dispersion parameter , and for the Gaussian, c ( τ ) = σ 2 ; but given our simplification above, we won't need the more general definition for the examples we will consider here. We then have:

p ( y ; μ ) = 1 2 π exp - 1 2 ( y - μ ) 2 = 1 2 π exp - 1 2 y 2 · exp μ y - 1 2 μ 2

Thus, we see that the Gaussian is in the exponential family, with

η = μ T ( y ) = y a ( η ) = μ 2 / 2 = η 2 / 2 b ( y ) = ( 1 / 2 π ) exp ( - y 2 / 2 ) .

There're many other distributions that are members of the exponential family: The multinomial (which we'll see later), the Poisson (for modelling count-data;also see the problem set); the gamma and the exponential (for modelling continuous, non-negative random variables, such as time-intervals); the beta and the Dirichlet(for distributions over probabilities); and many more. In the next section, we will describe a general “recipe” for constructing models in which y (given x and θ ) comes from any of these distributions.

Constructing glms

Suppose you would like to build a model to estimate the number y of customers arriving in your store (or number of page-views on your website) in any givenhour, based on certain features x such as store promotions, recent advertising, weather, day-of-week, etc. We know that the Poisson distributionusually gives a good model for numbers of visitors. Knowing this, how can we come up with a model for our problem? Fortunately,the Poisson is an exponential family distribution, so we can apply a Generalized Linear Model (GLM).In this section, we will we will describe a method for constructing GLM models for problems such as these.

More generally, consider a classification or regression problem where we would like to predict the value of some random variable y as a function of x . To derive a GLM for this problem, we will make the following threeassumptions about the conditional distribution of y given x and about our model:

  1. y x ; θ ExponentialFamily ( η ) . I.e., given x and θ , the distribution of y follows some exponential family distribution, with parameter η .
  2. Given x , our goal is to predict the expected value of T ( y ) given x . In most of our examples, we will have T ( y ) = y , so this means we would like the prediction h ( x ) output by our learned hypothesis h to satisfy h ( x ) = E [ y | x ] . (Note that this assumption is satisfied in the choices for h θ ( x ) for both logistic regression and linear regression.For instance, in logistic regression, we had h θ ( x ) = p ( y = 1 | x ; θ ) = 0 · p ( y = 0 | x ; θ ) + 1 · p ( y = 1 | x ; θ ) = E [ y | x ; θ ] .)
  3. The natural parameter η and the inputs x are related linearly: η = θ T x . (Or, if η is vector-valued, then η i = θ i T x .)

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
I only see partial conversation and what's the question here!
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
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
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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