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Φ = 1 m i = 1 m 1 { y ( i ) = 1 } μ 0 = i = 1 m 1 { y ( i ) = 0 } x ( i ) i = 1 m 1 { y ( i ) = 0 } μ 1 = i = 1 m 1 { y ( i ) = 1 } x ( i ) i = 1 m 1 { y ( i ) = 1 } Σ = 1 m i = 1 m ( x ( i ) - μ y ( i ) ) ( x ( i ) - μ y ( i ) ) T .

Pictorially, what the algorithm is doing can be seen in as follows:

two different data sets have a distribution circles in different quadrants with a line following y=-x separating them

Shown in the figure are the training set, as well as the contours of the two Gaussian distributions that have been fit to the data in each of thetwo classes. Note that the two Gaussians have contours that are the same shape and orientation, since they share a covariance matrix Σ , but they have different means μ 0 and μ 1 . Also shown in the figure is the straight line giving the decision boundary at which p ( y = 1 | x ) = 0 . 5 . On one side of the boundary, we'll predict y = 1 to be the most likely outcome, and on the other side, we'll predict y = 0 .

Discussion: gda and logistic regression

The GDA model has an interesting relationship to logistic regression. If we view the quantity p ( y = 1 | x ; Φ , μ 0 , μ 1 , Σ ) as a function of x , we'll find that it can be expressed in the form

p ( y = 1 | x ; Φ , Σ , μ 0 , μ 1 ) = 1 1 + exp ( - θ T x ) ,

where θ is some appropriate function of Φ , Σ , μ 0 , μ 1 . This uses the convention of redefining the x ( i ) 's on the right-hand-side to be n + 1 -dimensional vectors by adding the extra coordinate x 0 ( i ) = 1 ; see problem set 1. This is exactly the form that logistic regression—a discriminative algorithm—used to model p ( y = 1 | x ) .

When would we prefer one model over another? GDA and logistic regression will, in general, give different decision boundaries when trained on the same dataset. Which is better?

We just argued that if p ( x | y ) is multivariate gaussian (with shared Σ ), then p ( y | x ) necessarily follows a logistic function. The converse, however, is not true; i.e., p ( y | x ) being a logistic function does not imply p ( x | y ) is multivariate gaussian. This shows that GDA makes stronger modeling assumptions about the data than does logistic regression. It turns out that when these modelingassumptions are correct, then GDA will find better fits to the data, and is a better model. Specifically, when p ( x | y ) is indeed gaussian (with shared Σ ), then GDA is asymptotically efficient . Informally, this means that in the limit of very large training sets (large m ), there is no algorithm that is strictly better than GDA (in terms of, say, how accurately they estimate p ( y | x ) ). In particular, it can be shown that in this setting, GDA will be a better algorithm than logistic regression; and more generally,even for small training set sizes, we would generally expect GDA to better.

In contrast, by making significantly weaker assumptions, logistic regression is also more robust and less sensitive to incorrect modeling assumptions. There are many different sets of assumptions that would lead to p ( y | x ) taking the form of a logistic function. For example, if x | y = 0 Poisson ( λ 0 ) , and x | y = 1 Poisson ( λ 1 ) , then p ( y | x ) will be logistic. Logistic regression will also work well on Poisson data like this. But if we were to use GDA on such data—and fit Gaussian distributions tosuch non-Gaussian data—then the results will be less predictable, and GDA may (or may not) do well.

To summarize: GDA makes stronger modeling assumptions, and is more data efficient (i.e., requires less training data to learn “well”)when the modeling assumptions are correct or at least approximately correct. Logistic regression makes weaker assumptions, and is significantly more robust to deviationsfrom modeling assumptions. Specifically, when the data is indeed non-Gaussian, then in the limit of large datasets, logistic regression will almost always do better thanGDA. For this reason, in practice logistic regression is used more often than GDA. (Some related considerations about discriminative vs. generative models also apply forthe Naive Bayes algorithm that we discuss next, but the Naive Bayes algorithm is still considered a very good, and is certainly also a very popular, classification algorithm.)

Questions & Answers

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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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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