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Z ( θ ) = ξ exp i = 1 D θ i μ i ( ξ j )

Note that the sum over ξ in the partition function refers to the sum over all possible ξ , not just the ξ that have been observed. This fact makes computation of the partition function intractable and we must approximate it. Following a sampling-based learning technique, we conclude [link] :

ln Z ( θ ) ln 1 T t = 1 T exp i = 1 D ( θ i - θ i 0 ) μ i ( ξ t ) + ln Z ( θ 0 )

Where θ 0 is some set of parameters from which T samples are drawn (20 in our case). Since ln Z ( θ 0 ) is a constant, we can leave it out of the optimization's objective function and we solve the MLE problem via gradient ascent.

θ i t - 1 = j = 1 13 μ i ( ξ j ) - 13 t = 1 T μ i ( ξ t ) exp r = 1 D ( θ r t - 1 - θ r t - 2 ) μ r ( ξ t ) t = 1 T exp r = 1 D ( θ r t - 1 - θ r t - 2 ) μ r ( ξ t )
θ t = θ t - 1 + s * θ t - 1

Where s is some small step size. We update θ 0 on each iteration to be θ t - 2 . This is due to the fact that the partition function approximation is only reasonable in a neighborhood of θ 0 [link] . It follows that the ξ 's which are indexed by t are drawn from a model with parameters θ t - 2 , while the ξ 's indexed by j still represent the historical data.

Correcting for lack of data

Due to the small number of historical observations (13) and the large number of possible combinations for any edge ( 60 states * 60 states = 3 , 600 combinations), we must come up with a more concise way to learn the relationships between counties. To that end, we look not at the absolute voting percentages of counties but rather the difference in voting percentage between each pair of neighboring counties. This method has the added bonus of circumventing the problem of overall change that has affected every county. Unfortunately, there are still 119 possible differences that could occur (-59,-58,...,0,...58,59) and only 13 elections to determine the frequency with which each difference occurs. Therefore, we place each difference into a cluster, e.g. [-9,-6]. We use 11 clusters total and since the differences between counties are fairly consistent between years, the 13 observations should be sufficient for an approximation of the marginal probabilities for each edge. These approximation techniques do not affect the way we solve the problem via gradient ascent. However, once gradient ascent is finished we must convert our small θ into standard long form (as displayed in Section 2.1).

Performing map inference

Due to our approximation techniques in the learning process, we are confronted with a problem when attempting to predict the 2012 election. Since the entire model is based off relativity, any outcome for a particular county is equally likely as long as the rest of the model shifts with it. In order to ensure we do not get extremely low or high results, we must fix some subset of the counties as a starting point for the model. In order to do this, we utilize linear regression techniques (as discussed in the next section). Once the model is partially filled in, we solve the binary program stated above with our learned θ (in standard long form) via Gurobi Optimizer.

Multivariate regression

Multivariate Linear Regression is commonly used in social sciences as a means of predicting future outcomes based off of known data. It will provide us with a comparison as well as a starting off point for our Markov Random Field model. Our model will have Incumbent Party Vote % as the dependent variable. That is, if a Democratic president is currently in office, then we will be predicting the voting %'s earned by this year's Democratic Candidate.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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