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A brief discussion of PCA.

Principal component analysis

PCA is essentially just SVD. The only difference is that we usually center the data first using some grand mean before doing SVD. There are three perspectives of views for PCA. Each of them gives different insight on what PCA does.

Low-rank approximation

min Z 1 2 | | X Z | | F 2 s u b j e c t t o r a n k ( Z ) K

where Frobenius norm is a matrix version of sums of squared. This gives the interpretation of dimension reduction. Solution to the problem is: Z = i = 1 K U k d k V K T

We do lose some information when doing dimension reduction, but the majority of variance is explained in the lower-rank matrix (The eigenvalues give us information about how significant the eigenvector is. So we put the eigenvalues in the order of the magnitude of the eigenvectors, and discard the smallest several since the contribution of components along that particular eigenvector is less significant comparing that with a large eigenvalue). PCA guarantees the best rank-K approximation to X. The tuning parameter K can be either chosen by cross-validation or AIC/BIC. This property is useful for data visualization when the data is high dimensional.

Matrix factorization

minimize U , D , V { 1 2 X - U D V T F 2 } s u b j e c t t o U T U = I , V T V = I , D d i a g +

This gives the interpretation of pattern recognition. The first column of U gives the first major pattern in sample (row) space while the first column of V gives the first major pattern in feature space. This property is also useful in recommender systems (a lot of the popular algorithms in collaborative filtering like SVD++, bias-SVD etc. are based upon this “projection-to-find-major-pattern” idea).


max V K T X T X V K s u b j e c t t o V K T V K = 1 , V K T V j = 0

X T X here behaves like covariates for multivariate Gaussian. This is essentially an eigenvalue problem of covariance: X T X   =   V D 2 V T and X X T   =   U D 2 U T . Interpretation here is that we are maximizing the covariates in column and row space.

_PCA (Figure Credit: https://onlinecourses.science.psu.edu/stat857/node/35)

The intuition behind pca

The intuition behind PCA is as follows: The First PC (Principal Component) finds the linear combinations of variables that correspond to the direction with maximal sample variance (the major pattern of the dataset, the most spread out direction). Succeeding PCs then goes on to find direction that gives highest variance under the constraint of it being orthogonal (uncorrelated) to preceding ones. Geometrically, what we are doing is basically a coordinate transformation – the newly formed axes correspond to the newly constructed linear combination of variables. The number of the newly formed coordinate axes (variables) is usually much lower than the number of axes (variables) in the original dataset, but it’s still explaining most of the variance present in the data.

Another interesting insight

Another interesting insight on PCA is provided by considering its relationship to Ridge Regression (L2 penalty). The result given by Ridge Regression can be written like this:

Y ^ = X β ^ r = j = 1 p u j d j 2 d j 2 + λ u j T y

The term in the middle here, d j 2 d j 2 + λ , shrinks the singular values. For those major patterns with large singular values, lambda has little effect for shrinking; but for those with small singular values, lambda has huge effect to shrink them towards zero (not exactly zero, unlike lasso - L1 penalty, which does feature selection). This non-uniform shrinkage thus has a grouping effect. This is why Ridge Regression is often used when features are strongly correlated (it only captures orthogonal major pattern). PCA is really easy to implement - feed the data matrix(n*p) to the SVD command in Matlab, extract the PC loading(V) and PC score(U) vector and we will get the major pattern we want.

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, Comparison of three different matrix factorization techniques for unsupervised machine learning. OpenStax CNX. Dec 18, 2013 Download for free at http://cnx.org/content/col11602/1.1
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