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STOCK MARKET ANALYSIS And
PREDICTION
By:
Vivek Bhalgat
Vivek Bijlwan
(under Dr. Ratna Sanyal)
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SATYAM happenedRs. 430
Rs. 6.30 Rs. 117
In a span of 9 months , one could have made his money 18 times!!
OR
One could have cashed in at 430 , when others would sell at Rs. 6.30
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WhyWarren Buffett is the richest man
on Earth? In his own words
The basic ideas of investing are to look at
stocks as business, use the market'sfluctuations to your advantage..
So , what is a fluctuation ?
How to identify it?
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But How?
VALUE
INTRINSIC EXTRINSIC
Intrinsic value, or sometimes known as "Fundamental Value", is the value that remains in
an option when all of its extrinsic value has diminished due to Time Decay. It is the actual
value of a stock that has been built into the price of the option.
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ICAICA
Blind Signal Separation (BSS) or Independent Component Analysis (ICA) is theidentification & separation of mixtures of sources with little priorinformation.
Applications include:
Audio Processing
Medical data
Finance
Array processing (beamforming)
Coding
and most applications where Factor Analysis and PCA is currently used. While PCA seeks directions that represents data best in a |x0 - x|
2 sense,ICA seeks such directions that are most independent from each other.
We will concentrate on Time Series separation of Multiple Targets
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The simple Cocktail Party ProblemThe simple Cocktail Party Problem
Sources
Observations
s1
s2
x1
x2
Mixing matrix A
x =As
n sources, m=n observations
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MotivationMotivation
Get the Independent Signals out of the Mixture
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ICA Model (Noise Free)ICA Model (Noise Free)
Use statistical latent variables system(IID)
Random variable sk instead of time signal
xj = aj1s1 + aj2s2 + .. + ajnsn, for all j
x =As
ICs s are latent variables & are unknown AND Mixing matrixA isalso unknown
Task: estimate A and s using only the observeable random vector x
Lets assume that no. of ICs = no of observable mixtures
andA
is square and invertible So after estimating A, we can compute W=A-1and hence
s = Wx = A-1x
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IllustrationIllustration
2 ICs with distribution:
Zero mean and variance equal to 1
Mixing matrix A is
The edges of the parallelogram are in thedirection of the cols of A
So if we can Est joint pdf of x1 & x2 and then
locating the edges, we can Est A.
! 12
32
A
e
!otherwise
sifsp
i
i
0
3||32
1)(
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RestrictionsRestrictions
si are statistically independent
p(s1,s2) = p(s1)p(s2)
Nongaussian distributions
The joint density of unit
variance s1 & s2 is symmetric.So it doesnt contain anyinformation about thedirections of the cols of themixing matrix A. So A canntbe estimated.
If only one IC is gaussian, theestimation is still possible.
!2
exp2
1),(22
21
21
xxxxpT
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AmbiguitiesAmbiguities
Cant determine the variances (energies)of the ICs Both s & A are unknowns, any scalar multiple in one of the
sources can always be cancelled by dividing the correspondingcol of A by it.
Fix magnitudes of ICs assuming unit variance: E{si2}=1
Only ambiguity of sign remains
Cant determine the order of the ICs Terms can be freely changed, because both s and A areunknown. So we can call any IC as the first one.
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ICA Principal (NonICA Principal (Non--Gaussian is Independent)Gaussian is Independent)
Key to estimating A is non-gaussianity
The distribution of a sum of independent random variables tends toward a Gaussiandistribution. (By CLT)
f(s1) f(s2) f(x1) = f(s1 +s2)
Where w is one of the rows of matrix W.
y is a linear combination of si, with weights given by zi. Since sum of two indep r.v. is more gaussian than individual r.v., so zTs is moregaussian than either of si. AND becomes least gaussian when its equal to one of si.
So we could take w as a vector which maximizes the non-gaussianity of wTx.
Such a w would correspond to a z with only one non zero comp. So we get back the si.
szAswxwy TTT !!!
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Measures of NonMeasures of Non--GaussianityGaussianity
We need to have a quantitative measure of non-gaussianity for ICAEstimation.
Kurtotis : gauss=0 (sensitive to outliers)
Entropy : gauss=largest
Neg-entropy : gauss =0 (difficult to estimate)
Approximations
where v is a standard gaussian random variable and :
224 }){(3}{)( yEyEykurt !
! dyyfyfyH )(log)()(
)()()( yyyJ gauss !
_ a222 )(
48
1
12
1)( ykurtyEyJ !
_ a _ a? A2)()()( vGEyGEyJ }
)2/.exp()(
).cosh(log1)(
2uayG
yaa
yG
!
!
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Data Centering &WhiteningData Centering &Whitening
Centeringx =x E{x}
But this doesnt mean that ICA cannt estimate the mean, but it just simplifies theAlg.
ICs are also zero mean because of:E{s}= WE{x}
After ICA, add W.E{x} to zero mean ICs Whitening
We transform the xs linearly so that the x~ are white. Its done by EVD.x~= (ED-1/2ET)x = ED-1/2ET Ax = A~s
where E{xx~}= EDET
So we have to Estimate Orthonormal Matrix A~
An orthonormal matrix has n(n-1)/2 degrees of freedom. So for large dim A wehave to est only half as much parameters. This greatly simplifies ICA.
Reducing dim of data (choosing dominant Eig) while doing whitening alsohelp.
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RESULTS
Data taken
TCS at BSE for the past 400 days.
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Our Data sources : BSE and NSE
BSE
NSE
TCS
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Intrinsic & Extrinsic
TCS
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Their ICs
HCL Infosys
Wipro
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Correlation(TCS , Infosys)
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And the others
Correlation(TCS,Wipro) Correlation(Infosys,Wipro)
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With other sectors
Correlation(TCS, JK Cement) Correlation(TCS, Reliance)
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Work after Mid-Sem
Wavelet Transform :
Why Wavelet Transform?
Why not Fourier ? Time invariant
Why not Short term Fourier transform ? Heisenbergs Uncertainty
PrincipleWavelet Transform : Multi Resolution Signal Analysis
Unlike the STFT which has a constant resolution at all times and
frequencies, the WT has a good time and poor frequency resolution at
high frequencies, and good frequency and poor time resolution at low
frequencies
Analysis and Evaluation of Results
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http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html
http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb/
http://en.wikipedia.org/wiki/Independent_component_analy
sis Pierre Comon (1994): Independent Component Analysis: a
new concept?, Signal Processing, Elsevier, 36(3):287--314 (The
original paper describing the concept of ICA)
References: