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Ömer Nezi̇h Gerek Presentation1
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Transcript of Ömer Nezi̇h Gerek Presentation1
![Page 1: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/1.jpg)
Prediction / Co(variance – relation) Öngörü / Ko(varyans – relasyon)
Ömer Nezih Gerek
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Data Information
Data is useless in raw form.
Since you measure it, it must carry some
information!
Signatures (statistical)…
◦ 1. Model based
◦ 2. (non)Parametric
System, Prediction, etc.
Histogram, mean, etc.
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Non-parametric statistics
Histogram (a fair pdf estimate)
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Symmetric unimodal Skewed right Bi-modal
Multi-modal Skewed left Symmetric
N i( ) = count X k( ) = i{ }i=0,k=1
i=max_val , k=data_ size
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Things that we extract from pdf
Technically “everything”
All moments:
◦ from which, we can derive mean, variance,
etc..
But “not” the correlation characteristics
in a time series!
Pdf is stationary, and doesn’t care about
inter-symbol relations…
M x t( ) = E etx{ }
momn =¶n
¶t nM x t( )
t=0
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Sample statistics:
m :Mean (average) m = E x n[ ]{ }
s 2 :Variance s 2 = E x n[ ]- m( )2{ }
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What is this “correlation” thing
It is:
But this is an abstract def. Let’s estimate it
from data (with example )
Rx t( ) = E x t( )x t -t( ){ }
Rx 2( ) =1
Nx 2( )x 0( ) + x 3( )x 1( ) + x 4( )x 2( ) + ...{
...+ x N +1( )x N -1( ) + x N + 2( )x N( )}
t = 2
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Correlation somewhat depends on
the mean Let’s normalize it:
and call it “COVARIANCE”.
Cx t( ) = E x t( )- méë ùû x t -t( )- méë ùû{ }= E x t( )x t -t( ){ } - E x t( )m{ } - E x t -t( )m{ } + m2
= Rx t( ) - m2
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Rx n( ) :Autocorrelation
Rx n( ) = E x m[ ]x m+ n[ ]{ }
Cx n( ) :Autocovariance
Cx n( ) = E x m[ ]- m( ) x m+ n[ ]- m( ){ }Cx n( ) = E x m[ ]x m+ n[ ]{ }- m2
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Properties
Correlation and Covariance carries:
◦ time relation (single time difference), and
◦ info. regarding up to 2nd order monents: mean and variance.
Naturally, covariance decreases by distance…
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Example calculation:
2 3 1
× 2 3 1
------------------
2 3 1
6 9 3
4 6 2
------------------
2 9 14 9 2
R(0) R(1) R(2)
m = 2
C(0) = 10, C(1) = 5, C(2) = -2
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Covariance elements
is the “spectral
density!
◦ shows how much “power” the random signal
has at each “frequency”.
◦ remember “equalizers”, “woometers”, “radio
stations”…
C 0( ) = s 2 = E x t( ) - m( )2{ }
Sxx f( ) = FT C t( ){ }
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Remember the Fourier Transform
From time (x(t)) to frequency (X(f))
In case x(t) = R(t), X(f) will be Sx(f).
… and integral of power signal density is
power (which is always positive):
X f( ) = x t( )e- j 2p ft dt-¥
¥
ò
Sx f( )dff1
f2
ò ³ 0 Sx f( ) ³ 0
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Our source of info: Sx(f) & Rx(t)
We will extract significant information
from autocorrelation and power spectral
density:
◦ Best linear prediction (what will come in the
series?)
◦ The best linear model of the “process” that
produces a certain outcome
◦ The power of the process at different
frequencies.
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What happens to a random
signal when it passes through a
linear system?
Do you remember what happens to a
signal that passes through a linear system?
where H(f) is the frequency response.
Y f( ) = X f( ) × H f( )
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So, what happens to a random
signal when it passes through a
linear system?
Now, we don’t have the signal, x(t). We
only have statistical parameters, R(t) or
S(t):
where H(f) is the still frequency response.
Notice that Sy(f) is still positive
Sy f( ) = Sx f( ) × H f( )2
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implies Power at output:
Mean at output:
Variance at output:
Sy f( ) = Sx f( ) × H f( )2
Sy f( )dfò = Sx f( ) × H f( )2
dfò = Ry 0( )
my = mx × H 0( )
s y
2 = Ry 0( )- my
2
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So much math to do: Prediction
…x[1],x[2],…,x[n-2],x[n-1], what next?
A linear predictor “filters” incoming
samples to predict x[n]:
1, 2, 1, 2, 1, 2, 1, 2, x[n]=?
x[n] = 0*x[n-1] + 1*x[n-2]
◦ So our filter is: {0z-1 +1z-2}
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Series examples:
1, 2, 3, 4, 5, x[n] = ?
x[n] = 2*x[n-1] – x[n-2]
◦ So, our filter is: {2z-1 – 1z-2}
1, 1, 2, 4, 7, 13, 24, x[n] = ?
x[n] = x[n-1] + x[n-2] + x[n-2]
◦ So, our filter is: {1z-1 + 1z-2 + 1z-3}
But these series’ are too deterministic.
Besides, how long is our filter?
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If this is close to x[n]
Then this will be small!
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Ideal predictor:
Linear predictor:
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Stochastic series:
First order prediction:
Question: What is the optimum h1?
Answer: Such a value that minimizes:
x̂ n[ ] = h1 × x n-1[ ]
d n[ ] = x n[ ]- x̂ n[ ]
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Minimzation of
Equivalently:
Which has a power magnitude:
d n[ ] = x n[ ]- x̂ n[ ]
d n[ ] = x n[ ]- h1 × x n-1[ ]
s d
2 = E d2 n[ ]{ }
= E x n[ ]- h1 × x n-1[ ]( )2{ }
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Minimization (cont.)
Expanding:
s d
2 = E d2 n[ ]{ }
= E x n[ ]- h1 × x n-1[ ]( )2{ }
= E x2 n[ ]- 2h1 × x n[ ] × x n-1[ ]{ +h1
2 × x2 n-1[ ]} = E x2 n[ ]{ } - 2h1 ×E x n[ ] × x n-1[ ]{ }
+ h1
2 ×E x2 n-1[ ]{ }
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Minimization (cont.)
s d
2 = E x2 n[ ]{ } - 2h1 ×E x n[ ] × x n-1[ ]{ }
+ h1
2 ×E x2 n-1[ ]{ } = s x
2 - 2h1 ×Rx 1( ) + h1
2 ×s x
2
= 1+ h1
2 - 2h1r1éë ùûs x
2
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Minimization (cont.)
We have:
which can be minimized (according to h1)
by taking its derivative w.r.t h1 and
equating to zero!
s d
2 = 1+ h1
2 - 2h1r1éë ùûs x
2
¶
¶h1
s d
2 = 0
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Minimization (cont.)
¶
¶h1
s d
2 = 0
¶
¶h1
1+ h1
2 - 2h1r1éë ùûs x
2 = 0
2h1 - 2h1r1 = 0
Þ h1 = r1
The filter coefficient is the
same as the correlation
coefficient!
x̂ n[ ] = r1 × x n-1[ ]
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Optimum 1st order prediction
See that, the best prediction coefficient
depends on R(t):
Is this true for “longer” prediction filters?
Let’s take a look at 2nd order prediction
filter…
r1 =Rx 1( )s x
2: First correlation coefficient
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2nd order prediction
Question: What are optimum h1 and h2?
Answer: Values that minimize:
x̂ n[ ] = h1 × x n-1[ ]+ h2 × x n- 2[ ]
d n[ ] = x n[ ]- x̂ n[ ]
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Minimzation of
Equivalently:
Which has a power magnitude:
d n[ ] = x n[ ]- x̂ n[ ]
d n[ ] = x n[ ]- h1 × x n-1[ ]- h2 × x n- 2[ ]
s d
2 = E d2 n[ ]{ }
= E x n[ ]- h1 × x n-1[ ]- h2 × x n- 2[ ]( )2{ }
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Minimization (cont.)
Expanding:
s d
2 = E d2 n[ ]{ }
= E x n[ ]- h1 × x n-1[ ]- h2 × x n- 2[ ]( )2{ }
= E x2 n[ ]+ h1
2 × x2 n-1[ ]+ h2
2 × x2 n- 2[ ]{ - 2h1 × x n[ ]x n-1[ ]- 2h2 × x n[ ]x n- 2[ ]
+2h1 ×h2 × x n-1[ ] × x n- 2[ ]}
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Minimization (cont.)
Expanding:
s d
2 = E x2 n[ ]+ h1
2 × x2 n-1[ ]+ h2
2 × x2 n- 2[ ]{ - 2h1 × x n[ ]x n-1[ ]- 2h2 × x n[ ]x n- 2[ ]
+2h1 ×h2 × x n-1[ ] × x n- 2[ ]}
= E x2 n[ ]{ } + h1
2 ×E x2 n-1[ ]{ } + h2
2 ×E x2 n- 2[ ]{ } - 2h1 ×E x n[ ]x n-1[ ]{ } - 2h2 ×E x n[ ]x n- 2[ ]{ } + 2h1 ×h2 ×E x n-1[ ] × x n- 2[ ]{ }
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Minimization (cont.)
s d
2 = E x2 n[ ]{ } + h1
2 ×E x2 n-1[ ]{ } + h2
2 ×E x2 n- 2[ ]{ } - 2h1 ×E x n[ ]x n-1[ ]{ } - 2h2 ×E x n[ ]x n- 2[ ]{ } + 2h1 ×h2 ×E x n-1[ ] × x n- 2[ ]{ }= s x
2 + h1
2s x
2 + h2
2s x
2 - 2h1Rx 1( ) - 2h2Rx 2( ) + 2h1h2Rx 1( )
= s x
2 1+ h1
2 + h2
2 - 2h1r1 - 2h2r2 + 2h1h2r1éë ùû
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Minimization (cont.)
We now have:
which can be minimized (according to h1
and h2) by taking its derivative w.r.t h1,h2
and equating to zero!
s d
2 =s x
2 1+ h1
2 + h2
2 - 2h1r1 - 2h2r2 + 2h1h2r1éë ùû
¶
¶h1
s d
2 =¶
¶h2
s d
2 = 0
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Minimization (cont.)
¶
¶h1
s d
2 = 0 Þ h1 =r1 1- r2( )
1- r1
2( )
The filter coefficients have
somewhat changed…
¶
¶h2
s d
2 = 0 Þ h1 =r2 - r1
2( )1- r1
2( )
![Page 36: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/36.jpg)
Comparison of 1st and 2nd orders
s d,min,1
2 = 1- r1
2éë ùûs x
2
s d,min,2
2 = 1- r1
2 -r1
2 - r2( )1- r1
2( )
é
ë
êê
ù
û
úús x
2
1³ r1
2 ³ r2Þs d,min,1
2 ³s d,min,2
2
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By increasing the prediction window size,
prediction error “definitely” increases!
Note that 2nd and 1st order are the same
if:
s d,min,2
2 = 1- r1
2 -r1
2 - r2( )1- r1
2( )
é
ë
êê
ù
û
úús x
2 = 1- r1
2éë ùûs x
2
Þ r2 = r1
2
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A first order system property:
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Nth order prediction:
s d
2 = E d2 n[ ]{ }
= E x n[ ]- x̂ n[ ]( )2{ }
= E x n[ ]- hj x n- j[ ]j=1
N
åæ
èç
ö
ø÷
2ì
íï
îï
ü
ýï
ýï
![Page 40: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/40.jpg)
s d
2 = E x n[ ]- hj x n- j[ ]j=1
N
åæ
èç
ö
ø÷
2ì
íï
îï
ü
ýï
ýï
¶s d
2
¶hj
= E 2 x n[ ]- x̂ n[ ]( )¶
¶hj
- x̂ n[ ]( )ìíï
îï
üýï
ýï
¶
¶hj
- x̂ n[ ]( ) = x n- j[ ]
E x n[ ]- x̂ n[ ]( )x n- j[ ]{ } = E d n[ ]x n- j[ ]{ } = 0
Important observation : d n[ ] ^ x n- j[ ]
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Nth order prediction
E x n[ ]- hi x n- i[ ]i=1
N
åæ
èçö
ø÷x n- j[ ]
ìíî
üýý
= 0, "j
Rx j( ) - hi Rx j - i( )i=1
N
å = 0, "i
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Nth order prediction
In short:
or:
with:
rx = Rx ´ hopt
hopt = Rx
-1 ´ rx
s d,min
2 =s x
2 - rx
TRx
-1rx
Good
correlations
reduce the
error faster!
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Remarks
is not a cheap operation
There must be a good N estimate.
The result is only the best “linear” predictor.
Multidimensional extensions exist.
hopt = Rx
-1 ´ rx
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Graphically…
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Example
R(0) =1, R(1) = 0.9, R(2) = 0.81
What is the optimum 2-tap prediction filter?
hopt = 1 0.90.9 1
é
ëêù
ûú
-1
× 0.90.81
é
ëêù
ûú
= 5.263 -4.737-4.737 5.263
é
ëêù
ûú× 0.9
0.81é
ëêù
ûú= 0.9
0é
ëêù
ûú
hopt = R-1 ×r
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Example (cont.)
R(0) =1, R(1) = 0.9, R(2) = 0.81
Why is h=[0.9, 0] ?
R= 0.9R= 0.9
R= 0.9 ×0.9 = 0.81
Because
process is
1st order
x n[ ]x n-1[ ]x n- 2[ ]
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. . . . . . . . .
. . . x(n-1,m-1) x(n-1,m)
. . . x(n,m-1) x(n,m)
R(0) = RMS found...
R(1) = ave horiz-1( )R(2) = ave vert -1( )R(3) = ave diag-1( )
hopt = R-1 ×r same formula!...
"Template" and filter size may vary.
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Reasons for multidimensional
prediction: Image processing?
Consider solar radiation: We have relation to “last hour”, but;
◦ Don’t we have relation to “yesterday, same hour”?
◦ What about “last year, same hour”?
◦ What about “yesterday’s wind speed”?
◦ What about “last week’s electricity consumption”?
Do we need extremely long prediction size, N?
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The trick is to put “related” terms
near to each other
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instead of
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Then we can use 2D prediction
with similar correlation
definitions…
![Page 52: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/52.jpg)
… and achieve low prediction error
![Page 53: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/53.jpg)
Putting related items near each other
is good for other
methods, too…
![Page 54: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/54.jpg)
Other methods may include:
◦ Nonlinear prediction
◦ Neural networks
◦ Transformation (Fourier, wavelet, etc.)
◦ Adaptive methods
A case for solar radiation prediction:
![Page 55: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/55.jpg)
Correlation may include an
auxilliary signal Yielding a “hidden” process:
◦ Hidden Markov Model
We observe
pressure
We predict the
wind speed!
![Page 56: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/56.jpg)
Result is “wind measurement” using a barometer
The results are accurate enough to make RES sizing!
![Page 57: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/57.jpg)
Some examples with nonlinear prediction
(thanks to graduate students)
7860 7880 7900 7920 7940 7960 7980 8000 -2
0
2
4
6
8
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12
Saat
Rüzg
ar Hı
zı (m
/s)
Ölçülen Tahmin
7860 7880 7900 7920 7940 7960 7980 8000 0
1
2
3
4
5
Saat
Rüzg
ar Hı
zı (m
/s)
Ölçülen Tahmin
İzmir
Antalya
![Page 58: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/58.jpg)
and their distributions (İzmir)
0
2
4
6
8
10
12
14
16
18
20
Dağılım Yüzdeleri
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Rüzgar Hızı Durum Aralığı (m/s)
0
2
4
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8
10
12
14
16
18
20
Dağılım Yüzdeleri
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Rüzgar Hızı Durum Aralığı (m/s)
Real
Predicted
![Page 59: Ömer Nezi̇h Gerek Presentation1](https://reader034.fdocuments.co/reader034/viewer/2022042515/577c83621a28abe054b4ceae/html5/thumbnails/59.jpg)
Motto:
Know your math.
or, keep a SP guy around you