DTIC ADA494641: Ozturk Algorithm Augmentations

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AFRL-RI-RS-TR-2009-23
In-House Final Technical Report

January 2009

OZTURK ALGORITHM AUGMENTATIONS

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OZTURK ALGORITHM AUGMENTATIONS
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Steven Salerno
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14. ABSTRACT

In this effort, augmentations to the Ozturk Algorithm were investigated, including additional graphical representations
and determination of data correlation effects as introduced by data filtering. A graphical user interface was developed to
demonstrate the algorithm concepts.
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Ozturk Algorithm, Rank-ordered statistics
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std. Z39.18
Table of Contents 
1.0 Introduction ______________________________________________ 1 
1.1 Overview of the Ozturk Algorithm ___________________________ 1 
1.2 Summer Overview _________________________________________ 2 
2.0 PDF Identification & Analysis – Ozturk Algorithm _____________ 2 
2.1 Procedure – Ozturk Algorithm ______________________________ 3 
2.2 Extension of Features – Ozturk Algorithm _____________________ 7 
2.2.1 Extension – Plotting Techniques ____________________________ 8 
2.2.2 Extension – Correlation Test ______________________________ 13 
2.2.3 Extension – Multi-Distribution Identification ________________ 18 
3.0 GUI Implementation ______________________________________ 19 
3.1 GUI – Normality Test _____________________________________ 20 
3.2 GUI – Identification Charts ________________________________ 21 
3.3 GUI – Analysis with Identification Charts ____________________ 22 
3.4 GUI – Correlation Test ____________________________________ 22 
3.5 GUI – User Information Tabs ______________________________ 23 
4.0 Future Work ____________________________________________ 25 
4.1 Acknowledgements _______________________________________ 26 
4.2 References ______________________________________________ 27 

i
List of Figures 

Figure 2.1 – Vector Chart for Normal (n = 6) .................................................................... 4 
Figure 2.2 – Sample Data with Confidence Ellipses .......................................................... 6 
Figure 2.3 – Distribution Identification Chart (Old)[1] ........................................................ 8 
Figure 2.4 – Distribution Identification Chart (New) ......................................................... 9 
Figure 2.5 – 3-D Distribution Identification Chart ............................................................. 9 
Figure 2.6 – Multiple 3-D Distribution Identification Charts ........................................... 10 
Figure 2.7 – Regular ID Chart with Data .......................................................................... 11 
Figure 2.8 – Bivariate Histogram with Data[6] .................................................................. 11 
Figure 2.9 – Line Contour Chart with Data ...................................................................... 12 
Figure 2.11 – FFT of Uncorrelated/Correlated Data ........................................................ 15 
Figure 2.12 – Low pass FFT Real and Imaginary Plots ................................................... 16 
Figure 2.13 – Bandpass FFT Real and Imaginary Plots ................................................... 16 
Figure 2.14 – ROC Curve of Correlation Test .................................................................. 17 
Figure 2.15 – Laplace(0,0.625) v Normal(0,1) ................................................................. 18 
Figure 3.1 – Updated GUI Main Window ........................................................................ 20 
Figure 3.2 – GUI Normality Test Panel ............................................................................ 21 
Figure 3.3 – GUI Normality Test Plot .............................................................................. 21 
Figure 3.4 – GUI Identification Charts Panel ................................................................... 22 
Figure 3.5 – GUI Analysis Panel ...................................................................................... 22 
Figure 3.6 – GUI Correlation Test Plot Area .................................................................... 23 
Figure 3.7 – GUI User Information Tabs.......................................................................... 24 
ii
1.0 Introduction
This report covers work initiated in the Summer of 2006 and completed in the
Summer of 2008 as a summer engineering aide. Work in the applications of ranked-
ordered statistics as a method of signal data analysis was conducted. More specifically
work was conducted on the Ozturk algorithm, which is considered to have demonstrated
superior performance over histogram and simple parameter-based techniques [1-4]. The
work includes the research and implementation of new techniques based on the Ozturk
algorithm. A GUI was also created in order to house the previously done work, [5] and
the newly created implementation tools.

1.1 Overview of the Ozturk Algorithm
The Ozturk algorithm is an identification algorithm that allows for a variety of
tests that not only identify a best fit probability density function (PDF), but also analyzes
the sample data. The core algorithm was developed by Aydin Ozturk [3], and further
worked upon in concurrence with E.J. Dudewicz and others [4]. Prior work was carried
out with the help of Syracuse University and the only documented implementation of the
algorithm for real world use was associated with research conducted at the AFRL Sensors
Directorate at the Rome Research Site.
The basis of the algorithm deals with constructing a null-linked graph of vectors
from the rank ordered statistics of the sample data. The algorithm is scale and location
invariant, which makes it particularly attractive. From this graph one can easily identify
a distribution based on its endpoint, Q , and surrounding null points. From the ordered
n
statistics other tests can be carried out, including but not limiting to, testing for
Normality, parameter estimation, identifying the best fit distribution and the newest test,
the detection of correlation in Gaussian data. The power of the test comes from the
ability to use small sample sizes while still being able to correctly identify the best fit
distribution. The entirety of the algorithm will be discussed in Section 2.

1
1.2 Summer Overview
Much of the work done the Summer of 2006 was based on the previous work
done in the Summer of 2005. Since there was now a working GUI implementation of the
Ozturk algorithm, it was time to both test the algorithm’s capabilities and limits. In [1-4]
there has been no mention of the result of correlated data. This produces a potential
problem with real world data, since it is not always uncorrelated. When you have this
real world data, how do you determine whether or not it is correlated, and whether or not
the correlation is strong enough to affect the algorithm? There needs to be either a pre-
processing step or test to determine if the data is correlated, and if it is how strongly it is
correlated. The second major constraint is that, there are distributions that can mimic
other distributions if the parameters are set in a specific manner. In this case the
algorithm can only identify one distribution, while there may actually be multiple
distributions the data could be constructed from. Both of these issues were the basis for
most of the work, along with the full construction of a new Matlab GUI to provide the
user the ability to implement the new tools.

2.0 PDF Identification & Analysis – Ozturk Algorithm
Most statistical tests that are used presently test a data set for a specific
distribution. Often the test is designed specifically to assess the normality (Gaussian PDF
is assumed) of a data set. However when the solution does not match the hypothesis no
alternative possibility is given. This results in the tedious task of having to do multiple
statistical tests in order to identify the best fit distribution. Also a majority of the tests
start to lose their power when the sample size decreases. Both these common issues are
addressed through the Ozturk algorithm [1-4]. The algorithm uses its own test statistics
to create “linked vectors” to create a goodness-of-fit test for univariate and multivariate
distributions.

2
2.1 Procedure – Ozturk Algorithm
General reference is made to [2, 3]. Let X ≤ X ≤ … ≤ X be the sample
1:n 2:n n:n
ordered observations from a distribution with the cumulative distribution function (CDF)
F(x;μ, σ) = F{(x- μ)/σ} where μ and σ are the location and scale parameters, respectively.
For our case we use the Normal distribution as the null hypothesis; however any
distribution could be used as the null.
For illustration we will assume that X ≤ X ≤ … ≤ X is from a Normal
1:n 2:n n:n
distribution with an unknown mean μ and unknown variance σ2. To make the sample
linked vectors we need two main components, the length of the ιth vector, and the angle to
the horizontal axis θ. Let
Y =(X − X)/S 2.1
i:n i
where X = ∑X /n and S ={∑(X − X)2 /(n−1)}1/2. Y is now considered the
i:n i:n i:n
length of the vectors.
Now let m , m , …, m be equivalent to the expected order statistics of the
1:n 2:n n:n
standard Normal distribution, m = E((X - μ)/σ). The angle horizontal to the x-axis is θ
i:n i:n i
= πФ(m ) where π = 3.14159… and
i:n
m
i:n
Φ(m ) = ∫(2π)−1/2e(−t2/2)dt. 2.2
i:n
−∞
Since m < m < … < m and 0 < Ф(m ) < 1 then 0 < θ < θ < … < θ < π. θ is
1:n 2:n n:n i:n 1 2 n i:n
now considered the angle between the x-axis and the ιth vector.
To obtain the sample linked vectors we use the points Q = (U,V), where
i i i
i i
U = (∑|Y |Cosθ)/nand V = (∑|Y |Sinθ)/nare the coordinates of Q (i =
i j:n j i j:n j ι
j=1 j=1
1,…,n) and Q = (0,0). U and V can then be defined as the test statistics, as following
0 i:n i:n
n
U =1/n∑|Y |Cosθ , 2.3
i:n i:n i
i=1
n
V =1/n∑|Y |Sinθ . 2.4
i:n i:n i
i=1
3
To obtain the null linked vectors to construct the “known” of the null hypothesis
m΄ = E(|Y |), and E(m ) are used. To obtain the expected values equation 2.5 must be
i:n i:n i:n
numerically integrated with the specific n size of the data to generate the values.
n! ∞ ⎡1 ⎤k−1⎡1 ⎤n−k
E(x ) = ∫x −Φ(x) +Φ(x) φ(x)dx, 2.5
k|n (n−k)!(k −1)! ⎢⎣2 ⎥⎦ ⎢⎣2 ⎥⎦
−∞
x
where φ(x) =(2π)−1/2e−x2/2 and Φ(x) = ∫φ(x)dx.
0
Using the expected values of m and Y we obtain the point Q = (U ,V ) which
i:n i:n n n n
becomes the null-linked vectors endpoint that is the “known”, other extensions use this
information for analysis.
In a sample that we assume to be Normal, we would expect that the sample linked
vectors would closely follow the trajectory of the null-linked vectors, as in Figure 2.1.
vv nn== 66
QQ00QQnn
θθ66
VV
θθ55
θθ44
θθ33
θθ22
θθ11
uu
UU
Figure 2.1 – Vector Chart for Normal (n = 6)

However if the sample is not Normal the two paths should differ; how much they differ
by depends on the sample distribution. This use of the algorithm allows for an ad hoc
version (visual inspection) to be used and allow for distribution identification. This ad
hoc method can be used; however it is rarely done due to the fact that the algorithm
generates its own set of test statistics that can with far better accuracy provide an answer
to the best fit distribution of a data set in an automated fashion.
Because of the properties of the test statistics generated by the Ozturk algorithm,
we can generate a set of equations to model their behavior. Using m΄ = E(|Y |), we
i:n i:n
4
define the expected value for Q for the Normal distribution to be E(Q ) = E(U , V ) =
n n n n
n
(0,(1/n)∑m' Sinθ). Symmetric identities for the Normal distribution can be
i:n i
i=1
employed to see that E(U ) = 0, and a model can be fitted to V to obtain
n
E(V ) =0.326601+0.412921/n. 2.6
n
These models can also be applied to other symmetric distributions as well. It is seen that,
as long as a distribution is symmetric; E(U ) = 0. With this, models can be provided for
n
the E(V ) for any other symmetric distribution. Also using more properties of the test
n
statistics and the linked vectors we can obtain models for the variances of U and V
n n
Var(U )=σ2 =0.02123/n+0.01765/n2, 2.7
n u
Var(V )=σ2 =0.04427/n−0.0951/n2. 2.8
n v
For large values of n it can be shown that the distributions of U and V are both Normal.
n n
Since Q is the joint distribution of U and V , we can apply the bivariate normal
n n n
distribution to the Q endpoint. Using the exponent component of the general bivariate
n
normal we have
1 ⎡(U −μ)2 (U −μ)(V −μ) (V −μ)2⎤
⎢ n u −2ρ n u n v + n v ⎥, 2.9
2(1−ρ2) σ2 σσ σ2
⎣ ⎦
u u v v
where μ and μ are the means, σ 2 and σ 2 are the variance of U and V respectively, and ρ
1 2 1 2
is the coefficient of correlation between U and V . Under the assumption that μ is
n n u
always 0 and that value of ρ between U and V is 0, we obtain the simplified version of
n n
equation 2.9,
U2 (V −μ)2
n + n v =−2lnα, 2.10
σ2 σ2
u v
where α is the confidence level. For any given sample size, μ , σ 2, and σ 2 can be
v u v
obtained from equations 2.6, 2.7 and 2.8, respectively. With these values equation 2.10
can be solved for α, and return the specific α value for each set of endpoints of a given
data set. Setting equation 2.10 to a set of ellipses, generated by varying the value of α,
and putting it onto the null endpoint Q , we can create a set of 100(1- α) % confidence
n
ellipses around the Normal distribution. With these confidence ellipses we can now set
thresholds on the sample data in order to differentiate between potential Gaussian data, or
5