=Paper=
{{Paper
|id=Vol-2391/paper11
|storemode=property
|title=Solution for the problem of the parameters identification for autoregressions with multiple roots of characteristic equations
|pdfUrl=https://ceur-ws.org/Vol-2391/paper11.pdf
|volume=Vol-2391
|authors=Nikita Andriyanov,Maxim Sluzhivyi
}}
==Solution for the problem of the parameters identification for autoregressions with multiple roots of characteristic equations ==
https://ceur-ws.org/Vol-2391/paper11.pdf
Solution for the problem of the parameters identification for
autoregressions with multiple roots of characteristic equations
N A Andriyanov1,2, M N Sluzhivyi1
1
Ulyanovsk State Technical University, Severny Venets, 32, Ulyanovsk, Russia, 432027
2
Ulyanovsk Civil Aviation Institute, Mozhaiskogo, 8/8, Ulyanovsk, Russia, 432071
e-mail: nikita-and-nov@mail.ru
Abstract. When describing a real image using a mathematical model, the problem of model
parameters identification is of importance. In this case the identification itself is easier to
perform when a particular type of model is known. In other words, if there is a number of
models characterized by different properties, then if there is a correspondence with the type of
suitable images, then the model to be used can be determined in advance. Therefore, in this
paper, we do not consider the criteria for model selection, but perform the identification of
parameters for autoregressive models, including those with multiple roots of characteristic
equations. This is due to the fact that the effectiveness of identification is verified by the
images generated by this model. However, even using this approach where the model is
known, one must first determine the order of the model. In this regard, on the basis of Yule-
Walker equations, an algorithm for determining the order of the model is investigated, and the
optimal parameters of the model are also found. In this case the proposed algorithm can be
used when processing real images.
1. Introduction
Mathematical modeling is used in many areas of science and technology, including image processing.
In particular, methods focused on the description of images using models of random fields (RF), allow
the development of algorithms for parameter estimation, filtering, detection of anomalies in the
background of images and analyze them for a large number of simulated images.
There exist many images that are characterized by a smooth change in brightness [1,2]. Usually
such images have a slowly decreasing correlation function (CF) in a certain given neighborhood. The
first-order autoregressive (AR) image models [3,4] do not provide a strong correlation between pixels,
since its CF decreases exponentially. Meanwhile, separable RFs that can be generated by one-
dimensional autoregression with multiple roots of characteristic equations [5-6] are known. An
important feature of ARs with multiple roots is their considerable simplicity compared to arbitrary
high-order ARs. Also doubly stochastic models are used to describe real images [7-10]. However, if
the structure of the described image is not so complicated, then it is better to use simpler models. This
is due to the fact that the computational complexity of a double-stochastic model is much higher than,
for example, a model with multiple roots, for which the AR of any order can be described using a
single parameter.
At the same time, when describing images, for example, satellite images using mathematical
models, it is necessary to identify parameters of the model. At the same time, the model should be
selected in such a way that a compromise is found between its complexity and the similarity of the
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019)
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N A Andriyanov, M N Sluzhivyi
simulated image with the real one. The paper discusses the solution of the problem of the parameters
identification for AR models with multiple roots of characteristic equations, which, owing to the
separability of the CF, can significantly simplify this task. It should be noted that the application of the
parameter identification approach based on Yule-Walker equations to determining the order and
correlation parameters of AR models with multiple roots has a scientific novelty.
2. Brief overview of the parameters identification methods for a random processes
At present, the identification task for AR processes, the moving average (MA) and autoregression with
moving average model (ARMA) by single-channel observations without noise is the most well-
studied. Especially many methods have been developed for estimating the parameters of such
processes. The estimation methods are based either on the direct use of observations, or on the initial
calculation of the sample statistical characteristics (autocorrelations, spectral densities) from these
observations, and then using them to determine parameter estimates. The first group of methods
includes the least squares method (LSM) for the AR process [10,11], the maximum likelihood method
(ML) for the AR, MA, and ARMA processes, and various types and modifications of these methods
[12-14]. The second group is the Yule-Walker method for the AR process [13], the Box and Jenkins
correlation methods for the ARMA, MA [13] processes, the Lindberger method for the AR, MA, and
ARMA processes [15], the Durbin method for the MA process [16] , the Cleveland inverse
autocorrelation method for the ARMA process [17] and other methods [18]. In both groups, there are
methods in which parameter estimates are calculated by linear algorithms by solving a certain system
of linear equations for example, LSM and Yule-Walker method for the AR process, as well as
methods that use nonlinear methods for calculating estimates, which are reduced to numerical
minimization algorithms for a certain function of the parameters, for example, the ML method and the
Lindberger method for the APMA processes. The methods for determining the class of processes that
can be AR, MA, and ARMA are much less developed [19]. In the work of Kitler and Whitehead [20],
the problem of class determination is reduced to the problem of determining the order of the ARMA
process. Therefore, the solution of the determination order and class of the model task is of interest. In
the simplest case, such a task is reduced to a problem when it is necessary to determine its order using
the selected model of AR. So optimal identification is very important for satellite image processing
[21,22].
3. Autoregressions with multiple roots of characteristic equations
Using AR models with multiple roots of characteristic equations, it is possible to obtain realizations
of RFs that will be close in their properties to real images. In this case an important property of the
generated RF will be its quasi-isotropy. The general formula for models of different multiplicities can
be written as follows
N1 N2
xi , j = βξ i , j − ∑ ∑ α i1 , j1 xi −i1 , j − j1 , (1)
i1 = 0 j1 = 0
where N1 and N 2 characterize the multiplicity of the model; coefficients α i1 , j1 (α 0 ,0 = 0) are the
products of the corresponding coefficients of one-dimensional AR along the axes x and y
α i1 , j1 = α xi1 α yj1 . (2)
The coefficients of one-dimensional AR (2) can be obtained using expressions
α xi1 ( ρ x , N 1 ) = (−1) i1 +1 C Ni11 ρ xi1 , α yj1 ( ρ y , N 2 ) = (−1) j1 +1 C Nj12 ρ yj1 , (3)
n!
where С n = m!(n − m)! . Two-dimensional model's coefficient β is the normalized product of the
m
corresponding coefficients of one-dimensional AR along the axes x and y
σ
β = x βxβy , (4)
σξ
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1 1
− −
2 2
(1 − ρ x )
2 2 N1 −1
(1 − ρ y )
2 2 N 2 −1
.
where β x = N1 −1 , βy = N 2 −1
l 2 l 2
∑
l =0
(C N1 − l ρ x )
l
∑l =0
(C N 2 − l ρ y )
l
When identifying parameters, let’s approximate the CF of the initial data by the most appropriate
model. In order to obtain the CF models of arbitrary orders, it is necessary to use expressions for one-
dimensional CF of AR with multiple roots of characteristic equations [22]
m −1
ρ 2( m −l −1) ,
B x (k ) = σ x2 ∑ g (m, l , k ) (5)
l =0 (1 − ρ 2 ) 2 k −l −1
(m + k − 1)!(2m − l − 2)!
where g (m, l , k ) = . The variance of independent random values
l !(m − 1)!(m − l − 1)!(m + k − l − 1)!
ξi , i = 1,2,..., n , can be found for a given variance of simulated RF Bx (0) = σ x2
σ x2 (1 − ρ 2 ) 2 m −1
σ ξ2 = . (6)
∑ (C ρ )
m −1
l l 2
m −1
l =0
Thus, for spatial AR equation with characteristic roots of multiplicities (m1,m2) the expression for
CF can be written as
m1 −1
ρ x 2( m −l −1) m −1
1
ρ y 2( m −l −1)
2
B xy (k1 , k 2 ) = σ ∑ g (m1 , l , k1 ) ∑ g (m2 , l , k 2 ) (1 − ρ 2 ) 2k −l −1 .
2
2
(7)
(1 − ρ x ) 2 k −l −1 l =0
x 2 1 2
l =0 y
However, in the case of a model with multiple roots it is possible to separately carry out the
identification of parameters by row and column, using formula (5).
4. Identification of parameters based on theoretical values of correlation functions
To solve the problem of identification, we will use AR models of arbitrary order
xi = ρ1 xi −1 + ρ 2 xi − 2 + ... + ρ m xi − m + ξ i , i = 1,2,..., M , (8)
where m is order of AR model.
Choosing parameters ρ1 , ρ 2 ,..., ρ m it is possible to get a Gaussian RF {xi } , i = 1,2,..., M with a
variety of correlation properties. In this case to write values of CF you can use the following
expression
R x (k ) = ρ 1 R x (k − 1) + ρ 2 R x (k − 2) + ... + ρ m R x (k − m) , k > 0 . (9)
CF of models with multiple roots for different roots zν , ν = 1, 2, ..., m, of characteristic equations
described by expression
z m − ρ1 z m −1 − ρ 2 z m − 2 − ... − ρ m = 0 , (10)
under the condition of stability zν < 1 , v = 1,2,..., m , are represented by the following sum
R x (k ) = A1 z1 + A2 z 2 + ... + Am z m .
k k k
(11)
Substitution in (9) values k = 1, 2, … , m leads to the well-known Yule-Walker system of
equations, which, for example, for second-order systems takes the form
ρ 1 + ρ 2 R(1) = R(1),
(12)
ρ 1 R(1) + ρ 2 = R(2).
The solution of this system allows you to find the coefficients ρ1 , ρ 2 ,..., ρ m of equation (8)
based on predetermined or estimated values of CF Rx(1), Rx(2), …, Rx(m).
We will perform parameter identification for models with multiple roots of characteristic equations
of 1-4th orders. In this case the order can be identified if we take into account only the coefficients that
make some contribution to the model. Table 1 presents the results of identification of the correlation
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 81
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parameters for the values of CF of AR with multiple roots. By rows, the actual multiplicity is
presented, by columns — estimated parameters are presented. We assume ρ = 0.8 for models of all
orders. In the left column – the values found, in the right column - the real values.
Table 1. Identification of parameters based on the theoretical values of CF
m*=1 m*=2 m*=3 m*=4
ρ1 = 0.8, ρ1 = 0.8, ρ1 = 0.8, ρ1 = 0.8, ρ1 = 0.8, ρ1 = 0.8, ρ1 = 0.8, ρ1 = 0.8,
ρ 2 = 0, ρ 2 = 0, ρ 2 = 0, ρ 2 = 0, ρ 2 = 0, ρ 2 = 0, ρ 2 = 0, ρ 2 = 0,
m=1
ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 0,
ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0
ρ1 = 0.976, ρ1 = 1.6, ρ1 = 1.6, ρ1 = 1.6, ρ 1 = 1.6, ρ1 = 1.6, ρ 1 = 1.6, ρ1 = 1.6,
ρ 2 = 0, ρ 2 = −0.64, ρ 2 = −0.64, ρ 2 = −0.64, ρ 2 = −0.64, ρ 2 = −0.64, ρ 2 = −0.64, ρ 2 = −0.64,
m=2
ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 0, ρ 3 = 3 × 10 −14 , ρ 3 = 0, ρ 3 = 3 × 10 −14 , ρ 3 = 0,
ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0
ρ1 = 0.992, ρ 1 = 2.4, ρ 1 = 1.92, ρ 1 = 2.4, ρ 1 = 2.4, ρ 1 = 2.4, ρ1 = 2.4, ρ 1 = 2.4,
ρ 2 = 0, ρ 2 = −1.92, ρ 2 = −0.937, ρ 2 = −1.92, ρ 2 = −1.92, ρ 2 = −1.92, ρ 2 = −1.92, ρ 2 = −1.92,
m=3 ρ3 = 0, ρ 3 = 0.512, ρ 3 = 0, ρ 3 = 0.512, ρ 3 = 0.512, ρ 3 = 0.512, ρ3 = 0.512, ρ 3 = 0.512,
ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ4 = 0 ρ 4 = −1× 10−13 ρ4 = 0
ρ1 = 0.995, ρ 1 = 3.2, ρ 1 = 1.969, ρ 1 = 3.2, ρ1 = 2.837, ρ 1 = 3.2, ρ 1 = 3.2, ρ 1 = 3.2,
ρ 2 = 0, ρ 2 = −3.84, ρ 2 = −0.979, ρ 2 = −3.84, ρ 2 = −2.724, ρ 2 = −3.84, ρ 2 = −3.84, ρ 2 = −3.84,
m=4 ρ3 = 0, ρ 3 = 2.048, ρ3 = 0.886,
ρ 3 = 0, ρ 3 = 2.048, ρ 3 = 2.048, ρ 3 = 2.048, ρ 3 = 2.048,
ρ4 = 0 ρ 4 = −0.41 ρ4 = 0 ρ 4 = −0.41 ρ4 = 0 ρ 4 = −0.41 ρ 4 = −0.41 ρ 4 = −0.41
Analysis of table 1 shows that the values of the correlation coefficients are estimated the more
accurately, the higher the multiplicity is. If the estimated multiplicity exceeds the real value, then the
additional coefficients are either 0 or very close to it. Thus, the process of the model order
identification can be carried out first for some large multiplicity. If the resulting coefficients do not
have zero coefficients, then the calculation should be carried out at a higher multiplicity until we get
zero coefficients. If there are zero coefficients, the order corresponds to the number of the last
significant coefficient.
A similar relationship between correlation coefficients and CF values can also be obtained for the
two-dimensional case, i.e. images. The relation for CF values corresponding to expression (9) for the
three-point model, is written as
R(k1 , k 2 ) = ρ10 R(k1 − 1, k 2 ) + ρ 01 R(k1 , k 2 − 1) + ρ11 R(k1 − 1, k 2 − 1), k1 > 0, k 2 > 0. (13)
It is easy to verify that solving the two-dimensional Yule-Walker system of equations (13) for the
Habibi model CF will give correlation coefficients identical to the coefficients of the first-order two-
dimensional AR model. To increase the order of the AR, as in the one-dimensional case it is necessary
to increase the number of correlation coefficients. In this case, the RF model can be written as follows
mj mi
x i , j = ∑∑ ρ kl x i − k , j −l − ρ 00 x i , j + ξ i , j , i = 1, M 1 , j = 1, M 2 , (14)
l =0 k =0
where { xi , j } is RF implementation or simulated image; ρ kl are correlation coefficients for elements
lagging behind each other along the axes i and j by k and l pixels respectively; { ξ i , j } is two-
dimensional RF of independent Gaussian random variables with zero mean M { ξ i , j } = 0 and variance
mj mi
M { ξ i2, j } = σ ξ2 = [1 − ∑ ∑ ρ kl R(k , l ) + ρ 00 R(0,0)]σ x2 ; mi and mj are orders of the model; M1 and M2
l =0 k =0
are the image size.
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The number of components of the model, taking into account the random increment will be equal
to (mi + 1) × (m j + 1) . Using formulas (9) and (13), we can write the relation for calculating the CF
values
mj mi
R (k1 , k 2 ) = ∑∑ ρ kl R (k1 − k , k 2 − l ) − ρ 00 R (k1 , k 2 ) , k1 > 0, k 2 > 0 . (15)
l =0 k =0
The expression (15) can also be used for the case of non-separable CFs if the parameters of an
arbitrary AR RF are identified and the order is a priori unknown.
5. Parameter identification on the basis of real images CF
Let us identify the parameters of the model based on the proximity of the CF model and the given
data. In the first case, we will consider the AR with a separable CF and separately calculate the
coefficients for the row and column. In the second case, we use equations based on the expression
(15).
Let there be a real image represented as I (i, j ), i ∈1,..., M 1 , j ∈1,..., M 2 . Then its CF can be
expressed as follows
M1 M2
1
R I ( k1 , k 2 ) = ∑ ∑ ( I (i, j ) − m )( I (i − k , j − k ) − m ) ,
σ I2 i = k +1 j = k +1
I 1 2 I
(16)
1 2
where mI is average brightness over the entire image; σ I2 is brightness variance calculated over the
entire image.
Figure 1a and Figure 1b show the image to be investigated and its CF, respectively. Identification is
performed for the 4th order AR model.
- for RF having separable CF the results are as follows:
ρ10 = 1.098; ρ 20 = −0.39; ρ 30 = 0.364; ρ 40 = −0.082; ρ 01 = 0.828; ρ 02 = 0.0047; ρ 03 = 0.111; ρ 04 = 0.038 ;
ε = ( Rˆ − R) 2 = 0.387.
- for RF having non-separable CF the results are as follows:
ρ10 = 0.998; ρ 20 = −0.514; ρ 30 = 0.43; ρ 40 = −0.133; ρ 01 = 0.185; ρ11 = 0.152; ρ 21 = −0.194, ρ 31 = 0.142; ρ 41 = −0.092;
ρ 02 = −0.174; ρ12 = 0.309; ρ 22 = −0.265; ρ 32 = 0.217; ρ 42 = −0.097; ρ 03 = −0.048; ρ13 = 0.186; ρ 23 = −0.205;
ρ33 = 0.153, ρ 43 = −0.079; ρ 04 = −0.122; ρ14 = 0.267; ρ 24 = −0.242; ρ34 = 0.22; ρ 44 = −0.102; ε = ( Rˆ − R) 2 = 0.014 .
a) b)
Figure 1. The image (a) for which the adjustment of the parameters is carried out and its CF (b).
Analysis of the obtained values of the error variances shows that the use of the model with an non-
separable CF provides a greater proximity between the modeled and the real CF. This is explained by
the fact that for this model 24 correlation parameters were calculated, while for a model with a
separable CF only 8 parameters. At the same time, a sufficient proximity of the CF is provided,
especially in the neighborhood of zero. Therefore, it is advisable to use such models to reduce
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 83
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computational costs. Figure 2 shows the CF cross sections for the original image (presented by solid
line), as well as for the RF with separable CF (presented by dash-dotted line) and the RF with non-
separable CF (presented by dashed line).
Figure 2. CF cross sections of the original image and identifiable models.
Analysis of the presented curves shows that as k increases the discrepancy between the real and
simulated CF also increases. It is possible to achieve greater proximity by increasing the order of the
AR, but this leads to higher computational costs. Similar research was conducted with a sample of 100
images. Analysis of the results shows that the use of non-separable CF models provides the proximity
of CF to 10-15 times more than the use of separable CF models in 85% of cases. However, in 15% of
cases separable models were successfully used to describe the real image, which made it possible to
significantly reduce computational costs of image processing.
6. One dimensional example of calculating parameters
However, if a model with multiple roots is used, it is sufficient to calculate only the first correlation
coefficient, then calculate the correlation parameter using the first correlation coefficient and use
correlation parameter to find the remaining correlation coefficients according to the expression (3).
For example, if ρ1 = 3.6 and N=4 then it is easy to calculate parameter of the AR with multiple
roots of characteristic equations from the following equation
3.6 = (−1)1+1 С 41 ρ 1 , → ρ = 0.9 . (17)
After that we find the second, third and fourth correlation coefficients
ρ 2 = (−1)1+ 2 С 42 ρ 2 = −4.86,
ρ 3 = (−1)1+3 С 43 ρ 3 = 2.916, (18)
ρ 4 = (−1) 1+ 4
С ρ = −0.6561.
4
4
4
So it is quite a simple task to perform identification of parameters in this case.
7. Conclusion
In this paper a brief overview of the methods used to identify the parameters of the AR processes is
presented. Models with multiple roots and a method for determining the order of the model based on
the Yule-Walker equations are considered. It is shown that the proposed method enables to determine
the order of the model for simulated images with sufficient accuracy. A comparative analysis of the
identification of the parameters of models with a separable and non-separable CF of the real image is
performed. The analysis shows that models with a non-separable CF require more computational costs
at the same order of a model, however, they provide greater proximity of the CF in comparison with
separable CF models. The identification results may be used to describe real satellite images and for
the processing of such images.
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Acknowledgement
The work was supported by RFBR, Projects №17-01-00179, №18-31-00056.
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