=Paper=
{{Paper
|id=Vol-2005/paper-06
|storemode=property
|title=Digital signal processing under uncertainty conditions. Interval Approach
|pdfUrl=https://ceur-ws.org/Vol-2005/paper-06.pdf
|volume=Vol-2005
|authors=Sergey I. Kumkov
}}
==Digital signal processing under uncertainty conditions. Interval Approach==
https://ceur-ws.org/Vol-2005/paper-06.pdf
Digital signal processing
under uncertainty conditions.
Interval Approach
Sergey I. Kumkov1,2
1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch RAS,
Ekaterinburg, Russia
2
Ural Federal University, Ekaterinburg, Russia
kumkov@imm.uran.ru
Abstract. Digital signal processing under uncertainty conditions is con-
sidered. The signal represents an experimental chemical process, whose
parameters have to be estimated. Measurments both of the process and
its argument contain errors of bounded values. Sample of the process
measurements is very short and there is uncertainty of probability char-
acteristics of the errors. So, it is difficult to validate application of stan-
dard statistical methods to estimating the process parameters. An alter-
native is in application of the Interval Analysis methods. In the work,
these methods are used for constructing the information set of admissible
values of the process parameters and admissible tube of its dependencies.
Keywords: Digital signal, processing, algorithms, noised measurements,
two-dimensional uncertainies, interval analysis methods, set of admissi-
ble parameters, tube of admissible dependencies
1 Introduction
As a rule, in investigations of experimental chemical processes, data are obtained
with measuring errors. It is used to process such data by the standard methods
that are based on the mathematical statistics ideology [1], [2], [3].
But in practice, a sample of measurements is very short and the errors’
probability characteristics are unknown. Moreover, the errors can be not only
in the process measurements, but, also, in ones of the process’ argument, and
uncertainty of each measurement becomes two-dimensional. So, under these con-
ditions, it is difficult (or impossible) to validate application of standard methods.
As an alternative, application of the statistical methods can be completed by
using the Interval Analysis ones.
To do this, the process is described by some model fuction with a vector of
parameters; in our investigation, the linear dependence is used to describe the
process. The problem is formulated for estimation of admissible set of these pa-
rameters. Such a set is used to call the Information Set. It comprises only such
parameters of the model that are consistent with its description, accumulated
49
sample of measurements, and the given bounds on the measuring errors in the
process values and values of its argument. On the basis of the determined Infor-
mation Set, corresponding tube of the admissible dependencies of the process is
built.
The paper has the following structure. In Section 2, specific properties of
experimental data and difficulties of application of standard methods for their
procession are discussed, the typical model of a chemical process is introduced,
short description of the main Interval Analysis procedures used for constructing
the Information Set of process parameters is given, and problem of estimation is
formulated. In Section 3, results of processing real experimental data are given.
Here, results obtained by the interval approach are compared with ones calcu-
lated by formal aplication of standard least square means method. In Section 4,
conclusions are given on abilities of the suggested Interval Analysis approach
and its applications in addition to known estimation procedures [1], [2], [3].
2 Specific properties of experimental data.
Interval approach to estimating
the process parameters. Problem formulation
In practice of chemical experiments, a sample of measurements can be very short
and the errors’ probability characteristics are unknown. Moreover, the errors
can be not only in the process measurements, but, also, in ones of the process’
argument, and uncertainty of each measurement becomes two-dimensional. So, it
becomes difficult or even impossible to validate application of standard methods
to estimation of the process parameters. As an alternative, application of the
statistical methods can be completed by using the Interval Analysis ones.
Foundations of Interval Analysis and its applications to processing observa-
tions under uncertainty conditions were developed on the basis of the fundamen-
tal pioneering work of L.V. Kantorovich [4].
Nowadays, effective theoretical, applied, and numerical methods of the In-
terval Analysis have been created both abroad [5] and by Russian researchers
[6]. Special interval algorithms and software were developed for solving applied
problems of estimation of parameters for experimental chemical processes [8],
[9], [10], [11], [12].
Remind that the essence of the interval methods is in estimating the process
parameters under conditions of a short measurement sample, uncertainty of the
measuring errors probability characteristics, and under only interval bounding
onto the error values.
Introduce the following necessary definitions with using the standard on no-
tations in Interval Analysis [7].
The function describing the process has the form
F (x, A, B) = A + Bx, (1)
where x is the process’ argument; A, B are parameters to be estimated.
50
The uncertainty set of each measurement. Since absence of probability
characteristics of measuring errors, uncertainty of each measurement xn , Fn is
formalized as a rectangle H n with the left xn and right xn , lower F n and upper
F n boundaries
2a) {xn , Fn }, n = 1, N , H n :
2b) F n = [F n , F n ],
where F n = Fn − emax , F n = Fn + emax ; (2)
2c) xn = [xn , xn ],
where xn = xn − bmax , xn = xn + bmax ,
where emax , bmax are bounds onto maximal (on modulus) values of errors in the
process and its argument measurements.
The admissible value of the parameters vector (A, B) for the linear
model (1) is a pair
(A, B) : F (x, A, B) ∈ F n , for x ∈ xn , for all n = 1, N . (3)
and corresponding dependence F (n, A, B) is also called admissible.
The Information Set is a totality of all admissible values of the parameter
vector for model (1) satisfying the following system of interval inequalities:
I(A, B) = {(A, B) : F (x, A, B) ∈ F n , for x ∈ xn , for all n = 1, N }. (4)
The input sample (2a) is called consistent in the interval sense if by (4) there
exists at least one admissible value of the parameter vector and corresponding
admissible dependence.
The tube of admissible dependencies {T b(x)}, n = 1, N is a totality of
all admissible values of the dependence at the nth measuring of the process.
For the linear model (1) and the information set I(A, B), the tube boundaries
are calculated with taking into account the boundaries of the uncertainty sets
F n × xn (3)
T b(x) = min(A,B)∈I(A,B) F (x, A, B),
(5)
T b(x) = max(A,B)∈I(A,B) F (x, A, B).
Problem of estimation for the linear model (1) is formulated as follows:
by means of the Interval Analysis methods to construct the Information Set (4)
of the process parameters consistent with the given data (2) and to build the
tube (5) of the admissible dependencies of the process.
For the linear model (1), fast procedures for constructing the Information Set
(4) with exact description of its boundaries have been elaborated and applied to
solving many practical problems [8], [9], [10], [11], [12]. Due to direct using the
linearity of model (1), these procedures are more fast and give exact description
of the Information Set in comparison with even very powerful procedures of the
SIVIA-type [5].
51
Remark. As it will be shown below, formal application of the standard
Least Square Means method (LSQM) [2] and corresponding point-wise estimate
of the parameters (ASQ , BSQ ) for the linear model demonstrate to be useful for
analysis of the input sample (2) and for qualitative comparison with the results
on the basis of Interval Analysis.
3 Results of processing experimental data
The first example (Fig. 1) of experimental data is joined with investigating the
heat of fusion of cryolites (Institute of High Temperature Electrochemistry UrB
RAS, Ekaterinburg).
Figure 1a shows the sample of measurements (circles) and their two-dimen-
sional uncertainty sets built for bounds emax = 2 kJ mole−1 and bmax = 4 K
onto measuring errors of the process and its temperature argument. Remark
the difficult situation: there are only 7 measurements and their uncertainty sets
cross each other. At the left, there are four measurements with practically co-
inciding values of temperature. Point–dash lines correspond to the LSQM-line
and its rough standard tolerances ±3σ. The shadowed fragment is the tube of
admissible dependencies. The thick central line (Fig. 1a) marks the dependence
corresponding to the central point of the information set of admissible param-
eters (Fig. 1b), where, the cross marks the LSQM-point obtained by formal
application of standard LSQM-method.
In the next example (Fig. 2), the experimental data was obtainted in inves-
tigation of relative electric potential between Uranium and Gallium chlorides
in the process of treatment of nuclear waists (Institute of High Temperature
Electrochemistry UrB RAS, Ekaterinburg).
Figure 2a shows the sample of measurements (crosses) and their two-dimen-
sional uncertainty sets built for bounds emax = 0.0075 V and bmax = 5 K onto
measuring errors of the process and its temperature argument. Remark the diffi-
cult situation: there are only 10 measurements with coinciding values of the tem-
perature measurements and their uncertainty sets crossing each other. Moreover,
two measurements (at the right) and their uncertainty sets practicically coincide.
Point–dash lines correspond to the LSQM-line and its rough standard tolerances
±3σ. The shadowed fragment is the tube of admissible dependencies. The thick
central line (Fig. 2a) marks the dependence corresponding to the central point
of the information set of admissible parameters (Fig. 2b), where, the cross marks
the LSQM-point obtained by formal application of standard LSQM-method.
In the last example (Fig. 3), the experimental data was also obtainted in in-
vestigation of relative electric potential between Uranium and Gallium chlorides
in the process of treatment of nuclear waists (Institute of High Temperature
Electrochemistry UrB RAS, Ekaterinburg).
Figure 3a shows the sample of measurements (crosses) and their two-dimen-
sional uncertainty sets built for bounds emax = 0.0075 V and bmax = 5 K onto
measuring errors of the process and its temperature argument. Again remark
52
by IHTEC UrB RAS and IMM UrB RAS a)
32
Bounds onto measuring errors: Tube of admissible
_ 2 kJ mole-1
fusion heat + dependencies
temperature +_4 К
30
Dependence for the central point
of the Information Set
of admissible values of parameters
Heat of fusion / kJ mole-1
28
+3s
Measurements
26
Boxes
of uncertainty
24
22 _
LSQM-dependence,
20 _ 3s
T of fusion / K
1050 1100 1150 1200 1250 1300
_2.895
b)
Amax
Central point of
Information Set
_ 12.489
ASQ of parameters
Information Set
_ 14.797 of admissible
values
ACP LSQM-point
of parameters
A / kJ mole-1
_ 26.699
A min
0.02466 0.03248 0.03455 0.04444
-1 -1
Bmin BSQ BCP B / kJ mole K Bmax
Fig. 1. Processing data on heat of fusion of cryolites; a) input data and tube of admis-
sible dependencies; b) information set of admissible parameters
the difficult situation: there are only 10 measurements with coinciding values of
the temperature measurements and their uncertainty sets crossing each other.
Moreover, two measurements (at the right) and their uncertainty sets practi-
cically coincide. Point–dash lines correspond to the LSQM-line and its rough
standard tolerances ±3σ. The shadowed fragment is the tube of admissible de-
pendencies. The thick central line (Fig. 3a) marks the dependence corresponding
to the central point of the information set of admissible parameters (Fig. 3b),
where, the cross marks the LSQM-point obtained by formal application of stan-
dard LSQM-method.
Underline the sophisticated character of information sets of admissible va-
lues of parameters (Figs. 1b, 2b, and 3b). Such their detailed structure can not
be calculated by any standard method of processing the input data with two-
dimensional uncertainties in measurements.
53
by IHTEC UrB RAS and IMM UrB RAS a)
Approximating dependence 1st sample
E(T) = A + BT
Bounds onto measuring
errors:
_
potential +0.0075 V
_ К
temperature +5
Line for
central point
b)
Minimal outer box-estimate
Central point
CP
Approximating dependence E(T) = A + BT
CP
Fig. 2. Processing data on potential of Uranium–Gallium chlorides; the 1st sample;
a) input data and tube of admissible dependencies; b) information set of admissible
parameters
Besides the information set and the tube of admissible dependencies, the
described interval approach gives for practical using the following useful infor-
mation: the central “calibration” dependence (Figs. 1a, 2a, and 3a, the central
thick lines), the central estimate point (Acp , Bcp ), the minimal outer uncon-
ditional intervals [Amin , Amax ] and [B min , B max ] on parameters (Figs. 1b, 2b,
and 3b, the rectangles with boundaries in dashes), and LSQM estimate point
(ASQ , BSQ ).
54
Potential E, V by IHTEC UrB RAS and IMM UrB RAS a)
_ 2nd sample
2.32 Approximating dependence
E(T) = A + BT Sets of uncertainty
Bounds onto measuring of measurements
errors: LSQM-line
_
potential +0.0075 V
_ _ К
temperature +5
2.34
+3s
_ Line for
2.36 central point
Measurements
_
2.38
Tube of admissible
dependencies
_ 3s
s = 0.0052 V
_ T, K
2.40
723 745 771 800 823
A, V
_2.689 b)
Amax Approximating dependence
E(T) = A + BT
LSQM-point
ASQ Central point (CP)
_ 2.735 of information
set of admissible
Minimal outer
_ 2.741 box-estimate values of parameters
ACP
Information Set of admissible
values of parameters
A
_ 2.894
min
-4
B, V 10 /Т
+
4.242 4.865 4.898 5.555
Bmin BSQ BCP Bmax
Fig. 3. Processing data on potential of Uranium–Gallium chlorides; the 2nd sample;
a) input data and tube of admissible dependencies; b) information set of admissible
parameters
55
4 Conclusions
Digital signal procession is implemented on the basis of the Interval Analysis
methods. Parameters of noised chemical processes are estimated under condi-
tions of absence of probability data for the measuring errors and specific two-
dimensional uncertainty of each measurement. Such a case can not be treated
by any standard methods.
It was shown that under mentioned conditions Interval Analysis approach
gives sophisticated guaranteed estimation of the process parameters set and bet-
ter estimation of the tube of admissible dependencies.
Moreover, simulation results show that simultaneous using the interval and
standard statistical approaches complement each other and allows one to per-
form more detailed analysis and qualitative comparison of the estimation results.
Acknowledgments. The work was supported by RFBR, project no 18-01-
00410.
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