SLIDE 1 ONE ILL-POSED ESTIMATION PROBLEM OF EXPERIMENTAL PROCESS PARAMETERS. INTERVAL APPROACH
Institute of Mathematics and Mechanics UrB RAS, Ekaterinburg, Russia, kumkov@imm.uran.ru Ural Federal University, Ekaterinburg, Russia The 8th Small Workshop on Interval Methods (SWIM 2015) June 09 – 11, Charles University Prague, Czech Republic
SLIDE 2 The aim of this presentation is to demonstrate
- ne interesting practical problem of estimation
- f experimental process parameters under uncertainty
conditions when components of the parameter vector can be only estimated on the basis of the Interval Analysis approach and available a priori data
The work was supported by the RFBR Grant, proj. 15-01-07909.
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SLIDE 3
Topics of presentation Experimental process and its model. Measured information and its uncertainty. Interval approach and its peculiarities. Problem formulation and how to solve it ? Computation results. Conclusions. References. 3
SLIDE 4
Experimental process and its model Description of a reagent activity vs the temperature (similarly to [10,12]) has the form P(T, a, b, c) = T 2 a b/c, a > 0, b > 0, c > 0, (1) where T is the temperature (the argument), C◦; P(·) is the reagent activity, dimensionless value; a, b, and c are parameters (to be estimated) with dimensions: mole, 1/mole, and (C◦)2. 4
SLIDE 5
Measured information and its uncertainty
Results of the experiment are presented as the following collection (a sample with lenght N) of the reagent activity P measurements: {Tn, Pn}, n = 2, N, (2) where values Tn are supposed to be know exactly, but the activity values Pn are measured with error (noise) Pn = P ∗
n + en, |en| ≤ emax, n = 2, N, and for T1 = 0, P1 = 0,
(3) where Pn is a noised measurement; P ∗
n is unknown true value under
measuring; en is the error value in the nth measurement; emax is the bound onto the maximal (by modulus) value of the error. By physical reasoning, the conditional exact initial measurement at T1 = 0 is given zero.
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SLIDE 6 Conditions for estimation and a priori information
No probabilistic information on errors is known and the sample is dramatically short: N ≈ 5 ∼ 7 measurements only. In (1), parameters a, b and c are merged (“stuck”) that hampers estimation
- f their own admissible intervals without some additional information.
From theoretical estimations and previous experience, the following rough a priori constraints on possible values of the coefficients are given: aap = [aap, aap], bap = [bap, bap], cap = [cap, cap], 0 < aap < aap, 0 < bap < bap, 0 < cap < cap. (4) The LSQM-curve and pointwise estimation of parameters a, b, c and their practically meaningless “cloud-built” intervals are available by only formal application of standard statistical procedures [15–17].
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SLIDE 7
Interval approach and its essence
Ideas and methods of the Interval Analysis Theory and Applications arose from the fundamental, pioneer work by L.V. Kantorovich [1]. Nowadays, very effective developments of the theory and computational methods were created by many researchers, e.g. [2–4] and in Russia [5–8]. Special interval algorithms have been elaborated for estimating parameters of experimental chemical processes [9–14]. Remind that essence of this branch of numerical methods theory and application consists in estimation (or identification) of parameters under bounded errors (noises or perturbations) in the input information to be processed, and under complete absence of probabilistic characteristics of errors.
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SLIDE 8 The main definitions
Uncertainty set (interval) of each measurement (USM). It is the interval
- f values of measured process consistent with the measurement and the
error bound Hn = [hn, hn] : hn = Pn − emax, hn = Pn + emax, n = 2, N, (5) and for n = 1, H1 = H(0) = [0], trivially. Admissible value of the parameter vector and corresponding admissible curve (a, b, c) : P(Tn, a, b, c) ∈ Hn, for all n = 1, N. (6) Informational Set (InfSet) is a totality of admissible values of the parameters vector satisfying the system of interval inequalities (6) I(a, b, c) =
{
(a, b, c) : P(Tn, a, b, c) ∈ Hn, for all n = 1, N
}
. (7)
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SLIDE 9 Measurements and their uncertainty sets (USM)
e , =0.1 dim.less
max
0.1 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3
1
H
2
H
4
H
5
H
6
H
7
H 15 25 35 45 60 75 25 True ( ) model curvewith
_ _ _ _ _ _ _ _ _ _ _ _
true(model)values
_ _
noisedmeasurements LSQM-curve
3
H
n
H uncertaintyintervals
P,dim.less amd =2.0,mole bmd =1/140, cmd =100, )= P Tabc T ab c ( ,,, /
2
T,C C
2
()
1 _
mole
If the actual level of errors in the sample is lower the initially given a priori bound emax, the LSQM-curve and values of its parameters could be admissible.
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SLIDE 10 Problem formulation
Since of very short length of the measurements sample, absence of probabilistic characteristics of the errors, and measurements uncertainty, it is impossible to use (with any good reasoning) the standard statistical methods [15–17]).
It is necessary:
- n the basis of the Interval Analysis methods to built
the Informational set I(a, b, c) of admissible values (or the Set-membership) of coefficients a, b, and c consistent with the described data. 10
SLIDE 11 Applied procedures Direct set-estimation approach
There are several approaches to solve system (6) of the interval inequalities – classic linear programming methods [1, and many others], – parallelotopes Fiedler M., et al [2], Hansen [3], Jaulin, et al [4], Shary [5], – by the “stripes” method Shary&Sharaya [6], Sharaya [7], Zhilin [8]. More convenient and faster DIRECT method has been elaborated (see, Kumkov and with co-authors [9–14]) that gives exact estimation of the Informational set (7) on part of parameters for each node of the grid
- n other parameters. In the case under consideration, we represent the
set I(a, b, c) in the form of a collection of its cross-sections
{
Ia(b, c)
}
for nodes of the grid on the parameter a on its minimal outer interval a∗ of admissible values. It is performed in contrast, for example, to outer approximation
- f informational sets in the parallelotope approaches.
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SLIDE 12
Three successive auxiliary problems
The following auxiliary problems are solved. 1) Introducing the auxiliary merged parameter g = ab/c with g > 0, its corresponding informational interval g = [g, g] is calculated [10,12]. 2) Having the interval equation ad = g, where d = b/c, solve it w.r.t. the auxiliary parameter d as follows: d = g/aap. As a result in the plane a×d, we obtain the informational set I(a, d) with the curve (hyperbolic) lower Frd(a) and upper Frd(a) boundaries as a functions of the parameter a values from its a priori interval aap. 3) For each value a ∈ aap we have the interval d(a). So, it becomes possible to construct the informational set Ia(b, c) of admissible values for parameters b, c for each admissible value of the parameter a.
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SLIDE 13
Solution of the auxiliary Problem 1. Informational set of the “merged” parameter g = ab/c
7.70 3.13
1,2
G
1,3
G
1,4
G
1,5
G
1,6
G
1,7
G
ap
g =[0.73,2.37]
_ _
10
4 _
2.37 0.73 I g ()
ap
g g =1.43
md
_ _
10
4 _
_
_ _
10
4 _
g,1/() C
2
a) b) g =[1.34,1.59]
_ _
10
4 _
I g ()=
Partial informational intervals Resultantinformationalinterval Admissibleinputdata Inadmissibleinputdata
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SLIDE 14
Analysis of consistency of a priori data with the measured ones
Note that it is worthy to calculate the a priori interval gap of the parameter g and compare it with the obtained interval g for analysis of consistency of the a priori data (4) on parameters a, b, c with the given sample of measurements (2),(3).
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SLIDE 15 Solution of the auxiliary Problem 1. Tube
- f admissible dependencies
e , =0.1 dim.less
max
true(model)values
_ _
noisedmeasurements
n
H uncertaintyintervals
P,dim.less T,C 0.1 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3
1
H
2
H
4
H
5
H
6
H
7
H 15 25 35 45 60 75 25 )= P Tabc T g g=ab c ( ,,, , /
2
_ _ _ _ _ _ _ _ _ _ _ _
Tube
3
H Intervalofadmissiblevalues
dependencies g =[1.34,1.59]
_ _
10
4 _
15
SLIDE 16 Estimating from below the maximal value
- f the actual error in the sample
Thelimit curve-- “tube”
_ _ _ _ _ _ _ _ _ _ _ _
1
H
7
H*
6
H*
5
H*
4
H*
3
H*
2
H* P,dim.less 0.1 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 15 25 35 45 60 75 25 )= P Tabc T g g=ab c ( ,,, , /
2
Thelimitvalues: true(model)values
_ _
noisedmeasurements
n
H uncertaintyintervals
*
g =[1.34,1.59]
_ _
10
4 _ *
T,C
e , =0.1 dim.less
max Init
e7{
2
g =1.449
_ _
10
4 _ *
,1/( ) C !! g =1.428
2
_ _
10
4 _ ,1/(
) C
md
7
e e =0.0549 dim.less~max{
}
,
| |
n
n
e , =0.0457 dim.less
max
!!estimatefrombelow
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SLIDE 17 Solution of the auxiliary Problem 2. Informational set I(a, d) of parameters a, d for d = b/c
0.71 1.8 2.0 2.2 1.89
ap
a _
md
a
md
d
ap
a _ 0.88 0.48
ap
d _ 1.04 0.61 d _ d _ ) d,1/ mole ( (С ) 2
_ _
10 4
_
a,mole I a,d) ( d( =1.89) a for interval apriori =[
ap
a ]=[
ap
a _
ap
a _ 1.8,22],mole . , d a =/ g
d
)= Fr a (
_
g _ a /
d
)= Fr a (
_
g _ a /
U
ap
d d
ap
d _
ap
d d
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SLIDE 18
Analysis of consistency of a priori data with the measured ones In figure the a priori interval dap of the auxiliary parameter d is shown (the thick dash-dotted vertical segment) calculated by the a priori intervals bap and cap. The thick vertical line in dashes marks the outer interval in d of I(a, d) for the a priori interval aap. Comparison of these two intervals allows one to check out consistency of the a priori data (4) on parameters a, b, c with the given sample of measurements (2),(3). 18
SLIDE 19 Solution of Problem 2. Informational set I(a, d)
- f parameters a, d for more wide interval aap
0.71
md
d 0.48
ap
d _ 1.04
ap
d _ 0.39 d _ 1.26 d _ a, 1.89 2.0 a _* mole I a,d) ( a _
* md
a ) d,1/ mole ( (С ) 2
_ _
10 4
_
1.27 338 .
ap
a _
ap
a _
d
)= Fr a (
_
g _ a /
d
)= Fr a (
_
g _ a / d( =1.89) a formorewide interval apriori =[
ap
a ]=[
ap
a _
ap
a _ 1.27,338],mole . , I a,d) ( I a,d)= (
*
ap
d
U
I a,d) (
U
ap
d d d
ap
d
~ enhancing ~
enhancing
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SLIDE 20 Solution of Problem 3. Informational set Ia(b, c)
- f parameters b, c for fixed value a = 1.89, mole
1/160 1/140 1/120
md
b
md
c 130 b, 1/mole 100 80 c(b,1/d(1.89)) _ _ c(b,1/d(1.89)) _ _
ap
b _
ap
b _
ap
c _
ap
c _
ap
b _
ap
b _
_ _
ap
c _
ap
c _ , , ] ] [ [ c, (С )2 I b ( , ) c
1.89
forvalue=1.89mole a c=b / d() a
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SLIDE 21 Constructing the collection {Ia(b, c)} of cross-sections
- f the informational set I(a, b, c)
It is seen that the set Ia(b, c) is built by intersection of the rectangle bap×cap with the cone between the lower c(b, 1/d(a) and upper c(b, 1/d(a) rays for a ∈ aap (or from the enhanced
- ne a*) and b ∈ bap. Here, the set Ia(b, c) (shadowed five-apex
polygon) is shown for value a = 1.89, mole and corresponding interval d(1.89) by solution of Problem 2. 21
SLIDE 22 Informational set I(a, b, c) as a collection
- f its cross-sections {Ia(b, c)}
ap
b _
ap
b _
_ _
ap
c _
ap
c _ , , ] ] [ [ c, (С )2 I b (
{ }
, ) c
a
130 80
ap
c _
ap
c _ b,1/mole 1/160 1/120
ap
b _
ap
b _ 1.85 1.91 2.03 2.09 1.97 2.14 a, a 1.80
ap
_ a 2.20
ap
_ mole
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SLIDE 23 Conclusions In the considered ill-posed estimation problem with the “stuck” parameters and under absence of probabilistic characteristics
- f the measuring errors, the elaborated interval approach
allows one: to analyze consistency of the given sample of measurements itself; to analyze consistency of the given sample
- f measurements and the given a priori data, and to construct
the informational set of admissible values of parameters. Algorithms elaborated are simple in numeric implementation. In special cases, they can give exact estimations of the infor- mational set and are faster than usual interval approaches on the basis of parallelotopes. 23
SLIDE 24 References
- 1. Kantorovich L.V. On new approaches to computational methods and processing the
- bservations // Siberian mathematical journal. 1962, III, no. 5, pp. 701–709.
- 2. Fiedler M., Nedoma J., Ramik J., Rohn J., and K. Zimmermann. Linear optimization
problems with inexact data. Springer-Verlag., London. 2006.
- 3. Hansen E., G.W. Walster. Global Optimization using Interval Analysis. Marcel Dekker,
Inc., New York. 2004.
- 4. Jaulin L., Kieffer M., Didrit O., and E. Walter. Applied Interval Analysis. Springer-
Verlag, London. 2001.
- 5. Shary, S.P. Finite–Dimensional Interval Analysis. Electronic Book, 2014,
http://www.nsc.ru/interval/Library/InteBooks
- 6. Shary S.P. and I.A. Sharaya. Raspoznzvaniye razreshimosti interval’nykh uravneniyi i
ego prilozheniya k analizu dannykh // Vichslitelnye tekhnologii. (2013), 8, no. 3, pp.80– 109.
- 7. Sharaya I.A. Dopuskovoye mnozhestvo resheniyi interval’nykh lineyinykh system uravneniyi
so svyazannymi coeffitsientami // in Computational Mathematics, Proc. of XIV Baikal International Seminar-School “Methods of Optimization and Applications”. Irkutsk, Baikal, Russia, 2 – 8 July, 2008. Irkutsk, ISEM SO RAS. (2008), 3, pp.196–203.
- 8. Zhilin S.I. Simple method for outlier detection in fitting experimental data under
interval error // Chemometrics and Intelligent Laboratory Systems. (2007), 88, pp.6– 68.
- 9. Redkin A.A., Zaikov Yu.P., Korzun I.V., Reznitskikh O.G., Yaroslavtseva T.V., and
S.I. Kumkov. Heat Capacity of Molten Halides // J. Phys. Chem. B, (2015), 119: 509– 512.
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SLIDE 25 References
- 10. Kumkov S.I. and Yu.V. Mikushina. Interval Approach to Identification of Catalytic
Process Parameters // Reliable Computing. (2013), 19: 197–214.
- 11. Arkhipov P.A., Kumkov S.I., et.al. Estimation of plumbum Activity in Double systems
Pb–Sb and Pb–Bi // Rasplavy. (2012), no. 5, pp.43–52.
- 12. Kumkov S.I. and Yu.V. Mikushina. Interval Estimation of Activity Parameters of
Nano-Sized Catalysts // Proceedings of the All-Russian Scientific-Applied Conference “Statistics, Simulation, and Optimization”. The Southern-Ural State University, Chelyabinsk, Russia, November 28–December 2. (2011), pp. 141–146.
- 13. Kumkov S.I. Processing the experimental data on the ion conductivity of molten
electrolyte by the interval analysis methods // Rasplavy. (2010), no. 3, pp.86–96.
- 14. Potapov A.M., Kumkov S.I., and Y. Sato. Procession of Experimental Data on
Viscosity under One-Sided Character of Measuring Errors // Rasplavy (2010), no. 3,
- pp. 55–70.
- 15. GOST 8.207-76. The State System for Providing Uniqueness of Measuring. Direct
Measuring with Multiple Observation. Methods for Processing the Observation Results. –M.: Goststandart. Official Edition.
- 16. MI 2083-93. Recommendations. The State System for Providing Uniqueness of
- Measuring. Indirect Measuring. Determination of the Measuring Results and Estimation
- f their Errors. –M.: Goststandart. Official Edition.
- 17. R 40.2.028–2003. Recommendations. The State System for Providing Uniqueness of
- Measuring. Recommendations on Building the Calibration Characteristics. Estimation of
Errors (Uncertainties) of Linear Calibration Characteristics by Application of the Least Square Means Method. –M.: Goststandart. Official Edition.
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SLIDE 26
Thanks for attention 26
