Download Fuzzy information, likelihood, Bayes` theorem, and engineering

Institut f. Statistik u. Wahrscheinlichkeitstheorie
1040 Wien, Wiedner Hauptstr. 8-10/107
AUSTRIA
http://www.statistik.tuwien.ac.at
Fuzzy information, likelihood, Bayes’ theorem,
and engineering application
R. Viertl and O. Sunanta
Forschungsbericht SM-2014-1
November 2014
Kontakt: [email protected]
Reinhard Viertl () · Owat Sunanta
Department of Statistics and Probability Theory
Vienna University of Technology, Vienna, Austria
e-mail: [email protected]
Fuzzy Information, Likelihood, Bayes’ Theorem, and Engineering Application
1. Introduction
In solving an engineering problem, such as developing empirical models for manufacturing processes,
real data of continuous quantities are required. However, the information and/or data obtained from
different sources are often clouded by different kinds of uncertainty. The available data of manufacturing
problems are frequently imprecise and incomplete. To overcome such problems, fuzzy concepts along
with probability theory, e.g. Bayesian approach, can be adapted and applied to provide meaningful
description that bridges the gap between real data and empirical process models.
In standard Bayesian inference, a-priori distributions are standard probability distributions and the
observations are assumed to be numbers or vectors. Bayes’ theorem formulates the transition from the apriori distribution of the stochastic quantity, which describes the parameter of interest, to the a-posteriori
distribution. In case of continuous stochastic models X ~ f (·| ),  ⊝, based on observations x1,…, xn of
X, the transition from an a-priori density to an updated information concerning the distribution of the
stochastic quantity describing the parameter 𝜃̃ is given by the conditional density 𝜋(·|x1,…, xn) of 𝜃̃, i.e.
Bayes’ theorem:
𝜋(𝜃|𝑥1 , … , 𝑥𝑛 ) =
𝜋(𝜃)∙𝑙( ; 𝑥1 ,… , 𝑥𝑛 )
∫⊝ 𝜋(𝜃)∙𝑙( ; 𝑥1 ,…, 𝑥𝑛 )𝑑𝜃
,
where l( ; x1,…, xn) is the likelihood function defined on the parameter space ⊝.
However, the use of a-priori densities in form of classical probability densities has been criticized in some
situations. Moreover, real observations from continuous quantities are not precise numbers, but, more or
less, non-precise (also called fuzzy).
The first problem can be overcome by using a more general form of probability which is related to soft
computing, i.e. so-called fuzzy probability densities. The second problem can be solved by using general
fuzzy numbers and fuzzy vectors. See also related work in [4] and [7].
2. Probability based on densities
To overcome the deficiency of classical a-priori distributions in Bayesian inference in some cases, fuzzy
a-priori densities, which is a more general form of expressing a-priori information, seems to be
appropriate. In order to define fuzzy densities, a special form of general fuzzy numbers is necessary.
Definition: A general fuzzy number whose -cuts are non-empty compact intervals [a ,b] is called fuzzy
interval. For functions f *() defined on a set M, whose values f *(x) are fuzzy intervals, their -level
functions𝑓δ() and 𝑓δ̅ () are defined in the following way:
Let C[ f *(x)] = [a(x),b(x)] for all (0,1], the lower and upper -level functions are classical real-valued
functions defined by their values 𝑓δ(x) ≔ a (x) for all x  M and 𝑓δ̅ (x) ≔ b (x) for all xM.
Fuzzy densities are fuzzy valued function f *() defined on a measure space (M,,µ) possessing the
following properties:
( i ) f *(x) is fuzzy interval for all xM;
(ii) there exists g : M  [0,) which is a classical probability density on (M,𝒜,µ)
where 𝑓1 (x)  g(x)  𝑓1̅ (x) for all xM;
(iii) all -level functions 𝑓δ() and 𝑓δ̅ () are integrable functions with finite integral.
Probabilities of events A𝒜 based on a fuzzy probability density f *() are defined in the following way:
Let D be the set of all classical probability densities h(·) on (M,𝒜,µ) with 𝑓δ(x)  h(x)  𝑓δ̅ (x) for all x
M. The generalized probability P*(A) is a fuzzy interval, which is determined by the following family of
compact intervals B = [a ,b ] where
b ≔ sup{∫𝐴 ℎ(𝑥)𝑑(𝑥): hD}
for all (0,1].
a ≔ inf {∫𝐴 ℎ(𝑥)𝑑(𝑥): hD}
By applying the so-called construction lemma for general fuzzy numbers [5], the characterizing function
() of P*(A) is given by
(x) = sup{.𝟙[a ,b](x): [0,1]}
for all xℝ,
where 𝟙B() denotes the indicator function of the set B, and [a0 ,b0] ≔ℝ.
Fuzzy probability densities are a more general form of expressing a-priori information of the parameters 
in stochastic models f (| ),  ⊝.
3. Fuzzy data and likelihood function
Real observations of continuous stochastic quantities X are not precise numbers or vectors, whereas the
measurement results are more or less non-precise, also called fuzzy. This is often the case in dealing with
engineering problems. The best (to-date) mathematical description (see also [2] and [6]) of such
observations is explained by means of general fuzzy numbers 𝑥1∗ , … , 𝑥𝑛∗ with corresponding characterizing
functions 1(),…,n().
Remark 1: The fuzziness of an observation 𝑥𝑖∗ resolves the problem in standard continuous stochastic
models where observed data have zero probability.
For a fuzzy observation 𝑥𝑖∗ with characterizing function i () from a density f (), the probability of 𝑥𝑖∗ is
given by [8]
Prob(𝑥𝑖∗ ) = ∫ℝ i (x) f (x)𝑑𝜇(𝑥) > 0.
This makes it possible to define the likelihood function of independent fuzzy observations 𝑥1∗ , … , 𝑥𝑛∗ in a
natural way:
𝑙(𝜃; 𝑥1∗ , … , 𝑥𝑛∗ ) = ∏𝑛𝑖=1 𝑃𝑟𝑜𝑏( 𝑥𝑖∗ | ) = ∏𝑛𝑖=1 ∫ℝ 𝑖 (𝑥)𝑓(𝑥|𝜃)𝑑𝜇(𝑥)
for all 𝜃 ∈⊝
The likelihood function is, then, used in updating the a-priori density to yield the corresponding fuzzy aposteriori density
4. Generalized Bayes’ theorem
The standard Bayes’ theorem has to be generalized to handle fuzzy a-priori densities 𝜋*() on the
parameter space ⊝ and fuzzy data 𝑥1∗ , … , 𝑥𝑛∗ of parametric stochastic model X ~ f (| ),  ⊝. This is
possible by using the -level functions 𝜋() and 𝜋() of 𝜋*() along with defining the -level functions
𝜋(|𝑥1∗ , … , 𝑥𝑛∗ ) and 𝜋(|𝑥1∗ , … , 𝑥𝑛∗ ) of the fuzzy a-posteriori density in the following way:
𝜋δ (θ|𝑥1∗ , … , 𝑥𝑛∗ ) =
∗)
𝜋δ (𝜃)∙𝑙(𝜃; 𝑥1∗ ,…, 𝑥𝑛
𝜋δ (θ)+ 𝜋δ (θ)
∗ )𝑑𝜃
∙𝑙(𝜃;𝑥1∗ ,…, 𝑥𝑛
∫⊝
2
for all (0,1]
𝜋δ (θ|𝑥1∗ , … , 𝑥𝑛∗ ) =
Remark 2: The averaging
𝜋δ (θ)+ 𝜋δ (θ)
2
∗)
𝜋δ (𝜃)∙𝑙(𝜃; 𝑥1∗ ,…, 𝑥𝑛
𝜋δ (θ)+ 𝜋δ (θ)
∗ )𝑑𝜃
∙𝑙(𝜃; 𝑥1∗ ,… ,𝑥𝑛
∫⊝
2
is necessary in order to keep the sequential updating of standard
Bayes’ theorem.
Based on the fuzzy a-posteriori density 𝜋*(|𝑥1∗ , … , 𝑥𝑛∗ ), the fuzzy HPD-intervals of the parameter  ,
which provides valuable information on the unknown parameter , can be generated (see [5] for more
detail on the method).
5. Example: Manual Lapping of Valve Discs
Lapping is a low-speed, low-pressure abrading operation that accomplishes one or more of the following
results: extreme dimensional accuracy of lapped surface, e.g. flat or spherical, refinement of surface
finish, extremely close fit between mating surfaces, removal of minor damaged surface and subsurface
layers [3]. The most intriguing aspect of lapping lies in its abrading mechanism of random loose abrasive
particles, generally in form of abrasive compound, to refine and remove microscopic material. Lapping
involves a number of process parameters that influence the integrity of the lapped surface. In this
example, surface finish (roughness), which is a process parameter specific to manual flat lapping of valve
discs (Figure 1), is explored.
Figure 1: Seat Area of a Valve Disc
Valve Seat Diameter
Area to be lapped
Top view
Valve Seat Height
Front/Side view
5.1 Surface Finish (Roughness)
Surface roughness consists of fine irregularities in the surface texture, usually including those
resulting from the inherent actions of different manufacturing processes. Surface roughness can be
measured by a variety of methods and instruments, including using profilometer. Surface
roughness is commonly presented in terms of the arithmetic average (Ra) of the profile (absolute)
deviations from the mean line within the evaluation length (Figure 2).
Figure 2: A Profile of Surface Roughness (Ra)
Ra
Mean Line
Lapping is an ultra-fine finishing process and not meant to remove a significant amount of material.
Instead, lapping is a good option for correcting finished surfaces with minor imperfections. Hence,
the surfaces prior to lapping must have been pre-processed to a certain level of finish. In collecting
data for this example, the surfaces have already been prepared through the process of rough lapping
and have surface roughness approximately between 0.00031 mm and 0.00082 mm. After lapping,
based mainly on the capability of the lapping process itself and the equipment, the typical obtained
surface roughness produced is approximately between 0.00005 mm and 0.00050 mm [1].
5.2 Fuzzy Bayesian Inference
The obtained surface roughness (X ) is modeled with an exponential distribution and fuzzy gamma
density as a-priori density for the parameter  (0,).
𝑋~𝐸𝑥𝜃 ≙ 𝑓(𝑥|𝜃) =
1
𝑥
exp[− ] ∙ 𝟙(0,∞) (𝑥)
𝜃
𝜃
The a-priori density (·) for 𝜃̃ (mean value of obtained surface roughness) is assumed to be gamma
distributed. This assumption is intuitive and based on the common lapping process capability. In
other words, before actual lapping of the surfaces, the anticipated (obtained) surface roughness
measurements are expected to distribute with higher probability (density) around 0.00005 mm and
0.00050 mm. Different -level functions of the a-priori density are shown in Figure 3.
The obtained surface roughness data are collected and represented in form of fuzzy observations
𝑥𝑖∗ , i.e. D* = (𝑥1∗ , 𝑥2∗ ), with characterizing functions as depicted in Figure 4. The exponentially
distributed likelihood function is assumed here in order to obtain conjugate distributions for a-priori
and a-posteriori. The fuzzy observations are, then, used for updating the a-priori density to yield
the corresponding fuzzy a-posteriori density 𝜋*(|D* ) as shown in Figure 5.
Figure 3: Fuzzy a-priori density
[in 1000th mm]
Figure 4: Fuzzy sample
[in 1000th mm]
Figure 5: Fuzzy a-posteriori density
[in 1000th mm]
Figure 6: Fuzzy HPD Interval for 
[in 1000th mm]
The collected data of obtained surface roughness (between 0.000041 mm and 0.000128 mm)
support the prior assumption of lapping process capability. The a-posteriori density is more peaked
than the a-priori density, which may be interpreted in the following way: the collected data confirm
the assumption prior to the experiment. As shown in Figure 6, a 95% fuzzy HPD-interval of the
parameter , which is generated based on the fuzzy a-posteriori density 𝜋*(|D*) from Figure 5,
provides a fuzzy expected range of the obtained surface roughness.
6. Final Remark
As the demand for realistic models for representing the real observations increases, generalized concepts
for capturing fuzziness are necessary. A more general concept to take care of fuzzy a-priori information
and fuzzy data of an engineering problem, as opposed to the use of classical probability, is introduced in
form of fuzzy a-priori densities. This concept is more suitable in modeling the prior information, which is
usually uncertain, i.e. fuzzy. In this case, concepts of general fuzzy numbers and fuzzy vectors along with
their characterizing functions have been applied in capturing the imprecision of the collected continuous
quantities. The so-called fuzzy probability densities along with the -level functions are used to define
the fuzzy a-posteriori densities. The fuzzy observations are, then, used for updating the fuzzy a-priori
density to yield the corresponding fuzzy a-posteriori density. As a result, Bayes’ theorem is generalized
to model the imprecise data.
The generalized model is equipped with explainable mathematical grounds in capturing the variability
and imprecision of real observations. Fuzzy Bayesian inference provides promising results by the use of
fuzzy concepts together with the probabilistic models in analyzing a critical lapping parameter. The fuzzy
a-posteriori density and fuzzy HPD interval are useful in quantifying and broadening the understanding of
the real observations from actual lapping. These results are viable for further studies in modeling flat
lapping process.
References
[1] Groover, M. P. Fundamentals of Modern Manufacturing: Materials, Processes, and Systems, 4th
Edition, John Wiley & Sons, New Jersey, 2010.
[2] Klir, G., and Yuan, B. Fuzzy Sets and Fuzzy Logic-Theory and Applications, Prentice Hall,
Upper Saddle River, 1995.
[3] Lynah, P. R. Lapping. ASM Handbook, Vol. 16, ASM International, Ohio, pp. 492-505, 1990.
[4] Stein, M., Beer, M., and Kreinovich, V. Bayesian Approach for Inconsistent Information.
Information Sciences, Vol. 245, pp. 96–111, 2013.
[5] Viertl, R. Statistical Methods for Fuzzy Data, Wiley, Chichester, 2011.
[6] Wolkenhauer, O. Data Engineering-Fuzzy Mathematics in Systems Theory and Data Analysis,
Wiley, New York, 2001.
[7] Yang, C.C. Fuzzy Bayesian Inference. In Proceedings of Systems, Man, and Cybernetics, IEEE
International Conference, Vol. 3, pp. 2707-2712, 1997.
[8] Zadeh, L.A. Probability measures of fuzzy events. Journal of Mathematical Analysis and
Applications, Vol. 23, No. 2, pp. 421-427, 1968.
Bayes’ theorem, 2, 4
Bayesian inference, 9
exponential distribution, 6
fuzzy a-posteriori density, 4, 5, 6, 9
fuzzy a-priori densities, 2, 4
fuzzy Bayesian inference, 6, 9
fuzzy data, 3, 4, 9
fuzzy densities, 2
Fuzzy densities, 3
fuzzy HPD, 5, 9
fuzzy information, 2
gamma density, 6
lapping, 5, 6, 9
likelihood, 2, 3, 4, 6