Download Empirical riability functions based on fuzzy life time data

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Empirical riability functions
based on fuzzy life time data
M. Shafiq and R. Viertl
Forschungsbericht SM-2014-3
November 2014
Kontakt: [email protected]
1
Undefined 0 (0) 1
IOS Press
Journal of Intelligent & Fuzzy Systems
EMPIRICAL RELIABILITY FUNCTIONS
BASED ON FUZZY LIFE TIME DATA
Muhammad Shafiq ∗ , Reinhard Viertl
Institute of Statistics and Probability Theory
Vienna University of Technology, Vienna, Austria
Abstract. Reliability analysis is one of the significant applications of statistics. Many parametric and non-parametric
techniques are available for reliability estimation based on
precise life time observations, without considering the uncertainty of individual observations. But life time observations are not precise measurements but more or less fuzzy.
Therefore by considering life time measurements as precise
numbers we may lose information and get misleading results. This study is aimed to present some non-parametric estimates for reliability functions based on fuzzy life time data.
Keywords:, Characterizing function, Fuzzy number, Nonprecise data, Real measurement results
1. Introduction
Statistics is a technique of decision making, which
is generally based on data in the form of numbers or
vectors. These numbers either represent some quality
or measurement of some phenomena. During the data
collection process frequently we deal with variables
of continuous nature, e.g. measurement of blood pressure, life time of an object, recovery time of patient,
etc. These variables are very frequently measured as
precise numbers, whereas it is impossible to exactly
measure a continuous variable.
Statistical methods for precise life time data started in
the 20th century, and were comprehensively developed
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especially in the last five decades [1]. A significant
proportion of books and research papers have already
been written for life time data analysis, e.g. statistical
analysis for failure time data [5], statistical methods
for survival data analysis [7], regression modeling of
time to event data [3], statistical methods for reliability
[9], and for Bayesian reliability analysis [2].
All these publications consider life times as precise numbers and use stochastic models for inference.
It is worth mentioning that stochastic models cover
only random variation among the observations and ignore uncertainty of single observations. Keep in mind
that uncertainty of single observations is different from
random variation. Therefore by ignoring the fuzziness
of single observations we may lose information and
get misleading results. Therefore instead of standard
statistical tools the fuzzy number approaches are more
suitable, because they also consider the imprecision of
individual observations. Though some books and research papers are published dealing with the imprecision of observations, e.g. [15], [6], [8], [4], [11],
[14], but still in most of the publications it is ignored.
Concerning life time analysis, reliability estimation is
an important feature of the analysis. Reliability is the
measurement of probability that an instrument or person will survive for at least a specified time. Several
parametric and non-parametric approaches are available to estimate reliability or survival probabilities.
However these approaches are based on precise life
time observations. In [12] it is pointed out that life time
observations are not precise numbers but more or less
fuzzy.
1.1. Reliability Function
The reliability function is usually denoted by R(·),
which is defined as:
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0000-0000/0-1900/$00.00 R(x) = P r(X > x)
x ≥ 0.
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Where x is some specified time, and X is the stochastic quantity describing time of failure, and Pr stands
for probability. The value R(x) of the reliability function gives the probability that the unit will survive time
x or we can say that the event will occur after time x.
For the reliability function it is usually assumed that
R(0) = 1, and limx→∞ R(x) = 0 [7]. Keeping in
view fuzziness of life time observations in this study
some empirical reliability functions as generalized estimators for the reliability function R(x) = P r(X >
x) ∀ x ≥ 0, based on fuzzy life time observations
are presented.
2. Fuzzy Models for Life Times
According to [13] some concepts of fuzzy theory
are explained below
2.1. Fuzzy Numbers
Let x∗ represent a so-called fuzzy number which
is determined by the so-called characterizing function
ξ(·) which is a real function of one real variable satisfying the following three conditions:
1. ξ : R → [0 ; 1].
2. For all δ ∈ (0 ; 1] the so-called δ-cut Cδ (x∗ ) :=
{x ∈ R : ξ (x) ≥ δ} is a non-empty and finite
union of compact intervals [aδ,j ; bδ,j ], i.e.
Skδ
Cδ (x∗ ) = j=1
[aδ,j ; bδ,j ] 6= ∅.
3. ξ(·) has bounded support, i.e. supp[ξ(·)] :=
[x ∈ R : ξ (x) > 0 ] ⊆ [a ; b].
The set of all fuzzy numbers is represented by F(R).
If all δ-cuts of a fuzzy number are non-empty closed
bounded intervals, the corresponding fuzzy number is
called fuzzy interval.
2.4. Construction Lemma
Skδ
Let (Aδ ; δ ∈ (0 ; 1]) with Aδ = j=1
[aδ,j ; bδ,j ]
be a nested family of non-empty subsets of R. Then the
characterizing function of the generated fuzzy number
is given by
ξ (x) = sup {δ·1Aδ (x) : δ ∈ [0 ; 1]} ∀ x ∈ R [14].
2.5. Fuzzy Vectors
A n-dimensional fuzzy vector x∗ is determined by
its so-called vector characterizing function ζ(., ..., .)
which is a real function of n real variables x1 , x2 , ..., xn
obeying the following three conditions:
1. ζ : Rn → [0 ; 1].
2. For all δ ∈ (0 ; 1] the so-called δ-cut Cδ (x∗ ) :=
{x ∈ Rn : ζ(x) ≥ δ} is non-empty, bounded, and
a finite union of simply connected and closed
bounded sets.
3. The support of ζ(., ..., .) is a bounded set.
The set of all n-dimensional fuzzy vectors is denoted
by F(Rn ).
2.6. Extension Principle
This is the generalization of an arbitrary function
g : M → N for fuzzy argument value x∗ in M . Let
x∗ be a fuzzy element of M with membership function
µ : M → [0 ; 1], then the fuzzy value y ∗ = g(x∗ ) is
the fuzzy element y ∗ in N whose membership function ν(·) is defined by
sup {µ(x) : x ∈ M, g(x) = y} if ∃x : g(x) = y
ν(y) :=
0
if @x : g(x) = y
[6]
2.2. Lemma
For any characterizing function of a fuzzy number
the following holds
true:
ξ (x) = max δ·1Cδ (x∗ ) (x) : δ ∈ [0 ; 1] ∀x ∈ R.
For the proof see [13]
2.3. Remark
It should be noted that not all families (Aδ ; δ ∈
(0 ; 1]) of nested finite unions of compact intervals are
the δ-cuts of a fuzzy number. But the following construction lemma holds:
Let x1 , x2 , ..., xn be a random sample of size
n from a stochastic quantity X, then each xi is
an element of the observation space MX of X and
(x1 , x2 , ..., xn ) is an element of the sample space of
n
X, denoted by MX
. The sample space is the Cartesian
product MX × MX × ... × MX of n copies of MX .
But with fuzzy data the situation is different than for
precise observations, i.e. if we have fuzzy observations
x∗1 , x∗2 , ..., x∗n which are fuzzy elements of MX , then
(x∗1 , x∗2 , ..., x∗n ) is not a fuzzy element of the sample
n
space MX
.
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Shafiq & Viertl /
n
To obtain a fuzzy element of the sample space MX
∗
∗
∗
from a fuzzy sample x1 , x2 , ..., xn with corresponding characterizing functions ξ1 (·), ξ2 (·), ..., ξn (·) respectively, the so-called minimum t-norm from fuzzy
theory is applied.
For the vector-characterizing function of the combined fuzzy sample x∗ , applying the minimum t-norm,
i.e. ζ (x1 , x2 , ..., xn ) = min {ξ1 (x1 ), ξ2 (x2 ), ..., ξn (xn )}
n
∀ (x1 , x2 , ..., xn ) ∈ Rn , a fuzzy subset of MX
is obtained.
By this combination the δ-cuts of x∗ are obtained
through the Cartesian product of the δ-cuts Cδ (x∗i ), i =
1(1)n, i.e. Cδ [ζ(., ..., .)] = Cδ (x∗1 ) × Cδ (x∗2 ) × ... ×
Cδ (x∗n ) ∀ δ ∈ (0; 1] [13].
3.3. Interval-valued Estimation of the Reliability
Function
An interval-valued approach for the estimation of
the reliability function defined by [10] is the following:
The generalized estimation of R(·) is interval-valued
with upper and lower boundary functions RU (·) and
RL (·) based on the generalized empirical distribution
function FL (·) and FU (·) respectively
There are different methods to generalize the empirical reliability function.
3.1. Estimation of R(·) based on the Smoothed
Empirical Distribution Function
F̂nsm (x)
Rx
n
1 X −∞ ξi (t)dt
R∞
:=
n i=1 −∞
ξi (t)dt
Rsm (x) = 1 − F̂nsm (x)
∀x ≥ 0.
3.2. Estimation of R(·) based on the Adapted
Cumulative Sum
Rx
i=1 −∞ ξi (t)dt
Pn R ∞
i=1 −∞ ξi (t)dt
(
i−1
n
i−ξ(i) (x)
n
)
for x ∈ x : ξ(i) (x) ↑
x ∈ x : ξ(i) (x) = 1 ∨ ↓
f or
)
x ∈ x : ξ(i) (x) = 1 ∨ ξ(i) (x) ↑
,
f or x ∈ x : ξ(i) (x) ↓
where ↑ and, ↓ denotes non-decreasing and nonincreasing respectively, for characterizing functions of
the fuzzy observations. The corresponding intervalestimate for the generalized empirical reliability function is limited by the upper and lower boundary RU (·)
and RL (·) respectively, where
∀x ∈ R
The corresponding estimate for the reliability function
is
i−1+ξ(i) (x)
n
i
f or
n
FU (x) :=
FL (x) :=
3. Generalized Estimates for the Reliability
Function R(·)
(
RU (x) = 1 − FL (x)
∀x≥0
RL (x) = 1 − FU (x)
∀ x ≥ 0.
3.4. Estimation of R(·) based on the Fuzzy valued
Empirical Distribution Function
Let x∗1 , x∗2 , ..., x∗n be fuzzy
intervals
with corre
sponding δ-cuts Cδ (x∗i ) = xδ,i ; xδ,i
∀ δ ∈ (0 ; 1]
and i = 1(1)n.
The upper F̂(δ,U ) (·) and lower F̂(δ,L) (·) δlevel functions of the fuzzy empirical distribution
function are defined through the following equations:
Pn
Snad (x)
:=
∀x ∈ R
The corresponding estimate for the reliability function
is
n
F̂δ,U (x) =
1X
1(−∞; x] (xδ,i ) ∀ x ∈ R
n i=1
F̂δ,L (x) =
1X
1(−∞; x] (xδ,i ) ∀ x ∈ R.
n i=1
and
n
Rad (x) = 1 − Snad (x) ∀x ≥ 0.
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Shafiq & Viertl /
The corresponding estimate for the upper (Rδ,U )and
lower (Rδ,L ) δ-level functions of the fuzzy valued empirical reliability function are
∀x≥0
0.6
0.4
x [hours]
0
Rδ,L (x) = 1 − F̂δ,U (x)
0.2
Reliability
0.8
1
Rad (x)
0
Rδ,U (x) = 1 − F̂δ,L (x)
∀ x ≥ 0.
50
100
150
200
250
Fig. 3. Estimate Rad (·) based on the Adaptive Cumulative Sum for
the fuzzy sample from figure 1
0.6
0.4
RL (x)
0.2
Reliability
0.8
RU (x)
x [hours]
0
In figure 1 a sample of fuzzy life times is given. Different estimates of the reliability function corresponding to the different generalizations from subsections
3.1 to 3.4 are depicted in figures 2 to 5.
1
4. Illustrative Example
0
50
100
150
200
250
1
ξi(x)
0.2
0.4
0.6
0.8
Fig. 4. Upper (RU (·)) and Lower (RU (·)) Reliability Function
based on section 3.3 for the fuzzy sample from figure 1
0
x [hours]
0
50
100
150
200
250
Rδ,L (x), Rδ,U (x)
0.6
0.4
0.2
Reliability
0.8
1
Fig. 1. Characterizing functions of a fuzzy sample
0
x [hours]
0
50
100
150
200
250
1
Rsm (x)
0.6
0.4
0.2
x [hours]
0
Reliability
0.8
Fig. 5. Upper Rδ,U (·) and Lower Rδ,L (·) δ-level Reliability curves
based on Fuzzy Empirical Distribution Function from section 3.4 for
the fuzzy sample from figure 1
0
50
100
150
200
250
Fig. 2. Estimate Rsm (·) based on Smoothed Empirical Distribution
Function from section 3.1 for the fuzzy sample from figure 1
5. Conclusion
Life time data analysis became very popular in the
20th century, so far a lot of research has been done
dealing with precise life time observations. Many reliability estimation techniques are available dealing with
precise life time observations. In practical applications
it is impossible to measure a life time as precise number. In [12], it is shown that life time observations are
not precise numbers but fuzzy. Therefore for fuzzy life
time observation fuzzy number approaches are more
suitable than precise data analysis techniques. In this
study various generalized empirical reliability functions based on fuzzy life time data are described.
Shafiq & Viertl /
References
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