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A Bayesian Framework for Parameters Estimation in Complex System

ADRIANA CALAPOD1, LUIGE VLĂDĂREANU2, RADU ADRIAN MUNTEANU3, DAN

GEORGE TONŢ4, GABRIELA TONŢ4

1Agency of Environment Protection, Bihor County,

Dacia nr.25/A , Oradea, 410464, ROMÂNIA

http://www.apm-bihor.ro/

2Institute of Solid Mechanics of Romanian Academy

C-tin Mille 15, Bucharest 1, 010141, ROMÂNIA,

luigiv@arexim.ro, http://www.acad.ro

3Department of Electrical Measurements,

Faculty of Electrical Engineering,

Technical University Cluj Napoca,

Constantin Daicoviciu st. no 15, 400020 Cluj - Napoca, ROMÂNIA,

radu.a.munteanu@mas.utcluj.ro

4Department of Electrical Engineering, Measurements and Electric Power Use,

Faculty of Electrical Engineering and Information Technology

University of Oradea

Universităţii st., no. 1, zip code 410087, Oradea, ROMÂNIA,

gtont@uoradea.ro dtont@uoradea.ro, http://www.uoradea.ro

Abstract: -. The real-life complex development situations express that the methods applied to new product

development process content reliability risks which require assessment and quantification at the earliest stage,

extracting relevant information from the process. Reliability targets have to be realistic and systematically

defined, in a meaningful way for marketing, engineering, testing, and production. Potential problems

proactively identified and solved during design phase and products launched at or near planned reliability

targets eliminate extensive and prolonged improvement efforts after start on. Once in the market, products

standard procedures require monitoring of early signs of issues, allowing corrective action to be quickly taken.

Reliability validation before a product goes to market by the means of Bayesian statistical method because the

model has shorter confidence intervals than the classical statistical inference models, allowing a more accurate

decision-making process. The paper proposes the estimation of the shape parameters in a complex data

structures approached with exponential gamma distribution as model of life time, reliability and failure rate

functions. The numerical simulation performed in the case study validates the correctness of the proposed

methodology.

Key-Words: - failure, rate, Bayesian model, adequate function, distribution, simulation.

1 Introduction

The systematic reduction of product

development time and cost without risking or

sacrificing reliability includes procedures and

standards regarding the choice of types of

components of product, research and selection of

manufacturers, suppliers, purchasing specifications,

analysis of faults, the reliability of their introduction

into manufacturing and thereafter etc. Based on

experience with similar components, specifying

target reliability prediction is based on laboratory

tests and a large amount of data obtained in

operational regime. Variables influencing the failure

rate of components are:

1. criteria for failure: critical values of certain

accessible parameters of the component as is

considered defective (failure may be total, catalectic

or derived)

2. electrical constraints such as: current, power,

noise, etc..

3. thermal constraints which depend on the type of

the studied components

a. Passive components are characterized by ambient

temperature and the capsule or layer temperature

b. Active components are characterized by two

types of temperature: a junction and normalized

junction

4. constraints climate: humidity, pressure, dust,

altitude

5. mechanical constraints: shock, vibration

MATHEMATICAL METHODS AND APPLIED COMPUTING

ISSN: 1790-2769719ISBN: 978-960-474-124-3

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6. other factors: technology, construction, etc.

manufacturer. In reality because of the large number

of variables that influence the intensity of failure, it

is difficult to determine their influence. If the note

by R (t) the reliability or the likelihood of good

functioning at time t and dR crash probability

between time t and t + dt, the intensity of failure can

be considered as a function of time λ (t) or z (t ) that

vary by a curve in the "bathtub" result obtained by

accumulating a large volume of data and is

concordant with some general principles. The

behavior of dynamical systems can be characterized

using the so-called reliability bathtub curve, as

shown in Figure 1, i.e. the initial decreasing failure

rate period (infant-mortality, AB) is followed by a

relatively long constant failure rate period (useful

life), while the probability of failures sharply

increases toward the end of the system’s design life

(wear-out period).

2 Summarizing the modeling of

failure rate

A particularly problem in determining the reliability

of the elements is related to the complexity of

environmental factors, acting simultaneously. The

climatic factors, factors related to the chemical,

thermal, mechanical (shock, vibration) and the

variables that influence the rate of failure of

components have to be taken into account. The

component resistance and demands that characterize

the spectrum of operation, determines the intensity

of component failure in a given situation. It notes

that over time, the component with quality level Q2

will fail several times assuming its replacement,

unlike the component with quality level Q1 where

the constraints have not exceeded the supported

level. Even in the same batch, components don’t

have exactly the same quality regarding not only the

mechanical resistance to vibration, shock, pressure

or acceleration but temperature resistance, to

withstand the different voltage variations, resistance

to moisture, corrosion, radiation, etc. Appears

observable that the electronic component reliability

can not be expressed by a single numerical value

significantly, requesting a spectrum that cannot be

expressed by a single numerical value but by an

interval. Therefore, the usual models employed to

describe the behavior over time of various

applications based on

chemical and statistical findings are based on a large

number of experiments, having a predictive

character.

physical phenomena,

Among the best known laws of survival or mortality

is the Gompertz law (1825). Gompertz hypothesis

was that death may be caused by two distinct cases:

- By a hazard, accident, this could lead to the death

of a healthy person, regardless of age;

- By the force of mortality which consists in the

gradual weakening of the individual as the force of

mortality ρ in dt is proportional to ρ dρ.

Neglecting first situation, Gompertz equation can be

written under the form:

d

=

ρ

where:

ea⋅=ρ

If St is the number of survival to age at time t, and St

+ dt is the number of survival at time dt, the

mortality relative to the time interval dt is express

as:

k

dt

ρ

1

(1)

kx

t

t

S

dttSS

+⋅−

(2)

As a model of aging, taking into account chemical

reactions in the dielectric is adopted the Arrhenius

equation, which agree on the degradation rate:

KTE

eAv

where A is a constant;

ΔE - energy of activation (ie the corresponding

energy level that any molecule reach to enter the

reaction);

K - Boltzmann's constant

Considering the electrical solicitation (expressed by

a function S), can be accepted Eyring relationship:

/

Δ−

⋅=

(3)

)]([)exp(

1

T

d

cS

t

b

ATAv

a

+⋅

−

⋅⋅=

(4)

where A1, a, b, c, and d are constant.

For the low power applications, Arrhenius

relationship may be written as:

b

e

⋅=τα

[

T

e

R

0

the equation (6) may represent the general

degradation low of thermal insulation. At T =

constant, may be represented as:

T

−

(5)

]

d

t

R

− ⋅ ⋅τ−

=

exp

(6)

t

T

b

R

R

lnlnln ln

0

+

⎟

⎠

⎞

⎜

⎝

⎛

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

τ

(7)

MATHEMATICAL METHODS AND APPLIED COMPUTING

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For common dielectrics, ratio (R/R0) is permitted

0.5 and t = tlim, tlim, considered under the forms (8),

(9), (10):

T

b

R

R

t

+

⎭

⎬

⎫

⎩

⎨

⎧

−−=

τ

ln lim]) ln(ln[

0

lim

(8)

tbat/ lnlim

+=

(9)

......

32

lim

++++=

T

d

T

c

T

b

at

(10)

In the operating regime, for the electrical servo-

motors the temperature distribution in the winding is

exponential, ie f (T) = λ (exp [-λ (T-T0)]). The

distribution of failure times of winding can be

express as:

⎡

−

−

)(ln

att

⎥⎦

⎤

⎢⎣

−

−

=

0

2

ln

exp)(T

At

bb

tf

λ

λ

(11)

Symmetrically, the stabilized form of the aging law

of mechanical materials is reached if in the

Arrhenius equation is taken into account the effects

of mechanical demands on growth of aging rate:

gb

at

+=

lim

ln

T

σ⋅−

(12)

where: g is the influence coefficient for mechanical

load.

The time distribution function, damage due to

fatigue, can be written under the form:

⎡

−

Π

2t

s

where: m is the mean and s is standard deviation

referring to parameter b.

The law for mechanical sub-assemblies, as rolling

bearings, which failure is due to global and local

warming, is formalized as stated in equation (14):

⎥⎦

⎤

⎢⎣

−−

s

=

2

2

2

)limln(

exp)(

matTT

tf

(13)

3

2

0

0

)(

ln

m

k

T

b

at

δ δ −

+

+=

(14)

where: T0 is the ambient temperature (in Kelvin

degrees); k0 - factor heat sizing of functional

interfier

δ and δm - functional and minimal interfier

(acceptable).

Models widespread today are the following:

- Model of Bazovskz

- Tatar's model

- The type RADC TR - 67 - 108

- Model of MIL HDBK 217 B

Bazovskz's model, based on Arrhenius's law which

assumes that chemical reaction rate in solution

doubled for an increase in temperature to 100C,

concludes under the form:

⎞

⎜⎜

⎝

V

()

12

1

2

12

θ−θ

λλ⋅

⎟⎟

⎠

⎛

=

K

V

n

(15)

where:

λ1 is failure rate for the voltage V1 and the

temperature in Celsius;

θ1 and θ2 temperature in Celsius degrees;

λ2, mechanical constraints

Exponent n and the value of K, variation factor of

the failure rate for a change in temperature with 10C

should be determined in operational conditions for

each type of component.

Tatar's model is an exponential model described

through equation (16):

(

bSbTb

b21

exp

++⋅= λλ

)

ST

(16)

bTbS

5

2

4

2

3

++

Model RADC TR 67-108 (Rome Air Development

Center Reliability notebook) for condensers is

presented under general form as eq. (17):

⎤

⎢

⎣

where: S is nominal voltage;

T is ambient temperature in 0K.

λb, NS, H, NT, G are constant characteristics for

each type of component; e.g, for the Tantalum

capacitors with solid electrolyte: λb = 1.3 x 10-8; NS

= 0.52, H = 3, NT = 358, G = 8

A similar, but more comprehensive model for

condensers is the MILHDB 217 B described by the

equation:

G

NT

T

H

b

e

NS

S

⎟

⎠

⎞

⎜

⎝

⎛

⋅

⎥

⎦

⎥

⎢

⎡

+

⎟⎠

⎞

⎜⎝

⎛

=

1

λλ

(17)

G

NT

T

H

b

e

NS

K

⎟

⎠

⎞

⎜

⎝

⎛

⎥

⎦

⎥

⎤

⎢

⎣

⎢

⎡

+

⎟

⎠

⎞

⎜

⎝

⎛

=

1

ρ

λ

(18)

The doubts regarding dependency of the failure rate

on temperature question the validity of the models

used, in particular MILHDB 217. The Arrhenius

formula, that relates physical and chemical process

rates to temperature, has been used to describe the

relationship between temperature and time to failure

for electronic components. This formula is the basis

of methods for predicting the reliability of electric

systems. For the majority of modern electronic

components, the failure mechanisms are not

activated or accelerated by temperature increase.

The materials and processes are stable up to

temperatures higher of those recommended for use.

MATHEMATICAL METHODS AND APPLIED COMPUTING

ISSN: 1790-2769721ISBN: 978-960-474-124-3

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This inadvertence may be induced by the

observation that a large proportion of components

was observed to fail at higher temperatures. The

current data do not show such a relationship [5].

For quantitative assessment of failure rate are

employed two models:

- linear regression if model is linear or linear by

logarithming;

- adjustment of gradient developed by Fletcher and

Powell if RADC model is not linear.

These models consist of minimizing the sum of the

squares differences between estimated values λi and

n

λλ

adjusted values

))(( ,

i

1

2'

i

'∑

=

−

i

i

λ

, apply allowed

by above mentioned models.

Important issues are related to share data.

Qualitative variables

manufacturers, types of use, etc.) requiring a

particular distinction between the variables made by

constant term iteration.

The analysis made concludes that the charges which

cumulate a small number of hours comparing to

MTBF of the component lead to erroneous results

and hence the accuracy of models is found

depending on the quantity of available data. . From

this standpoint it is recognized that RADC 67-108

reflect better the average behavior of the

components.

The lack of any precise formula linking specific

environments to failure rates, even though empirical

relationships have been established between certain

failure rates and specific stresses, (voltage and

temperature).

The omission of some factors that affect reliability,

as:

− transient over-stress;

− temperature cycling;

− control of assembly;

− test and maintenance,

remain shortcomings reported in the dedicated

literature.

A more profound research into the significance of

the rate of failure for a component raised and other

issues related to the validity of some multipliers

used in the models. Adverse effects of increasing

device complexity have generally been counteracted

by process improvement.

Therefore, before presenting the model developed

for the failure rate of components are considered

necessary clarification and theoretical findings

regarding the practical failure rate.

treat (technology,

3. Dynamical model for failure rates

via Bayesian theorem

Assuming that each component is characterized by a

resistance degree to a constrain with determined

nature (electricity, heating etc.), Bayes' theorem is

operating by potential resilience for i component, i =

1, 2, ... i, ... n, γi,j , where n is the components

number of the batch subject to spectrum j of

solicitation. Generalizing, we can consider all the j

constraints simultaneously, j=1, 2, ... k, ....

Under these assumptions it can be assumed that the

component failure phenomenon is actually the result

of two factors independent:

− resistance ri, j of the component to all the j

constraints;

− the constraints spectrum of the component.

For i-1, i, i +1 components with different levels of

resistance ri-1 <ri <ri+1 and subjected to the same set

of demands j, defected at different times ti-1 <ti<ti+1

In reality the problem is complex because of

repeated constrain and a combination of different

nature to whom the components are submitted. A

deeper comprehension of a multidimensional

spectrum requires that the phenomenon of catalectic

failure to be considered random, undetermined, thus

statistical and probabilistic approached. Such a

model-based technique is provided by through

Bayes' theorem and the resulting consequences.

Given a random variable x whose probability

distribution depends on a set of parameters P = (P1,

P2, ... Pp). Exact values of the parameters are not

known with certainty, Bayesian reasoning assigns a

probability distribution of the various possible

values of these parameters that are considered as

random variables. Bayes' theory is generally

expressed through probabilistic statements as

following:

/(

)()/(

BP

)(

)

ABP

APBAP

×=

(19)

P (A / B) is the probability of A given the event B

occurs or the posteriori probability. Using Bayes'

theory may be recurring, that if exist an a priori

distribution (P (A)) and a series of tests with

experimental results B1, B2,…Bn..., expressed

according to successive equations:

)(

)/(

P)(

)/(

P

)(),/(

)

B

(

P

)/(

P

)()/(

2

2

B

1

1

B

21

1

1

B

1

ABPA

APBBAP

ABP

APBAP

=

=

(20)

MATHEMATICAL METHODS AND APPLIED COMPUTING

ISSN: 1790-2769722ISBN: 978-960-474-124-3

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)(

)/(

P

) ,...,/(),...,/(

12121

n

n

B

nn

ABP

BBBAPBBBAP

−=

−

A posteriori distribution is used as the test results

are known, being obtained as a new function a

priori. The start of operations sequences in the

Bayesian method regards the transformation γ .

In any transformation is looking to find invariant

terms. The determination of λi, invariants, Bayesian

distribution is gamma. Admitting that for each

individual component i of a batch, a resistance r i, j

for the requests spectrum j, it is recognized that

there is a statistical distribution g(ri,j) of the

quantities ri,j. As a consequence of univocal

relationship ri, j→ λi, the distribution g(ri, j) implies a

distribution h(λi) of the components failure

intensities of considered batch. The distribution h

(λi) is a random distribution of values and reflects

the statistical distribution λ of components that

survive from batch at time t. The components tend

to defect faster, the mean distribution h(λi) move to

the left as in fig. 1.

Fig.1. Distribution h(λi) depending on statistical

distribution ˝λ˝

With these assumptions we define:

f (t) - unconditioned distribution of failure times

taken over all components of a given batch;

f (t/λ) - probability density of failure time of

components subject to failure rate

Based on these definitions we can write the

distribution f(t) expression in the terms of Bayesian

method, corresponding to a limit likelihood the of

the failure time taken to fall throughout the λ values,

i.e. a priori distribution between zero and infinity:

∫

0

Moving in time to the left of the distribution

() th

/

λ

an be measured by the ratio between the

distributions posteori and a priori

()

[]

()

[]

( )

MothoM

λλ

/

∞

=

)()/()(

λ

d

λλ

htftf

(21)

( )

λ

t

MthM

β

λ

+==

1

/

'

(22)

Also (

posteriori distribution:

t β+

1

) characterize the time restriction a

( )

( )

λ

t β+

β

β

αβ

αβ

σ

λσ

==

+

+

1

=

1

1

////

(23)

The aging phenomenon has a general model given

by the equation:

( )

e

t

β+

1

If it accepts the general pattern given by the relation

(24) the obtain curve is a three-dimensional diagram

(fig.2).

The period of decline that characterizes hyperbolic

useful life of components is followed by an

exponential growth that

components at time t in operational regime.

(

α

)

ut

tf

β

+

=

1

(24)

characterizes aging

Fig.2 Failure rate depending on a posteriori

distribution

4. Case study. Experimental results

In order to establish the period of time in which by

an artificial aging of a batch of components is

achieved a desired level of reliability, we assume

the condition:

(

λ=

ktz

where:

*

λ is the reached target of components

failure rate in operational conditions;

k is acceleration coefficient;

xt is artificial aging duration.

A method for setting α and β parameters based on

available information, for a situation where prior

information is limited to the estimated failure rate

consider the empirical matching of random variables

moments, consisting of observations performed at

theoretical moments calculated using unconditioned

probability density. The results show that in some

cases reach an inadequate sensitivity distribution

from the experimental results so that the uncertainty

is either over or under estimated.

Assuming a distribution of gamma a priori:

()

*

1

β+

k

)

*

x

(25)

1

λ

β

xt

α

=

+

, ()β

1

α+

>

*

λ (26)

MATHEMATICAL METHODS AND APPLIED COMPUTING

ISSN: 1790-2769723ISBN: 978-960-474-124-3