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ECE695: Reliability Physics of Nano-Transistors
Lecture 32: Physical vs. Empirical distribution
Muhammad Ashraful Alam
[email protected]
Alam ECE 695
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Copyright 2013
This material is copyrighted by M. Alam under the
following Creative Commons license:
Conditions for using these materials is described at
http://creativecommons.org/licenses/by-nc-sa/2.5/
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Alam ECE 695
Outline
1. Physical Vs. empirical distribution
2. Properties of classical distribution function
3. Moment-based fitting of data
4.
Conclusions
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Data vs. Hypothesis
Outliers identified
(box-plot, Chauvenet)
Trend identified using median
based plotting, stem-leaf
histogram
CDF plotted using
Kaplan-Meier formula
Non-parametric bootstrap to
identify parameter
uncertainty
Empirical reliability
(Hypothesis testing)
Statistical reliability
(Series/parallel systems)
Physical Reliability
(Distribution function,
prediction of an analytical
model
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Statistical Distribution is Physical
Experiments
Weibull
Log-normal
Normal
exponential
Fish-in-a-river
(Inverse exp.)
Fermi distribution
Bose-Einstein Dist
(coin-flip)
unknown
Flutter of
Saturn probe
(unknown)
If a problem can be mapped into one of the well known family,
large number of results are available.
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Outline
1. Physical Vs. empirical distribution
2. Parametric Vs. non-parametric fits
3. Estimating various distribution functions
4.
Conclusions
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Choosing distribution function
People choose functions that describe wide range of phenomena
 Normal: After all, everything eventually becomes normal (not
really!) Distribution of last resort.
 Log-Normal: A variant of normal distribution that seems
to describe many reliability problems phenomenologically (correlated
processes, such as electromigration in interconnects, shunt distribution
in solar cells)
 Weibull: Many physical systems are described by it.
In the limiting case, it becomes Exponential distribution (extreme value
problems such as thin oxide breakdown)
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Two parameter family: Normal distribution
Binomial distribution, Poisson
distribution, chi-square,
student-t distribution …
 {t − µ}2 
1
f (t ; µ ,σ )=
⋅ exp  −

2
2σ 
σ 2π

1 + erf ( z
F (t ) =
Φ (σ −1 ( t − µ ) ) Φ ( z ) =

MATpd = fitdist(data,'Normal')
CDF
PDF
µ =average, σ =standard deviation
2)  / 2
http://en.wikipedia.org/wiki/Normal_distribution
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Two parameter family: log-normal distribution
2

ln(t ) − ln( µ )} 
1
{
(PDF) f (t ; µ =
,σ )
⋅ exp  −

2
2σ
t × σ 2π


µ =average, σ =standard deviation
 −1 t 
1 + erf ( z
(CDF) F (t ) =
Φ  σ ln  Φ ( z ) =

µ

σ
2)  / 2
ln(t2 t1 ) / Φ −1 ( F (t2 )) − Φ −1 ( F (t1 ) 
ln(
t2 @ F 0.5=
/ t1 @ F 0.159)
=
 − σ 2 {ln(t µ )}2 2 
exp
2 1


λ (t ) =
π tσ erf 0.5 ln(t µ ) σ
{
}
http://en.wikipedia.org/wiki/Log-normal_distribution
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Two parameter family: Weibull distribution
Abrahmi recystallization (1905)
β β −1 −( t α )β
(α , β > 0)
f (t ;α , β ) = β ⋅ t ⋅ e
α
1 − exp ( −(t α ) β )
F (t ) =
4
3.5
f(t,α,β)
3
2.5
β =shape parameter, α =scale parameter
1.5
λ (t ) =
β −1
βt
αβ
β=0.1
β=1
β=10
2
1
β=1 …. Exponential distribution
0.5
0
0
0.2 0.4 0.6 0.8 1
Time (t)
Memory-less distribution, constant hazard rate
1.2 1.4 1.6 1.8 2
β=2 …. Rayleigh distribution
Light scattering, Corrosion in contacts, Failure rate increases with time
http://en.wikipedia.org/wiki/Weibull_distribution
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Empirical statistical distributions
Normal
1
e
2πσ
PDF
−
Log Normal
( t − µ )2
2σ
1
e
tσ 2π
2
−
Weibull
(ln t − ln µ )2
2σ
f(t)
f(t)
β
α
2
t 
⋅ 
α 
β −1
( )
⋅e α
− t
β
f(t)
Asymmetric
β=1
σ
β=3
PDF
μ
CDF
1 1
+ erf
2 2
t
t
t − µ 
σ 2 


t
1 1
 ln t − ln µ 
+ erf 

2 2
 σ 2 
(
α)
1− e
− t
β
ln(-ln(1-F(t)))
F(t)
F(t)
β=1
CDF
β=3
t
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t
t
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Empirical statistical distributions
Normal
1
e
2πσ
PDF
−
Log Normal
( t − µ )2
2σ
1
e
tσ 2π
2
f(t)
−
Weibull
(ln t − ln µ )2
2σ
β
α
2
t 
⋅ 
α 
β −1
( )
⋅e α
− t
f(t)
f(t)
β=1
Asymmetric
σ
β=3
PDF
μ
t
t
t
ln(-ln(1-F(t)))
F(t)
F(t)
β=1
CDF
β=3
t
t
Moment 1st
µ
µ ⋅e
Moment 2nd
σ2
(
σ2
−
σ
t
2
α
2
)
σ2
2µ ⋅ e − 1 ⋅ e
α 2 1+
1
β
2
1
− α 2 Γ 2 (1 + )
β
β12
β
Definitions of distribution functions
Name
Prob. distribution
cumulative PDF
survival function
hazard rate
cum. hazard rate
average hazard
Symbol
f (t ; α, β,...)
Expression
f (t ; α, β,...)
t
F (t )
∫
R( t )
1 − F (t )
λ(t )
f (t )
1 − F (t )
H (t )
λ c (t )
f (t ' )dt '
−∞
∫
t
o
λ(t ' )dt '
t
1/ t ∫ λ(t ' )dt '
o
These functions are used in difference fields in different ways …
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Example: Derivation of hazard function
H(t) … Probability that having survived till time t (event A),
it fails within time (t+dt) (event B)
B includes A ….(failed before)
P( A * B) P( B)
P( B A) =
=
=
P( A)
P ( A)
f (t )dt
f (t )
⇒ h(t ) =
R(t )
R(t )
Integrated Hazard till time t ….
t
H (t ) =
∫ h(u)du =
− ln R(t )
0
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Discrete Transform: because data is discrete
Fi =
=
fi
=
λi
i −α
n − 2α + 1
dFi Fi +1 − Fi
=
=
dt
ti +1 − ti
fi
=
1 − Fi
1
( n − 2α + 1)( ti +1 − ti )
1
( n − i − α + 1)( ti +1 − ti )
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Transformation among reliability functions
F (t )
f (t )
f (t )
dF (t )
dt
f (t )
F (t ) ∫ot f (t ' )dt '
R( t )
∞
∫
'
f (t )dt
t
λ(t ) ∫
F (t )
1 − F (t )
'
f (t )
∞
t
'
f (t )dt
(∫
H (t )
− ln
λ c (t )
1
− ln
t
∞
−
'
'
f (t )dt
t
(∫
∞
t
'
)
f (t ' )dt '
λ(t )
R(t )
d λ c (t )  −λ ( t ) t
λ ( t ') dt ' − H ( t ) dH ( t ) 
∫
λ
+
t
0
e
c

e
λ ( t )e
dt

dt 
1 − R(t )
λ ( t ') dt '
∫
0
1− e
R( t )
− ln(1 − F (t )) − ln R(t )
−
− ln R(t )
t
t
t
c
1 − e − H (t )
1 − e −λc ( t ) t
λ ( t ') dt '
∫
0
e
e − H (t )
e −λc ( t ) t
λ(t )
dH (t )
dt
λc + t
H (t )
t λ c (t )
H (t )
t
λ c (t )
−
−
∫
t
o
)
λ c (t )
dR(t )
−
dt
d ln(1 − F (t )) d ln R(t )
−
dt
dt
− ln(1 − F (t ))
t
H (t )
t
λ(t ' )dt '
t
1/ t ∫ λ(t ' )dt '
o
HW: Derive a few reliability functions yourself ...
d λ c (t )
dt
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Series and parallel systems: how to use the
distribution functions
1 − Fs (t ) = [1 − F1 (t ) ] × [1 − F2 (t ) ] × ...
Fs (t ) = F1 (t ) × F2 (t ) × ...
Rs (t ) = R1 (t ) × R2 (t ) × ...
1 − Rs (t ) = [1 − R1 (t ) ] × [1 − R2 (t ) ] × ...
dF dt −dR dt
d
λ=
=
= − ln R
R(t )
dt
1 − F (t )
Advantage of redundant system ..
λ s (t ) = λ1 (t ) + λ 2 (t ) + ...
λ i (t ) 1 + F + F 2 + ...F n −1
=
λ s (t )
nF n −1
R1, R2, … may have different distributions
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Outline
1. Physical Vs. empirical distribution
2. Properties of classical distribution function
3. Moment-based fitting of data
4.
Conclusions
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Matching moments to distributions
Of 60 oxides, 7 failed in 1000 hrs
0.14
Rank Lifetime
Fi=
(i-0.3)/(n+0.4)
1
181
0.012
2
299
0.028
3
389
0.045
4
430
0.061
5
535
0.078
6
610
0.094
0.02
7
805
0.111
0
0.12
CDF
0.1
Measured Data
Log -normal distribution
Weibull distribution
Weibull Distribution Parameters
When t=α, ln(1-F(t))=-1, F(t)=0.632, a=2990
β estimated using parameter fitting as 1.56
0.08
0.06
0.04
Good match
at high (F)
10 2.3 10 2.4
10 2.5
10 2.6 102.7 10 2.8 10 2.9
Time
Log-Normal Distribution Parameters
s=ln(T50%/T15.9%), σ=ln(3600/980)=1.30
μ=ln(T50%)=ln(3600)=8.19
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Problem of matching the moments
Log-normal distribution is considerably optimistic
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Problem of matching distribution: BFRW
P ( x,=
t 0)
= δ ( x − x0 )
∂P
∂2P
=D 2
∂t
∂x
P ( x, t )
t
(4π Dt )
P=
( x 0,=
t) 0
−1 2
e

− ( x − x0 )2 4 Dt
L
−e
1 ⇒ f (t ) =
∫ f (τ )dτ + ∫ P( x, t )dx =
0
0
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− ( x + x0 )2 4 Dt
x0
4π Dt
3
e


− x02 4 Dt
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Match apparently reasonable, but wrong
t k −1e − t θ
=
fG (t ) =
Tavg kθ
k
Γ(k )θ
Monte Carlo
Monte
Carlo
1000 Samples
1000,
10000, 50000
-4
10
Gamma
distribution
PDF
-6
10
Log-normal
distribution
-8
10
PDF =
x0
4π Dt 3
e − x0
2
4 Dt
→ t −3/2
-10
10
2
10
4
6
10
10
Time to reach the fall
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10
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Conclusions
1.
Once the data is plotted using the principles discussed in lecture, 31
one should develop a statistical model for the phenomena involved.
2.
If unsuccessful, one should choose functions with least number of
variables that described the system. In reliability, 2-parameter family
such as log-normal, Weibull distributions are most popular, because
they have been seen to give good results for correlated and weakestlink problems.
3.
Reliability is an extreme value problem, therefore one should pay
particular attention to tails of the distribution and choose sample
size accordingly.
4.
Moment-based methods are popular, but cannot distinguish between
the tails of the distribution (reflects very high moments).
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References
D. C. Hoaglen, F. Mosteller, and J.W. Tukey,
“Understanding Robust and Exploratory Data
Analysis”, Wiley Interscience, 1983. Explains the
importance of Median based analysis when the
dataset is small and the quality cannot be
guaranteed.
AT&T, “Statistical Quality Control Handbook”.
Joan Fisher Box, “R. A. Fisher and the Design of
Experiments, 1922-1926”, The American
Statistician, vol. 34, no. 1, pp. 1-7, Feb. 1980.
Linda C. Wolsterholme, “Reliability Modeling – A
Statistical Approach, Chapman Hall, CRC, 1999.
Chapter 1-7 has excellent summary of ‘Goodness of
Fit” analysis.
R. H. Myers and D.C. Montgomery, “Response
Surface Methodology”, Wiley Interscience, 2002.
This book discusses design of experiment in great
detail.
An excellent textbook that covers many topics
discussed in this Lectures is Applied Statstics and
Proability for Engineers, 3rd Edision, D.C.
Montgomery and G. C. Runger, Wiley, 2003.
F. Yates, “Sir Ronald Fisher and the Design of
Experiments”, Biometrics, vol. 20, no. 2, In
Memoriam: Ronald Aylmer Fisher, 1890-1962., pp.
307-321, (Jun. 1964.
Ranjith Roy, “A primer on the Taguchi Method”,
Van Nostrand Reinhold International Co. Ltd.,
1990.
Lloyd W Condra, “Reliability Improvement with
design of experiments”, Marcel Dekker Inc.,
1993.
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