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ECE 510 Lecture 4 Reliability Plotting
T&T 6.1-6
Scott Johnson
Glenn Shirley
Functional Forms
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 2
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 3
Reliability Functional Forms
• Choose functional form for model to fit data
Data
Model (functional form)
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 4
A Function Bestiary – Bestiary: A medieval collection of stories providing physical and
allegorical descriptions of real or imaginary animals
• Continuous distributions
– Normal
– Exponential
– Lognormal
– Weibull
– Gamma
– Beta
• Discrete distributions
– Hypergeometric
– Binomial
– Poisson
• Statistical distributions
– Chi-square
– Student’s t
– F
Most common for reliability
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 5
Normal Distribution
• Using Excel:
– PDF = NORMDIST(x,μ,σ,FALSE)
– CDF = NORMDIST(x,μ,σ,TRUE)
• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = x
– σ = 1/slope
– μ = x-intercept = – (y-intercept) / slope
μ = mean σ = standard deviation
uniform rand is CDF where
)(normal rand CDFNORMSINV
2
2
1
2
1
x
exf
2
2
1
2
1
xx
exdxF
σ2 = variance
2xe
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 6
Normal Distribution
• Using Excel:
– PDF = NORMDIST(x,μ,σ,FALSE)
– CDF = NORMDIST(x,μ,σ,TRUE)
• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = x
– σ = 1/slope
– μ = x-intercept
μ = mean σ = standard deviation
uniform rand is CDF where
)(normal rand CDFNORMSINV
Normal Distribution
0
0.1
0.2
0.3
0.4
0.5
-3 -2 -1 0 1 2 3x value
PD
F
Normal Distribution
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
x value
CD
F
2
2
1
2
1
x
exf
2
2
1
2
1
xx
exdxF
σ2 = variance
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Normal Distribution Reliability Plots
Reliabilty Function F(t) = CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
F(t
)
Survival Function S(t) = 1-F(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
S(t
)
PDF f(t) = d/dt CDF
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5 6 t
f(t)
Failure rate = h(t) = f(t)/S(t)
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 t
h(t
)
Cumulative Hazard Function H(t)
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 t
H(t
)
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Use of Normal Distributions
• Most measurement error
• Sum of random things is normal
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 9
Exponential Distribution
• Using Excel:
– PDF = λ*EXP(-λx)
– CDF = 1-EXP(-λx)
• Plot using:
– y-axis = “exbit” = -LN(1-CDF)
– x-axis = x
– λ = slope
λ = scale factor
uniform rand is CDF where
)1ln(lexponentia rand
CDF
xexf
xexF 1
te
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 10
Exponential Distribution
• Using Excel:
– PDF = λ*EXP(-λx)
– CDF = 1-EXP(-λx)
• Plot using:
– y-axis = “exbit” = -LN(1-CDF)
– x-axis = x
– λ = slope
λ = scale factor
uniform rand is CDF where
)1ln(lexponentia rand
CDF
xexf
xexF 1
Exponential Distribution
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6x value
PD
F
Exponential Distribution
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4
x value
CD
F
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Exponential Reliability Plots
Reliabilty Function F(t) = CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
F(t
)
Survival Function S(t) = 1-F(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
S(t
)
PDF f(t) = d/dt CDF
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0 1 2 3 4 5 6 t
f(t)
Failure rate = h(t) = f(t)/S(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
h(t
)
Cumulative Hazard Function H(t)
0
1
2
3
4
5
6
0 1 2 3 4 5 6 t
H(t
)
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Use of Exponential Distributions
• Constant fail rate
– No “memory” of the past; no age
– Radioactive decay
– Soft errors, external environment
• Easy to calculate
– MTTF = 1/λ
– Median time to fail from so 5.01 50
50 t
etF
2ln50 t
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Exercise 4.1 • Given an exponential fail distribution with
khr
%04.0
what is the probability of failure within 15,000 hours of use? What is the MTTF?
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Solution 4.1
• Convert to “pure” units
khr
%04.0
hour
fails4000000.0
then evaluate the fail function at 15,000 hours
%6.0006.011 000,154000000.0 eetF t
The MTTF is even easier
hours 000,500,21
MTTF
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LogNormal Distribution
• Using Excel:
– PDF = NORMDIST(ln(t),ln(t50),σ,FALSE)/t
– CDF = NORMDIST(ln(t),ln(t50),σ,TRUE)
• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = ln(t)
– σ = 1/slope
– ln(t50) = x-intercept
t50 = median time to fail σ = standard deviation
uniform rand is CDF where
expnormal rand CDFNORMSINV
2
50lnln
2
1
2
1
tt
et
tf
2
50lnln
2
1
2
1
ttt
et
tdtF
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LogNormal Distribution
• Using Excel:
– PDF = NORMDIST(ln(t),ln(t50),σ,FALSE)/t
– CDF = NORMDIST(ln(t),ln(t50),σ,TRUE)
• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = ln(t)
– σ = 1/slope
– ln(t50) = x-intercept
t50 = median time to fail σ = standard deviation
uniform rand is CDF where
expnormal rand CDFNORMSINV
2
50lnln
2
1
2
1
tt
et
tf
2
50lnln
2
1
2
1
ttt
et
tdtF
LogNormal Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 2 4 6 8 10x value
PD
F
LogNormal Distribution
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
x value
CD
F
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Lognormal Reliability Plots
Mostly decreasing failure rate: IM-type mechanism
Reliabilty Function F(t) = CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
t
F(t
)
Failure rate = h(t) = f(t)/S(t)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6t
h(t
)
Survival Function S(t) = 1-F(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
t
S(t
)Cumulative Hazard Function H(t)
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6t
H(t
)
PDF f(t) = d/dt CDF
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6t
f(t)
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Use of Lognormal Distributions
Le
SB eI ~
11 tt RR
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Weibull Distribution
• Using Excel: – PDF = WEIBULL(x,β,α,FALSE) – CDF = WEIBULL(x,β,α,TRUE) =1-EXP(-((x/α)^β)) – Note γ=0 in Excel
• Plot using: – y-axis = weibit = ln(-ln(1-CDF)) – x-axis = ln(x) – β = slope
– α = exp(-intercept/slope)
xxxf exp
1
xxF exp1
β = shape parameter
α = scale parameter
γ = location parameter
uniform rand is CDF where
1ln Weibullrand1 CDF
Note: α and β are often
swapped in meaning!
Excel swaps them (below). T&T use βm and αc.
t
e
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 20
Weibull Distribution
• Using Excel: – PDF = WEIBULL(x,β,α,FALSE) – CDF = WEIBULL(x,β,α,TRUE) =1-EXP(-((x/α)^β)) – Note γ=0 in Excel
• Plot using: – y-axis = weibit = ln(-ln(1-CDF)) – x-axis = ln(x) – β = slope
– α = exp(-intercept/slope)
xxxf exp
1
Weibull Distribution
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
x value
CD
F
Weibull Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5x value
PD
F
xxF exp1
β = shape parameter
α = scale parameter
γ = location parameter
uniform rand is CDF where
1ln Weibullrand1 CDF
Note: α and β are often
swapped in meaning!
Excel swaps them (below). T&T use βm and αc.
alpha beta
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 21
Weibull Reliability Plots Weibull, β=0.5 (<1) Weibull, β=1.5 (>1)
Increasing failure rate: Wearout (WO) type mechanism
Decreasing failure rate: Infant Mortality (IM) type mechanism
Reliabilty Function F(t) = CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
F(t
)
Survival Function S(t) = 1-F(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
S(t
)
PDF f(t) = d/dt CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
f(t)
Failure rate = h(t) = f(t)/S(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
h(t
)
Cumulative Hazard Function H(t)
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 t
H(t
)
Reliabilty Function F(t) = CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
F(t
)
Survival Function S(t) = 1-F(t)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
S(t
)
PDF f(t) = d/dt CDF
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 t
f(t)
Failure rate = h(t) = f(t)/S(t)
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6 t
h(t
)
Cumulative Hazard Function H(t)
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 t
H(t
)
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 22
Use of Weibull Distributions
• When fail is caused by the worst of many items
• When it fits the data well
Weibull
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Main Reliability Functions
t
e
2
t
e
t
e
2ln
t
e
Multiple Mechanisms
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Multiple Mechanisms
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 25
ththth
tFtFtStStF
tStStS
tot
tot
tot
21
2121
21
1
Survivals multiply, hazard rates add:
Exercise 4.2
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 26
tthUseful: for the Weibull, from T&T table 4.3:
Plot both the hazard rate h(t) (like above) and the fail function F(t).
Hand fit 2 Weibull distributions to the human mortality data like this:
Solution 4.2
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 27
tth
2
2
1
11
tt
eetF thth 21
Reliability Plotting
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 28
Reliability Plotting
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 29
• Note straight lines (dotted, each Weibull)
Probit Plot
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 30
• Our eyes detect straight lines
NORMSINV(CDF)
Excel NORMxxx Functions
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 31
• Probit = NORMSINV(CDF) • CDF = NORMSDIST(Probit)
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
Probit
CD
F
Probit Plots in Excel
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 32
• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = x
– σ = 1/slope
– μ = x-intercept = – (y-intercept) / slope
Probit Plots in Excel
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 33
probits data
• Plot using:
y-axis = probit = NORMSINV(CDF)
x-axis = x
σ = 1/slope
μ = x-intercept = – (y-intercept) / slope
Uncertainties in Probit Plots
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 34
“Exbit” Plots
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 35
• Plot using:
– y-axis = “exbit” = -LN(1-CDF)
– x-axis = x
– λ = slope
• Note that “exbit” is not a standard name
“exbits” data
Weibit Plots
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 36
• Plot using: – y-axis = Weibit = ln(-ln(1-CDF)) – x-axis = ln(x) – β = slope
– α = exp(-intercept/slope)
• Note that “Weibit” is a standard name
Weibit ln data
Lognormal Probit Plot
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 37
• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = ln(t)
– σ = 1/slope
– ln(t50) = x-intercept
probits ln data
The Graph Paper Method
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 38
Exponential (semi-log) Normal
Perc
ent
Perc
ent
More Graph Paper
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 39
Lognormal Weibull
Perc
ent
Perc
ent
Exercise 4.3
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• Make probit, “exbit”, Weibit, and lognormal probit plots
• Determine parameters for each plot
• Look at all 4 data sets (0 – 3)
• Determine which type each distribution is
– Give the parameters for each correct distribution
Solution 4.3
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 41
Data0 – exponential
λ = 3.12 Data1 – normal
µ = –0.06 σ = 0.88 Data2 – Weibull
α = 3.22 β = 1.97
Data3 – lognormal µ =0.88 σ = 0.67
JMP Plots
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 42
• JMP versions of probit, “exbit”, and Weibit plots
Truncated Distributions
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 43
CDF, Probit Scale
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
Value
Pro
bit
Cumulative Distribution Function (CDF)
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3Value
CD
F
Histogram
0
5
10
15
20
25
30
35
40
45
-3
-2.5 -2
-1.5 -1
-0.5 0
0.5 1
1.5 2
2.5
Value (Min of Range)
Nu
mb
er
of O
ccu
ran
ce
s
Top-Truncated Distributions
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 44
4.0
3.0
Count
Rank
4.0
3.0
MissingCount
Rank
Note Adj CDF doesn’t reach 1
Exercise 4.4
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 45
• Make a truncated probit plot of the data on tab Ex8.
• Find the mean and standard deviation of the original distribution as best you can.
Solution 4.4
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 46
Data Censoring
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 47
• Missing data is called “censored”
– Type I, time censored
• Exact times to fail up to time t; no data after
– Type II, fail count censored
• Exact times to fail for the first r units to fail; no data after
– Multicensored or readout
• Have a time interval within which each unit failed up to tmax; no data after
The End
16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 48
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