ECE 510 Lecture 4 Reliability Plotting
T&T 6.1-6
Scott Johnson
Glenn Shirley
Functional Forms
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Reliability Functional Forms
• Choose functional form for model to fit data
Data
Model (functional form)
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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
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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
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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
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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
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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
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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
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
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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
)
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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
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ththth
tFtFtStStF
tStStS
tot
tot
tot
21
2121
21
1
Survivals multiply, hazard rates add:
Exercise 4.2
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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
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tth
2
2
1
11
tt
eetF thth 21
Reliability Plotting
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Reliability Plotting
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• Note straight lines (dotted, each Weibull)
Probit Plot
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• Our eyes detect straight lines
NORMSINV(CDF)
Excel NORMxxx Functions
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• 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
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• Plot using:
– y-axis = probit = NORMSINV(CDF)
– x-axis = x
– σ = 1/slope
– μ = x-intercept = – (y-intercept) / slope
Probit Plots in Excel
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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
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“Exbit” Plots
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• 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
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• 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
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• 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
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Exponential (semi-log) Normal
Perc
ent
Perc
ent
More Graph Paper
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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
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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
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• JMP versions of probit, “exbit”, and Weibit plots
Truncated Distributions
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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
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4.0
3.0
Count
Rank
4.0
3.0
MissingCount
Rank
Note Adj CDF doesn’t reach 1
Exercise 4.4
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• 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
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Data Censoring
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• 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
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