ECE 510 Lecture 4 Reliability Plottingweb.cecs.pdx.edu/~cgshirl/Documents/QRE_ECE510... · 16 Jan 2013 ECE 510 S.C.Johnson, C.G.Shirley 4 A Function Bestiary – Bestiary: A medieval

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


Reliability Functional Forms

• Choose functional form for model to fit data

Data

Model (functional form)


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


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


Normal Distribution

• Using Excel:

– PDF = NORMDIST(x,μ,σ,FALSE)

– CDF = NORMDIST(x,μ,σ,TRUE)

• Plot using:


– x-axis = x

– σ = 1/slope

– μ = x-intercept

μ = mean σ = standard deviation


)(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


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

)


Use of Normal Distributions

• Most measurement error

• Sum of random things is normal


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


)1ln(lexponentia rand

CDF

xexf

xexF 1

te



• Using Excel:

– PDF = λ*EXP(-λx)

– CDF = 1-EXP(-λx)

• Plot using:


– x-axis = x

– λ = slope

λ = scale factor


)1ln(lexponentia rand

CDF

xexf

xexF 1


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6x value

PD

F


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

x value

CD

F


Exponential Reliability Plots


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 t

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)


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 t

h(t

)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 t

H(t

)


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


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?


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


LogNormal Distribution

• Using Excel:

– PDF = NORMDIST(ln(t),ln(t50),σ,FALSE)/t

– CDF = NORMDIST(ln(t),ln(t50),σ,TRUE)

• Plot using:


– x-axis = ln(t)

– σ = 1/slope

– ln(t50) = x-intercept

t50 = median time to fail σ = standard deviation


expnormal rand CDFNORMSINV

2

50lnln

2

1

2

1

tt

et

tf

2

50lnln

2

1

2

1

ttt

et

tdtF



• Using Excel:

– PDF = NORMDIST(ln(t),ln(t50),σ,FALSE)/t

– CDF = NORMDIST(ln(t),ln(t50),σ,TRUE)

• Plot using:


– x-axis = ln(t)

– σ = 1/slope


t50 = median time to fail σ = standard deviation


expnormal rand CDFNORMSINV

2

50lnln

2

1

2

1

tt

et

tf

2

50lnln

2

1

2

1

ttt

et

tdtF


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


0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

x value

CD

F


Lognormal Reliability Plots

Mostly decreasing failure rate: IM-type mechanism


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

t

F(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

)


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)


Use of Lognormal Distributions

Le

SB eI ~

11 tt RR


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


1ln Weibullrand1 CDF

Note: α and β are often

swapped in meaning!

Excel swaps them (below). T&T use βm and αc.

t

e



• 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


xxxf exp

1


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

x value

CD

F


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


1ln Weibullrand1 CDF

Note: α and β are often

swapped in meaning!

Excel swaps them (below). T&T use βm and αc.

alpha beta


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


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 t

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)


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 t

h(t

)


0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 t

H(t

)


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 t

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)


0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 t

h(t

)


0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 t

H(t

)


Use of Weibull Distributions

• When fail is caused by the worst of many items

• When it fits the data well

Weibull


Main Reliability Functions

t

e

2

t

e

t

e

2ln

t

e

Multiple Mechanisms


Multiple Mechanisms


ththth

tFtFtStStF

tStStS

tot

tot

tot

21

2121

21

1

Survivals multiply, hazard rates add:

Exercise 4.2


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


tth

2

2

1

11

tt

eetF thth 21

Reliability Plotting


Reliability Plotting


• Note straight lines (dotted, each Weibull)

Probit Plot


• Our eyes detect straight lines

NORMSINV(CDF)

Excel NORMxxx Functions


• 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


• Plot using:


– x-axis = x

– σ = 1/slope

– μ = x-intercept = – (y-intercept) / slope

Probit Plots in Excel


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


“Exbit” Plots


• Plot using:


– x-axis = x

– λ = slope

• Note that “exbit” is not a standard name

“exbits” data

Weibit Plots


• Plot using: – y-axis = Weibit = ln(-ln(1-CDF)) – x-axis = ln(x) – β = slope


• Note that “Weibit” is a standard name

Weibit ln data

Lognormal Probit Plot


• Plot using:


– x-axis = ln(t)

– σ = 1/slope


probits ln data

The Graph Paper Method


Exponential (semi-log) Normal

Perc

ent

Perc

ent

More Graph Paper


Lognormal Weibull

Perc

ent

Perc

ent

Exercise 4.3


• 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


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


• JMP versions of probit, “exbit”, and Weibit plots

Truncated Distributions


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


4.0

3.0

Count

Rank

4.0

3.0

MissingCount

Rank

Note Adj CDF doesn’t reach 1

Exercise 4.4


• 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


Data Censoring


• 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
