Some engineering problems
that Tony has helped me to solve
Tim Davis
(Tony’s PhD student 1983-1991)
“Competing risk survival analysis – theory & industrial
applications”: PhD thesis, University of Birmingham
1
• Discrete failure
times
• Several failure
types (causes)
• censoring
2
Note: the use of the word “Survival” rather than “Reliability“ reflecting our initial appeal to the
literature in the medical field
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hazard functions
(cause specific hazard)
The ordinary (or in this context non-specific) hazard function is then
𝑎
𝒉 𝒕 = 𝐥𝐢𝐦+ 𝐏𝐫(𝒕 ≤ 𝑻 < 𝒕 + ∆ |𝑻 ≥ 𝒕)
∆→𝟎
𝒉 𝒕𝒊 = 𝐏𝐫(𝑻 = 𝒕𝒊 |𝑻 ≥ 𝒕)
ℎ 𝑡 =
ℎ𝑗 (𝑡)
𝑗=1
𝑘
𝐿=
Pr(𝑇 = 𝑡𝑖 )𝑑𝑖 . Pr(𝑇 ≥ 𝑡𝑖 )𝑐𝑖
𝑖=0
𝑑𝑖
giving ℎ(𝑡𝑖 )=
𝑛𝑖
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 𝑎𝑡 𝑡𝑖
=
𝑛𝑢𝑚𝑏𝑒𝑟 "𝑎𝑡 𝑟𝑖𝑠𝑘" 𝑎𝑡 𝑡𝑖
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The Survivor function
(continuous time)
S 𝑡 =
𝑢𝑖≤𝑡 [1
Discrete equivalent? 𝐹𝑗 𝑡 =
− ℎ𝑗 𝑢𝑖 ]
𝑢𝑖≤𝑡 [1
(discrete time)
− ℎ𝑗 (𝑢𝑖 )],
but in general….𝑆(𝑡) ≠
𝑎
𝑗=1 𝐹𝑗 (𝑡)
specifically when there are tied
lifetimes for different fail types
5
The conventional likelihood function for competing
risks
𝒂
𝒌𝒋
𝒂
(𝑷𝒓 𝑻 = 𝒕𝒋𝒊 , 𝑪 = 𝒋 )𝒅𝒋𝒊
𝑳=
𝒋=𝟏 𝒊=𝟏
𝑷𝒓(𝑻 ≥ 𝒕𝒌𝒊 )𝒄𝒋𝒊
𝒌=𝟏
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Discrete fail times
But in discrete time in general (specifically when there are tied lifetimes for different fail types
𝑎
𝑆(𝑡) ≠
𝐹𝑗 (𝑡)
𝑗=1
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Maximizing the conventional likelihood
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Revised likelihood function
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Revised likelihood function
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Maximizing the revised likelihood
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Hazard functions of tire failures
ℎ 𝑡 = Pr{𝑇 ∈ [𝑡, 𝑡 + ∆𝑡)|𝑇 ≥ 𝑡}
𝑡
𝐻 𝑡 =
ℎ 𝑢 𝑑𝑢
0
Estimate by
𝑑𝑖
ℎ 𝑡𝑖 = ; 𝐻 𝑡 =
𝑛𝑖
𝑛𝑖 = 𝑁 −
𝑖:𝑡𝑖 <𝑡
ℎ𝑗 𝑡 = Pr{𝑇 ∈ 𝑡, 𝑡 + ∆𝑡 , 𝐶 = 𝑗|𝑇 ≥ 𝑡}
4
Estimate by
ℎ𝑗 𝑡𝑖 = 𝑑𝑗𝑖 /𝑛𝑖
𝑑𝑖
𝑛𝑖
(𝑑𝑖 + 𝑟𝑖 )
𝑖:𝑡𝑖 <𝑡
Inflation pressure
reduced
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The 2000/01 Firestone tire crisis
(Tony’s “extended phenotype” i.e the long reach of
Tony’s
influence)
• In 2000, it was reported in the US media that people
•
•
•
•
•
•
had been killed (~300 in total) in roll-over accidents
involving tread separations .
All the accidents involved certain Firestone tires
Most of the accidents involved Ford Explorers
In September 2000, Firestone recalled some (~5m)
of the suspect tires
In May 2001, Ford recalled another ~20m tires that,
it was determined (based on my work), might also fail.
Several trips to Washington DC during the crisis, and legal depositions & taking the
witness stand for 1½ days in the high profile court case followed.
This crisis was my “Challenger accident”. A heady mix of science, ethics, legal
wrangles, and politics. The science & ethics won.
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The hazard function (again)
hazard analysis  Increasing Failure Rate. Note differences
between factory of origin for the same tire type.
Subject Tires
(colour relates to factory)
Cumulative hazard x10-6
500
450
400
350
300
250
200
150
100
Other tires
50
(colour relates to brand)
0
0
1
2
3
4
5
6
Tire age (years)
7
8
9
10
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Developing a lab test to mimic the field
Use of Factorial design to develop a lab test to
replicate the failure mode, and the relative failure
frequency
Standard
Load
1785lbs
Pressure
32psi
30psi
Ambient
Temp.
1500lbs
26psi
100oF
22psi
18psi
1300lbs
70oF
= no tread separation
= tread separation
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The 2000/01 Firestone tire crisis
The recall decision was made to replace 20 million tires
($3Bn) before the authorities asked us (Ford) to do it.
Engineering Analysis Report and Initial Decision
regarding
EA0023: Firestone Wilderness AT Tires
U.S. Department of Transportation
National Highway Safety Administration
Safety Assurance
Office of Defect Investigation
October 2001
“… the set of cumulative
hazard function curves for the
recalled tires… demonstrate
that if they are not removed
from service, the focus tires
from these plants – … will
experience a similar increase
in tread separation failures
over the next few years.…”
NHTSA report available at
www.nhtsa.gov/nhtsa/annou
nce/press/Firestone/
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Engine Modelling
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Engine Modelling
Engine modeling involves predicting how an engine performs (in terms
of torque, yT, or emissions, yE) as a result of changing load (xL), RPM (xR),
spark advance (xS) air & fuel mixture (xA), amount of exhaust gas
recycled (xE), etc. It is an important activity in Engine Mapping.
There are two possible ways to view the (empirical) model:
Either as a single response function, written as
yT = f(xL, xR, xS, xA, xE)
{# of parameters = # of coefficients}
Or as a “two-stage” response function, written as
1st stage: yT = fs(xS; b1, b2, b3,…); fs(.) are know as “spark sweeps”
2nd stage: 𝜷 = 𝒈(𝑥𝐿, 𝑥𝑅 , 𝑥𝐴, 𝑥𝐸 ) {# of parameters < # of coefficients}
Hence, the 2-stage approach reduces model complexity (design
parsimony) – hence less prediction error.
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𝑦𝑇
Engine mapping with Spark Sweeps
Residual Plot
𝑅2 ≈ 0.98
3
2
Residual
1
0
𝑦𝑇 = 𝑓(𝑥𝐿, 𝑥𝑅 , 𝑥𝑆, 𝑥𝐴, 𝑥𝐸 )
# of parameters = # of coefficients
-1 0
10
20
30
40
50
40
50
-2
-3
Spark advance (deg BTDC)
𝑥𝑠
𝑦𝑇
Residual Plot
3
1st stage: 𝑦𝑇 = 𝑓𝑠(𝑥𝑆; 𝜷)
2nd stage: 𝜷 = 𝒈(𝑥𝐿, 𝑥𝑅 , 𝑥𝐴, 𝑥𝐸 )
# of parameters < # of coefficients
𝑥𝑠
Residual
2
1
0
-1
0
10
20
30
-2
-3
Spark advance (deg BTDC)
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fitted max
true max
45
Torque (lb.ft)
40
35
30
error
25
A Quadratic Spline provides a solution
for the form of the response function:𝑦𝑇 𝛽0 + 𝛽𝐿 − (𝑥𝑆 − 𝑘)2 [if 𝑥𝑆 < 𝑘]
𝑦𝑇 = 𝛽0 + 𝛽𝐿 + 𝑥𝑆 − 𝑘)2 if [𝑥𝑆 > 𝑘]
𝑘=min spark advance for best torque
b0=Torque when 𝑥𝑆 = 𝑘
𝛽 = D in 𝑦𝑇 when 𝑆 is set 1o < (>) k
20
15
10
0
10
20
30
40
Spark advance (deg BTDC)
50
fitting a cubic response function
𝑦𝑇 =𝛽0 + 𝛽1𝑥𝑆 + 𝛽2𝑥𝑆2 + 𝛽3𝑥𝑆3
doesn’t fix this. And in any case, we know from
basic physics that sweeps only have one
turning point.
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Modelling structure
Error around a single observation in a within a spark sweep = 𝜎
Error between sweeps=𝛾 variance matrix (G)
Covariance matrix of the estimated coefficients(𝜷)of a sweep = S
Variance matrix for 2nd stage model=𝜎 2 S+ G [Cov(𝜎, G)=0]
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Flying paper helicopters
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