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2015 ARS, North America, Tucson2015 ARS, North America, TucsonRed Room, Session #15Red Room, Session #15
Current Time:
09:40 PM
2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test
Rob Schubert, Shure Inc.Jeff Whalen, Shure Inc.
Alexander Ho, Shure Inc.
Begins at 3:30 PM, Thursday, June 4thBegins at 3:30 PM, Thursday, June 4th
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IntroductionIntroduction
Rob Schubert - Corporate Quality/Reliability Engineer at Shure Inc.
“The Most Trusted Audio Brand Worldwide”
Industry: Consumer and professional audio electronics
Founded: 1925
Products: Microphones, wireless microphone systems, headphones and earphones, mixers, conferencing systems
All which use cables!
Rob Schubert, Shure Inc. Slide Number: 3Session #15Red Room
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AgendaAgenda Basic definitions 5 min Shure’s cable flex test 5 min Weibull comparison with generated data 10 min Weibull comparison with real data which fits 3 parameter 10 min Weibull comparison with real data which doesn’t quite fit 10 min Advantages of each 5 min Summary 5 min Questions 10 min
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DefinitionsDefinitions
• Shape = Slope = β : slope of the line in a probability plot
• Scale = η = shifts the data “out”
• Threshold = Location = γ : forced to zero in 2 parameter Weibull
• Anderson Darling test = How well your data fits a PDF - Smaller fits better
• Likelihood ratio test = if small (<.05), 3 parameter fits better
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DefinitionsDefinitions
Shape = Slope = β : slope of the line in a probability plot
Rob Schubert, Shure Inc. Slide Number: 6Session #15Red Room
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DefinitionsDefinitions
100000100001000100
99
908070605040
30
20
10
5
3
2
1
Dat a
Perc
ent
Effect of Scale Parameter on Weibull plot
Scale = η = Shifts the data “out”
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DefinitionsDefinitions
Threshold = Location = γ : forced to zero in 2 parameter Weibull
(not represented on probability plot)
Rob Schubert, Shure Inc. Slide Number: 8Session #15Red Room
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Shure’s cable flex testShure’s cable flex test
Checking for continuity on 20 stationsFailure mode – open on any conductorEach unit is independently measured
every millisecond and cycles are recorded
Rob Schubert, Shure Inc. Slide Number: 9Session #15Red Room
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Shure’s cable flex testShure’s cable flex test
Rob Schubert, Shure Inc. Slide Number: 10Session #15Red Room
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Shure’s cable flex testShure’s cable flex test
Rob Schubert, Shure Inc. Slide Number: 11Session #15Red Room
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Shure’s cable flex testShure’s cable flex test
Based on MIL-DTL-915G
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Shure’s cable flex requirementsShure’s cable flex requirements
Purpose: to ensure cables last in the field
how can we measure that?
how many samples should we use?
Originally based on 3 parameter WeibullLater changed to 2 parameter Weibull Now re-investigating 3 parameter Weibull
Which is correct?
Rob Schubert, Shure Inc. Slide Number: 13Session #15Red Room
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100001000100
99.9
95
80
50
20
5
2
1
0.1
cable 1
Perc
ent
100001000100
99.9
95
80
50
20
5
2
1
0.1
cable 1 - Threshold
Perc
ent
Goodness of Fit Test
WeibullAD = 0.999 P-Value = 0.012
3-Parameter WeibullAD = 0.232 P-Value > 0.500
Probabi l i t y Plot for cable 1
Weibull - 95% CI 3-Parameter Weibull - 95% CI
150 data points for 1 single cable (Cable #1)150 data points for 1 single cable (Cable #1)
Distribution Shape Scale Threshold L10Weibull 2.306 9313 4050
3-Parameter 1.710 6996 1994 3946
3 parameter Weibull fits better
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3 parameter Weibull – Cable #13 parameter Weibull – Cable #1Even with 150 points, the parameters have fairly wide confidence bandsEven with 150 points, the parameters have fairly wide confidence bands
shape scale threshold
estimate 1.710 6996 1994.3
lower bound 1.484 6245 1695.1upper bound 1.972 7836 2293.6
100001000100
99.999
9080706050403020
10
532
1
0.1
cable 1 - Threshold
Perc
ent
AD* 0.331
Shape 1.71046Scale 6995.97Thres 1994.34Mean 8233.89StDev 3756.91Median 7640.96I QR 5091.21Failure 150Censor 0
Table of Statistics
Probabi l i t y Plot for cable 1
Complete Data - ML Estimates3-Parameter Weibull - 95% CI
Rob Schubert, Shure Inc. Slide Number: 15Session #15Red Room
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Let’s take out the real world variability and Let’s take out the real world variability and look at some ideal datalook at some ideal data
Rob Schubert, Shure Inc. Slide Number: 16Session #15Red Room
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200 Generated points with same shape, scale and 200 Generated points with same shape, scale and threshold threshold
100001000
99.9
95
80
50
20
5
2
1
0.1
generat ed dat a
Perc
ent
100001000100
99.9
95
80
50
20
5
2
1
0.1
generat ed dat a - Threshold
Perc
ent
Goodness of Fit Test
WeibullAD = 0.510 P-Value = 0.210
3-Parameter WeibullAD = 0.118 P-Value > 0.500
Probabi l i t y Plot for generated data
Weibull - 95% CI 3-Parameter Weibull - 95% CI
Distribution Shape Scale Threshold L103-Parameter generated 1.710 6995 1994 -
Resulted in:Weibull 2.436 9698 38503-Parameter 1.802 7428 1967 4099
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3 parameter Weibull on generated data3 parameter Weibull on generated data
100001000100
99.999
9080706050403020
10
532
1
0.1
generat ed dat a - Threshold
Perc
ent
AD* 0.179
Shape 1.80269Scale 7428.49Thres 1967.72Mean 8573.30StDev 3792.23Median 8029.53I QR 5182.39Failure 200Censor 0
Table of Statistics
Probabi l i t y Plot for generated data
Complete Data - ML Estimates3-Parameter Weibull - 95% CI
shape scale threshold
estimate 1.80 7428 1967
lower bound 1.55 6617 1483
upper bound 2.10 8339 2452
Rob Schubert, Shure Inc. Slide Number: 18Session #15Red Room
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Let’s take sets of this data:Let’s take sets of this data:
ETC.ETC.
Then randomize…
Continue until 1000 sets are made:3 samples, 5 samples, 10 samples, 20 samples and 40 samples
All in all, 78000 data points analyzed per cable
Then randomize…
Rob Schubert, Shure Inc. Slide Number: 19Session #15Red Room
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15 Let’s first look at Anderson DarlingLet’s first look at Anderson Darling
Rob Schubert, Shure Inc. Slide Number: 20Session #15Red Room
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Anderson Darling by sample sizeAnderson Darling by sample size
(Statistically this is ±3-5%)
Rob Schubert, Shure Inc. Slide Number: 21Session #15Red Room
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Distribution of differences in the Distribution of differences in the Anderson Darling statisticAnderson Darling statistic
3 parameterlower
2 parameter lower
Se
ts o
f dat
a
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Let’s review the effect of sample Let’s review the effect of sample size on thresholdsize on threshold
Rob Schubert, Shure Inc. Slide Number: 23Session #15Red Room
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Threshold by sample sizeThreshold by sample size
(Statistically this is ±3-5%)
Rob Schubert, Shure Inc. Slide Number: 24Session #15Red Room
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Distribution of differences in ThresholdDistribution of differences in ThresholdS
ets
of d
ata
Rob Schubert, Shure Inc. Slide Number: 25Session #15Red Room
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What about confidence intervals of What about confidence intervals of those thresholds?those thresholds?
Rob Schubert, Shure Inc. Slide Number: 26Session #15Red Room
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40 samples – Threshold confidence intervals40 samples – Threshold confidence intervals
As threshold increases, confidence interval goes down
40003000200010000
16000
14000
12000
10000
8000
6000
4000
2000
0
40 samples t hreshold
40 s
ampl
e co
nf r
ange
Rob Schubert, Shure Inc. Slide Number: 27Session #15Red Room
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15 What about slope? Let’s review What about slope? Let’s review
slopeslope
Rob Schubert, Shure Inc. Slide Number: 28Session #15Red Room
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Difference in slope (2 vs 3 parameter)Difference in slope (2 vs 3 parameter)
2 param. Slope
steeper
3 param. Slope
steeperSe
ts o
f dat
a
Rob Schubert, Shure Inc. Slide Number: 29Session #15Red Room
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1000 different generated Weibull sets of 40 samples 1000 different generated Weibull sets of 40 samples with different slope, scale and thresholdwith different slope, scale and threshold
50.037.525.012.50.0-12.5-25.0
Median
Mean
5.55.04.54.03.53.0
1st Quartile 1.7403Median 3.41723rd Quartile 6.3986Maximum 48.5880
4.6248 5.3782
3.1649 3.6993
5.7641 6.2974
A-Squared 68.33P-Value < 0.005
Mean 5.0015StDev 6.0189Variance 36.2270Skew ness 1.6762Kurtosis 10.9488N 983
Minimum -32.8531
Anderson-Darling Normality Test
95% Confidence I nterv al for Mean
95% Confidence I nterv al for Median
95% Confidence I nterv al for StDev9 5 % Conf idence I n t er v a ls
Summary for di f ference
2 param. Slope
steeper
3 param. Slope
steeper
2 parameter slope is generally steeper
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Do not ignore the WarningsDo not ignore the Warnings
* WARNING * Variance/Covariance matrix of estimated parameters does not exist. The threshold parameter is assumed fixed when calculating confidence intervals.
* WARNING * Convergence has not been reached for the log-likelihood criterion.
* WARNING * Newton-Raphson algorithm has not converged after 20 iterations.
* WARNING * Convergence has not been reached for the parameter estimates criterion.
40 samples 20 samples 10 samples 5 samples 3 samplesWarnings 3 131 1193 countless countless
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15 Back to real data! Cable #1Back to real data! Cable #1
Rob Schubert, Shure Inc. Slide Number: 32Session #15Red Room
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3 parameter Weibull – Cable #13 parameter Weibull – Cable #1
shape scale threshold
Estimate 1.710 6996 1994.3
lower bound 1.484 6245 1695.1upper bound 1.972 7836 2293.6
100001000100
99.999
9080706050403020
10
532
1
0.1
cable 1 - Threshold
Perc
ent
AD* 0.331
Shape 1.71046Scale 6995.97Thres 1994.34Mean 8233.89StDev 3756.91Median 7640.96I QR 5091.21Failure 150Censor 0
Table of Statistics
Probabi l i t y Plot for cable 1
Complete Data - ML Estimates3-Parameter Weibull - 95% CI
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Anderson Darling by sample size (Cable #1)Anderson Darling by sample size (Cable #1)
Rob Schubert, Shure Inc. Slide Number: 34Session #15Red Room
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Anderson Darling – compared to generatedAnderson Darling – compared to generatedDoes not fit 2 parameterDoes not fit 2 parameter
Cable 1 = real data
Rob Schubert, Shure Inc. Slide Number: 35Session #15Red Room
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Anderson Darling – compared to generatedAnderson Darling – compared to generatedDoes not fit 3 parameterDoes not fit 3 parameter
Cable 1 = real data
Order r
ever
sed fo
r clar
ity
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Anderson Darling – compared to generatedAnderson Darling – compared to generatedLikelihood ratio testLikelihood ratio test
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Anderson Darling – compared to generatedAnderson Darling – compared to generated3 parameter scores lower than 2 parameter3 parameter scores lower than 2 parameter
Rob Schubert, Shure Inc. Slide Number: 38Session #15Red Room
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Threshold by sample size (Cable #1)Threshold by sample size (Cable #1)
Rob Schubert, Shure Inc. Slide Number: 39Session #15Red Room
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Threshold – compared to generated:Threshold – compared to generated:Above upper confidence limitAbove upper confidence limit
Cable 1 = real data
Rob Schubert, Shure Inc. Slide Number: 40Session #15Red Room
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Threshold – compared to generated:Threshold – compared to generated:Below lower confidence limit (and positive)Below lower confidence limit (and positive)
Cable 1 = real data
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Threshold – compared to generated:Threshold – compared to generated:NegativeNegative
Cable 1 = real data
Order r
ever
sed fo
r clar
ity
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Threshold – compared to generated:Threshold – compared to generated:Within confidence limitsWithin confidence limits
Cable 1 = real data
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Median of positive results - Delta to Median of positive results - Delta to “true” “true” (150 – 200 data points)(150 – 200 data points)
Order r
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sed fo
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Let’s try a cable that doesn’t quite fit Let’s try a cable that doesn’t quite fit either Weibulleither Weibull
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100001000
99.9
95
80
50
20
5
2
1
0.1
cable 2
Perc
ent
1000100
99.9
95
80
50
20
5
2
1
0.1
cable 2 - Threshold
Perc
ent
Goodness of Fit Test
WeibullAD = 3.961 P-Value < 0.010
3-Parameter WeibullAD = 0.940 P-Value = 0.019
Probabi l i t y Plot for cable 2
Weibull - 95% CI 3-Parameter Weibull - 95% CI
170 data points for Cable #2170 data points for Cable #2
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3 parameter Weibull – cable #23 parameter Weibull – cable #2
shape scale thresholdestimate 2.00 965 968.9
lower bound 1.73 864 906.9upper bound 2.31 1078 1030.8
1000100
99.999
9080706050403020
10
532
1
0.1
Cable 2 - Threshold
Perc
ent
AD* 1.034
Shape 1.99766Scale 965.093Thres 968.853Mean 1824.16StDev 447.561Median 1772.17I QR 619.268Failure 170Censor 0
Table of Statistics
Probabi l i t y Plot for Cable 2
Complete Data - ML Estimates3-Parameter Weibull - 95% CI
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Anderson Darling by sample sizeAnderson Darling by sample size
Rob Schubert, Shure Inc. Slide Number: 48Session #15Red Room
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Anderson Darling – compare to previous:Anderson Darling – compare to previous:Does not fit 2 parameterDoes not fit 2 parameter
Cable 1 & 2 = real data
Rob Schubert, Shure Inc. Slide Number: 49Session #15Red Room
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Anderson Darling – compare to previous:Anderson Darling – compare to previous:Does not fit 3 parameterDoes not fit 3 parameter
Order r
ever
sed fo
r clar
ity
Cable 1 & 2 = real data
Rob Schubert, Shure Inc. Slide Number: 50Session #15Red Room
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Anderson Darling – compare to previous:Anderson Darling – compare to previous:Likelihood ratio testLikelihood ratio test
Cable 1 & 2 = real data
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Anderson Darling – compare to previous:Anderson Darling – compare to previous:3 parameter scores lower than 2 parameter3 parameter scores lower than 2 parameter
Cable 1 & 2 = real data
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Threshold by sample size (Cable #2)Threshold by sample size (Cable #2)
Rob Schubert, Shure Inc. Slide Number: 53Session #15Red Room
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Threshold – compare to generated:Threshold – compare to generated:Above upper confidence limitAbove upper confidence limit
Rob Schubert, Shure Inc. Slide Number: 54Session #15Red Room
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Threshold – compare to generated:Threshold – compare to generated:Below lower confidence limitBelow lower confidence limit
Order r
ever
sed fo
r clar
ity
Rob Schubert, Shure Inc. Slide Number: 55Session #15Red Room
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Threshold – compare to generated:Threshold – compare to generated:NegativeNegative
Order r
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Threshold – compare to generated:Threshold – compare to generated:Within Confidence limitsWithin Confidence limits
Rob Schubert, Shure Inc. Slide Number: 57Session #15Red Room
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Median of positive results - Delta to “true” Median of positive results - Delta to “true” (150 – 200 data points)(150 – 200 data points)
negative
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Advantages of the 2 parameter Advantages of the 2 parameter WeibullWeibullWorks with a very small set of dataNegative threshold doesn’t occur (As
seen, negative threshold makes little sense)
Simpler to understand for reliability engineers, most common distribution used (little need to discuss)
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Advantages of the 3 parameter Advantages of the 3 parameter WeibullWeibullSimple specification setting (can set
threshold as a minimum life)May be more accurate, but needs
significant data to actualize
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““Schubert’s Six” rules Schubert’s Six” rules 3 parameter vs 2 parameter3 parameter vs 2 parameter 3 and 5 samples are unusable (10 is barely
acceptable) with 3 parameter. 3 parameter slope is less steep. Negative threshold means you should take more
samples. Positive threshold generally measures high, but
asymptotically approaches the true answer with more samples.
Confidence interval of threshold is smaller as threshold gets larger.
Do not ignore the *Warnings*!
Rob Schubert, Shure Inc. Slide Number: 61Session #15Red Room
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Shure’s directionShure’s direction
Specify minimum flex life and test via 3 parameter Weibull (all cases, regardless of fit)
Set specifications by use case with a 25% safety factor
Test with a minimum of 20 samples, retest (add data) if threshold is negative or if calculation shows warnings
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Where to Get More InformationWhere to Get More Information
MIL-DTL-915G www.shure.com (product line) Presentation, Data, Macros, etc. available here:https://www.dropbox.com/sh/5qyncmbshmc4af1/AAD-EZcTF0jvcPxBQZUB-UCua?dl=0
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Authors/ContributorsAuthors/Contributors
Rob SchubertCorporate Quality/Reliability Engineer
Shure Inc – 8 years
Certified Reliability Engineer (ASQ)
Master’s in Acoustical Engineering, Penn State
Thesis: Use of Multiple-Input Single-Output Methods to Increase Reliability of Measurements of Road Noise in Automobiles
Previous work experience: Ford (13 years) - Quality/Reliability Engineer, 6 Sigma Black belt, Noise & Vibration Engineer
Contributors
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QuestionsQuestions
Thank you for your attention.
Do you have any questions?