2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test

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

Rob Schubert, Shure Inc. Slide Number: 2Session #15Red Room

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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!


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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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Shape = Slope = β : slope of the line in a probability plot


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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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Threshold = Location = γ : forced to zero in 2 parameter Weibull

(not represented on probability plot)


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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


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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?


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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


Complete Data - ML Estimates3-Parameter Weibull - 95% CI


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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


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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


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


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

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AD* 0.179


Table of Statistics

Probabi l i t y Plot for generated data



estimate 1.80 7428 1967

lower bound 1.55 6617 1483

upper bound 2.10 8339 2452


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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…


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15 Let’s first look at Anderson DarlingLet’s first look at Anderson Darling


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Anderson Darling by sample sizeAnderson Darling by sample size

(Statistically this is ±3-5%)


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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

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Let’s review the effect of sample Let’s review the effect of sample size on thresholdsize on threshold


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Threshold by sample sizeThreshold by sample size

(Statistically this is ±3-5%)


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Distribution of differences in ThresholdDistribution of differences in ThresholdS

ets

of d

ata


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What about confidence intervals of What about confidence intervals of those thresholds?those thresholds?


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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


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15 What about slope? Let’s review What about slope? Let’s review

slopeslope


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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

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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


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3 parameter Weibull – Cable #13 parameter Weibull – Cable #1


Estimate 1.710 6996 1994.3


100001000100

99.999

9080706050403020

10

532

1

0.1

cable 1 - Threshold

Perc

ent

AD* 0.331


Table of Statistics




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Anderson Darling by sample size (Cable #1)Anderson Darling by sample size (Cable #1)


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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


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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

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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


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Threshold by sample size (Cable #1)Threshold by sample size (Cable #1)


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Threshold – compared to generated:Threshold – compared to generated:Above upper confidence limitAbove upper confidence limit

Cable 1 = real data


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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

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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)

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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

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20

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2

1

0.1

cable 2

Perc

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1000100

99.9

95

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50

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2

1

0.1

cable 2 - Threshold

Perc

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WeibullAD = 3.961 P-Value < 0.010

3-Parameter WeibullAD = 0.940 P-Value = 0.019



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


1000100

99.999

9080706050403020

10

532

1

0.1

Cable 2 - Threshold

Perc

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AD* 1.034


Table of Statistics

Probabi l i t y Plot for Cable 2



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Anderson Darling by sample sizeAnderson Darling by sample size


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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


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Anderson Darling – compare to previous:Anderson Darling – compare to previous:Does not fit 3 parameterDoes not fit 3 parameter

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Anderson Darling – compare to previous:Anderson Darling – compare to previous:Likelihood ratio testLikelihood ratio test



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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



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Threshold by sample size (Cable #2)Threshold by sample size (Cable #2)


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Threshold – compare to generated:Threshold – compare to generated:Above upper confidence limitAbove upper confidence limit


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Threshold – compare to generated:Threshold – compare to generated:Below lower confidence limitBelow lower confidence limit

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Threshold – compare to generated:Threshold – compare to generated:NegativeNegative

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Threshold – compare to generated:Threshold – compare to generated:Within Confidence limitsWithin Confidence limits


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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*!


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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

http://www.shure.com/

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

[email protected]

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

mailto:[email protected]


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QuestionsQuestions

Thank you for your attention.

Do you have any questions?

2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test

Engineering

Transcript of 2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test