Surviving Survival Forecasting of Product Failure

Transcription

1 Surviving Survival Forecasting of Product Failure Ryan Carr Advisory Statistical Data Scientist SAS #AnalyticsX

2 Agenda Survival Model Concepts Censoring & time Alignment Preparing the data for analysis Parametric Models Exponential Log-linear Weibull 2p Weibull 3p Generalized Gamma Process to Forecast % failure at fixed points from in-service date Process to Forecast weekly failure based on in-service dates Conclusion

3 Survival Model Concepts Right Censored Unit 1 1 Mar Jun 16 Unit 1 Failed after 16 weeks Unit 2 1 Mar 16 Today The failure time for Unit 2 is considered censored since it did not fail during our study period

4 Survival Model Concepts Time Aligned Unit 1 Unit 3 1 Mar Mar 16 Today Unit 1 was placed into service the first week Unit 2 was not placed into service until 4 weeks later Unit 1 Unit 3 Week 0 Week 0 Week 20 Time align each unit by using relative times (hours, days, weeks ) rather than absolute times.

5 Survival Model Concepts Preparing the data In Service Data Returns by Week FOR each in-service date Two ways to get 1 week in service.

6 Survival Model Concepts Preparing the data Censored / time Aligned Data Censored replacements are actually units still in service this row has only been in service 1 week before the end of the study. These 44 units actually failed only 1 week after being placed in service. It is the sum of any failing 1 week after any in-service date

7 Survival Model Concepts Preparing the data Forecast Template For forecasting (after the model is fit) We focus back on only those units still in service.

8 Comparing Models Distributions Exponential Weibull 2p Weibull 3p Lognormal Generalized Gamma SOME of the Relationships among the distributions: Exponential is Weibull 2p with Scale=1 Weibull 2p is Generalized Gamma with Shape=1 Weibull 3p is Weibull 2p with an offset parameter LogNormal is Generalized Gamma with Shape=0 NOTE: distribution information from and

9 Comparing Models Exponential CDF G t = exp( α t ) ods output ParameterEstimates = exp_pe2 ; proc lifereg data=returns_censored outest=pe_exponential ; model WeeksInService*censor(1)= / distribution=exponential ; weight replacements ; output out=resid_exponential sres=sresiduals ; probplot / hlower=.05 ; inset ; run; α = 10443

10 Comparing Models Weibull 2p CDF G t = exp( α t γ ) ods output ParameterEstimates = w2p_pe2 ; proc lifereg data=returns_censored outest=pe_weibull2p ; model WeeksInService*censor(1)= / distribution=weibull ; weight replacements ; output out=resid_weibull2p sres=sresiduals ; probplot / hlower=0.05 ; inset ; α = 1800 γ = 1.374

11 Comparing Models Weibull 3p CDF G t = exp( α t δ γ ) ods output ParmEst = pe_w3p ; proc reliability data=returns_censored ; freq replacements ; distribution W3 ; probplot WeeksInService*Censor(1) ; α = 3382 γ = δ = The Weibull 3p model is a generalization of the Weibull 2p model where a location or offset parameter is added. This offset represents the minimum time to event.

12 Comparing Models LogNormal CDF ods output ParameterEstimates = ln_pe2 ; proc lifereg data=returns_censored outest=pe_lognormal ; model WeeksInService*censor(1)= / distribution=lognormal ; weight replacements ; output out=resid_lognormal sres=sresiduals ; probplot / hlower=0.05 ; inset ; run; μ = σ = 2.443

13 Comparing Models Generalized Gamma CDF ods output ParameterEstimates = gg_pe2 ; proc lifereg data=returns_censored inest=in_estw outest=pe_ggamma ; model WeeksInService*censor(1)= / distribution=gamma ; weight replacements ; output out=resid_ggamma sres=sresiduals ; probplot ; inset ; run; a = d= p=

14 Generalized Gamma Setting initial parameters proc lifereg data=returns_censored outest=out_estw noprint ; model WeeksInService*censor(1)= / distribution=weibull maxiter=5000 ; weight replacements ; data in_estw ; set out_estw ; _dist_ = "Gamma" ; _shape1_ = 1 ; * Weibull 2p * ; proc lifereg data=returns_censored inest=in_estw outest=pe_ggamma ; model WeeksInService*censor(1)= / distribution=gamma maxiter=10000 ; weight replacements ; output out=resid_ggamma sres=sresiduals ; probplot ; inset ; run; NOTE: The Generalized Gamma is a fairly complex distribution and may have convergence problems in maximum likelihood parameter estimation Two steps to help with convergence are: 1) Start parameter search at a reasonable position like the Weibull 2p estimates 2) Set the maximum iterations to a higher number

15 Applying Models to forecasts Have models Have parameter estimates How do we apply these to get estimated future values?

16 Applying Models Points in time Exponential CDF G t = exp( α t ) α = Direct Formula ods output ParameterEstimates = exp_pe2 ; proc sql ; select estimate into :alpha from exp_pe2 where parameter ='Weibull Scale' ; 0.25% Chance of failure by week 26 data prob_failure ; do WeeksInService = 4, 13, 26 ; cdf = 1 - exp(- WeeksInService/&alpha.) ; output ; end ;

17 Applying Models Points in time Weibull 2p CDF G t = exp( α t γ ) α = 1800 γ = Direct Formula proc sql ; select put(estimate, 15.10) as estimate into :alpha from w2p_pe2 where parameter ='Weibull Shape' ; select put(estimate, 15.10) as estimate into :gamma from w2p_pe2 where parameter ='Weibull Scale' ; 0.30% Chance of failure by week 26 data prob_failure ; do WeeksInService = 4, 13, 26 ; cdf = (1- (exp(-((weeksinservice)/&gamma.)**&alpha.)) ) ; output ; end ;

18 Applying Models Points in time Weibull 3p Direct Formula CDF G t = exp( α t δ γ ) α = 3382,γ = 1.195, δ = select estimate into :delta from pe_w3p where parameter ='Weibull Threshold' ; 0.28% Chance of failure by week 26 data prob_failure ; do WeeksInService = 4, 13, 26 ; cdf = (1- (exp(-((weeksinservice-&delta.)/&gamma.)**&alpha.)) ) ; output ; end ;

19 Applying Models Points in time LogNormal CDF μ = 9.999, σ = CDF Function proc lifereg data=returns_censored outest=pe_lognormal ;... proc sql ; select intercept, _scale_ into :mu, :sigma from pe_lognormal; data prob_failure ; do WeeksInService = 4, 13, 26 ; cdf2 = cdf('lognormal', WeeksInService, &mu., &sigma.) ; output ; end ; 0.29% Chance of failure by week 26

20 Applying Models Points in time Generalized Gamma CDF a = 9.838, d= , p= Lifereg maxiter=0 proc lifereg outest=pe_ggamma ; data pfail_in ; censor = 0 ; replacements = 1 ; do WeeksInService = 4, 13, 26 ; output ; end ; 0.25% Chance of failure by week 26 proc lifereg data=pfail_in inest=pe_ggamma noprint ; model WeeksInService*censor(1)= / distribution=gamma maxiter=0 ; output out=prob_failurel cdf=cdf ;

21 Applying Models Weekly Returns Generate periods / weeks for projection Apply models to get probability of failure each week Determine units still in service from each source (ship week) For each source (ship week) Apply weekly failure rates. Remove units from service for next week Predict next week s failure Align returns by source (ship week) and summarize expected returns each future week.

22 Applying Models Weekly Returns Generate periods / weeks for projection data forecastin ; fcstrange = &numperiods. ; do WeeksInService = 1 to fcstrange ; replacements=1 ; censor = 0 ; output ; end ; ForecastIn

23 Applying Models Weekly Returns Apply models to get probability of failure each week proc lifereg data=forecastin inest=pe_ggamma noprint ; model WeeksInService*censor(1)= / distribution=gamma maxiter=0 ; weight replacements ; output out=predcdf cdf=cdf ; data predpct ; set predcdf (keep=weeksinservice CDF) ; prevcdf = lag(cdf) ; if _n_ = 1 then prevcdf = 0 ; retn_pct = cdf - prevcdf ;

24 Applying Models Weekly Returns Determine units still in service from each source (ship week) proc sql ; create table UnitsInService as select ship_week, WeeksInService, censor, field_pop from ShippedStillInField a where censor = 1 group by WeeksInService, censor ;

25 Applying Models Weekly Returns For each source (ship week) Apply weekly failure rates. Remove units from service for next week Predict next week s failure proc transpose data=predpct out=predpctt prefix=retnpct ; var retn_pct ; data fct ; set UnitsInService ; if _n_ = 1 then set predpctt (drop=_name_) ; retain retnpct: ; array retnpct(*) retnpct: ; array forc[&numperiods.] ; offset = WeeksInService - 1 ; do i = 1 to (&numperiods.-offset) ; forc[i] = round(field_pop * retnpct(i+offset), 1) ; field_pop = field_pop - forc[i] ; end ;

26 Applying Models Weekly Returns Align returns by source (ship week) and summarize expected returns each future week. data forecast; merge shipsum fct_sort; by descending WeeksInService ; proc print data=forecast noobs label ; id ship_week ; var field_pop forc1-forc26 ; sum field_pop forc1-forc26 ; label field_pop="units in Service" ; format field_pop forc: comma9. ;

27 Applying Models Weekly Returns Comparing results with graphs Generalized Gamma Weibull 3p

28 Applying Models Summary Distribution Formula CDF() LIFEREG Exponential Yes Yes Yes Weibull2p Yes Yes Yes Weibull3p Yes Yes With mods LogNormal Yes Yes Generalized Gamma Yes NOTE: searching the internet for applications of the Generalized Gamma in SAS leads to many unanswered questions. The few answers I could find focused on implementation of the partial gamma function via SAS IML. The use of LIFEREG with maxiter=0 as a means of forecasting with an existing model and new data was not directly documented.

29 Conclusion selecting model Can structure comparisons leveraging relationships of models and nested ML Lognormal, Exponential and Weibull 2p are all instances of Generalized Gamma But Weibull 3p is not? Could structure comparison using RMSE of actual vs predicted Ultimately Test against reality Understand essentially, all models are wrong, but some are useful George E. P. Box. Use simplest projection method(s)

30 #AnalyticsX