Predicting failures of a part using historical failure data

Question

Suppose I want to forecast the number of failures of part $A$ operating in my system. In my repertoire, I have historical failure numbers for part $A$ as well as the number of units of part $A$ that were operating in the system when the part failed. The objective is to forecast failures from this historical data and determine what year inventory for part $A$ will be fully depleted. We assume that one failure depletes one unit of inventory.

Data example:

Years	Failures	In Service
2017	100	2000
2018	200	3000
2019	150	2500
2020	250	3300

One challenge of the data is that the in service number changes over time. For some years, fewer parts may be operating in the system than other years. Furthermore, failures for part $A$ are not necessarily linear. Current analysis simply takes an average of the last three years, e.g. $(200 + 150 + 250)/3 = 200$, and repeats this number in future forecasts (e.g. for 2021 we assume $200$ failures, in 2022 we assume $200$ failures, etc..) for parts that do not behavior linearly and fitting a simple linear regression line on parts where the failure history does behave linearly to estimate a "failure growth rate", which corresponds to the slope of this line.

I wish to enhance this analysis by incorporating the changing in service numbers and providing actual robust models for nonlinear historical failure data. The straightforward method seems to be to weigh the yearly historical failure numbers by the number of units in service that year, and then running a prediction model on this data (using all the tools in a data scientist's toolkit).

However, I am concerned with what happens to the in service number once part failures are replaced by newer parts. If failed parts are being replaced by newer parts, then does this not completely invalidate the trained prediction model since the newer parts won't necessarily follow the same distribution of the older parts with historical data? If this is the case, then is it even useful to use historical data of failures over theoretical approaches for forecasting future failures?

Answer 1

Engineering failure-time analysis is very similar to demographic analysis of human survival. In both cases there is typically an early failure rate (infant mortality) and an aging process. In engineering circles the aging process is typically fit to a Weibull distribution, and for analysis of units in service it would be the aging portion of the data that would be important (as opposed to the warranty claims for early failures.)

If you plot your data as Failures versus numbers at risk ("In_Service") then you get:

dat <- read.table(text="Years   Failures    In_Service
  2017  100 2000
  2018  200 3000
  2019  150 2500
  2020  250 3300", head=TRUE)
plot(Failures~In_Service, data=dat)

To my eyes this suggest a rising probability of failure over time which is what a Weibull model could handle. Unfortunately this is only the ascending portion and doesn't capture the failure rates over a longer time frame. your data is grouped so you would need to find a modeling framework that would allow counts to be the dependent variable with time as the independent variable weighted by the numbers in service.

Digression: Lets set it up as a life-table:

So 2000-100 were in service at end of 2017, so 1100 were new to service in 2018 and 1900 were at least a year old. and 3000 -(2000-200) were in-service at end of 2018 with 1900 at least 2 years and 1100 only a year old and 2500 -(2000-200) were in-service at end of 2018 with 1900 at least 2 years and 1100 only a year old ooooops ... those numbers appear to imply an additional process besides "birth and death".

So you cannot argue that a life-table analysis is accurate for this data. #------------------------

So these might be some of the questions and perspectives that come up in a statistical consultation. You would be asked whether you have any real data and then discussion could commence about how accurate that data might be and whether it might be refined to allow better statistical models to be applied.

Answer 2

This sounds like a Poisson or negative binomial process. You could model this using a generalized linear model accounting for the exposure of each part. Newly added parts would have shorter exposure, older parts would have longer exposure. To investigate a time trend you could include time as a factor in the model. This could be performed in SAS Proc Genmod. Here is a link to a paper on constructing prediction intervals for future observations based on historical data in non-normal models such as Poisson and negative binomial.

Johnson, G.S. (2021). Tolerance and Prediction Intervals for Non-normal Models. Researchgate.net.