A Sample Test | Component Lifetime

[Joris Kievits]

Hi,

I'm currently working on a project for which I have to determine the Life-Time of a certain mechanical component within a certain confidence interval. By sampling a small number (let's say n = 10) of these components and measuring the number of hours until failure, I want to determine this confidence interval.
I currently have one main question:

How can I determine the 'trueness' of the mean of these 10 components after determining the standard deviation and mean from the test? (If I were to measure just 2 times but the deviation is extremely small then I would still have a huge uncertainty to my mean right?)

I'd greatly appreciate it if someone could help me figure this out!

 

[Dale]

I tend to like Bayesian statistical methods, so that is what I would use here. If you are modeling your failure time as exponential, meaning that you have a constant hazard rate $\lambda$ then a conjugate prior/posterior for $\lambda$ is the Gamma distribution with $\alpha$ equal to the number of observations and $\beta$ equal to the sum of the failure times for previous observations

 

[Joris Kievits]

Hi Dave,

Thanks for responding so quickly! Could you please elaborate a bit more, maybe using an example? I'm not familiar with Bayesian methods and I'd hate to misinterpret your answer..

Thanks either way!

 

[Dale]

Sure. Suppose you model a set of components whose true constant failure rate is $\lambda$ = 0.01/day meaning that the mean time to failure is 100 days. Since we assume a constant failure rate (no burn-in or wear) we can use an exponential distribution to model the days to failure of any given component. So suppose you take 10 such components and you use them until they all fail, then you might have data that looks like this:
{98.0327, 364.585, 135.671, 69.0766, 113.28, 15.9393, 64.3092, 118.104, 15.5044, 71.9093}

So if you don't know $\lambda$ then the Bayesian approach would be to treat it as a random variable, take this data, and compute a probability distribution function for P($\lambda$ | data). It turns out that the best PDF to use for $\lambda$ is the gamma distribution (see https://en.wikipedia.org/wiki/Exponential_distribution#Bayesian_inference ). The gamma distribution is a pretty common distribution, but not as common as the normal distribution, so you may not have worked with it before. It has two parameters $\alpha$ and $\beta$, and the data allows us to set $\alpha$ and $\beta$ so that we get $\alpha = n = 10$ and $\beta = n \bar{x} = \Sigma x = 1066.41$. So then we can plot this PDF if you like, or you can summarize it with mean $\alpha/\beta = 0.0094$ and standard deviation $\sqrt{\alpha}/\beta=0.0029$. You can construct 95% credible intervals and so forth as well.

 

[Joris Kievits]

Hi Dave,

I'm trying to implement this into an Excel file, so that I can update it while testing. However, I seem to get extremely low values for the Gamma distribution. I'm also not able to totally wrap my head around the prior/posterior story so that might be what I've done wrong. Is there any way that you could show me the exact calculations for a data set of for example: { 8.4, 8.9, 7.3, 9.0, 6.9 } hours of operation until failure?
I'd understand if it would be too much trouble...
Thanks for the help either way!

 

[Dale]

So, for that data $\alpha = n = 5$ and $\beta = \Sigma x = 40.5$. So the mean failure rate is $\alpha/\beta = 0.12$ failures per hour. That makes sense because they typically failed in less than 10 hours per failure, so that is more than 0.1 failures per hour.

If you are evaluating the PDF, then the PDF of this gamma distribution is equal to 7.9 at 0.1. The CDF is equal to 0.38 at the same 0.1 value. If you find that your excel sheet is giving you far different numbers (like 3.8E-14 and 7.6E-16 for the PDF and CDF) then try using $1/\beta$ instead of $\beta$. Unfortunately, some software packages use $\beta$ and some use $1/\beta$ and often there is no way to know which is used without just trying it both ways and seeing which makes sense.

 

[Joris Kievits]

Hi Dave,

It was indeed the case that I needed to use $1/\beta$! Thanks for the clear explanation! However, I have some questions with respect to Bayes and the "trueness" of this PDF. Is this accounted for in the Gamma distribution? Or does the value, that I get from the Gamma distribution, still have to be modified using some "trueness error"? And you told me that I could model the time to failure as an exponential distribution, is this the Gamma distribution or should I implement the mean of the Gamma distribution, with an amount of uncertainty, into another exponential distribution? Thanks for all the help so far and I'm sorry for going on and on...

 

[Dale]

Hi Dave,

I am Dale, not Dave

I have some questions with respect to Bayes and the "trueness" of this PDF. Is this accounted for in the Gamma distribution? Or does the value, that I get from the Gamma distribution, still have to be modified using some "trueness error"?

I don’t know what you mean by “trueness”.

And you told me that I could model the time to failure as an exponential distribution, is this the Gamma distribution or should I implement the mean of the Gamma distribution, with an amount of uncertainty, into another exponential distribution?

So the idea is that we are modeling the time to failure as an exponential distribution.

The exponential distribution has a single parameter, $\lambda$, which is the failure rate. We don’t know that failure rate, so we treat it as a random variable too, in this case as a gamma distributed variable. The gamma distribution has two parameters, $\alpha$ and $\beta$, which are determined from the data.

So if we are doing a Monte Carlo simulation of failure times you would first simulate a failure rate with the gamma distribution and then you would use that rate as the parameter to simulate an exponential distribution to get the failure time. Then to get your next Monte Carlo draw you would simulate a new gamma and a new exponential. This process accounts for both the inherent randomness of failure times as well as the lack of knowledge of the exact failure rate.

 

[Joris Kievits]

Hi Dale (my apologies for mixing up your name),

By "trueness" I mean the amount of uncertainty associated with a small number of tests. If I were to perform 3 tests but the values would all be very close to each other I'd have a small standard deviation but I would expect that there is more uncertainty to the mean than just this small standard deviation... do you know if this is true?

About the exponential distribution, what value from the Gamma distribution should I use as lambda? I've attached a two plots, one of the measurements and one of the Gamma distribution (for n=10) I got out of Excel. Does it look correct to you?

 

[Dale]

By "trueness" I mean the amount of uncertainty associated with a small number of tests. If I were to perform 3 tests but the values would all be very close to each other I'd have a small standard deviation but I would expect that there is more uncertainty to the mean than just this small standard deviation... do you know if this is true?

This is automatically accounted for in the Bayesian method. There $\alpha=n$ and a small $\alpha$ gives a broad distribution for the failure rate.

About the exponential distribution, what value from the Gamma distribution should I use as lambda?

All of them!

If you are doing a Monte Carlo simulation then you start with a draw from the gamma distribution and only then make a draw from the exponential distribution. If you are doing something else then you would marginalize over $\lambda$

I've attached a two plots, one of the measurements and one of the Gamma distribution (for n=10) I got out of Excel. Does it look correct to you?

I am not on my main computer, but visually yes it looks good.