Auto-Correlated Data and Capability Index

S

StatMan45

#1
What pre-processing method(s) would you recommend to calculate the capability for a data set that follows an ARMA(2,2) process? Since there is significant positive correlation the variance will be underestimated and therefore will produce control limits that are too small. This data comes from a catalyst temperature with an upper limit only. The temperature of the catalytic bed is determined by the reaction rate of SO2 -> S03. The concentration of SO2 is diluted to reduce the reaction and hence the temperature. The dilution is via a control loop, hence the ARMA behavior.

Thanks
 
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Stijloor

Staff member
Super Moderator
#2
What pre-processing method(s) would you recommend to calculate the capability for a data set that follows an ARMA(2,2) process? Since there is significant positive correlation the variance will be underestimated and therefore will produce control limits that are too small. This data comes from a catalyst temperature with an upper limit only. The temperature of the catalytic bed is determined by the reaction rate of SO2 -> S03. The concentration of SO2 is diluted to reduce the reaction and hence the temperature. The dilution is via a control loop, hence the ARMA behavior.

Thanks
Comments anyone?

Thank you.

Stijloor.
 

bobdoering

Stop X-bar/R Madness!!
Trusted Information Resource
#3
It sounds to me that you have a closed loop automated adjustment, which generally will generate a sawtooth chart and continuous uniform distribution (note - not discreet uniform distribution and not independent data, so typical capability calculations are not applicable.) For this (if that is true):

Capability= (tolerance)/(process variation)

or...for this case:

Capability=(USL-LSL)/(UAL-LAL)

where the adjustment limits (AL) are the upper and lower values where the closed loop system makes its adjustment.
 
S

StatMan45

#4
Thanks for the reply. Yes, the process is under closed loop control but it is not a 'bang-bang' or on-off control as might be found with a home thermostat. It is either a proportional-integral or proportional only control so it will not have any action limits in the traditional sense. Since it is a digital controller it does have and update rate but it is fast enough to not be of concern.

The distribution is not uniform as you might suspect if the controller were being used in an on-off type mode. The distribution is left skewed and 'Johnson' transformable.

I guess the goal here would be to answer the question "What is the probability of exceeding the upper spec limit given the process." Usually this means estimating the distributional parameters and using them to answer the question.

If I do this with the Johnson transformation should I relate the probability to a sigma of a normal distribution?

Thanks
 

bobdoering

Stop X-bar/R Madness!!
Trusted Information Resource
#5
If I do this with the Johnson transformation should I relate the probability to a sigma of a normal distribution?[/FONT
I would say if you have a good fit, then that would be a good estimate of the capability.

Automotive understands that the common capability indices are for bilateral normal distributions, and everything else has to be handled in a different manner. That can be construed to mean "as the customer wishes" and most will not have any idea what to do except for transformation. So, that being the case, it should be as good of an estimate as any other single number to represent the ratio of the variation to the specifications of a process over time!

Good luck!
 

Miner

Forum Moderator
Staff member
Admin
#6
Transformation of the data is one option. Another option is:

  • Replace the mean in the Cp/Cpk formulae with the median
  • Replace 6s with the difference between the 0.99865 and 0.00135 percentiles
  • Replace 3s with the difference between the 0.99865 and 0.5 percentiles or the difference between the 0.5 and 0.00135 percentiles depending on whether you are calculating CPu or Cpl.
 
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