Combining Random Errors

[danderso8]

I'm working on a problem at work that has me a little stumped...

We have a medical system that we would like to minimize positional errors on. The treatment process in question is repeated up to twenty (but sometimes as few as one or two) times per patient. There are several sources of error that are (almost completely) unrelated to one another.

All the sources of error can be grouped into 3 main types of error:

Daily errors
Constant errors
Universal errors

Daily errors are those that will randomly vary from treatment to treatment.

Constant errors will randomly vary from patient to patient, but for a given patient, the value will be the same across all treatments.

Universal errors will vary randomly from installation to installation, but will remain constant across a large group of patients.


The question is, how do I add up these errors to come up with the expected total error? I know that for uncorrelated random errors, I can add them in quadrature, but my thoughts are waffling back and forth on whether that is valid for these errors. One point to think about is that across a larger number of treatments, the Daily errors would tend to average out, but the others would remain constant for a patient.

Thanks for any feedback,

dan

[alThor]

You need to specify how you measure "error". If it is something like variance, then you add as usual (assuming independence like you did).

[rufus]

Um. It sounds like you have calibration issues.

[danderso8]

Hi Althor,

Let me rephrase the question...

You have a watermelon catapult. Your target is a turtle two miles away.

there are lots of parts to the catapult...rubber bands, launch arms, etc. These parts you have measured to the best of your ability, but there is still some error associated with those measurements. You run your calculations of how far the melon will fly using these measurements, but realize that you will be off by a bit because of the imperfections in the measurements. These errors will be constant across a series of launches.

There are aspects that vary with each launch...how much tension on the rubber bands, exact size and shape of the melon, temperature, wind, etc.

Also varying with each launch is the position of the target. The turtle randomly walks around a little bit, but stays relatively close to the bullseye you have previously painted on the ground. Ideally, we would like to hit the turtle directly, cause he's lazy and doesn't want to walk to pick up his lunch. But we can only aim at the bullseye.


Some turtles are not very hungry and require only one or two watermelons. Others are ravenous and need twenty melons for lunch. It takes long enough to reload the catapult that we can't expect the turtle's location during subsequent shots to be related to his location to pick up the first watermelon. (i.e., his location is completely randomized on each shot)

So the question is, how do the varying errors add with the constant errors across a series of launches?

What is the distance we will expect the turtle to have to walk to get his melon for a single shot? What is the average & SD for him to walk for a group of shots?

I don't know if that clarifies it much, but it sure is more fun now. [img]/images/graemlins/smile.gif[/img]

--Dan

[danderso8]

[ QUOTE ]
Um. It sounds like you have calibration issues.

[/ QUOTE ]

Hi Useless,

thanks for the bump.

[Giraffe]

If the three error sources are independent and approximately normally distributed, then you can add them as usual (that is, add their squares and take the square root). The different categorizations and such don't matter.