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Subject: Re: How to combine two probability density functions (PDFs)?
Date: Mon, 30 Aug 2010 19:55:05 +0000 (UTC)
Organization: US Geological Survey
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"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i59len$b87$1@fred.mathworks.com>...
> "Ryan " <rgold@usgs.gov> wrote in message <i5994v$koj$1@fred.mathworks.com>...
> > My expectation is that the age I seek is UNIFORMLY distributed.
> > 
> > Here's a description of a scenario: There is a stratigraphic horizon which I want to date (e.g., represent an earthquake in the geologic record). Below that horizon, I have dated deeper layer of sediment that provides a maximum constraint on the age of the earthquake. Similarly, above the earthquake horizon, I have dated a shallower layer of sediment that provides a minimum constraint on the age of the earthquake. The question: how do I describe the age of the earthquake based on these two boundaries?
> > 
> > Here is a modified scenario, from the one I previously described: Maximum age constraint = 10000 (yr) with a 1-sigma value of 400 (yr) [normally distributed] / Minimum age constraint = 5000 (yr) and a 1-sigma value of 200 (yr) [normally distributed].
> > 
> > My intuition is that I lack sufficient knowledge about sedimentation rates, erosion rates, etc., to state with confidence where the earthquake event that occurs between the boundaries is closer to the maximum or minimum bracket. Thus, I expect to see a uniformly distributed zone in the region between the constraints (e.g., flat PDF from 6000-8500 yr, in the example above), with tails that taper toward the boundaries.
> > 
> > Many thanks for additional input!
> > 
> > Sincerely,
> > Ryan
> - - - - - - - -
>   We can make the assumption that before we are given that the three events are in sequence, their a priori distributions are all mutually independent.  Given that the earthquake occurs at age t, then the probability that this lies between the ages of the other two events is:
> 
>  p(t) = normcdf(t,5000,200)*(1-normcdf(t,10000,400));
> 
> This is just the product of the probabilities, given that the earthquake is at time t, that the "10000" layer age was greater than t and the "5000" layer age was less than t. 
> 
>   I claim that p(t) is proportional to your desired pdf quantity, but to make this a valid probability density it should be normalized by being divided by its integral w.r. to t from -inf to +inf, since you have assumed a uniform a priori distribution for the earthquake age.  Offhand I cannot think of any fast way to do this integration, so you might have to resort to using one of matlab's numerical quadrature routines to get the normalizing factor.
> 
>   A plot of p(t) supports your claim that there is a "plateau" in between the earlier and later ages.  Given all the assumptions above, the same formula would still be valid if these distributions are adjusted so that the tails overlap by appreciable amounts.
> 
>   Even if sedimentation rates are uncertain, I am surprised that no account is being taken of the thickness of the sediment layers in this problem.  I would think the age of the earthquake would still be correlated to a significant degree with the respective thicknesses above and below to the dated layers.  However, I have to admit my knowledge of geology is rather minimal.  (My brother who is in fact a geologist can testify to that fact.)
> 
> Roger Stafford

Hi Roger,

Many thanks for the constructive suggestion. I think the solution works well (i.e., it produces a PDF that is consistent with my expectations).

Regarding your inquiry into sedimentation rates, you are correct that we are able to sometimes able to use information such as sample position, sediment type, and sediment package thickness to gain insight into the age of an earthquake, relative to the bracketing ages. Unfortunately, I lack that information for this particular project.

Thanks for your help.

Sincerely,
Ryan