From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
Subject: Re: Help With Statistics on "Compressed Data"
Keywords: statistics, compressed data,
Reply-To: thompson@atlas.socsci.umn.edu
Organization: Economics Department, University of Minnesota
Date: Tue, 10 Nov 1992 00:12:24 GMT

ekdug@b56vxg.kodak.com (Linda Stustman) writes:

>I'm looking for a way to estimate an upper bound on the standard deviation
>of a stream of plant data.  The complication is that the data is coming from 
>a process data base that has a type of compression applied to it.

>Simply put, data from the process is generated ever 30 minutes (an analysis
>by a gas chromatograph).  The process data base compares the new value with
>the previous one and only records the new value (with an associated time
>stamp) if the absolute value of the difference between the two readings is
>greater than a fixed threshold.

>In practice, this means 3 to 6 values a day are recorded for the variable,
>out of the 48 analyses that are actually done.  My problem is to come up
>with a way of providing a reasonable estimate for the standard deviation
>of the analysis values that uses the information present in the recorded
>values, but also includes the information that the other 42 to 45 analyses
>varied less that the threshold value.

>My only idea on how to attack the problem (to date) is to assume that the 
>range of +/- the threshold value corresponds to +/- 3 sigma of a normally
>distributed variable.  Then, I could generate the appropriate number of 
>"missing" values from a random normal distribution, add the recorded 
>values and calculate a standard deviation from the resulting augmented
>"data" set.  Perhaps arguments could be made for treating the +/- range
>of the threshold value as +/- 2 or +/- 4 sigma (and does anyone have any
>comments?).

This does not sound like a very good idea to me.  Here are the random
thoughts on which this opinion is based:

It is not clear to me exactly what model you have in mind here.  Are
you proposing to assume that the mean of the process is unchanged from
analysis to analysis?  What about the variance?  Is that also to be
assumed not changing over time?  Is there any autocorrelation in
either the basic values that are begin measured or the measurement
errors?  Is your initial value sampled unconditionally, or are you
extracting a subsequence from a measurement process that has been
running for some time?  No good statistical answer to your question
can be obtained without (at least implicitly) answering these
questions.

Another important question: Is the threshhold value known?  If so then
it certainly would not be appropriate to fix it at +/- K standard
deviations for any value of K, since this is equivalent to saying that
you already know the standard deviation!

Suppose that you assume that the analysis at time t produced a value
x(t), that the x(t) values are independent normals with mean M and
variance V, that x(t) is observed if and only if 

	|x(t) - x(t-1)| > E,

and that the aim of the analysis is to estimate M and V given
knowledge of E and a sample (t1,x(t1)), ..., (tn,x(tn)) of n data
points corresponding to the points at which the threshhold was met.

Then it seems that you have a fairly standard, if nonlinear, maximum
likelihood problem.  I haven't checked, but I suspect that all
parameters are identified.  In fact, I think that this remains true
even when the threshold E is also treated as a parameter.  Your main
difficulty in this scenario would be to calculate the likelihood,
since the censoring introduces a rather messy dependency among the
observations.

One simplification might be to first work conditionally on the
sequence of dummy variables for whether or not an observation was
taken.  Clearly you will not be able to estimate any of the parameters
M, V or E from this data alone.  However, I think that you can
estimate the ratio E to Sqrt(V) (the standard deviation) from this
data alone.

--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209