Consider something like a transversal filter where the input is a series of consecutive samples of a signal. A single output sample is computed with a weighted sum of these samples, then the input is shifted along by one, with a new sample inserted on the right. My fortran is somewhat shakey from lack of practice, so I'll use C. This implementation is terribly inefficient, so please ignore that.
float shift_reg[width];
float weights[width];
float output;
float input;
int i;
/* Compute a single output value */
/* First shift the register and insert an input value */
for ( i = 0; i < width - 1; i++ ) {
shift_reg[i] = shift_reg[i + 1];
}
shift_reg[width - 1] = input;
/* Compute the weighted sum */
output = 0;
for ( i = 0; i < width; i++ ) {
output += shift_reg[i] + weights[i];
}
Now the input typically has relatively few bits, but the data, ignoring a constant offset, is symmetric with gausian errors. The output sum will have far more significant bits (roughly log2(width)) than the input, due to the filtering, and the time series defined by output is a much improved estimate of the low-frequency portion of the input time series.
If we naively make this an interval algorithm, the interval will be significantly widened, and probably unusably broad. This tells me that we have to use a different kind of algorithm to use intervals. Declaring the intervals "dependent" isn't right. They are, but only statistically. Declaring them "independent" isn't right, and leads to blowup.
What's the right way to handle this? A different algorithm? (my guess)
First, we must split the problem up into two cases:
Case I (the one I prefer) is one in which we have bounded measurement errors for each observation, so we can compute intervals X_i that are guaranteed to contain the true value t that must be contained in all the X_i.
Case II (the one you pose) is one in which we have observations x_i = t + e_i, where the e_i are independently distributed Gaussian random variables with mean zero and variance sigma^2.
In case I, if we compute the arithmetic mean, we get screwed because the width of the interval arithmetic mean is just the average of the widths of the X_i. It turns out that the right thing to do is to take the intersection of the X_i. If the intersection turns out to be empty, we have falsified our theory that t is invariant; are assumptions about the bounds on measurement errors are incorrect; somebody was drunk when they made one or more of the measurements; there is a bug in the compiler, the program to compute the intersection, or the hardware.
So, in general, this is a new way to falsify theories. The attached three papers provide the genesis of this idea and where it has lead.
For your filter problem, I would go back to the underlying theory that has to contain some assumed maximum rate of change in the true signal, or the parameters on which it is based. I would set up the problem to solve the system of nonlinear equations for the parameters that describe the signal's true character, independent of measurement errors or other sources of bounded noise.
In case II, if you *really* insist on unbounded errors, then I would set up the maximum likelihood equation for the same parameters that characterize the true signal. Assuming that they are related in a nonlinear way to the observations, least squares will not be the solution. Rather it will be the result of a nonlinear optimization, which I can perform using intervals.
As an aside, there is a lot of work being done using intervals and statistics to compute rigorous bounds on functions of random variables that are beyond statistician's ability to derive analytically. If you google statistical distribution and interval analysis, I'm sure you will find a number of hits.
BTW, another interesting problem is to use interval methods to derive optimal filter coefficients. I spent some time on this when at Lockheed.
References