|laziness, impatience, and hubris|
Estimating continuous functionsby zdog (Priest)
|on Mar 29, 2004 at 06:50 UTC||Need Help??|
I'm presently working on a programming project (of which the programming language shall remain nameless :-), and I've reached a point where my education and/or my creativity does not suffice to come to a satisfactory solution.
Dear friends! Say you are given a table with 4 columns. Each column represents a variable in a continuous mathematical function. So, say the function takes 3 independent variables and its result is the dependent variable. The function is one-to-one-to-one-to-one, or three-to-one, or something like that, so if you are given any three of the variables, you should be able to solve for a specific value for the 4th variable. Unfortunately, this function is not known to us. Instead, we are given a table containing a limited set of input and results of the function.
In case that description is not clear, hopefully this example will clarify. Say the table looks something like this:
We do not know the function, but we do know that for those values of x, we get those values of y. However, we do not know what we get for y if x were to equal 3.5. Not necessarily an accurate estimate, but surely not an unreasonable one, of the resulting value is 4.5 because the value of x was directly in between 3 and 4 so we take the value in the middle of 4 and 5 for the y value. Unfortunately, as we add more independent variables, the relationship between them becomes increasingly obfuscated given such a data set.
So I think you can guess where I am going with this ...
Does anyone know of a method for coming up with such reasonable estimates for those more complex functions and data tables, if such a method that would be reasonably implementable in a language like Perl exists? The value 1 can be an estimate for any set of values, but certainly something that tends to approach reality is favorable. This problem busted my brain for a few hours before I decided to ask someone else.
One thing I tried is if one of the given points (given by a set of inputed variables) from the table is not close enough to the required point, the midpoints between the points are calculated and those become treated as "given" points. This is repeated until something close arises. This wouldn't be bad if the number of midpoints didn't multiply so quickly. I tried limiting the number of "given" midpoints for each loop to those that are closest, but then issues with the domain arose so that the desired point was no longer within the "given" set.
Basically, this comes down to drawing or estimating the curve of a continuous function given a set of points. The example I gave is a reasonable solution to the problem in two dimensions. Is there any good or decent way to expand this to more dimensions? 3, 4, 5, and so on? Visually, it doesn't seem like too difficult of a problem, but mathematically, especially in the realm of programming, it may be ..
Thanks for any thoughts, ideas, or insights!
Zenon Zabinski | zdog | firstname.lastname@example.org