I am trying to write a Java program for generating a forecast using
exponential smoothing as described here:
https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm. As
described at the linked document, exponential smoothing uses a dampening
factor "alpha". The document further goes on to say that an optimal value
for "alpha" can be found using the "Marquardt procedure", which I take as
referring to the Levenberg-Marquardt algorithm. From the linked document,
it seems that the problem of finding the optimal "alpha" is treated as a
least-squares problem and fed into the optimizer, with an initial guess for
"alpha".
After extensive web search I could not find any ready example of using the
Levenberg-Marquardt algorithm to find "alpha" for this kind of a problem,
with any programming language. So, I dug into the Javadocs and test cases
for the class *LevenbergMarquardtOptimizer* to see if I could come up with
a solution of my own. My program is given an array of values, say *[9, 8,
9, 12, 10, 12, 11, 7, 13, 9, 11, 10]*, and an initial guess for "alpha",
say *0.2*. I have been able to determine that this information needs to be
converted into a *LeastSquaresProblem*, for which I have done the following
so far:
1. Set the input array as the *target*;
2. Set the starting point *start *as the initial value of alpha (*{ 0.2
}*);
3. Set *weight* to *[1, 1 - alpha, (1 - alpha)^2, ...]*; and
4. Set the optimization function of the model to return smooth values
for each of the input values.
I am now unsure how the Jacobian should be calculated. I would like to know
if I have approached the problem correctly so far, and how to calculate the
Jacobian. I have not been able to find any material on the web or printed
form that describes the procedure for finding the Jacobian for a problem
like this.
Any help or pointers will be greatly appreciated.
