Hello.
On Thu, 17 May 2018 09:14:13 +0530, Manish Java wrote:
Dear Josef
My sincere thanks for your reply and your insights into using the
Levenberg-Marquardt algorithm. I already have a method in place for
finding
the optimal "alpha" using a linear optimization algorithm. I was
therefore
looking add a non-linear optimization strategy for users who would
want to
use it.
For univariate optimization, there is
http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/optim/univariate/BrentOptimizer.html
I will use the linear optimization algorithm I have for the time
being and
see if I can incorporate some other type of non-linear optimization
algorithm, such as some form of gradient descent.
If it is a least-squares problem, it should be possible to
use either "LevenbergMarquardtOptimizer" or "GaussNewtonOptimizer":
http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/fitting/leastsquares/package-frame.html
The unit tests in the code repository should put you on track.
There is also the userguide:
http://commons.apache.org/proper/commons-math/userguide/leastsquares.html
[While working your way to implementing the code for getting the
solution,
you could even contribute another example to that page.]
If you can't make it work, don't hesitate to ask more.
HTH,
Gilles
Once again, many thanks for your kind reply and insights into the
approach
I shared.
Kind regards
~ Manish
On Wed, May 16, 2018 at 9:54 PM, <josef.v...@uni-ulm.de> wrote:
Dear Manish,
There are a few issues with your approach:
first: as far as I understand, the LevenbergMarquardtOptimizer is
designed
to optimize more than one parameter. In the present case alpha is
the sole
parameter to be optimized. There is no Jacobian for the
one-dimensional
case.
In the web-side that you mentioned, they recommend to use a proper
starting value for the smoothed sequence as additional unknown
parameter.
In your case it would be something around 9. With a second parameter
to be
optimized, you could formally use the LevenbergMarquardtOptimizer.
second: you alpha-value should be restrained to stay in the range of
0<alpha<1.0.
You can achieve this by using a fit parameter x, and in the routine
that
calculates the predicted sequence you have to first convert x to
alpha
using for example the formula alpha= exp(x)/(1+exp(x)).
third: your calculation for the weights does not reflect the
weight-formula they recommend on the web-side.
fourth: The jacobian is the derivative of predicted sequence values
against the parameters,i.e against alpha and the startvalue
mentioned
above. However, in your case the 'weights', which correspond to the
inverse
of the variance or standard deviation of the observations (aka
measurement
error) also depend on alpha, and hence you do not have a
straightforward
least squares problem, were the weights (or measurement errors or
observational errors) are assumed to be constant.
For your case, I would use a standard optimizer like 'Powell' or
'BobyQA'
and in the 'value'-function I would calculate the 'Residual sum of
squares'
based on observations, sequence- and weight formulas.
Good luck, Josef
Zitat von Manish Java <manish.in.j...@gmail.com>:
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.
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