Hi Andrea,
Le 06/12/2013 10:02, andrea antonello a écrit :
> Hi Ted,
> thanks for the reply.
>
>> How would you like to handle the fact that you may have an infinite number
>> of solutions? Will you be happy with any of them? Or do you somehow want
>> to find all of them?
>
> I am afraid my math knowledge goes only to a certain point here (too
> few it seems).
> In my usecase I am trying to define a terrain model from sparse points
> (lidar elevation points).
> So my idea was to get a set of points (given X, Y, Z) to define the
> parameters of the equation and then be able to calculate an elevation
> (Z) for any position I need interpolated.
>
> I would then expect to have an exact value once the parameters are defined.
>
> I am not sure how to find the parameters though...
For this use case, I would suggest to use any kind of multivariate
optimizer. Your Z variable is linear in 6 unknown parameters (a, b, c,
d, e and f). What you want is find values for these 6 parameters such
that the quadratic sum of residuals is minimum, i.e. you want to minimize:
sum (Z_i - Z(X_i, Y_i))^2
Where X_i, Y_i, Z_i are the coordinates for point i and Z(X_i, Y_i) is
the theoretical value from you model equation, which depends on the 6
parameters.
As Z is linear in the 6 parameters, the sum above is quadratic, so your
problem is a least squares problem. Take a look at the
fitting.leastsquares package. You have the choice among two solvers
there: Gauss-Newton and Levenberg-Marquardt. As you problem is simple
(linear Z), both should give almost immediate convergence, pick up one
as you want. Start with something roughly along this line (of course,
you should change the convergence threshold to something meaningful for
your data):
// your points
final double[] xArray = ...;
final double[] yArray = ...;
final double[] zArray = ...;
// start values for a, b, c, d, e, f, all set to 0.0
double[] initialGuess = new double[6];
// model: this computes the theoretical z values, given a current
// estimate for the parameters a, b, c ...
MultivariateVectorFunction model =
new MultivariateVectorFunction() {
double[] value(double[] parameters) {
double[] computedZ = new double[zArray.length];
for (int i = 0; i < computedZ.length; ++i) {
double x = xArray[i];
double y = yArray[i];
computedZ[i] = parameter[0] * x * x +
parameter[1] * y * y +
parameter[2] * x * y +
parameter[3] * x +
parameter[4] * y +
parameter[5];
}
return computedZ;
}
};
// jacobian: this computes the Jacobian of the model
MultivariateMatrixFunction jacobian =
new MultivariateMatrixFunction() {
double[][] value(double[] parameters) {
double[][] jacobian =
new double[zArray.length][parameters.length];
for (int i = 0; i < computedZ.length; ++i) {
double x = xArray[i];
double y = yArray[i];
jacobian[i][0] = x * x;
jacobian[i][1] = y * y;
jacobian[i][2] = x * y;
jacobian[i][3] = x;
jacobian[i][4] = y;
jacobian[i][5] = 1;
}
return jacobian;
}
};
// convergence checker
ConvergenceChecker<PointVectorValuePair> checker =
new SimpleVectorValueChecker(1.0e-6, 1.0e-10);
// configure optimizer
LevenbergMarquardtOptimizer optimizer =
new LevenbergMarquardtOptimizer().
withModelAndJacobian(model, jacobian).
withTarget(zArray).
withStartPoint(initialGuess).
withConvergenceChecker(checker);
// perform fitting
PointVectorValuePair result = optimizer.optimize();
// extract parameters
double[] parameters = result.getPoint();
BEware I just wrote the previous code in the mail, it certainly has
syntax errors and missing parts, it should only be seen as a skeleton.
best regards,
Luc
>
> Cheers,
> Andrea
>
>
>
>>
>>
>>
>> On Thu, Dec 5, 2013 at 5:31 AM, andrea antonello <[email protected]
>>> wrote:
>>
>>> Hi, I need a hint about how to solve an equation of the type:
>>> Z = aX^2 + bY^2 + cXY + dX + eY + f
>>>
>>> Is an example that handles such a case available? A testcase code maybe?
>>>
>>> Thanks for any hint,
>>> Andrea
>>>
>>> PS: is there a way to search the mailinglist for topics? I wasn't able
>>> to check if this had already been asked.
>>>
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