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From andrea antonello <andrea.antone...@gmail.com>
Subject Re: [math] solving bivariate quadratic equations
Date Fri, 06 Dec 2013 14:26:37 GMT
```Hi Luc,
thanks a ton for your example code, much appreciated.

It took me a little while to understand that this works only with the
development version :)

I was able to get it working on a small sample of data with results in
the expected range.
I didn't check yet really on the result validity because I got a NPE
because of missing weights.
In the testcaes I then saw:
.withWeight(new DiagonalMatrix(weights))
where the weights are calculated for each point as 1/Z_i.

I didn't expect to have to use weights for this calculation, is there
a way to define the weights properly (for dummies like me)?

Thanks,
Andrea

On Fri, Dec 6, 2013 at 10:58 AM, Luc Maisonobe <luc@spaceroots.org> wrote:
> Hi Andrea,
>
> Le 06/12/2013 10:02, andrea antonello a écrit :
>> Hi Ted,
>>
>>> 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
>
>  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 <andrea.antonello@gmail.com
>>>> 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
>>>>
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>
>
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