Re: [math] Struggling with least squares

[Gilles]

Hello.

On Thu, 27 Dec 2018 07:32:30 -0600 (CST), David Tinker wrote:

Hi Guys. I am struggling to use the least squares optimizer to fit a 2 variable non-linear function to a curve of observed data points. I am pretty

sure I am doing something stupid because I don't know the maths. My

MultivariateJacobianFunction moves away from my starting values on the first iteration but then converges back to the starting values. My variables are

CP and W' and this is what happens:

cp 250.0 W' 24000.0 (starting values)
cp 197.17292724155843 W' 5627.212534968825
cp 233.7927945744999 W' 18662.916420529345
cp 245.72618039512386 W' 22592.92114117481
cp 248.91747806252718 W' 23643.613738664597
cp 249.99999974080222 W' 23999.9999146684
...
cp 249.99999993520052 W' 23999.99997866709
optimum {250; 24,000}

Any ideas?

What happens when different starting points?

Gilles

Here is the code:

public class LeastSquaresExample {

    private static final double DELTA = 0.000001; // for calculating
derivatives

    public static void main(String[] args) {
        Vector2D[] observedPoints = new Vector2D[3];

observedPoints[0] = new Vector2D(388, 250); // maps power in watts

to time in seconds
        observedPoints[1] = new Vector2D(368, 450);
        observedPoints[2] = new Vector2D(321, 780);

        int pmax = 961;

        MultivariateJacobianFunction fn = point -> {
            double cp = point.getEntry(0);
            double wPrime = point.getEntry(1);
            System.out.println("cp " + cp + " W' " + wPrime);

RealVector value = new ArrayRealVector(observedPoints.length);

            RealMatrix jacobian = new
Array2DRowRealMatrix(observedPoints.length, 2);
            for (int i = 0; i < observedPoints.length; i++) {
                double p = observedPoints[i].getX();
                value.setEntry(i, morton3pTime(cp, wPrime, pmax, p));

// each row in the jacobian is a measurement and cols are

partial derivatives wrt cp(0) and wPrime(1)
                double a, b;
                a = morton3pTime(cp - DELTA, wPrime, pmax, p);
                b = morton3pTime(cp + DELTA, wPrime, pmax, p);
                jacobian.setEntry(i, 0, (a - b) / (DELTA * 2));

                a = morton3pTime(cp, wPrime - DELTA, pmax, p);
                b = morton3pTime(cp, wPrime + DELTA, pmax, p);
                jacobian.setEntry(i, 1, (a - b) / (DELTA * 2));
            }
            return new Pair<>(value, jacobian);
        };

        double[] target = new double[observedPoints.length];
        for (int i = 0; i < observedPoints.length; i++) target[i] =
observedPoints[i].getY();

        LeastSquaresProblem problem = new LeastSquaresBuilder()
                .start(new double[]{250.0, 24000.0})
                .model(fn)
                .target(target)
                .maxEvaluations(1000)
                .maxIterations(1000)
                .build();
        LeastSquaresOptimizer.Optimum optimum = new
LevenbergMarquardtOptimizer().optimize(problem);
        RealVector pt = optimum.getPoint();
        System.out.println("optimum " + pt);
    }

private static double morton3pTime(double cp, double wPrime, double

pmax, double p) {
        return wPrime / (p - cp) + wPrime / (cp - pmax);
    }
}

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