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From Vincent Wolowski <vwolow...@gmail.com>
Subject [math] fitting hill equation & parameter estimation
Date Mon, 14 May 2012 16:43:48 GMT
Hi all,

I am having some troubles fitting data to a hill equation
in order to estimate the dissociation constant.
I would be thankful for your advise.

The details:

The hill equation is the following:
x^h / (x^h + Kd^h) * s

An example of the data is the following:
x, y
1.0, 2.5118
5.0, 6.8069
25.0, 16.4898
125.0, 24.8289
625.0, 18.2857

In R I used the nls package and I have done the following which works fine:

ints_values <- c(2.5118,  6.8069,   16.4898,    24.8289,   18.2857)
datapoints <- data.frame(x = c(1,5,25,125,625), intensity = ints_values)
model <- nls(intensity ~ (x^h / (x^h + Kd^h) * s),
             data = datapoints,
             start = list(Kd=90, h=1.8, s=0.8),
             upper = c(1000,2,1),
             lower = c(1,1,0.4),
             algorithm = "port",
             trace = TRUE)
res <- coef(model)[1]
print(res) // shows Kd

I tried the following with the Apache commons math lib (see below) but
the fitting does not work.
Is it the right approach to use the LevenbergMarquardtOptimizer here?

...
CurveFitter fitter = new CurveFitter(new LevenbergMarquardtOptimizer());
...
fitter.addObservedPoint(1.0, 2.5118);
fitter.addObservedPoint(5.0, 6.8069);
...
ParametricUnivariateFunction hill_equation = new HillFunction();

// 0:h=1.8, 1:Kd=90, 2:s=0.8
double[] initialguess = new double[3];
initialguess[0] = 1.8;
initialguess[1] = 90.0;
initialguess[2] = 2.0;

double[] parameter_estimations = fitter.fit(hill_equation, initialguess);

private static class HillFunction implements ParametricUnivariateFunction {
        public double value(double x, double[] params)
        {
            double h = params[0];
            double Kd = params[1];
            double s = params[2];
            return (Math.pow(x,h) / (Math.pow(x,h) + Math.pow(Kd,h)) * s);
        }

        public double[] gradient(double x, double[] params)
        {
        	double h = params[0];
        	double Kd = params[1];
        	double s = params[2];
        	double[] gradientVector = new double[params.length];
        	gradientVector[0] = // differentiate hill equation with respect to h
        	gradientVector[1] = // differentiate hill equation with respect to Kd
        	gradientVector[2] = // differentiate hill equation with respect to s
                return gradientVector;
        }
    }

Also tried to apply a DifferentiableMultivariateVectorFunction with an
Jacobian matrix, but no fitting either.

Best regards,
Vincent

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