public void minimize_test()
{
#region doc_minimize
// Example from https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm
// In this example, the Gauss–Newton algorithm will be used to fit a model to
// some data by minimizing the sum of squares of errors between the data and
// model's predictions.
// In a biology experiment studying the relation between substrate concentration [S]
// and reaction rate in an enzyme-mediated reaction, the data in the following table
// were obtained:
double[][] inputs = Jagged.ColumnVector(new [] { 0.03, 0.1947, 0.425, 0.626, 1.253, 2.500, 3.740 });
double[] outputs = new[] { 0.05, 0.127, 0.094, 0.2122, 0.2729, 0.2665, 0.3317 };
// It is desired to find a curve (model function) of the form
//
// rate = \frac{V_{max}[S]}{K_M+[S]}
//
// that fits best the data in the least squares sense, with the parameters V_max
// and K_M to be determined. Let's start by writing model equation below:
LeastSquaresFunction function = (double[] parameters, double[] input) =>
{
return((parameters[0] * input[0]) / (parameters[1] + input[0]));
};
// Now, we can either write the gradient function of the model by hand or let
// the model compute it automatically using Newton's finite differences method:
LeastSquaresGradientFunction gradient = (double[] parameters, double[] input, double[] result) =>
{
result[0] = -((-input[0]) / (parameters[1] + input[0]));
result[1] = -((parameters[0] * input[0]) / Math.Pow(parameters[1] + input[0], 2));
};
// Create a new Gauss-Newton algorithm
var gn = new GaussNewton(parameters: 2)
{
Function = function,
Gradient = gradient,
Solution = new[] { 0.9, 0.2 } // starting from b1 = 0.9 and b2 = 0.2
};
// Find the minimum value:
gn.Minimize(inputs, outputs);
// The solution will be at:
double b1 = gn.Solution[0]; // will be 0.362
double b2 = gn.Solution[1]; // will be 0.556
#endregion
Assert.AreEqual(0.362, b1, 1e-3);
Assert.AreEqual(0.556, b2, 3e-3);
}
static void Main(string[] args)
{
var point1 = new Vector <double>(0d);
var point2 = new Vector <double>(2d);
var point3 = new Vector <double>(3d);
var point4 = new Vector <double>(1d);
var functional = new L2Functional <double>((point1, 1d), (point2, 9d), (point3, 16d), (point4, 4d));
var optimizer = new GaussNewton <double>(1000, 1e-14);
var value = optimizer.Minimize(functional, new PolynomialFunction <double>(), new Vector <double>(new[] { 1d, 0d, 1d }));
Console.WriteLine(JsonSerializer.Serialize(value));
}
