/// <summary>
/// Demonstrates how to solve systems of non-linear
/// equations using the LMA
/// </summary>
static void SolveExamples(LMBackend solverType)
{
var solver = new LMSolver(optimizerBackend: solverType);
// 1st example:
// Aim: Find (x, y) for which
// y = -0.5x² + 3.0 &&
// y = exp(-x)
// is true.
// Transform to a minimization problem:
// r[0] = 0 = y + 0.5x² - 3.0
// r[1] = 0 = y - exp(-x)
Action <double[], double[]> f = (p, r) => {
// Here: p[0] = x; p[1] = y
var x = p[0];
var y = p[1];
r[0] = y + 0.5 * x * x - 3.0;
r[1] = y - Math.Exp(-x);
};
// -> Find (x, y) that minimizes sum(r²) using LMA
// first solution
var result1 = solver.Solve(f, new[] { -2.0, 0.0 });
// second solution
var result2 = solver.Solve(f, new[] { 2.0, 0.0 });
Console.WriteLine("Find (x, y) for which");
Console.WriteLine(" y = -0.5x² + 3.0 &&");
Console.WriteLine(" y = exp(-x)");
Console.WriteLine("is true.");
Console.WriteLine("1st solution: x = {0}, y = {1}", result1.OptimizedParameters[0], result1.OptimizedParameters[1]);
Console.WriteLine("2nd solution: x = {0}, y = {1}", result2.OptimizedParameters[0], result2.OptimizedParameters[1]);
Console.WriteLine();
// 2nd Example:
// Solve
// y = -x² + 6 &&
// y = -2x - 2
result1 = solver.Solve((p, r) => {
r[0] = p[1] + p[0] * p[0] - 6.0; // 0 = y + x² - 6
r[1] = p[1] + 2.0 * p[0] + 2.0; // 0 = y + 2x + 2
}, new[] { 0.0, 0.0 });
result2 = solver.Solve((p, r) => {
r[0] = p[1] + p[0] * p[0] - 6.0;
r[1] = p[1] + 2.0 * p[0] + 2.0;
}, new[] { 10.0, 0.0 });
Console.WriteLine("Solve");
Console.WriteLine(" y = -x² + 6 &&");
Console.WriteLine(" y = -2x - 2");
Console.WriteLine("1st solution: x = {0}, y = {1}", result1.OptimizedParameters[0], result1.OptimizedParameters[1]);
Console.WriteLine("2nd solution: x = {0}, y = {1}", result2.OptimizedParameters[0], result2.OptimizedParameters[1]);
}
/// <summary>
/// Demonstrantes the usage of the generic minimization API
/// LMSolver.Minimize
/// </summary>
static void GenericMinimizationExamples(LMBackend solverType)
{
var solver = new LMSolver(optimizerBackend: solverType);
// grid points
var xs = new[] { -1.0, -1.0, 1.0, 1.0 };
var zs = new[] { -1.0, 1.0, -1.0, 1.0 };
// data points/samples
var ys = new[] { 0.0, 1.0, 1.0, 2.0 };
// Aim: fit a regression model to the samples
// Model: y' = β0 + β1 * x + β2 * z
// Regressors: x and z (provided as grid points)
// Unknown paramters: β0, β1, β2
// Design matrix X = [1 xs zs] (4x3 matrix)
// Parameter vector ε = [β0 β1 β2]^T
// Residue: ε = y - y' = y - X β
// Find β' = argmin_β sum(ε²)
var fit = solver.Minimize((p, r) => {
// compute the residue vector ε = y - y' based on the
// current parameters p (== β0, β1, β2)
var β0 = p[0];
var β1 = p[1];
var β2 = p[2];
for (int i = 0; i < ys.Length; ++i)
{
r[i] = ys[i] - (β0 + β1 * xs[i] + β2 * zs[i]);
}
}, new[] { 0.0, 0.0, 0.0 }, // initial guess for β'
ys.Length); // number of samples
Console.WriteLine();
Console.WriteLine("Aim: fit a linear 2D regression model to four samples");
Console.WriteLine(" Model: y' = β0 + β1 * x + β2 * z");
Console.WriteLine(" Regressors: x and z (grid points)");
Console.WriteLine(" Unknown paramters: β0, β1, β2");
Console.WriteLine(" Design matrix X = [1 xs zs] (4x3 matrix)");
Console.WriteLine(" Parameter vector ε = [β0 β1 β2]^T");
Console.WriteLine(" Residue: ε = y - y' = y - X β");
Console.WriteLine(" Find β' = argmin_β sum(ε²)");
Console.WriteLine("Fit: y' = {0} + {1} x + {2} z", fit.OptimizedParameters[0], fit.OptimizedParameters[1], fit.OptimizedParameters[2]);
}
/// <param name="ftol">Relative error desired in the sum of squares.
/// Termination occurs when both the actual and
/// predicted relative reductions in the sum of squares
/// are at most ftol.</param>
/// <param name="xtol">Relative error between last two approximations.
/// Termination occurs when the relative error between
/// two consecutive iterates is at most xtol.</param>
/// <param name="gtol">Orthogonality desired between fvec and its derivs.
/// Termination occurs when the cosine of the angle
/// between fvec and any column of the Jacobian is at
/// most gtol in absolute value.</param>
/// <param name="epsilon">Step used to calculate the Jacobian, should be
/// slightly larger than the relative error in the
/// user - supplied functions.</param>
/// <param name="stepbound">Used in determining the initial step bound. This
/// bound is set to the product of stepbound and the
/// Euclidean norm of diag*x if nonzero, or else to
/// stepbound itself. In most cases stepbound should lie
/// in the interval (0.1,100.0). Generally, the value
/// 100.0 is recommended.</param>
/// <param name="patience">Used to set the maximum number of function evaluations
/// to patience * (number_of_parameters + 1)</param>
/// <param name="scaleDiagonal">If 1, the variables will be rescaled internally. Recommended value is 1.</param>
/// <param name="verbose">true: print status messages to stdout</param>
/// <param name="optimizerBackend">NativeLmmin: use native C implementation of lmmin</param>
public LMSolver(double ftol = defaultTolerance,
double xtol = defaultTolerance,
double gtol = defaultTolerance,
double epsilon = defaultTolerance,
double stepbound = 100.0,
int patience = 100,
bool scaleDiagonal = true,
bool verbose = false,
LMBackend optimizerBackend = LMBackend.NativeLmmin)
{
this.Ftol = ftol;
this.Xtol = xtol;
this.Gtol = gtol;
this.Epsilon = epsilon;
this.InitialStepbound = stepbound;
this.Patience = patience;
this.ScaleDiagonal = scaleDiagonal;
this.VerboseOutput = verbose;
this.OptimizerBackend = optimizerBackend;
}
/// <summary>
/// Curve fitting using the Fit... convenience functions
/// </summary>
static void CurveFittingExamples(LMBackend solverType)
{
var solver = new LMSolver(optimizerBackend: solverType);
// 1st example: fit a model that is non-linear in its parameters
// y' = p1 * sin(p2 * t + p3) + p4
// to some noisy data (t, y)
// => minimize sum((y - y')²)
//
// for generating some noise
var rand = new Random(12345);
// sample points (here: equidistant)
var ts = Enumerable.Range(0, 100).Select(i => i * 0.01).ToArray();
// signal: "measured" values at sample points
var sins = Enumerable.Range(0, 100)
.Select(i => 2.0 * Math.Sin(3.0 * ts[i] + 0.25) + rand.NextDouble() * 0.02)
.ToArray();
// FitCurve evaluates the model p[0] * Math.Sin(p[1] * t + p[2]) + p[3]
// for each t and computes the residual based on the corresponding y value
var fit1D = solver.FitCurve((t, p) => p[0] * Math.Sin(p[1] * t + p[2]) + p[3], new[] { 1.0, 1.0, 1.0, 1.0 }, ts, sins);
Console.WriteLine();
Console.WriteLine("Aim: fit a sine wave to noisy data");
Console.WriteLine("Fit: y' = {0} sin({1} x + {2}) + {3}", fit1D.OptimizedParameters[0], fit1D.OptimizedParameters[1], fit1D.OptimizedParameters[2], fit1D.OptimizedParameters[3]);
// 2nd example: fit a linear plane to some sample points
// (cf. GenericMinimizationExamples for doing the same with
// the generic Minimization API)
// grid points
var xs = new[] { -1.0, -1.0, 1.0, 1.0 };
var zs = new[] { -1.0, 1.0, -1.0, 1.0 };
// data points/samples
var ys = new[] { 0.0, 1.0, 1.0, 2.0 };
var fitPlane = solver.FitSurface((x, z, p) => p[0] + p[1] * x + p[2] * z, new[] { 0.0, 0.0, 0.0 }, xs, zs, ys);
Console.WriteLine();
Console.WriteLine("Aim: fit a plane to four data points");
Console.WriteLine("Fit: y = {0} + {1} x + {2} z", fitPlane.OptimizedParameters[0], fitPlane.OptimizedParameters[1], fitPlane.OptimizedParameters[2]);
}
