C# Multi Variable Curve Fitting Example

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using System;

using CenterSpace.NMath.Core;
using CenterSpace.NMath.Analysis;

namespace CenterSpace.NMath.Analysis.Examples.CSharp
{
  class MultiVariableCurveFittingExample
  {
    /// <summary>
    /// The MultiVariableFunctionFitter&lt;T&gt; Needs a parameterized function
    /// and a set of data points. One way to specify the parameterized function,
    /// and optionally its gradient with respect to the parameters, is to
    /// implement an instance of the abstract class DoubleParameterizedFunctional.
    /// You must overwrite the Evaluate() method which computes and returns the
    /// parameterized function value at a specified set of parameters and 
    /// point. It is optional to overwrite the GradientWithRespectToParams() method.
    /// If you do not overwrite it a numerical approximation using finite differences
    /// will be used to approximate the gradient if it is needed.
    /// 
    /// Here the parameterized function we are defining is a real valued function
    /// of two variables, x0 and x1, and three parameters, p0, p1, and p2, defined
    /// by the formula:
    /// 
    /// p0*x1*x0^2 + p1*sin(x0) + p2*x1^3
    /// 
    /// </summary>
    class ParameterizedFunction : DoubleParameterizedFunctional
    {
      /// <summary>
      /// Creates an instance of our parameterized function. We must 
      /// initialize the base class with the dimension of our functions
      /// domain. Since our function is a function of two variables 
      /// we initialize the base class with 2.
      /// </summary>
      public ParameterizedFunction()
        : base( 2 )
      {
        ;
      }

      /// <summary>
      /// Override the abstract evaluate function.
      /// </summary>
      /// <param name="parameters">The parameter values.</param>
      /// <param name="x">The point to evaluate at.</param>
      /// <returns>The value of the parameterized function at the given
      /// point and parameters.</returns>
      public override double Evaluate( DoubleVector parameters, DoubleVector x )
      {
        return parameters[0] * x[1] * Math.Pow( x[0], 2.0 ) + parameters[1] * Math.Sin( x[0] ) + parameters[2] * Math.Pow( x[1], 3.0 );
      }

      /// <summary>
      /// Since the gradient of our function is rather easy to derive, we will
      /// override the GradientWithRespectToParams() function. Remember, this is
      /// the vector of partials with respect to the parameters, NOT the variables.
      /// </summary>
      /// <param name="parameters">Evaluate the gradient at these parameter values.</param>
      /// <param name="x">Evaluate the gradient at this point.</param>
      /// <param name="grad">Place the value of the gradient in this vector.</param>
      /// <remarks>Note how this function does not return the gradient as a new
      /// vector, but places the gradient value in a vector supplied by the 
      /// calling routine. This is for optimization purposes. The curve fitter uses 
      /// a optimization algorithm that will most likely be iterative, and thus may 
      /// need to evaluate the gradient many times. Having the vector 
      /// passed in to the routine allows the calling code to allocate space for the 
      /// gradient once and reuse it on successive calls, thus avoiding the potential 
      /// of allocating a large number of small objects on the managed heap.</remarks>
      public override void GradientWithRespectToParams( DoubleVector parameters, DoubleVector x, ref DoubleVector grad )
      {
        grad[0] = x[0] * x[0] * x[1];
        grad[1] = Math.Sin( x[0] );
        grad[2] = Math.Pow( x[1], 3 );
      }
    }

    /// <summary>
    /// A .NET example in C# showing how to fit a generalized multivariable function to a set 
    /// of points.
    /// </summary>
    /// <remarks>
    /// Uses the trust-region algorithm.
    /// </remarks>
    static void Main( string[] args )
    {
      // Class MultiVariableFunctionFitter fits a parameterized multivariable function to a
      // set of points. In the space of the function parameters, beginning at a specified
      // starting point, the Fit() method finds a minimum (possibly local) in the sum of
      // the squared residuals with respect to the data. Fit() uses a nonlinear least
      // squares minimizer specified as a generic.

      // For example, here is dataset from the Matlab docs, which fits a function
      // z = f(x, y) to three-dimensional data describing a surface  
      // http://www.mathworks.com/support/solutions/data/1-17YMU.html?solution=1-17YMU

      // Since the domain of the function has two dimensions, we use a two-column matrix to
      // hold the x,y data.
      var xyValues = new DoubleMatrix( 10, 2 );
      xyValues[Slice.All, 0] = new DoubleVector( "3.6 7.7 9.3 4.1 8.6 2.8 1.3 7.9 10.0 5.4" );
      xyValues[Slice.All, 1] = new DoubleVector( "16.5 150.6 263.1 24.7 208.5 9.9 2.7 163.9 325.0 54.3" );

      var zValues = new DoubleVector( "95.09 23.11 60.63 48.59 89.12 76.97 45.68 1.84 82.17 44.47" );

      // Published starting guess in the space of the function parameters.
      var start = new DoubleVector( "10 10 10" );

      // Construct a curve fitting object for our function, then perform the fit. We will use the
      // TrustRegionMinimizer implementation of the non-linear least squares minimizer to find the optimal
      // set of parameters. 
      var f = new ParameterizedFunction();
      var fitter = new MultiVariableFunctionFitter<TrustRegionMinimizer>( f );
      DoubleVector solution = fitter.Fit( xyValues, zValues, start );

      Console.WriteLine();

      // Display the results
      Console.WriteLine( "Fit #1" );
      Console.WriteLine( "Matlab solution: " + new DoubleVector( "0.0074 -19.9749 -0.0000" ) );
      Console.WriteLine( "NMath solution: " + solution );
      Console.WriteLine( "NMath residual: " + fitter.Minimizer.FinalResidual );
      Console.WriteLine();

      // The parameterized function used by the fitter may also be specified using a delegate.
      // here we define a delegate for the same function
      // p0*x1*x0^2 + p1*sin(x0) + p2*x1^3
      int xDimension = 2; // The dimension of the domain of f.
      Func<DoubleVector, DoubleVector, double> fdelegate = delegate( DoubleVector p, DoubleVector x )
      {
        return p[0] * x[1] * Math.Pow( x[0], 2.0 ) + p[1] * Math.Sin( x[0] ) + p[2] * Math.Pow( x[1], 3.0 );
      };

      // The delegate for the parameterized function may be used directly in MultiVariableFunctionFitter
      // constructors, or may be wrapped by the DoubleVectorParameterizedDelegate, which implements 
      // DoubleParameterizedFunctional. Here we do the latter.
      // Note that we do not supply the gradient with respect
      // to parameters here. The gradient will be computed using a finite difference algorithm if
      // needed.
      fitter.Function = new DoubleVectorParameterizedDelegate( fdelegate, xDimension );
      // Perform the fit and display the results
      solution = fitter.Fit( xyValues, zValues, start );
      Console.WriteLine( "Fit #1 (Repeated without user specified Partial Derivatives)" );
      Console.WriteLine( "NMath solution: " + solution );
      Console.WriteLine( "NMath residual: " + fitter.Minimizer.FinalResidual );
      Console.WriteLine();

      // Now let's perform the fit again using some random data. First we generate
      // 50 random x,y points in range (0,10).
      xyValues = new DoubleMatrix( 50, 2, new RandGenUniform( 0, 10 ) );

      // The target solution.
      var target = new DoubleVector( "1 2 3" );

      // When calculating the z values, we add some noise, so the points
      // don't lie exactly on the target surface.
      zValues = new DoubleVector( 50, new RandGenUniform( -1, 1 ));
      for ( int i = 0; i < zValues.Length; i++ )
      {
        zValues[i] += fdelegate( target, xyValues.Row( i ) );
      }

      // Perform the fit and display the results
      solution = fitter.Fit( xyValues, zValues, start );
      Console.WriteLine( "Fit #2" );
      Console.WriteLine( "Target solution: " + target );
      Console.WriteLine( "Actual solution: " + solution );
      Console.WriteLine( "Residual: " + fitter.Minimizer.FinalResidual );
      Console.WriteLine();

      Console.WriteLine();
      Console.WriteLine( "Press Enter Key" );
      Console.Read();

    }  // Main

  }  // class

} // namespace

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