nonlin_05.cc

这是一个从音频信号里提取特征参量的程序

// file: $isip/class/numeric/NonlinearOptimization/nonlin_05.cc
// version: $Id: nonlin_05.cc,v 1.6 2001/11/13 15:30:15 gao Exp $
//

// isip include files
//
#include "NonlinearOptimization.h"

// method: levMarqTemplate
//
// arguments:
//  TVector& params: (input/output) parameter vector to optimize
//  TIntegral& chi_square: (output) converged chi-square estimate
//  const TVector& x: (input) measured x data
//  const TVector& y: (input) measured y data
//  const TVector& stddev: (input) the standard deviation of the error in
//                         measuring y(i)
//  boolean (*opt_func)(TIntegral&,
//                      TVector&,
//                      const TIntegral,
//                      const TVector&): (input) function whose parameters we
//                                       are optimizing
//  TIntegral convergence: (input) iterations end when the chi-square
//                         estimate decreases by less than convergence
//                         percent
//
// return: boolean value indicating status
//
// this method implements the Levenberg-Marquardt method of
// least-squares optimization of nonlinear functions. we follow closely
// the derivation provided in:
// 
//  W. Press, S. Teukolsky, W. Vetterling, B. Flannery,
//  Numerical Recipes in C, Second Edition, Cambridge University Press,
//  New York, New York, USA, pp. 681-685, 1995.
//
template <class TMatrix, class TVector, class TIntegral>
boolean NonlinearOptimization::levMarqTemplate(TVector& params_a,
					       TIntegral& chi_square_a,
					       const TVector& x_a,
					       const TVector& y_a,
					       const TVector& stddev_a,
			     boolean (*opt_func_a)(TIntegral&,
						   TVector&,
						   const TIntegral,
						   const TVector&),
					       TIntegral convergence_a) {
  
  // verify the internal parameters
  //
  long params_length = params_a.length();
  if (params_length <= 0) { 
    return Error::handle(name(), L"levMarqTemplate", ERR_LEV_MARQ,
			 __FILE__, __LINE__);
  }
  
  // verify the function pointer
  //
  if (opt_func_a == NULL) {
    return Error::handle(name(), L"levMarqTemplate", ERR_LEV_MARQ,
			 __FILE__, __LINE__);
  }
  
  // check vector lengths
  //
  long length = x_a.length();
  if ((y_a.length() != length) || (stddev_a.length() != length)) {
    return Error::handle(name(), L"levMarqTemplate", ERR_LEV_MARQ,
			 __FILE__, __LINE__);
  }
  
  // if no data to be optimized, just return 
  //
  if (length == 0) {
    return true;
  }
  
  // setup the internal data sizes
  //
  TMatrix alpha(params_length, params_length, Integral::SYMMETRIC);
  
  TVector beta(params_length);
  
  // set the initial lambda parameter:
  //  lambda is the deviation parameter that tells the algorithm how
  //  much to adjust the parameters on each iteration. As the Chi
  //  square errors decrease, the lambda value is decreased so the
  //  gradient descent does not jump around the minimum. If the Chi
  //  square error is very large, the lambda value is increased so
  //  that the estimates take a large jump on the next iteration.
  //
  TIntegral lambda = LEVMARQ_INIT_LAMBDA;
  
  // precompute the inverse variance
  //
  TVector inv_variance(length);
  inv_variance.square(stddev_a);
  inv_variance.inverse();
  
  // store the current parameter set
  //
  TVector tmp_params(params_a);
  
  // run the initial iteration to get initial chi-square measurements
  // step 1 on pp. 684
  //
  if (!levMarqChiSquare(chi_square_a, alpha, beta, x_a, y_a,
			inv_variance, tmp_params, opt_func_a)) {
    return Error::handle(name(), L"levMarqTemplate", ERR_LEV_MARQ,
			 __FILE__, __LINE__);
  }

  // copies of alpha and beta
  //
  TMatrix tmp_alpha(alpha);
  TVector tmp_beta(beta);    

  // loop until convergence
  //
  TIntegral new_chi_square = chi_square_a;
  boolean converged = false;
  boolean is_plateau = false;

  while (!converged) {
    
    // modify the alpha matrix (eq. 15.5.13)
    //
    scaleDiagonal(alpha, (1 + lambda));
    
    // solve the set of linear equations (eq. 15.5.14) for a candidate
    // shift in the parameters. after this operation, tmp_params
    // contains the candidate shift. step 3 on pp. 684
    //
    LinearAlgebra::linearSolve(tmp_params, alpha, beta);
    
    // test the candidate shift
    //
    tmp_params.add(params_a);
    if (!levMarqChiSquare(new_chi_square, alpha, beta, x_a, y_a,
			  inv_variance, tmp_params, opt_func_a)) {
      return Error::handle(name(), L"levMarqTemplate", ERR_LEV_MARQ,
			   __FILE__, __LINE__);
    }

    // see if the chi-square measurement decreased. steps 4 and 5 pp. 684
    //
    if (new_chi_square > chi_square_a) {
      
      // reset the parameters
      //
      alpha.assign(tmp_alpha);
      beta.assign(tmp_beta);
      tmp_params.assign(params_a);
      
      // increase lambda to cause a steeper descent and reset the
      // test parameters
      //
      lambda *= 10.0;
    }
    else {
      
      // accept the current solution and update the convergence values
      //	
      lambda *= 0.1;
      params_a.assign(tmp_params);
      
      // set the new alpha and beta
      //
      tmp_alpha.assign(alpha);
      tmp_beta.assign(beta);	

      // check for equality. if we are exactly equal then we'll give another
      // chance to converge.
      //
      if (chi_square_a == new_chi_square) {
	if (is_plateau) {
	  converged = true;
	}
	is_plateau = true;
      }

      // check the percent difference for convergence
      //
      else {
	is_plateau = false;
	if (Integral::almostEqual(chi_square_a, new_chi_square,
				  convergence_a)) {
	  converged = true;
	}
      }
      
      // update the chi-square estimate
      //
      chi_square_a = new_chi_square;
    }
  }
  return true;
}

// explicit instantiations
//
template boolean
NonlinearOptimization::levMarqTemplate<MatrixFloat,
  VectorFloat, float>(VectorFloat&, float& chi_square_a,
		      const VectorFloat& x_a, const VectorFloat& y_a,
		      const VectorFloat& stddev_a,
		      boolean (*opt_func_a)(float&, VectorFloat&,
					    const float,
					    const VectorFloat&),
		      float convergence_a);

template boolean
NonlinearOptimization::levMarqTemplate<MatrixDouble,
  VectorDouble, double>(VectorDouble&, double& chi_square_a,
		      const VectorDouble& x_a, const VectorDouble& y_a,
		      const VectorDouble& stddev_a,
		      boolean (*opt_func_a)(double&, VectorDouble&,
					    const double,
					    const VectorDouble&),
		      double convergence_a);


// method: levMarqChiSquare
//
// arguments:
//  TIntegral& chi_square: (output) the error estimate of the input data
//                         given the model parameters defined by:
//       chi_square   =      { y(i) - y_est(x(i); params) } ^ 2
//                       sum ------------------------------------
//                        i               stddev(i)^2
//  TMatrix& alpha: (output) is a modification of the Hessian (2nd
//                  derivative) matrix with respect to the parameters
//                  being optimized as shown on page 684 of the reference.
//                  alpha(k,l) is the 2nd derivative of the Chi square
//                  merit function w.r.t (params(k) * params(l))
//  TVector& beta: (output) gradient of the Chi square merit function w.r.t
//                          params(k). As the estimation converges,
//                          beta(k) should approach 0.0
//  const TVector& x: (input) measured x data
//  const TVector& y: (input) measured y data
//  const TVector& inv_variance: (input) 1 / (stddev^2)
//  const TVector& params: (input) test parameters
//  boolean (*opt_func)(TIntegral&,
//                      TVector&,
//                      const TIntegral,
//                      const TVector&): (input) function whose parameters we
//                                       are optimizing
//
// return: boolean value indicating status
//
// this method measures the chi-square merit of a set of parameters as a
// parametric fit of the input (x,y) data to a functional form (opt_func)
// according to the derivation provided in:
//
//  W. Press, S. Teukolsky, W. Vetterling, B. Flannery,
//  Numerical Recipes in C, Second Edition, Cambridge University Press,
//  New York, New York, USA, pp. 681-685, 1995.
//
template <class TMatrix, class TVector, class TIntegral>
boolean NonlinearOptimization::levMarqChiSquare(TIntegral& chi_square_a,
						TMatrix& alpha_a,
						TVector& beta_a,
						const TVector& x_a,
						const TVector& y_a,
						const TVector& inv_variance_a,
						const TVector& params_a,
					boolean (*opt_func_a)(TIntegral&,
							      TVector&,
							      const TIntegral,
							      const TVector&)
						) {
  
  // check the internal data
  //
  if (opt_func_a == NULL) {
    return Error::handle(name(), L"levMarqChiSquare", ERR_LEV_MARQ,
			 __FILE__, __LINE__);
  }
  
  // declare local variables
  //
  long num_params = params_a.length();
  long num_points = x_a.length();
  TIntegral y_tmp = 0;
  TVector y_diff(num_points);
  TVector derivatives(num_params);
  
  // initialize outputs
  //
  chi_square_a = 0;
  
  alpha_a.changeType(Integral::SYMMETRIC);
  alpha_a.setDimensions(num_params, num_params);
  alpha_a.assign(0.0);
  
  beta_a.setLength(num_params);
  beta_a.assign(0.0);
  
  // loop over all data points and accumulate the chi-squared statistic
  // a side effect of this is updating the alpha and beta parameters
  //
  for (long i = 0; i < num_points; i++) {
    
    // call the user-supplied function and compute the sample error
    //
    (*opt_func_a)(y_tmp, derivatives, (TIntegral)x_a(i), params_a);
    y_diff(i) = y_a(i) - y_tmp;
    
    // update the alpha values (eq. 15.5.11) and beta values (eq. 15.5.6 and
    // eq. 15.5.8). note that alpha_d is symmetric so only alpha_d(k,l),
    // k <= l, need be computed
    //
    for (long k = 0; k < num_params; k++) {
      
      beta_a(k) += y_diff(i) * derivatives(k) * inv_variance_a(i);
      
      for (long l = 0; l <= k; l++) {
	alpha_a.setValue(k, l, alpha_a.getValue(k, l) +
			 (derivatives(k) * derivatives(l) *
			  inv_variance_a(i)));
      }
    }
  }
  
  // update the chi square estimate
  //
  y_diff.square();
  y_diff.mult(inv_variance_a);
  chi_square_a = y_diff.sum();
  
  // exit gracefully
  //
  return true;
}

// explicit instantiations
//
template 
boolean NonlinearOptimization::levMarqChiSquare<MatrixFloat,
  VectorFloat, float>(float& chi_square_a, MatrixFloat& alpha_a,
		      VectorFloat& beta_a, const VectorFloat& x_a,
		      const VectorFloat& y_a,
		      const VectorFloat& inv_variance_a,
		      const VectorFloat& params_a,
		      boolean (*)(float&, VectorFloat&, const float,
		      const VectorFloat&));
  

  template 
boolean NonlinearOptimization::levMarqChiSquare<MatrixDouble,
  VectorDouble, double>(double& chi_square_a,
			MatrixDouble& alpha_a,
			VectorDouble& beta_a,
			const VectorDouble& x_a,
			const VectorDouble& y_a,
			const VectorDouble& inv_variance_a,
			const VectorDouble& params_a,
			boolean (*opt_func_a)(double&,
					      VectorDouble&,
					      const double,
					      const VectorDouble&));

// method: scaleDiagonal
//
// arguments:
//  TMatrix& mat: (input/output) matrix whose diagonal we want to scale
//  const TIntegral& scale: (input) value to scale the diagonal by
//
// return: boolean value indicating status
//
// this method scales each element of the matrix diagonal by the
// scale value
//
template <class TMatrix, class TIntegral>
boolean NonlinearOptimization::scaleDiagonal(TMatrix& mat_a,
					     const TIntegral& scale_a) {
  
  // multiply each element of the diagonal of the matrix by
  // the scale
  //
  long dim = mat_a.getNumRows();
  for (long i = 0; i < dim; i++) {
    mat_a.setValue(i, i, (mat_a(i, i) * scale_a));
  }
  
  // exit gracefully
  //
  return true;
}

// explicit instantiations
//
template boolean
NonlinearOptimization::scaleDiagonal<MatrixFloat, float>(MatrixFloat&,
							 const float&);
template boolean
NonlinearOptimization::scaleDiagonal<MatrixDouble, double>(MatrixDouble&,
							 const double&);