C/C++ Interactive Guide/c-cplusplus-interactive-guide.iso/c_ref/csource4/209_01/simplexr.doc

Text File | 1990-03-05 | 9.8 KB | 335 lines

SIMPLEXR.DOC       VERS:- 01.00  DATE:- 09/26/86  TIME:- 10:02:52 PM

notes on the simplex method of function minimization,
    by use of the Nelder-Mead algorithm.







        By J. A. Rupley, Tucson, Arizona


         NOTES ON THE ALGORITHM OF THE SIMPLEX FITTING PROGRAM


        A. THE SIMPLEX METHOD OF FUNCTION MINIMIZATION


             The fitting of a model with variable parameters to a set of
        data points is handled by minimizing an error function, such as
        the sum of squares of the differences between observed values and
        values calculated according to the model.

             We describe here a simplex approximation procedure for
        estimating the parameter values that give a function minimum.
        The strategy is that of Nelder and Mead, and their article
        (Computer Journal, vol 7, p 308, 1965) should be consulted for
        more information than is given below.

             A simplex is constructed in parameter space, with vertices
        that are arbitrary but not too far from the point at which the
        function is a minimum and that describe a volume that includes
        the minimum point.

             The position of each vertex in parameter space defines (or
        equally, is defined by) a set of parameter values, for which one
        calculates the function value for that vertex.  The simplex has
        (M + 1) vertices, where M is the number of free parameters.

             In successive iterations the vertex with the highest
        function value is moved, to obtain a new vertex position of
        smaller function value.  The movement is directed with respect to
        the center of the reduced simplex, which is the simplex less the
        highest vertex.  The movements can be the following:
             reflection;
             expansion beyond the reflected position;
             reflection after failed expansion;
             contraction toward the center of the reduced simplex from
                  the original position;
             contraction from the reflected position;
             if none of these operations gives a lower function value
                  then the entire simplex is shrunk about the lowest
                  vertex.

             Exit from the iterative minimization is if the fractional
        RMS deviation of the function values at the vertices is less than
        a test value and if the centroid of the simplex is within two RMS
        deviations of the mean of the vertices.  The default test value

        is 1E-8 * the mean function value.

             Thus at exit from the minimization, the centroid of the
        simplex gives an arbitrarily close approximation of the parameter
        values at the function minimum.


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             The routine that calculates the function values must be
        adapted to the model to be optimized.

             Entry of the data and starting parameter values can be at
        compilation, by initializing the data and parameter arrays, or by
        reading a text file.

             Results should be written to a file, for convenience of
        review and for recovery from crashes.















































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        B. THE QUADRATIC FIT TO THE LEAST SQUARES FUNCTION SURFACE


             Standard deviations of the parameters are calculated by
        fitting a quadratic function to the surface about the minimum in
        parameter space and then using the properties of this function to
        calculate the variance-covariance matrix for the parameters.  The
        strategy is a modification of that of Nelder and Mead (1965).

             The contracted simplex obtained through the previous
        minimization process is used to define a new coordinate system of
        M oblique axes in parameter space.  The origin of the axes is at
        the centroid of the simplex.  Each axis corresponds to a free
        parameter.  The unit value (scale) for each axis is set in the
        new system at the average of the deviations of the vertices of
        the simplex from the centroid value.

             In effect, one constructs in the new coordinate system a new
        simplex, based on the one obtained by the previous minimization
        process.  The (M + 1) vertices of the new construct are at the
        centroid (the zero of the new system) and at the unit points on
        each free parameter axis.  As before, the values of the least
        squares function at these vertices describe a surface.  It is
        this least squares surface to which a quadratic function is to be
        fit.


             If necessary, the scales of the axes (the unit values) in
        the new coordinate system can be adjusted to give better behavior
        of the surface in the vicinity of the minimum.

             The diagonal matrix Q transforms the new coordinate system
        back to the original system.  The elements of Q are for each
        parameter the (adjusted) average differences between the vertices
        and the centroid of the original simplex.

             The function value, y, in the region near the function
        minimum generally can be approximated in the new coordinate
        system by the quadratic vector function:

              y  =  a(o)   +   2 a'.x   +   x'.B.x                   (1)

             The elements of the vector x are the values of the M free
        parameters at a point in parameter space.  The scalar a(o), the
        vector a and the matrix B are determined by the shape of the
        surface y near the minimum and can be calculated with the use of
        simple numerical approximations, evaluated at x = 0 (the position
        of the centroid of the original simplex):








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                a(0)   =   y(0)

                ai     =   0.5 * (dy/dxi)
                       =   0.25 * (yi - y-i)

                bij    =   0.5 * (d2y/dxi.dxj)
                       =   0.25 * (yij + y-i-j + 2 * y(0)
                                           - y-i - y-j - yi - yj)

                bii    =   0.5 * (d2y/dxi2)
                       =   0.5 * (yi + y-i - 2 * y(0))

             The point x = 0 (the centroid position) may not be the true
        position of the function minimum, although it is assumed to be
        close to it.  A refined estimate of the minimum position, based
        on the quadratic approximation of equation (1) above, is given by

        x(min):

             xmin = - B-1.a


             The function minimum at x(min) is given by y(min) :

             ymin = a(0) - a'.B-1.a


             The position of the minimum in the original coordinate
        system is obtained by back transformation of x(min) with the
        matrix Q, giving the point p(min):

             pmin = p(0) - Q'.B-1.a

             As a test of the quality of the quadratic approximation (1),
        there should be close agreement between the function values
        y(min), y(p(min)), and y(centroid), and between the sets of
        parameter values defining the points p(min) and the centroid.

             The variance-covariance matrix C is given by

             C = Q'.B-1.Q

             A diagonal element of C, multiplied by the mean square error
        (MSE), gives the variance of the corresponding parameter:

             var(i) = cii * MSE

             MSE = ymin/(n - m)

        where the divisor (n - m) = # degrees freedom: n = # of data
        points used in calcn of ymin, and m = # of free parameters; and
        ymin = the sum of residuals squared





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             The standard deviation is the square root of the variance:

             std-dev(i) = sqrt(var(i))

             The quadratic fit can fail, giving negative values of the

        diagonal elements of the variance-covariance matrix or a value of
        y(p(min)) > y(centroid).  Failure can come from too tight a
        simplex, resulting in too small values of the B matrix elements,
        or from too large a simplex, resulting in non-quadratic behavior
        of the least squares function within the region defined by the
        simplex.  The problem of too tight a simplex does not arise if
        the precision of the floating point routines of the program is
        sufficiently high (e.g., 14 digits).  The problem of non-quadratic
        behavior commonly arises in the early stages of the fitting, when
        the minimum has not yet been approached closely.  This can be
        taken into account by modifying the fitting algorithm, or one can
        be patient.  If a parameter moves so close to a bound that
        expansion of the simplex toward the bound is not possible, then
        that parameter should be fixed.

             It may be useful to cycle between short sessions of simplex
        fitting and quadratic approximation, to more rapidly approach the
        lsq function minimum.  Toward this end, the presumed more
        accurate approximation of the function minimum given by p(min) is
        used to replace the previously obtained centroid value of the
        simplex.  That is, the simplex returned to the simplex fitting
        routine would have M vertices at the "unit" positions on the
        parameter axes and the (M+1)th vertex at p(min), not at the
        "origin", ie the old centroid value.



























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