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### 2.8 An older function for curve fitting

This was a popular function for curve fitting and has been enhanced to honour constraints. `nonlin_curvefit` ( see nonlin_curvefit) does now the same job if used with the default backend, and should be prefered due to its more powerful interface. The statistics returned by `leasqr` can also (and partially better) be computed with `curvefit_stat` ( see curvefit_stat). There are currently two things which still only `leasqr` does:

- internally providing a function for plotting fits during optimization,
- returning a pre-computed matrix for determining confidence regions.

Function File: leasqr (x, y, pin, F)
Function File: leasqr (x, y, pin, F, stol)
Function File: leasqr (x, y, pin, F, stol, niter)
Function File: leasqr (x, y, pin, F, stol, niter, wt)
Function File: leasqr (x, y, pin, F, stol, niter, wt, dp)
Function File: leasqr (x, y, pin, F, stol, niter, wt, dp, dFdp)
Function File: leasqr (x, y, pin, F, stol, niter, wt, dp, dFdp, options)
Function File: [f, p, cvg, iter, corp, covp, covr, stdresid, Z, r2] = leasqr (…)

  Levenberg-Marquardt nonlinear regression.

  Input arguments:

  x

    Vector or matrix of independent variables.

  y

    Vector or matrix of observed values.

  pin

    Vector of initial parameters to be adjusted by leasqr.

  F

    Name of function or function handle. The function must be of the form `y = f(x, p)`, with y, x, p of the form y, x, pin.

  stol

    Scalar tolerance on fractional improvement in scalar sum of squares, i.e., `sum ((wt .* (y-f))^2)`. Set to 0.0001 if empty or not given;

  niter

    Maximum number of iterations. Set to 20 if empty or not given.

  wt

    Statistical weights (same dimensions as y). These should be set to be proportional to `sqrt (y) ^-1`, i.e., the covariance matrix of the data is assumed to be proportional to diagonal with diagonal equal to `(wt.^2)^-1`. The constant of proportionality will be estimated. Set to `ones (size (y))` if empty or not given.

  dp

    Fractional increment of p for numerical partial derivatives. Set to `0.001 * ones (size (pin))` if empty or not given.

    - dp(j) > 0 means central differences on j-th parameter p(j).
    - dp(j) < 0 means one-sided differences on j-th parameter p(j).
    - dp(j) = 0 holds p(j) fixed, i.e., leasqr won’t change initial guess: pin(j)
  dFdp

    Name of partial derivative function in quotes or function handle. If not given or empty, set to `dfdp`, a slow but general partial derivatives function. The function must be of the form `prt = dfdp (x, f, p, dp, F [,bounds])`. For backwards compatibility, the function will only be called with an extra ’bounds’ argument if the ’bounds’ option is explicitly specified to leasqr (see dfdp.m).

  options

    Structure with multiple options. The following fields are recognized:

    `fract_prec`

      Column vector (same length as pin) of desired fractional precisions in parameter estimates. Iterations are terminated if change in parameter vector (chg) relative to current parameter estimate is less than their corresponding elements in ’fract_prec’, i.e., `all (abs (chg) < abs (options.fract_prec .* current_parm_est))` on two consecutive iterations. Defaults to `zeros (size (pin))`.

    `max_fract_change`

      Column vector (same length as pin) of maximum fractional step changes in parameter vector. Fractional change in elements of parameter vector is constrained to be at most ’max_fract_change’ between sucessive iterations, i.e., `abs (chg(i)) = abs (min([chg(i), options.max_fract_change(i) * current param estimate]))`. Defaults to `Inf * ones (size (pin))`.

    `inequc`

      Cell-array containing up to four entries, two entries for linear inequality constraints and/or one or two entries for general inequality constraints. Initial parameters must satisfy these constraints. Either linear or general constraints may be the first entries, but the two entries for linear constraints must be adjacent and, if two entries are given for general constraints, they also must be adjacent. The two entries for linear constraints are a matrix (say m) and a vector (say v), specifying linear inequality constraints of the form ‘m.’ * parameters + v >= 0’. If the constraints are just bounds, it is suggested to specify them in ’options.bounds’ instead, since then some sanity tests are performed, and since the function ’dfdp.m’ is guarantied not to violate constraints during determination of the numeric gradient only for those constraints specified as ’bounds’ (possibly with violations due to a certain inaccuracy, however, except if no constraints except bounds are specified). The first entry for general constraints must be a differentiable vector valued function (say h), specifying general inequality constraints of the form ‘h (p[, idx]) >= 0’; p is the column vector of optimized paraters and the optional argument idx is a logical index. h has to return the values of all constraints if idx is not given, and has to return only the indexed constraints if idx is given (so computation of the other constraints can be spared). If a second entry for general constraints is given, it must be a function (say dh) which returnes a matrix whos rows contain the gradients of the constraint function h with respect to the optimized parameters. It has the form jac_h = dh (vh, p, dp, h, idx[, bounds]); p is the column vector of optimized parameters, and idx is a logical index — only the rows indexed by idx must be returned (so computation of the others can be spared). The other arguments of dh are for the case that dh computes numerical gradients: vh is the column vector of the current values of the constraint function h, with idx already applied. h is a function h (p) to compute the values of the constraints for parameters p, it will return only the values indexed by idx. dp is a suggestion for relative step width, having the same value as the argument ’dp’ of leasqr above. If bounds were specified to leasqr, they are provided in the argument bounds of dh, to enable their consideration in determination of numerical gradients. If dh is not specified to leasqr, numerical gradients are computed in the same way as with ’dfdp.m’ (see above). If some constraints are linear, they should be specified as linear constraints (or bounds, if applicable) for reasons of performance, even if general constraints are also specified.

    `bounds`

      Two-column-matrix, one row for each parameter in pin. Each row contains a minimal and maximal value for each parameter. Default: [-Inf, Inf] in each row. If this field is used with an existing user-side function for ’dFdp’ (see above) the functions interface might have to be changed.

    `equc`

      Equality constraints, specified the same way as inequality constraints (see field ’options.inequc’). Initial parameters must satisfy these constraints. Note that there is possibly a certain inaccuracy in honoring constraints, except if only bounds are specified. Warning: If constraints (or bounds) are set, returned guesses of corp, covp, and Z are generally invalid, even if no constraints are active for the final parameters. If equality constraints are specified, corp, covp, and Z are not guessed at all.

    `cpiv`

      Function for complementary pivot algorithm for inequality constraints. Defaults to cpiv_bard. No different function is supplied.

    For backwards compatibility, options can also be a matrix whose first and second column contains the values of `fract_prec` and `max_fract_change`, respectively.

  Output:

  f

    Column vector of values computed: f = F(x,p).

  p

    Column vector trial or final parameters, i.e, the solution.

  cvg

    Scalar: = 1 if convergence, = 0 otherwise.

  iter

    Scalar number of iterations used.

  corp

    Correlation matrix for parameters.

  covp

    Covariance matrix of the parameters.

  covr

    Diag(covariance matrix of the residuals).

  stdresid

    Standardized residuals.

  Z

    Matrix that defines confidence region (see comments in the source).

  r2

    Coefficient of multiple determination, intercept form.

  Not suitable for non-real residuals.

  References: Bard, Nonlinear Parameter Estimation, Academic Press, 1974. Draper and Smith, Applied Regression Analysis, John Wiley and Sons, 1981.

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