nmrglue.leastsqbound
Constrained multivariate least-squares optimization
This module is imported as nmrglue.leastsqbound and can be called as such.
Developer Functions
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nmrglue.analysis.leastsqbound.leastsqbound(func, x0, args=(), bounds=None, Dfun=None, full_output=0, col_deriv=0, ftol=1.49012e-08, xtol=1.49012e-08, gtol=0.0, maxfev=0, epsfcn=0.0, factor=100, diag=None)[source]
Bounded minimization of the sum of squares of a set of equations.
x = arg min(sum(func(y)**2,axis=0))
y
Parameters : | func : callable
should take at least one (possibly length N vector) argument and
returns M floating point numbers.
x0 : ndarray
The starting estimate for the minimization.
args : tuple
Any extra arguments to func are placed in this tuple.
bounds : list
(min, max) pairs for each element in x, defining
the bounds on that parameter. Use None for one of min or
max when there is no bound in that direction.
Dfun : callable
A function or method to compute the Jacobian of func with derivatives
across the rows. If this is None, the Jacobian will be estimated.
full_output : bool
non-zero to return all optional outputs.
col_deriv : bool
non-zero to specify that the Jacobian function computes derivatives
down the columns (faster, because there is no transpose operation).
ftol : float
Relative error desired in the sum of squares.
xtol : float
Relative error desired in the approximate solution.
gtol : float
Orthogonality desired between the function vector and the columns of
the Jacobian.
maxfev : int
The maximum number of calls to the function. If zero, then 100*(N+1) is
the maximum where N is the number of elements in x0.
epsfcn : float
A suitable step length for the forward-difference approximation of the
Jacobian (for Dfun=None). If epsfcn is less than the machine precision,
it is assumed that the relative errors in the functions are of the
order of the machine precision.
factor : float
A parameter determining the initial step bound
(factor * || diag * x||). Should be in interval (0.1, 100).
diag : sequence
N positive entries that serve as a scale factors for the variables.
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Returns : | x : ndarray
The solution (or the result of the last iteration for an unsuccessful
call).
cov_x : ndarray
Uses the fjac and ipvt optional outputs to construct an
estimate of the jacobian around the solution. None if a
singular matrix encountered (indicates very flat curvature in
some direction). This matrix must be multiplied by the
residual standard deviation to get the covariance of the
parameter estimates – see curve_fit.
infodict : dict
a dictionary of optional outputs with the key s:
- 'nfev' : the number of function calls
- 'fvec' : the function evaluated at the output
- 'fjac' : A permutation of the R matrix of a QR
factorization of the final approximate
Jacobian matrix, stored column wise.
Together with ipvt, the covariance of the
estimate can be approximated.
- 'ipvt' : an integer array of length N which defines
a permutation matrix, p, such that
fjac*p = q*r, where r is upper triangular
with diagonal elements of nonincreasing
magnitude. Column j of p is column ipvt(j)
of the identity matrix.
- 'qtf' : the vector (transpose(q) * fvec).
mesg : str
A string message giving information about the cause of failure.
ier : int
An integer flag. If it is equal to 1, 2, 3 or 4, the solution was
found. Otherwise, the solution was not found. In either case, the
optional output variable ‘mesg’ gives more information.
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Notes
“leastsq” is a wrapper around MINPACK’s lmdif and lmder algorithms.
cov_x is a Jacobian approximation to the Hessian of the least squares
objective function.
This approximation assumes that the objective function is based on the
difference between some observed target data (ydata) and a (non-linear)
function of the parameters f(xdata, params)
func(params) = ydata - f(xdata, params)
so that the objective function is
min sum((ydata - f(xdata, params))**2, axis=0)
params
Contraints on the parameters are enforced using an internal parameter list
with appropiate transformations such that these internal parameters can be
optimized without constraints. The transfomation between a given internal
parameter, p_i, and a external parameter, p_e, are as follows:
With min and max bounds defined
p_i = arcsin((2 * (p_e - min) / (max - min)) - 1.)
p_e = min + ((max - min) / 2.) * (sin(p_i) + 1.)
With only max defined
p_i = sqrt((max - p_e + 1.)**2 - 1.)
p_e = max + 1. - sqrt(p_i**2 + 1.)
With only min defined
p_i = sqrt((p_e - min + 1.)**2 - 1.)
p_e = min - 1. + sqrt(p_i**2 + 1.)
These transfomations are used in the MINUIT package, and described in
detail in the section 1.3.1 of the MINUIT User’s Guide.
References
- James and M. Winkler. MINUIT User’s Guide, July 16, 2004.