scipy.optimize.leastsq¶

scipy.optimize.leastsq(func, x0, args=(), Dfun=None, full_output=0, col_deriv=0, ftol=1.49012e-08, xtol=1.49012e-08, gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None)[source]¶

Minimize the sum of squares of a set of equations.

x = arg min(sum(func(y)**2,axis=0))
         y

Parameters:

Parameters:	func : callable should take at least one (possibly length N vector) argument and returns M floating point numbers. It must not return NaNs or fitting might fail. x0 : ndarray The starting estimate for the minimization. args : tuple, optional Any extra arguments to func are placed in this tuple. Dfun : callable, optional 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, optional non-zero to return all optional outputs. col_deriv : bool, optional non-zero to specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation). ftol : float, optional Relative error desired in the sum of squares. xtol : float, optional Relative error desired in the approximate solution. gtol : float, optional Orthogonality desired between the function vector and the columns of the Jacobian. maxfev : int, optional The maximum number of calls to the function. If Dfun is provided then the default maxfev is 100(N+1) where N is the number of elements in x0, otherwise the default maxfev* is 200(N+1). epsfcn* : float, optional A variable used in determining a suitable step length for the forward- difference approximation of the Jacobian (for Dfun=None). Normally the actual step length will be sqrt(epsfcn)x If epsfcn is less than the machine precision, it is assumed that the relative errors are of the order of the machine precision. factor* : float, optional A parameter determining the initial step bound (`factor * \|\| diag * x\|\|`). Should be in interval `(0.1, 100)`. diag : sequence, optional N positive entries that serve as a scale factors for the variables.
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 variance 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 fjacp = qr, 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.

func : callable

should take at least one (possibly length N vector) argument and returns M floating point numbers. It must not return NaNs or fitting might fail.

x0 : ndarray

The starting estimate for the minimization.

args : tuple, optional

Any extra arguments to func are placed in this tuple.

Dfun : callable, optional

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, optional

non-zero to return all optional outputs.

col_deriv : bool, optional

non-zero to specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation).

ftol : float, optional

Relative error desired in the sum of squares.

xtol : float, optional

Relative error desired in the approximate solution.

gtol : float, optional

Orthogonality desired between the function vector and the columns of the Jacobian.

maxfev : int, optional

The maximum number of calls to the function. If Dfun is provided then the default maxfev is 100*(N+1) where N is the number of elements in x0, otherwise the default maxfev is 200*(N+1).

epsfcn : float, optional

A variable used in determining a suitable step length for the forward- difference approximation of the Jacobian (for Dfun=None). Normally the actual step length will be sqrt(epsfcn)*x If epsfcn is less than the machine precision, it is assumed that the relative errors are of the order of the machine precision.

factor : float, optional

A parameter determining the initial step bound (factor * || diag * x||). Should be in interval (0.1, 100).

diag : sequence, optional

N positive entries that serve as a scale factors for the variables.

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 variance 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.

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