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def least_squares(fun, x0, jac='2-point', bounds=(-inf, inf), method='trf', ftol=1e-08, xtol=1e-08, gtol=1e-08, x_scale=1.0, loss='linear', f_scale=1.0, diff_step=None, tr_solver=None, tr_options=None, jac_sparsity=None, max_nfev=None, verbose=0, args=(), kwargs=None)
Solve a nonlinear least-squares problem with bounds on the variables.
Given the residuals f(x) (an m-D real function of n real
variables) and the loss function rho(s) (a scalar function), `least_squares`
finds a local minimum of the cost function F(x)::
minimize F(x) = 0.5 * sum(rho(f_i(x)**2), i = 0, ..., m - 1)
subject to lb <= x <= ub
The purpose of the loss function rho(s) is to reduce the influence of
outliers on the solution.
Parameters
----------
fun : callable
Function which computes the vector of residuals, with the signature
``fun(x, *args, **kwargs)``, i.e., the minimization proceeds with
respect to its first argument. The argument ``x`` passed to this
function is an ndarray of shape (n,) (never a scalar, even for n=1).
It must allocate and return a 1-D array_like of shape (m,) or a scalar.
If the argument ``x`` is complex or the function ``fun`` returns
complex residuals, it must be wrapped in a real function of real
arguments, as shown at the end of the Examples section.
x0 : array_like with shape (n,) or float
Initial guess on independent variables. If float, it will be treated
as a 1-D array with one element. When `method` is 'trf', the initial
guess might be slightly adjusted to lie sufficiently within the given
`bounds`.
jac : {'2-point', '3-point', 'cs', callable}, optional
Method of computing the Jacobian matrix (an m-by-n matrix, where
element (i, j) is the partial derivative of f[i] with respect to
x[j]). The keywords select a finite difference scheme for numerical
estimation. The scheme '3-point' is more accurate, but requires
twice as many operations as '2-point' (default). The scheme 'cs'
uses complex steps, and while potentially the most accurate, it is
applicable only when `fun` correctly handles complex inputs and
can be analytically continued to the complex plane. Method 'lm'
always uses the '2-point' scheme. If callable, it is used as
``jac(x, *args, **kwargs)`` and should return a good approximation
(or the exact value) for the Jacobian as an array_like (np.atleast_2d
is applied), a sparse matrix (csr_matrix preferred for performance) or
a `scipy.sparse.linalg.LinearOperator`.
bounds : 2-tuple of array_like or `Bounds`, optional
There are two ways to specify bounds:
1. Instance of `Bounds` class
2. Lower and upper bounds on independent variables. Defaults to no
bounds. Each array must match the size of `x0` or be a scalar,
in the latter case a bound will be the same for all variables.
Use ``np.inf`` with an appropriate sign to disable bounds on all
or some variables.
method : {'trf', 'dogbox', 'lm'}, optional
Algorithm to perform minimization.
* 'trf' : Trust Region Reflective algorithm, particularly suitable
for large sparse problems with bounds. Generally robust method.
* 'dogbox' : dogleg algorithm with rectangular trust regions,
typical use case is small problems with bounds. Not recommended
for problems with rank-deficient Jacobian.
* 'lm' : Levenberg-Marquardt algorithm as implemented in MINPACK.
Doesn't handle bounds and sparse Jacobians. Usually the most
efficient method for small unconstrained problems.
Default is 'trf'. See Notes for more information.
ftol : float or None, optional
Tolerance for termination by the change of the cost function. Default
is 1e-8. The optimization process is stopped when ``dF < ftol * F``,
and there was an adequate agreement between a local quadratic model and
the true model in the last step.
If None and 'method' is not 'lm', the termination by this condition is
disabled. If 'method' is 'lm', this tolerance must be higher than
machine epsilon.
xtol : float or None, optional
Tolerance for termination by the change of the independent variables.
Default is 1e-8. The exact condition depends on the `method` used:
* For 'trf' and 'dogbox' : ``norm(dx) < xtol * (xtol + norm(x))``.
* For 'lm' : ``Delta < xtol * norm(xs)``, where ``Delta`` is
a trust-region radius and ``xs`` is the value of ``x``
scaled according to `x_scale` parameter (see below).
If None and 'method' is not 'lm', the termination by this condition is
disabled. If 'method' is 'lm', this tolerance must be higher than
machine epsilon.
gtol : float or None, optional
Tolerance for termination by the norm of the gradient. Default is 1e-8.
The exact condition depends on a `method` used:
* For 'trf' : ``norm(g_scaled, ord=np.inf) < gtol``, where
``g_scaled`` is the value of the gradient scaled to account for
the presence of the bounds [STIR]_.
* For 'dogbox' : ``norm(g_free, ord=np.inf) < gtol``, where
``g_free`` is the gradient with respect to the variables which
are not in the optimal state on the boundary.
* For 'lm' : the maximum absolute value of the cosine of angles
between columns of the Jacobian and the residual vector is less
than `gtol`, or the residual vector is zero.
If None and 'method' is not 'lm', the termination by this condition is
disabled. If 'method' is 'lm', this tolerance must be higher than
machine epsilon.
x_scale : array_like or 'jac', optional
Characteristic scale of each variable. Setting `x_scale` is equivalent
to reformulating the problem in scaled variables ``xs = x / x_scale``.
An alternative view is that the size of a trust region along jth
dimension is proportional to ``x_scale[j]``. Improved convergence may
be achieved by setting `x_scale` such that a step of a given size
along any of the scaled variables has a similar effect on the cost
function. If set to 'jac', the scale is iteratively updated using the
inverse norms of the columns of the Jacobian matrix (as described in
[JJMore]_).
loss : str or callable, optional
Determines the loss function. The following keyword values are allowed:
* 'linear' (default) : ``rho(z) = z``. Gives a standard
least-squares problem.
* 'soft_l1' : ``rho(z) = 2 * ((1 + z)**0.5 - 1)``. The smooth
approximation of l1 (absolute value) loss. Usually a good
choice for robust least squares.
* 'huber' : ``rho(z) = z if z <= 1 else 2*z**0.5 - 1``. Works
similarly to 'soft_l1'.
* 'cauchy' : ``rho(z) = ln(1 + z)``. Severely weakens outliers
influence, but may cause difficulties in optimization process.
* 'arctan' : ``rho(z) = arctan(z)``. Limits a maximum loss on
a single residual, has properties similar to 'cauchy'.
If callable, it must take a 1-D ndarray ``z=f**2`` and return an
array_like with shape (3, m) where row 0 contains function values,
row 1 contains first derivatives and row 2 contains second
derivatives. Method 'lm' supports only 'linear' loss.
f_scale : float, optional
Value of soft margin between inlier and outlier residuals, default
is 1.0. The loss function is evaluated as follows
``rho_(f**2) = C**2 * rho(f**2 / C**2)``, where ``C`` is `f_scale`,
and ``rho`` is determined by `loss` parameter. This parameter has
no effect with ``loss='linear'``, but for other `loss` values it is
of crucial importance.
max_nfev : None or int, optional
Maximum number of function evaluations before the termination.
If None (default), the value is chosen automatically:
* For 'trf' and 'dogbox' : 100 * n.
* For 'lm' : 100 * n if `jac` is callable and 100 * n * (n + 1)
otherwise (because 'lm' counts function calls in Jacobian
estimation).
diff_step : None or array_like, optional
Determines the relative step size for the finite difference
approximation of the Jacobian. The actual step is computed as
``x * diff_step``. If None (default), then `diff_step` is taken to be
a conventional "optimal" power of machine epsilon for the finite
difference scheme used [NR]_.
tr_solver : {None, 'exact', 'lsmr'}, optional
Method for solving trust-region subproblems, relevant only for 'trf'
and 'dogbox' methods.
* 'exact' is suitable for not very large problems with dense
Jacobian matrices. The computational complexity per iteration is
comparable to a singular value decomposition of the Jacobian
matrix.
* 'lsmr' is suitable for problems with sparse and large Jacobian
matrices. It uses the iterative procedure
`scipy.sparse.linalg.lsmr` for finding a solution of a linear
least-squares problem and only requires matrix-vector product
evaluations.
If None (default), the solver is chosen based on the type of Jacobian
returned on the first iteration.
tr_options : dict, optional
Keyword options passed to trust-region solver.
* ``tr_solver='exact'``: `tr_options` are ignored.
* ``tr_solver='lsmr'``: options for `scipy.sparse.linalg.lsmr`.
Additionally, ``method='trf'`` supports 'regularize' option
(bool, default is True), which adds a regularization term to the
normal equation, which improves convergence if the Jacobian is
rank-deficient [Byrd]_ (eq. 3.4).
jac_sparsity : {None, array_like, sparse matrix}, optional
Defines the sparsity structure of the Jacobian matrix for finite
difference estimation, its shape must be (m, n). If the Jacobian has
only few non-zero elements in *each* row, providing the sparsity
structure will greatly speed up the computations [Curtis]_. A zero
entry means that a corresponding element in the Jacobian is identically
zero. If provided, forces the use of 'lsmr' trust-region solver.
If None (default), then dense differencing will be used. Has no effect
for 'lm' method.
verbose : {0, 1, 2}, optional
Level of algorithm's verbosity:
* 0 (default) : work silently.
* 1 : display a termination report.
* 2 : display progress during iterations (not supported by 'lm'
method).
args, kwargs : tuple and dict, optional
Additional arguments passed to `fun` and `jac`. Both empty by default.
The calling signature is ``fun(x, *args, **kwargs)`` and the same for
`jac`.
Returns
-------
result : OptimizeResult
`OptimizeResult` with the following fields defined:
x : ndarray, shape (n,)
Solution found.
cost : float
Value of the cost function at the solution.
fun : ndarray, shape (m,)
Vector of residuals at the solution.
jac : ndarray, sparse matrix or LinearOperator, shape (m, n)
Modified Jacobian matrix at the solution, in the sense that J^T J
is a Gauss-Newton approximation of the Hessian of the cost function.
The type is the same as the one used by the algorithm.
grad : ndarray, shape (m,)
Gradient of the cost function at the solution.
optimality : float
First-order optimality measure. In unconstrained problems, it is
always the uniform norm of the gradient. In constrained problems,
it is the quantity which was compared with `gtol` during iterations.
active_mask : ndarray of int, shape (n,)
Each component shows whether a corresponding constraint is active
(that is, whether a variable is at the bound):
* 0 : a constraint is not active.
* -1 : a lower bound is active.
* 1 : an upper bound is active.
Might be somewhat arbitrary for 'trf' method as it generates a
sequence of strictly feasible iterates and `active_mask` is
determined within a tolerance threshold.
nfev : int
Number of function evaluations done. Methods 'trf' and 'dogbox' do
not count function calls for numerical Jacobian approximation, as
opposed to 'lm' method.
njev : int or None
Number of Jacobian evaluations done. If numerical Jacobian
approximation is used in 'lm' method, it is set to None.
status : int
The reason for algorithm termination:
* -1 : improper input parameters status returned from MINPACK.
* 0 : the maximum number of function evaluations is exceeded.
* 1 : `gtol` termination condition is satisfied.
* 2 : `ftol` termination condition is satisfied.
* 3 : `xtol` termination condition is satisfied.
* 4 : Both `ftol` and `xtol` termination conditions are satisfied.
message : str
Verbal description of the termination reason.
success : bool
True if one of the convergence criteria is satisfied (`status` > 0).
See Also
--------
leastsq : A legacy wrapper for the MINPACK implementation of the
Levenberg-Marquadt algorithm.
curve_fit : Least-squares minimization applied to a curve-fitting problem.
Notes
-----
Method 'lm' (Levenberg-Marquardt) calls a wrapper over least-squares
algorithms implemented in MINPACK (lmder, lmdif). It runs the
Levenberg-Marquardt algorithm formulated as a trust-region type algorithm.
The implementation is based on paper [JJMore]_, it is very robust and
efficient with a lot of smart tricks. It should be your first choice
for unconstrained problems. Note that it doesn't support bounds. Also,
it doesn't work when m < n.
Method 'trf' (Trust Region Reflective) is motivated by the process of
solving a system of equations, which constitute the first-order optimality
condition for a bound-constrained minimization problem as formulated in
[STIR]_. The algorithm iteratively solves trust-region subproblems
augmented by a special diagonal quadratic term and with trust-region shape
determined by the distance from the bounds and the direction of the
gradient. This enhancements help to avoid making steps directly into bounds
and efficiently explore the whole space of variables. To further improve
convergence, the algorithm considers search directions reflected from the
bounds. To obey theoretical requirements, the algorithm keeps iterates
strictly feasible. With dense Jacobians trust-region subproblems are
solved by an exact method very similar to the one described in [JJMore]_
(and implemented in MINPACK). The difference from the MINPACK
implementation is that a singular value decomposition of a Jacobian
matrix is done once per iteration, instead of a QR decomposition and series
of Givens rotation eliminations. For large sparse Jacobians a 2-D subspace
approach of solving trust-region subproblems is used [STIR]_, [Byrd]_.
The subspace is spanned by a scaled gradient and an approximate
Gauss-Newton solution delivered by `scipy.sparse.linalg.lsmr`. When no
constraints are imposed the algorithm is very similar to MINPACK and has
generally comparable performance. The algorithm works quite robust in
unbounded and bounded problems, thus it is chosen as a default algorithm.
Method 'dogbox' operates in a trust-region framework, but considers
rectangular trust regions as opposed to conventional ellipsoids [Voglis]_.
The intersection of a current trust region and initial bounds is again
rectangular, so on each iteration a quadratic minimization problem subject
to bound constraints is solved approximately by Powell's dogleg method
[NumOpt]_. The required Gauss-Newton step can be computed exactly for
dense Jacobians or approximately by `scipy.sparse.linalg.lsmr` for large
sparse Jacobians. The algorithm is likely to exhibit slow convergence when
the rank of Jacobian is less than the number of variables. The algorithm
often outperforms 'trf' in bounded problems with a small number of
variables.
Robust loss functions are implemented as described in [BA]_. The idea
is to modify a residual vector and a Jacobian matrix on each iteration
such that computed gradient and Gauss-Newton Hessian approximation match
the true gradient and Hessian approximation of the cost function. Then
the algorithm proceeds in a normal way, i.e., robust loss functions are
implemented as a simple wrapper over standard least-squares algorithms.
.. versionadded:: 0.17.0
References
----------
.. [STIR] M. A. Branch, T. F. Coleman, and Y. Li, "A Subspace, Interior,
and Conjugate Gradient Method for Large-Scale Bound-Constrained
Minimization Problems," SIAM Journal on Scientific Computing,
Vol. 21, Number 1, pp 1-23, 1999.
.. [NR] William H. Press et. al., "Numerical Recipes. The Art of Scientific
Computing. 3rd edition", Sec. 5.7.
.. [Byrd] R. H. Byrd, R. B. Schnabel and G. A. Shultz, "Approximate
solution of the trust region problem by minimization over
two-dimensional subspaces", Math. Programming, 40, pp. 247-263,
1988.
.. [Curtis] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
sparse Jacobian matrices", Journal of the Institute of
Mathematics and its Applications, 13, pp. 117-120, 1974.
.. [JJMore] J. J. More, "The Levenberg-Marquardt Algorithm: Implementation
and Theory," Numerical Analysis, ed. G. A. Watson, Lecture
Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977.
.. [Voglis] C. Voglis and I. E. Lagaris, "A Rectangular Trust Region
Dogleg Approach for Unconstrained and Bound Constrained
Nonlinear Optimization", WSEAS International Conference on
Applied Mathematics, Corfu, Greece, 2004.
.. [NumOpt] J. Nocedal and S. J. Wright, "Numerical optimization,
2nd edition", Chapter 4.
.. [BA] B. Triggs et. al., "Bundle Adjustment - A Modern Synthesis",
Proceedings of the International Workshop on Vision Algorithms:
Theory and Practice, pp. 298-372, 1999.
Examples
--------
In this example we find a minimum of the Rosenbrock function without bounds
on independent variables.
>>> import numpy as np
>>> def fun_rosenbrock(x):
... return np.array([10 * (x[1] - x[0]**2), (1 - x[0])])
Notice that we only provide the vector of the residuals. The algorithm
constructs the cost function as a sum of squares of the residuals, which
gives the Rosenbrock function. The exact minimum is at ``x = [1.0, 1.0]``.
>>> from scipy.optimize import least_squares
>>> x0_rosenbrock = np.array([2, 2])
>>> res_1 = least_squares(fun_rosenbrock, x0_rosenbrock)
>>> res_1.x
array([ 1., 1.])
>>> res_1.cost
9.8669242910846867e-30
>>> res_1.optimality
8.8928864934219529e-14
We now constrain the variables, in such a way that the previous solution
becomes infeasible. Specifically, we require that ``x[1] >= 1.5``, and
``x[0]`` left unconstrained. To this end, we specify the `bounds` parameter
to `least_squares` in the form ``bounds=([-np.inf, 1.5], np.inf)``.
We also provide the analytic Jacobian:
>>> def jac_rosenbrock(x):
... return np.array([
... [-20 * x[0], 10],
... [-1, 0]])
Putting this all together, we see that the new solution lies on the bound:
>>> res_2 = least_squares(fun_rosenbrock, x0_rosenbrock, jac_rosenbrock,
... bounds=([-np.inf, 1.5], np.inf))
>>> res_2.x
array([ 1.22437075, 1.5 ])
>>> res_2.cost
0.025213093946805685
>>> res_2.optimality
1.5885401433157753e-07
Now we solve a system of equations (i.e., the cost function should be zero
at a minimum) for a Broyden tridiagonal vector-valued function of 100000
variables:
>>> def fun_broyden(x):
... f = (3 - x) * x + 1
... f[1:] -= x[:-1]
... f[:-1] -= 2 * x[1:]
... return f
The corresponding Jacobian matrix is sparse. We tell the algorithm to
estimate it by finite differences and provide the sparsity structure of
Jacobian to significantly speed up this process.
>>> from scipy.sparse import lil_matrix
>>> def sparsity_broyden(n):
... sparsity = lil_matrix((n, n), dtype=int)
... i = np.arange(n)
... sparsity[i, i] = 1
... i = np.arange(1, n)
... sparsity[i, i - 1] = 1
... i = np.arange(n - 1)
... sparsity[i, i + 1] = 1
... return sparsity
...
>>> n = 100000
>>> x0_broyden = -np.ones(n)
...
>>> res_3 = least_squares(fun_broyden, x0_broyden,
... jac_sparsity=sparsity_broyden(n))
>>> res_3.cost
4.5687069299604613e-23
>>> res_3.optimality
1.1650454296851518e-11
Let's also solve a curve fitting problem using robust loss function to
take care of outliers in the data. Define the model function as
``y = a + b * exp(c * t)``, where t is a predictor variable, y is an
observation and a, b, c are parameters to estimate.
First, define the function which generates the data with noise and
outliers, define the model parameters, and generate data:
>>> from numpy.random import default_rng
>>> rng = default_rng()
>>> def gen_data(t, a, b, c, noise=0., n_outliers=0, seed=None):
... rng = default_rng(seed)
...
... y = a + b * np.exp(t * c)
...
... error = noise * rng.standard_normal(t.size)
... outliers = rng.integers(0, t.size, n_outliers)
... error[outliers] *= 10
...
... return y + error
...
>>> a = 0.5
>>> b = 2.0
>>> c = -1
>>> t_min = 0
>>> t_max = 10
>>> n_points = 15
...
>>> t_train = np.linspace(t_min, t_max, n_points)
>>> y_train = gen_data(t_train, a, b, c, noise=0.1, n_outliers=3)
Define function for computing residuals and initial estimate of
parameters.
>>> def fun(x, t, y):
... return x[0] + x[1] * np.exp(x[2] * t) - y
...
>>> x0 = np.array([1.0, 1.0, 0.0])
Compute a standard least-squares solution:
>>> res_lsq = least_squares(fun, x0, args=(t_train, y_train))
Now compute two solutions with two different robust loss functions. The
parameter `f_scale` is set to 0.1, meaning that inlier residuals should
not significantly exceed 0.1 (the noise level used).
>>> res_soft_l1 = least_squares(fun, x0, loss='soft_l1', f_scale=0.1,
... args=(t_train, y_train))
>>> res_log = least_squares(fun, x0, loss='cauchy', f_scale=0.1,
... args=(t_train, y_train))
And, finally, plot all the curves. We see that by selecting an appropriate
`loss` we can get estimates close to optimal even in the presence of
strong outliers. But keep in mind that generally it is recommended to try
'soft_l1' or 'huber' losses first (if at all necessary) as the other two
options may cause difficulties in optimization process.
>>> t_test = np.linspace(t_min, t_max, n_points * 10)
>>> y_true = gen_data(t_test, a, b, c)
>>> y_lsq = gen_data(t_test, *res_lsq.x)
>>> y_soft_l1 = gen_data(t_test, *res_soft_l1.x)
>>> y_log = gen_data(t_test, *res_log.x)
...
>>> import matplotlib.pyplot as plt
>>> plt.plot(t_train, y_train, 'o')
>>> plt.plot(t_test, y_true, 'k', linewidth=2, label='true')
>>> plt.plot(t_test, y_lsq, label='linear loss')
>>> plt.plot(t_test, y_soft_l1, label='soft_l1 loss')
>>> plt.plot(t_test, y_log, label='cauchy loss')
>>> plt.xlabel("t")
>>> plt.ylabel("y")
>>> plt.legend()
>>> plt.show()
In the next example, we show how complex-valued residual functions of
complex variables can be optimized with ``least_squares()``. Consider the
following function:
>>> def f(z):
... return z - (0.5 + 0.5j)
We wrap it into a function of real variables that returns real residuals
by simply handling the real and imaginary parts as independent variables:
>>> def f_wrap(x):
... fx = f(x[0] + 1j*x[1])
... return np.array([fx.real, fx.imag])
Thus, instead of the original m-D complex function of n complex
variables we optimize a 2m-D real function of 2n real variables:
>>> from scipy.optimize import least_squares
>>> res_wrapped = least_squares(f_wrap, (0.1, 0.1), bounds=([0, 0], [1, 1]))
>>> z = res_wrapped.x[0] + res_wrapped.x[1]*1j
>>> z
(0.49999999999925893+0.49999999999925893j)
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