Fit the model using a regularized maximum likelihood.
The regularization method AND the solver used is determined by the
argument method.
Parameters: | start_params : array-like, optional
Initial guess of the solution for the loglikelihood maximization.
The default is an array of zeros.
method : ‘l1’ or ‘l1_cvxopt_cp’
maxiter : Integer or ‘defined_by_method’
Maximum number of iterations to perform.
If ‘defined_by_method’, then use method defaults (see notes).
full_output : bool
Set to True to have all available output in the Results object’s
mle_retvals attribute. The output is dependent on the solver.
See LikelihoodModelResults notes section for more information.
disp : bool
Set to True to print convergence messages.
fargs : tuple
Extra arguments passed to the likelihood function, i.e.,
loglike(x,*args)
callback : callable callback(xk)
Called after each iteration, as callback(xk), where xk is the
current parameter vector.
retall : bool
Set to True to return list of solutions at each iteration.
Available in Results object’s mle_retvals attribute.
alpha : non-negative scalar or numpy array (same size as parameters)
The weight multiplying the l1 penalty term
trim_mode : ‘auto, ‘size’, or ‘off’
If not ‘off’, trim (set to zero) parameters that would have been
zero if the solver reached the theoretical minimum.
If ‘auto’, trim params using the Theory above.
If ‘size’, trim params if they have very small absolute value
size_trim_tol : float or ‘auto’ (default = ‘auto’)
For use when trim_mode == ‘size’
auto_trim_tol : float
For sue when trim_mode == ‘auto’. Use
qc_tol : float
Print warning and don’t allow auto trim when (ii) (above) is
violated by this much.
qc_verbose : Boolean
If true, print out a full QC report upon failure
|
Notes
Extra parameters are not penalized if alpha is given as a scalar.
An example is the shape parameter in NegativeBinomial nb1 and nb2.
Optional arguments for the solvers (available in Results.mle_settings):
'l1'
acc : float (default 1e-6)
Requested accuracy as used by slsqp
'l1_cvxopt_cp'
abstol : float
absolute accuracy (default: 1e-7).
reltol : float
relative accuracy (default: 1e-6).
feastol : float
tolerance for feasibility conditions (default: 1e-7).
refinement : int
number of iterative refinement steps when solving KKT
equations (default: 1).
Optimization methodology
With
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the negative log likelihood, we solve the convex but
non-smooth problem
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via the transformation to the smooth, convex, constrained problem
in twice as many variables (adding the “added variables”
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)
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subject to
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With
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the derivative of
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in the
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parameter direction, theory dictates that, at the
minimum, exactly one of two conditions holds:
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