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fit.fisher

Fisher information of a parameter fit.

The profile likelihood of sbmlsim.fit.identifiability scans a parameter and re-optimizes the others, which is exact and costs a simulation per point. The Fisher information is the local alternative: the curvature of the cost at the optimum, from which the standard errors, the correlations of the parameters and the directions the data does not constrain follow, at the cost of one jacobian.

For a fit which minimizes cost = 0.5 * Σ r² with the weighted residuals r, the Gauss-Newton approximation of the Hessian is J' J with the jacobian J = ∂r/∂θ, which is the Fisher information matrix of the estimate. Its inverse, scaled by the variance of the residuals, is the covariance of the parameters:

FIM = J' J,   cov = σ² (J' J)⁻¹,   σ² = 2 cost / (n - k)

The two analyses answer different questions and disagree where the cost is not a quadratic: the Fisher information is a local statement about the optimum, the profile likelihood follows the cost until it rises by the threshold. A parameter which the Fisher information calls determined and the profile likelihood calls non-identifiable is a parameter whose cost is flat away from the optimum, which is what the profile is for.

FisherInformation dataclass 

FisherInformation(
    opid,
    sid,
    pids,
    values,
    scale,
    matrix,
    cost,
    n,
    alpha=0.95,
    rank_tolerance=DEFAULT_RANK_TOLERANCE,
    units=list(),
)

The Fisher information of a parameter set on a problem.

Attributes:

Name Type Description
opid str

id of the optimization problem.
sid str

id of the parameter set.
pids list[str]

parameters, in the order of the matrix.
values ndarray

values of the parameters, in the units of the model.
scale ParameterScaleType

space the matrix is expressed in, i.e. the space the optimizer searches, see ParameterScaleType.
matrix ndarray

the Fisher information J' J.
cost float

cost of the parameter set.
n int

number of data points of the fit.
alpha float

confidence level of the intervals.

k property 

k

Get the number of parameters.

sigma2 property 

sigma2

Get the variance of the residuals, 2 cost / (n - k).

A fit with as many parameters as data points has no degrees of freedom left and no variance to estimate, which is nan.

eigenvalues property 

eigenvalues

Get the eigenvalues of the Fisher information, largest first.

The matrix is symmetric and positive semi-definite, so the eigenvalues are real and not negative up to rounding. A small eigenvalue is a direction in parameter space the data does not constrain.

condition_number property 

condition_number

Get the ratio of the largest to the smallest eigenvalue.

A large condition number is a problem which is sloppy: the data determines some combinations of the parameters much better than others.

rank property 

rank

Get the number of directions the data constrains.

is_identifiable property 

is_identifiable

Check whether the data constrains every direction locally.

covariance property 

covariance

Get the covariance of the parameters in the scaled space.

cov = σ² (J' J)⁻¹, with the pseudo-inverse for a matrix which is rank deficient, i.e. for a problem which is locally not identifiable. The covariance of such a problem is not a covariance, its entries of the unconstrained directions are arbitrary; is_identifiable says whether it can be read. The warning about it is logged once per information.

standard_errors property 

standard_errors

Get the standard error of every parameter in the scaled space.

correlation property 

correlation

Get the correlation of the parameters.

Two parameters which are correlated to ±1 are the pair the data only determines together, i.e. the fit can trade one against the other.

summary_df property 

summary_df

Get the parameters with their errors and intervals as a table.

confidence_intervals 

confidence_intervals()

Get the confidence interval of every parameter.

The interval is θ ± t · SE in the space the optimizer searches, with the quantile of the t distribution of n - k degrees of freedom, and is transformed back into the units of the model. On a logarithmic scale the interval is therefore not symmetric around the value.

Returns:

Type Description
tuple[ndarray, ndarray]

The lower and the upper bound in the units of the model.

to_dict 

to_dict()

Convert to a dictionary of JSON serializable values.

jacobian 

jacobian(problem, x, step=DEFAULT_STEP)

Get the jacobian of the weighted residuals by central differences.

Parameters:

Name Type Description Default
problem OptimizationProblem

initialized optimization problem.

 required
x ndarray

parameters in the space the optimizer searches.

 required
step float

relative step of the differences.

 DEFAULT_STEP

Returns:

Type Description
ndarray

The jacobian with one row per residual and one column per parameter.

fisher_information 

fisher_information(
    problem,
    settings,
    parameter_set,
    alpha=0.95,
    step=DEFAULT_STEP,
    rank_tolerance=DEFAULT_RANK_TOLERANCE,
)

Calculate the Fisher information of a parameter set on a problem.

Parameters:

Name Type Description Default
problem OptimizationProblem

optimization problem, initialized with the settings.

 required
settings FitSettings

settings the parameters were fitted with.

 required
parameter_set ParameterSet

parameters to evaluate the information at, i.e. the result of a fit.

 required
alpha float

confidence level of the intervals.

 0.95
step float

relative step of the finite differences of the jacobian.

 DEFAULT_STEP
rank_tolerance float

eigenvalues below this fraction of the largest one are a direction the data does not constrain.

 DEFAULT_RANK_TOLERANCE

Returns:

Type Description
FisherInformation

The Fisher information with the errors, the correlations and the
FisherInformation

directions the data does not constrain.