Proportional Intensity Regression Model

class surpyval.recurrent.regression.proportional_intensity.ProportionalIntensityModel

Bases: RecurrenceSimulationMixin, LikelihoodInferenceMixin

Model to provide methods and attributes when using a fitted proportional intensity model.

Simulation reuses the shared RecurrenceSimulationMixin (seeding, max_events backstop and the count/time-terminated drivers); the only addition here is the per-item covariate vector Z, which the simulation entry points take and thread through to the sampler. When the model was fitted by maximum likelihood it also carries the likelihood-inference behaviour (log_likelihood, aic, bic, standard_errors) from LikelihoodInferenceMixin.

Examples

>>> from surpyval.datasets import load_rossi_static
>>> from surpyval.recurrent import CrowAMSAA
>>> from surpyval.recurrent import ProportionalIntensityNHPP
>>> data = load_rossi_static()
>>> x = data['week'].values
>>> c = data['arrest'].values
>>> Z = data[["fin", "age", "race", "wexp", "mar", "paro", "prio"]].values
>>> model = ProportionalIntensityNHPP.fit(x, Z, c, dist=CrowAMSAA)
>>> type(model)
surpyval.recurrent.regression.proportional_intensity.ProportionalIntensityModel
>>> model.cif([1, 2, 3], Z.mean(axis=0))
array([0.00625402, 0.04304137, 0.13302238])
cif(x, Z)

Compute the cumulative incidence function of the model with the parameters found by the fit method.

Parameters

  • x (array_like) – The times to compute the CIF at.

  • Z (array_like) – The covariates for the item.

cif_cb(x, Z, alpha_ci=0.05, bound='two-sided')

Confidence bounds on the fitted CIF at x for covariates Z, from the delta method.

The variance of the fitted CIF is propagated from the joint covariance of the base-rate parameters and covariate coefficients (the inverse observed information) through the CIF’s gradient, and the bounds are computed on the log scale so they cannot go negative.

Parameters

  • x (array_like) – Values at which to compute the confidence bounds.

  • Z (array_like) – The covariates for the item.

  • alpha_ci (float, optional) – The total tail probability of the bound(s). Default is 0.05.

  • bound ({'two-sided', 'lower', 'upper'}, optional) – Two-sided bounds are returned as an (len(x), 2) array with columns [lower, upper]; one-sided bounds have the shape of x.

Returns

The confidence bounds on the CIF.

Return type

numpy array

count_terminated_simulation(events, Z, items=1, seed=None)

Simulate count-terminated recurrence data based on the fitted model.

Parameters

  • events (int) – Number of events to simulate per sequence.

  • Z (array_like) – Covariate vector applied to every simulated sequence.

  • items (int, optional) – Number of items (or sequences) to simulate. Default is 1.

  • seed (int or numpy.random.Generator, optional) – Seed for a reproducible simulation.

Returns

An NonParametricCounting model built from the simulated data.

Return type

NonParametricCounting

count_terminated_simulation_data(events, Z, items=1, seed=None)

Simulate count-terminated recurrence data and return the raw events. Like count_terminated_simulation() but yields the simulated RecurrentEventData rather than the fitted MCF.

covariance()

Approximate parameter covariance matrix, ordered to match parameter_names. Computed as the inverse of the numerical Hessian of the negative log-likelihood at the MLE.

iif(x, Z)

Compute the instantaneous incidence function of the model with the parameters found by the fit method.

Parameters

  • x (array_like) – The times to at which to compute the iif.

  • Z (array_like) – The covariates for the item.

mcf(x, Z, items=1000, seed=None)

Estimate the mean cumulative function at x for covariates Z by simulating items time-terminated sequences out to max(x).

param_cb(name, alpha_ci=0.05, bound='two-sided')

Confidence bound(s) on a fitted parameter, mirroring the univariate Parametric.param_cb API.

Wald bounds from the observed information, computed on a transformed scale chosen from the parameter’s bounds so the result respects its support: log scale for one-sided-bounded parameters (e.g. a positive rate), logit scale for interval-bounded parameters (e.g. a repair efficiency in (0, 1)), and the natural scale for unbounded ones.

Parameters

  • name (str) – The parameter to bound; one of parameter_names.

  • alpha_ci (float, optional) – The total tail probability of the bound(s). Default is 0.05.

  • bound ({'two-sided', 'lower', 'upper'}, optional) – Two-sided bounds are returned as [lower, upper].

Returns

The confidence bound(s) on the parameter.

Return type

numpy array

plot(ax=None, plot_bounds=True, confidence=0.95)

PLots the CIF of the model against the data used to fit it.

To do this, the plot method takes the average of the covariates, and uses them to calculate the CIF of the model. This is then plotted against the non-parametric MCF of the raw data. That is, the raw MCF is created without considering the covariates. A delta-method confidence band is drawn around the fitted CIF.

Parameters

  • ax (matplotlib.axes.Axes, optional) – The axes to plot the data on. If None, the current axes will be used.

  • plot_bounds (bool, optional) – Whether to draw the confidence band around the fitted CIF. Default is True.

  • confidence (float, optional) – The confidence level of the band. Default is 0.95.

Returns

ax – The axes the data was plotted on.

Return type

matplotlib.axes.Axes

standard_errors()

Standard errors of the fitted parameters (the square roots of the diagonal of covariance()), ordered to match parameter_names. Entries are NaN where the variance is non-positive, which typically indicates a boundary optimum.

time_terminated_simulation(T, Z, items=1, tol=1e-08, max_events=10000, seed=None)

Simulate time-terminated recurrence data based on the fitted model.

Parameters

  • T (float) – Time termination value.

  • Z (array_like) – Covariate vector applied to every simulated sequence.

  • items (int, optional) – Number of items (or sequences) to simulate. Default is 1.

  • tol (float, optional) – Interarrival times below this value end a sequence early (a possible asymptote). Default is 1e-8.

  • max_events (int, optional) – Hard per-sequence event cap that guarantees termination. Default is 10000.

  • seed (int or numpy.random.Generator, optional) – Seed for a reproducible simulation.

Returns

An NonParametricCounting model built from the simulated data.

Return type

NonParametricCounting

Warning

A sequence is terminated early and right-censored at its last event if an interarrival time falls below tol or it reaches max_events before T. A warning is raised in either case.

time_terminated_simulation_data(T, Z, items=1, tol=1e-08, max_events=10000, seed=None)

Simulate time-terminated recurrence data and return the raw events. Like time_terminated_simulation() but yields the simulated RecurrentEventData rather than the fitted MCF.