Tweedie family with variance power estimated by profile likelihood.
TweedieEstimated(
p_range=(1.01, 1.99),
n_grid=20,
)
TweedieEstimated behaves exactly like Tweedie (log link, V(mu) = mu^p, compound Poisson-Gamma density for 1 < p < 2) except that the variance power p is not fixed at construction time. Instead, GAM.fit() performs a profile-likelihood grid search: the model is refit at each candidate p in p_range, and the value minimizing AIC is retained. This is useful when the appropriate degree of “Poisson-ness” versus “Gamma-ness” in claims, rainfall, or other zero-inflated positive data is not known in advance. Users typically construct this family via the tw() convenience function rather than instantiating TweedieEstimated directly.
Parameters
p_range: tuple of float = (1.01, 1.99)-
(p_min, p_max) range to search over. Both endpoints should lie strictly within (1, 2) for the compound Poisson-Gamma case (the typical use case for insurance-style data with structural zeros).
n_grid: int = 20-
Number of candidate p values in the initial grid search over
p_range. A finer grid gives a more precise estimate of p at the cost of additional model fits.
Notes
Once fitted, the estimated value of p is available via the inherited p property, and p_estimated reports whether estimation has completed. The link, variance function, and deviance are identical to those of Tweedie:
g(\mu) = \log(\mu), \qquad V(\mu) = \mu^{p}.
See Tweedie for the full deviance and log-likelihood formulas.
Examples
Fit a GAM letting Whittaker choose the Tweedie variance power automatically:
import numpy as np
import whittaker as wk
rng = np.random.default_rng(0)
n = 300
x = np.linspace(0, 2 * np.pi, n)
mu = np.exp(1.0 + 0.5 * np.sin(x))
y = np.array([rng.gamma(2.0, m / 2.0) if rng.random() > 0.3 else 0.0 for m in mu])
data = {"x": x, "y": y}
model = wk.GAM("y ~ s(x)", family=wk.tw())
model.fit(data, method="REML")
print(model.family)
print(model.summary())
Tweedie(p=1.0100, link='log', estimated=True)
GAM fit summary
============================================================
Formula: y ~ s(x)
Family: Tweedie(p=1.0100, link='log', estimated=True)
Inference: REML
Observations: 300
Coefficients: 10
Parametric coefficients:
Term Estimate Std.Err t value p-value
------------------------ ---------- ---------- ---------- ----------
(Intercept) 0.6950 0.0587 11.847 < 1e-16
Approximate significance of smooth terms:
Term EDF Ref.df Chi.sq p-value
------------------------ ------ ------ ---------- ----------
s(x) 3.98 4 33.944 7.65e-07
Total EDF: 4.98
Scale est: 1.968344
Deviance: 580.6941
Null dev: 656.3635
Dev. expl: 11.5%
GCV score: 2.001593
AIC: 1198.39
BIC: 1216.84
Attributes
|
Name
|
Description
|
|
p_estimated
|
Whether the variance power p has completed profile-likelihood estimation.
|
p_estimated
Whether the variance power p has completed profile-likelihood estimation.
False immediately after construction, when p is only a provisional midpoint of p_range. Set to True by _set_p once GAM.fit() has run its grid search over p_range and selected the AIC-minimizing value, at which point the inherited p property reports the estimated value rather than the placeholder.