Inverse Gaussian family with log link.
The Inverse Gaussian family models strictly positive, continuous responses whose variance grows even faster than in the Gamma case, producing heavier right tails. It arises naturally as a first-passage-time distribution (e.g. the time for a diffusion process to reach a threshold) and is a common choice for lifetime, duration, and other highly skewed positive data. InverseGaussian is the Tweedie special case with variance power p = 3 (compare Tweedie and tw()), implemented directly here for exact deviance and log-likelihood computations rather than the saddlepoint approximation used by the general Tweedie family. The log link is used for the same reasons as in Gamma: it keeps fitted values positive and gives coefficients a multiplicative interpretation.
Notes
The link is the natural logarithm:
g(\mu) = \log(\mu)
The variance function grows with the cube of the mean:
V(\mu) = \mu^{3},
making this family appropriate when large means are associated with disproportionately large variability. The deviance is
D(y, \hat\mu) = \sum_i \frac{(y_i - \hat\mu_i)^2}{\hat\mu_i^2\, y_i} .
Examples
Fit a GAM to heavy-tailed positive data with a smooth trend:
import numpy as np
import whittaker as wk
rng = np.random.default_rng(0)
n = 200
x = np.linspace(0, 2 * np.pi, n)
mu = np.exp(1.0 + 0.4 * np.sin(x))
scale = 0.5
lam = mu / scale
y = rng.wald(mu, lam)
data = {"x": x, "y": y}
model = wk.GAM("y ~ s(x)", family=wk.InverseGaussian())
model.fit(data, method="REML")
print(model.summary())
GAM fit summary
============================================================
Formula: y ~ s(x)
Family: InverseGaussian(link='log')
Inference: REML
Observations: 200
Coefficients: 10
Parametric coefficients:
Term Estimate Std.Err t value p-value
------------------------ ---------- ---------- ---------- ----------
(Intercept) 1.0954 0.0555 19.735 < 1e-16
Approximate significance of smooth terms:
Term EDF Ref.df Chi.sq p-value
------------------------ ------ ------ ---------- ----------
s(x) 4.19 5 32.744 4.231e-06
Total EDF: 5.19
Scale est: 0.198556
Deviance: 38.6806
Null dev: 45.4465
Dev. expl: 14.9%
GCV score: 0.203847
AIC: 783.07
BIC: 800.19
Attributes
|
Name
|
Description
|
|
scale_known
|
Whether the dispersion parameter is fixed. Always False for InverseGaussian.
|
scale_known
Whether the dispersion parameter is fixed. Always False for InverseGaussian.
The dispersion phi is estimated from the data during fitting rather than fixed, analogous to how Gamma and Gaussian estimate their own dispersion parameters. This affects how GAM.fit() and GAM.summary() treat the scale.
Methods
|
Name
|
Description
|
|
deviance()
|
Total Inverse Gaussian deviance, the (weighted) sum of unit_deviance.
|
|
initialize()
|
Starting values for mu: y nudged away from zero.
|
|
link()
|
Apply the log link: \eta = \log(\mu).
|
|
link_derivative()
|
Derivative of the log link: g'(\mu) = 1/\mu.
|
|
link_inverse()
|
Apply the inverse log link: \mu = e^{\eta}.
|
|
log_likelihood()
|
Inverse Gaussian log-likelihood evaluated at dispersion scale.
|
|
simulate()
|
Simulate Inverse Gaussian (Wald) response values with mean mu and dispersion scale.
|
|
unit_deviance()
|
Per-observation Inverse Gaussian deviance contributions.
|
|
variance()
|
Inverse Gaussian variance function: V(\mu) = \mu^3.
|
deviance()
Total Inverse Gaussian deviance, the (weighted) sum of unit_deviance.
deviance(
y,
mu,
*,
weights=None,
)
Parameters
y: NDArray-
Observed response values (positive), shape (n,).
mu: NDArray-
Fitted conditional mean values, shape (n,).
weights: NDArray | None = None-
Optional prior weights, shape
(n,).
Returns
float-
The total (weighted) deviance.
initialize()
Starting values for mu: y nudged away from zero.
Since the log link requires strictly positive mu, values are pushed away from zero to avoid log(0) on the first P-IRLS iteration.
Parameters
y: NDArray-
Observed response values (positive), shape
(n,).
Returns
NDArray-
Starting values for
mu, shape (n,).
link()
Apply the log link: \eta = \log(\mu).
Parameters
mu: NDArray-
Conditional mean values, shape
(n,). Must be positive.
Returns
NDArray-
Linear predictor values \eta = \log(\mu), shape
(n,).
link_derivative()
Derivative of the log link: g'(\mu) = 1/\mu.
Parameters
mu: NDArray-
Conditional mean values, shape
(n,).
Returns
NDArray-
Derivative values 1/\mu, shape
(n,).
link_inverse()
Apply the inverse log link: \mu = e^{\eta}.
The linear predictor is clipped to [-30, 30] before exponentiating to guard against overflow while fitting.
Parameters
eta: NDArray-
Linear predictor values, shape
(n,).
Returns
NDArray-
Conditional mean values \mu = e^{\eta}, shape
(n,).
log_likelihood()
Inverse Gaussian log-likelihood evaluated at dispersion scale.
log_likelihood(
y,
mu,
scale,
*,
weights=None,
)
Parameters
y: NDArray-
Observed response values (positive), shape (n,).
mu: NDArray-
Fitted conditional mean values, shape (n,).
scale: float-
Dispersion parameter \phi.
weights: NDArray | None = None-
Optional prior weights, shape
(n,).
Returns
float-
The total log-likelihood.
simulate()
Simulate Inverse Gaussian (Wald) response values with mean mu and dispersion scale.
simulate(
mu,
scale,
rng,
)
Draws are generated via Michael, Schucany & Haas’s transformation of a chi-squared variate, then adjusted with a uniform random variable to select between the two candidate roots.
Parameters
mu: NDArray-
Mean (fitted values), shape (n,).
scale: float-
Dispersion parameter \phi.
rng: np.random.Generator-
A
numpy.random.Generator instance.
Returns
NDArray-
Simulated response values, shape
(n,).
unit_deviance()
Per-observation Inverse Gaussian deviance contributions.
Computes d_i = (y_i - \hat\mu_i)^2 / (\hat\mu_i^2\, y_i).
Parameters
y: NDArray-
Observed response values (positive), shape (n,).
mu: NDArray-
Fitted conditional mean values, shape
(n,).
Returns
NDArray-
Per-observation deviance contributions, shape
(n,).
variance()
Inverse Gaussian variance function: V(\mu) = \mu^3.
The variance grows with the cube of the mean, giving this family the heaviest right tail among Whittaker’s built-in continuous positive families.
Parameters
mu: NDArray-
Conditional mean values, shape
(n,).
Returns
NDArray-
Variance values \mu^3, shape
(n,).