Gaussian (Normal) family with identity link.
The Gaussian family models a continuous, unbounded response with constant variance. It is the default family in Whittaker and corresponds to classical (penalized) least-squares regression: with the identity link, P-IRLS converges in a single step since the working response and weights do not depend on the current fit. Use it whenever the response is real-valued, approximately symmetric, and its spread does not depend systematically on its mean — for example, physical measurements, log-transformed sizes, or residual-like quantities. If the variance grows with the mean, or the response is a count, proportion, or strictly positive quantity, consider Poisson, Binomial, Gamma, or another family instead.
Notes
The canonical (and only supported) link is the identity function:
g(\mu) = \mu
so the linear predictor eta is directly on the response scale and no back-transformation is needed for predictions. The variance function is constant in the mean,
V(\mu) = 1, \qquad \operatorname{Var}(Y) = \phi \, V(\mu) = \sigma^2,
which is what makes the Gaussian family the special case in which ordinary least squares and maximum-likelihood estimation coincide. The deviance is the residual sum of squares:
D(y, \hat\mu) = \sum_i (y_i - \hat\mu_i)^2 .
Examples
Fit a GAM with a smooth term to noisy sine-wave data using the (default) Gaussian family:
import numpy as np
import whittaker as wk
rng = np.random.default_rng(0)
n = 200
x = np.linspace(0, 2 * np.pi, n)
y = np.sin(x) + rng.normal(0, 0.3, n)
data = {"x": x, "y": y}
model = wk.GAM("y ~ s(x)", family=wk.Gaussian())
model.fit(data, method="REML")
print(model.summary())
GAM fit summary
============================================================
Formula: y ~ s(x)
Family: Gaussian(link='identity')
Inference: REML
Observations: 200
Coefficients: 10
Parametric coefficients:
Term Estimate Std.Err t value p-value
------------------------ ---------- ---------- ---------- ----------
(Intercept) 0.0046 0.0205 0.224 0.8233
Approximate significance of smooth terms:
Term EDF Ref.df Chi.sq p-value
------------------------ ------ ------ ---------- ----------
s(x) 7.31 8 1279.244 < 1e-16
Total EDF: 8.31
Scale est: 0.083835
Deviance: 16.0702
Null dev: 123.9418
Dev. expl: 87.0%
GCV score: 0.087470
AIC: 80.11
BIC: 107.52
Methods
|
Name
|
Description
|
|
deviance()
|
Total deviance: the (weighted) residual sum of squares \sum_i (y_i - \hat\mu_i)^2.
|
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initialize()
|
Starting values for mu: a copy of the observed response y.
|
|
link()
|
Apply the identity link: \eta = \mu.
|
|
link_derivative()
|
Derivative of the identity link: g'(\mu) = 1.
|
|
link_inverse()
|
Apply the inverse identity link: \mu = \eta.
|
|
log_likelihood()
|
Gaussian log-likelihood evaluated at variance scale.
|
|
simulate()
|
Simulate Gaussian response values N(mu, scale).
|
|
variance()
|
Constant variance function: V(\mu) = 1.
|
deviance()
Total deviance: the (weighted) residual sum of squares \sum_i (y_i - \hat\mu_i)^2.
deviance(
y,
mu,
*,
weights=None,
)
Parameters
y: NDArray-
Observed response values, shape (n,).
mu: NDArray-
Fitted conditional mean values, shape (n,).
weights: NDArray | None = None-
Optional prior weights, shape
(n,).
Returns
float-
The total (weighted) residual sum of squares.
initialize()
Starting values for mu: a copy of the observed response y.
Appropriate for the Gaussian family since the identity link places no constraint on the valid range of mu.
Parameters
y: NDArray-
Observed response values, shape
(n,).
Returns
NDArray-
Starting values for
mu, shape (n,).
link()
Apply the identity link: \eta = \mu.
The Gaussian family uses the identity link, so this is a no-op that returns mu unchanged.
Parameters
mu: NDArray-
Conditional mean values, shape
(n,).
Returns
NDArray-
Linear predictor values \eta = \mu, shape
(n,).
link_derivative()
Derivative of the identity link: g'(\mu) = 1.
Parameters
mu: NDArray-
Conditional mean values, shape
(n,).
Returns
NDArray-
An array of ones, shape
(n,).
link_inverse()
Apply the inverse identity link: \mu = \eta.
Returns eta unchanged, since the identity link is self-inverse.
Parameters
eta: NDArray-
Linear predictor values, shape
(n,).
Returns
NDArray-
Conditional mean values \mu = \eta, shape
(n,).
log_likelihood()
Gaussian log-likelihood evaluated at variance scale.
log_likelihood(
y,
mu,
scale,
*,
weights=None,
)
Computes \ell_i = -\tfrac{1}{2}\log(2\pi\phi) - \tfrac{(y_i - \mu_i)^2}{2\phi} for each observation, where \phi is scale, and sums (optionally weighted) over observations.
Parameters
y: NDArray-
Observed response values, shape (n,).
mu: NDArray-
Fitted conditional mean values, shape (n,).
scale: float-
Residual variance \phi = \sigma^2.
weights: NDArray | None = None-
Optional prior weights, shape
(n,).
Returns
float-
The total log-likelihood.
simulate()
Simulate Gaussian response values N(mu, scale).
simulate(
mu,
scale,
rng,
)
Parameters
mu: NDArray-
Mean (fitted values), shape (n,).
scale: float-
Residual variance sigma^2.
rng: np.random.Generator-
A
numpy.random.Generator instance.
Returns
NDArray-
Simulated response values, shape
(n,).
variance()
Constant variance function: V(\mu) = 1.
The Gaussian variance does not depend on the mean, so Var(Y) = phi * V(mu) = phi (the residual variance sigma^2).
Parameters
mu: NDArray-
Conditional mean values, shape
(n,).
Returns
NDArray-
An array of ones, shape
(n,).