Functional Generalized Linear Model

Functional GLM extends scalar-on-function regression to exponential-family responses. Instead of a Gaussian error term, the response may be binary (binomial), a count (Poisson), a positive continuous value (gamma), or unrestricted Gaussian. fdars.regression.functional_glm first projects each observed curve onto a small number of functional principal components (FPCA), then fits a standard GLM on the resulting scores — one scalar score per FPC per subject. The coefficient function \(\hat\beta(t)\) is reconstructed back to the original domain by recombining the FPC loadings.

Theory

Let \(X_1(t), \dots, X_n(t)\) be \(n\) functional observations on grid \(t_1, \dots, t_m\), and let \(y_1, \dots, y_n\) be scalar responses from a specified exponential-family distribution. The model proceeds in two stages:

Stage 1 — FPCA projection. Decompose the functional data into n_comp leading functional principal components \(\phi_1(t), \dots, \phi_K(t)\) and form the score matrix \(S \in \mathbb{R}^{n \times K}\) where \(s_{ik} = \int X_i(t)\,\phi_k(t)\,dt\). Each curve is now represented by \(K\) scalar scores.

Stage 2 — GLM in score space. Fit a GLM on the scores:

\[ g\!\bigl(\mathbb{E}[y_i \mid S_i]\bigr) \;=\; \alpha + S_i^\top \gamma, \]

where \(g(\cdot)\) is the link function for the chosen family and \(\gamma \in \mathbb{R}^K\) are scalar coefficients. The coefficient function on the original domain is

\[ \hat\beta(t) \;=\; \sum_{k=1}^{K} \hat\gamma_k\,\phi_k(t). \]

The GLM is fit by iteratively reweighted least squares (IRLS) with a maximum of max_iter iterations and convergence tolerance tol on the deviance change.

Link functions by family

Family	Link	\(g(\mu)\)	Notes
"gaussian"	identity	\(\mu\)	Reduces to ordinary functional linear model
"binomial"	logit	\(\log(\mu / (1-\mu))\)	Binary or proportion response in \([0, 1]\)
"poisson"	log	\(\log(\mu)\)	Count response; response must be non-negative
"gamma"	inverse (canonical)	\(1/\mu\)	Positive continuous response; NOT log-link

Gamma family uses the inverse canonical link, not log

fdars.regression.functional_glm with family="gamma" uses the inverse canonical link \(g(\mu) = 1/\mu\), not the log-link that R's glm(..., family=Gamma) defaults to. Results will differ from R if you assume a log-link. If you need a log-link for a Gamma response, transform the response before calling functional_glm with family="gaussian", or interpret the inverse-link coefficients appropriately.

AIC is not comparable to R glm() AIC

The AIC returned by functional_glm is computed from the score-space GLM log-likelihood, which treats the FPC scores as fixed predictors. This is not the same quantity as R's glm() AIC, which is based on the full-data likelihood. The AIC value here is useful for comparing models with different n_comp or family choices within fdars, but should not be compared numerically to AIC from R or other software.

Parameters

Parameter	Type	Default	Description
data	ndarray (n, m)	—	Functional data matrix; rows are observations
response	ndarray (n,)	—	Scalar response vector
family	str	"gaussian"	Exponential family: "gaussian", "binomial", "poisson", "gamma"
n_comp	int	3	Number of FPC components for the FPCA projection
scalar_covariates	ndarray (n, q) or None	None	Additional scalar predictors to include in the GLM alongside the FPC scores
max_iter	int	25	Maximum IRLS iterations
tol	float	1e-6	Convergence tolerance on deviance change

Returns a dict:

Key	Shape / Type	Description
"intercept"	float	GLM intercept on the link scale
"beta_t"	(m,)	Functional coefficient \(\hat\beta(t)\) on the original domain
"beta_se"	(m,)	Pointwise standard errors of \(\hat\beta(t)\)
"gamma"	(K,)	Score-space GLM coefficients (\(K\) = actual n_comp used)
"fitted_values"	(n,)	Fitted responses on the response scale (inverse-linked)
"linear_predictors"	(n,)	Linear predictors \(\hat\eta_i = \alpha + S_i^\top \hat\gamma\)
"ncomp"	int	Actual number of FPC components used
"coefficients"	(K+1,)	Full coefficient vector (intercept + score coefficients)
"std_errors"	(K+1,)	Standard errors of coefficients
"log_likelihood"	float	Log-likelihood of the fitted score-space GLM
"deviance"	float	Residual deviance
"iterations"	int	Number of IRLS iterations taken
"aic"	float	AIC from the score-space GLM (not comparable to R glm AIC — see note above)
"bic"	float	BIC from the score-space GLM
"family"	str	Family string echoed back

import numpy as np
from docs_fig import fig, render
import fdars.regression as reg

rng = np.random.default_rng(1)
n, m = 30, 60
t = np.linspace(0, 1, m)
X = np.array([np.sin(2 * np.pi * t * (1 + 0.3 * rng.normal())) + rng.normal(0, 0.1, m)
              for _ in range(n)])

# --- Family 1: Binomial (binary response) ---
logit_true = X @ np.sin(2 * np.pi * t) / m
prob_true = 1 / (1 + np.exp(-3 * logit_true))
y_bin = rng.binomial(1, prob_true).astype(float)

res_bin = reg.functional_glm(X, y_bin, family="binomial", n_comp=3)
beta_bin = np.asarray(res_bin["beta_t"])

# --- Family 2: Poisson (count response, y >= 0) ---
# True log-rate is a linear functional of X; exp() ensures y >= 0.
log_rate_true = X @ np.cos(2 * np.pi * t) / m
y_poi = rng.poisson(np.exp(1.5 * log_rate_true)).astype(float)

res_poi = reg.functional_glm(X, y_poi, family="poisson", n_comp=3)
beta_poi = np.asarray(res_poi["beta_t"])

f, (a0, a1) = fig(1, 2, figsize=(12.0, 3.8))

a0.plot(t, beta_bin, color="#3f51b5", lw=2.2)
a0.set(title="Functional GLM (binomial) — β(t)",
       xlabel="t", ylabel="β(t)")

a1.plot(t, beta_poi, color="#e8710a", lw=2.2)
a1.set(title="Functional GLM (poisson) — β(t)",
       xlabel="t", ylabel="β(t)")

print(render(f))
print(f"binomial  deviance={res_bin['deviance']:.3f}  aic={res_bin['aic']:.3f}  "
      f"family={res_bin['family']}")
print(f"poisson   deviance={res_poi['deviance']:.3f}  aic={res_poi['aic']:.3f}  "
      f"family={res_poi['family']}")
print("FDARS_FENCE_OK")

binomial deviance=34.332 aic=42.332 family=binomial poisson deviance=25.588 aic=78.876 family=poisson FDARS_FENCE_OK

References

1. Cardot, H., Ferraty, F., and Sarda, P. (1999). "Functional linear model." Statistics and Probability Letters, 45(1), 11–22. — FPC-score representation in functional regression.
2. James, G. M. (2002). "Generalized linear models with functional predictors." Journal of the Royal Statistical Society, Series B, 64(3), 411–432. — functional GLM methodology underlying functional_glm.
3. Wood, S. N. (2017). Generalized Additive Models: An Introduction with R, 2nd ed. CRC Press. — Chapter 3: GLM theory and IRLS algorithm.
4. Ramsay, J. O., and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer. — Chapter 15: principal components basis for functional regression.