Concurrent (Varying-Coefficient) Regression

Concurrent regression — also called the varying-coefficient model — extends functional regression by letting each predictor's effect vary smoothly over the domain. Where ordinary linear regression assigns a single scalar coefficient to each predictor, concurrent regression assigns an entire coefficient function \(\beta_k(t)\). At each point \(t\) the model behaves like a local ordinary regression, but the coefficients change fluidly as \(t\) progresses.

fdars.regression.concurrent_regression estimates one smooth coefficient curve per predictor using local kernel regression: the bandwidth controls how quickly the coefficients are allowed to change, and the kernel controls the weighting of neighboring time points.

Theory

Given \(p\) functional predictors \(X^{(1)}(t), \dots, X^{(p)}(t)\) and a functional response \(Y(t)\), all observed at the same \(m\) grid points for \(n\) subjects, the model is

\[ Y_i(t) \;=\; \beta_0(t) \;+\; \sum_{k=1}^{p} \beta_k(t)\,X_i^{(k)}(t) \;+\; \varepsilon_i(t), \]

where \(\beta_0(t)\) is a time-varying intercept, \(\beta_1(t), \dots, \beta_p(t)\) are time-varying coefficient functions, and \(\varepsilon_i(t)\) is a zero-mean error process. At each grid point \(t_j\) the function solves a weighted least-squares system with kernel weights centred at \(t_j\):

\[ K_h(t_j - t_s) \;=\; K\!\left(\frac{t_j - t_s}{h}\right), \]

where \(h\) is the bandwidth and \(K\) is the chosen kernel ("gaussian", "epanechnikov", or "tricube"). The resulting \(\hat\beta(t)\) is a smooth estimate of how the coefficient evolves over the domain.

Parameters

Parameter	Type	Default	Description
predictors	list[ndarray (n, m)]	—	List of \(p\) predictor matrices; each row is one subject's curve
response	ndarray (n, m)	—	Functional response matrix
argvals	ndarray (m,) or None	None	Evaluation grid; None → uniform grid on \([0, 1]\)
bandwidth	float	0.2	Kernel bandwidth; must be positive
kernel	str	"gaussian"	Kernel: "gaussian", "epanechnikov", or "tricube"

Returns a dict:

Key	Shape	Description
"beta_curve"	(p, m)	Time-varying coefficient curves — one row per predictor
"intercept"	(m,)	Time-varying intercept function \(\hat\beta_0(t)\)
"fitted"	(n, m)	Fitted response curves \(\hat Y_i(t)\)
"residuals"	(n, m)	Residual curves \(Y_i(t) - \hat Y_i(t)\)
"argvals"	(m,)	Evaluation grid used (echoed back)

beta_curve shape: (p, m) — predictors × grid

res["beta_curve"] has shape (p, m), where p = len(predictors) and m is the number of grid points. This is not (n, m) — confusing the two is the most common transposition error when working with this function. Row k of beta_curve is the coefficient curve for the k-th predictor, evaluated at every grid point.

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

rng = np.random.default_rng(0)
n, m = 20, 50
t = np.linspace(0, 1, m)
# Two synthetic predictor curves + a response
x1 = np.array([np.sin(2 * np.pi * t) + rng.normal(0, 0.1, m) for _ in range(n)])
x2 = np.array([np.cos(2 * np.pi * t) + rng.normal(0, 0.1, m) for _ in range(n)])
y  = x1 * np.sin(2 * np.pi * t) + x2 * 0.5 + rng.normal(0, 0.05, (n, m))

res = reg.concurrent_regression([x1, x2], y, t)
beta = np.asarray(res["beta_curve"])  # shape (2, m) — p=2 predictors

f, ax = fig(figsize=(8.0, 3.8))
ax.plot(t, beta[0], color="#3f51b5", lw=2.2, label="β₁(t) — sin predictor")
ax.plot(t, beta[1], color="#e8710a", lw=2.2, label="β₂(t) — cos predictor")
ax.set(title="Concurrent regression — estimated coefficient curves",
       xlabel="t", ylabel="β(t)")
ax.legend(fontsize=9)
print(render(f))
print(f"beta_curve shape: {beta.shape}  (p=2 predictors × m={m} grid points)")
print("FDARS_FENCE_OK")

beta_curve shape: (2, 50) (p=2 predictors × m=50 grid points) FDARS_FENCE_OK

Caveats and interpretation

Bandwidth selection

The bandwidth h is the most consequential tuning choice. There is no built-in cross-validation in concurrent_regression — the user supplies the bandwidth directly.

Bandwidth trades bias for variance

• Small bandwidth (e.g. bandwidth=0.05): the kernel weights few neighbours at each \(t\), so the estimated \(\hat\beta(t)\) tracks local fluctuations closely — resulting in wiggly, high-variance coefficient curves. On small samples this produces noisy estimates that are hard to interpret.
• Large bandwidth (e.g. bandwidth=0.5): the kernel pools many neighbours, smoothing over local structure — resulting in over-smoothed, biased estimates that miss true variation in \(\beta(t)\).
• Practical starting point: try bandwidth=0.2 (the default) on a normalised argvals grid and inspect the shape of res["beta_curve"]. If the curves look spiky, widen; if they look flat when you expect variation, narrow.
• Manual CV: to select bandwidth systematically, evaluate held-out prediction error (functional_mse) over a grid of candidate bandwidths. fdars does not provide fregre_np_cv for the concurrent model, so this must be coded manually.

Kernel choice

The three kernels differ in their support and decay:

Kernel	Support	Decay	When to prefer
"gaussian" (default)	infinite (global)	exponential	Smooth data; all observations contribute with decaying weight
"epanechnikov"	compact \([\!-1, 1\!]\)	quadratic	Faster computation; observations beyond bandwidth contribute nothing
"tricube"	compact \([\!-1, 1\!]\)	cubic	Similar to Epanechnikov; slightly smoother drop-off

For most functional datasets with smooth \(\beta(t)\), the choice of kernel has less impact than the choice of bandwidth. Start with "gaussian".

Model scope: local-at-each-t, not global

Concurrent regression is a varying-coefficient (local) model — it does not fit a single global relationship between predictor and response. At each grid point \(t_j\) it runs an independent weighted regression using only the kernel-weighted neighbourhood of \(t_j\). This means:

• The model can capture coefficient functions that reverse sign or change shape over the domain.
• It cannot borrow information across widely separated regions of \(t\) (unlike a basis-regression approach).
• Residuals \(\varepsilon_i(t)\) are not assumed uncorrelated across \(t\) — the model makes no statement about temporal dependence of errors.

References

1. Hastie, T., and Tibshirani, R. (1993). "Varying-coefficient models." Journal of the Royal Statistical Society, Series B, 55(4), 757–796. — foundational paper on the varying-coefficient model.
2. Fan, J., and Zhang, W. (1999). "Statistical estimation in varying coefficient models." Annals of Statistics, 27(5), 1491–1518. — local polynomial estimation of time-varying coefficients.
3. Ramsay, J. O., and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer. — Chapter 14: concurrent regression and the functional linear model.