kernel_ridge – Kernel Ridge Regression – closed-form non-linear ridge in the dual.#

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Operational op under axis family, sub-layer L4_A_model_selection, layer l4. Standalone callable: mf.functions.kernel_ridge_fit.

Function signature#

mf.functions.kernel_ridge_fit(
    X: np.ndarray | pd.DataFrame,
    y: np.ndarray | pd.Series,
) -> KernelRidgeFitResult

Parameters#

name

type

default

constraint

description

X

`np.ndarray

pd.DataFrame`

—

—

y

`np.ndarray

pd.Series`

—

—

Returns#

KernelRidgeFitResult — frozen dataclass with fit results.

Attribute

Type

Description

.alpha

float

Ridge regularisation strength.

.kernel

str

Kernel type (e.g. rbf, linear).

.n_features_in_

int

Number of input features.

.predict(X)

np.ndarray

Predictions for new data X, shape (n_samples,).

.summary()

str

Table: alpha, kernel, feature count.

Behavior#

Ridge regression with a non-linear kernel: ŷ(x) = Σ_i α_i K(x, x_i) + b where the dual coefficients α = (K + λ I)⁻¹ y are recovered in closed form. Operational v0.9.1 dev-stage v0.9.0F (audit-fix). Surfaces as a first-class L4 family because Coulombe / Surprenant / Leroux / Stevanovic (2022 JAE) ‘How is Machine Learning Useful for Macroeconomic Forecasting?’ Eq. 16 / §3.1.1 uses KRR as the headline non-linearity feature in the macro horse race.

Tunable params: alpha (= ridge penalty λ; default 1.0); kernel (‘rbf’ default / ‘linear’ / ‘poly’ / ‘sigmoid’ / ‘laplacian’ / ‘chi2’ – any sklearn-supported kernel); gamma (RBF bandwidth, default sklearn auto = 1/n_features); degree (poly kernel only, default 3); coef0 (poly / sigmoid, default 1.0).

Distinct from svr_rbf (ε-insensitive loss, sparsity in support vectors) and from ridge (linear). The dual representation also pairs with the L7 dual_decomposition op for kernel-weighted training-target attribution.

When to use

Non-linear macro forecasting baselines; KRR vs SVR-RBF / RF ablations; replicating Coulombe et al. (2022) Feature 1 nonlinearity test.

In recipe context#

Set params.family = "kernel_ridge" in the relevant layer to activate this op within a recipe:

# Layer L4 recipe fragment
params:
  family: kernel_ridge

References#

  • macroforecast design Part 2, L4: ‘forecasting model is the layer where every authoring iteration ends – pick family, tune, repeat.’

  • Saunders, Gammerman & Vovk (1998) ‘Ridge Regression Learning Algorithm in Dual Variables’, ICML.

  • Coulombe, Leroux, Stevanovic & Surprenant (2022) ‘How is Machine Learning Useful for Macroeconomic Forecasting?’, Journal of Applied Econometrics 37(5): 920-964 – Eq. 16 + §3.1.1.