KernelBasis

Kernel feature transformer against the training basis

from _api_doc_utils import *

1 Where it fits

Group: Transforms

KernelBasis stores a training design and transforms new rows into kernel similarities against that basis. For a Gaussian kernel, the transformed feature for training row \(j\) is

\[ \phi_j(x) = \exp\{-\|x-x_j\|^2/(2h^2)\}. \]

The resulting feature matrix can be fed into any downstream regression estimator.

2 Python API

Constructor: cm.KernelBasis

Use KernelBasis(kernel='gaussian', bandwidth=0.5, coef0=1.0, degree=2.0). fit(x) stores training rows. transform(x) returns kernel features. summary() reports the chosen kernel and basis dimensions.

print(inspect.signature(cm.KernelBasis))
(kernel='gaussian', bandwidth=0.5, coef0=1.0, degree=2.0)
cls = cm.KernelBasis
display(HTML(html_table(["Public method"], public_methods(cls))))
Public method
fit(self, /, x)
fit_transform(self, /, x)
summary(self, /)
transform(self, /, x)

3 Minimal example

rng=np.random.default_rng(23)
x=rng.normal(size=(80,2)); y=np.sin(x[:,0])+rng.normal(scale=.1,size=80)
basis=cm.KernelBasis(kernel="gaussian", bandwidth=0.8); z=basis.fit_transform(x)
reg=cm.Ridge(penalty=0.1); reg.fit(z,y)
print(basis.summary())
print(reg.predict(basis.transform(x[:3])))
{'kernel': 'gaussian', 'n_train': 80, 'n_features': 2, 'bandwidth': 0.8, 'coef0': 1.0, 'degree': 2.0, 'diagonal': array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])}
[ 0.53255368 -0.02676873  0.27494531]

4 summary() contract

The table below is generated by fitting the live class in this repository and then inspecting summary(). Shapes are shown because most values are plain NumPy arrays or scalars.

rng=np.random.default_rng(123); x=rng.normal(size=(50,2)); model=cm.KernelBasis(kernel='gaussian',bandwidth=.8); model.fit(x)
summary = model.summary()
display(HTML(html_table(["summary() key", "shape"], summary_shape_rows(summary))))
summary() key shape
kernel ()
n_train ()
n_features ()
bandwidth ()
coef0 ()
degree ()
diagonal (50,)