Radial Basis Function Kernel Calculator

Calculator inputs

Enter vectors directly, paste datasets, or upload numeric CSV files.

Calculation mode

Machine-learning preset

Decimal precision

Vector X

Use commas, spaces, semicolons, or tabs.

Vector Z

Feature names

Missing-value handling

Gamma input mode

Gamma

Logarithmic gamma slider: 0.5

Sigma

Numerical tolerance

Feature scaling

Custom feature means

Custom standard deviations

Distance mode

Feature weights

Training dataset

Each line must contain the same feature count.

Test dataset

Training sample labels

Test sample labels

Numeric target labels for kernel alignment

Upload training CSV

A first header row is allowed. Maximum size: 2 MB.

Upload test CSV

Gamma comparison values

Log sweep minimum

Log sweep maximum

Log sweep points

Generate logarithmically spaced gamma comparison values

Maximum dataset rows

Normalise square kernel matrix

Centre square kernel matrix

Formula used

K(x, z) = exp(−γ ||x − z||²)

The squared Euclidean distance measures separation between vectors. Gamma controls how rapidly similarity decreases. Identical vectors normally return a value of one.

K(x, z) = exp(−||x − z||² / (2σ²)), γ = 1 / (2σ²)

How to use

Enter two vectors with matching feature counts.
Select gamma, sigma, auto, or scale mode.
Choose scaling and distance behaviour.
Paste datasets or upload CSV files.
Calculate and inspect matrices, diagnostics, and charts.
Copy, print, or export the results.

Example data

Input	Example	Purpose
Vector X	1, 2, 3	Reference sample
Vector Z	2, 3, 5	Comparison sample
Gamma	0.5	Kernel width control
Scaling	Standardisation	Balances feature magnitudes
Dataset row	4, 2, 1	Pairwise matrix sample

Machine-learning guidance

Small gammaProduces wider similarity regions and smoother boundaries.

Large gammaProduces narrow similarity regions and more local behaviour.

Feature scalingPrevents large-range features from dominating distances.

Model tuningEvaluate gamma through validation rather than fixed labels.

Frequently asked questions

What does an RBF kernel measure?

It converts distance into a similarity score. Nearby samples receive larger values. Distant samples approach zero.

Why do identical vectors return one?

Their squared distance equals zero. The exponent therefore equals zero. The exponential of zero is one.

How does gamma affect results?

Gamma controls the similarity radius. Larger values create narrower influence. Smaller values create broader influence.

What is sigma?

Sigma expresses Gaussian kernel width. It is inversely related to gamma. Larger sigma values create wider kernels.

Should features be scaled?

Scaling is usually important for RBF models. Large numeric ranges can dominate distances. Compare several scaling methods carefully.

What does gamma auto mean?

Auto uses one divided by feature count. It is a simple baseline. Validation may find a better value.

What does gamma scale mean?

Scale also considers dataset variance. It adapts gamma to feature spread. Constant data falls back safely.

Is every RBF matrix positive semidefinite?

Standard Gaussian RBF matrices should be positive semidefinite. Floating-point error can create tiny negative eigenvalues. Tolerance handles those cases.

Can this replace model validation?

No calculator replaces proper validation. Use cross-validation for model selection. Treat similarity labels as descriptive guidance.