What Is the Tikhonov Regularization (Ridge Regression) Parameter Calculator?
Tikhonov regularization introduces a trade-off parameter lambda to stabilize solutions of Fredholm integral equations and collinear regressions.
It modifies the Moore-Penrose pseudo-inverse into a damped filter.
How Does the Tikhonov Regularization (Ridge Regression) Parameter Calculator Work?
Sums the data residual squared error and the weighted solution energy penalty.
Evaluates the spectral filter factor fi = si^2 / (si^2 + lambda).
Diagnoses whether the current lambda produces under-regularized noise amplification or over-smoothed bias.
Tikhonov Regularization (Ridge Regression) Parameter Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
Tikhonov objective function balancing data fidelity against solution roughness or magnitude.
How to Use the Tikhonov Regularization (Ridge Regression) Parameter Calculator
- Enter the characteristic singular value or eigenvalue square.
- Specify the trial regularization parameter lambda.
- Provide current residual and solution norm squares from optimization iterations.
Step-by-Step Example Calculation
Geophysical Gravity Inversion
Input Values:
Understanding Your Result
A filter factor close to 1.0 indicates full signal transmission without damping.
Filter factors below 0.5 indicate significant attenuation of unstable spectral components.
Factors That Affect the Result
- Noise level: Higher noise in measurement vector b mandates larger lambda to maintain numerical stability.
- L-curve criterion: The optimal lambda lies at the corner of the log(||x||) vs log(||Ax-b||) L-curve.
When Should You Use This Calculator?
- Solving ill-posed tomography, deconvolution, seismic inversion, and biomedical image reconstruction problems.
- Training ridge regression machine learning models under severe feature multicollinearity.
Assumptions & Limitations
- Assumes quadratic L2 penalty; does not enforce sparsity (unlike L1 Lasso).
- Assumes linear forward operator A.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Standard mathematical Tikhonov formulation exact for linear least-squares systems.