ML EPS REG

ML_EPS_REG = [real]
Default: ML_EPS_REG = 1E-14 

Description: Initial value for the threshold of the eigenvalues of the covariance matrix in the evidence approximation.

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This threshold is used to determine which eigenvalues $\lambda _{k}$ of the covariance matrix $\mathbf {\Phi } ^{\mathrm {T} }\mathbf {\Phi } /\sigma _{\mathrm {v} }^{2}$ are used in the optimization of the regularization parameters $\sigma _{\mathrm {w} }^{2}$ and $\sigma _{\mathrm {v} }^{2}$ determined by the following equations

$\sigma _{\mathrm {w} }^{2}={\frac {|\mathbf {\bar {w}} |^{2}}{\gamma }},$

$\sigma _{\mathrm {v} }^{2}={\frac {|\mathbf {T} -\mathbf {\phi } \mathbf {\bar {w}} |^{2}}{M-\gamma }},$

$\gamma =\sum \limits _{k=1}^{N_{\mathrm {B} }}{\frac {\lambda _{k}}{\lambda _{k}+1/\sigma _{\mathrm {w} }^{2}}}$.

All eigenvalues satisfying $\lambda _{i}/\lambda _{\mathrm {max} }$ > ML_EPS_REG are included in the above equations, whereas the other eigenvalues are disregarded.

The smaller the value of ML_EPS_REG, the smaller the error of the fit can potentially become. But at the same time, the danger of over fitting potentially increases. We determined empirically that the default of ML_EPS_REG=1E-14 is fairly safe. If a user desires higher accuracy of the force field, the user can try to set ML_EPS_REG=1E-15 (lower values are not recommended), but the user should very carefully monitor the convergence of the calculations as described below.

If at any point during iterating, the approximation of the square of the quadratic norm of errors (eight column of REGR/REGRF in ML_LOGFILE) becomes too large (more than 1.2 times larger than in previous iterations), then ML_EPS_REG is automatically doubled. Whenever the regression doesn't converge within 10 steps, ML_EPS_REG is increased by a factor of 4. This counter is also reset whenever ML_EPS_REG is changed. The maximum allowed iteration depths is 50 (the iteration number is the second entry of REGR/REGRF in ML_LOGFILE). When this value is reached, no force field is created and there is most likely something seriously wrong in the calculation.

The seventh entry of REGR/REGRF in the ML_LOGFILE shows the ratio of the regularization ($\sigma _{v}^{2}/\sigma _{w}^{2}$) and the largest eigenvalue. Usually this number is a number with many varying digits. If this number becomes a "well rounded" number (e.g. 1.00000000E-14), this is an indication that the cap for the current ML_EPS_REG is reached. That means that regularization becomes crucial.

Related Tags and Sections

ML_LMLFF, ML_IALGO_LINREG, ML_IREG, ML_SIGV0, ML_SIGW0

Examples that use this tag

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