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This may be overcome in the numerical scheme by regularisation: replacing the condition (20) with (c=c_{0} tilde{x})), where (c_{0} tilde {x})) is a function which is compatible with the boundary condition (19) at (tilde{x}=0).
Finally, some methods aim to reduce the amount of parameter learning to avoid overfitting, achieved by regularisation techniques modifying the training objective function or limiting the parameter learning cycles (Duda et al, 2001).
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Regression under these ill-posed circumstances is typically handled by a regularisation parameter, which shrinks the regression coefficients by penalising their size.
In the case where experimental duration is short, and small covariance windows are therefore used, we show that accurate power estimation can be achieved by matrix regularisation.
This could potentially be reduced by increasing the regularisation weight.
This could potentially be reduced by decreasing the regularisation weight.
It is possible that the performance of GN and NNLS could be improved by tuning the regularisation parameters for each instance of noise separately, but this is not practical in an experimental situation as one does not know the true image.
The parameter λ in the LM method is the damping factor, which changes at each iteration Full inversion of the smoothing step is expensive, so we replace the regularisation step by the semi-implicit AOS method (Eq. (20)), which can be solved efficiently using the Thomas algorithm.
As we try to estimate a core field model with a relatively complex behaviour in time up to SH degree 18, the model is not fully resolved by the data and needs regularisation.
The regularisation was achieved by minimising the integral over the sphere of the strength of the field model component perpendicular to main field direction i.e. for the WDMAM the CM4-1990.
Note also the regions of nightside data (eclipse), the circled areas to the left of the figures, which generally show less disturbance; this is not imposed by the model or any regularisation; rather, it is simply a result of the data itself and thus another indicator of the ability of the model to describe the observed disturbances.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com