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These model patterns are compared to the convection predicted by mapping the magnetopause electric field to the ionosphere using the Moscow State University's (MSU) paraboloid model of the magnetosphere.
Thus, each time-series profile is represented as the sum of sine waves with various phases and amplitudes, and the magnitudes of the contributing model patterns are a unique "footprint" for each profile.
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The common feature of both model patterns is the presence of the negative disturbances in the northern and southern polar caps and the positive disturbances at equator.
The model patterns were presented within the box at the model area.
The model pattern was shifted in time to find the phase at which the sum of squares of residuals was minimized.
A common modelling pattern is the use of matrices of decision variables.
The model derived expression patterns are termed model_target_exp.
For the one-parameter models, climate response patterns are consistent with the results of the correlation analysis (Additional file 1: Figures S3 S6).
The use of random seasonal components is an advantage over traditional fixed effect models, since temporal patterns are not assumed to repeat precisely the same way.
Whether this variation is the result of unmeasured soil environmental variables or null model (random) patterns is a subject for further research.
When considering the results from best-fit models or from the 95% CI set of models, several patterns were discerned.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com