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We investigate which physical properties are most predictive of the position of local star forming galaxies on the BPT diagrams, by means of different Machine Learning (ML) algorithms. Exploiting the large statistics from the Sloan Digital Sky Survey (SDSS), we define a framework in which the deviation of star-forming galaxies from their median sequence can be described in terms of the relative variations in a variety of observational parameters. We train artificial neural networks (ANN) and random forest (RF) trees to predict whether galaxies are offset above or below the sequence (via classification), and to estimate the exact magnitude of the offset itself (via regression). We find, with high significance, that parameters primarily associated to variations in the nitrogen-over-oxygen abundance ratio (N/O) are the most predictive for the [N ii]-BPT diagram, whereas properties related to star formation (like variations in SFR or EW(H α)) perform better in the [S ii]-BPT diagram. We interpret the former as a reflection of the N/O–O/H relationship for local galaxies, while the latter as primarily tracing the variation in the effective size of the S+ emitting region, which directly impacts the [S ii] emission lines. This analysis paves the way to assess to what extent the physics shaping local BPT diagrams is also responsible for the offsets seen in high redshift galaxies or, instead, whether a different framework or even different mechanisms need to be invoked.

The relationships between stellar mass, gas-phase metallicity and star-formation rate (i.e. the mass–metallicity, MZR, and the fundamental metallicity relation, FMR) in the local Universe are revisited by fully anchoring the metallicity determination for SDSS galaxies on the Te abundance scale defined exploiting the strong-line metallicity calibrations presented by Curti et al. Self-consistent metallicity measurements allow a more unbiased assessment of the scaling relations involving M, Z and SFR, which provide powerful constraints for the chemical evolution models. We parametrize the MZR with a new functional form that allows us to better characterize the turnover mass. The slope and saturation metallicity are in good agreement with previous determinations of the MZR based on the Te method, while showing significantly lower normalization compared to those based on photoionization models. The Z–SFR dependence at fixed stellar mass is also investigated, being particularly evident for highly star-forming galaxies, where the scatter in metallicity is reduced up to a factor of ${\sim}30{{\ \rm per\ cent}}$. A new parametrization of the FMR is given by explicitly introducing the SFR dependence of the turnover mass into the MZR. The residual scatter in metallicity for the global galaxy population around the new FMR is 0.054 dex. The new FMR presented in this work represents a useful local benchmark to compare theoretical predictions and observational studies (of both local and high-redshift galaxies) whose metallicity measurements are tied to the abundance scale defined by the Te method, hence allowing proper assessment of its evolution with cosmic time.