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Fig. 9

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Correlation between the halo mass M200c and t1/2 across different RF and CNN models. The first and second rows show the correlation trends for the six RF models discussed in Section 3, while the third row corresponds to the three CNN models described in Section 4. In each panel, the shaded red region represents the 16th-84th percentile range of the ML predictions, and the shaded gray region shows the same range for the actual simulation values. The solid red and dotted black lines indicate the median t1/2 in logarithmic mass bins for the ML models and the simulation, respectively. Pearson correlation coefficients are annotated to quantify the strength of the relationship between M200c and the predicted formation times. The solid green line gives the t1/2 - M200c relation based on formation time values calculated using the analytical formalism based on EPS theory as given in Correa et al. (2015).

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