We investigate timing and spectral characteristics of the transient X-ray pulsar 2S 1417$-$624 during its 2018 outburst with emph{NICER} follow up observations. We describe the spectra with high-energy cut-off and partial covering fraction absortion (PCFA) model and present flux-dependent spectral changes of the source during the 2018 outburst. Utilizing the correlation-mode switching of the spectral model parameters, we confirm the previously reported sub-critical to critical regime transitions and we argue that secondary transition from the gas-dominated to the radiation pressure-dominated disc do not lead to significant spectral changes below 12 keV. Using the existing accretion theories, we model the spin frequency evolution of 2S 1417$-$624 and investigate the noise processes of a transient X-ray pulsar for the first time using both polynomial and luminosity-dependent models for the spin frequency evolution. For the first model, the power density spectrum of the torque fluctuations indicate that the source exhibits red noise component ($Gamma sim -2$) within the timescales of outburst duration which is typical for disc-fed systems. On the other hand, the noise spectrum tends to be white on longer timescales with high timing noise level that indicates an ongoing accretion process in between outburst episodes. For the second model, most of the red noise component is eliminated and the noise spectrum is found to be consistent with a white noise structure observed in wind-fed systems.
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