Discrete Sampling of Extreme Events Modifies Their Statistics.

Area of Science:

• Statistical Physics
• Stochastic Processes
• Time Series Analysis

Background:

• Extreme value (EV) statistics are crucial for analyzing correlated systems in fields like meteorology and seismology.
• Understanding the impact of discrete sampling on these statistics is vital for accurate data interpretation.
• Continuous stochastic processes are often analyzed using discrete measurements, raising questions about data fidelity.

Purpose of the Study:

• To investigate how discrete sampling affects the extreme value (EV) distribution of correlated stochastic processes.
• To determine the conditions under which discrete sampling alters EV statistics compared to continuous data.
• To analyze the influence of potential landscapes and sampling rates on EV distribution convergence.

Main Methods:

• Modeling correlated random variables using Langevin dynamics for a particle in a potential field.
• Analyzing the equilibrium measure of the stochastic process.
• Comparing EV distributions of discretely sampled data with the full continuous dataset.
• Investigating processes whose equilibrium measures belong to Gumbel, Fréchet, and Weibull attractors.

Main Results:

• For potentials growing faster than linearly, discrete sampling causes EV distributions to diverge from continuous data, converging to that of independent and identically distributed variables.
• For sublinear potentials, the long-time limit of discrete sampling yields the same EV statistics as continuous data.
• The study demonstrates that EV statistics are highly sensitive to the data sampling rate.

Conclusions:

• Discrete sampling can fundamentally alter extreme value statistics in correlated systems, depending on the system's potential landscape.
• The choice of sampling rate is critical for accurately characterizing extreme events in discretely measured continuous processes.
• These findings have significant implications for data analysis in various scientific disciplines relying on extreme value theory.