TY - JOUR

T1 - Failure of maximum likelihood methods for chaotic dynamical systems

AU - Judd, Kevin

PY - 2007

Y1 - 2007

N2 - The maximum likelihood method is a basic statistical technique for estimating parameters and variables, and is the starting point for many more sophisticated methods, like Bayesian methods. This paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system, because there are trajectories that appear to be far more (infinitely more) likely than truth. This failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems. The reason for the failure is rather simple; in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed, and hence their likelihood dominates truth. The residuals of these truth-dominating trajectories are not consistent with the noise distribution; they would typically have too small standard deviation and many outliers, and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based.

AB - The maximum likelihood method is a basic statistical technique for estimating parameters and variables, and is the starting point for many more sophisticated methods, like Bayesian methods. This paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system, because there are trajectories that appear to be far more (infinitely more) likely than truth. This failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems. The reason for the failure is rather simple; in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed, and hence their likelihood dominates truth. The residuals of these truth-dominating trajectories are not consistent with the noise distribution; they would typically have too small standard deviation and many outliers, and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based.

U2 - 10.1103/PhysRevE.75.036210

DO - 10.1103/PhysRevE.75.036210

M3 - Article

SN - 1539-3755

VL - 75

SP - online - approx 5-20pp

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

IS - 3

ER -