Platzer Paul, Chapron Bertrand
Laboratoire d'Océanographie Physique et Spatiale (LOPS), Ifremer, 1625 route de Sainte-Anne, 29280 Plouzané, Bretagne France.
J Stat Phys. 2025;192(2):34. doi: 10.1007/s10955-025-03416-x. Epub 2025 Feb 15.
For any multi-fractal dynamical system, a precise estimate of the local dimension is essential to infer variations in its number of degrees of freedom. Following extreme value theory, a local dimension may be estimated from the distributions of pairwise distances within the dataset. For absolutely continuous random variables and in the absence of zeros and singularities, the theoretical value of this local dimension is constant and equals the phase-space dimension. However, due to uneven sampling across the dataset, practical estimations of the local dimension may diverge from this theoretical value, depending on both the phase-space dimension and the position at which the dimension is estimated. To explore such variations of the estimated local dimension of absolutely continuous random variables, approximate analytical expressions are derived and further assessed in numerical experiments. These variations are expressed as a function of 1. the random variables' probability density function, 2. the threshold used to compute the local dimension, and 3. the phase-space dimension. Largest deviations are anticipated when the probability density function has a low absolute value, and a high absolute value of its Laplacian. Numerical simulations of random variables of dimension 1 to 30 allow to assess the validity of the approximate analytical expressions. These effects may become important for systems of moderately-high dimension and in case of limited-size datasets. We suggest to take into account this source of local variation of dimension estimates in future studies of empirical data. Implications for weather regimes are discussed.
对于任何多重分形动力系统而言，精确估计局部维度对于推断其自由度数量的变化至关重要。根据极值理论，可以从数据集中成对距离的分布来估计局部维度。对于绝对连续随机变量，并且在不存在零值和奇点的情况下，该局部维度的理论值是恒定的，并且等于相空间维度。然而，由于数据集采样不均匀，局部维度的实际估计可能会偏离该理论值，这取决于相空间维度和估计维度的位置。为了探索绝对连续随机变量估计局部维度的这种变化，推导了近似解析表达式，并在数值实验中进一步评估。这些变化表示为以下三个因素的函数：1. 随机变量的概率密度函数；2. 用于计算局部维度的阈值；3. 相空间维度。当概率密度函数的绝对值较低且其拉普拉斯算子的绝对值较高时，预计会出现最大偏差。对1到30维随机变量的数值模拟可以评估近似解析表达式的有效性。对于中等高维系统和有限大小数据集的情况，这些影响可能会变得很重要。我们建议在未来对经验数据的研究中考虑这种维度估计局部变化的来源。并讨论了其对天气状况的影响。
