Abstract
We exploit ideas of nonlinear dynamics and statistical physics in a complex nondeterministic dynamical setting using the Ruelle-Takens embedding. We present some new insights on the quality of the prediction in the laminar regime and we exhibit the data collapse of the predicted relative first difference fluctuations to the universal Bramwell-Hodsworth-Pinton distribution. Hence, the nearest neighbour method of prediction acts as a filter that does not eliminate the randomness, but exhibits its universal character.

Abstract
We study the European river Danube and the South American river Negro daily water levels. We present a fit for the Negro daily water level period and standard deviation. Unexpectedly, we discover that the river Negro and Danube are mirror rivers in the sense that the daily water levels fluctuations histograms are close to the universal non-parametric BHP and reversed BHP, respectively. Hence, the probability of a certain positive fluctuation range in the river Negro is, approximately, equal to the probability of the corresponding symmetric negative fluctuation range in the river Danube.

Abstract
We consider the alpha re-scaled Standard & Poor's 100 (SP100) daily index positive returns r(t)(alpha) and negative returns (-r(t))(alpha) that we call, after normalization, the alpha positive fluctuations and alpha negative fluctuations, respectively. We use the Kolmogorov-Smirnov statistical test as a method to find the values of alpha that optimize the data collapse of the histogram of the alpha fluctuations with the truncated Bramwell-Holdsworth-Pinton (BHP) probability density function (pdf) and the truncated generalized log-normal pdf f(LN) that best approximates the truncated BHP pdf. The optimal parameters we found are alpha(+)(BHP) = 0.52, alpha(-)(BHP) = 0.48, alpha(+)(LN) = 0.52 and alpha(-)(LN) = 0.50. Using the optimal alpha's, we compute analytical approximations of the probability distributions of the normalized positive and negative SP100 index daily returns r(t). Since the BHP pdf appears in several other dissimilar phenomena, our result reveals a universal feature of the stock exchange markets.

Abstract
The Box-Cox (BC) transformation is widely used in data analysis for achieving approximate normality in the transformed scale. The transformation is only possible for non-negative data. This positiveness requirement implies a truncation to the distribution on the transformed scale and the distribution in the transformed scale is truncated normal. This fact has consequences for the estimation of the parameters specially if the truncated probability is high. In the seminal paper Box and Cox proposed to estimate parameters using the normal distribution which in practice means to ignore any consequences of the truncation on the estimation process. In this work we present the framework for exact likelihood estimation on the PN distribution to which we call method m1 and how to calculate the parameters estimates using consistent estimators. We also present a pseudo-Likelihood function for the same model not taking into account truncation and allowing to replace parameters µ and s for their estimates. We call m2 to this estimation method. We conclude that for cases where the truncated probability is low both methods give good estimation results. However for larger values of the truncated probability the m2 method does not present the same efficiency. © 2024 American Institute of Physics Inc.. All rights reserved.