autores contribuíram igualmente na escrita e elaboração do presente trabalho.Físicas e físicos têm começado a trabalhar em áreas onde é necessária a análise de sinais ruidosos. Nessas áreas, tais como a Economia, a Neurociência e a Física, a noção de causalidade deve ser interpretada como uma medida estatística. Introduzimos ao leitor leigo a causalidade de Granger entre duas séries temporais e ilustramos como calculá-la: um sinal X "Granger-causa" Y se a observação do passado de X aumenta a previsibilidade do futuro de Y em comparação com o que é possível prever apenas pela observação do passado de Y . Em outras palavras, para haver causalidade de Granger entre dois sinais basta que a informação do passado de um melhore a previsão do futuro de outro, mesmo na ausência de mecanismos físicos de interação. Apresentamos a derivação da causalidade de Granger nos domínios do tempo e da frequência e damos exemplos numéricos através de um método nãoparamétrico no domínio da frequência. Métodos paramétricos são abordados no Apêndice. Discutimos limitações e aplicações desse método e outras alternativas para medir causalidade. Palavras-chave: Causalidade de Granger, processo autoregressivo, causalidade de Granger condicional, estimação não-paramétrica Physicists are starting to work in areas where noisy signal analysis is required. In these fields, such as Economics, Neuroscience, and Physics, the notion of causality should be interpreted as a statistical measure. We introduce to the lay reader the Granger causality between two time series and illustrate ways of calculating it: a signal X "Granger-causes" a signal Y if the observation of the past of X increases the predictability of the future of Y when compared to the same prediction done with the past of Y alone. In other words, for Granger causality between two quantities it suffices that information extracted from the past of one of them improves the forecast of the future of the other, even in the absence of any physical mechanism of interaction. We present derivations of the Granger causality measure in the time and frequency domains and give numerical examples using a non-parametric estimation method in the frequency domain. Parametric methods are addressed in the Appendix. We discuss the limitations and applications of this method and other alternatives to measure causality.
We propose and demonstrate an original geometric argument for the ancient Babylonian square root method, which is analyzed and compared to the Newton-Raphson method. Based on simple geometry and algebraic analysis the former original iterated map is derived and reinterpreted. Time series, fixed points, stability analysis and convergence schemes are studied and compared for both methods, in the approach of discrete dynamical systems.
In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.
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