A perspective is taken on the intangible complexity of economic and social systems by investigating the dynamical processes producing, storing and transmitting information in financial time series. An extensive analysis based on the moving average cluster entropy approach has evidenced market and horizon dependence in highest-frequency data of real world financial assets. The behavior is scrutinized by applying the moving average cluster entropy approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). An extensive set of series is generated with a broad range of values of the Hurst exponent H and of the autoregressive, differencing and moving average parameters p , d , q . A systematic relation between moving average cluster entropy and long-range correlation parameters H, d is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter d ≃ 0.05 , d ≃ 0.15 and d ≃ 0.25 are consistent with moving average cluster entropy results obtained in time series of DJIA, S&P500 and NASDAQ. The findings clearly point to a variability of price returns, consistently with a price dynamics involving multiple temporal scales and, thus, short- and long-run volatility components. An important aspect of the proposed approach is the ability to capture detailed horizon dependence over relatively short horizons (one to twelve months) and thus its relevance to define risk analysis indices.

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Accurate estimates of the urban fractal dimension D f are obtained by implementing the detrended moving average algorithm on high-resolution multi-spectral satellite images from the WorldView2 (WV2) database covering the largest European cities. Fractal dimension D f varies between 1.65 and 1.90 with high values for highly urbanised urban sectors and low ones for suburban and peripheral ones. Based on recently proposed models, the values of the fractal dimension D f are checked against the exponents β s and β i of the scaling law Y ∼ N β , respectively for socio-economic and infrastructural variables Y, with N the population size. The exponents β s and β i are traditionally derived as if cities were zero-dimensional objects, with the relevant feature Y related to a single homogeneous population value N, thus neglecting the microscopic heterogeneity of the urban structure. Our findings go beyond this limit. High sensitive and repeatable satellite records yield robust local estimates of the urban scaling exponents. Furthermore, the work discusses how to discriminate among different scaling theories, shedding light on the debated issue of scaling phenomena contradictory perspectives and pave paths to a more systematic adoption of the complex system science methods to urban landscape analysis.

Despite half a century of research, there is still no general agreement about the optimal approach to build a robust multi-period portfolio. We address this question by proposing the detrended cluster entropy approach to estimate the weights of a portfolio of high-frequency market indices. The information measure gathered from the markets produces reliable estimates of the weights at varying temporal horizons. The portfolio exhibits a high level of diversity, robustness and stability as not affected by the drawbacks of traditional mean-variance approaches.