Approximate Entropy and Sample Entropy are two algorithms for determining the regularity of series of data based on the existence of patterns. Despite their similarities, the theoretical ideas behind those techniques are different but usually ignored. This paper aims to be a complete guideline of the theory and application of the algorithms, intended to explain their characteristics in detail to researchers from different fields. While initially developed for physiological applications, both algorithms have been used in other fields such as medicine, telecommunications, economics or Earth sciences. In this paper, we explain the theoretical aspects involving Information Theory and Chaos Theory, provide simple source codes for their computation, and illustrate the techniques with a step by step example of how to use the algorithms properly. This paper is not intended to be an exhaustive review of all previous applications of the algorithms but rather a comprehensive tutorial where no previous knowledge is required to understand the methodology.

Randomness has been mathematically defined and quantified in time series using algorithms such as Approximate Entropy (ApEn). Even though ApEn is independent of any model and can be used with any time series, as the markets have different statistical values, it cannot be applied directly to make comparisons between series of financial data. In this paper, we develop further the use of Approximate Entropy to quantify the existence of patterns in evolving data series, defining a measure to allow comparisons between time series and epochs using a maximum entropy approach. We apply the methodology to the stock markets as an example of its application, showing that the number of patterns changed for the six analyzed markets depending on the economic situation, in agreement with the Adaptive Markets Hypothesis.

The energy balance of the Earth is controlled by the shortwave and longwave radiation emitted to space. Changes in the thermodynamic state of the system over time affect climate and are noticeable when viewing the system as a whole. In this paper, we study the changes in the complexity of climate in the last four decades using data from the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2). First, we study the complexity of the shortwave and longwave radiation fields independently using Approximate Entropy and Sample Entropy, observing that the rate of complexity change is faster for shortwave radiation. Then, we study the causality of those changes using Transfer Entropy to capture the non-linear dynamics of climate, showing that the changes are mainly driven by the variations in shortwave radiation. The observed behavior of climatic complexity could be explained by the changes in cloud amount, and we research that possibility by investigating its evolution from a complexity perspective using data from the International Satellite Cloud Climatology Project (ISCCP).

The location of the Earth Polychromatic Imaging Camera (EPIC) aboard the Deep Space Climate Observatory (DSCOVR) offers a global view of the Earth and captures its atmosphere at different locations and time of the day. In this paper, we take advantage of that unique feature to study the daytime variability of cloud fraction in a seasonal and zonal context. We observe that the ensemble behavior over ocean has a distinctive convex shape, with higher cloud fractions at early morning and late afternoon, while no such pattern is seen over land. Unique perspectives are obtained by analyzing the cloud fraction of the globe as a whole and separately for each hemisphere, and by studying the effect of the viewing zenith angle on cloud fraction retrievals.

Knowledge of the daytime variability of cloud fraction is pivotal for the accurate determination of the atmosphere's energy balance. Yet, polar orbiting satellites are unable to fully capture daytime variability and individual geostationary satellites only provide measurements for portions of the Earth. In addition, diurnal cloud fraction variations analyzed with General Circulation Models provide results that cannot be reconciled with observations in many situations. Here, we show that the Earth Polychromatic Imaging Camera (EPIC) aboard the Deep Space Climate Observatory (DSCOVR) satellite presents a unique opportunity to examine changes in daytime global cloud fraction resolved at one‐hour frequency. For the first time, we characterize the global daytime variability of clouds with a single sensor, presenting evidence that the fraction of liquid clouds reaches its maximum over continental areas and its minimum over ocean around noon. We compare our results with previous global and regional studies and examine implications for the Earth's energy budget.

One of the largest uncertainties in climate sensitivity predictions is the influence of clouds. While some aspects of cloud formation and evolution are well understood, others such as the diurnal variability of their heights remains largely unexplored at global scales. Aiming to fill that fundamental gap in cloud knowledge, this paper studies the daytime evolution of cloud top height using the EPIC instrument aboard the DSCOVR satellite, complemented by coincident cloud height retrievals by GOES-R’s ABI instrument. Both datasets indicate that cloud height exhibits a minimum around midday for low clouds with amplitudes between 250 and 600 m depending on the season. The two datasets also agree that high clouds exhibit a contrasting behavior with steady increase of cloud height from morning to evening. We investigate dependences on the type of underlying surface, finding that the amplitude of the diurnal cycles is weaker over ocean than over land for both EPIC and ABI retrievals. We also find a positive correlation between cloud fraction and height over ocean which turns negative over land for low clouds, while for high clouds the correlation is largely positive.