Muller Plot
A Muller plot combines information about the succession of different OTUs (genotypes, phenotypes, species, …) and information about dynamics of their abundances (populations or frequencies) over time. Muller plots may be used to visualize evolutionary dynamics. They may be also employed in the study of diversity and its dynamics; that is, how diversity emerges and how changes over time. An example of a Muller plot (produced by the MullerPlot package in R) showing the evolutionary dynamics of an artificial community They are called Muller plots in honor of Hermann Joseph Muller, who used them to explain his idea of Muller’s ratchet. …

DE-LSTM
We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with underlying nonlinear dynamics. The deep learning model aims to approximate the probability density function of a stochastic process via numerical discretization and the underlying nonlinear dynamics is modeled by the Long Short-Term Memory (LSTM) network. After the numerical discretization by a softmax function, the function estimation problem is solved by a multi-label classification problem. A penalized maximum log likelihood method is proposed to impose smoothness in the predicted probability distribution. It is shown that LSTM is a state space model, where the internal dynamics consists of a system of relaxation processes. A sequential Monte Carlo method is outlined to compute the time evolution of the probability distribution. The behavior of DE-LSTM is investigated by using the Ornstein-Uhlenbeck process and noisy observations of Mackey-Glass equation and forced Van der Pol oscillators. While the probability distribution computed by the conventional maximum log likelihood method makes a good prediction of the first and second moments, the Kullback-Leibler divergence shows that the penalized maximum log likelihood method results in a probability distribution closer to the ground truth. It is shown that DE-LSTM makes a good prediction of the probability distribution without assuming any distributional properties of the noise. For a multiple-step forecast, it is found that the prediction uncertainty, denoted by the 95% confidence interval, does not grow monotonically in time. For a chaotic system, Mackey-Glass time series, the 95% confidence interval first grows, then exhibits an oscillatory behavior, instead of growing indefinitely, while for the forced Van der Pol oscillator, the prediction uncertainty does not grow in time even for 3,000-step forecast. …

Analytic Hierarchy Process (AHP)
The analytic hierarchy process (AHP) is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It was developed by Thomas L. Saaty in the 1970s and has been extensively studied and refined since then. It has particular application in group decision making, and is used around the world in a wide variety of decision situations, in fields such as government, business, industry, healthcare, shipbuilding and education. Rather than prescribing a ‘correct’ decision, the AHP helps decision makers find one that best suits their goal and their understanding of the problem. It provides a comprehensive and rational framework for structuring a decision problem, for representing and quantifying its elements, for relating those elements to overall goals, and for evaluating alternative solutions. Users of the AHP first decompose their decision problem into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently. The elements of the hierarchy can relate to any aspect of the decision problem—tangible or intangible, carefully measured or roughly estimated, well or poorly understood—anything at all that applies to the decision at hand. Once the hierarchy is built, the decision makers systematically evaluate its various elements by comparing them to each other two at a time, with respect to their impact on an element above them in the hierarchy. In making the comparisons, the decision makers can use concrete data about the elements, but they typically use their judgments about the elements’ relative meaning and importance. It is the essence of the AHP that human judgments, and not just the underlying information, can be used in performing the evaluations. The AHP converts these evaluations to numerical values that can be processed and compared over the entire range of the problem. A numerical weight or priority is derived for each element of the hierarchy, allowing diverse and often incommensurable elements to be compared to one another in a rational and consistent way. This capability distinguishes the AHP from other decision making techniques. In the final step of the process, numerical priorities are calculated for each of the decision alternatives. These numbers represent the alternatives’ relative ability to achieve the decision goal, so they allow a straightforward consideration of the various courses of action. …