01/01/2016
In the health informatics era, modeling longitudinal data remains problematic. The issue is method: health data are highly nonlinear and dynamic, multilevel and multidimensional, comprised of multiple major/minor trends, and causally complex—making curve fitting, modeling, and prediction difficult. The current study is fourth in a series exploring a case‐based density (CBD) approach for modeling complex trajectories, which has the following advantages: it can (1) convert databases into sets of cases (k dimensional row vectors; i.e., rows containing k elements); (2) compute the trajectory (velocity vector) for each case based on (3) a set of bio‐social variables called traces; (4) construct a theoretical map to explain these traces; (5) use vector quantization (i.e., k‐means, topographical neural nets) to longitudinally cluster case trajectories into major/minor trends; (6) employ genetic algorithms and ordinary differential equations to create a microscopic (vector field) model (the inverse problem) of these trajectories; (7) look for complex steady‐state behaviors (e.g., spiraling sources, etc) in the microscopic model; (8) draw from thermodynamics, synergetics and transport theory to translate the vector field (microscopic model) into the linear movement of macroscopic densities; (9) use the macroscopic model to simulate known and novel case‐based scenarios (the forward problem); and (10) construct multiple accounts of the data by linking the theoretical map and k dimensional profile with the macroscopic, microscopic and cluster models. Given the utility of this approach, our purpose here is to organize our method (as applied to recent research) so it can be employed by others.
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01/01/2016
Probability distributions have proven effective at modeling diversity in complex systems. The two most common are the Gaussian normal and skewed‐right. While the mechanics of the former are well‐known; the latter less so, given the significant limitations of the power‐law. Moving past the power‐law, we demonstrate that there exists, hidden‐in‐full‐view, a limiting law governing the diversity of complexity in skewed‐right systems; which can be measured using a case‐based version C c of Shannon entropy, resulting in a 60/40 rule. For our study, given the wide range of approaches to measuring complexity (i.e., descriptive, constructive, etc), we examined eight different systems, which varied significantly in scale and composition (from galaxies to genes). We found that skewed‐right complex systems obey the law of restricted diversity; that is, when plotted for a variety of natural and human‐made systems, as the diversity of complexity → ∞ (primarily in terms of the number of types; but also, secondarily, in terms of the frequency of cases) a limiting law of restricted diversity emerges, constraining the majority of cases to simpler types. Even more compelling, this limiting law obeys a scale‐free 60/40 rule: when measured using C c , 60%(or more) of the cases in these systems reside within the first 40% (or less) of the lower bound of equiprobable diversity types—with or without long‐tail and whether or not the distribution fits a power‐law. Furthermore, as an extension of the Pareto Principle, this lower bound accounts for only a small percentage of the total diversity; that is, while the top 20% of cases constitute a sizable percentage of the total diversity in a system, the bottom 60% are highly constrained. In short, as the central limit theorem governs the diversity of complexity in normal distributions, restricted diversity seems to govern the diversity of complexity in skewed‐right distributions.
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12/01/2015
Allostatic load (AL) is a complex clinical construct, providing a unique window into the cumulative impact of stress. However, due to its inherent complexity, AL presents two major measurement challenges to conventional statistical modeling (the field's dominant methodology): it is comprised of a complex causal network of bioallostatic systems, represented by an even larger set of dynamic biomarkers; and, it is situated within a web of antecedent socioecological systems, linking AL to differences in health outcomes and disparities. To address these challenges, we employed case‐based computational modeling (CBM), which allowed us to make four advances: (1) we developed a multisystem, 7‐factor (20 biomarker) model of AL's network of allostatic systems; (2) used it to create a catalog of nine different clinical AL profiles (causal pathways); (3) linked each clinical profile to a typology of 23 health outcomes; and (4) explored our results (post hoc) as a function of gender, a key socioecological factor. In terms of highlights, (a) the Healthy clinical profile had few health risks; (b) the pro‐inflammatory profile linked to high blood pressure and diabetes; (c) Low Stress Hormones linked to heart disease, TIA/Stroke, diabetes, and circulation problems; and (d) high stress hormones linked to heart disease and high blood pressure. Post hoc analyses also found that males were overrepresented on the High Blood Pressure (61.2%), Metabolic Syndrome (63.2%), High Stress Hormones (66.4%), and High Blood Sugar (57.1%); while females were overrepresented on the Healthy (81.9%), Low Stress Hormones (66.3%), and Low Stress Antagonists (stress buffers) (95.4%) profiles. © 2015 Wiley Periodicals, Inc. Complexity 21: 291–306, 2016
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03/01/2015
This article introduces a new case‐based density approach to modeling big data longitudinally, which uses ordinary differential equations and the linear advection partial differential equations (PDE) to treat macroscopic, dynamical change as a transport issue of aggregate cases across continuous time. The novelty of this approach comes from its unique data‐driven treatment of cases: which are K dimensional vectors; where the velocity vector for each case is computed according to its particular measurements on some set of empirically defined social, psychological, or biological variables. The three main strengths of this approach are its ability to: (1) translate the data driven, nonlinear trajectories of microscopic constituents (cases) into the linear movement of macroscopic trajectories, which take the form of densities; (2) detect the presence of multiple, complex steady state behaviors, including sinks, spiraling sources, saddles, periodic orbits, and attractor points; and (3) predict the motion of novel cases and time instances. To demonstrate the utility of this approach, we used it to model a recognized cohort dynamic: the longitudinal relationship between a country's per capita gross domestic product (GDP) and its longevity rates. Data for the model came from the widely used Gapminder dataset. Empirical results, including the strength of the model's fit and the novelty of its results (particularly on a topic of such extensive study) support the utility of our new approach. © 2014 Wiley Periodicals, Inc. Complexity 20: 45–57, 2015
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11/01/2012
Recently, the continuity equation (also known as the advection equation) has been used to study stability properties of dynamical systems, where a linear transfer operator approach was used to examine the stability of a nonlinear equation both in continuous and discrete time (Vaidya and Mehta, IEEE Trans Autom Control 2008, 53, 307–323; Rajaram et al., J Math Anal Appl 2010, 368, 144–156). Our study, which conducts a series of simulations on residential patterns, demonstrates that this usage of the continuity equation can advance Haken's synergetic approach to modeling certain types of complex, self‐organizing social systems macroscopically. The key to this advancement comes from employing a case‐based approach that (1) treats complex systems as a set of cases and (2) treats cases as dynamical vsystems which, at the microscopic level, can be conceptualized as k dimensional row vectors; and, at the macroscopic level, as vectors with magnitude and direction, which can be modeled as population densities. Our case‐based employment of the continuity equation has four benefits for agent‐based and case‐based modeling and, more broadly, the social scientific study of complex systems where transport or spatial mobility issues are of interest: it (1) links microscopic (agent‐based) and macroscopic (structural) modeling; (2) transforms the dynamics of highly nonlinear vector fields into the linear motion of densities; (3) allows predictions to be made about future states of a complex system; and (4) mathematically formalizes the structural dynamics of these types of complex social systems.
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