Dynamical density delay maps: simple, new method for visualising the behaviour of complex systems

Background. Physiologic signals, such as cardiac interbeat intervals, exhibit complex fluctuations. However, capturing important dynamical properties, including nonstationarities may not be feasible from conventional time series graphical representations. Methods. We introduce a simple-to-implement visualisation method, termed dynamical density delay mapping (``D3-Map'' technique) that provides an animated representation of a system's dynamics. The method is based on a generalization of conventional two-dimensional (2D) Poincar� plots, which are scatter plots where each data point, x(n), in a time series is plotted against the adjacent one, x(n+1). First, we divide the original time series, x(n) (n=1,..., N), into a sequence of segments (windows). Next, for each segment, a three-dimensional (3D) Poincar� surface plot of x(n), x(n+1), hx(n),x(n+1) is generated, in which the third dimension, h, represents the relative frequency of occurrence of each (x(n),x(n+1)) point. This 3D Poincar\'e surface is then chromatised by mapping the relative frequency h values onto a colour scheme. We also generate a colourised 2D contour plot from each time series segment using the same colourmap scheme as for the 3D Poincar\'e surface. Finally, the original time series graph, the colourised 3D Poincar\'e surface plot, and its projection as a colourised 2D contour map for each segment, are animated to create the full ``D3-Map.'' Results. We first exemplify the D3-Map method using the cardiac interbeat interval time series from a healthy subject during sleeping hours. The animations uncover complex dynamical changes, such as transitions between states, and the relative amount of time the system spends in each state. We also illustrate the utility of the method in detecting hidden temporal patterns in the heart rate dynamics of a patient with atrial fibrillation. The videos, as well as the source code, are made publicly available. Conclusions. Animations based on density delay maps provide a new way of visualising dynamical properties of complex systems not apparent in time series graphs or standard Poincar\'e plot representations. Trainees in a variety of fields may find the animations useful as illustrations of fundamental but challenging concepts, such as nonstationarity and multistability. For investigators, the method may facilitate data exploration

Mortgage Modification and Strategic Behavior: Evidence from a Legal Settlement with Countrywide

We investigate whether homeowners respond strategically to news of mortgage modification programs. We exploit plausibly exogenous variation in modification policy induced by U.S. state government lawsuits against Countrywide Financial Corporation, which agreed to offer modifications to seriously delinquent borrowers with subprime mortgages throughout the country. Using a difference-in-difference framework, we find that Countrywide’s relative delinquency rate increased thirteen percent per month immediately after the program’s announcement. The borrowers whose estimated default rates increased the most in response to the program were those who appear to have been the least likely to default otherwise, including those with substantial liquidity available through credit cards and relatively low combined loan-to-value ratios. These results suggest that strategic behavior should be an important consideration in designing mortgage modification programs.

Asymmetric properties of long-term and total heart rate variability

We report on two new physiological phenomena: the long-term and total heart rate asymmetry, which describe a significantly larger contribution of heart rate accelerations to long-term and total heart rate variability. In addition to the existing pair of indices, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SD1}}_{\rm d}, {\text {SD1}}_{\rm a},$$\end{document} which are based on partitioning short-term variance, we introduce two other pairs of descriptors based on partitioning long-term (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SD2}}_{\rm d}, {\text {SD2}}_{\rm a}$$\end{document}) and total (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SDNN}}_{\rm d}, {\text {SDNN}}_{\rm a}$$\end{document}) heart rate variability. The new asymmetric descriptors are used to analyze RR intervals time series derived from the 30-min ECG recordings of 241 healthy subjects resting in supine position. It is shown that both new types of asymmetry are present in 76% of the subjects. The new phenomena reported here are real physiological findings rather than artifacts of the method since they vanish after data shuffling