The method Dynamical Density Delay Maps was proposed in:

Burykin A, Costa MD, Citi L, and Goldberger AL "Dynamical density delay maps: simple, new method for visualising the behaviour of complex systems." BMC Medical Informatics and Decision Making, 14.1 (2014): 6.

Please cite the above publication when referencing this material, and also include the standard citation for PhysioNet: Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE. "PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals," Circulation 101(23):e215-e220 [Circulation Electronic Pages; http://circ.ahajournals.org/content/101/23/e215.full]; 2000 (June 13).

## Description

This directory contains Matlab functions for visualizing the behavior of complex systems by means of the dynamical density delay mapping ("D3-Map") technique . This simple-to-implement visualization method provides an animated representation of a system's dynamics. The method is based on a generalization of conventional two-dimensional (2D) Poincaré 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) Poincaré surface plot of `x(n), x(n + 1), h[x(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 Poincaré surface is then chromatized by mapping the relative frequency `h ` values onto a color scheme. We also generate a colorized 2D contour plot from each time series segment using the same colormap scheme as for the 3D Poincaré surface. Finally, the original time series graph, the colorized 3D Poincaré surface plot, and its projection as a colorized 2D contour map for each segment, are animated to create the full "D3-Map".

This visualization technique can be applied to cardiac interval time series, to uncover complex dynamical changes, e.g. transitions between sleep stages, or to detect hidden temporal patterns, e.g, RR patterns in atrial fibrillation.

- D3M-2Dfun.m generates colorized contour Poincaré plots
- D3M-3Dfun.m generates colorized 3D Poincaré surface maps
- dscatter2.m is a support function for creating a scatter plot colored by density
- D3Mdemo.m is a demo script to show how to generate videos with D3M-3Dfun.m

The functions have been tested on Matlab R2013b.

## Sample Input

The folders includes 3 cardiac interval time series for testing the functions:

- Sleep_RR.dat ASCII file with the cardiac interbeat (RR) intervals obtained from the electrocardiographic recording of a healthy subject during sleep (~6 h)
- Typical_AF.dat ASCII file with the cardiac interbeat (RR) intervals obtained from the electrocardiographic recording of a subject with atrial fibrillation
- Atypical_AF.dat ASCII file with the cardiac interbeat (RR) intervals obtained from the electrocardiographic recording of a subject with atrial fibrillation, whose contour and 3D maps show atypical (i.e., periodic) dynamical patterns

## Acknowledgments

This package was developed at the Wyss Institute at Harvard by A. Burykin, L. Citi, T. Silva, M.D. Costa and A.L. Goldberger. S. Mariani and T. Henriques contributed to the modification and packaging of the software and the creation of this tutorial.

Users of our software should cite: Burykin A, Costa MD, Citi L, and Goldberger AL, "Dynamical density delay maps: simple, new method for visualising the behaviour of complex systems." BMC Medical Informatics and Decision Making, 14.1 (2014): 6.