# Student Feature - Joshua Garland

By Invited Student Contributor

Joshua Garland is an Omidyar Postdoctoral Fellow at the Santa Fe Institute. He received his Ph. D. from the University of Colorado (supervised by Elizabeth Bradley), an M.S. in Applied Mathematics also from the University of Colorado, and a dual B.S. in Mathematics and Computer Science from Colorado Mesa University.

His research asks “When perfect isn’t possible, how can we adapt mathematics to describe and understand the world around us?” In the study of complex adaptive systems, the available data often falls far short of the demands of the theory. In mathematics, we often say “assume we have an infinite noise-free time series, then…” But in studying complex systems, we often ask “I have 347 noisy observations that I collected over three years in a jungle. What can we learn from this information?” Garland’s research aims to develop rigorous models that bridge the gap between theory and observation in imperfect systems like the climate, traded financial markets, and the human heart.

Garland first became interested in complex systems and all their unique challenges early in his graduate studies while teaching at the Santa Fe Institute’s Complex System Summer School. He was invited to assist with the Nonlinear Dynamics portion of the course but chose to attend the other lectures as well and found himself entranced. What he discovered was a fascinating and interwoven world of scientific problems that were imperfect in nature (ill-sampled, noisy and complicated) but that nonetheless required rigorous mathematics to solve. This was in stark contrast to the perfect world of pure (often linear and static) mathematics he grew up in, where tools and theory were bountiful but perfection (e.g. smoothness, stationarity, and independence) was simply assumed. In trying to answer questions about and understand these complex systems, Garland knew the theories and methods he grew up with had tremendous potential, yet these systems, the observations of the systems, and even the fundamental understanding of these systems were imperfect. As he grappled with this discontinuity he began to wonder what was possible mathematically when perfection was simply not an option.

The first area he attempted to bridge this gap for was in nonlinear time-series analysis. Time-series data is a critical element in the study of complex systems because it captures the dynamics of those systems--and also allows one to build predictive and explanatory models of those dynamics. It is rare, however, that one can measure all of the system's state variables; rather, one has one measurement, or maybe a few: phytoplankton populations, deuterium content of an ice core, instructions per cycle of a running computer, or volatility of a financial security. To construct a useful model from a scalar time series like this, one must find a way to reverse the projection operation that is effected by the measurement of that single quantity, often with little or no knowledge of the underlying dynamics. In the dynamical systems literature there is a powerful technique, delay-coordinate reconstruction, that completely inverts this lossy projection as long as certain mathematical assumptions are satisfied. This method has been used to develop many effective models, both predictive and explanatory, but the theoretical requirements can be burdensome.

In Garland’s recent work [2,3]--in an attempt to bridge the gap between mathematical rigor and practical utility--he has demonstrated that in the case of predictive modeling, a complete inversion of this projection is often not necessary and can be overkill. In this minimalist view, a reduced-order reconstruction is created in as few dimensions as possible--often far fewer than the mathematical machinery would suggest. This reduced-order method allows for partial reconstruction of the systems dynamics: topologically incorrect, perhaps, but adequate to model the future state of the system. Although this approach violates the most basic tenets of delay-coordinate reconstruction, it produces surprisingly accurate predictions while being computationally efficient, requires far less data than traditional approaches, and is quite robust to noise. This strategy is an effective trade-off between mathematical rigor (models that are correct but intractable) and practical effectiveness. Understanding when, why, and how such a reduced-order model is effective--and when this trade-off in accuracy and rigor is worthwhile--is a key aspect of Garland’s current research. This will most likely require new mathematical tools in time-series analysis, computational topology, and information theory.

While developing this theory, Garland studied a wide variety of complex systems, including but not limited to traded financial markets, cardiac failure mechanisms and treatments, influence propagation on social networks, the evolution of history, and the climate. What fascinated him about all these systems was the obvious but illusive mechanisms that seemed to underpin and tie them all together. His main research efforts focus on gaining insight into these universalities through information mechanics (e.g. production, storage, and transmission of information), and then in turn leveraging this knowledge to gain a deeper understanding of systems like the climate, traded financial markets, and the human heart.

“I believe thoroughly understanding the information mechanics of complex systems is a crucial first step in developing rigorous mathematics to describe the imperfect world around us,” he says.

The following is a brief snapshot into this primary research endeavor through one particular example system - the climate:

Coaxing ancient ice records into revealing their knowledge of abrupt climate change, super volcanoes, and the dawn of human civilization.

The climate is a complex, high-dimensional, adaptive dynamical system whose regime shifts have critical implications. One potential proxy for understanding this system is through the study of ancient paleoclimate records such as ice cores. It seems that locked in these records are valuable information about Earth’s past (and future) climate system but understanding what information is available in these records--let alone extracting it--is not well understood. Alongside collaborators from the University of Colorado at Boulder, Garland is studying the information mechanics of ancient stable water isotopes records in order to understand how information is produced, transmitted and processed by the Earth’s climate system. Garland and collaborators hope that a combination of new mathematical theory, analysis, and synthesis will grant them the ability to answer big questions such as: What information can we extract from these records? Are extreme climate events and information mechanics associated? Does the past climate record inform us about the future climate? Is the climate a coupled communication system? Preliminary results on this project [1] suggest that answering such questions is indeed possible. However, due to massive data challenges, including nonlinear temporal spacing, multiple coupled noise sources, and missing data, there is much work to be done.

References

[1] J. Garland, T.R Jones, E. Bradley, R.G. James and J.W.C. White “A First Step Toward Quantifying the Climate's Information Production Over the Last 68,000 Years,” IDA-13 (Proceedings of the 15th International Symposium on Intelligent Data Analysis), Springer Lecture Notes in Computer Science Volume 9897. Stockholm, Sweden. October 2016.

[2] J. Garland, R.G. James, and E. Bradley, “Leveraging information storage to select forecast optimal parameters for delay-coordinate reconstructions," Physical Review E 93:022221 doi:

10.1103/PhysRevE.93.022221 (2016).

[3] J. Garland, E. Bradley, “Prediction in Projection," CHAOS 25 :123108 doi:10.1063/1.4936242

(2015).