REU Computational Dynamics and Topology 2023

Research plan:

1. Develop Topological Data Analysis (TDA) methods to detect early warning signal of financial bubbles.

   1. Develop computer code in Python/C++/Julia/Java for the application of TDA to time series, one-dimensional and multi-dimensional. Translate and improve existing R code.
   2.  Use machine learning (ML) to fit LPPLS model to financial time series; improve existing methods to segment time-series into positive and negative bubbles, and/or develop new ones.
   3. Develop methods to optimize the choice of TDA parameters (embedding dimension, delay, sliding window) for a given time series. See [1].
      1. For a LPPLS model with give parameters, how the TDA parameters must be chosen
      2. For a given financial time series, which characteristics of the time series should be considered in order to choose suitable TDA parameters.
   4. Consider other models besides the LPPLS model and apply a similar analysis.
   5. Apply the TDA method to real-world financial time-series, e.g., to banking data.

2. Detect bifurcations of chaotic attractors through TDA.

   1. Consider the discrete Lorentz model. When one parameter varies and others are fixed, the resulting attractor undergoes bifurcations: the attractor changes from a fixed point to a pair of periodic orbits to a chaotic attractor, which becomes more and more chaotic as the parameter keeps changing. Use TDA to detect such bifurcations.
   2. Use Lyapunov exponents to measure the chaoticity of the attractor as it undergoes bifurcations. See [2].

3. Optimize piezoelectric energy harvesting devices.

   1. Measure the ‘chaoticity’ of the models (e.g., via Lyapunov exponents) and find quantitative relations between chaos and system's performance.
   2. Understand the relation between excitation frequency bandwidth, synchronization and resonances. In particular, investigate whether resonances are 'bad' for the system’s efficiency. For instance, investigate whether coupling beams through springs worsens the system's performance, in the sense that it yields to more resonances.  See [3].

 

Faculty adviser:

Dr. Marian Gidea

PhD student co-adviser:

Atreish Ramlakhan

Students:

Tamar Leiser, Semyon Lomasov, Yedidya Moise, Jacob Stein

References: 

[1] Henry DI Abarbanel, Reggie Brown, John J Sidorowich, and Lev Sh Tsimring. The analysis of observed chaotic data in physical systems. Reviews of modern physics, 65(4):1331, 1993.

[2] Gonchenko, S.V.; Ovsyannikov, I.I.; Sim_o, C.; Turaev, D. (2005) Three-DimensionalHenon-Like Maps and Wild Lorenz-Like Attractors. International Journal of Bifurcation and Chaos, Vol. 15, No. 11, 3493-3508.

[3] Jerzy Margielewicz, Damian Gąska, Grzegorz Litak, Piotr Wolszczak , Daniil Yurchenko. Nonlinear dynamics of a new energy harvesting system with quasi-zero stiffness, Applied Energy, Volume 307, 1 February 2022

 

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Previous REU's

REU on Stochastic Processes and Financial Time Series 2021

Presentation

The program consists of 2 parts. Both parts of the program will be entirely online.

1) Undergraduate summer workshop June 1, 2021 - June 11, 2021 

This is a 2-week intense summer workshop on financial time series and stochastic processes. The workshop has daily lectures in three courses. The first course is "Financial Surveillance via Change-Point Detection" by prof. Pablo Roldan. The second course is "Stochastic Interacting Particle Systems" by prof. Peter Nandori. The third course is "Topological Data Analysis of Time Series" by prof. Marian Gidea. See the abstract for each course below. All lectures will be online. Lecture times are 10 am - 11 am, 11:15 am - 12:15 pm, 2 pm - 3 pm Eastern time on every weekday between June 1st and June 11th. 

2) Hands-on research experience for undergraduates 

Motivated students who completed the summer workshop will have the opportunity to join a research projects in one of the topics discussed in the workshop. The research will be carried out during a period of one month following the workshop. A limited number of stipends is available for students participating in the research. Students will present their results at the Yeshiva Mathematical Physics seminar.

Research Outcomes

Documents

• Siegfried E. and Roldan P.: Change Point Detection improved by pre-processing via block sample variance. Preprint, October 2021.

• Han Zhang and Qian Zhou: Change Point Detection methods applied to Financial Time Series. Research report, October 2021.

Presentations

The following 15-minutes presentations took place at the Yeshiva Mathematical Physics seminar on October 6th, 2021.

Webpage: https://www.yu.edu/ug/math/colloquia-seminars

• Vatsal Strivastava: Applications of TDA in Knot Theory

• Elisheva Siegfried: Change Point Detection improved by pre-processing via block sample variance

• Han Zhang and Qian Zhou: Change Point Detection methods applied to Financial Time Series

• Youngmin Ko: Capturing Patterns in Equity Price Change via TDA

• Noah Bergam:  Story Arcs and Plot Holes: Topological Insights on Vector-Embedded Language

Workshop Courses

Stochastic interacting particle systems (10-11am)

Lecturer: Peter Nandori, peter.nandori@yu.edu

Abstract

On the atomic level, physical materials are quantized yet on the macroscopic level, they look continuous. It is an important challenge in mathematical physics to provide feasible mathematical models for systems that are quantized microscopically but macroscopically obey some basic laws of physics, such as the heat equation. There is a vast scientific literature on this subject, in particular the challenge has been solved for many stochastic microscopic models, also known as stochastic interacting particle systems. In this minicourse, we will present some of these results. In particular, we will focus on one of the simplest techniques which is based on duality of Markov chains. We will also present interesting open problems.

Bibliography

• Li, Y., Nandori, P., Young, L-S. Local thermal equilibrium for certain stochastic models of heat transport Journal of Statistical Physics, 163, 1: 61-91, 2016.

• Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999).

• Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65-74 (1982).

• Nandori, P., Local equilibrium in inhomogeneous stochastic models of heat transport
  Journal of Statistical Physics, 164, 2: 410-437, 2016.

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Financial Surveillance via Change-Point Detection (11:15-12:15pm)

Lecturer: Pablo Roldan, pablo.roldan@yu.edu

Abstract

The world’s history of economic crises, including the recent COVID-related downturn around March 2020, provides graphic evidence of the importance of efficient methods for continuous financial surveillance toward better active risk management. In this course, we consider the problem of “real-time” detection of crashes in “live”-monitored financial time series. This problem will be approached using statistical time-series analysis. In particular, we will discuss Change-Point Detection methods, which try to identify times when the probability distribution of a stochastic process or time series changes. Our working assumption is that, whenever a financial crash happens, the underlying distribution of the observable time-series changes drastically. In general, the problem concerns both detecting whether a change has occurred (or possibly several changes) and identifying the time when they might have occurred. Of course, we would like to detect these structural breaks efficiently, i.e. as promptly as possible, and without raising many “false alarms”. We plan to 1. Review the most useful Change-Point Detection methods for this particular problem; and 2. Apply them to a concrete financial time-series, such as the evolution of the SP 500 stock index, or the price of Bitcoin.

 

Software

This will be an eminently hands-on course. We will use the R environment for statistical computing to apply CPD techniques to financial problems. Before the course, you should install the following software at home:

1. R 3.0.1+

2. RStudio IDE

3. Package 'cpm' (you can install this package from within RStudio)

Bibliography

• Nicholas A. James and David S. Matteson: ecp: An R Package for Nonparametric Multiple
  Change Point Analysis of Multivariate Data. Journal of Statistical Software, December 2014, Volume 62, Issue 7. https://www.jstatsoft.org/article/view/v062i07

• Gordon J. Ross: Parametric and Nonparametric Sequential Change Detection in R: The cpm Package. Journal of Statistical Software, August 2015, Volume 66, Issue 3. https://www.jstatsoft.org/article/view/v066i03

• H. Vincent Poor and Olympia Hadjiliadis: Quickest Detection. Cambridge University Press, 2009.

• Alexander Tartakovsky, Igor Nikiforov and Michèle Basseville: Sequential Analysis. Hypothesis Testing and Changepoint Detection. CRC Press, 2015.

Lecture Notes (updated daily)

Please see the REU dropbox folder.

 

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Topological Data Analysis of Time Series (2-3pm)

Lecturer: Marian Gidea, marian.gidea@yu.edu

Abstract

Topological Data Analysis (TDA) is a new way to make sense out of complex data, complementary to Statistics. Whereas the statistical approach aims to approximate a data set by a probability density function, the TDA approach describes the data set as a geometric shape, using tools from topology. This course will provide a hands-on introduction to TDA. We will use computer software to analyze time series obtained from both chaotic and stochastic processes. In particular, we will use TDA for early detection of ‘critical transitions’, when the underlying system switches abruptly from one state to another, radically different state. As an application, we will investigate crashes in the financial markets.

Software

R, R Studio, TDA package and/or Python

Bibliography

• Computational Topology: An Introduction, New ed., by Herbert Edelsbrunner and John Harer, American Mathematical Society (December 8, 2009) edelcomp.pdf

• Algebraic Topology for Computer Vision by Daniel Freedman and Chao Chen. (PDF) Algebraic topology for computer vision (researchgate.net)

Courses Schedule

June 1: 

9:45 - 10: Opening remarks

10 - 11 Nandori lecture 1

11:15 - 12:15 Roldan lecture 1

2 - 3 Gidea lecture 1

3 - 4 social hour