## Summer School 2018

The 2018 LML Summer School will be taking place from **Monday 23rd July to Friday 17th August 2018.** Successful applicants will carry out a four-week research project under supervision at LML. Travel to and from London, accommodation, and a stipend of £1,000 will be provided.

There are 8 projects available to apply for this year. Further below you can find more details on each individual project including project backgrounds, goals and references for further information.

**Applications are now closed for the 2018 LML Summer School.**

**LML Summer School Projects 2018**

- Regularized portfolio optimization.
- Random search on networks with a resetting procedure.
- Risk and return: the gambler’s approach to evaluating the skill of natural hazard forecasts.
- Evaluation of model performance in climate and earthquake hazard prediction.
- Anomalous diffusion in physics and biology.
- From microscopic to macroscopic noise: the dynamics of transitions around noisy networks.
- Local dimension analysis of conceptual models of atmospheric jet at mid-latitude.
- Ergodicity in a simple random dynamical system.

## Projects – Further Information

**Project 1**

** Supervisors: Imre Kondor and Fabio Caccioli**

** Title: Regularized portfolio optimization**

Portfolio optimization is a textbook problem in financial mathematics, and it refers to the search for an investment strategy that is optimal, where optimal means that it minimizes risk for a given expected profit.

A challenge of portfolio optimization in practice is that, as the number of investment opportunities increases, the amount of data needed to find a reliable solution to the optimization problem quickly becomes large, and one is often confronted with large sample to sample fluctuations.

In fact, the optimization of large portfolios is characterized by a phase transition at a critical value r_c of the ratio between N (the number of assets) and T (the number of data points) beyond which the estimation error diverges and the optimization problem becomes meaningless.

The introduction of a regularizer into the optimization problem is a method to reduce sample fluctuations and improve the estimation of risk.

By means of numerical experiments performed on synthetic data in a variety of controlled settings, this project will look at the performance of L_1 regularization in the context of mean-variance portfolio optimization, and clarify the effect of averaging over different samples drawn from the same distribution compared to the effect of cross-validation on a given, unique sample. There is a phase transition taking place in this system at r_c=2, a fact which empirical studies apparently failed to notice. The understanding of this fact is one of the possible questions arising in this context.

**Main goal**

To reproduce in a controlled setting results analogous to those of Brodie et al on sparse portfolio optimization, and characterize the relationship between sparsity and estimation error.

**References**

[1] Brodie et al. “Sparse and stable Markowitz portfolios” , PNAS 106, 12267–12272 (2009)

[2] Pantaleo et al. “When do improved covariance matrix estimators enhance portfolio optimization? An empirical comparative study of nine estimators”, Quantitative Finance 11, 1067- 1080 (2011)

[3] Pafka, Kondor “Noisy covariance matrix and portfolio optimization” EPJB 27, 277–280 (2002)

[4] Kondor, Pafka, “Noisy covariance matrix and portfolio optimization II” Physica A 319 487-494 (2003)

[5] Markowitz “Portfolio selection” The journal of finance (1952)

[6] Kondor, Papp, Caccioli arXiv:1709.08755 (2017)

**Project 2**

**Supervisors: Colm Connaughton and Fernando Metz**

**Title: Random search on networks with a resetting procedure**

Several systems in nature can be seen as large assemblies of elementary units interconnected through the links of a network. Examples are abundant in physics, biology and economy, including the World Wide Web (WWW), transportation networks, social networks, interbank networks, networks of ecological associations, gene regulatory networks, etc. The navigation and search of information on these systems are very important issues with many practical applications. The designing of efficient algorithms to retrieve information from a networked structure is a challenging scientific problem, due to the nontrivial connectivity properties of many real-world networks.

The basic strategy to explore a network, unveil its topology and retrieve information from a particular node, consists in performing a random walk through the different nodes of the network. At each time step, a random walker located at a certain node follows randomly one of its links and reach a given neighbour. The process is iterated until the desired information (a target node) is retrieved. This procedure is at the core of the PageRank algorithm used by the search engine Google to establish a ranking of websites associated to a certain information.

Search with resetting is a usual procedure in everyday life. Suppose one is looking for a face in the crowd: after searching unsuccessfully for a while, a natural tendency is to return to the starting point and restart the search. The main goal of the present project is to explore the effect of such resetting procedure on the performance of searching algorithms on networks.

**Main goal**

The main goal is to study the problem of a single particle performing a random walk on a network with a rate of resetting, i.e., the particle has a probability to return to its initial position. The student will use numerical simulations and analytical approaches to study this problem on simple models of networks.

**References**

[1] Dynamical processes on complex networks, Alain Barrat, Marc Barthélemy and Alessandro Vespignani, Cambridge University Press, 2008.

[2] Random walks on graphs: ideas, techniques and results, R. Burioni and D. Cassi, J. Phys. A: Math. Gen. 38, R45 (2005).

[3] Diffusion with stochastic resetting, M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. 106, 160601 (2011).

**Project 3**

**Supervisors: Jiancang Zhuang and Max Werner**

**Title: Risk and return: the gambler’s approach to evaluating the skill of natural hazard forecasts**

Assessing the ‘goodness’ of forecasts or predictions of natural hazards is of both scientific and practical value. A fundamental pillar of science is the testability of theories and models against observations: the strength of predictions informs our confidence in models and hypotheses. From a practical standpoint, the accuracy and reliability of a forecast is critical for robust decision-making. In this project, we will explore the gambler’s approach to assessing the ‘goodness’ of forecasts and predictions of natural hazards [1]: the approach uses analogies to gambling [2-3] and benefits from language and concepts that are often intuitive to non-experts. The gambling scores have also, however, drawn fire in the literature for some undesirable properties [5-6].

**Main goal**

The challenge is to resolve the debate about the utility of the gambling skill scores. For that purpose, the student will assess the ‘goodness’ of a set of probabilistic earthquake forecasts for different tectonic regions, and contrast the gambling approach with other skill scores (such as likelihood scores). Depending on interest, the student may pursue theoretical properties of the gambling scores, extend their application to other natural hazard forecasts, or prototype the scores for automated testing of earthquake forecasts and predictions in the global Collaboratory for the Study of Earthquake Predictability [7]. We seek students with an interest and background in statistics, maths, or other quantitative discipline.

**References**

[1] Zechar, J. D. (2010, Evaluating earthquake predictions and earthquake forecasts: a guide for students and new researchers, Community Online Resource for Statistical Seismicity Analysis, doi:10.5078/corssa-77337879. Available at http://www.corssa.org/export/sites/corssa/.galleries/articles-pdf/zechar.pdf

[2] Zhuang J. (2010) Gambling scores for earthquake predictions and forecasts. Geophysical Journal International. 181: 382-390.

[3] Zechar, J. D., and Zhuang, J. (2014) A parimutuel gambling perspective to compare probabilistic seismicity forecasts. Geophysical Journal International, 199, 60-68.

[5] G. Molchan, L. Romashkova (2011) Gambling score in earthquake prediction analysis, Geophysical Journal International, 184, 3, 1445–1454.

[6] G. Molchan, L. Romashkova, A. Peresan (2017) On some methods for assessing earthquake predictions, Geophysical Journal International, 210, 3, 1474–1480,

[7] T. H. Jordan (2006) Earthquake Predictability, Brick by Brick. Seismological Research Letters, 77 (1): 3–6. doi: https://doi.org/10.1785/gssrl.77.1.3

**Project 4**

**Supervisors: Max Werner and Erica Thompson**

**Title: Evaluation of model performance in climate and earthquake hazard prediction**

Reliable forecasts of natural hazards – such as climate change, earthquakes, droughts, or windstorms – can provide critical information for reducing disaster risk, building resilience and effecting behavioural change. Reliable hazard predictions, however, are challenging: geosystems are complex, multi-scale and nonlinear; knowledge of these systems is often partial; models are necessarily simplified; critical data are often lacking; and the system state may only rarely repeat. Under these conditions, how do different disciplines go about building confidence in hazard predictions, and what is the role of different types of forecast uncertainties?

**Main goal**

In this project, the student will contrast model evaluation practices in the climate change and earthquake forecasting communities [1, 2]. A specific objective is to contrast forecast skill scores between the two fields [3, 4] and to implement promising metrics from one field to assess probabilistic forecasts of the other field. What properties of evaluation metrics are desirable? Depending on interest and time, a second effort involves comparing approaches to dealing with forecast uncertainty: What types of uncertainties are represented (aleatory, epistemic, radical, deep, etc.), and how (probability distributions, ranges, scenarios, visually, graphically, textually)? How does uncertainty affect model evaluation? The goal is to transplant good ideas from one discipline to another.

**References**

[1] Joliffe, I. T., and Stephenson, D. B. (2012), Forecast Verification: A Practitioner’s Guide in Atmospheric Science, 2nd Ed., Wiley-Blackwell, Oxford. Available at http://onlinelibrary.wiley.com/book/10.1002/9781119960003

[2] Jordan, T. H. (2006) Earthquake Predictability, Brick by Brick. Seismological Research Letters, 77 (1): 3–6. doi: https://doi.org/10.1785/gssrl.77.1.3

[3] Smith, L.A., Suckling, E.B., Thompson, E.L. et al. (2015), Towards improving the framework for probabilistic forecast evaluation, Climatic Change, 132(1), 31-45. https://doi.org/10.1007/s10584-015-1430-2

[4] Zechar, J. D. (2010), Evaluating earthquake predictions and earthquake forecasts: a guide for students and new researchers, Community Online Resource for Statistical Seismicity Analysis, doi:10.5078/corssa-77337879. Available at http://www.corssa.org/export/sites/corssa/.galleries/articles-pdf/zechar.pdf

**Project 5**

**Supervisors: Yuzuru Sato, Rainer Klages and Nicholas Moloney**

**Title: Anomalous diffusion in physics and biology**

This project is at the interface between applied mathematics, statistical physics and biological dynamics. Depending on the scientific interests of the student, it may focus on one of the following two sub-projects:

- Crawling amoeba: statistical data analysis and mathematical modeling: We analyse experimental data given in terms of space-time trajectories of crawling amoeba with the aim to construct a simple mathematical model to reproduce the data. For this purpose we employ a random dynamical systems approach based on extreme value statistics [4]. First results indicate the occurrence of weak anomalous diffusion that should be further analysed.

- Lévy walks in weakly chaotic dynamical systems: A Lévy walk is a generalisation of a simple random walk on the line that generates diffusion which is faster than Brownian motion [1]. Such type of motion is widely observed in nature [2] but is also generated by deterministic, so-called ‘weakly’ chaotic dynamical systems [3]. This sub-project aims at exploring Lévy-type diffusion and in particular how it changes in the latter class of maps under control parameter variation.

**Main goal**

The student will learn methods of stochastic and dynamical systems theory by matching analytical results to data from own computer simulations and/or experi- ments, respectively.

**References**

[1] J. Klafter, I. M. Sokolov, First steps in random walks (Oxford, 2011)

[2] V. Zaburdaev, S. Denisov, J. Klafter, Rev.Mod.Phys. 87, 483 (2015)

[3] G. Zumofen, J. Klafter, Phys.Rev.E 47, 851 (1993)

[4] V. Lucarini et al., Extremes and Recurrence in Dynamical Systems (Wiley, 2016)

**Project 6**

**Supervisors: Claire Postlethwaite and Rosemary Harris**

**Title: From microscopic to macroscopic noise: the dynamics of transitions around noisy networks.**

Systems that evolve continuously with time can often be described by differential equations, and solutions of these can display a variety of behaviours, for instance, they may be at equilibrium, or be periodic in time. A heteroclinic network [1] is a type of solution consisting of a set of dynamical states connected by trajectories. Often the states are equilibrium solutions, but they could equally well be periodic orbits or even chaotic sets. Near a heteroclinic network, the dynamics are typically intermittent: the system spends long periods of time near the states in the network, with rapid transitions between them as the trajectory travels around the network.

Heteroclinic networks can be found in mathematical models describing a diverse range of physical systems; they arise naturally in applications such as population dynamics, fluid mechanics and game theory. Heteroclinic networks are also particularly suited to the modelling of cognitive functions due to the sequential nature of the dynamics as the trajectory explores the network. However, current theory is not yet fully developed for many of these situations, and in particular, physical systems are often affected by the presence of stochastic noise.

It is well known that stochastic noise plays a fundamental role in modifying the qualitative behaviour of dynamical systems [2], but the effects of noise on network attractors are often counter-intuitive and not well understood. For instance, one might expect that noise would increase switching between different parts of the network, but in some circumstances the reverse can occur [3]. A particularly surprising phenomena is the appearance of long-time correlations, or memory, in the sequence of transitions between states when microscopic noise is added to the system [3, 4].

**Main goal:**

The main goal of this project is to investigate how the addition of microscopic noise to heteroclinic network attractors affects the sequence of transitions made by a trajectory around the network. We are particularly interested in understanding when the sequence of transitions made around a network is Markovian, when it is not Markovian, and what conditions need to be met for each case.

**References:**

[1] V. Kirk, M. Silber, (1994) A competition between heteroclinic cycles. Nonlinearity, 7, 1605–1621.

[2] M.I. Friedlin, A.D. Wentzell, (2012) Random Perturbations of Dynamical Systems. Springer Series of Comprehensive Studies in Mathematics, 260.

[3] D. Armbruster, V. Kirk, E. Stone, (2003) Noisy heteroclinic networks. Chaos, 13(1), 71–79.

**Project 7**

**Supervisors: Davide Faranda, Yuzuru Sato and Nicholas Moloney**

**Title: Local dimension analysis of conceptual models of atmospheric jet at mid-latitude**

Atmospheric flows are characterized by chaotic attractors: geometric objects which host the trajectory of motions. For decades such attractors have been considered high dimensional and their properties have been computed only for the average stationary state. Recent results have shown that 1) low dimensional attractors of experimental turbulent flows can be found if one consider global observables and lump turbulence contributions in a stochastic forcing, 2) extreme weather events correspond to special regions of the attractor (unstable fixed points, high and low dimensional manifolds). Our preliminary results suggest that the position of the atmospheric jet at mid-latitudes contain essential information on mid-latitude dynamics. We have therefore derived empirically low dimensional models of the atmospheric circulation from the analysis of the jet position. The validity of these models need yet to be tested.

**Main goal**

Investigate numerically the dynamical properties of conceptual models for the atmospheric mid-latitude jet. Compare these results with those obtained by the direct analysis of climate data of the jet position in the north and southern hemisphere for the last 40 years.

**Background knowledge
**Background of dynamical systems, stochastic process, fluid mechanics, and basic computer skills are preferred.

**References:**

[1] Davide Faranda, Yuzuru Sato, Brice Saint-Michel, Cecile Wiertel-Gasquet, Vincent Padilla, Berengere Dubrulle, & Francois Daviaud. Stochastic chaos in a turbulent swirling flow. Physical review letters, 119, 014502, 2017.

[2] Nicholas Moloney, Davide Faranda, Yuzuru Sato, An overview of the extremal index, to be submitted, 2017

**Project 8**

**Supervisors: Yuzuru Sato, Jeroen Lamb and Ole Peters**

**Title: Ergodicity in a simple random dynamical system**

We will investigate the following repeated gamble: suppose that we start with the initial condition of $1 of wealth. Upon tossing a fair coin, we increase the current wealth by 50% on heads, or we decrease it by 40% on tails. The process is subsequently repeated. This gamble has the following interesting properties:

• The expectation value of your wealth grows exponentially at 5% per round.

• If you play many rounds, your wealth will shrink exponentially at about 5% per round. This property has been called non-ergodicity because the behavior of the expectation value of wealth does not reflect the temporal behavior of wealth.

**Main goal**

We want to cast the gamble in the language of random dynamical systems and understand precisely how the notion of ergodicity in that field is related to the phenomenon described above.

**References**

[1] Peters, Ole, and Murray Gell-Mann. “Evaluating gambles using dynamics.” Chaos: An Interdisciplinary Journal of Nonlinear Science 26.2 (2016): 023103. https://doi.org/10.1063/1.4940236

[2] Ashwin, Peter, Philip J. Aston, and Matthew Nicol. “On the unfolding of a blowout bifurcation.” Physica D: Nonlinear Phenomena 111.1 (1998): 81-95. http://www.sciencedirect.com/science/article/pii/S0167278997800061

[3] Jeroen Lamb, Lecture note at Imperial College London, http://rdstcc.blogspot.co.uk