Topics 2021/2022

On Neural Network models in integral form with uncertainty, supervisor Stefania Tomasiello

  Neural networks (NNs) are well-known computing schemes mainly used for forecasting and classification problems, but also employed to solve ordinary or partial differential equations. They are usually modeled as a system of differential equations [1] or integro-differential equations for the recurrent schemes [2, 3]. Neural networks models in integral form (as a system of integral equations) have not been extensively investigated, though they exhibit some interesting properties [4]. The aim of this thesis is to investigate the approximation ability of a new NN scheme with uncertainty in integral form, formally and numerically through some benchmark problems. In particular, as an application example, the credit risk assessment will be considered [5], in order to predict whether a customer will be solvent or not.

References

[1] S. Haykin, Neural Networks and Learning Machines, Pearson College, 3rd ed. 2008

[2] B. de Vries, J.C. Principe, A theory for neural networks with time delays, in: Proceedings: Conference on Advances in Neural Information Processing Systems (NIPS-3), 1990, pp. 162–168.

[3] F. Colace, V. Loia, S. Tomasiello, Revising Recurrent Neural Networks from a Granular perspective, Applied Soft Computing, 2019, 82, 105535

[4] A. Nordbo, J. Wyller, G.T. Einevoll, Neural network firing-rate models on integral form: Effects of temporal coupling kernels on equilibrium-state stability, Biological Cybernetics, 2007, 97 (3), 195–209.

[5] M. Corazza et al. Design of adaptive Elman networks for credit risk assessment, Quantitative Finance, 2020, in press

Machine learning for fractional partial differential equations, supervisors Stefania Tomasiello, Urve Kangro

There has been a growing interest recently on the application of neural networks-like approaches for the numerical solution of partial differential equations (PDEs), e.g. [1, 2] even with fractional derivatives [3].  Such approaches seem to overcome the typical shortcomings of the classical grid-based techniques, e.g. a suitable discretization, especially in complex domains.  

The aim of this thesis is to investigate a neural network-like approach for the numerical solution of a class of fractional PDEs, and in particular the space fractional Black–Scholes equation  for pricing European options, which has been recently considered in [4] by using a finite difference scheme.

References

[1] V. Dwivedi, B. Srinivasan, Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations,  Neurocomputing 391 (2020) 96–118

[2] Q. Wei, Y. Jiang,  J. Z. Y. Chen, Machine-learning solver for modified diffusion equations, PHYSICAL REVIEW E 98 (2018) 053304

[3] H. Qu, Z. She, X. Liu, Neural network method for solving fractional diffusion equations, Applied Mathematics and Computation 391 (2021) 125635

[4]  K. S. Patel , M. Mehra, Fourth order compact scheme for space fractional advection–diffusion reaction equations with variable coefficients, Journal of Computational and Applied Mathematics 380 (2020) 112963

[5] M. Raissi, G. E.Karniadakis, Hidden physics models: Machine learning of nonlinear partial differential equations, J. Comput. Phys. 357 (2018) 125–14

New metaheuristics for portfolio optimization, supervisors Stefania Tomasiello, ...

 Metaheuristics are algorithms to tackle optimization problems and have been widely used during the last decades to support complex decision-making in several fields, such as logistics, bioinformatics, finance. One of the financial applications is portfolio optimization [1], which aims to find the best allocation of  resources for a set of assets.  While the current literature still shows interest in this kind of application [2, 3], a comparative analysis among different metaheuristics seems to be missing.

The aim of this thesis is to discuss a comparison among different metaheuristics (i.e. population-based and non-population-based algorithms) by considering the latest algorithms, such as the String Theory Algorithm (STA) [4]. STA is a nature-inspired meta-heuristic, based on a theory from modern physics (i.e. String Theory), where the elemental objects are strings, whose vibrations determine the particles properties. Some benchmark datasets will be considered for the comparative purposes (e.g. [2]).

References
[1] A. Soler-Dominguez, A. A. Juan, R. Kizys, 1A Survey on Financial Applications of Metaheuristics, ACM Computing Surveys, 50(1),  15:1-19, 2017.

[2]  M. Dhaini,  N. Mansour,  Squirrel search algorithm for portfolio optimization, Expert Systems With Applications, 178, 114968, 2021.

[3] T. E. Simos, S. D. Mourtas, V. N. Katsikis, Time-varying Black–Litterman portfolio optimization using a bio-inspired approach and neuronets, Applied Soft Computing, 112,  107767, 2021

[4] L. Rodriguez, O. Castillo, M. Garcia, J. Soria, A new meta-heuristic optimization algorithm based on a paradigm from physics: string theory, Journal of Intelligent & Fuzzy Systems, 41 (1) 1657-1675, 2021

Option pricing in case of stochastic volatility, supervisor Toomas Raus

One assumption to find the price of option is the assumption of the behavior of the underlying asset (eg share) price. In the most common case, it is assumed that the price of the underlying asset behaves according to the geometric Brown movement in which the volatility of the underlying asset is constant. However, this assumption may not be valid in practice and volatility behavior can often be described as a random process. The master's thesis is intended to provide an overview of option valuation in case of stochastic volatility. Various numerical methods based on the binomial method proposed in the literature are examined in more detail. It is planned to draw up the corresponding program and carry out numerical experiments.

Aegridade kointegratsioon ja veaparandusmudel, juhendaja Toomas Raus

Lihtsustatult öeldes on kaks mittestatsionaarset aegrida omavahel kointegeeritud siis, kui leidub nende aegridade selline lineaarne kombinatsioon, mis on statsionaarne aegrida. Näiteks käituvad valuutakursside või erinevad energiakandjate hinnad vastavalt juhusliku ekslemise protsessile (olles seega mittestatsionaarsed) , kuid kointegratsiooni korral on nende aegridade käitumine pikaajaliselt omavahel kooskõlas. Kui kaks aegrida on kointegreeritud, siis saame esitada nende aegridade nii pikaajalise ja lühiajalise seose nn.  veaparandusmudeli kujul. Magistritöös on kavas anda teoreetiline ülevaade aegridade kointegratsioonist ning töö praktilises osas uurida  valuutakursside või erinevate energiakandjate hinna kointegratsiooni. 

Credit scoring as classification problem, supervisor Kalev Pärna

A typical problem faced by credit institutions (e.g. banks) is to decide whether or not to extend a loan to a new loan applicant. The decision is based on the estimation of the default risk of the applicant or, in other words,  how large is the possibility that the applicant will have difficulties in paying back the loan.  This is typical classification problem: one has to classify a new loan applicant between two classes, "Yes" (meaning positive decision) and "No" (negative decision).  A widely used method for estimation of the default risk is logistic regression, allowing to obtain a formula for calculation of the probability of the default on the basis of several dependent variables characterizing the applicant (monthly income, age, gender, family status, education, etc). However,  some machine learning methods like classification trees (CART) or k-nearest neighbors (k-NN) can also be applied to achieve the same goal.

The student interested in the topic  will analyze a real data set with the purpose to evaluate and compare different classification methods. The proper use of dichotomous and nominal input variables will be of special interest in this work.

Overview and simulation study of performance of extensions of kNN, supervisor Raul Kangro

Modelling complex seasonality’s, supervisor Märt Möls

Time series data are sometimes affected by multiple cycles of different length - there can be a weekly cycle (better sales on Fridays), monthly pattern (better sales in the beginning of the month as people have more cash after payday) and the effects of calendar seasonality (more tourists during summer so better sales) might be present also. How to model multiple seasonality’s in one model? In this thesis one could compare for example TBATS models (which allow multiple seasonality’s) to alternative approaches.

Pseudo-observations in predicting life expectancy, supervisor Anastassia Kolde

In life-insurance business it is essential for the company to estimate the life expectancy of a client. One way is to use only historic data but in that case we are omitting data of those people who are currently alive in our client base. Latter issue brings us to censoring and to overcome this issue pseudo-observations have been proposed. Student will give an overview of pseudo-observations and run a simulation study in order to compare survival analysis method with conventional regression based analysis models or random forest.