Abstract
Two distinguished properties of the F-transform: the best approximation in a local sense and the reduction in dimension imply the fact that the F-transform has many successful applications. We show that the technique of F-transform fully agrees with the technique of dimensionality reduction, based on Laplacian eigenmaps. To justify this claim, we characterize the processed by the F-transform data in terms of the adjacency graph that reflects their (data) intrinsic geometry.
In the second part, we give an overview of the F-transform applications to mathematical finance: we discussed the estimation of market volatility and the numerical scheme and solution to the one-factor Black-Scholes partial differential equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003)
Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River (1999)
Holčapek, M., Tichý, T.: Nonparametric regression via higher degree F-transform for implied volatility surface estimation. In: Proceedings of Mathematical Methods in Economics, Liberec, pp. 277–282 (2016)
Holčapek, M., Števuliáková, P., Perfilieva, I.: F-transform method for option pricing. In: Proceedings of the 8th International Scientific Conference Managing and Modelling of Financial Risks, Ostrava, Czech Republic, VŠB (2016)
Holčapek, M., Valášek, R.: Numerical solution of partial differential equations with the help of fuzzy transform technique. In: Proceedings of the IEEE International Conference on Fuzzy Systems, FUZZ-IEEE, Naples, Italy (2017)
Khastan, A., Perfilieva, I., Alijani, Z.: A new fuzzy approximation method to Cauchy problems by fuzzy transform. Fuzzy Sets Syst. 288, 75–89 (2016)
Luce, R.D.: Several possible measures of risk. Theor. Dec. 12(3), 217–228 (1980)
Novák, V., Perfilieva, I., Dvořák, A.: Insight into Fuzzy Modeling. Wiley, Hoboken (2016)
Perfilieva, I.: Fuzzy transform: application to the Reef growth problem. In: Demicco, R.V., Klir, G.J. (eds.) Fuzzy Logic in Geology, pp. 275–300. Academic Press, Amsterdam (2003)
Perfilieva, I.: Fuzzy transform: theory and application. Fuzzy Sets Syst. 157, 993–1023 (2006)
Perfilieva, I., Danková, M., Bede, B.: Towards a higher degree F-transform. Fuzzy Sets Syst. 180, 3–19 (2011)
Perfilieva, I., Števuliáková, P., Valášek, R.: F-transform-based shooting method for nonlinear boundary value problems. Soft Comput. 21(13), 3493–3502 (2017)
Perfilieva, I., Števuliáková, P., Valášek, R.: F-transform for numerical solution of two-point boundary value problem. Iran. J. Fuzzy Syst. (2017, in press)
Scholkopf, B., Smola, A., Mulller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10(5), 1299–1319 (1998)
Troiano, L., Mejuto, E., Kriplani, P.: An alternative estimation of market volatility based on fuzzy transform. In: Proceedings of IFSA-SCIS 2017, Otsu, Shiga, Japan (2017)
Acknowledgment
This work was supported by the project LQ1602 IT4Innovations excellence in science. The additional support was also provided by the Czech Science Foundation (GAČR) through the project of No. 16-09541S.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Perfilieva, I. (2018). Dimensionality Reduction by Fuzzy Transforms with Applications to Mathematical Finance. In: Anh, L., Dong, L., Kreinovich, V., Thach, N. (eds) Econometrics for Financial Applications. ECONVN 2018. Studies in Computational Intelligence, vol 760. Springer, Cham. https://doi.org/10.1007/978-3-319-73150-6_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-73150-6_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73149-0
Online ISBN: 978-3-319-73150-6
eBook Packages: EngineeringEngineering (R0)