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A hybrid form of fractal-type functions in data fitting and parameters search by sequential quadratic programming

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Abstract

We consider a hybrid form of combinations of regular continuous functions and irregular fractal interpolation functions in data fitting problems. Three types of models are considered in this paper. We apply a sequential quadratic programming method, SLSQP, to find the values of hyperparameters in these models that minimize the given empirical error. Two examples are given to show the results. The advantage of our approach is that it is not limited to types and construction methods of fractal functions.

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Data availability

The datasets generated during the current study are available in the following.

Example 4.1: https://www.investing.com/crypto/bitcoin/btc-usd-historical-data

Example 4.2: https://www.investing.com/indices/nq-100-historical-data

References

  1. M.F. Barnsley, Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. M.F. Barnsley, Fractals Everywhere (Academic Press, New York, 1988)

    MATH  Google Scholar 

  3. M. F. Barnsley, J. Elton, D. Hardin, P. Massopust, Hidden variable fractal interpolation functions. SIAM J. Math. Anal. 20, 1218–1242 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. F. Barnsley, P. R. Massopust, Bilinear fractal interpolation and box dimension. J. Approx. Theory 192, 362–378 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. P. T. Boggs, J. W. Tolle, Sequential quadratic programming. Acta Numer. 4, 1–51 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. A. K. B. Chand, G. P. Kapoor, Generalized cubic spline fractal interpolation functions. SIAM J. Numer. Anal 44, 655–676 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. A. K. B. Chand, M. A. Navascués, Natural bicubic spline fractal interpolation. Nonlinear Anal. 69, 3679–3691 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Chandra, S. Abbas, Box dimension of mixed Katugampola fractional integral of two-dimensional continuous functions. Fract. Calc. Appl. Anal. 25, 1022–1036 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Chandra, S. Abbas, On fractal dimensions of fractal functions using function spaces. Bull. Aust. Math. Soc. 106, 470–480 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Drakopoulos, P. Manousopoulos, One dimensional fractal interpolation: Determination of the vertical scaling factors using convex hulls, in Topics on Chaotic Systems: Selected Papers from Chaos 2008 International Conference (2009)

  11. P. E. Gill, E. Wong, Sequential quadratic programming methods, in Mixed Integer Nonlinear Programming, ed. by J. Lee, S. Leyffer. The IMA Volumes in Mathematics and its Applications, vol 154 (Springer, New York, NY, 2012), pp. 147–224

  12. D. Kraft, A software package for sequential quadratic programming. Technical Report. DFVLR-FB 88-28, DLR German Aerospace Center-Institute for Flight Mechanics Koln, Germany (1988)

  13. D.-C. Luor, Fractal interpolation functions for random data sets. Chaos Solitons Fract. 114, 256–263 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. P. Manousopoulos, V. Drakopoulos, T. Theoharis, Parameter identification of 1D fractal interpolation functions using bounding volumes. J. Comput. Appl. Math. 233, 1063–1082 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. P. Manousopoulos, V. Drakopoulos, T. Theoharis, Parameter identification of 1D recurrent fractal interpolation functions with applications to imaging and signal processing. J. Math. Imaging Vision 40, 162–170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. A. Marvasti, W. Strahle, Fractal geometry analysis of turbulent data. Signal Process 41, 191–201 (1995)

    Article  MATH  Google Scholar 

  17. P. R. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets (Academic Press, San Diego, 1994)

    MATH  Google Scholar 

  18. P. R. Massopust, Interpolation and Approximation with Splines and Fractals (Oxford University Press, New York, 2010)

    MATH  Google Scholar 

  19. D. S. Mazel, M. H. Hayes, Using iterated function systems to model discrete sequences. IEEE Trans. Signal Process 40, 1724–1734 (1992)

    Article  ADS  MATH  Google Scholar 

  20. M. A. Navascués, V. Sebastián, Generalization of Hermite functions by fractal interpolation. J. Approx. Theory 131, 19–29 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Viswanathan, A. K. B. Chand, Fractal rational functions and their approximation properties. J. Approx. Theory 185, 31–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The financial support was provided by Ministry of Science and Technology, R.O.C. under Grant MOST 110-2115-M-214-002.

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Authors and Affiliations

  1. Department of Data Science and Analytics, I-Shou University, No. 1, Sec. 1, Syuecheng Road., Dashu District, Kaohsiung City, 84001, Taiwan, ROC

    Dah-Chin Luor

  2. Department of Applied Science, School of Academic Studies, R.O.C. Naval Academy, No. 669, Junxiao Road, Zuoying District, Kaohsiung City, 81345, Taiwan, ROC

    Chiao-Wen Liu

Authors
  1. Dah-Chin Luor
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  2. Chiao-Wen Liu
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Contributions

Both authors contributed to the study conception and design. Material preparation and analysis were performed by [Dah-Chin Luor]. Data collection, experiment design, and program coding were performed by [Chiao-Wen Liu]. The first draft of the manuscript was written by [Dah-Chin Luor] and both authors commented on previous versions of the manuscript. Both authors read and approved the final manuscript

Corresponding author

Correspondence to Dah-Chin Luor.

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Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Framework of Fractals in Data Analysis: Theory and Interpretation. Guest editors: Santo Banerjee, A. Gowrisankar.

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Luor, DC., Liu, CW. A hybrid form of fractal-type functions in data fitting and parameters search by sequential quadratic programming. Eur. Phys. J. Spec. Top. 232, 969–978 (2023). https://doi.org/10.1140/epjs/s11734-023-00776-x

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