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Applications of fractal interpolants in kernel regression estimations

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  • Liu, Chiao-Wen
  • Luor, Dah-Chin

Abstract

Generating a function from its finite set of samples is widely studied in data fitting problems. The theory of nonparametric modeling has been developed by many researchers, and several types of estimators have been established in the literature. On the other hand, the fractal theory provides new technologies for making complicated curves and fitting experimental data, and fractal interpolation functions are established as generalizations of traditional interpolation techniques. The aim of this paper is to investigate applications of (smooth) fractal interpolation functions in nonparametric kernel regressions. We discuss a key sufficient condition in detail in the construction of (smooth) fractal interpolation functions. Then the Priestley–Chao estimator is considered, and we establish its fractal perturbation. The raw data we consider in our example is the Crude Oil WTI Futures daily highest price from 2021/10/13 to 2022/7/29. We take 11 samples from the raw data to construct a fractal perturbation of the Priestley–Chao regression function. SLSQP is applied to find the optimal values of hyperparameters so that the given empirical mean squared error is minimum. We see that nonparametric kernel regressions for finite sample data are widely used in applications, and we can decrease the empirical mean squared error further by considering fractal perturbations of these regression functions.

Suggested Citation

  • Liu, Chiao-Wen & Luor, Dah-Chin, 2023. "Applications of fractal interpolants in kernel regression estimations," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    • Handle: RePEc:eee:chsofr:v:175:y:2023:i:p1:s0960077923008147
    DOI: 10.1016/j.chaos.2023.113913
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    References listed on IDEAS

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    1. Srijanani Anurag Prasad, 2021. "Super Coalescence Hidden-Variable Fractal Interpolation Functions," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(03), pages 1-9, May.
    2. Viswanathan, P., 2022. "A revisit to smoothness preserving fractal perturbation of a bivariate function: Self-Referential counterpart to bicubic splines," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    3. Luor, Dah-Chin, 2020. "On the distributions of fractal functions that interpolate data points with Gaussian noise," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    4. Luor, Dah-Chin, 2018. "Fractal interpolation functions for random data sets," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 256-263.
    5. Tyada, K.R. & Chand, A.K.B. & Sajid, M., 2021. "Shape preserving rational cubic trigonometric fractal interpolation functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 866-891.
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