Abstract
This article shows how the consumption rate function of a certain customer can be restored from a sequence of discrete purchases. For this, purchases are considered as integrals of the unknown consumption function with some uncertainties. To restore the consumption rate function, it is required to solve the problem of restoring the function from the sequence of integrals. Recovery takes place in the form of a cubic integral smoothing spline. Such integral spline is based on value-second derivative representation. A method for constructing such spline is briefly described. It is shown how to predict future rare events in the economy, based on the restored process parameters, such as the consumption rate.
The choice of the best smoothing parameter \(\alpha \) for cubic integral smoothing spline is considered. This article provides an equation for calculating the cross-validation score \(CV(\alpha )\). Equation for calculating the cross-validation score \(CV(\alpha )\) is adjusted due to the problem of spline identification at the ends of the interval. Oversmoothing problem and optimization of cross-validation are discussed. The problem of application of the L-curve method for cubic integral smoothing spline is considered. L-curve construction stumbles upon the problem of non-monotonicity of residuals and penalty with decreasing smoothing parameter \(\alpha \). Two options for minimizing the sum of residuals and the penalty are considered as analogous to the L-curve method. Extensive simulations have been performed to determine the best way to select the smoothing parameter. It has been shown that due to the peculiarity of the input data, the cross-validation method works only in half of the cases. Recommendations are given for choosing the best smoothing parameter for cubic integral smoothing spline.
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The reported study was funded by RFBR according to the research project N019-010-00154.
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Korablev, Y.A. Restoration of the Product Consumption Rate with Integral Cubic Smoothing Spline, Study of the Best Smoothing Parameter Choice. Acta Appl Math 180, 8 (2022). https://doi.org/10.1007/s10440-022-00509-7
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DOI: https://doi.org/10.1007/s10440-022-00509-7
Keywords
- Consumption rate
- Rare events
- Integral spline
- Smoothing
- Value-second derivative representation
- Smoothing parameter
- Cross-validation
- L-curve
- Tikhonov’s \(\alpha \) regularization parameter