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Power Curve Modeling and Analysis
Published in Yu Ding, Data Science for Wind Energy, 2019
The spline-based regression is to use piecewise polynomials to model a nonlinear response. One of the popular spline functions used is the cubic spline [86]; see Fig. 5.5, middle panel, for an illustration. A cubic spline partitions the input domain into a few segments, which is in fact an action of binning, and models each segment using a cubic polynomial. In order to produce a smooth, coherent model for the whole domain, a cubic spline imposes continuity and smoothness constraints at the partition points, known as knots. In Fig. 5.5, two knots are used and denoted as ξ1 and ξ2, respectively. Although ξ1 and ξ2 partition the input domain in Fig. 5.5 into three roughly equal parts, knots in general do not have to be evenly spaced. Each cubic polynomial is specified by four parameters, producing a total of 12 parameters for the three piecewise cubic polynomials. The constraints imposed at the partition points, however, reduce the number of actual parameters that need to be estimated. For the cubic spline in Fig. 5.5, there are three constraints at each knot, which require, respectively, the equality of the function value, that of its first-order derivative and that of the second-order derivative, at each of the partition points. With the six constraints considered, the number of actual parameters to be estimated for the cubic spline is six.
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Published in Yeong Koo Yeo, Chemical Engineering Computation with MATLAB®, 2020
Table 2.17 shows experimental data on a pressure drop (kPa) according to flow rates (liter/sec) in a filter. Perform cubic spline interpolation using the built-in function spline.
Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
All splines considered on this page are cubic splines; they are all piecewise cubic functions. However, anyone who says “cubic spline” usually means a special cubic spline with continuous first and second derivatives. The cubic spline is given by the function values in the nodes and derivative values on the edges of the interpolation interval (either of the first or second derivatives). If the exact values of the first derivative in both boundaries are known, the spline is called a clamped spline, or spline with exact boundary conditions. This spline has interpolation error O(h4).If the value of the first (or second) derivative is unknown, we can set the so-called natural boundary conditions S″(A) = 0, S″(B) = 0. Thus, we get a natural spline with interpolation error O(h2). The closer to the boundary nodes, the greater the error becomes. In the inner nodes the interpolation accuracy is much better.One more boundary condition we can use when boundary derivatives are unknown is the parabolically terminated spline. The boundary interval is represented as the second (instead of the third) degree polynomial (for inner intervals, third degree polynomials are still used). In a number of cases this provides better accuracy than natural boundary conditions.We can also set periodic boundary conditions (this kind of conditions is used to model periodic functions).
An Advanced Statistical Approach to Data-Driven Earthquake Engineering
Published in Journal of Earthquake Engineering, 2020
Ikkyun Song, In Ho Cho, Raymond K. W. Wong
On one hand, cubic spline is a curve constructed by combining a number of cubic polynomial sections. Those sections join at a certain point, called “knot,” of which location should be pre-selected for the cubic spline basis. The cubic polynomial sections are joined such that the entire spline becomes continuous up to second derivative. Although somewhat different from the practical regression splines [see Wood, 2006], to help grasp a sense of relevant mathematical forms, some cubic spline functions [Gu, 2013] are given by
Optimal sensors placement for structural health monitoring based on system identification and interpolation methods
Published in Journal of the Chinese Institute of Engineers, 2021
In Limongelli (2003), the cubic spline interpolation is used to interpolate the signal without setting sensors on the floor, while still creating the simulated response. The advantages of cubic spline interpolation functions are high stability, smoothness of the interpolated curve, and simplicity of calculation. This study uses this interpolation method to replace the number of sensors’ measurements and simplify the setting work.
Physical fitness spurts in pre-adolescent boys and girls: Timing, intensity and sequencing
Published in Journal of Sports Sciences, 2022
Sara Pereira, Carla Santos, Go Tani, Duarte Freitas, Fernando Garbeloto, Eduardo Guimarães, Leah E. Robinson, Adam Baxter-Jones, Peter T. Katzmarzyk, José Maia
We used the original methodology by Van’t Hof et al. (1976) to estimate age-at-peak MGS, which has been extended and applied to Belgium boys (Beunen & Malina, 1988), Spanish boys and girls, (Yague, 1998) and Brazilian boys and girls (Silva et al., 2019). This methodology is valid for estimating the age-at-peak MGS if the following conditions are satisfied: (i) the presence of a real spurt, i.e., an increase in velocity, (ii) a specific slowdown of growth velocity after the peak, and (iii) the presence of reduced measurement error in all variables, i.e., reliable data. Failure to meet these assumptions implied that the MGS was not identified. Since each child has their own growth velocity curve as well as in all PF tests, mean constant curves were obtained for data aligned by age-at-peak MGS. Thus, the possibility to compute mean PF velocity values before and after age-at-peak MGS, i.e., −18, −12, −6, and +6, +12, +18 months from the mid-growth maximum velocity which is marked as 0. A generalization of the previous methodology was developed by a mathematician at the University of Porto, who developed the software to conduct all analyses. This methodology has been previously shown to be successful in identifying gross motor coordination and PF spurts in Portuguese children (Dos Santos et al., 2019; Guimarães et al., 2020; Pereira et al., in press). Mean velocity curves, also called mean constant curves, were developed and defined in terms of time, i.e., months before and after age-at-peak MGS. Although measurements were obtained each year, the method allows the estimation of individual velocities every six months. Graphical data were displayed using a cubic spline procedure as implemented in GraphPad Prism v.8.2.1. (Malina et al., 2004). A cubic spline employs interpolating cubic polynomials, which use information from neighbouring points to obtain a degree of global smoothness. The cubic spline was chosen over other curve fitting procedures because it maintains the integrity of the data without transforming or modifying the underlying growth characteristics. To test for differences in age-at-MGS as well as in PF spurts’ intensities between boys and girls, a Student t-test was used with a significant level of 5%, and these analyses were done in IBM-SPSS v.27.