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Linear Smoothers
Published in Taylor Arnold, Michael Kane, Bryan W. Lewis, A Computational Approach to Statistical Learning, 2019
Taylor Arnold, Michael Kane, Bryan W. Lewis
Many derivations of smoothing splines begin with a version of the Lagrange form shown in Equation 4.57, but with a Lagrange parameter on the integral smoothing term instead of the least squares term. But the derivation is effectively the same. You can see this, for instance, in the interesting cases when z = 0 and μ > 0 in Equation 4.57 by multiplying through by λ=1/μ. $ \lambda = 1/\mu . $ B-splines can be used to derive smoothing splines in a similar way and are often used in software implementations. The banded matrix structure noted above is a consequence of the derivative continuity condition on f in Equation 4.56, and B-spline implementations similarly yield banded systems. However, they exhibit potentially better numerical stability in edge cases.
Smoothing Scatterplots
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
We now turn our attention to another method called smoothing splines. The word spline comes from the engineering community, where draftsmen used long thin strips of wood called splines. They used these splines to draw a smooth curve between points; different global curves are produced when the positions of the points are changed. As we will see, a smoothing spline is a solution to a constrained optimization problem, where we optimally trade off fidelity of the fit to the data with smoothness of the estimate. To set the stage, we will first briefly describe parametric spline regression models, where a piecewise polynomial model is used to find the fit between the predictor variable and the response. This will be followed by a discussion of how these ideas can be extended to provide scatterplot smooths based on splines.
Hybridizing moiré with analytical and numerical techniques: differentiating moiré measured displacements to obtain strains
Published in C A Walker, Handbook of Moiré Measurement, 2003
One-dimensional splines are effective when the situation under consideration involves differentiating a single measured displacement, particularly if slopes are only needed in one, consistent direction. ‘Smoothing splines’ enable measured data to be smoothed as well as differentiated. While one-dimensional splines can be used to differentiate multiple displacements, i.e. both in-plane displacement components of a plane-stress problem, the situation is now more labour-intensive and cross-derivatives (say shear strains) are often less reliable. In [1–3] one-dimensional ‘smoothing’ cubic splines are used to smooth and differentiate moiré data. Differentiating moiré fringes by traditional methods is most reliable when the differentiation direction is fairly normal to the fringes or, at least, when it cuts across many fringes. Nevertheless, the chain rule of differential calculus can be combined with one-dimensional splines to compute accurate derivatives in directions of relatively low fringe density [3].
Phase I Monitoring of Spatial Surface Data from 3D Printing
Published in Technometrics, 2018
As a side note, besides the kernel smoothing procedure considered here, there are several other smoothing approaches in the literature, including regression and smoothing splines, and so forth (see Qiu 2005, sec. 2.5). Smoothing splines may not be appropriate for the current problem because of its extensive computation. As pointed out earlier, a typical observed surface in the current problem has several hundred thousand observations. Its smoothing spline estimator would be difficult to compute in such cases. To use a regression spline approach, appropriate knots or basis functions need to be determined in advance. Usually, the knots should be chosen at places of a given surface with large curvature. This task, together with determination of the number of knots, is often difficult to achieve. As a comparison, the LCK estimator defined in (4) is easy to compute. The estimator is computed from all observations in the local neighborhood O(x, y) only, the number of which is usually much smaller than the total number of observations in the entire surface since the bandwidth h is often chosen small. That is the main reason why it is adopted here.
Real-time crash prediction for a long low-traffic volume corridor using corrected-impurity importance and semi-parametric generalized additive model
Published in Journal of Transportation Safety & Security, 2022
Arash Khoda Bakhshi, Mohamed M. Ahmed
The terms of additive in GAM comes from adding smoothing functions that are driven using the data. GAM estimates parameters using a double loop of iteration. The inner one estimates the parameters of the smoothing functions, and the outer one is used for convergence of GAM parameters (Jones & Almond, 1992). Equation 2 explains the fundament of GAM, where is the smoothing function for the kth nonlinear variable, and other variables are as same as equation 1. In this study, Smoothing Splines were used as smoothing functions. The next section briefly reviews Smoothing Splines.