This paper describes the technique of thin plate or laplacian smoothing splines and its application to continentwide interpolation of climate data down to resolutions of less than 1 km. The smoothing spline that the customer likes is a cubic spline, but sas supports thin plate splines and penalized bsplines, just to name two others. It was reasoned that this was due to short range correlation in the. Instead of radial basis functions we present a method based on the use of polynomials with local support defined on finite element grids. As the smoothing parameter varies from 0 to 1, the smoothing spline varies, from the leastsquares approximation to the data by a linear polynomial when p is 0, to the thin plate spline interpolant to the data when p is 1. The problem is that i get negative values for rainfall which is impossible is.
Thin plate splines 4 comparing kriging and spline interpolation 5 references d g rossiter cu empirical interpolation. A spline is one way to fit a smooth curve to twodimensional data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the. Thin plate splines tps are a spline based technique for data interpolation and smoothing. Creating highresolution digital elevation model using. Thin plate splines, with the degree of smoothing determining by minimising generalised cross validation, can estimate this smooth function in two ways. Thin plate smoothing spline interpolation of daily rainfall for new zealand using a climatological rainfall surface andrew tait, roddy henderson, richard turner and xiaogu zheng national institute of water and atmospheric research, private bag 14901, kilbirnie, wellington, new zealand received 6 june 2005 revised 5 march 2006 accepted 10. Abstract two dimensional thin plate smoothing splines were used to interpolate 100 daily rainfall values, with the degree of data smoothing determined by. The thin plate spline is the 2d generalization of the cubic spline. The thinplate smoothing spline model is a mathematically elegant method for surface estimations that has been progressively developed over the last decade. Pdf the underlying formal connections between thin plate splines and kriging are examined from the point of view of enhancing usage of both methods find, read and cite all the research you.
Thinplate splines have the edge in terms of mse over the cubic regression spline, but are much more costly to set up the basis function for. In this paper, we carry over the concept of the thin plate spline from the plane to the sphere. Approximation methods for thin plate spline mappings and. Thinplate splines are a well established technique for the interpolation and smoothing of scattered data. Smoothing splines are function estimates, obtained from a set of noisy observations of the target, in order to balance a measure of goodness of fit of to with a derivative based measure of the smoothness of. Monte carlo simulation applied in the confidence analysis of the elevation model is also discussed. However, the traditional formulation of the method leads to large, dense and often ill. The thin plate spline approach can be generalized to smoothing with respect to more than two dimensions and to other orders of differentiation in the penalty.
A major problem in the application of thin plate splines is that their computation involves the solution of a linear system that is illconditioned for large data sets. In chapter 4 the procedure of the dem creation is presented. Thin plate splines have the edge in terms of mse over the cubic regression spline, but are much more costly to set up the basis function for. This method is more efficient when dealing with large data sets as the resulting. Thinplate smoothing spline modeling of spatial climate data.
This is a special case of the spatial process estimate. The algorithm is based on a weighted sum of locally defined thin plate splines, and yields an interpolation function which is differentiable. Contribute to oltthinplatespline development by creating an account on github. Initial application of the procedure splina to fit a second order thin plate smoothing spline to the data set yielded exact interpolation. However, the traditional formulation of the method leads to. This study presents a method for estimating daily rainfall on a 0. May 11, 2016 hello, i am using thin plate spline to interpolation rainfall data. Smoothing splines natural cubic splines smoothing splines i fixthelocationsofkknots atquantilesofx. Id be interested to see whether the effects of the two splines are really that different given the uncertainty in their estimation at end points of a variable, esp if the data is patchy. Introduction the problem of constructing interpolation functions for sets of scattered data in two independent variables has been treated in many papers. Standard thin plate splines require the solution of a dense linear system of equations whose size increases with the. Approximation of a thin plate spline smoother using.
Compute thin plate spline interpolation and deformation on a 3d surface. This lecture will discuss the thin plate spline, and other isotropic methods for smoothing in two and sometimes more dimensions. In other words smoothing noisy data values observed at n distinct points on a. Opencv suitable implementation for thin plate spline. Opencv suitable implementation for thin plate spline warping. Three ways to add a smoothing spline to a scatter plot in.
The gammod procedure uses thinplate regression splines wood 2003 to construct spline basis expansions. Approximatethinplatesplinemappings cornell university. Probability distribution function pdf of daily rainfall from rams x 20 km. A major problem in the application of thinplate splines is that their computation involves the solution of a linear system that is. Pdf interpolation of rainfall data with thin plate smoothing. This can be useful in its own right, shows how the mathematics work, and allows easy visualization. Recall that natural cubic splines are piecewise cubic polynomial and exact.
Pdf on thin plate splines and kriging researchgate. In n dimensions, the idea of thinplate splines is to choose a function fx that exactly. A summary of the basic methodology, with climate interpolation principally in mind, can be found in hutchinson 1991. The existence of a solution to our smoothing problem. Using the curve fitting app or the fit function, you can fit cubic spline interpolants, smoothing splines, and thin plate splines. Pdf the thin plate spline tps is an interpolation approach that has been. Key features of thin plate smoothing spline analyses are their robustness and operational simplicity. Three ways to add a smoothing spline to a scatter plot in sas. Finite element thinplate splines finite element thinplate spline tpsfem is a method that combines the.
The thin plate smoothing spline techniques involved in this study, and the results of their application to the ar1 parameters, are discussed in the following sections. The thin plate spline method is often used to fit data in high dimensions. For a simple example showing how to use splines to perform interpolation, see cubic spline interpolation. To develop the spherical tps, we utilize the idea of an elastic shell that is attributed with the bending energy and the. The problem is that i get negative values for rainfall which is impossible is that normal. An important diagnostic associated with thin plate smoothing spline analyses is the signal of the fitted spline, as estimated by the trace of the influence matrix. The interpolation and smoothing of scattered data is a. I implemented the thin plate spline algorithm see also this description in order to interpolate scattered data using python my algorithm seems to work correctly when the bounding box of the initial scattered data has an aspect ratio close to 1. Thin plate splines are a well established technique for the interpolation and smoothing of scattered data. The thin plate spline tps is an interpolation approach that has been developed to investigate a frequently occurring problem in geosciences. Abstract thin plate smoothing splines incorporating varying degrees of topographic dependence were used to interpolate 100 daily rainfall values, with the. Robust point matching rpm is a common extension and shortly known as the tpsrpm algorithm.
Thin plate smoothing spline interpolation of daily rainfall for new. It is now well known that there is a unique interpolating minimizer of f uit is the thinplate spline. They are an important special case of a polyharmonic spline. Splines can be used to smooth noisy data and perform interpolation. The smoothing spline that the customer likes is a cubic spline, but sas supports thinplate splines and penalized bsplines, just to name two others.
I am using thin plate spline to interpolation rainfall data. Fits a thin plate spline surface to irregularly spaced data. Thin plate spline thin plate spline tps is a technique for interpolating and smoothing surface over scattered data 1. Thinplate smoothing spline modeling of spatial climate.
Using the curve fitting app or the fit function, you can fit cubic spline interpolants, smoothing splines, and thinplate splines. Interpolation of rainfall data with thin plate smoothing. In this example, you fit a thin plate spline curve to data in the miningx data set. I am trying to understand thin plate spline interpolation of scattered data.
A new smoothing method is proposed which can be viewed as a finite element thin plate spline. Thin plate spline interpolation, one of the main algorithms for multivariable scattered data approximation, was introduced by duchon in 56 and has since generated a vast amount of research on its theory and applications for a comprehensive survey, see wendland 18. Smoothing splines, local regression, and gams reading. The existence of a solution to our smoothing problem is demonstrated. Interpolation of rainfall data with thin plate smoothing splines. As i understand it tps is just a special case of radial basis function interpolation. Data mining and analysis jonathan taylor nov 5, 2018 slide credits.
They were introduced to geometric design by duchon. As the dimension increases there are some restrictions on the smallest order of differential that can be used, 1 but actually duchons original paper, 9 gives slightly more. Thinplate splines 4 comparing kriging and spline interpolation 5 references d g rossiter cu empirical interpolation. Thin plate spline interpolation problem geonet, the esri.
This occurred with both transformed and untransformed data. Ultrasonic elastogram generation by 2d thin plate smoothing. Thinplate splines are an attractive method for interpolating and smoothing arbitrarily spaced points in the plane. However, scaling one of the data points coordinates changes the interpolation result. Smooth interpolation of scattered data by local thin plate.
The method is based on first order techniques similar to mixed finite element techniques for the biharmonic equation. Partial thin plate splines partial thin plate splines have been described in detail by wahba 1990. Compared to thinplate smoothing splines, thinplate regression splines produce fewer. The thin plate spline tps is a commonly used basis function for representing coordinate mappings from r2 to r2. Thin plate smoothing spline tpss is mainly 2d generalization of cubic spline 18. Tpsfem smoother sx is represented as a linearcombination of piecewise linear basis functions. Thin plate smoothing spline interpolation of daily rainfall for new zealand using a climatological rainfall surface andrew tait, roddy henderson, richard turner and xiaogu zheng national institute of water and atmospheric research, private bag 14901, kilbirnie, wellington, new zealand received 6 june 2005 revised 5 march 2006. Interpolating mean rainfall using thin plate smoothing splines. The thin plate smoothing spline model is a mathematically elegant method for surface estimations that has been progressively developed over the last decade. They include thin plate splines wahba and wendelberger 1980 as a special case. To develop the spherical tps, we utilize the idea of an elastic shell that is attributed with the bending. Aug 26, 2006 traditional thin plate splines use radial basis functions and require the solution of a dense linear system of equations whose size is proportional to the number of data points.
In 5we discuss the disadvantages of using smoothing splines for interpolation. Error indicators and adaptive refinement of finite element. Pdf interpolation of rainfall data with thin plate. Jul 31, 2006 thin plate splines are an attractive method for interpolating and smoothing arbitrarily spaced points in the plane. The thinplate regression splines are based on thinplate smoothing splines duchon 1976, 1977.
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