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Polynomial Interpolation
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
Splines of other orders may be defined as well. The piecewise linear interpolants of the previous sections are considered linear splines and in those cases in which a thrice continuously differentiable interpolant is needed a quintic spline, with degree five polynomial sections interpolated to the data {(x0,y0),…,(xn,yn)}, may be used. However, linear and cubic splines are by far the most commonly used types of splines. We may also use other boundary conditions (for example, if the function is known to have some property such as periodicity).
Time series analysis and forecasting
Published in Amithirigala Widhanelage Jayawardena, Fluid Mechanics, Hydraulics, Hydrology and Water Resources for Civil Engineers, 2021
Amithirigala Widhanelage Jayawardena
Spline, in the ordinary sense, refers to a flexible strip used in drafting to draw a smooth curve through a set of points. In the mathematical sense too it has the same meaning. Spline methods can be used for interpolation as well as for regression. The basic objective in spline interpolation is to connect a set of data points by a smooth curve via a combination of several piecewise low order curves. The interpolation (or regression) functions can be polynomial, sinusoidal, exponential or their combinations. The cubic spline which consists of N polynomial functions each of which has order not greater than three is by far the most widely used. It is of the form y=ai+bix+cix2+dix3
Fundamentals to Geometric Modeling and Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
The Bézier curve is another type of spline. Given four control points as shown in Figure 4.15 (a), p0, p1, p2 and p3. Suppose we interpolate the two endpoints with a cubic polynomial p(u), then we have () p0=p(0), () p3=p(1).
Spatial interpolation based on previously-observed behavior: a framework for interpolating spaceborne GNSS-R data from CYGNSS
Published in Journal of Spatial Science, 2023
Although not frequently employed in satellite data analyses, spatial interpolation is commonly used with other types of geospatial data that are often collected at point locations like soil moisture (e.g. (Yao et al. 2013)), precipitation (e.g. (Tait et al. 2006)), temperature (e.g. (Stahl et al. 2006)), aerosols (e.g. (Pfister et al. 2005)), and inundation extent (e.g. (Bales and Wagner 2009)). There are myriad spatial interpolation techniques. Some techniques, like linear, cubic, or spline interpolations assign smoothly-varying functions to interpolate between known data points. Inverse distance weighting (IDW) is another commonly-used spatial interpolation technique, in which interpolated points are calculated using a weighted average of neighboring observations, with the closest neighbors receiving the highest weights. Other interpolation methods, like the linear regression technique, use empirical relationships between the variable of interest and ancillary data to predict the variable at unsampled locations. For example, observed precipitation from rain gauges might be regressed against topographic variables like elevation, and then precipitation at unsampled locations will be interpolated using this regression (Yao et al. 2013).
Anisotropic spline approximation with non-uniform B-splines
Published in Applicable Analysis, 2018
B-splines are used in different fields of applications such as Computer-aided design and finite-element analysis. In this context, lately, tensor product splines became important due to their good approximation properties over or boxes and their efficient implementation.[1,2] However, over arbitrary domains tensor product spline approximation cannot be used directly since some of those properties get lost. The concept of diversification of uniform tensor product B-splines over subsets is crucial for the construction [3] of a spline space such that an anisotropic error estimate with respect to the sup-norm can be obtained. Diversification itself is not connected to the uniformity of tensor product B-splines but the methods for proving the error estimate are tied to it. The aim of this paper is a generalisation of the results in [3] in two different ways. We adapt the construction to arbitrary knot sequences and, at the same time, consider the error with respect to the -norm for .
Spatial Distribution and Potential Ecological Risk Assessment of Trace Metals in Reclaimed Mine Soils in Abuakwa South Municipal, Ghana
Published in Soil and Sediment Contamination: An International Journal, 2023
Douglas Siaw Baah, Gordon Foli, Emmanuel Gikunoo, Solomon S. R. Gidigasu
The data obtained contained both the coordinates of the locations and also the concentrations of the various trace metals such as As, Cr, Ni, and Pb. Both data were imported into the ARC GIS software specifically ARC Map. Spline interpolation within the spatial analyst tools was used to generate the various concentrations of each trace metal. Spline interpolation makes use of a mathematical function that minimizes the overall surface curvature resulting in a smooth surface that passes exactly through the input points. Due to the curvature and continuous nature of the soil sample, spline interpolation was best for such depiction, hence its use. Maps of the various concentrations generated were produced as an outcome and presented in Figure 5.