Interior Knots Research Articles

Efficient Generalized Ridge Regression

Abstract The original ridge estimator of the unknown p×1 vector of β-coefficients in a linear model used a single scalar, k, to determine a point on a shrinkage path of finite length that extends from the Ordinary Least Squares estimator, ^β 0, to the shrinkage terminus (usually ^β ≡ 0). Generalized ridge estimators use two or more parameters to determine not only the shape of their shrinkage path but also a specific point on that path. The efficient generalized ridge regression path proposed here is a continuous two-piece linear function that (1) starts at ^β 0, the Best Linear Unbiased Estimator, (2) has its interior knot at the ^ β-estimator with Maximum Likelihood of achieving overall minimum MSE risk under normal distribution-theory, and (3) ends at the shrinkage terminus. This new path is efficient in the senses that it is the shortest path and, at least when p > 2, essentially the only known shrinkage path that always contains the ^ β-vector that is most likely to be optimally biased. Functions in R-packages freely distributed via CRAN perform the calculations and produce the graphics used here to illustrate shrinkage concepts by interpreting ridge Trace diagnostic plots. These new concepts and visual tools provide invaluable data-analytic insights and improved self-confidence to applied researchers and data scientists fitting linear models when p, the number of non-constant X-predictor variables in the model, is strictly less than the number of observations available.

Formula omitted] Hermite interpolation by planar PH B-spline curves with shape parameter

We solve the problem of G2∕C1 Hermite interpolation (i.e. interpolation of prescribed boundary points as well as first derivatives and curvatures at these points) by planar quintic Pythagorean Hodograph B-spline curves with one free interior knot which acts as a shape parameter. We present conditions on the data ensuring the existence of solutions. Finally, we illustrate the influence of the interior knot on the shape of the resulting interpolant and on the values of the absolute rotation index or the bending energy.

Orthogonal polynomials, biorthogonal polynomials and spline functions

A sequence of continuous real functions ( f m ) m ∈ N 0 , with exponential decay at infinity that implies their Fourier-Laplace transforms F m ( i z ) are analytic on | ℜ ( z ) | < B , z ∈ C , generates a sequence of polynomials ( Q k ) k ∈ N 0 , that are biorthogonal to the distributional derivatives μ m : = ( − 1 ) m f m ( m ) , m ∈ N 0 . The generating function is a generalized Taylor series expansion of the Fourier-Laplace kernel e x z in terms of μ ˆ m ( i z ) = z m F m ( i z ) , m ∈ N 0 . If μ m = ω P m , where P m are polynomials of degree m and ω is a weight function, Q m is a constant multiple of P m and they are orthogonal polynomials. In the case where f m are uniform or quasi-uniform B -splines, the corresponding biorthogonal polynomials exhibit properties reminiscent of Appell sequences, while in the case where f m are refinable functions they are eigenfunctions of the adjoint of the corresponding refinement operators. For nonuniform B -splines, a suitable choice of the knot sequence defines ( f m ) m ∈ N 0 in which the corresponding polynomials Q k reduce to the Jacobi polynomials and the biorthogonal spline functions μ m reduce to the corresponding weighted Jacobi polynomials, if there are no interior knots.

Performance evaluation of regression splines for propensity score adjustment in post-market safety analysis with multiple treatments

ABSTRACTObservational studies provide a core resource in assessing post-market drug safety and effectiveness. Propensity scores are a predominant method for confounding adjustment to achieve unbiased estimation of average treatment effects in observational data. However, the use of propensity score methods has been limited to comparing two treatment groups, while medical situations frequently present with multiple treatment options. Inverse probability of treatment weighting (IPTW) is a popular propensity score adjustment method, but its performance degrades with decreased positivity leading to extreme weights, a problem that can be amplified with multiple treatment groups. Meanwhile, regression on a spline of the propensity score has shown favorable performance compared to other propensity score methods in recent studies involving two treatments. This project utilizes a simulation study to compare IPTW and propensity score splines as adjustment methods in a three-treatment setting. We test a variety of spline methods, including natural cubic splines with varying numbers of interior knots, and thin-plate regression splines. We vary several parameters across simulations, including the degree of propensity score overlap among treatment groups, treatment prevalence, outcome prevalence, and true marginal relative risk. We assess methods based on their bias, root mean squared error, and coverage of the true marginal relative risk across simulations. We find that all methods perform similarly well when there is good propensity score distribution overlap. However, with even moderate decrease in overlap or low outcome prevalence, IPTW produces more biased estimates and higher variance than propensity score splines. Low treatment prevalence or unequal treatment prevalences across groups also worsens IPTW performance. Overall, a natural cubic spline with a relatively small number of interior knots provides good performance across a range of simulations.

180 Comprehensive Study of Posttraumatic Cerebral Energy Metabolism

INTRODUCTION: Previous studies have documented dysfunctional cerebral metabolism following traumatic brain injury (TBI), characterized by reduction in cerebral metabolic rates (CMRs) of glucose and oxygen. In our largest series to date, here, we provide further evidence and time courses of these metabolic changes. METHODS: CMRs for oxygen, glucose, and lactate were determined in 74 TBI subjects over 6 postinjury days by a modified Kety-Schmidt method and in 35 normal subjects. Comparison to normal, and time courses, utilized a mixed-effects model, accounting for repeated measurements in the trauma group. Time courses are represented as a restricted cubic spline with 3 interior knots placed at the quartiles of postinjury hours. A random intercept for subject was used as the random-effects structure. RESULTS: Post-TBI CMRO2 (0.599 vs 1.391; P CONCLUSION: In our largest series of patients to date, we have demonstrated that posttraumatic cerebral metabolism is characterized by depressed glucose and oxygen metabolism that is persistent for at least 6 days postinjury. There is mismatch between glucose and oxygen utilization, indicated by diminished metabolic ratio, and frequent lactate uptake. Further study is required to fully characterize the dysfunctional metabolism, which may be a source of further secondary injury in the early postinjury period. Language: en

Generation of spherical non-uniform rational basis spline curves and its application in five-axis machining

A method of generating spherical non-uniform rational basis spline curves based on De Boor’s algorithm is presented in this article. The spherical curve preserves many good properties from non-uniform rational basis spline curves in Euclidean space, such as local modification property, convex hull property, rotation invariant property, knot insertion property, and so on. Construction of closed spherical non-uniform rational basis spline curve will be discussed too. Furthermore, a progressive iterative approximation scheme is proposed to approximate the given points on unit sphere using spherical non-uniform rational basis spline curves. The convergence and curve continuity will be discussed. Since the analytical expression of the spherical non-uniform rational basis spline curve is messy, numerical differentiation is used to test its continuity at interior knots. Finally, several examples are presented to verify the effectiveness of the proposed method. The proposed method is applied on tool path generation of five-axis machining to avoid interference by adjusting fewer directions and obtain smooth tool path simultaneously.

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C 2 Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C 2 quintic bases on uniform triple knots are constructed for both open and closed C 2 curves, and are used to derive simple explicit formulae for the B–spline control points of C 2 PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C 2 to C 1. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.

A plug-in the number of knots selector for polynomial spline regression

A plug-in the number of interior knots (NIKs) selector is proposed for polynomial spline estimation in nonparametric regression. The existence and properties of the optimal NIKs for spline regression are established by minimising the weighted mean integrated squared error. We obtain plug-in formulae for the optimal NIKs based on the theoretical results of asymptotic optimality, and develop strategies for choosing the NIKs of the spline estimator. The proposed NIKs selection method is tested on our simulated data with quite satisfactory performance, and is illustrated by analysing a fossil data set.

Cam Curve Synthesis Method Based on Classical Splines

This paper proposes a cam curve synthesis method based on classical splines. The cam curve is described by the classical spline of order 6. Corresponding equations and boundary conditions are given to satisfy the specific requirements of the cam design problem. Then, a universal computation method is given and an example to illustrate the effectiveness of the proposed method is presented. It is shown that excellent design result can be gained easily, and the designer can adjust the interpolation values of the interior knots of the spline to change the cam curve in a local influence without any violation of the specific constraints.

Blind watermarking of NURBS curves and surfaces

This paper presents two watermarking methods for NURBS curves and surfaces. Both methods are blind, shape-preserving and data amount-preserving. These features are often required in watermarking of CAD models. The first method is based on the replacement of exterior knots of NURBS. Watermarks are embedded into the cross-ratio of four knots in the knot vector(s) and the method is robust to affine transformation, Möbius reparameterization, and interior knot insertion and removal operations. The second method is based on reparameterization using Möbius transformation. Watermarks are embedded into the ratio of two knot intervals selected from the knot vector(s) and the method is robust to affine transformation and linear reparameterization. The capacity of the first method is m−1 for a degree m NURBS curve and m+n−2 for a degree m×n NURBS surface, and the capacity of the second method is 1 for a curve and 2 for a surface. Experiments have been conducted to demonstrate the shape-preserving, data amount-preserving properties and robustness.

Predicting board strength by X-ray scanning of logs: The impact of different measurement concepts

The objective of this study was to compare the individual board strength predictions from an X-ray log scanner by using either two or four X-ray directions. The benefit of applying traceability between log and board was also studied. In total, 119 Norway spruce [Picea abies (L.) Karst.] sawlogs were scanned by an X-ray log scanner at the log sorting station of a sawmill and sawn into two centre pieces per log. Individual board traceability was enabled by following the rotational position of the log in the scanner and at the succeeding sawing. All boards were graded by a commercial strength grading machine before destructive testing was done. The resulting data were used to derive variables for building multivariate partial least squares strength prediction models. In the modelling a hierarchical modelling approach was used, where annual ring width, dry density and elasticity were also modelled. For all concepts studied the models’ fit was similar. Only minor benefits could be found when using four directions and traceability compared with two directions and no traceability. One conclusion is that the result for traceability, from four directions in particular, is more sensitive for the interior knot reconstruction result. The strength prediction was on the same R 2 level as for the strength grading machine.

Interlacing property for B-splines

We prove that the zeros of the derivatives of any order of a B-spline are increasing functions of its interior knots. We then prove that if the interior knots of two B-splines interlace, then the zeros of their derivatives of any order also interlace. The same results are obtained for Chebyshevian B-splines.

G1 continuity conditions of adjacent NURBS surfaces

For the special NURBS surfaces such as bicubic or biquartic B-spline surfaces with single interior knots, some results of the G1 continuity conditions for two adjacent surfaces appeared recently. B...

Formula omitted] continuity conditions of adjacent NURBS surfaces

For the special NURBS surfaces such as bicubic or biquartic B-spline surfaces with single interior knots, some results of the G 1 continuity conditions for two adjacent surfaces appeared recently. But for the NURBS surfaces with arbitrary degrees and generally structured knots, the G 1 continuity conditions are still unsolved. Therefore, this paper pays attention to studying this problem. In this paper, we obtain the necessary and sufficient conditions of G 1 continuity, and present two kinds of sufficient conditions of G 1 continuity for two adjacent NURBS surfaces with arbitrary degrees and generally structured knots. In particular, we obtain the intrinsic conditions for the control points and weights of the common boundary curve (so-called the uniform continuity conditions of G 1 continuity). It should be pointed out that this case is unique in the discussion of G 1 continuity of adjacent NURBS surfaces. At the end, we give the algorithm for constructing adjacent G 1 NURBS surfaces.

Reconstruction of convergent [formula omitted] smooth B-spline surfaces

Recently, there have been improvements on reconstruction of smooth B-spline surfaces of arbitrary topological type, but the most important problem of smoothly stitching B-spline surface patches (the continuity problem of B-spline surface patches) in surface reconstruction has not been solved in an effective way. Therefore, the motivation of this paper is to study how to better improve and control the continuity between adjacent B-spline surfaces. In this paper, we present a local scheme of constructing convergent G 1 smooth bicubic B-spline surface patches with single interior knots over a given arbitrary quad partition of a polygonal model. Unlike previous work which only produces (non-controllable) toleranced G 1 smooth B-spline surfaces, our algorithm generates convergent G 1 smooth B-spline surfaces, which means the continuity of the B-spline surfaces tends to G 1 smoothness as the number of control points increases. The most important feature of our algorithm is, in the meaning of convergent approximation order, the ability to control the continuity of B-spline surfaces within the given tolerance and capture the geometric details presented by the given data points.

Regression splines for threshold selection with application to a random-effects logistic dose-response model

A flexible parametric method is proposed for smoothing the dose effect in a general, threshold dose-response model with random effects. This method uses a combination of linear B-splines in which an interior knot, regarded as a free parameter for a piece-wise linear spline, indicates a break point in the dose-response curve. The knot will be estimated and may be interpreted as a threshold value. The proposed methodology is applied to data from a developmental toxicity study conducted by the National Toxicology Program.

A practical construction of G1 smooth biquintic B-spline surfaces over arbitrary topology

The motivation of this paper is to develop a local scheme of constructing G 1 smooth B-spline surfaces with single interior knots over arbitrary topology. In this paper, we obtain the conditions of G 1 continuity between two adjacent biquintic B-spline surfaces with interior single knots. These conditions are directly represented by the relevant control points of the two B-spline surfaces. By utilizing these G 1 conditions, we develop the first local scheme of constructing G 1 smooth biquintic B-spline surfaces with interior single knots for arbitrary topological type. The high complexity of deriving the local G 1 scheme is well overwhelmed. The biquintic is the lowest degree for which there exists a local scheme of constructing G 1 smooth B-spline surfaces with interior single knots over arbitrary topology.

Theory Of A B-Spline Basis Function

This paper extends the work of a previous paper of the author. A theoretical argument is provided to justify the heuristic algorithm used in the former paper. On the basis of the theory one derives, the previous algorithm can be further simplified. In the simplified basis function algorithm, the regular basis function (where $N_i^1(t)$ is 1 for $t_i \\le t \\lt t_{i + 1}$ and zero elsewhere) can be used for all cases except the case of the last point of a clamped B-spline where the basis function is modified to $N_{i,1} (t)$ where is 1 for $t_i \\le t \\le t_{i + 1}$ and zero elsewhere. Under this simplified algorithm, the knots ( i.e. , $t_{0}$ , $t_{1}, \\ldots, t_{n+k}$ ) are a k -extended partition in the interior of the knot vector with a possibility that two ends of the knot vector could be a $(k + 1)$ -extended partition (case of clamped B-spline). It is shown that given a set of $(n + 1)$ control points, $V_{0}$ , $V_{1}, \\ldots, V_{n}$ , the order of k , and the knots $(t_{0}, t_{1}, \\ldots, t_{n+k})$ , the B-spline $P(t) = \\sum_{i = 0}^{n} N_{i}^{k}(t)V_{i}$ is a continuous function for $t \\in [t_{k - 1}, t_{n + 1}]$ and maintains the partition of unity. This algorithm circumvents the problem of generating a spike at the last point of a clamped B-spline. The constraint of having k -extended partition interior knots overcomes the problem of disconnecting the B-spline at the k repeated knot.

Models of Knots and Log Geometry of Young Pinus sylvestris Sawlogs Extracted from Computed Tomographic Images

The value of solid wood products is largely determined by the sizes, types and distribution of the knots in the products. Hence, there is a great interest in describing the internal knot structure of individual logs. The Swedish Stem Bank has been extensively used for modelling the interior knot structure of Scots pine (Pinus sylvestris L.) and for simulating the outcome of sawing operations. The stem bank holds parametric descriptions, extracted from computed tomographic (CT) imagery, of mature trees. A method for extracting parametric descriptions, in compliance with the stem bank, from young Scots pine sawlogs is presented in this study. A key step in the algorithm is the use of an artificial neural network to find knots in the CT images. The accuracy of the extracted descriptions was evaluated by comparing the size and position of knots measured on 10 real boards with corresponding boards simulated based on the description. The study showed that the number of knots on the real boards was well predicted (R 2=0.90). The differences in tangential and longitudinal position were 0.3±3.6 mm and 1.6±4.2 mm, respectively. The differences in tangential and longitudinal diameter were 0.6±4.0 mm and −0.6±3.9 mm, respectively. Knot diameters were more accurately predicted on boards distant from the pith than on boards close to pith.

Knot Identification from CT Images of Young Pinus sylvestris Sawlogs Using Artificial Neural Networks

The value of solid wood products is to a large extent determined by the sizes, types and distribution of the knots in the products. Hence there is a great interest in describing the internal knot structure of individual logs. The Swedish Stem Bank has been extensively used for modelling the interior knot structure of Scots pine ( Pinus sylvestris L.) and for simulating the outcome of sawing operations. The stem bank holds parametric descriptions, extracted from computer tomography (CT) imagery, of mature trees. To enlarge the stem bank with trees from younger stands, a better method for extracting the knot properties from the CT images is needed. In this study, artificial neural networks were used for segmenting and classifying knots in transverse CT images of a 30-year-old Scots pine. The cross-validated prediction rate of correctly classified pixels was 95.9% - 1.2%. Classified knots were distinctly separated. Misclassifications were mainly located in the border areas between knots and clear