Automatic, computer aided geometric design of free-knot, regression splines

A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality

Geometrically designed, variable knot regression splines: variation diminish optimality of knots

A new method for Computer Aided Geometric Design of variable knot regression splines, named GeDS, has recently been introduced by Kaishev et al. (2006). The method utilizes the close geometric relationship between a spline regression function and its control polygon, with vertices whose y-coordinates are the regression coefficients and whose x-coordinates are certain averages of the knots, known as the Greville sites. The method involves two stages, A and B. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order (n > 2) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible, and finally the LS estimates of the regression coefficients of this curve are found. In Kaishev et al. (2006) the implementation of stage A has been thoroughly addressed and the pointwise asymptotic properties of the GeD spline estimator have been explored and used to construct asymptotic confidence intervals. In this paper, the focus of the attention is at giving further insight into the optimality properties of the knots of the higher order spline curve, obtained in stage B so that it is nearly a variation diminishing (shape preserving) spline approximation to the linear fit of stage A. Error bounds for this approximation are derived. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS estimator is compared with other existing variable knot spline methods and smoothing techniques and is shown to perform very well, producing nearly optimal spline regression models. It is fast and numerically efficient, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved

Geometrically designed, variable know regression splines: asymptotics and inference

A new method for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on the property that the spline regression function, viewed as a parametric curve, has a control polygon and, due to the shape preserving and convex hull properties, closely follows the shape of this control polygon. The latter has vertices, whose x-coordinates are certain knot averages, known as the Greville sites and whose y-coordinates are the regression coefficients. Thus, manipulation of the position of the control polygon and hence of the spline curve may be interpreted as estimation of its knots and coefficients. These geometric ideas are implemented in the two stages of the GeDS estimation method. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order (n > 2) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible and finally, the LS estimates of the regression coefficients of this curve are found. To implement stage A, an automatic adaptive knot location scheme for generating linear spline fits is developed. At each step of stage A, a knot is placed where a certain bias dominated measure is maximal. This stage is equipped with a novel stopping rule which serves as a model selector. The optimal knots defined in stage B ensure that the higher order spline curve is nearly a variation diminishing (i.e., shape preserving) spline approximation to the linear fit of stage A. Error bounds for this approximation are derived in Kaishev et al. (2006). The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. Large sample properties of the GeDS estimator are also explored, and asymptotic normality is established. Asymptotic conditions on the rate of growth of the knots with the increase of the sample size, which ensure that the bias is of negligible magnitude compared to the variance of the GeD estimator, are given. Based on these results, pointwise asymptotic confidence intervals with GeDS are also constructed and shown to converge to the nominal coverage probability level for a reasonable number of knots and sample sizes

Geometrically designed, variable knot regression splines

A new method of Geometrically Designed least squares (LS) splines with variable knots, named GeDS, is proposed. It is based on the property that the spline regression function, viewed as a parametric curve, has a control polygon and, due to the shape preserving and convex hull properties, it closely follows the shape of this control polygon. The latter has vertices whose x-coordinates are certain knot averages and whose y-coordinates are the regression coefficients. Thus, manipulation of the position of the control polygon may be interpreted as estimation of the spline curve knots and coefficients. These geometric ideas are implemented in the two stages of the GeDS estimation method. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order (n > 2) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible and finally, the LS estimates of the regression coefficients of this curve are found. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline coefficients. Numerical examples are provided and further supplemental materials are available online

Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy

The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlev\'e equation. Moreover we prove that these are unique in certain sectors near infinity.Comment: 13 pages, Late

Averaging facial expression over time

The visual system groups similar features, objects, and motion (e.g., Gestalt grouping). Recent work suggests that the computation underlying perceptual grouping may be one of summary statistical representation. Summary representation occurs for low-level features, such as size, motion, and position, and even for high level stimuli, including faces; for example, observers accurately perceive the average expression in a group of faces (J

Vector-soliton collision dynamics in nonlinear optical fibers

We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schroedinger equations. We study a low-dimensional model system of Hamiltonian ODE derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured. In certain bands of initial velocities the solitons are initially captured, but separate after passing each other twice, a phenomenon known as the two-bounce or two-pass resonance. We derive an analytic formula for the critical velocity. Using matched asymptotic expansions for separatrix crossing, we determine the location of these "resonance windows." Numerical simulations of the ODE models show they compare quite well with the asymptotic theory.Comment: 32 pages, submitted to Physical Review

Extending the Lee Carter Model: a Three-way Decomposition

In this paper, we focus on a Multidimensional Data Analysis approach to the Lee-Carter (LC) model of mortality trends . In particular, we extend the bilinear LC model and specify a new model based on a three-way structure, which incorporates a further component in the decomposition of the log-mortality rates. A multi-way component analysis is performed using the Tucker 3 model. The suggested methodology allows us to obtain combined estimates for the three modes: i) time, ii) agegroups and iii) different populations. From the results obtained by the Tucker 3 decomposition, we can jointly compare, in both a numerical and graphical way, the relationships among all three modes and obtain a time series component as a leading indicator of the mortality trend for a group of populations. Further, we carry out a correlation analysis of the estimated trends in order to assess the reliability of the results of the three-way decomposition. The model’s goodness of fit is assessed using an analysis of the residuals. Finally, we discuss how the synthesised mortality index can be used to build concise projected life tables for a group of populations. An application which compares ten European countries is used to illustrate the approach and provide a deeper insight into the model and its implementation

The Composition of Titan's Lower Atmosphere and Simple Surface Volatiles as Measured by the Cassini-Huygens Probe Gas Chromatograph Mass Spectrometer Experiment

The Cassini-Huygens Probe Gas Chromatograph Mass Spectrometer (GCMS) determined the composition of the Titan atmosphere from ~140km altitude to the surface. After landing, it returned composition data of gases evaporated from the surface. Height profiles of molecular nitrogen (N2), methane (CH4) and molecular hydrogen (H2) were determined. Traces were detected on the surface of evaporating methane, ethane (C2H6), acetylene (C2H2), cyanogen (C2N2) and carbon dioxide (CO2). The methane data showed evidence that methane precipitation occurred recently. The methane mole fraction was (1.48+/-0.09) x 10(exp -2) in the lower stratosphere (139.8 km to 75.5 km) and (5.65+/-0.18) x 10(exp -2) near the surface (6.7 km to the surface). The molecular hydrogen mole fraction was (1.01+/-0.16) x 10(exp -3) in the atmosphere and (9.90+/-0.17) x 10(exp -4) on the surface. Isotope ratios were 167.7+/-0.6 for N-14/N-15 in molecular nitrogen, 91.1+/-1.4 for C-12/C-13 in methane and (1.35+/-0.30) x 10(exp -4) for D/H in molecular hydrogen. The mole fractions of Ar-36 and radiogenic Ar-40 are (2.1+/-0.8) x 10(exp -7) and (3.39 +/-0.12) x 10(exp -5) respectively. Ne-22 has been tentatively identified at a mole fraction of (2.8+/-2.1) x 10(exp -7) Krypton and xenon were below the detection threshold of 1 x 10(exp -8) mole fraction. Science data were not retrieved from the gas chromatograph subsystem as the abundance of the organic trace gases in the atmosphere and on the ground did not reach the detection threshold. Results previously published from the GCMS experiment are superseded by this publication