339 research outputs found
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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
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Automated Graduation using Bayesian Trans-dimensional Models
This paper presents a new method of graduation which uses parametric formulae together with Bayesian reversible jump Markov chain Monte Carlo methods. The aim is to provide a method which can be applied to a wide range of data, and which does not require a lot of adjustment or modification. The method also does not require one particular parametric formula to be selected: instead, the graduated values are a weighted average of the values from a range of formulae. In this way, the new method can be seen as an automatic graduation method which we believe that in many cases can be applied without any adjustments and provide satisfactory graduated values
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Modelling Claims Run-off with Reversible Jump Markov Chain Monte Carlo Methods
In this paper we describe a new approach to modelling the development of claims run-off triangles. This method replaces the usual adhoc practical process of extrapolating a development pattern to obtain tail factors with an objective procedure. An example is given, illustrating the results in a practical context, and the WinBUGS code is supplied
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Entropy, longevity and the cost of annuities
This paper presents an extension of the application of the concept of entropy to annuity costs. Keyfitz (1985) introduced the concept of entropy, and analysed this in the context of continuous changes in life expectancy. He showed that a higher level of entropy indicates that the life expectancy has a greater propensity to respond to a change in the force of mortality than a lower level of entropy. In other words, a high level of entropy means that further reductions in mortality rates would have an impact on measures like life expectancy. In this paper, we apply this to the cost of annuities and show how it allows the sensitivity of the cost of a life annuity contract to changes in longevity to be summarized in a single figure index
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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
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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
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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
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Geometrically Designed Variable Knot Splines in Generalized (Non-)Linear Models
In this paper we extend the GeDS methodology, recently developed by Kaishev et al. (2016) for the Normal univariate spline regression case, to the more general GNM (GLM) context. Our approach is to view the (non-)linear predictor as a spline with free knots which are estimated, along with the regression coefficients and the degree of the spline, using a two stage algorithm. In stage A, a linear (degree one) free-knot spline is fitted to the data applying iteratively re-weighted least squares. In stage B, a Schoenberg variation diminishing spline approximation to the fit from stage A is constructed, thus simultaneously producing spline fits of second, third and higher degrees. We demonstrate, based on a thorough numerical investigation that the nice properties of the Normal GeDS methodology carry over to its GNM extension and GeDS favourably compares with other existing spline methods. The proposed GeDS GNM(GLM) methodology is extended to the multivariate case of more than one independent variable by utilizing tensor product splines and their related shape preserving variation diminishing property
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