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

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

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 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

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

Double Chain Ladder and Bornhuetter-Ferguson

In this article we propose a method close to Double Chain Ladder (DCL) introduced by Martínez-Miranda, Nielsen, and Verrall (2012a). The proposed method is motivated by the potential lack of stability of the DCL method (and of the classical Chain ladder method [CLM] itself). We consider the implicit estimation of the underwriting year inflation in the CLM method and the explicit estimation of it in DCL. This may represent a weak point for DCL and CLM because the underwriting year inflation might be estimated with significant uncertainty. A key feature of the new method is that the underwriting year inflation can be estimated from the less volatile incurred data and then transferred into the DCL model. We include an empirical illustration that illustrates the differences between the estimates of the IBNR and RBNS cash flows from DCL and the new method. We also apply bootstrap estimation to approximate the predictive distributions

An Integrated Assessment of the Horticulture Sector in Northern Australia to Inform Future Development

The horticulture sector in northern Australia, covering north of Western Australia (WA), Northern Territory (NT), and north Queensland (QLD), contributes $1.6 billion/year to the Australian economy by supplying diverse food commodities to meet domestic and international demand. To date, the Australian Government has funded several studies on developing the north’s agriculture sector, but these primarily focused on land and water resources and omitted an integrated, on-ground feasibility analysis for including farmers’/growers’ perspectives. This study is the first of its kind in the north for offering a detailed integrated assessment, highlighting farmers’ perspectives on the current state of the north’s horticulture sector, and related challenges and opportunities. For this, we applied a bottom-up approach to inform future agriculture development in the region, involving a detailed literature review and conducting several focus group workshops with growers and experts from government organisations, growers’ associations, and regional development agencies. We identified several key local issues pertaining to crop production, availability of, and secure access to, land and water resources, and workforce and marketing arrangements (i.e., transport or processing facilities, export opportunities, biosecurity protocols, and the role of the retailers/supermarkets) that affect the economic viability and future expansion of the sector across the region. For example, the availability of the workforce (skilled and general) has been a challenge across the north since the start of the COVID-19 pandemic in 2020. Similarly, long-distance travel for farm produce due to a lack of processing and export facilities in the north restricts future farm developments. Any major investment should be aligned with growers’ interests. This research highlights the importance of understanding and incorporating local growers’ and researchers’ perspectives, applying a bottom-up approach, when planning policies and programs for future development, especially for the horticulture sector in northern Australia and other similar regions across the globe where policy makers’ perspectives may differ from farmers

Hospitalizations for Pandemic (H1N1) 2009 among Maori and Pacific Islanders, New Zealand

Community transmission of influenza A pandemic (H1N1) 2009 was followed by high rates of hospital admissions in the Wellington region of New Zealand, particularly among Maori and Pacific Islanders. These findings may help health authorities anticipate the effects of pandemic (H1N1) 2009 in other communities

Estimation of Mortalities

If a linear regression is fit to log-transformed mortalities and the estimate is back-transformed according to the formula Ee^Y = e^{\mu + \sigma^2/2} a systematic bias occurs unless the error distribution is normal and the scale estimate is gauged to normal variance. This result is a consequence of the uniqueness theorem for the Laplace transform. We determine the systematic bias of minimum-L_2 and minimum-L_1 estimation with sample variance and interquartile range of the residuals as scale estimates under a uniform and four contaminated normal error distributions. Already under innocent looking contaminations the true mortalities may be underestimated by 50% in the long run. Moreover, the logarithmic transformation introduces an instability into the model that results in a large discrepancy between rg_Huber estimates as the tuning constant regulating the degree of robustness varies. Contrary to the logarithm the square root stabilizes variance, diminishes the influence of outliers, automatically copes with observed zeros, allows the `nonparametric' back-transformation formula E Y^2 = \mue^2 + \sigma^2, and in the homoskedastic case avoids a systematic bias of minimum-L_2 estimation with sample variance. For the company-specific table 3 of [Loeb94], in the age range of 20-65 years, we fit a parabola to root mortalities by minimum-L_2 , minimum-L_1, and robust rg_Huber regression estimates, and a cubic and exponential by least squares. The fits thus obtained in the original model are excellent and practically indistinguishable by a \chi^2 goodness-of-fit test. Finally , dispensing with the transformation of observations, we employ a Poisson generalized linear model and fit an exponential and a cubic by maximum likelihood