Sensitivity analysis in optimized parametric curve fitting

Purpose – Curve fitting from unordered noisy point samples is needed for surface reconstruction in many applications -- In the literature, several approaches have been proposed to solve this problem -- However, previous works lack formal characterization of the curve fitting problem and assessment on the effect of several parameters (i.e. scalars that remain constant in the optimization problem), such as control points number (m), curve degree (b), knot vector composition (U), norm degree (k), and point sample size (r) on the optimized curve reconstruction measured by a penalty function (f) -- The paper aims to discuss these issues -- Design/methodology/approach - A numerical sensitivity analysis of the effect of m, b, k and r on f and a characterization of the fitting procedure from the mathematical viewpoint are performed -- Also, the spectral (frequency) analysis of the derivative of the angle of the fitted curve with respect to u as a means to detect spurious curls and peaks is explored -- Findings - It is more effective to find optimum values for m than k or b in order to obtain good results because the topological faithfulness of the resulting curve strongly depends on m -- Furthermore, when an exaggerate number of control points is used the resulting curve presents spurious curls and peaks -- The authors were able to detect the presence of such spurious features with spectral analysis -- Also, the authors found that the method for curve fitting is robust to significant decimation of the point sample -- Research limitations/implications - The authors have addressed important voids of previous works in this field -- The authors determined, among the curve fitting parameters m, b and k, which of them influenced the most the results and how -- Also, the authors performed a characterization of the curve fitting problem from the optimization perspective -- And finally, the authors devised a method to detect spurious features in the fitting curve -- Practical implications – This paper provides a methodology to select the important tuning parameters in a formal manner -- Originality/value - Up to the best of the knowledge, no previous work has been conducted in the formal mathematical evaluation of the sensitivity of the goodness of the curve fit with respect to different possible tuning parameters (curve degree, number of control points, norm degree, etc.

Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory

Good estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losse

Comparison of methods for estimating continuous distributions of relaxation times

The nonparametric estimation of the distribution of relaxation times approach is not as frequently used in the analysis of dispersed response of dielectric or conductive materials as are other immittance data analysis methods based on parametric curve fitting techniques. Nevertheless, such distributions can yield important information about the physical processes present in measured material. In this letter, we apply two quite different numerical inversion methods to estimate the distribution of relaxation times for glassy \lila\ dielectric frequency-response data at 225 \kelvin. Both methods yield unique distributions that agree very closely with the actual exact one accurately calculated from the corrected bulk-dispersion Kohlrausch model established independently by means of parametric data fit using the corrected modulus formalism method. The obtained distributions are also greatly superior to those estimated using approximate functions equations given in the literature.Comment: 4 pages and 4 figure

The Construction of Curves and Surfaces Using Numerical Optimization Techniques

Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions

PLTTER user's guide

The PLTTER graphics system, which is part of CDDMS is discussed. CDDMS is a comprehensive system for data basing and subsequent plotting of data acquired during wind tunnel tests or from computational flow analyses. The PLTTER is a system which creates report-quality plots of data which is stored on a CDDMS data base. The Requests file system allows plot-controlling information to be arranged in the way which is most appropriate for any application. The PLTTER system features many capabilities which are especially useful when plotting wind tunnel data. The PLTTER offers a variety of page formats, different grid options and parametric curve fitting algorithms, and a powerful legend capability to identify relevant information about individual curves. One or more plots on a page can be suppressed if desired so that an established page format can be maintained. Final plot output may be standard Versatec plots, QMS Laser printer plots, or microfiche

Modelling of catastrophic losses

Catastrophe modelling is a risk management tool that uses computer technology to help insurers, reinsurers and risk managers better assess the potential losses caused by natural and man-made catastrophes. The models use historical disaster information to simulate the characteristics of potential catastrophes and to determine the potential losses cost. The aim of this paper is to describe parametric curve-fitting methods for modelling extreme historical losses. Article summarizes relevant theoretical results above Extreme value theory (EVT) and Excess over Threshold Method (EOT) and provide example of their application to Danish data on large fire insurance losses. Application of these methods is not possible without appropriate software packages. Article refers to these options too

Interpretation of sedimentary (sub)populations extracted from grain size distributions of Central European loess-paleosol series

Grain size proxies of aeolian dust deposits have widely been applied in environmental and sedimentary studies. However, large body of research papers are not taking into consideration that a complex grain size distribution curve cannot be an indicator of a single one environmental factor (e.g. wind speed/strength, transportation distance, aridity). The aim of the present paper is to discuss the main differences of frequently used statistical methods and to provide possible interpretations of the results by applying these various approaches on the high-resolution loess-paleosol profile of Dunaszekcső, South Hungary (Central Europe). Beside single statistical descriptors (mean, median, mode) of grain size and simple indices of size-fraction ratios (U-ratio, Grain Size Index), some more complex algorithms were also used in our paper. The applied parametric curve-fitting, end-member modelling and hierarchical cluster analysis techniques are using the whole spectrum of the measured grain size distributions and provide a more reliable and more representative results even in case of small scale variations. According to our findings, approaches which provide direct linkage among simple statistical descriptors and single atmospheric or other environmental elements are rather oversimplified as properties aeolian dust deposits are influenced by the integrated effects of several concurrent processes. Differences of more complex decomposition methods arise from the different approach and scope. End-members are determined from the unmixing based on the covariance structure of the whole grain size data-series of the section, while the parametric curve-fitting is based on the one-by-one deconvolution of the grain size distribution curves. End-members of loess-paleosol samples are regarded as representation of the average dust grain size distribution of various temporal sediment clusters of seasonal or other short-term intervals, while (sub)populations by parametric curve-fitting are proposed to illustrate process-related elements of background and dust storm depositional components for each sample. Results of cluster analysis represent similar grouping conditions as end-member modelling with a reduced sedimentary and genetically meaning

Small Sample Stochastic Tail Modeling: Tackling Sampling Errors and Sampling Bias by Pivot-Distance Sampling and Parametric Curve Fitting Techniques

We describe two original open source software applications that have been developed to aid model efficiency studies: (1) CSTEP (Cluster Sampling for Tail Estimation of Probability) for reducing sampling error through variations of distance sampling and cluster/pivot processes; and (2) AMOOF2 (Actuarial Model Outcome Optimal Fit Version 2.0) for mitigating small sample bias in parametric, time-efficient probability density function fitting. CSTEP uses the scenario reduction method of representative scenarios to sample scenarios from a population of stochastic scenarios to obtain a sample-run distribution of a financial outcome that can be analyzed by AMOOF2 to fit the optimal probability density function