Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

Determinants of extended door-to-needle time in acute ischemic stroke and its influence on in-hospital mortality: results of a nationwide Dutch clinical audit

Background Intravenous thrombolysis (IVT) plays a prominent role in the treatment of acute ischemic stroke (AIS). The sooner IVT is administered, the higher the odds of a good outcome. Therefore, registering the in-hospital time to treatment with IVT, i.e. the door-to-needle time (DNT), is a powerful way to measure quality improvement. The aim of this study was to identify determinants that are associated with extended DNT. Methods Patients receiving IVT in 2015 and 2016 registered in the Dutch Acute Stroke Audit were included. DNT and onset-to-door time (ODT) were dichotomized using the median (i.e. extended DNT) and the 90th percentile (i.e. severely extended DNT). Logistic regression was performed to identify determinants associated with (severely) extended DNT/ODT and its effect on in-hospital mortality. A linear model with natural spline was used to investigate the association between ODT and DNT. Results Included were 9518 IVT treated patients from 75 hospitals. Median DNT was 26 min (IQR 20–37). Determinants associated with a higher likelihood of extended DNT were female sex (OR 1.17, 95% CI 1.05–1.31) and admission during off-hours (OR 1.12, 95% CI 1.01–1.25). Short ODT correlated with longer DNT, whereas longer ODT correlated with shorter DNT. Young age (OR 1.38, 95% CI 1.07–1.76) and admission to a comprehensive stroke center (OR 1.26, 1.10–1.45) were associated with severely extended DNT, which was associated with in-hospital mortality (OR 1.54, 95%CI 1.19–1.98). Conclusions Even though DNT in the Netherlands is short compared to other countries, lowering the DNT may be achievable by focusing on specific subgroups.

Modeling Student Mathematics Achievement in Senior High School Based on Selection Results Using Gee 2 Method with Natural Spline

Every school has a vision and mission to become the superior institution so that it can compete and gain trust from the public. To achieve that, one of the efforts of the school is doing the selection of new students at the beginning of each academic year. In Lumajang region, admission of new students (PPDB) are selected using several components, such as national test scores (NUN) and Mapping/Placement Test(MP). This research explores the best model of the relationship between selection components (and other conditions of students at the time of selection) and academic achievement during high school (in the form semester mathematics grade) starting from Semester 1 till 5 at 3 schools in Lumajang regions. We apply Generalized Estimating Equation order 2 (GEE2) with Natural Spline. The results show that (i) the three schools, have a different model and PGRI has the highest mean, followed by SMA1and SMA3, as shown by significant negative estimates of the coefficients. (i) Although it is relatively small, distance from school has a negative contribution to the mathematics grade as shown by negative (but significant) coefficient; (ii) The Junior High School NUN has nonlinear (and nonparametric) contribution as shown by the graphical representation and coefficient of the natural spline. (iii) A score of Placement Test contributes positively and significantly to the semester mathematics grade.