B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.

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Flexible smoothing with B-splines and penalties

Principles and Practice of Structural Equation Modeling (2nd ed.)

미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

This paper surveys locally weighted learning, a form of lazy learning and memory-based learning, and focuses on locally weighted linear regression. The survey discusses distance functions, smoothing parameters, weighting functions, local model structures, regularization of the estimates and bias, assessing predictions, handling noisy data and outliers, improving the quality of predictions by tuning fit parameters, interference between old and new data, implementing locally weighted learning efficiently, and applications of locally weighted learning. A companion paper surveys how locally weighted learning can be used in robot learning and control.

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Locally Weighted Learning

There has been major progress in recent years in data-based bandwidth selection for kernel density estimation. Some “second generation” methods, including plug-in and smoothed bootstrap techniques, have been developed that are far superior to well-known “first generation” methods, such as rules of thumb, least squares cross-validation, and biased cross-validation. We recommend a “solve-the-equation” plug-in bandwidth selector as being most reliable in terms of overall performance. This article is intended to provide easy accessibility to the main ideas for nonexperts.

https://www.stat.washington.edu/courses/stat527/s13/readings/Jones_etal_JASA_1996.pdf

A Brief Survey of Bandwidth Selection for Density Estimation

In this paper a review will be given of some methods available for modelling relationships between categorical response variables and explanatory variables. These methods are all classed under the name polytomous logistic regression (PLR). Models for PLR will be presented and compared; model parameters will be tested and estimated by weighted least squares and by likelihood. Usually, software is needed for computation, and available statistical software is reported.

An industrial problem is solved to some extent as an example to illustrate the use of PLR. The paper is concluded by a discussion on the various PLR-methods and some topics that need a further study are mentioned.

Polytomous logistic regression

We consider local polynomial fitting for estimating a regression function and its derivatives nonparametrically. This method possesses many nice features, among which automatic adaptation to the boundary and adaptation to various designs. A first contribution of this paper is the derivation of an optimal kernel for local polynomial regression, revealing that there is a universal optimal weighting scheme. Fan (1993, Ann. Statist., 21, 196-216) showed that the univariate local linear regression estimator is the best linear smoother, meaning that it attains the asymptotic linear minimax risk. Moreover, this smoother has high minimax risk. We show that this property also holds for the multivariate local linear regression estimator. In the univariate case we investigate minimax efficiency of local polynomial regression estimators, and find that the asymptotic minimax efficiency for commonly-used orders of fit is 100% among the class of all linear smoothers. Further, we quantify the loss in efficiency when going beyond this class.

Local polynomial regression: Optimal kernels and asymptotic minimax efficiency

Usually children learn the basic principles of number and arithmetic by the help of finger-based representations. However, whether the reliance on finger-based representations is only beneficial or whether it may even become detrimental is the subject of an ongoing debate between neuro-cognitive and mathematics education researchers. From the neuro-cognitive perspective finger counting provides multi-sensory input conveying both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis enhances mathematical skills. From this neuro-cognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, mathematics education research recommends fostering mental numerical representations so as to induce children to abandon finger-counting. More precisely mathematics education recommends moving from finger counting to concrete structured representations and then, finally, to mental representations of numbers. Taken together, there is obviously an important debate between the neuro-cognitve and mathematics education research concerning the benefits or detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be presented and discussed.

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Effects of Finger Counting on Numerical Development – The Opposing Views of Neurocognition and Mathematics Education

Given data from a sample of noisy curves, we consider a nonlinear parametric regression model with unknown model function. An iterative algorithm for estimating individual parameters as well as the model function is introduced under the assumption of a certain shape invariance: the individual regression curves are obtained from a common shape function by linear transformations of the axes. Our algorithm is based on least-squares methods for parameter estimation and on nonparametric kernel methods for curve estimation. Asymptotic distributions are derived for the individual parameter estimators as well as for the estimator of the shape function. An application to human growth data illustrates the method.

/pdf/model-estimation-in-nonlinear-regression-under-shape-5cce91w8vo.pdf

Model Estimation in Nonlinear-regression Under Shape Invariance

Data are abundant, quantitative information about the state of society and the wider world is around us more than ever. Paradoxically, recent trends in the public discourse point towards a post-factual world that seems content to ignore or misrepresent empirical evidence. As statistics educators we are challenged to promote understanding of statistics about society. In order to re-root public debate to be based on facts instead of emotions and to promote evidence-based policy decisions, statistics education needs to embrace two areas widely neglected in secondary and tertiary education: understanding of multivariate phenomena and the thinking with and learning from complex data.
First published May 2017 at Statistics Education Research Journal Archives

/pdf/statistical-literacy-for-active-citizenship-a-call-for-data-1o7s3s8gwg.pdf

Joachim Engel

Papers

Polytomous logistic regression

Local polynomial regression: Optimal kernels and asymptotic minimax efficiency

Effects of Finger Counting on Numerical Development – The Opposing Views of Neurocognition and Mathematics Education

Model Estimation in Nonlinear-regression Under Shape Invariance

Statistical literacy for active citizenship: a call for data science education