Statistical Idea Research Articles

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-Form Estimation of Nonlinear Dynamics

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of C^{infty } bump functions of varying support sizes.We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at https://github.com/MathBioCU/WENDy.

On Adaptive Estimation of Linear Functionals from Observations against White Noise

We consider the problem of adaptive estimation of a linear functional of an unknown multivariate vector from its observations against white Gaussian noise. As a family of estimators for the functional, we use those generated by projection estimators of the unknown vector, and the main problem is to select the best estimator in this family. The goal of the paper is to explain and mathematically justify a simple statistical idea used in adaptive (i.e., observation-based) choice of the best estimator of a linear functional from a given family of estimators. We also discuss generalizations of the considered statistical model and the proposed estimation method, which allow to cover a broad class of statistical problems.

A Moving Updated Statistical Prediction Model for Summer Rainfall in the Middle-Lower Reaches of the Yangtze River Valley

AbstractBecause summer rainfall in the middle-lower reaches of the Yangtze River valley has remarkable interannual and decadal variability and because the precursors that modulate the interannual rainfall change with the decadal variation of the background state, a new model that employs a novel statistical idea is needed to yield an accurate prediction. In this study, the interannual rainfall model (IAM) and the decadal rainfall model (DM) were constructed. Moving updating of the IAM with the latest data within an optimal length of training period (20 yr) can partially offset the effect of decadal change of precursors in IAM. To predict the interannual rainfall of 2001–13 for validation, 13 regression models were fitted with precursors that change every 4–5 yr, from the preceding winter North Atlantic Ocean sea surface temperature anomaly (SSTA) dipole to the Mascarene high, followed by the East Asia sea level pressure anomaly (SLPA) dipole and the preceding autumn North Pacific SSTA dipole. The moving updated model demonstrated high skill in predicting interannual rainfall, with a correlation coefficient of 0.76 and a hit rate of 76.9%. The DM was linked to the April SLPA in the central tropical Pacific Ocean, and it maintained good performance in the testing period, with a correlation coefficient of 0.77 and a root-mean-square error (RMSE) of 7.7%. The statistical model exhibited superior capability even when compared with the best forecast by the Climate Forecast System, version 2 (CFSv2), initiated in early June, as indicated by increased correlation coefficient from 0.62 to 0.75 and reduced RMSE from 12.3% to 10.7%.

Eugenics and the normal body: the role of visual images and intelligence testing in framing the treatment of people with disabilities in the early twentieth century

This article examines how the emergence of a statistical concept of the normal at the end of the nineteenth century led to the development of a theory of eugenics, and examines the cultural pathways by which this theory came to shape both the public perception and institutional treatment of people now understood as disabled. The statistical idea of the normal and the theory of eugenics were developed simultaneously in the work of one man: Francis Galton, cousin of Charles Darwin and often identified as the first ‘social Darwinist.’ Beginning with an analysis of the role of composite photography in Galton’s research, this article traces the historical co-emergence of normality and eugenics along two key lines of development: the use of new visual imagining techniques in the public sphere as a means by which to popularize these ideas amongst a general audience, and their application in the institutional and legal treatment of disabled people in the first three decades of the twentieth century. In so doing, the article intends to provide a careful history of the development and application of eugenical thinking and practice in the Anglophone world over this period of emergence and influence.

The Gravitation of Market Prices as A Stochastic Process

AbstractThe theory of value has been based ever since Adam Smith on the idea that the market prices of commodities, those at which actual trade takes place, gravitate around a central position known as natural prices. This article seeks to develop a statistical idea of the process in question and suggests in particular that market prices can be said to gravitate around natural prices if the probability of their means being very close to natural prices after t observations tends to 1 as t tends to infinity. A set of possible conditions leading to that result is also presented.

TEACHING FOR STATISTICAL LITERACY: UTILISING AFFORDANCES IN REAL-WORLD DATA

It is widely held that context is important in teaching mathematics and statistics. Consideration of context is central to statistical thinking, and any teaching of statistics must incorporate this aspect. Indeed, it has been advocated that real-world data sets can motivate the learning of statistical principles. It is not, however, a straightforward task to take a real-world example and incorporate it into a lesson that will teach important statistical principles. This paper considers issues involved in using real data to exemplify statistical ideas and examines pre-service teachers’ attempts to design teaching activities using such data. Pre-service teachers were supplied with a topical data set and asked to plan lessons that would teach some key statistical idea to year 6 students. The lessons were analysed using a hierarchy for teaching statistical literacy, and great variation was found in the level of statistical thinking demanded in the planned lessons. Teachers who had completed a preliminary activity helping them to think carefully about what might be taught from real data in general produced lessons with stronger statistical content. A key requirement for having lessons with deep consideration of statistical ideas is to identify the actual affordances for teaching contained within a data set; the planning process then benefits from explicit attention to making that content evident in the teaching activities.

Lexical ambiguity: making a case against spread

AbstractWe argue for decreasing the use of the word spread when describing the statistical idea of dispersion or variability in introductory statistics courses. In addition, we argue for increasing the use of the word variability as a replacement for spread.

2A2-A5 企業で必要とされている統計的思考と学校教育への期待(科学教育における統計的思考の育成,学会企画課題研究,次世代の科学力を育てる-社会とのグラウンディングを求めて-)

2A2-A5 企業で必要とされている統計的思考と学校教育への期待(科学教育における統計的思考の育成,学会企画課題研究,次世代の科学力を育てる-社会とのグラウンディングを求めて-)

Application of statistical idea and the triple-type theory in the scientific research of biomedicine

Application of statistical idea and the triple-type theory in the scientific research of biomedicine

Wigner-Yanase skew information and uncertainty relations.

The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.

Statistical Sampling and Data Collection Activities

One way that students learn about sampling is by actually collecting data. Collecting data gives them a feel for the practical struggles and small decisions needed in real data gathering, and it illustrates many important ideas in sampling. Another way of learning about sampling is to discuss surveys reported in the popular press. Reading and critiquing unusual news stories is fun, and real survey findings emphasize the importance of the statistical idea being illustrated. We have developed several demonstrations and examples of survey sampling to use in the classroom, and we use these activities regularly in our introductory statistics courses for college students who have completed high school algebra. We have also had success introducing the activities to eighth-grade students in New York City public schools, and since these activities are in the spirit of the NCTM's Standards for statistics for grades 9–12, they would be particularly useful in the high school setting.

Subjective risk measures: Bayesian predictive scenarios analysis

In this paper we study methods for measuring risk. First, we introduce a conditional risk measure and point out that it is a coherent risk measure. Using the Bayesian statistical idea a subjective risk measure is defined. In some special cases, closed form expressions for the risk measures can be obtained. The credibility theory can be used to relax the strong assumptions on the model and prior distributions, and to obtain approximated risk measure formulas. Applications in both finance and insurance are discussed.

Binomial-Beta Hierarchical Models for Ecological Inference

The authors develop binomial-beta hierarchical models for ecological inference using insights from the literature on hierarchical models based on Markov chain Monte Carlo algorithms and King's ecological inference model. The new approach reveals some features of the data that King's approach does not, can be easily generalized to more complicated problems such as general R × C tables, allows the data analyst to adjust for covariates, and provides a formal evaluation of the significance of the covariates. It may also be better suited to cases in which the observed aggregate cells are estimated from very few observations or have some forms of measurement error. This article also provides an examples of a hierarchical model in which the statistical idea of “borrowing strength” is used not merely to increase the efficiency of the estimates but to enable the data analyst to obtain estimates.

Neural-network design for small training sets of high dimension

We introduce a statistically based methodology for the design of neural networks when the dimension d of the network input is comparable to the size n of the training set. If one proceeds straightforwardly, then one is committed to a network of complexity exceeding n. The result will be good performance on the training set but poor generalization performance when the network is presented with new data. To avoid this we need to select carefully the network architecture, including control over the input variables. Our approach to selecting a network architecture first selects a subset of input variables (features) using the nonparametric statistical process of difference-based variance estimation and then selects a simple network architecture using projection pursuit regression (PPR) ideas combined with the statistical idea of slicing inverse regression (SIR). The resulting network, which is then retrained without regard to the PPR/SIR determined parameters, is one of moderate complexity (number of parameters significantly less than n) whose performance on the training set can be expected to generalize well. The application of this methodology is illustrated in detail in the context of short-term forecasting of the demand for electric power from an electric utility.

Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth‐grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add‐them‐all‐up‐and‐divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.

Influence of particle—craze interactions on the sub-critical fracture of core—shell HIPS

High impact polystyrene (HIPS) particles with sizes well below 1 μm have been demonstrated to interact with growing crazes giving rise to some local, non-catastrophic fibril texture damage, which increases with increasing particle size and volume fraction of the second phase. The influence of such damage on the sub-critical fracture behaviour of some core-shell HIPSs, which have been well characterized from the morphological and structural points of view, is investigated. A statistical craze fibril fracture model is adopted that considers a two-parameter Weibull distribution and a Paris law for the sub-critical crack advance is derived. The meaning and dependence of the Weibull parameters on the structural and morphological characteristics of the considered materials are discussed. The proposed approach is not limited to the statistical idea and can, in principle, be used to derive the Paris behaviour in a larger class of materials.

Research into Practice: What Do Children Understand about Average?

The statistical idea we come across most frequently is the idea of average. Children in fourth grade and beyond fairly easily learn to apply the algorithm for finding the mean, but what do they understand about the mean as a statistical idea?

Binary classification by stochastic neural nets

We classify points in R(d) (feature vector space) by functions related to feedforward artificial neural networks. These functions, dubbed "stochastic neural nets", arise in a natural way from probabilistic as well as from statistical considerations. The probabilistic idea is to define a classifying bit locally by using the sign of a hidden state-dependent noisy linear function of the feature vector as a new (d+1)th coordinate of the vector. This (d+1)-dimensional distribution is approximated by a mixture distribution. The statistical idea is that the approximating mixtures, and hence the a posteriori class probability functions (stochastic neural nets) defined by them, can be conveniently trained either by maximum likelihood or by a Bayes criterion through the use of an appropriate expectation-maximization algorithm.

A statistical view of heavy metal pollution index of river sediment.

How to set up a standard method for judging as to whether or not river sediment is polluted by heavy metal is essential to the definition of polluted sediments. In a previous report, the authors proposed a pollution index to determine the degree of pollution and how it was applied to the sediment of first class rivers in Japan. The fundamental philosophy of this method is based on the statistical idea that the upper 10% of the distribution formed by the normalized group of upstream sediment falls within the critical region. In this case the distribution of pollution indices must be normal. In the present paper they propose a new pollution index, which is distributed as noncentral chi-square in most of the cases of practical application.

The theory of solution hardening

Abstract The glide plane of a dislocation contains a concentration c of obstacles, which we take to be attractive potential wells each made of four parabolic arcs of width w, exerting a maximum attractive force fm. If the line tension of the dislocation is T, the nature of its motion is controlled by the parameter β = f m/4Tcw 2. When β is large the motion occurs in the Friedel limit. A length of dislocation line interacting with three consecutive obstacles A, B, C breaks away from B and moves until it interacts with a fourth obstacle D. The portions of dislocation line outside the segment AC do not move appreciably during this process. If the Burgers vector of the dislocation is 6, the critical resolved shear stress τc is given in this limit by bτc = (f m 3c/2T)1/2. In the Labusch limit, where β is small, we find, in agreement with Labusch, that the motion of a dislocation segment through a well is then adiabatic and non-dissipative. The segment of dislocation which moves coherently is in contact with many obstacles. Labusch assumes that the obstacles are statistically aggregated into clusters, and shows that, if the clusters are large enough, one obtains a flow stress roughly β−1/6 times that predicted by Friedel. We believe that his argument is not self-consistent, because the dislocation then interacts coherently with only three consecutive clusters. By using an earlier statistical idea, due to Mott, and applying it to a simple model due to Lowell, we give an analysis which clarifies the physical processes occurring. While the advance of a segment of dislocation which is interacting with a large number n of obstacles is reversible, it does not occur under zero force, but under a force of order ± n 1/2/m. In regions where this force opposes the motion of the dislocation, it determines the flow stress; in regions where this force aids the motion, both this force and the force on the dislocation produced by the external stress lead to an acceleration of the dislocation and hence to dissipation.