Jump Detection in Regression Surfaces

Abstract

We consider the problem of locating jumps in regression surfaces. A jump detection algorithm is suggested based on local least squares estimation. This method requires O(Nk) computations, where N is the sample size and k is the window width of the neighborhood. This property makes it possible to handle large data sets. The conditions imposed on the jump location curves, the jump surfaces, and the noise are mild. We demonstrate this method in detail with some numerical examples.     

Fisher information in window censored renewal process data and its applications

Suppose we have a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded. Assuming a parametric model for the renewal time distribution with parameter θ, we obtain the likelihood of the observed data and describe the exact and asymptotic behavior of the Fisher information (FI) on θ contained in this window censored renewal process. We illustrate our results with exponential, gamma, and Weibull models for the renewal distribution. We use the FI matrix to determine optimal window length for designing experiments with recurring events when the total time of observation is fixed. Our results are useful in estimating the standard errors of the maximum likelihood estimators and in determining the sample size and duration of clinical trials that involve recurring events associated with diseases such as lupus.     

Stability of Gabor frames with arbitrary sampling points

Summary We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S₀(R^d) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S₀(R^d) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.     

Letter-Value Plots: Boxplots for Large Data

Boxplots are useful displays that convey rough information about the distribution of a variable. Boxplots were designed to be drawn by hand and work best for small datasets, where detailed estimates of tail behavior beyond the quartiles may not be trustworthy. Larger datasets afford more precise estimates of tail behavior, but boxplots do not take advantage of this precision, instead presenting large numbers of extreme, though not unexpected, observations. Letter-value plots address this problem by including more detailed information about the tails using “letter values,” an order statistic defined by Tukey. Boxplots display the first two letter values (the median and quartiles); letter-value plots display further letter values so far as they are reliable estimates of their corresponding quantiles. We illustrate letter-value plots with real data that demonstrate their usefulness for large datasets. All graphics are created using the R package lvplot, and code and data are available in the supplementary materials.     

Bayesian Calibration and Uncertainty Analysis for Computationally Expensive Models Using Optimization and Radial Basis Function Approximation

We presenta Bayesian approach to model calibration when evaluation of the model is computationally expensive. Here, calibration is a nonlinear regression problem: given a data vector Y corresponding to the regression model f(β), find plausible values of β. As an intermediate step, Y and f are embedded into a statistical model allowing transformation and dependence. Typically, this problem is solved by sampling from the posterior distribution of β given Y using MCMC. To reduce computational cost, we limit evaluation of f to a small number of points chosen on a high posterior density region found by optimization.Then,we approximate the logarithm of the posterior density using radial basis functions and use the resulting cheap-to-evaluate surface in MCMC.We illustrate our approach on simulated data for a pollutant diffusion problem and study the frequentist coverage properties of credible intervals. Our experiments indicate that our method can produce results similar to those when the true “expensive” posterior density is sampled by MCMC while reducing computational costs by well over an order of magnitude.     

Perishable Inventory System at Service Facilities with N Policy

Abstract

This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N (< M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.     

Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field X _i , i ∈ Z ^d from the density is contaminated by measurement errors. In particular, the observations Y _i , i ∈ Z ^d are such that Y _i = X _i + ε _i , where the errors ε _i are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Robustness of Tail Index Estimation

Abstract

The implementation of the Hill estimator, which estimates the heaviness of the tail of a distribution, requires a choice of the number of extreme observations in the tails, r from a sample of size n where 2 ≤ r + 1 ≤ n. This article is concerned with a robust procedure of choosing an optimal r. Thus, an estimation procedure, δ _s , based on the idea of spacing statistics, H^(r) is developed. The proposed decision rule for choosing r under the squared error loss is found to be a simple function of the sample size. The proposed rule is then illustrated across a wide range of data, including insurance claims, currency exchange rate returns, and city size.     

A multiple window scan statistic for time series models

In this article we extend the results derived for scan statistics in Wang and Glaz (2014) for independent normal observations. We investigate the performance of two approximations for the distribution of fixed window scan statistics for time series models. An R algorithm for computing multivariate normal probabilities established in Genz and Bretz (2009) can be used along with proposed approximations to implement fixed window scan statistics for ARMA models. The accuracy of these approximations is investigated via simulation. Moreover, a multiple window scan statistic is defined for detecting a local change in the mean of a Gaussian white noise component in ARMA models, when the appropriate length of the scanning window is unknown. Based on the numerical results, for power comparisons of the scan statistics, we can conclude that when the window size of a local change is unknown, the multiple window scan statistic outperforms the fixed window scan statistics.     

The Plot-Data Interface in Statistical Graphics

Abstract

Statistical software systems include modules for manipulating data sets, model fitting, and graphics. Because plots display data, and models are fit to data, both the model-fitting and graphics modules depend on the data. Today's statistical environments allow the analyst to choose or even build a suitable data structure for storing the data and to implement new kinds of plots. The multiplicity problem caused by many plot varieties and many data representations is avoided by constructing a plot-data interface. The interface is a convention by which plots communicate with data sets, allowing plots to be independent of the actual data representation. This article describes the components of such a plot-data interface. The same strategy may be used to deal with the dependence of model-fitting procedures on data.     

SinaPlot: An Enhanced Chart for Simple and Truthful Representation of Single Observations Over Multiple Classes

Recent developments in data-driven science have led researchers to integrate data from several sources, over diverse experimental procedures, or databases. This alone poses a major challenge in truthfully visualizing data, especially when the number of data points varies between classes. To aid the representation of datasets with differing sample size, we have developed a new type of plot overcoming limitations of current standard visualization charts. SinaPlot is inspired by the strip chart and the violin plot and operates by letting the normalized density of points restrict the jitter along the x-axis. The plot displays the same contour as a violin plot but resembles a simple strip chart for a small number of data points. By normalizing jitter over all classes, the plot provides a fair representation for comparison between classes with a varying number of samples. In this way, the plot conveys information of both the number of data points, the density distribution, outliers and data spread in a very simple, comprehensible, and condensed format. The package for producing the plots is available for R through the CRAN network using base graphics package and as geom for ggplot through ggforce. We also provide access to a web-server accepting excel sheets to produce the plots (http://servers.binf.ku.dk:8890/sinaplot/).     

Uniform asymptotic joint normality of sample quantities in censored cases

Summary The asymptotic joint distribution of an increasing number of sample quantiles as the sample size increases, when the underlying sample is censored, is shown to be asymptotically uniformly (or type (B) _d ) normally distributed under fairly general conditions. The discussions for uncensored cases have been given by [4].     

FIRE OCCURRENCE PATTERNS AT LANDSCAPE LEVEL: BEYOND POSITIONAL ACCURACY OF IGNITION POINTS WITH KERNEL DENSITY ESTIMATION METHODS

ABSTRACT. Based on kernel density estimation methods, this paper introduces an alternative approach of fire occurrence modeling that addresses the inherent positional inaccuracies of recorded wildland fire ignition points. These observations, recorded in longitude and latitude using only degrees and first minutes, contain positional inaccuracies of about ± 700 to ± 925 meters in x and y axes. Kernel density estimation was applied to these historical fire observations recorded between 1985 and 1995 in Halkidiki peninsula, Greece, as well as, to simulated inaccurate points into which positional inaccuracies of the same magnitude were randomly introduced. Substantial differences were observed when a regular grid of quadrants was superimposed over the two point distributions. Although, at higher grid resolution these mismatches were minimized, the problem of generalization appeared. Contrar‐ily, the concept of “moving window” assisted to retain high grid resolution and minimize the effect of inaccurate point observations. In addition, the kernel approach, which considers also the relative position of points within the “moving window,” produced more realistic estimates.     

Empirical Bayes detection of a change in distribution

The problem of detection of a change in distribution is considered. Shiryayev (1963, Theory Probab. Appl., 8, pp. 22–46, 247–264 and 402–413; 1978, Optimal Stopping Rules, Springer, New York) solved the problem in a Bayesian framework assuming that the prior on the change point is Geometric (p). Shiryayev showed that the Bayes solution prescribes stopping as soon as the posterior probability of the change having occurred exceeds a fixed level. In this paper, a myopic policy is studied. An empirical Bayes stopping time is investigated for detecting a change in distribution when the prior is not completely known.Research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant GP 7987.     

Prediction of the Maximum Size in Wicksell's Corpuscle Problem, II

This is a continuing paper of the authors (1998, Ann. Inst. Statist. Math., 50, 361–377). In the Wicksell corpuscle problem, the maximum size of random spheres in a volume part is to be predicted from the sectional circular distribution of spheres cut by a plane. The size of the spheres is assumed to follow the three-parameter generalized gamma distribution. Prediction methods based on the moment estimation are proposed and their performances are evaluated by simulation. For a practically probable case, one of these prediction methods is as good as a method previously proposed by the authors where the two shape parameters are assumed to be known.     

Exploring Multidimensional Data with the Flipped Empirical Distribution Function

Abstract

This article introduces a new form of empirical distribution function (EDF) called the flipped empirical distribution function (FEDF), to represent univariate data graphically. Because the plot shows the location of individual points, it may be useful when we need to manipulate specific data points as with dynamic graphics. The article introduces several methods to explore multidimensional data using the FEDF. They are called a parallel FEDF, an FEDF scatterplot matrix, and an FEDF starplot. Usefulness of these plots in exploring multidimensional data becomes more prominent when they are implemented with the methods of dynamic graphics such as selecting, deleting, linking, locating, and identifying a group of data points.     

Scatterplots for Logistic Regression

Abstract

We present a method for graphically displaying regression data with Bernoulli responses. The method, which is based on the use of grayscale graphics to visualize contributions to a likelihood function, provides an analog of a scatterplot for logistic regression, as well as probit analysis. Furthermore, the method may be used in place of a traditional scatterplot in situations where such plots are often used.     

On Generating Random Intervals and Hyperrectangles

Abstract

We look at the problem of generating a random hyperrectangle in a unit hypercube such that each point of the hypercube has probability p of being covered. For random intervals of [0, 1], we compare various methods based either on the distribution of the length or on the distribution of the midpoint. It is shown that no constant length solution exists. Nice versatility is achieved when a random interval is chosen from among the spacings defined by a Dirichlet process. A small simulation is included.     

Tests for specific nonparametric relations between two distribution functions with applications

Let (X,Y) be a random vector and let G and H be the marginal distributions of X and Y, respectively. In this paper, we propose two tests, one of Kolmogorov‐Smirnov type and the other of Wilcoxon type, for the null hypothesis Ψ(G) = H against the alternative Ψ(G) < H, where Ψ() is a function such that Ψ(G) is a distribution function. The tests are based on the empirical distribution functions of the observations on X and Y, which are dependent. We obtain their asymptotic null distributions. A suspected relationship between the distribution functions of two dependent outcomes can be specified as a hypothesis to be tested in examples like the load sharing models, record values, and auction bidding models. As an application, we consider in detail the problem of testing the effect of load sharing in two component parallel systems.     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.