Keith Ord

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• Ph.D. (London)
• Professor Emeritus at Georgetown University

Art’s many contributions to the quantitative modeling of spatial processes provide a solid foundation for generations of researchers to come. In this review, we take a roughly chronological path through Art’s published research, starting with his work on spatial point processes, which demonstrated the frequent pattern of inhibition at a very local...

Exponential smoothing has been one of the most popular forecasting methods used to support various decisions in organizations, in activities such as inventory management , scheduling, revenue management, and other areas. Although its relative simplicity and transparency have made it very attractive for research and practice, identifying the underly...

‘Infectious Disease Control’ examines the historical development of approaches to the geographical control, elimination, and eradication of infectious diseases. It begins at a local spatial scale, seven centuries ago, among the plague-ridden lazarettos of Venice. It ends at the global scale with twenty-first-century developments in the Internet-bas...

‘Global Origins and Dispersals’ addresses two key questions. First, how do epidemic diseases emerge and can their geographical origins be traced to any particular part of the world? Second, why do more infectious diseases appear to be emerging in recent decades and how far does this crudescence relate to the unprecedented changes in the global envi...

‘Pandemics, I: Pandemics in History’ surveys the historical geography of pandemic events. Special attention is paid to those diseases that have manifested as worldwide epidemics and to which the term global pandemic is commonly applied, namely plague, cholera, influenza, and the human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/A...

A Geography of Infection explores the distinctive spatial patterns and processes by which infectious diseases spread from place to place and can grow from local and regional epidemics into global pandemics. The book focuses initially on the local scale of doctors’ practices and small islands where epidemic outbreaks are slight in the numbers infect...

‘Epidemics as Diffusion Waves’ establishes the parameters of the book. It begins by defining the basic building blocks of the study: infection, contagion, and disease. Key epidemiological concepts and terms are introduced and defined, and classic compartmental ( susceptible → infective → recovered ; SIR ) approaches to epidemic modelling are outlin...

‘Epidemics in Small Communities’ develops an understanding of the ways in which an infection can achieve community circulation in small geographical areas. It begins by examining the local disease records of two doctors in general practice in the United Kingdom (William Pickles of Wensleydale and Edgar Hope-Simpson of Cirencester) to shed light on...

‘Pandemics, II: COVID-19’ explores the spatial and temporal patterns of infection, illness, and death due to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the causative agent of coronavirus disease 2019 (COVID-19). It covers the period from the putative beginning of the COVID-19 pandemic in late 2019 to the turn of 2021. Consecu...

This note provides an evaluation of the contributions of the M5 Competition to the construction of prediction intervals. We consider the choice of criteria used in the evaluations, the relative performance of designed and benchmark methods and the take-home lessons both for statistical forecasters and for those interested in forecasting retail sale...

The goal is to predict the final extent of the Ebola epidemic in West Africa, 2014–2015, well before its end. Our models are based on the nature of the reported data, and the social, medical, and technological conditions that existed in real time during the course of the epidemic. The spatial and temporal nature of Ebola transmission is considered....

A framework for the forecasting of composite time series, such as market shares, is proposed. Based on Gaussian multi-series innovations state space models, it relies on the log-ratio function to transform the observed shares (proportions) onto the real line. The models possess an unrestricted covariance matrix, but also have certain structural ele...

Privatization and fiscal deficits have been linked theoretically as emerging market countries completed transitions from command to market-based economies. This study examines the joint relationships among relative fiscal deficits, privatization, and exogenous factors for twenty-five Central and Eastern European emerging market countries. Pooled re...

This article has no abstract.

In keeping with the Annals of Regional Science 50-year tradition of emphasizing spatial analytic contributions, a new statistic, H i , is introduced as an extension of the recent work on map pattern analysis using local spatial criteria. In conjunction with local statistics for the mean level of a spatial process, H i tests for local spatial hetero...

Organizations with large-scale inventory systems typically have a large proportion of items for which demand is intermittent and low volume. We examine various different approaches to demand forecasting for such products, paying particular attention to the need for inventory planning over a multi-period lead-time when the underlying process may be...

Outliers in time series have the potential to affect parameter estimates and forecasts when using exponential smoothing. The aim of this study is to show the way in which important types of outliers can be incorporated into linear innovations state space models for exponential smoothing methods. The types of outliers include an additive outlier, a...

The commentary recommends a plot of the probability of a positive outcome as a tool for improving the interpretation of regression models in a forecasting context.

This paper is concerned with identifying an effective method for forecasting the lead time demand of slow-moving inventories. Particular emphasis is placed on prediction distributions instead of point predictions alone. It is also placed on methods which work with small samples as well as large samples in recognition of the fact that the typical ra...

Intervention analysis was employed to determine the existence of political business cycles in the United States and the United Kingdom; the two economic variables tested were unemployment and disposable personal income. The political intervention variables were the party in power, the timing of the elections, the incumbent running for re-election,...

The statistics Gi(d) and Gi*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are deri...

A basic assumption of least squares regression is that the error terms in the regression model are independent under the null hypothesis. To test whether this assumption has been met in regression models of time series data with normally distributed error terms and given regression structure, the d statistic of Durbin and Watson [4] is often used....

The paper explores the long-term income elasticity of consumer and mortgage credit growth since World War II. It also examines other economic factors, to determine whether recent credit use is anomalous. Two-stage least squares show consumer credit income elasticity to be slightly below 1.0, taking other factors into account. A vector autoregressiv...

The online auction market has been growing at a spectacular rate. Most auctions are open-bid auctions where all the participants know the current highest bid. This knowledge has led to a phenomenon known as sniping, whereby some bidders may wait until the last possible moment before bidding, thereby depriving other bidders of the opportunity to res...

Improvements in both technology and statistical understanding have led to considerable advances in spatial model building over the past 40 years, yet major challenges remain both in model specification and in ensuring that the underlying statistical assumptions are validated. The basic concept in such modeling efforts is that of spatial dependence,...

Compositional time series are formed from measurements of proportions that sum to one in each period of time. We might be interested in forecasting the proportion of home loans that have adjustable rates, the proportion of nonagricultural jobs in manufacturing, the proportion of a rock's geochemical composition that is a specific oxide, or the prop...

Organizations with large-scale inventory systems typically have a large proportion of items for which demand is intermittent and low volume. We examine different approaches to forecasting for such products, paying particular attention to the need for inventory planning over a multi-period lead-time when the underlying process may be non-stationary....

The most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non-negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non-negative data. We explore exponential smoothing st...

Although the involvement of common childhood infections in the aetiology of acute appendicitis has long been conjectured, supporting evidence is largely restricted to a disparate set of clinical case reports. A systematic population-based analysis of the implied comorbid associations is lacking in the literature. Drawing on a classic epidemiologica...

This article outlines the context in geography and statistics in the mid 1960s, at the height of geography's so-called “quantitative revolution,” that led us into a long-term collaboration about spatial statistics, which has continued in surges and lulls for some 40 years. We focus upon problems in spatial autocorrelation, including the measurement...

Using an innovations state space approach, it has been found that the Akaike information criterion (AIC) works slightly better, on average, than prediction validation on withheld data, for choosing between the various common methods of exponential smoothing for forecasting. There is, however, a puzzle. Should the count of the seed states be incorpo...

The future of the global industry lies in the continuous improvement of both products and processes, a renewed commitment to competition, and an aggressive approach to satisfying customers needs in quality, quantity, and timing. In quality management, the degree of customer satisfaction for a given product may be measured in the form of the loss to...

A new approach is proposed for forecasting a time series with multiple seasonal patterns. A state space model is developed for the series using the innovations approach which enables us to develop explicit models for both additive and multiplicative seasonality. Parameter estimates may be obtained using methods from exponential smoothing. The propo...

Proceedings from a Symposium, of the same name, held on the campus of Syracuse University from March to June, 1989. Chapters by the following authors: L. Anselin, P. Doreian, D. A. Griffith, R. P. Haining, K. V. Mardia, R. J. Martin, J. K. Ord, J. H. P. Paelinck, S. Richardson, B. D. Ripley, A. Sen, G. J. G. Upton, D. Wartenberg.

Statistical process control (SPC) has evolved beyond its classical applications in manufacturing to monitoring economic and social phenomena. This extension has required the consideration of autocorrelated and possibly non-stationary time series. Less attention has been paid to the possibility that the variance of the process may also change over t...

In Chap. 2, state space models were introduced for all 15 exponential smoothing methods. Six of these involved only linear relationships, and so are “linear innovations state space models”. In this chapter, we consider linear innovations state space models, including the six linear models of Chap. 2, but also any other models of the same form. The...

For any innovations state space model, the initial (seed) states and the parameters are usually unknown, and therefore must be estimated. This can be done using maximum likelihood estimation, based on the innovations representation of the probability density function. In Chap. 3 we outlined transformations (referred to as “general exponential smoot...

In 1900, Louis Bachelier published the findings of his doctoral research on stock prices; his empirical results indicated that stock prices behaved like a random walk. However, this study was overlooked for the next 50 years. Then, in 1953, Maurice Kendall published his analysis of stock market prices in which he suggested that price changes were e...

Time series are often formed from counts. The number of accidents per month at an intersection, the number of cardiac cases per day presenting at an emergency center, the number of power failures each month in a geographical region, and the weekly demand for a slow moving inventory are all examples of time series of counts. Such data are non-negati...

Exponential smoothing was used in Chap. 5 to generate the one-step-ahead prediction errors needed to evaluate the likelihood function when estimating the parameters of an innovations state space model. It relied on a fixed seed state vector to initialize the associated recurrence relationships, something that was rationalized by recourse to a finit...

Since the pioneering work of Brown (1959), it has been a common practice to use exponential smoothing methods to forecast demand in computerized inventory control systems. It transpired that exponential smoothing often produced good point forecasts. However, the methods proposed to measure the risk associated with the predictions typically ignored...

In exponential smoothing methods, the m seasonal components are combined with level and trend components to indicate changes to the time series that are caused by seasonal effects. It is sometimes desirable to report the value of these m seasonal components, and then it is important for them to make intuitive sense. For example, in the additive sea...

Although exponential smoothing methods have been around since the 1950s, a modeling framework incorporating stochastic models, likelihood calculations, prediction intervals, and procedures for model selection was not developed until relatively recently, with the work of Ord et al. (1997) and Hyndman et al. (2002). In these (and other) papers, a cla...

In this chapter we consider a broader class of innovations state space models, which enables us to examine multiplicative structures for any or all of the trend, the seasonal pattern and the innovations process. This general class was introduced briefly in Sect. 2.5.2. As for the linear models introduced in the previous chapter, this discussion wil...

One important step in the forecasting process is the selection of a model that could have generated the time series and would, therefore, be a reasonable choice for producing forecasts and prediction intervals. As we have seen in Chaps. 2–4, there are many specific models within the general innovations state space model (2.12). There are also many...

Point forecasts for each of the state space models were given in Table 2.1 (p. 18). It is also useful to compute the associated prediction distributions and prediction intervals for each model. In this chapter, we discuss how to compute these distributions and intervals.

The purpose of this chapter is to examine the links between the (linear) innovations state space models and autoregressive integrated moving average (ARIMA) models, frequently called “Box-Jenkins models” because Box and Jenkins (1970) proposed a complete methodology for identification, estimation and prediction with these models. We will show that...

In this chapter, we discuss some of the mathematical properties of the linear innovations state space models described in Chap. 3. These results are based on Hyndman et al. (2008). We provide conditions that ensure the model is of minimal dimension (Sect. 10.1) and conditions that guarantee the model is stable (Sect. 10.2). We will see that the non...

Up to this point in the book, we have considered models based upon a single series. However, in many applications, additional information may be available in the form of input or regressor variables; the name may be rather opaque, but we prefer it to the commonly-used but potentially misleading description of independent variables. We then refer to...

The primary purpose of this book is to demonstrate that the innovations form of the state space model provides a simple but flexible approach to forecasting time series. However, for reasons that are not completely clear, the innovations form has been largely over-shadowed in the literature by another version of the state space model that has multi...

Time series arise in many different contexts including minute-by-minute stock prices, hourly temperatures at a weather station, daily numbers of arrivals at a medical clinic, weekly sales of a product, monthly unemployment figures for a region, quarterly imports of a country, and annual turnover of a company. That is, time series arise whenever som...

We consider the properties of nonlinear exponential smoothing state space models under various assumptions about the innovations, or error, process. Our interest is restricted to those models that are used to describe non-negative observations, because many series of practical interest are so constrained. We first demonstrate that when the innovati...

INTRODUCTIONTHE CHARACTERISTICS OF FORECASTING COMPETITIONSTHE OBJECTIVES OF FORECASTING COMPETITIONSOBJECTIONS TO FORECASTING COMPETITIONSCONCLUSIONS FROM FORECASTING COMPETITIONSMULTIVARIATE INFORMATION SETSCONCLUSIONS

This paper treats some of the important considerations in constructing an analytical model for the distribution of demand during lead time. It presents a formal model that can be developed along one of two lines. The first has order size and order intensity leading to a compound distribution of period demand, then period demand and lead time giving...

Heretofore, the Poisson and the Laplace distributions have been used to model demand during lead time for slow-moving items. In this paper, we present a Poisson-like distribution called the Hermite. The advantage of the Hermite is that it is as simple to use as the Poisson and the Laplace are. Moreover, the Hermite is the exact distribution of dema...

In general linear modeling, an alternative to the method of least squares (LS) is the least absolute deviations (LAD) procedure. Although LS is more widely used, the LAD approach yields better estimates in the presence of outliers. In this paper, we examine the performance of LAD estimators for the parameters of the first-order autoregressive model...

Modern databases enable the estimation of conditional distributions for Y given X in a specified interval. Confidentiality requirements typically mean that individual observations cannot be released and output may be restricted to Y values for X within specified intervals. The width of such an interval could have a major impact upon the quality of...

Increased reliance upon outsourcing has made the issue of vendor selection even more critical to the success of the modern manufacturing organization. The usual performance measure on which selection is based has been the distribution of the vendor’s delivery lead-time (LT), often as characterized by the mean and variance. In this paper, we show th...

By adopting a real options framework we develop a production control model that jointly incorporates process and market uncertainties. In this model, process uncertainty is defined by random fluctuations in the outputs' yield and market risk through demand uncertainty for the output. In our approach, production outputs represent commodities or item...

Total Quality Management represents a set of ideas that we are happy to discuss in the classroom but less willing to practice. Personal experiences demonstrate how these concepts can improve an instructor's performance.

The state space approach to modelling univariate time series is now widely used both in theory and in applications. However, the very richness of the framework means that quite different model formulations are possible, even when they purport to describe the same phenomena. In this paper, we examine the single source of error [SSOE] scheme, which h...

Three general classes of state space models are presented, using the single source of error formulation. The first class is the standard linear model with homoscedastic errors, the second retains the linear structure but incorporates a dynamic form of heteroscedasticity, and the third allows for non-linear structure in the observation equation as w...

A new approach is proposed for forecasting a time series with multiple seasonal patterns. A state space model is developed for the series using the single source of error approach which enables us to develop explicit models for both additive and multiplicative seasonality. Parameter estimates may be obtained using methods adapted from general expon...

Exponential smoothing is often used to forecast lead-time demand (LTD) for inventory control. In this paper, formulae are provided for calculating means and variances of LTD for a wide variety of exponential smoothing methods. A feature of many of the formulae is that variances, as well as the means, depend on trends and seasonal effects. Thus, the...

Charles Holt's classic paper on exponentially weighted moving averages appeared as Report ONR 52 from the Office of Naval Research in 1957. Although widely cited, the full version has remained unpublished. This note provides some context for looking back to ONR 52, which then follows in its complete form. We indicate that the paper still has some l...

Whenever an unusual event disrupts the structural patterns of a time series, one of the aims of a forecaster is to model the effects of that event, with a view to establishing a new basis for forecasting. Intervention analysis has long been the method of choice for such adjustments, but it is often represented as a procedure for dealing with events...

By adopting a real options framework we develop a production control model that jointly incorporates process and market uncertainties. In this model, process uncertainty is defined by random fluctuations in the outputs’ yield and market risk through demand uncertainty for the output. In our approach, production outputs represent commodities or items...

30/01/2002 Exponential Smoothing_122401.docExponential Smoothing for Inventory Control: Mean and Variances of Lead-time Demand Exponential smoothing is often used to forecast lead-time demand for inventory control. In this paper, formulae are provided for calculating means and variances of lead-time demand for a wide variety of exponential smoothin...

Exponential smoothing, often used for sales forecasting in inventory control, has always been rationalized in terms of statistical models that possess errors with constant variances. It is shown in this paper that exponential smoothing remains the appropriate approach under more general conditions where the variances are allowed to grow and contrac...

The main objective of this paper is to provide analytical expression for forecast variances that can be used in prediction intervals for the exponential smoothing methods. These expressions are based on state space models with a single source of error that underlie the exponential smoothing methods. In cases where an ARIMA model also underlies an e...

Exponential smoothing (ES) forecasting methods are widely used but are often dis-cussed without recourse to a formal statistical framework. This paper reviews and compares a variety of potential models for ES. As well as autoregressive integrated moving average and structural models, a promising class of dynamic non-linear state space models is des...

A fundamental concern of spatial analysts is to find patterns in spatial data that lead to the identification of spatial autocorrelation or association. Further, they seek to identify peculiarities in the data set that signify that something out of the ordinary has occurred in one or more regions. In this paper we provide a statistic that tests for...

This paper shows that Medical Innovation (Coleman, Katz and Menzel 1966) and several subsequent studies analyzing the diffusion of the drug tetracycline have confounded social contagion with marketing effects. First, we describe the medical community's understanding of tetracycline and how the drug was marketed at the time the Medical Innovation da...

The problem of constructing prediction intervals for linear time series (ARIMA) models is examined. The aim is to find prediction intervals that incorporate an allowance for sampling error associated with parameter estimates. The effect of constraints on parameters arising from stationarity and invertibility conditions is also incorporated. Two new...

A new class of models for data showing trend and multiplicative seasonality is presented. The models allow the forecast error variance to depend on the trend and/ or the seasonality. It can be shown that each of these models has the same updating equations and forecast functions as the multiplicative Holt-Winters method, regardless of whether the e...

This paper investigates the interaction between the economics of production and imperfections in the production process. Specifically, this paper is the first to devise a model in an attempt to provide managers with guidelines to choose the appropriate production run times to buffer against both the production of defective items and stoppages occur...

This paper describes the M3-Competition, the latest of the M-Competitions. It explains the reasons for conducting the competition and summarizes its results and conclusions. In addition, the paper compares such results / conclusions with those of the previous two M-Competitions as well as with those of other major empirical studies. Finally, the im...

This paper shows how two different theoretical approaches to model innovation adoption, network threshold modeling and random utility modeling, both result in discrete-time hazard models, a popular statistical modeling approach to study innovation adoption and diffusion. The paper emphasizes the need for careful situational analysis prior to model...

We show how one can specify micro-level hazard models of adoption that separately represent the awareness and evaluation/choice stage of the adoption decision process even when one observes only the final outcome, i.e. adoption or not. These models can be derived from random utility theory. Using a small simulation study, we show that the two-stage...

Artificial neural networks (ANNs) are an information processing paradigm inspired by the way the brain processes information. Using neural networks requires the investigator to make decisions concerning the architecture or structure used. ANNs are known to be universal function approximators and are capable of exploiting nonlinear relationships bet...

We consider an inventory model where two suppliers are used concurrently to replenish one stock item. The unit demand is assumed to be normally distributed and the supplier lead times are mixtures of Erlang distributions. The lead time distributions need not be identical. This paper presents closed form expressions to evaluate the means and varianc...

This paper describes the M3-Competition, the latest of the M-Competitions. It explains the reasons for conducting the competition and summarizes its results and conclusions. In addition, the paper compares such results / conclusions with those of the previous two M-Competitions as well as with those of other major empirical studies. Finally, the im...