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Introduction

**Skills and Expertise**

## Publications

Publications (96)

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...

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...

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...

The vector innovations structural time series framework is proposed as a way of modelling a set of related time series. As with all multivariate approaches, the aim is to exploit potential inter- series dependencies to improve the fit and forecasts. The model is based around an unobserved vector of components representing features such as the level...

An innovations state space modeling framework is introduced for forecasting complex seasonal time series such as those with multiple seasonal periods, high-frequency seasonality, non-integer seasonality, and dual-calendar effects. The new framework incorporates Box–Cox transformations, Fourier representations with time varying coefficients, and ARM...

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 Beveridge-Nelson vector innovations structural time series framework is a new formulation that decomposes a set of variables into their permanent and transitory components. The proposed framework is flexible, modelling inter-series relationships and common features in a simple manner. In particular, it is shown that this new specification is si...

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...

It is a common practice to complement a forecasting method such as simple exponential smoothing with a monitoring scheme to detect those situations where forecasts have failed to adapt to structural change. It will be suggested in this paper that the equations for simple exponential smoothing can be augmented by a common monitoring statistic to pro...

This paper concerns the forecasting of seasonal intraday time series. An extension of Holt-Winters exponential smoothing has been proposed that smoothes an intraday cycle and an intraweek cycle. A recently proposed exponential smoothing method involves smoothing a different intraday cycle for each distinct type of day of the week. Similar days are...

This paper provides a comparison of a parameter-driven and an observation-driven discrete state space model. The two models are shown to have non-overlapping feasible regions for dispersion and first-order autocorrelation, with the region for the parameter-driven model being much larger than that of the observation-driven model, as well as providin...

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...

Damped trend exponential smoothing has previously been established as an important forecasting method. Here, it is shown to have close links to simple exponential smoothing with a smoothed error tracking signal. A special case of damped trend exponential smoothing emerges from our analysis, one that is more parsimonious because it effectively relie...

The problem of computing estimates of the state vector when the Kalman filter is seeded with an arbitrarily large variance is considered. To date the response in the literature has been the development of a number of relatively complex hybrid filters, usually involving additional quantities and equations over and above the conventional filter. We s...

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...

This paper has a focus on non-stationary time series formed from small non-negative integer values which may contain many zeros and may be over-dispersed. It describes a study undertaken to compare various suitable adaptations of the simple exponential smoothing method of forecasting on a database of demand series for slow moving car parts. The met...

This paper compares two alternative models for autocorrelated count time series. The first model can be viewed as a 'single source of error' discrete state space model, in which a time-varying parameter is specified as a function of lagged counts, with no additional source of error introduced. The second model is the more conventional 'dual source...

I assess the role of wealth and systemic risk in explaining future asset returns. I show that the residuals of the trend relationship among asset wealth and human wealth predict both stock returns and government bond yields. Using data for a set of industrialized countries, I find that when the wealth-to-income ratio falls, investors demand a highe...

This paper considers Beveridge-Nelson decomposition in a context where the permanent and transitory components both follow a Markov switching process. Our approach incorporates Markov switching into a single source of error state-space framework, allowing business cycle asymmetries and regime switches in the long run multiplier.

This paper considers Beveridge-Nelson decomposition in a context where the permanent and transitory components both follow a Markov switching process. Our approach insorporates Markov switching into a single source of error state-space framework, allowing business cycle asymmetries and regime switches in the long-run multiplier.

The paper examines the performance of four multivariate volatility models, namely CCC, VARMA-GARCH, DCC and BEKK, for the crude oil spot and futures returns of two major benchmark international crude oil markets, Brent and WTI, to calculate optimal portfolio weights and optimal hedge ratios, and to suggest a crude oil hedge strategy. The empirical...

Applications of exponential smoothing to forecasting time series usually rely on three basic methods: simple exponential smoothing, trend corrected exponential smoothing and a seasonal variation thereof. A common approach to selecting the method appropriate to a particular time series is based on prediction validation on a withheld part of the samp...

These notes supplement Gardner's comprehensive update of his 1985 state of the art paper on the exponential smoothing methods of forecasting. A general definition of exponential smoothing, accommodating both linear and nonlinear versions of this method, is presented. Damped trend corrected exponential smoothing, augmented by a long-run growth rate,...

It is a common practice to complement a forecasting method such as simple exponential smoothing with a monitoring scheme to detect those situations where forecasts have failed to adapt to structural change. It will be suggested in this paper that the equations for simple exponential smoothing can be augmented by a common monitoring statistic to pro...

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...

An approach to exponential smoothing that relies on a linear single source of error state space model is outlined. A maximum likelihood method for the estimation of associated smoothing parameters is developed. Commonly used restrictions on the smoothing parameters are rationalised. Issues surrounding model identification and selection are also con...

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...

In the exponential smoothing approach to forecasting, restrictions are often imposed on the smoothing parameters which ensure that certain components are exponentially weighted averages. In this paper, a new general restriction is derived on the basis that the one-step ahead prediction error can be decomposed into permanent and transient components...

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...

A Kalman filter for application to stationary or non-stationary time series is proposed. A major feature is a new initialisation method to accommodate non-stationary time series. The filter works on time series with missing values at any point of time including the initialisation phase. It can also be used where a state space model does not satisfy...

Traditional computerised inventory control systems usually rely on exponential smoothing to forecast the demand for fast moving inventories. Practices in relation to slow moving inventories are more varied, but the Croston method is often used. It is an adaptation of exponential smoothing that (1) incorporates a Bernoulli process to capture the spo...

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...

We provide a new approach to automatic forecasting based on an extended range of exponential smoothing methods. Each method in our taxonomy of exponential smoothing methods provides forecasts that are equivalent to forecasts from a state space model. This equivalence allows: (1) easy calculation of the likelihood, the AIC and other model selection...

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 parsimonious method of exponential smoothing is introduced for time series generated from a combination of local trends and local seasonal effects. It is compared with the additive version of the Holt-Winters method of forecasting on a standard collection of real time series. Copyright © 2001 by John Wiley & Sons, Ltd.

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...

We examine Italian inflation rates and the Phillips curve with a very long-run perspective, one that covers the entire existence of the Italian lira from political unification (1861) to Italy's entry in the European Monetary Union (end of 1998). We first study the volatility, persistence and stationarity of the Italian inflation rate over the long...

The basic ideals underlying the Kalman filter are outlined in this paper without direct recourse to the complex formulae normally associated with this method. The novel feature of the paper is its reliance on a new algebraic system based on the first two moments of the multivariate normal distribution. The resulting framework lends itself to an obj...

A new simple formula is found to correct the underestimation of the standard deviation for total lead time demand when using simple exponential smoothing. The traditional formula for the standard deviation of lead time demand is to multiply the standard deviation for the one-period-ahead forecast error (estimated by using the residuals) by the squa...

The focus of this paper is on the relationship between the exponential smoothing methods of forecasting and the integrated autoregressive-moving average models underlying them. In this paper we derive, for the first time, the general linear relationship between their parameters. A method, suitable for implementation on computer, is proposed to dete...

We examine Italian inflation rates and the Phillips curve with a very long-run perspective, one that covers the entire existence of the Italian lira from political unification (1861) to Italy's entry in the European Monetary Union (end of 1998). We first study the volatility, persistence and stationarity of the Italian inflation rate over the long...

The innovations representation for a local linear trend can adapt to long run secular and short term transitory effects in the data. This is illustrated by the theoretical power spectrum for the model which may possess considerable power at frequencies that might be associated with cycles of several years' duration. Whilst advantageous for short te...

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

The problem considered in this paper is how to find reliable prediction intervals with simple exponential smoothing and trend corrected exponential smoothing. Methods for constructing prediction intervals based on linear approximation and bootstrapping are proposed.

A class of nonlinear state-space models, characterized by a single source of randomness, is introduced. A special case, the model underpinning the multiplicative Holt-Winters method of forecasting, is identified. Maximum likelihood estimation based on exponential smoothing instead of a Kalman filter, and with the potential to be applied in contexts...

Exponential smoothing methods are often used to forecast demand in computerized inventory control systems. These methods, by themselves, are rather ad hoc, but they can be given a proper statistical foundation by setting up a class of structural time series models. The purpose of the paper is to highlight the potential role of these models in inven...

The problem of controlling inventories with intermittent demands is considered. A method for determining re-order levels consistent with a specified customer service level is proposed. The distinguishing feature is the use of a probability distribution with a spike at zero to represent the relative frequency of periods with no transactions.

The paper outlines a finite sample version of exponential smoothing, and proposes a formula for estimating the smoothing parameter. The resulting method, which can be implemented on a recursive basis over time, is compared with alternative approaches, such as progressive numerical optimization of the smoothing parameter and adaptive forecasting on...

A dynamic, linear model for the analysis of univariate time series is proposed. It encompasses many of the common statistical models as special cases such as multiple regression, exponential smoothing and mixed autoregressive‐moving average processes. its distinguishing feature is that it relies on only one primary source of randomness. It therefor...

Application of inventory theory often rely on the normal and negative exponential distributions to represent the lead time demand of fast and slow moving items respectively. Yet it is now accepted that both distributions, when taken together, are incapable of adequately describing the demand characteristics of all items found in the typical invento...

This paper considers the problem of obtaining estimates of seasonal and trend components of time series. It proposes a simple computational procedure for finding those estimates which minimize the sum of absolute errors for an additive time series model. It also examines a multiplicative model with a compound relative error criterion. The proposed...

This paper investigates the role of safety stocks in fixed order quantity-reorder point inventory systems. A common confusion over the notion of a shortage probability is shown to lead to methods which encourage excessive safety stocks. An alternative approach, proposed by R. G. Brown, is validated under quite general demand conditions. When combin...

Continuous review stock systems are considered in this paper. They are modeled using discrete time dynamic programming, which in contrast to previous continuous time formulations of the problem, yields a reasonable algorithm for computational purposes. The model is used to establish the optimal forms of stationary ordering policies under various co...

It is often believed that the square root formula called the EOQ only applies to situations where customers demand small quantities at a fairly constant rate. This note shows that essentially the same formula can be derived for "lumpy" customer orders occurring at a variable rate. Also, as the EOQ is derived assuming that a fixed quantity is always...

In this paper a periodic review stock system using the so-called (S, s) ordering policy is considered. New and more accurate approximations are found for calculating the optimal values of the reorder point s and the minimum order quantity \Delta = S - s. When demands are normally distributed it is shown that the six cost and distribution parameters...

Most continuous time inventory models which allow for the stochastic nature of demands usually include a delivery lag. This disguises a close link between deterministic and stochastic formulations of the inventory problem. By assuming that there is no delivery lag a stochastic version of the classical economic lot size model is developed. It yields...

Crow-fly distance between the depot and delivery points is usually used when considering the problem of locating a depot. Instead, we assume that vans can only travel along a rectangular system of roads. This leads us to a much simpler solution.