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... The coefficients $(') are estimated using a discounted least squares approach in which is minimized; w is a discount coefficient such that 0 < o <1. It can be shown, see Brown Ledolter & Box (1978) and McKenzie (1976). ...

... Collecting coefficients of Bi leads to the result in (6.14). This result can be used to specify the ARIMA model that is implied by Winters' additive method; also see McKenzie (1976). Substituting the restrictions in (6.12) into (6.14), ...

The simplifying operators in ARIMA (autogressive integrated moving average) models determine the form of the corresponding forecast functions. For example, regular differences imply polynomial trends and seasonal differences certain periodic functions. The same functions also arise in the context of many other forecast procedures, such as regressions on time, exponential smoothing and Kalman filtering. In this paper we describe how the various methods update the coefficients in these forecast functions and discuss their similarities and differences. In addition, we compare the forecasts from seasonal ARIMA models and the forecasts from Winters' additive and multiplicative smoothing methods. /// L'application d'opérateurs simplifiant les modèles ARIMA détermine du même coup le genre de fonctions de prédiction qui en résulte. Par exemple, l'utilisation de différences finies basées sur des intervalles réguliers génère des fonctions polynomiales, tandis que les différences finies éliminant les comportements saisonniers génèrent des fonctions à caractère périodique. Ces mêmes fonctions se retrouvent dans d'autres méthodes de prédiction, telles que les régressions sur le temps, la graduation exponentielle et le filtre de Kalman. Cet article présente donc une description de l'influence des différentes méthodes sur le processus de révision des coefficients des fonctions de prédiction. Une analyse comparative de ces méthodes est aussi inclue. De plus, nous comparons les prédictions d'un modèle ARIMA saisonnier et celles des modèles (additifs et multiplicatifs) de graduation de Winters.

... The coefficients $(') are estimated using a discounted least squares approach in which is minimized; w is a discount coefficient such that 0 < o <1. It can be shown, see Brown Ledolter & Box (1978) and McKenzie (1976). ...

... Collecting coefficients of Bi leads to the result in (6.14). This result can be used to specify the ARIMA model that is implied by Winters' additive method; also see McKenzie (1976). Substituting the restrictions in (6.12) into (6.14), ...

Seasonal Series Globally Constant Seasonal Models Locally Constant Seasonal Models Winters Seasonal Forecast Procedures Seasonal Adjustment

... The moving average parameters of which are related to the smoothing parameters. Turning to the seasonal model (SHW), it is optimal for a certain ARIMA process derived by McKenzie (1976) [9]. However, the generating process is very complex. ...

... Commonly used exponential methods such as simple exponential smoothing, double exponential smoothing, and Holt's exponential method are optimal if the underlying time series follow specific ARIMA models; see [20], [21], lent to forecasts generated by the ARIMA(0,1,1) model with moving average parameter Â D ! D 1 ˛. ...

George Edward Pelham Box was born on October 19, 1919 in Gravesend, Kent, UK and died on March 28, 2013 in Madison, Wisconsin, USA. George Box made significant contributions to many fields of statistics including design of experiments and response surface methodology, evolutionary operation, statistical inference, robustness, Bayesian methods, time series analysis and forecasting, and quality improvement. Our paper discusses his contributions to time series analysis and forecasting. His work in this area started in collaboration with Gwilym Jenkins in the early 1960s and continued over the next several decades. His contributions include the classic and extraordinarily influential book ‘Time Series Analysis: Forecasting and Control’ written with Gwilym Jenkins and first published by Holden Day in 1970. Subsequent contributions to time series analysis include joint work with George Tiao, Gregory Reinsel, Daniel Pena, and many former graduate students. His work provided a unified framework for carrying out time series analysis in practice and laid the foundation for many new developments in the field. Copyright © 2014 John Wiley & Sons, Ltd.

... On se limite au cas univarié. Les résultats 3.19, 3.21, 3.23 et 3.25 ontétéétablis par McKenzie (1974McKenzie ( , 1976) avec toutefois la réserve mentionnée dans la Remarque 1 qui suit la Proposition 3.19. Les démonstrations sont toutefois différentes. ...

Cette monographie a été publiée pour la première fois en 1985 par les Presses de l'Université de Montréal (C.P. 6128, succ. ``A'', Montréal (Qubec), Canada, H3C 3J7). Elle avait été réalisée dans la collection Séminaire de mathématique supérieures, Séminaire scientifique OTAN (NATO Advanced Study Institute) par le Département de mathématiques et de statistique - Université de Montréal.
Il s'agissait de mes notes de cours à la vingt-et-unième session du Séminaire de mathématiques supérieures/Séminaire scientifique OTAN (ASI 82/62), tenue au Département de mathématiques et de statistique de l'Université de Montréal du 26 juillet au 13 août 1982. Cette session avait pour titre général ``Analyse des données'' et tait place sous les auspices de l'Organisation du Trait de l'Atlantique Nord, du ministre de l'éducation du Québec, du Conseil de recherches en sciences naturelles et en génie du Canada et de l'Université de Montréal.

... The advantages of the ARIMA modeling for signal extraction are, first, that it permits more flexibility than the fixed filters used by exponential smoothing procedures, as the Holt-Winters algorithm, and second, that the use of a statistical model offers a systematic framework for analysis. For example, the Holt-Winters equations in (2.2) are optimal to estimate the components of a series which follows a MA(13) model after beign differenced as 12 y t (McKenzie, 1976). Later, Newbold (1988) showed that the parameters of the MA(13) model satisfy the following relationships: However, ARIMA models still have some disadvantages when trying to estimate the unobserved components of a time series. ...

In this paper we model the monthly number of passengers flying with the Spanish airline IBERIA from January 1985 to December 1992 and predict future values of the series up to October 1994. This series is characterized by strong seasonal variations and by having an upward trend which has a rupture during 1990 with the slope changing to be negative. We compare observed values with predictions made by a deterministic components model, the Holt-Winters exponential smoothing filter, an ARIMA model and a structural time series model. As expected, we show that the deterministic components model is too rigid in the presence fo breaks in trends although surprisingly the within-sample fit is better than for any of the other models considered. With respect to Holt-Winters predictions, they fail because they are not able to acommodate outliers. Finally, ARIMA and structural models are shown to have very similar prediction performance, being very flexible to predict reasonably well when there are changes in trend and outliers.

... The reduced form is an ARIMA(O, {1, m}, m + 1) model, where {1, m} denotes differences of order 1 and m. This particular ARIMA model underlies the additive version of the Holt-Winters method (McKenzie 1976; Roberts 1982 ). The associated invertibility conditions have been derived by Archibald (1990). ...

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 involving non-Gaussian disturbances, is considered. A method for computing prediction intervals is proposed and evaluated on both simulated and real data.

In der einschlägigen deutschsprachigen Literatur stehen sich zahlreiche Prognoseformeln recht beziehungslos gegenüber, so beispielsweise die Verfahren von HOLT (1957), MUTH (1960), BROWN & MEYER (1961), D’ESOPO (1961), BOX & JENKINS (1962), NERLOVE & WAGE (1964) und HARRISON (1967).

A concerted effort is made to emphasize the need for the added stroke of Interpretation to the Box-Jenkins iterative cycle of identification, estimation and verification in time series modelling. This aspect is of paramount importance for both successful model-building and communication between the analyst and manager. Apart from discussions of how the various elements of a Box-Jenkins fit can be explained and a section listing six basic ways by which mixed models can be interpreted in terms of more easily justified simple models, an attempt is made to reconcile the relatively unfamiliar Box-Jenkins models with the well-understood classical decomposition of a series into trend, seasonal and irregular. This reconciliation is illustrated with an example. It is believed that the Box-Jenkins approach generally produces forecasts superior to those from other extrapolatory procedures; and this paper is intended to help statisticians understand more fully, and communicate more successfully, the results they obtain using this powerful and versatile methodology.

A new estimation procedure called sequential discount estimation is introduced in order to provide both linear and non-linear predictors. This is obtained by revising the classical locally linear model representations generalising the exponential weighted regression and the discount weighted estimation models. This allows a complete on-line learning of the unknown components and can accommodate multiplicative and bilinear predictor models in a simple analytical manner. The principles of parsimony and operational simplicity are preserved. Special canonical representations and some limiting results are discussed. Finally the method is applied to two real data sets.

In recent years considerable interest has centred on a family of moving average models proposed by Box and Jenkins (1970) for the description of time series which contain a seasonal component. A method is given in the paper for estimating the parameters of the seasonal models based on a principle due to Walker (1961) for the estimation of non‐zero members of the autocorrelation function. An illustration of the method is given.

The Holt-Winters method has been widely used to forecast a seasonal time series in application fields as a nonparametric forecasting technique. We investigate the asymptotic forecast errors of the Holt-Winters method. For that purpose, we show that the nonlinear least squares estimates of the smoothing parameters included in the smoothing algorithm hold strong convergence properties under suitable conditions. Then we show the mean squared errors and the limiting distributions of the forecast errors for some stochastic processes. Finally, numerical studies are performed to evaluate the forecasting performance of the Holt-Winters method.

A procedure for deriving the variance of the forecast error for Winters' additive seasonal forecasting system is given. Both point and cumulative T-step ahead forecasts are dealt with. Closed form expressions are given in the cases when the model is (i) trend-free and (ii) non-seasonal. The effects of renormalization of the seasonal factors is also discussed. The fact that the error variance for this system can be infinite is discussed and the relationship of this property with the stability of the system indicated. Some recommendations are given about what to do in these circumstances.

Autoregressive models for spatial interaction have been proposed by several authors (Whittle [15] and Mead [11], for example). In the past, computational difficulties with the ML approach have led to the use of alternative estimators. In this article, a simplified computational scheme is given and extended to mixed regressive-autoregressive models. The ML estimator is compared with several alternatives.

The Holt-Winters forecasting procedure is a simple widely used projection method which can cope with trend and seasonal variation. However, empirical studies have tended to show that the method is not as accurate on average as the more complicated Box-Jenkins procedure. This paper points out that these empirical studies have used the automatic version of the method, whereas a non-automatic version is also possible in which subjective judgement is employed, for example, to choose the correct model for seasonality. The paper re-analyses seven series from the Newbold-Granger study for which Box-Jenkins forecasts were reported to be much superior to the (automatic) Holt-Winters forecasts. The series do not appear to have any common properties, but it is shown that the automatic Holt-Winters forecasts can often be improved by subjective modifications. It is argued that a fairer comparison would be that between Box-Jenkins and a non-automatic version of Holt-Winters. Some general recommendations are made concerning the choice of a univariate forecasting procedure. The paper also makes suggestions regarding the implementation of the Holt-Winters procedure, including a choice of starting values.

Recently published books on time series are briefly reviewed. Then a critical assessment is made of recent practical developments in time-series analysis, including the treatment of trend and seasonality, the Box-Jenkins forecasting procedure, adaptive filtering, Bayesian forecasting, spectral analysis, autoregressive spectrum estimation, the use of the Fast Fourier Transform and the identification of linear systems. The gas furnace data of Box and Jenkins are re-examined with somewhat surprising results. Finally, the relationship between time and spatial processes is briefly considered.

The Holt-Winters forecasting procedure is a variant of exponential smoothing which is simple, yet generally works well in practice, and is particularly suitable for producing short-term forecasts for sales or demand time-series data. Some practical problems in implementing the method are discussed, including the normalization of seasonal indices, the choice of starting values and the choice of smoothing parameters. There is an important distinction between an automatic and a non-automatic approach to forecasting and detailed suggestions are made for implementing Holt-Winters in both ways. The question as to what underlying model, if any, is assumed by the method is also addressed. Some possible areas for future research are then outlined.

This paper is a critical review of exponential smoothing since the original work by Brown and Holt in the 1950s. Exponential smoothing is based on a pragmatic approach to forecasting which is shared in this review. The aim is to develop state-of-the-art guidelines for application of the exponential smoothing methodology. The first part of the paper discusses the class of relatively simple models which rely on the Holt-Winters procedure for seasonal adjustment of the data. Next, we review general exponential smoothing (GES), which uses Fourier functions of time to model seasonality. The research is reviewed according to the following questions. What are the useful properties of these models? What parameters should be used? How should the models be initialized? After the review of model-building, we turn to problems in the maintenance of forecasting systems based on exponential smoothing. Topics in the maintenance area include the use of quality control models to detect bias in the forecast errors, adaptive parameters to improve the response to structural changes in the time series, and two-stage forecasting, whereby we use a model of the errors or some other model of the data to improve our initial forecasts. Some of the major conclusions: the parameter ranges and starting values typically used in practice are arbitrary and may detract from accuracy. The empirical evidence favours Holt's model for trends over that of Brown. A linear trend should be damped at long horizons. The empirical evidence favours the Holt-Winters approach to seasonal data over GES. It is difficult to justify GES in standard form–the equivalent ARIMA model is simpler and more efficient. The cumulative sum of the errors appears to be the most practical forecast monitoring device. There is no evidence that adaptive parameters improve forecast accuracy. In fact, the reverse may be true.

Forecasting methods are reviewed. They may be classified into univariate, multivariate and judgemental methods, and also by whether an automatic or non-automatic approach is adopted. The choice of ‘best’ method depends on a wide variety of considerations. The use of forecasting competitions to compare the accuracy of univariate methods is discussed. The strengths and weaknesses of different univariate methods are compared, both in automatic and non-automatic mode. Some general recommendations are made as well as some suggestions for future research.

We present a method for investigating the evolution of trend and seasonality in an observed time series. A general model is fitted to a residual spectrum, using components to represent the seasonality. We show graphically how well the fitted spectrum captures the evidence for evolving seasonality associated with the different seasonal frequencies. We apply the method to model two time series and illustrate the resulting forecasts and seasonal adjustment for one series. Copyright © 2000 John Wiley & Sons, Ltd.

Although the basic principles of exponential smoothing and discounted least squares are easily understood, the full power of the technique is only rarely exploited. The reason for this failure lies in the complexity of the standard procedures. Often they require fairly complex mathematical models and use a variety of cumbersome algebraic manipulations. An alternative formulation for exponential smoothing is presented. It simplifies these procedures and allows an easier use of the full range of models. This new formulation is obtained by considering the relationship between general exponential smoothing (GES) and the well-known ARMA process of Box and Jenkins. The three commonest seasonal models have only recently been considered for GES systems. They are discussed in some detail here. The computational requirements of the GES and equivalent ARMA procedures are reviewed and some recommendations for their application are made. The initialization of GES forecasting systems and the important problem of model selection is also discussed. A brief illustrative example is given.

This paper presents expressions for the variance of the forecast error for arbitrary lead times for both the additive and multiplicative Holt-Winters seasonal forecasting models. It is shown that even when the smoothing constants are chosen to have values between zero and one, when the period is greater than four, the variance may not be finite for some values of the smoothing constants. In addition, the regions where the variance becomes infinite are almost the same for both models. These results are of importance for practitioners, who may choose values for the smoothing constants arbitrarily, or by searching on the unit cube for values which minimize the sum of the squared errors when fitting the model to a data set. It is also shown that the variance of the forecast error for the multiplicative model is nonstationary and periodic.

This paper considers univariate online electricity demand forecasting for lead times from a half-hour-ahead to a day-ahead. A time series of demand recorded at half-hourly intervals contains more than one seasonal pattern. A within-day seasonal cycle is apparent from the similarity of the demand profile from one day to the next, and a within-week seasonal cycle is evident when one compares the demand on the corresponding day of adjacent weeks. There is strong appeal in using a forecasting method that is able to capture both seasonalities. The multiplicative seasonal ARIMA model has been adapted for this purpose. In this paper, we adapt the Holt–Winters exponential smoothing formulation so that it can accommodate two seasonalities. We correct for residual autocorrelation using a simple autoregressive model. The forecasts produced by the new double seasonal Holt–Winters method outperform those from traditional Holt–Winters and from a well-specified multiplicative double seasonal ARIMA model.

A large number of statistical forecasting procedures for univariate time series have been proposed in the literature. These range from simple methods, such as the exponentially weighted moving average, to more complex procedures such as Box–Jenkins ARIMA modelling and Harrison–Stevens Bayesian forecasting. This paper sets out to show the relationship between these various procedures by adopting a framework in which a time series model is viewed in terms of trend, seasonal and irregular components. The framework is then extended to cover models with explanatory variables. From the technical point of view the Kalman filter plays an important role in allowing an integrated treatment of these topics.

Prediction interval formulae are derived for the Holt-Winters forecasting procedure with an additive seasonal effect. The formulae make no assumptions about the ‘true’ underlying model. The results are contrasted with those obtained from various alternative approaches to the calculation of prediction intervals. Some large discrepancies are noted and it is suggested that the formulae presented here should be preferred to those which depend on an inappropriate deterministic model or which depend on invalid generalised approximations which take no account of the particular properties of the given series. Results for cumulative forecasts and for a damped trend model are also given. For completeness we also give results for one- and two-parameter exponential smoothing. Finally, we make some general comments as to why prediction intervals tend to be too narrow in practice.

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 exponential smoothing method, there is an equivalent state space model with the same variance expression. We also discuss relationships between these new ideas and previous suggestions for finding forecast variances and prediction intervals for the exponential smoothing methods.

In Winters’ seasonal exponential smoothing methods, a time series is decomposed into: level, trend and seasonal components, that change over time. The seasonal factors are initialized so that their average is 0 in the additive version or 1 in the multiplicative version. Usually, only one seasonal factor is updated each period, and the average of the seasonal factors is no longer 0 or 1; the ‘seasonal factors’ no longer meet the usual meaning of seasonal factors. We provide an equivalent reformulation of previous equations for renormalizing the components in the additive version. This form of the renormalization equations is then adapted to new renormalization formulas for the multiplicative Winters’ method. For both the standard and renormalized equations we make a minor change to the seasonal equation. Predictions from our renormalized smoothing values are the same as for the original smoothed values. The formulas can be applied every period, or when required. However, we recommend renormalization every time period. We show in the multiplicative version that the level and trend should be adjusted along with the seasonal component.

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 error variation in the model is constant or not. While the point forecasts from the different models are identical, the prediction intervals will, of course, depend on the structure of the error variance and so it is essential to be able to choose the most appropriate form of model. Two methods for making this choice are presented and examined by simulation.

In the additive Holt-Winters' seasonal exponential smoothing model, it is theoretically possible for smoothing parameters in the usual (0, 1) interval to produce ‘non-invertible’ models. A sample of 406 monthly series reveal that this is a real concern. In the multiplicative model, reasonable estimation procedures produce smoothing constants outside the additive invertible region. When this occurs, the impact on forecasts of values in the distant past is much larger than for recent values and forecasts are poor.Results from the 406 series show that the problem can be avoided and forecasts improved if a subset of the additive invertible region is used as the parameter space.

After a critical review of a number of published methods for short-term sales forecasting, two adaptive methods for seasonal sales forecasting have been developed. The conclusions reached are that for most industrial purposes short-term sales forecasting methods should contain as few parameters as possible, that Brown's one-parameter method is to be recommended for non-seasonal forecasting and that the two-parameter seasonal techniques developed in this paper offer improved seasonal forecasting methods.

A step-by-step account is given of a Box-Jenkins analysis of some sales figures showing high multiplicative seasonal variation. Various practical problems are encountered and discussed. A critical appraisal is made of the Box-Jenkins procedure and some general remarks are made on short-term sales forecasting.

The growing use of computers for mechanized inventory control and production planning has brought with it the need for explicit forecasts of sales and usage for individual products and materials. These forecasts must be made on a routine basis for thousands of products, so that they must be made quickly, and, both in terms of computing time and information storage, cheaply; they should be responsive to changing conditions. The paper presents a method of forecasting sales which has these desirable characteristics, and which in terms of ability to forecast compares favorably with other, more traditional methods. Several models of the exponential forecasting system are presented, along with several examples of application.

Results of a simulation study of the short-range forecasting effectiveness of exponentially smoothed and selected Box-Jenkins models for sixty-three monthly sales series are presented. Brown's exponentially smoothed constant, linear and six- and eight-term harmonic models are compared with two seasonal factor models and ten Box-Jenkins models. The seasonal factor models are Winters' three parameter model and a single parameter model developed by the author. The forecasting errors of the best of the Box-Jenkins models that were tested are either approximately equal to or greater than the errors of the corresponding exponentially smoothed models. Test results indicate that the four exponentially smoothed seasonal models yield approximately equivalent accuracy for most data series. If a sharply defined pattern exists, the seasonal factor models can yield smaller forecast errors than the harmonic models. If no seasonal pattern exists, the errors of the exponentially smoothed models with seasonal components are only slightly greater than those of the constant and linear models.