Forecasting Time Series With Complex Seasonal Patterns Using Exponential Smoothing

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 ARMA error correction. Likelihood evaluation and analytical expressions for point forecasts and interval predictions under the assumption of Gaussian errors are derived, leading to a simple, comprehensive approach to forecasting complex seasonal time series. A key feature of the framework is that it relies on a new method that greatly reduces the computational burden in the maximum likelihood estimation. The modeling framework is useful for a broad range of applications, its versatility being illustrated in three empirical studies. In addition, the proposed trigonometric formulation is presented as a means of decomposing complex seasonal time series, and it is shown that this decomposition leads to the identification and extraction of seasonal components which are otherwise not apparent in the time series plot itself.

[1]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[2]  John F. Magee,et al.  Production planning and inventory control , 1958 .

[3]  R. Brown.  Statistical forecasting for inventory control , 1960 .

[4]  Peter R. Winters,et al.  Forecasting Sales by Exponentially Weighted Moving Averages , 1960 .

[5]  J. Muth.  Optimal Properties of Exponentially Weighted Forecasts , 1960 .

[6]  E. J. Hannan,et al.  The Estimation of Seasonal Variation in Economic Time Series , 1963 .

[7]  Robert Goodell Brown,et al.  Smoothing, forecasting and prediction of discrete time series , 1964 .

[8]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[9]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[10]  Julius Shiskin,et al.  The X-11 variant of the census method II seasonal adjustment program , 1965 .

[11]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[12]  E. J. Hannan,et al.  The Seasonal Adjustment of Economic Time Series , 1970 .

[13]  W. R. Christiaanse.  Short-Term Load Forecasting Using General Exponential Smoothing , 1971 .

[14]  J. Scott Armstrong,et al.  Long-Range Forecasting: From Crystal Ball to Computer , 1981 .

[15]  George E. P. Box,et al.  Deterministic and Forecast‐Adaptive Time‐Dependent Models , 1978 .

[16]  Chris Chatfield,et al.  The Holt-Winters Forecasting Procedure , 1978 .

[17]  Steven C. Wheelwright,et al.  Forecasting methods and applications. , 1979 .

[18]  Andrew Harvey,et al.  An Algorithm for Exact Maximum Likelihood Estimation of Autoregressive–Moving Average Models by Means of Kaiman Filtering , 1980 .

[19]  Robert J. Genetski,et al.  Long-Range Forecasting: From Crystal Ball to Computer , 1981 .

[20]  Robert L. Winkler,et al.  The accuracy of extrapolation (time series) methods: Results of a forecasting competition , 1982 .

[21]  S. A. Roberts.  A General Class of Holt-Winters Type Forecasting Models , 1982 .

[22]  Forecasting for inventory control. , 1982, Hospital materiel management quarterly.

[23]  Robert L. Winkler,et al.  The Accuracy of Extrapolation (Time Series) Methods , 1982 .

[24]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[25]  Guy Melard,et al.  Algorithm AS197: A fast algorithm for the exact likelihood of autoregressive-moving average models , 1984 .

[26]  E. McKenzie.  General exponential smoothing and the equivalent arma process , 1984 .

[27]  Everette S. Gardner,et al.  Exponential smoothing: The state of the art , 1985 .

[28]  E. S. Gardner,et al.  Forecasting Trends in Time Series , 1985 .

[29]  Ralph D. Snyder,et al.  Recursive Estimation of Dynamic Linear Models , 1985 .

[30]  Arnold L. Sweet,et al.  Computing the variance of the forecast error for the holt‐winters seasonal models , 1985 .

[31]  Robert Kohn,et al.  Computing the likelihood and its dierivatives for a gaussian ARMA model , 1985 .

[32]  R. Kohn,et al.  Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions , 1985 .

[33]  E. McKenzie,et al.  Error analysis for winters' additive seasonal forecasting system , 1986 .

[34]  Alan Pankratz,et al.  Forecasts of power‐transformed series , 1987 .

[35]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[36]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[37]  Irma J. Terpenning,et al.  STL : A Seasonal-Trend Decomposition Procedure Based on Loess , 1990 .

[38]  Leonard J. Tashman,et al.  Automatic forecasting software: A survey and evaluation☆ , 1991 .

[39]  Robert Fildes,et al.  Forecasting, Structural Time Series Models and the Kalman Filter: Bayesian Forecasting and Dynamic Models , 1991 .

[40]  P. D. Jong.  The Diffuse Kalman Filter , 1991 .

[41]  J. Keith Ord,et al.  Automatic forecasting using explanatory variables: A comparative study , 1991 .

[42]  C. Chatfield,et al.  Prediction intervals for multiplicative Holt-Winters , 1991 .

[43]  Spyros Makridakis,et al.  Accuracy measures: theoretical and practical concerns☆ , 1993 .

[44]  C. Chatfield.  Prediction Intervals , 1993, Introduction to Meta‐Analysis.

[45]  C. Chatfield,et al.  The M2-competition: A real-time judgmentally based forecasting study , 1993 .

[46]  A. Harvey,et al.  Forecasting Hourly Electricity Demand Using Time-Varying Splines , 1993 .

[47]  Jeff Harrison,et al.  Applied Bayesian Forecasting and Time Series Analysis , 1994 .

[48]  B. Hansen,et al.  Are Seasonal Patterns Constant Over Time? A Test for Seasonal Stability , 1995 .

[49]  J. Ord,et al.  Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models , 1997 .

[50]  Adrian Pagan,et al.  Seasonal Integration and the Evolving Seasonals Model , 1996 .

[51]  Siem Jan Koopman,et al.  The modeling and seasonal adjustment of weekly observations , 1997 .

[52]  J. Fox.  Applied Regression Analysis, Linear Models, and Related Methods , 1997 .

[53]  Steven C. Wheelwright,et al.  Forecasting: Methods and Applications, 3rd Ed , 1997 .

[54]  F P Wheeler,et al.  Bayesian Forecasting and Dynamic Models (2nd edn) , 1998, J. Oper. Res. Soc..

[55]  David F. Findley,et al.  New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program , 1998 .

[56]  David R. Anderson,et al.  Model Selection and Multimodel Inference , 2003 .

[57]  R. Lawton.  How should additive Holt–Winters estimates be corrected? , 1998 .

[58]  Siem Jan Koopman,et al.  Signal extraction and the formulation of unobserved components models , 2000 .

[59]  A. Koehler,et al.  Forecasting for Inventory Control with Exponential Smoothing Associate Professor , 1999 .

[60]  Tommaso Proietti,et al.  Comparing seasonal components for structural time series models , 2000 .

[61]  D. Katz,et al.  American Statistical Association , 2022, The SAGE Encyclopedia of Research Design.

[62]  Guy Melard,et al.  Automatic ARIMA modeling including interventions, using time series expert software , 2000 .

[63]  Chris Chatfield,et al.  Time‐series forecasting , 2000 .

[64]  Spyros Makridakis,et al.  The M3-Competition: results, conclusions and implications , 2000 .

[65]  Georges A. Darbellay,et al.  Forecasting the short-term demand for electricity: Do neural networks stand a better chance? , 2000 .

[66]  J. Ord,et al.  A New Look at Models For Exponential Smoothing , 2001 .

[67]  Siem Jan Koopman,et al.  Time Series Analysis by State Space Methods , 2001 .

[68]  Denise R. Osborn,et al.  The Econometric Analysis of Seasonal Time Series , 2001 .

[69]  Craig B. Borkowf,et al.  Time-Series Forecasting , 2002, Technometrics.

[70]  M. Khan,et al.  Detection and Forecasting of Islamic Calendar Effects in Time series Data , 2002 .

[71]  Diego J. Pedregal,et al.  An unobserved component model for multi-rate forecasting of telephone call demand: the design of a forecasting support system , 2002 .

[72]  Baki Billah,et al.  Empirical information criteria for time series forecasting model selection , 2005 .

[73]  Anne B. Koehler,et al.  Forecasting for Inventory Control with Exponential Smoothing , 2002 .

[74]  Rob J Hyndman,et al.  A state space framework for automatic forecasting using exponential smoothing methods , 2002 .

[75]  Andrew Harvey,et al.  Forecasting, structural time series models and the Kalman filter: Selected answers to exercises , 1990 .

[76]  David R. Anderson,et al.  Model selection and multimodel inference : a practical information-theoretic approach , 2003 .

[77]  James W. Taylor.  Exponential smoothing with a damped multiplicative trend , 2003 .

[78]  J. W. Taylor,et al.  Short-term electricity demand forecasting using double seasonal exponential smoothing , 2003, J. Oper. Res. Soc..

[79]  Reconstructing the Kalman Filter for Stationary and Non Stationary Time Series , 2003 .

[80]  Anne B. Koehler,et al.  Normalization of seasonal factors in Winters’ methods , 2003 .

[82]  Roger Koenker,et al.  SparseM: A Sparse Matrix Package for R , 2003 .

[83]  J. Geweke,et al.  Bayesian Forecasting , 2004 .

[84]  Rob J. Hyndman,et al.  The interaction between trend and seasonality , 2004 .

[85]  S. Koopman,et al.  Measuring Asymmetric Stochastic Cycle Components , 2005 .

[86]  Rob J. Hyndman,et al.  Time Series Forecasting: The Case for the Single Source of Error State Space , 2005 .

[87]  Anne B. Koehler,et al.  Time Series Forecasting: The Case for the Single Source of Error State Space Approach , 2005 .

[88]  Rob J Hyndman,et al.  Prediction intervals for exponential smoothing using two new classes of state space models 30 January 2003 , 2003 .

[89]  Kai Ming Lee,et al.  Measuring Asymmetric Stochastic Cycle Components in U.S. Macroeconomic Time Series , 2005 .

[90]  P. McSharry,et al.  A comparison of univariate methods for forecasting electricity demand up to a day ahead , 2006 .

[91]  Rob J Hyndman,et al.  Another look at measures of forecast accuracy , 2006 .

[92]  P. Young,et al.  Modulated cycles, an approach to modelling periodic components from rapidly sampled data , 2006 .

[93]  Rob J Hyndman.  Automatic time series forecasting , 2006 .

[94]  E. S. Gardner.  EXPONENTIAL SMOOTHING: THE STATE OF THE ART, PART II , 2006 .

[95]  Jonathan Weinberg,et al.  Bayesian Forecasting of an Inhomogeneous Poisson Process With Applications to Call Center Data , 2007 .

[96]  J. Randall Brown,et al.  Rational Arithmetic Mathematica Functions to Evaluate the One-sided One-sample K-S Cumulative Sample Distribution , 2007 .

[97]  A state space model for exponential smoothing with group seasonality , 2007 .

[98]  James W. Taylor,et al.  Forecasting daily supermarket sales using exponentially weighted quantile regression , 2007, Eur. J. Oper. Res..

[99]  F Sa Su,et al.  Forecasting Time Series with Multiple Seasonal Patterns , 2007 .

[100]  Haipeng Shen,et al.  Interday Forecasting and Intraday Updating of Call Center Arrivals , 2008, Manuf. Serv. Oper. Manag..

[101]  Everette S. Gardner,et al.  Exponential smoothing in the telecommunications data , 2008 .

[102]  Rob J Hyndman,et al.  Automatic Time Series Forecasting: The forecast Package for R , 2008 .

[103]  Rob J Hyndman,et al.  The admissible parameter space for exponential smoothing models , 2008 .

[104]  James W. Taylor,et al.  A Comparison of Univariate Time Series Methods for Forecasting Intraday Arrivals at a Call Center , 2008, Manag. Sci..

[105]  Rob J Hyndman,et al.  Forecasting with Exponential Smoothing: The State Space Approach , 2008 .

[106]  James W. Taylor.  An evaluation of methods for very short-term load forecasting using minute-by-minute British data , 2008 .

[107]  Haipeng Shen,et al.  FORECASTING TIME SERIES OF INHOMOGENEOUS POISSON PROCESSES WITH APPLICATION TO CALL CENTER WORKFORCE MANAGEMENT , 2008, 0807.4071.

[108]  M. Riani,et al.  Transformations and seasonal adjustment , 2009 .

[109]  Amir F. Atiya,et al.  A new Bayesian formulation for Holt's exponential smoothing , 2009 .

[110]  Rob J. Hyndman,et al.  EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA , 2009 .

[111]  Ralph D. Snyder,et al.  Forecasting intraday time series with multiple seasonal cycles using parsimonious seasonal exponential smoothing , 2009 .

[112]  Andrew Metcalfe,et al.  Time Series Data , 2009 .

[113]  James W. Taylor.  Exponentially weighted methods for forecasting intraday time series with multiple seasonal cycles , 2010 .

[114]  De Livera,et al.  Exponentially weighted methods for multiple seasonal time series , 2010 .

[115]  Rob J Hyndman,et al.  The vector innovations structural time series framework , 2010 .

[116]  James W. Taylor,et al.  Triple seasonal methods for short-term electricity demand forecasting , 2010, Eur. J. Oper. Res..

[117]  D. Livera,et al.  Automatic forecasting with a modified exponential smoothing state space framework , 2010 .

[118]  FORECASTING FOR INVENTORY CONTROL , 2012 .