Application of Exponential Smoothing to Forecasting a Time Series
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Design of Observational Studies (Springer Series in Statistics)
Springer Series in Statistics Advisors: P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, S. Zeger For other titles published in this series, go to http://www.springer.com/series/692 Paul R. Rosenbaum Design of Observational Studies 123 Paul R. Rosenbaum Statistics Department Wharton School University of Pennsylvania Philadelphia, PA 19104-6340 USA [email protected] ISBN 978-1-4419-1212-1 e-ISBN 978-1-4419-1213-8 DOI 10.1007/978-1-4419-1213-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009938109 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is a part of Springer Science+Business Media (www.springer.com). For Judy “Simplicity of form is not necessarily simplicity of experience.” Robert Morris, writing about art. “Simplicity is not a given. It is an achievement.” William H. -
Trend Analysis and Forecasting the Spread of COVID-19 Pandemic in Ethiopia Using Box– Jenkins Modeling Procedure
International Journal of General Medicine Dovepress open access to scientific and medical research Open Access Full Text Article ORIGINAL RESEARCH Trend Analysis and Forecasting the Spread of COVID-19 Pandemic in Ethiopia Using Box– Jenkins Modeling Procedure Yemane Asmelash Introduction: COVID-19, which causes severe acute respiratory syndrome, is spreading Gebretensae 1 rapidly across the world, and the severity of this pandemic is rising in Ethiopia. The main Daniel Asmelash 2 objective of the study was to analyze the trend and forecast the spread of COVID-19 and to develop an appropriate statistical forecast model. 1Department of Statistics, College of Natural and Computational Science, Methodology: Data on the daily spread between 13 March, 2020 and 31 August 2020 were Aksum University, Aksum, Ethiopia; collected for the development of the autoregressive integrated moving average (ARIMA) 2 Department of Clinical Chemistry, model. Stationarity testing, parameter testing and model diagnosis were performed. In School of Biomedical and Laboratory Sciences, College of Medicine and Health addition, candidate models were obtained using autocorrelation function (ACF) and partial Sciences, University of Gondar, Gondar, autocorrelation functions (PACF). Finally, the fitting, selection and prediction accuracy of the Ethiopia ARIMA models was evaluated using the RMSE and MAPE model selection criteria. Results: A total of 51,910 confirmed COVID-19 cases were reported from 13 March to 31 August 2020. The total recovered and death rates as of 31 August 2020 were 37.2% and 1.57%, respectively, with a high level of increase after the mid of August, 2020. In this study, ARIMA (0, 1, 5) and ARIMA (2, 1, 3) were finally confirmed as the optimal model for confirmed and recovered COVID-19 cases, respectively, based on lowest RMSE, MAPE and BIC values. -
19 Apr 2021 Model Combinations Through Revised Base-Rates
Model combinations through revised base-rates Fotios Petropoulos School of Management, University of Bath UK and Evangelos Spiliotis Forecasting and Strategy Unit, School of Electrical and Computer Engineering, National Technical University of Athens, Greece and Anastasios Panagiotelis Discipline of Business Analytics, University of Sydney, Australia April 21, 2021 Abstract Standard selection criteria for forecasting models focus on information that is calculated for each series independently, disregarding the general tendencies and performances of the candidate models. In this paper, we propose a new way to sta- tistical model selection and model combination that incorporates the base-rates of the candidate forecasting models, which are then revised so that the per-series infor- arXiv:2103.16157v2 [stat.ME] 19 Apr 2021 mation is taken into account. We examine two schemes that are based on the pre- cision and sensitivity information from the contingency table of the base rates. We apply our approach on pools of exponential smoothing models and a large number of real time series and we show that our schemes work better than standard statisti- cal benchmarks. We discuss the connection of our approach to other cross-learning approaches and offer insights regarding implications for theory and practice. Keywords: forecasting, model selection/averaging, information criteria, exponential smooth- ing, cross-learning. 1 1 Introduction Model selection and combination have long been fundamental ideas in forecasting for business and economics (see Inoue and Kilian, 2006; Timmermann, 2006, and ref- erences therein for model selection and combination respectively). In both research and practice, selection and/or the combination weight of a forecasting model are case-specific. -
Exponential Smoothing – Trend & Seasonal
NCSS Statistical Software NCSS.com Chapter 467 Exponential Smoothing – Trend & Seasonal Introduction This module forecasts seasonal series with upward or downward trends using the Holt-Winters exponential smoothing algorithm. Two seasonal adjustment techniques are available: additive and multiplicative. Additive Seasonality Given observations X1 , X 2 ,, X t of a time series, the Holt-Winters additive seasonality algorithm computes an evolving trend equation with a seasonal adjustment that is additive. Additive means that the amount of the adjustment is constant for all levels (average value) of the series. The forecasting algorithm makes use of the following formulas: at = α( X t − Ft−s ) + (1 − α)(at−1 + bt−1 ) bt = β(at − at−1 ) + (1 − β)bt−1 Ft = γ ( X t − at ) + (1 − γ )Ft−s Here α , β , and γ are smoothing constants which are between zero and one. Again, at gives the y-intercept (or level) at time t, while bt is the slope at time t. The letter s represents the number of periods per year, so the quarterly data is represented by s = 4 and monthly data is represented by s = 12. The forecast at time T for the value at time T+k is aT + bT k + F[(T +k−1)/s]+1 . Here [(T+k-1)/s] is means the remainder after dividing T+k-1 by s. That is, this function gives the season (month or quarter) that the observation came from. Multiplicative Seasonality Given observations X1 , X 2 ,, X t of a time series, the Holt-Winters multiplicative seasonality algorithm computes an evolving trend equation with a seasonal adjustment that is multiplicative. -
Exponential Smoothing Statlet.Pdf
STATGRAPHICS Centurion – Rev. 11/21/2013 Exponential Smoothing Statlet Summary ......................................................................................................................................... 1 Data Input........................................................................................................................................ 2 Statlet .............................................................................................................................................. 3 Output ............................................................................................................................................. 4 Calculations..................................................................................................................................... 6 Summary This Statlet applies various types of exponential smoothers to a time series. It generates forecasts with associated forecast limits. Using the Statlet controls, the user may interactively change the values of the smoothing parameters to examine their effect on the forecasts. The types of exponential smoothers included are: 1. Brown’s simple exponential smoothing with smoothing parameter . 2. Brown’s linear exponential smoothing with smoothing parameter . 3. Holt’s linear exponential smoothing with smoothing parameters and . 4. Brown’s quadratic exponential smoothing with smoothing parameter . 5. Winters’ seasonal exponential smoothing with smoothing parameters and . Sample StatFolio: expsmoothing.sgp Sample Data The data -
Public Health & Intelligence
Public Health & Intelligence PHI Trend Analysis Guidance Document Control Version Version 1.5 Date Issued March 2017 Author Róisín Farrell, David Walker / HI Team Comments to [email protected] Version Date Comment Author Version 1.0 July 2016 1st version of paper David Walker, Róisín Farrell Version 1.1 August Amending format; adding Róisín Farrell 2016 sections Version 1.2 November Adding content to sections Róisín Farrell, Lee 2016 1 and 2 Barnsdale Version 1.3 November Amending as per Róisín Farrell 2016 suggestions from SAG Version 1.4 December Structural changes Róisín Farrell, Lee 2016 Barnsdale Version 1.5 March 2017 Amending as per Róisín Farrell, Lee suggestions from SAG Barnsdale Contents 1. Introduction ........................................................................................................................................... 1 2. Understanding trend data for descriptive analysis ................................................................................. 1 3. Descriptive analysis of time trends ......................................................................................................... 2 3.1 Smoothing ................................................................................................................................................ 2 3.1.1 Simple smoothing ............................................................................................................................ 2 3.1.2 Median smoothing .......................................................................................................................... -
Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models
mathematics Article Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models Oscar Trull 1,* , Juan Carlos García-Díaz 1 and Alicia Troncoso 2 1 Department of Applied Statistics and Operational Research and Quality, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Department of Computer Science, Pablo de Olavide University, 41013 Sevilla, Spain; [email protected] * Correspondence: [email protected] Received: 8 January 2020; Accepted: 13 February 2020; Published: 18 February 2020 Abstract: The Holt–Winters models are one of the most popular forecasting algorithms. As well-known, these models are recursive and thus, an initialization value is needed to feed the model, being that a proper initialization of the Holt–Winters models is crucial for obtaining a good accuracy of the predictions. Moreover, the introduction of multiple seasonal Holt–Winters models requires a new development of methods for seed initialization and obtaining initial values. This work proposes new initialization methods based on the adaptation of the traditional methods developed for a single seasonality in order to include multiple seasonalities. Thus, new methods to initialize the level, trend, and seasonality in multiple seasonal Holt–Winters models are presented. These new methods are tested with an application for electricity demand in Spain and analyzed for their impact on the accuracy of forecasts. As a consequence of the analysis carried out, which initialization method to use for the level, trend, and seasonality in multiple seasonal Holt–Winters models with an additive and multiplicative trend is provided. Keywords: forecasting; multiple seasonal periods; Holt–Winters, initialization 1. Introduction Exponential smoothing methods are widely used worldwide in time series forecasting, especially in financial and energy forecasting [1,2]. -
LECTURE 2 MOVING AVERAGES and EXPONENTIAL SMOOTHING OVERVIEW This Lecture Introduces Time-Series Smoothing Forecasting Methods
Business Conditions & Forecasting – Exponential Smoothing Dr. Thomas C. Chiang LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecture introduces time-series smoothing forecasting methods. Various models are discussed, including methods applicable to nonstationary and seasonal time-series data. These models are viewed as classical time-series model; all of them are univariate. LEARNING OBJECTIVES • Moving averages • Forecasting using exponential smoothing • Accounting for data trend using Holt's smoothing • Accounting for data seasonality using Winter's smoothing • Adaptive-response-rate single exponential smoothing 1. Forecasting with Moving Averages The naive method discussed in Lecture 1 uses the most recent observations to forecast future ˆ values. That is, Yt+1 = Yt. Since the outcomes of Yt are subject to variations, using the mean value is considered an alternative method of forecasting. In order to keep forecasts updated, a simple moving-average method has been widely used. 1.1. The Model Moving averages are developed based on an average of weighted observations, which tends to smooth out short-term irregularity in the data series. They are useful if the data series remains fairly steady over time. Notations ˆ M t ≡ Yt+1 - Moving average at time t , which is the forecast value at time t+1, Yt - Observation at time t, ˆ et = Yt − Yt - Forecast error. A moving average is obtained by calculating the mean for a specified set of values and then using it to forecast the next period. That is, M t = (Yt + Yt−1 + ⋅⋅⋅ + Yt−n+1 ) n (1.1.1) M t−1 = (Yt−1 +Yt−2 + ⋅⋅⋅+Yt−n ) n (1.1.2) Business Conditions & Forecasting Dr. -
A Simple Method to Adjust Exponential Smoothing Forecasts for Trend and Seasonality
Southern Methodist University SMU Scholar Historical Working Papers Cox School of Business 1-1-1993 A Simple Method to Adjust Exponential Smoothing Forecasts for Trend and Seasonality Marion G. Sobol Southern Methodist University Jim Collins Southern Methodist University Follow this and additional works at: https://scholar.smu.edu/business_workingpapers Part of the Business Commons This document is brought to you for free and open access by the Cox School of Business at SMU Scholar. It has been accepted for inclusion in Historical Working Papers by an authorized administrator of SMU Scholar. For more information, please visit http://digitalrepository.smu.edu. A SIMPLE METHOD TO ADJUST EXPONENTIAL SMOOTHING FORECASTS FOR TREND AND SEASONALITY Working Paper 93-0801* by Marion G. Sobol Jim Collins Marion G. Sobol Jim Collins Edwin L. Cox School of Business Southern Methodist University Dallas, Texas 75275 * This paper represents a draft of work in progress by the authors and is being sent to you for information and review. Responsibility for the contents rests solely with the authors and may not be reproduced or distributed without their written consent. Please address all correspondence to Marion G. Sobol. Abstract Early exponential smoothing techniques did not adjust for seasonality and trend. Later Holt and Winters developed methods to include these factors but they require the arbitrary choice of three smoothing constants '( , ~ and t. Our technique requires only alpha and an estimate of trend and seasonality as done for decomposition analysis. Since seasonality and trend are relatively constant why keep reestimating them - just include them. Moreover this technique is easy to perfonn on a standard spreadsheet. -
3. Regression & Exponential Smoothing
3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z1, z2, . , zn 1. Models the observation zt as a function of time as zt = f(t, β) + εt where f(t, β) is a function of time t and unknown coefficients β, and εt are uncorrelated errors. 1 ∗ Examples: – The constant mean model: zt = β + εt – The linear trend model: zt = β0 + β1t + εt – Trigonometric models for seasonal time series 2π 2π z = β + β sin t + β cos t + ε t 0 1 12 2 12 t 2. A time Series modeling approach, the observation at time t is modeled as a linear combination of previous observations ∗ Examples: P – The autoregressive model: zt = j≥1 πjzt−j + εt – The autoregressive and moving average model: p q P P zt = πjzt−j + θiεt−i + εt j=1 j=1 2 Discounted least squares/general exponential smoothing n X 2 wt[zt − f(t, β)] t=1 • Ordinary least squares: wt = 1. n−t • wt = w , discount factor w determines how fast information from previous observations is discounted. • Single, double and triple exponential smoothing procedures – The constant mean model – The Linear trend model – The quadratic model 3 3.2 Constant Mean Model zt = β + εt • β: a constant mean level 2 • εt: a sequence of uncorrelated errors with constant variance σ . If β, σ are known, the minimum mean square error forecast of a future observation at time n + l, zn+l = β + εn+l is given by zn(l) = β 2 2 2 • E[zn+l − zn(l)] = 0, E[zn+1 − zn(l)] = E[εt ] = σ • 100(1 − λ) percent prediction intervals for a future realization are given by [β − µλ/2σ; β + µλ/2σ] where µλ/2 is the 100(1 − λ/2) percentage point of standard normal distribution. -
Forecasting by Extrapolation: Conclusions from Twenty-Five Years of Research
University of Pennsylvania ScholarlyCommons Marketing Papers Wharton Faculty Research November 1984 Forecasting by Extrapolation: Conclusions from Twenty-five Years of Research J. Scott Armstrong University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/marketing_papers Recommended Citation Armstrong, J. S. (1984). Forecasting by Extrapolation: Conclusions from Twenty-five Years of Research. Retrieved from https://repository.upenn.edu/marketing_papers/77 Postprint version. Published in Interfaces, Volume 14, Issue 6, November 1984, pages 52-66. Publisher URL: http://www.aaai.org/AITopics/html/interfaces.html The author asserts his right to include this material in ScholarlyCommons@Penn. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/marketing_papers/77 For more information, please contact [email protected]. Forecasting by Extrapolation: Conclusions from Twenty-five Years of Research Abstract Sophisticated extrapolation techniques have had a negligible payoff for accuracy in forecasting. As a result, major changes are proposed for the allocation of the funds for future research on extrapolation. Meanwhile, simple methods and the combination of forecasts are recommended. Comments Postprint version. Published in Interfaces, Volume 14, Issue 6, November 1984, pages 52-66. Publisher URL: http://www.aaai.org/AITopics/html/interfaces.html The author asserts his right to include this material in ScholarlyCommons@Penn. This journal article is available at ScholarlyCommons: https://repository.upenn.edu/marketing_papers/77 Published in Interfaces, 14 (Nov.-Dec. 1984), 52-66, with commentaries and reply. Forecasting by Extrapolation: Conclusions from 25 Years of Research J. Scott Armstrong Wharton School, University of Pennsylvania Sophisticated extrapolation techniques have had a negligible payoff for accuracy in forecasting. -
Exponential Smoothing
Key Concepts: Week 3 Lesson 1: Exponential Smoothing Learning Objectives Understand how exponential smoothing treats old and new information differently Understand how changing the alpha or beta smoothing factors influences the forecasts Able to apply the techniques to generate forecasts in spreadsheets Summary of Lesson Exponential smoothing, as opposed to the three other time series models we have discussed (Cumulative, Naïve, and Moving Average), treats data differently depending on its age. The idea is that the value of data degrades over time so that newer observations of demand are weighted more heavily than older observations. The weights decrease exponentially as they age. Exponential models simply blend the value of new and old information. We have students create forecast models in spreadsheets so they understand the mechanics of the model and hopefully develop a sense of how the parameters influence the forecast. The alpha factor (ranging between 0 and 1) determines the weighting for the newest information versus the older information. The “α” value indicates the value of “new” information versus “old” information: As α → 1, the forecast becomes more nervous, volatile and naïve As α → 0, the forecast becomes more calm, staid and cumulative α can range from 0 ≤ α ≤ 1, but in practice, we typically see 0 ≤ α ≤ 0.3 The most basic exponential model, or Simple Exponential model, assumes stationary demand. Holt’s Model is a modified version of exponential smoothing that also accounts for trend in addition to level. A new smoothing parameter, β, is introduced. It operates in the same way as the α. We can also use exponential smoothing to dampen trend models to account for the fact that trends usually do not remain unchanged indefinitely as well as for creating a more stable estimate of the forecast errors.