Forecasting Using Exponential Smoothing Methods : Simple, Holt’s, Holt-Winters
Maryanto
2025-04-07
1 Introduction
In this session, we will learn about time series analysis and perform forecasting using several exponential smoothing methods, including: - Simple Exponential Smoothing - Holt’s Linear Trend Method - Holt-Winters Method
1.1 Data Description
The dataset we will use contains annual sales data from the year 2000 to 2016. Based on this historical data, the objective of this analysis is to: Forecast sales for the next two years, specifically for 2017 and 2018.
1.2 Library
# import libs
library(tidyverse)
library(lubridate)
library(forecast)
library(TTR)
library(fpp)
library(tseries)
library(TSstudio)
library(padr)1.3 Data
year <- c(2000,2001,2002,2003,2004,2005,2006,2007,
2008,2009,2010,2011,2012,2013,2014,2015,
2016)
sales <- c(156, 161, 189, 182, 224, 258, 283, 325,
332, 388, 475, 502, 537, 584, 631,
704, 689)
df <- data.frame(year, sales)
df# create ts object
df_ts <- ts(data = df$sales, start = 2000, frequency = 1)
df_ts#> Time Series:
#> Start = 2000
#> End = 2016
#> Frequency = 1
#> [1] 156 161 189 182 224 258 283 325 332 388 475 502 537 584 631 704 689# visualize the data
df_ts %>%
autoplot()
Between the years 2000 and 2015, the sales showed a positive upward trend. However, after 2015, the sales began to show a declining trend.
2 Forecasting Model
2.1 Exponential Smoothing
2.1.1 SES
Simple Exponential Smoothing (SES) is used when data has no trend or seasonality.
# Apply SES
ses_model <- ses(df_ts, h = 2)
ses_model#> Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
#> 2017 689.0015 632.7974 745.2056 603.0448 774.9582
#> 2018 689.0015 609.5209 768.4821 567.4464 810.5566plot(ses_model)
accuracy(ses_model)#> ME RMSE MAE MPE MAPE MASE ACF1
#> Training set 31.35529 41.19581 33.94409 8.12971 8.839146 0.9412573 -0.01177232.1.2 Holt’s
Use when data has a trend but no seasonality.
holt_model <- holt(df_ts, h = 2)
holt_model#> Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
#> 2017 731.895 694.600 769.1900 674.8572 788.9328
#> 2018 766.887 718.803 814.9711 693.3488 840.4253plot(holt_model)
accuracy(holt_model)#> ME RMSE MAE MPE MAPE MASE
#> Training set -0.3203326 25.44848 19.17602 -1.728756 6.103914 0.531744
#> ACF1
#> Training set -0.0038153172.1.3 Holt-Winters’
Use when data has both trend and seasonality.
# If you have monthly data with seasonality
df_monthly <- ts(sales, start = 2000, frequency = 12)
hw_model <- hw(df_monthly, seasonal = "additive", h = 12)
hw_model#> Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
#> Jun 2001 756.9330 724.8649 789.0012 707.88909 805.9770
#> Jul 2001 791.0206 737.5729 844.4683 709.27940 872.7618
#> Aug 2001 825.1081 733.1534 917.0629 684.47544 965.7409
#> Sep 2001 859.1957 718.1915 1000.1999 643.54844 1074.8429
#> Oct 2001 893.2832 695.3114 1091.2551 590.51142 1196.0551
#> Nov 2001 927.3708 665.7519 1188.9897 527.25919 1327.4824
#> Dec 2001 961.4583 630.2579 1292.6587 454.93103 1467.9856
#> Jan 2002 995.5459 589.3472 1401.7445 374.31868 1616.7731
#> Feb 2002 1029.6334 543.4112 1515.8557 286.02069 1773.2462
#> Mar 2002 1063.7210 492.7616 1634.6804 190.51395 1936.9280
#> Apr 2002 1097.8085 437.6557 1757.9614 88.19191 2107.4251
#> May 2002 1131.8961 378.3112 1885.4810 -20.61253 2284.4047plot(hw_model)
2.2 Mov AVG & Trend Extrapolation
2.2.1 Moving Average (Simple)
# Moving average (e.g., 2-period)
ma2 <- SMA(df_ts, n = 2)
ma2#> Time Series:
#> Start = 2000
#> End = 2016
#> Frequency = 1
#> [1] NA 158.5 175.0 185.5 203.0 241.0 270.5 304.0 328.5 360.0 431.5 488.5
#> [13] 519.5 560.5 607.5 667.5 696.5plot(df_ts, type = "l")
lines(ma2, col = "blue", lwd = 2)
2.2.2 Trend Extrapolation
model_linear <- lm(sales ~ year)
summary(model_linear)#>
#> Call:
#> lm(formula = sales ~ year)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -57.412 -21.652 1.132 18.676 63.804
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -74211.725 3328.981 -22.29 0.000000000000651 ***
#> year 37.152 1.658 22.41 0.000000000000603 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 33.49 on 15 degrees of freedom
#> Multiple R-squared: 0.971, Adjusted R-squared: 0.9691
#> F-statistic: 502.2 on 1 and 15 DF, p-value: 0.0000000000006034# Predict future years
future <- data.frame(year = c(2017, 2018))
predict(model_linear, newdata = future)#> 1 2
#> 723.7794 760.93142.3 Autocorrelation & ACF
Use ACF (Autocorrelation Function) to detect patterns
acf(df_ts, main = "ACF Plot of Time Series")
Interpretation:
- If significant spikes (outside blue bounds) → data is autocorrelated
- Lag 1 spike: short-term dependence
- Slowly declining spikes: possible trend
- Repeating pattern: seasonality
2.4 Forecast Accuracy Measures
2.4.1 Trend Extrapolation
accuracy(model_linear)#> ME RMSE MAE MPE MAPE
#> Training set 0.000000000000002507632 31.45557 25.26759 0.7846039 9.524381
#> MASE
#> Training set 0.1538372.4.2 SES
accuracy(ses_model)#> ME RMSE MAE MPE MAPE MASE ACF1
#> Training set 31.35529 41.19581 33.94409 8.12971 8.839146 0.9412573 -0.01177232.4.3 Holt’s
accuracy(holt_model)#> ME RMSE MAE MPE MAPE MASE
#> Training set -0.3203326 25.44848 19.17602 -1.728756 6.103914 0.531744
#> ACF1
#> Training set -0.0038153172.4.4 Holt-Winter
accuracy(hw_model)#> ME RMSE MAE MPE MAPE MASE
#> Training set 5.132598 25.02288 19.90842 2.400402 5.609929 0.04457775
#> ACF1
#> Training set 0.009726304Based on the accuracy measures above, it can be concluded that the Holt-Winters model is the most suitable for forecasting the next two years (2017 and 2018), as it has the lowest RMSE value (24.74) compared to the other models.