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### Triple Exponential Smoothing - Excel : SAS : R

In the previous article of this series, we explained Double Exponential Smoothing method, also called Holt's method for time series forecasting. Now we are taking up TES also called Holt Winter's Method.
Once we are done with this article we are all set to learn ARIMA.

You HAVE TO read the following articles before reading this one, else it would not make much sense.

Time Series Forecasting - Part 1 (Overview)

Time Series Forecasting - Part 2 (SES)

Time Series Forecasting - Part 3 (DES)

### Triple Exponential Smoothing (TES)

The method is used for a time series with SEASONALITY and a TREND, such series are generally interesting as we would learn to deal with trend as well as Seasonality, so let's learn it.

### TES using Excel

Please download the file to understand the calculation of TES better .

TES Demo File

Mathematical Equations for TES

In TES method, first we need to check the presence of seasonality and trend in the time series and then we check the nature of trend. The trend can be either ADDITIVE or MULTIPLICATIVE in mature. We have illustrated the same in sheet  "Additive or Multiplicative". We need to use the equation according to nature of the trend.

You can see that, calculations of Level, Slope (Trend) and seasonality for first observations i.e. in cells F3G3 and H3 of sheet TES Calculation are exceptional, second equation onward equations as shown below have been used.

In the forecasting window, formula is different, as the last observation's Level and Trend are used for forecasting along with Seasonality of the last occurrence of respective month.

The value of α, β and 𝜸 lies between 0 and 1.

Rest remains same, you need to check MAPE for accuracy of forecast.

### TES using SAS

Download the csv file for using in SAS and R codes :

TES sales.csv

/* First import the time series data*/

Proc import datafile = "G:\AA\Time Series\TES sales.csv"
out = data_1     dbms = csv replace; Run;

/*  Then use the Proc Forecast*/

PROC FORECAST DATA =  data_1
METHOD =  ADDWINTERS /* for Additive and METHOD = WINTERS for Multiplicative Series  */
TREND= 3                         /* fit a linear trend model for TES */
INTERVAL = month         /* frequency of input time series*/
SEASONS = 12                   /* Use this option for Holts Winter Method*/
nsstart = 2                     /* give number of season for special seasonal effect*/
WEIGHT = (0.7, 0.1, 0.4)   /*In TES method  Alpha , Beta and gamma VALUE between  0 to 1 */
LEAD = 3                         /* number of forecast */
OUT = pred_results         /* FORECASTE VALUE WITH ACTUAL VALUE*/
OUTFULL
OUTRESID
OUTEST = model_stats;   /* MODEL STATS VALUE WITH MAPE VALUE*/
VAR Sales;
RUN;

Run the code and check the results.

### TES using R

# First import the time series data
setwd("G:\\AA\\Time Series")
data = read.csv("TES sales.csv")

# let's make the series a time series compatible to R
ts = ts(data\$Sales, start=c(2010, 1), end=c(2012,12), frequency=12)

# Lets decompose the series and see if any trend or seasonality is there.
fit = stl(ts, s.window="period")
plot(fit)

# and we get these plots  >>>>>>>>>>>>>>>>

# we need to install and use the package forecast
if(!require(forecast)) install.packages("forecast")
library(forecast)

# let's now learn plotting classical seasonality plot
seasonplot(ts)

# In these plots, we see if the series move together or not. If they do, they are seasonal, well in our case, definitely there is definitely a seasonality. Not showing here as we have shown these plots earlier.

# now let use TES to forecast
fit_1 = HoltWinters(ts)

forecast(fit_1, 3)
plot(forecast(fit_1, 3))
# and it is done !

In the subsequent articles of the series, we would cover other forecasting techniques enlisted in the first article of the series, till then ...

Enjoy reading our other articles and stay tuned with us.

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The article has been written by Suyash Nigam with Rajat Agarwal as a co-author.

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