es() - Exponential Smoothing

Ivan Svetunkov

2024-04-01

es() is a part of smooth package and is a wrapper for the ADAM function with distribution="dnorm". It implements Exponential Smoothing in the ETS form, selecting the most appropriate model among 30 possible ones.

We will use some of the functions of the greybox package in this vignette for demonstrational purposes.

Let’s load the necessary packages:

require(smooth)
require(greybox)

The simplest call for the es() function is:

ourModel <- es(BJsales, h=12, holdout=TRUE, silent=FALSE)

## Forming the pool of models based on... ANN , AAN , Estimation progress:    33 %44 %56 %67 %78 %89 %100 %... Done!

ourModel

## Time elapsed: 0.21 seconds
## Model estimated using es() function: ETS(AMdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 237.8564
## Persistence vector g:
##  alpha   beta 
## 0.9984 0.2184 
## Damping parameter: 0.9012
## Sample size: 138
## Number of estimated parameters: 6
## Number of degrees of freedom: 132
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 487.7127 488.3540 505.2763 506.8560 
## 
## Forecast errors:
## ME: 2.888; MAE: 3.029; RMSE: 3.734
## sCE: 15.244%; Asymmetry: 88.8%; sMAE: 1.332%; sMSE: 0.027%
## MASE: 2.543; RMSSE: 2.434; rMAE: 0.977; rRMSE: 0.975

In this case function uses branch and bound algorithm to form a pool of models to check and after that constructs a model with the lowest information criterion. As we can see, it also produces an output with brief information about the model, which contains:

1. How much time was elapsed for the model construction;
2. What type of ETS was selected;
3. Values of persistence vector (smoothing parameters);
4. What type of initialisation was used;
5. How many parameters were estimated (standard deviation is included);
6. Cost function type and the value of that cost function;
7. Information criteria for this model;
8. Forecast errors (because we have set holdout=TRUE).

The function has also produced a graph with actual values, fitted values and point forecasts.

If we need prediction interval, then we can use the forecast() method:

plot(forecast(ourModel, h=12, interval="prediction"))

The same model can be reused for different purposes, for example to produce forecasts based on newly available data:

es(BJsales, model=ourModel, h=12, holdout=FALSE)

## Time elapsed: 0 seconds
## Model estimated using es() function: ETS(AMdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 255.555
## Persistence vector g:
##  alpha   beta 
## 0.9984 0.2184 
## Damping parameter: 0.9012
## Sample size: 150
## Number of estimated parameters: 1
## Number of degrees of freedom: 149
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 513.1099 513.1370 516.1206 516.1883

We can also extract the type of model in order to reuse it later:

modelType(ourModel)

## [1] "AMdN"

This handy function also works with ets() from forecast package.

If we need actual values from the model, we can use actuals() method from greybox package:

actuals(ourModel)

## Time Series:
## Start = 1 
## End = 138 
## Frequency = 1 
##   [1] 200.1 199.5 199.4 198.9 199.0 200.2 198.6 200.0 200.3 201.2 201.6 201.5
##  [13] 201.5 203.5 204.9 207.1 210.5 210.5 209.8 208.8 209.5 213.2 213.7 215.1
##  [25] 218.7 219.8 220.5 223.8 222.8 223.8 221.7 222.3 220.8 219.4 220.1 220.6
##  [37] 218.9 217.8 217.7 215.0 215.3 215.9 216.7 216.7 217.7 218.7 222.9 224.9
##  [49] 222.2 220.7 220.0 218.7 217.0 215.9 215.8 214.1 212.3 213.9 214.6 213.6
##  [61] 212.1 211.4 213.1 212.9 213.3 211.5 212.3 213.0 211.0 210.7 210.1 211.4
##  [73] 210.0 209.7 208.8 208.8 208.8 210.6 211.9 212.8 212.5 214.8 215.3 217.5
##  [85] 218.8 220.7 222.2 226.7 228.4 233.2 235.7 237.1 240.6 243.8 245.3 246.0
##  [97] 246.3 247.7 247.6 247.8 249.4 249.0 249.9 250.5 251.5 249.0 247.6 248.8
## [109] 250.4 250.7 253.0 253.7 255.0 256.2 256.0 257.4 260.4 260.0 261.3 260.4
## [121] 261.6 260.8 259.8 259.0 258.9 257.4 257.7 257.9 257.4 257.3 257.6 258.9
## [133] 257.8 257.7 257.2 257.5 256.8 257.5

We can also use persistence or initials only from the model to construct the other one:

# Provided initials
es(BJsales, model=modelType(ourModel),
   h=12, holdout=FALSE,
   initial=ourModel$initial)

## Time elapsed: 0.02 seconds
## Model estimated using es() function: ETS(AMdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 255.2509
## Persistence vector g:
##  alpha   beta 
## 0.9719 0.2835 
## Damping parameter: 0.8698
## Sample size: 150
## Number of estimated parameters: 4
## Number of degrees of freedom: 146
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 518.5019 518.7777 530.5444 531.2355

# Provided persistence
es(BJsales, model=modelType(ourModel),
   h=12, holdout=FALSE,
   persistence=ourModel$persistence)

## Time elapsed: 0.02 seconds
## Model estimated using es() function: ETS(AMdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 256.1161
## Persistence vector g:
##  alpha   beta 
## 0.9984 0.2184 
## Damping parameter: 0.944
## Sample size: 150
## Number of estimated parameters: 4
## Number of degrees of freedom: 146
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 520.2321 520.5080 532.2747 532.9658

or provide some arbitrary values:

es(BJsales, model=modelType(ourModel),
   h=12, holdout=FALSE,
   initial=200)

## Time elapsed: 0.03 seconds
## Model estimated using es() function: ETS(AMdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 255.3409
## Persistence vector g:
##  alpha   beta 
## 0.9882 0.2671 
## Damping parameter: 0.8925
## Sample size: 150
## Number of estimated parameters: 5
## Number of degrees of freedom: 145
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 520.6818 521.0984 535.7349 536.7788

Using some other parameters may lead to completely different model and forecasts (see discussion of the additional parameters in the online textbook about ADAM):

es(BJsales, h=12, holdout=TRUE, loss="MSEh", bounds="a", ic="BIC")

## Time elapsed: 0.84 seconds
## Model estimated using es() function: ETS(MAN)
## Distribution assumed in the model: Normal
## Loss function type: MSEh; Loss function value: 0.0018
## Persistence vector g:
##  alpha   beta 
## 1.5365 0.0002 
## 
## Sample size: 138
## Number of estimated parameters: 4
## Number of degrees of freedom: 134
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 1021.898 1022.199 1033.607 1034.348 
## 
## Forecast errors:
## ME: -1.396; MAE: 1.604; RMSE: 1.809
## sCE: -7.369%; Asymmetry: -84.7%; sMAE: 0.705%; sMSE: 0.006%
## MASE: 1.346; RMSSE: 1.179; rMAE: 0.517; rRMSE: 0.472

You can play around with all the available parameters to see what’s their effect on the final model.

In order to combine forecasts we need to use “C” letter:

es(BJsales, model="CCN", h=12, holdout=TRUE)

## Time elapsed: 0.23 seconds
## Model estimated: ETS(CCN)
## Loss function type: likelihood
## 
## Number of models combined: 10
## Sample size: 138
## Average number of estimated parameters: 6.2884
## Average number of degrees of freedom: 131.7116
## 
## Forecast errors:
## ME: 2.861; MAE: 3.007; RMSE: 3.706
## sCE: 15.102%; sMAE: 1.323%; sMSE: 0.027%
## MASE: 2.524; RMSSE: 2.416; rMAE: 0.97; rRMSE: 0.967

Model selection from a specified pool and forecasts combination are called using respectively:

# Select the best model in the pool
es(BJsales, model=c("ANN","AAN","AAdN","MNN","MAN","MAdN"),
   h=12, holdout=TRUE)

## Time elapsed: 0.11 seconds
## Model estimated using es() function: ETS(AAdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 238.0241
## Persistence vector g:
## alpha  beta 
## 0.955 0.296 
## Damping parameter: 0.8456
## Sample size: 138
## Number of estimated parameters: 6
## Number of degrees of freedom: 132
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 488.0481 488.6893 505.6116 507.1914 
## 
## Forecast errors:
## ME: 2.807; MAE: 2.966; RMSE: 3.65
## sCE: 14.815%; Asymmetry: 87.5%; sMAE: 1.305%; sMSE: 0.026%
## MASE: 2.489; RMSSE: 2.379; rMAE: 0.957; rRMSE: 0.952

# Combine the pool of models
es(BJsales, model=c("CCC","ANN","AAN","AAdN","MNN","MAN","MAdN"),
   h=12, holdout=TRUE)

## Time elapsed: 0.11 seconds
## Model estimated: ETS(CCN)
## Loss function type: likelihood
## 
## Number of models combined: 6
## Sample size: 138
## Average number of estimated parameters: 6.6291
## Average number of degrees of freedom: 131.3709
## 
## Forecast errors:
## ME: 2.83; MAE: 2.983; RMSE: 3.673
## sCE: 14.936%; sMAE: 1.312%; sMSE: 0.026%
## MASE: 2.504; RMSSE: 2.394; rMAE: 0.962; rRMSE: 0.959

Now we introduce explanatory variable in ETS:

x <- BJsales.lead

and fit an ETSX model with the exogenous variable first:

es(BJsales, model="ZZZ", h=12, holdout=TRUE,
   xreg=x)

## Time elapsed: 1.01 seconds
## Model estimated using es() function: ETSX(AAdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 237.5664
## Persistence vector g (excluding xreg):
##  alpha   beta 
## 0.9321 0.3047 
## Damping parameter: 0.8768
## Sample size: 138
## Number of estimated parameters: 7
## Number of degrees of freedom: 131
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 489.1327 489.9943 509.6235 511.7460 
## 
## Forecast errors:
## ME: 2.872; MAE: 2.994; RMSE: 3.696
## sCE: 15.161%; Asymmetry: 89.9%; sMAE: 1.317%; sMSE: 0.026%
## MASE: 2.514; RMSSE: 2.409; rMAE: 0.966; rRMSE: 0.965

If we want to check if lagged x can be used for forecasting purposes, we can use xregExpander() function from greybox package:

es(BJsales, model="ZZZ", h=12, holdout=TRUE,
   xreg=xregExpander(x), regressors="use")

## Time elapsed: 0.5 seconds
## Model estimated using es() function: ETSX(ANN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 252.7445
## Persistence vector g (excluding xreg):
## alpha 
##     1 
## 
## Sample size: 138
## Number of estimated parameters: 6
## Number of degrees of freedom: 132
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 517.4889 518.1301 535.0524 536.6322 
## 
## Forecast errors:
## ME: 2.738; MAE: 2.983; RMSE: 3.648
## sCE: 14.453%; Asymmetry: 82.7%; sMAE: 1.312%; sMSE: 0.026%
## MASE: 2.504; RMSSE: 2.378; rMAE: 0.962; rRMSE: 0.952

We can also construct a model with selected exogenous (based on IC):

es(BJsales, model="ZZZ", h=12, holdout=TRUE,
   xreg=xregExpander(x), regressors="select")

## Time elapsed: 0.99 seconds
## Model estimated using es() function: ETS(AMdN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 237.8564
## Persistence vector g:
##  alpha   beta 
## 0.9984 0.2184 
## Damping parameter: 0.9012
## Sample size: 138
## Number of estimated parameters: 6
## Number of degrees of freedom: 132
## Information criteria:
##      AIC     AICc      BIC     BICc 
## 487.7127 488.3540 505.2763 506.8560 
## 
## Forecast errors:
## ME: 2.888; MAE: 3.029; RMSE: 3.734
## sCE: 15.244%; Asymmetry: 88.8%; sMAE: 1.332%; sMSE: 0.027%
## MASE: 2.543; RMSSE: 2.434; rMAE: 0.977; rRMSE: 0.975

Finally, if you work with M or M3 data, and need to test a function on a specific time series, you can use the following simplified call:

es(Mcomp::M3$N2457, silent=FALSE)

This command has taken the data, split it into in-sample and holdout and produced the forecast of appropriate length to the holdout.