es() is a part of smooth package. It allows constructing Exponential Smoothing (also known as ETS), selecting the most appropriate one among 30 possible ones, including exogenous variables and many more.
In this vignette we will use data from Mcomp package, so it is advised to install it. We also use some of the functions of the greybox package.
Let’s load the necessary packages:
require(smooth)
require(greybox)
require(Mcomp)You may note that Mcomp depends on forecast package and if you load both forecast and smooth, then you will have a message that forecast() function is masked from the environment. There is nothing to be worried about - smooth uses this function for consistency purposes and has exactly the same original forecast() as in the forecast package. The inclusion of this function in smooth was done only in order not to include forecast in dependencies of the package.
The simplest call of this function is:
es(M3$N2457$x, h=18, holdout=TRUE, silent=FALSE)## Forming the pool of models based on... ANN, ANA, AAN, Estimation progress: 100%... Done!## Time elapsed: 0.47 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.145
## Initial values were optimised.
## 3 parameters were estimated in the process
## Residuals standard deviation: 0.413
## Cost function type: MSE; Cost function value: 1288657.07
##
## Information criteria:
## AIC AICc BIC BICc
## 1645.978 1646.236 1653.702 1654.292
## Forecast errors:
## MPE: 26.3%; sCE: -1919.1%; Bias: 87%; MAPE: 39.8%
## MASE: 2.944; sMAE: 120.1%; RelMAE: 1.258; sMSE: 242.7%
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:
holdout=TRUE).The function has also produced a graph with actuals, fitted values and point forecasts.
If we need prediction intervals, then we run:
es(M3$N2457$x, h=18, holdout=TRUE, intervals=TRUE, silent=FALSE)## Forming the pool of models based on... ANN, ANA, AAN, Estimation progress: 100%... Done!## Time elapsed: 0.45 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.145
## Initial values were optimised.
## 3 parameters were estimated in the process
## Residuals standard deviation: 0.413
## Cost function type: MSE; Cost function value: 1288657.07
##
## Information criteria:
## AIC AICc BIC BICc
## 1645.978 1646.236 1653.702 1654.292
## 95% parametric prediction intervals were constructed
## 72% of values are in the prediction interval
## Forecast errors:
## MPE: 26.3%; sCE: -1919.1%; Bias: 87%; MAPE: 39.8%
## MASE: 2.944; sMAE: 120.1%; RelMAE: 1.258; sMSE: 242.7%
Due to multiplicative nature of error term in the model, the intervals are asymmetric. This is the expected behaviour. The other thing to note is that the output now also provides the theoretical width of prediction intervals and its actual coverage.
If we save the model (and let’s say we want it to work silently):
ourModel <- es(M3$N2457$x, h=18, holdout=TRUE, silent="all")we can then reuse it for different purposes:
es(M3$N2457$x, model=ourModel, h=18, holdout=FALSE, intervals="np", level=0.93)## Time elapsed: 0.08 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.145
## Initial values were provided by user.
## 1 parameter was estimated in the process
## 2 parameters were provided
## Residuals standard deviation: 0.431
## Cost function type: MSE; Cost function value: 1965686.226
##
## Information criteria:
## AIC AICc BIC BICc
## 1994.861 1994.897 1997.606 1997.690
## 93% nonparametric prediction intervals were constructedWe can also extract the type of model in order to reuse it later:
modelType(ourModel)## [1] "MNN"This handy function, by the way, also works with ets() from forecast package.
We can then use persistence or initials only from the model to construct the other one:
es(M3$N2457$x, model=modelType(ourModel), h=18, holdout=FALSE, initial=ourModel$initial, silent="graph")## Time elapsed: 0.02 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.151
## Initial values were provided by user.
## 2 parameters were estimated in the process
## 1 parameter was provided
## Residuals standard deviation: 0.432
## Cost function type: MSE; Cost function value: 1965400.549
##
## Information criteria:
## AIC AICc BIC BICc
## 1996.845 1996.952 2002.334 2002.589es(M3$N2457$x, model=modelType(ourModel), h=18, holdout=FALSE, persistence=ourModel$persistence, silent="graph")## Time elapsed: 0.02 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.145
## Initial values were optimised.
## 2 parameters were estimated in the process
## 1 parameter was provided
## Residuals standard deviation: 0.432
## Cost function type: MSE; Cost function value: 1965686.226
##
## Information criteria:
## AIC AICc BIC BICc
## 1996.861 1996.968 2002.351 2002.605or provide some arbitrary values:
es(M3$N2457$x, model=modelType(ourModel), h=18, holdout=FALSE, initial=1500, silent="graph")## Time elapsed: 0.02 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.15
## Initial values were provided by user.
## 2 parameters were estimated in the process
## 1 parameter was provided
## Residuals standard deviation: 0.433
## Cost function type: MSE; Cost function value: 1968545.705
##
## Information criteria:
## AIC AICc BIC BICc
## 1997.028 1997.136 2002.518 2002.773Using some other parameters may lead to completely different model and forecasts:
es(M3$N2457$x, h=18, holdout=TRUE, cfType="aTMSE", bounds="a", ic="BIC", intervals=TRUE)## Time elapsed: 0.45 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.08
## Initial values were optimised.
## 3 parameters were estimated in the process
## Residuals standard deviation: 0.42
## Cost function type: aTMSE; Cost function value: 246.291
##
## Information criteria:
## AIC AICc BIC BICc
## 815.4061 815.6641 823.1302 823.7205
## 95% parametric prediction intervals were constructed
## 72% of values are in the prediction interval
## Forecast errors:
## MPE: 33.3%; sCE: -2194.4%; Bias: 90.4%; MAPE: 43.3%
## MASE: 3.232; sMAE: 131.9%; RelMAE: 1.381; sMSE: 277.6%You can play around with all the available parameters to see what’s their effect on final model.
In order to combine forecasts we need to use “C” letter:
es(M3$N2457$x, model="CCN", h=18, holdout=TRUE, silent="graph")## Estimation progress: 10%20%30%40%50%60%70%80%90%100%... Done!## Time elapsed: 0.72 seconds
## Model estimated: ETS(CCN)
## Initial values were optimised.
## Residuals standard deviation: 1408.59
## Cost function type: MSE
##
## Information criteria:
## (combined values)
## AIC AICc BIC BICc
## 1647.337 1647.651 1654.083 1654.546
## Forecast errors:
## MPE: 27.8%; sCE: -1977.3%; Bias: 88.4%; MAPE: 40.5%
## MASE: 3.005; sMAE: 122.6%; RelMAE: 1.284; sMSE: 249.9%Model selection from a specified pool and forecasts combination are called using respectively:
es(M3$N2457$x, model=c("ANN","AAN","AAdN","ANA","AAA","AAdA"), h=18, holdout=TRUE, silent="graph")## Estimation progress: 17%33%50%67%83%100%... Done!## Time elapsed: 0.97 seconds
## Model estimated: ETS(ANN)
## Persistence vector g:
## alpha
## 0.158
## Initial values were optimised.
## 3 parameters were estimated in the process
## Residuals standard deviation: 1439.368
## Cost function type: MSE; Cost function value: 2007704.532
##
## Information criteria:
## AIC AICc BIC BICc
## 1688.987 1689.245 1696.711 1697.301
## Forecast errors:
## MPE: 25.3%; sCE: -1880.4%; Bias: 86%; MAPE: 39.4%
## MASE: 2.909; sMAE: 118.7%; RelMAE: 1.243; sMSE: 238.1%es(M3$N2457$x, model=c("CCC","ANN","AAN","AAdN","ANA","AAA","AAdA"), h=18, holdout=TRUE, silent="graph")## Estimation progress: 17%33%50%67%83%100%... Done!## Time elapsed: 1.05 seconds
## Model estimated: ETS(CCC)
## Initial values were optimised.
## Residuals standard deviation: 1386.689
## Cost function type: MSE
##
## Information criteria:
## (combined values)
## AIC AICc BIC BICc
## 1689.857 1690.146 1696.984 1697.488
## Forecast errors:
## MPE: 17.1%; sCE: -1568.3%; Bias: 77.7%; MAPE: 37.3%
## MASE: 2.658; sMAE: 108.4%; RelMAE: 1.135; sMSE: 206.7%Now let’s introduce some artificial exogenous variables:
x <- cbind(rnorm(length(M3$N2457$x),50,3),rnorm(length(M3$N2457$x),100,7))and fit a model with all the exogenous first:
es(M3$N2457$x, model="ZZZ", h=18, holdout=TRUE, xreg=x)## Time elapsed: 0.62 seconds
## Model estimated: ETSX(MNN)
## Persistence vector g:
## alpha
## 0.157
## Initial values were optimised.
## 5 parameters were estimated in the process
## Residuals standard deviation: 0.412
## Xreg coefficients were estimated in a normal style
## Cost function type: MSE; Cost function value: 1253061.655
##
## Information criteria:
## AIC AICc BIC BICc
## 1647.261 1647.920 1660.134 1661.642
## Forecast errors:
## MPE: 20.3%; sCE: -1723.6%; Bias: 82.3%; MAPE: 39.7%
## MASE: 2.829; sMAE: 115.4%; RelMAE: 1.208; sMSE: 229%or construct a model with selected exogenous (based on IC):
es(M3$N2457$x, model="ZZZ", h=18, holdout=TRUE, xreg=x, xregDo="select")## Time elapsed: 0.6 seconds
## Model estimated: ETSX(MNN)
## Persistence vector g:
## alpha
## 0.156
## Initial values were optimised.
## 4 parameters were estimated in the process
## Residuals standard deviation: 0.41
## Xreg coefficients were estimated in a normal style
## Cost function type: MSE; Cost function value: 1254635.772
##
## Information criteria:
## AIC AICc BIC BICc
## 1645.383 1645.817 1655.681 1656.676
## Forecast errors:
## MPE: 20.9%; sCE: -1749%; Bias: 82.6%; MAPE: 39.8%
## MASE: 2.853; sMAE: 116.4%; RelMAE: 1.219; sMSE: 232.2%or the one with the updated xreg:
ourModel <- es(M3$N2457$x, model="ZZZ", h=18, holdout=TRUE, xreg=x, updateX=TRUE)If we want to check if lagged x can be used for forecasting purposes, we can use xregExpander() function from greybox package:
es(M3$N2457$x, model="ZZZ", h=18, holdout=TRUE, xreg=xregExpander(x), xregDo="select")## Time elapsed: 0.9 seconds
## Model estimated: ETSX(MNN)
## Persistence vector g:
## alpha
## 0.156
## Initial values were optimised.
## 4 parameters were estimated in the process
## Residuals standard deviation: 0.41
## Xreg coefficients were estimated in a normal style
## Cost function type: MSE; Cost function value: 1254635.772
##
## Information criteria:
## AIC AICc BIC BICc
## 1645.383 1645.817 1655.681 1656.676
## Forecast errors:
## MPE: 20.9%; sCE: -1749%; Bias: 82.6%; MAPE: 39.8%
## MASE: 2.853; sMAE: 116.4%; RelMAE: 1.219; sMSE: 232.2%If we are confused about the type of estimated model, the function formula() will help us:
formula(ourModel)## [1] "y[t] = l[t-1] * exp(a1[t-1] * x1[t] + a2[t-1] * x2[t]) * e[t]"A feature available since 2.1.0 is fitting ets() model and then using its parameters in es():
etsModel <- forecast::ets(M3$N2457$x)
esModel <- es(M3$N2457$x, model=etsModel, h=18)The point forecasts in the majority of cases should the same, but the prediction intervals may be different (especially if error term is multiplicative):
forecast(etsModel,h=18,level=0.95)## Point Forecast Lo 95 Hi 95
## Aug 1992 8523.456 853.30277 16193.61
## Sep 1992 8563.040 719.69262 16406.39
## Oct 1992 8602.625 587.42532 16617.82
## Nov 1992 8642.209 456.39433 16828.02
## Dec 1992 8681.794 326.50223 17037.09
## Jan 1993 8721.379 197.65965 17245.10
## Feb 1993 8760.963 69.78442 17452.14
## Mar 1993 8800.548 -57.19924 17658.29
## Apr 1993 8840.132 -183.36139 17863.63
## May 1993 8879.717 -308.76695 18068.20
## Jun 1993 8919.302 -433.47621 18272.08
## Jul 1993 8958.886 -557.54529 18475.32
## Aug 1993 8998.471 -681.02653 18677.97
## Sep 1993 9038.055 -803.96882 18880.08
## Oct 1993 9077.640 -926.41794 19081.70
## Nov 1993 9117.225 -1048.41679 19282.87
## Dec 1993 9156.809 -1170.00570 19483.62
## Jan 1994 9196.394 -1291.22258 19684.01forecast(esModel,h=18,level=0.95)## Point forecast Lower bound (2.5%) Upper bound (97.5%)
## Aug 1992 8523.456 3598.669 19791.18
## Sep 1992 8563.040 3667.669 20479.91
## Oct 1992 8602.625 3732.632 21507.95
## Nov 1992 8642.209 3682.506 21676.92
## Dec 1992 8681.794 3743.457 22592.11
## Jan 1993 8721.379 3608.492 23560.33
## Feb 1993 8760.963 3679.910 23715.88
## Mar 1993 8800.548 3698.535 24697.60
## Apr 1993 8840.132 3751.812 26030.47
## May 1993 8879.717 3841.542 26579.54
## Jun 1993 8919.302 3690.208 26788.83
## Jul 1993 8958.886 3883.779 27851.64
## Aug 1993 8998.471 3956.799 28866.52
## Sep 1993 9038.055 3919.546 30025.27
## Oct 1993 9077.640 3914.626 31414.41
## Nov 1993 9117.225 3968.082 32157.44
## Dec 1993 9156.809 3930.952 32994.69
## Jan 1994 9196.394 4074.940 33534.01Finally, 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(M3$N2457, intervals=TRUE, silent=FALSE)## Forming the pool of models based on... ANN, ANA, AAN, Estimation progress: 100%... Done!## Time elapsed: 0.4 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.151
## Initial values were optimised.
## 3 parameters were estimated in the process
## Residuals standard deviation: 0.434
## Cost function type: MSE; Cost function value: 1965396.467
##
## Information criteria:
## AIC AICc BIC BICc
## 1998.844 1999.061 2007.079 2007.592
## 95% parametric prediction intervals were constructed
## 50% of values are in the prediction interval
## Forecast errors:
## MPE: -127.6%; sCE: 1618.3%; Bias: -92.4%; MAPE: 129.2%
## MASE: 2.278; sMAE: 93.4%; RelMAE: 1.895; sMSE: 115.4%
This command has taken the data, split it into in-sample and holdout and produced the forecast of appropriate length to the holdout.