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:
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:
## Forming the pool of models based on... ANN, ANA, AAN, Estimation progress: 100%... Done!
## Time elapsed: 0.41 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.1452
## Initial values were optimised.
##
## Loss function type: MSE; Loss function value: 0.1653
## Error standard deviation: 0.4129
## Sample size: 97
## Number of estimated parameters: 3
## Number of degrees of freedom: 94
## Information criteria:
## AIC AICc BIC BICc
## 1645.978 1646.236 1653.702 1654.292
##
## Forecast errors:
## MPE: 26.3%; sCE: 1919.1%; Bias: 86.9%; MAPE: 39.8%
## MASE: 2.944; sMAE: 120.1%; sMSE: 242.7%; rMAE: 1.258; rRMSE: 1.367
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 actual values, fitted values and point forecasts.
If we need prediction interval, then we run:
## Forming the pool of models based on... ANN, ANA, AAN, Estimation progress: 100%... Done!
## Time elapsed: 0.3 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.1452
## Initial values were optimised.
##
## Loss function type: MSE; Loss function value: 0.1653
## Error standard deviation: 0.4129
## Sample size: 97
## Number of estimated parameters: 3
## Number of degrees of freedom: 94
## Information criteria:
## AIC AICc BIC BICc
## 1645.978 1646.236 1653.702 1654.292
##
## 95% parametric prediction interval was constructed
## 72% of values are in the prediction interval
## Forecast errors:
## MPE: 26.3%; sCE: 1919.1%; Bias: 86.9%; MAPE: 39.8%
## MASE: 2.944; sMAE: 120.1%; sMSE: 242.7%; rMAE: 1.258; rRMSE: 1.367
Due to multiplicative nature of error term in the model, the interval are asymmetric. This is the expected behaviour. The other thing to note is that the output now also provides the theoretical width of prediction interval and its actual coverage.
If we save the model (and let’s say we want it to work silently):
we can then reuse it for different purposes:
## Time elapsed: 0.03 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.1452
## Initial values were provided by user.
##
## Loss function type: MSE; Loss function value: 0.1838
## Error standard deviation: 0.4306
## Sample size: 115
## Number of estimated parameters: 1
## Number of provided parameters: 2
## Number of degrees of freedom: 114
## Information criteria:
## AIC AICc BIC BICc
## 1994.861 1994.897 1997.606 1997.690
##
## 93% nonparametric prediction interval was constructed
We can also extract the type of model in order to reuse it later:
## [1] "MNN"
This handy function, by the way, also works with ets() from forecast package.
If we need actual values from the model, we can use actuals()
method from greybox
package:
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 1983 2158.1 1086.4 1154.7 1125.6 920.0 2188.6 829.2 1353.1 947.2 1816.8
## 1984 1783.3 1713.1 3479.7 2429.4 3074.3 3427.4 2783.7 1968.7 2045.6 1471.3
## 1985 1821.0 2409.8 3485.8 3289.2 3048.3 2914.1 2173.9 3018.4 2200.1 6844.3
## 1986 3238.9 3252.2 3278.8 1766.8 3572.8 3467.6 7464.7 2748.4 5126.7 2870.8
## 1987 3220.7 3586.0 3249.5 3222.5 2488.5 3332.4 2036.1 1968.2 2967.2 3151.6
## 1988 3894.1 4625.5 3291.7 3065.6 2316.5 2453.4 4582.8 2291.2 3555.5 1785.0
## 1989 2102.9 2307.7 6242.1 6170.5 1863.5 6318.9 3992.8 3435.1 1585.8 2106.8
## 1990 6168.0 7247.4 3579.7 6365.2 4658.9 6911.8 2143.7 5973.9 4017.2 4473.0
## 1991 8749.1
## Nov Dec
## 1983 1624.5 868.5
## 1984 2763.7 2328.4
## 1985 4160.4 1548.8
## 1986 2170.2 4326.8
## 1987 1610.5 3985.0
## 1988 2020.0 2026.8
## 1989 1892.1 4310.6
## 1990 3591.9 4676.5
## 1991
We can then use persistence or initials only from the model to construct the other one:
## Time elapsed: 0.01 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.1509
## Initial values were provided by user.
##
## Loss function type: MSE; Loss function value: 0.1838
## Error standard deviation: 0.4324
## Sample size: 115
## Number of estimated parameters: 2
## Number of provided parameters: 1
## Number of degrees of freedom: 113
## Information criteria:
## AIC AICc BIC BICc
## 1996.845 1996.952 2002.334 2002.589
## Time elapsed: 0.01 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.1452
## Initial values were optimised.
##
## Loss function type: MSE; Loss function value: 0.1838
## Error standard deviation: 0.4325
## Sample size: 115
## Number of estimated parameters: 2
## Number of provided parameters: 1
## Number of degrees of freedom: 113
## Information criteria:
## AIC AICc BIC BICc
## 1996.861 1996.968 2002.351 2002.605
or provide some arbitrary values:
## Time elapsed: 0.01 seconds
## Model estimated: ETS(MNN)
## Persistence vector g:
## alpha
## 0.1498
## Initial values were provided by user.
##
## Loss function type: MSE; Loss function value: 0.184
## Error standard deviation: 0.4328
## Sample size: 115
## Number of estimated parameters: 2
## Number of provided parameters: 1
## Number of degrees of freedom: 113
## Information criteria:
## AIC AICc BIC BICc
## 1997.028 1997.136 2002.518 2002.773
Using some other parameters may lead to completely different model and forecasts:
## Time elapsed: 0.31 seconds
## Model estimated: ETS(ANN)
## Persistence vector g:
## alpha
## 0.0798
## Initial values were optimised.
##
## Loss function type: aTMSE; Loss function value: 39565651.9
## Error standard deviation: 1466.912
## Sample size: 97
## Number of estimated parameters: 3
## Number of degrees of freedom: 94
## Information criteria:
## AIC AICc BIC BICc
## 1974.076 1974.736 1985.865 1986.455
##
## 95% parametric prediction interval was constructed
## 44% of values are in the prediction interval
## Forecast errors:
## MPE: 33.4%; sCE: 2196.8%; Bias: 90.4%; MAPE: 43.4%
## MASE: 3.235; sMAE: 132%; sMSE: 278%; rMAE: 1.382; rRMSE: 1.463
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:
## Estimation progress: 10%20%30%40%50%60%70%80%90%100%... Done!
## Time elapsed: 0.53 seconds
## Model estimated: ETS(CCN)
## Initial values were optimised.
##
## Loss function type: MSE
## Error standard deviation: 1404.465
## Sample size: 97
## Information criteria:
## (combined values)
## AIC AICc BIC BICc
## 1646.545 1646.941 1654.794 1655.304
##
## Forecast errors:
## MPE: 27.4%; sCE: 1963.2%; Bias: 88.2%; MAPE: 40.4%
## MASE: 2.992; sMAE: 122.1%; sMSE: 248.5%; rMAE: 1.278; rRMSE: 1.383
Model selection from a specified pool and forecasts combination are called using respectively:
## Estimation progress: 17%33%50%67%83%100%... Done!
## Time elapsed: 0.57 seconds
## Model estimated: ETS(ANN)
## Persistence vector g:
## alpha
## 0.1582
## Initial values were optimised.
##
## Loss function type: MSE; Loss function value: 2007704.5316
## Error standard deviation: 1439.368
## Sample size: 97
## Number of estimated parameters: 3
## Number of degrees of freedom: 94
## 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%; sMSE: 238.1%; rMAE: 1.243; rRMSE: 1.354
## Estimation progress: 17%33%50%67%83%100%... Done!
## Time elapsed: 0.59 seconds
## Model estimated: ETS(CCC)
## Initial values were optimised.
##
## Loss function type: MSE
## Error standard deviation: 1386.675
## Sample size: 97
## Information criteria:
## (combined values)
## AIC AICc BIC BICc
## 1689.864 1690.147 1696.984 1697.488
##
## Forecast errors:
## MPE: 17.1%; sCE: 1568.3%; Bias: 77.7%; MAPE: 37.3%
## MASE: 2.658; sMAE: 108.4%; sMSE: 206.7%; rMAE: 1.135; rRMSE: 1.261
Now let’s introduce some artificial exogenous variables:
and fit a model with all the exogenous first:
## Time elapsed: 1.28 seconds
## Model estimated: ETSX(MMdN)
## Persistence vector g:
## alpha beta
## 0.0508 0.0000
## Damping parameter: 0.9429
## Initial values were optimised.
## Xreg coefficients were estimated in a normal style
##
## Loss function type: MSE; Loss function value: 0.155
## Error standard deviation: 0.4087
## Sample size: 97
## Number of estimated parameters: 7
## Number of provided parameters: 1
## Number of degrees of freedom: 90
## Information criteria:
## AIC AICc BIC BICc
## 1647.757 1649.015 1663.745 1665.880
##
## Forecast errors:
## MPE: 35%; sCE: 2291.8%; Bias: 90.1%; MAPE: 46.5%
## MASE: 3.401; sMAE: 138.8%; sMSE: 296.8%; rMAE: 1.453; rRMSE: 1.511
or construct a model with selected exogenous (based on IC):
## Time elapsed: 0.95 seconds
## Model estimated: ETSX(MNN)
## Persistence vector g:
## alpha
## 0.1442
## Initial values were optimised.
## Xreg coefficients were estimated in a normal style
##
## Loss function type: MSE; Loss function value: 0.1623
## Error standard deviation: 0.4114
## Sample size: 97
## Number of estimated parameters: 4
## Number of degrees of freedom: 93
## Information criteria:
## AIC AICc BIC BICc
## 1646.218 1646.652 1656.516 1657.511
##
## Forecast errors:
## MPE: 25.4%; sCE: 1912.5%; Bias: 83.9%; MAPE: 42.1%
## MASE: 3.029; sMAE: 123.6%; sMSE: 247.9%; rMAE: 1.294; rRMSE: 1.381
or the one with the updated xreg:
If we want to check if lagged x can be used for forecasting purposes, we can use xregExpander()
function from greybox
package:
## Time elapsed: 1.21 seconds
## Model estimated: ETSX(MNN)
## Persistence vector g:
## alpha
## 0.146
## Initial values were optimised.
## Xreg coefficients were estimated in a normal style
##
## Loss function type: MSE; Loss function value: 0.1568
## Error standard deviation: 0.4066
## Sample size: 97
## Number of estimated parameters: 5
## Number of degrees of freedom: 92
## Information criteria:
## AIC AICc BIC BICc
## 1644.885 1645.545 1657.759 1659.267
##
## Forecast errors:
## MPE: 25.4%; sCE: 1904.5%; Bias: 83%; MAPE: 41.4%
## MASE: 3.007; sMAE: 122.6%; sMSE: 247.9%; rMAE: 1.284; rRMSE: 1.381
If we are confused about the type of estimated model, the function formula()
will help us:
## [1] "y[t] = l[t-1] * b[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()
:
The point forecasts in the majority of cases should the same, but the prediction interval may be different (especially if error term is multiplicative):
## 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 19684.01
## Point forecast Lower bound (2.5%) Upper bound (97.5%)
## Aug 1992 9359.395 3676.176 19888.65
## Sep 1992 9549.279 3676.700 20612.51
## Oct 1992 9784.531 3677.698 21260.02
## Nov 1992 9979.559 3724.459 21905.25
## Dec 1992 10192.384 3715.061 22616.11
## Jan 1993 10425.218 3760.431 23544.45
## Feb 1993 10618.467 3783.103 24289.44
## Mar 1993 10852.680 3788.223 25064.03
## Apr 1993 11077.307 3822.658 25902.19
## May 1993 11306.703 3839.004 26561.27
## Jun 1993 11560.723 3876.534 27683.31
## Jul 1993 11772.357 3875.137 28162.45
## Aug 1993 12015.456 3887.544 28973.42
## Sep 1993 12265.828 3918.511 29836.86
## Oct 1993 12515.615 3975.508 30802.29
## Nov 1993 12786.734 4003.705 31702.98
## Dec 1993 13025.802 4007.593 32598.17
## Jan 1994 13301.721 4051.399 33737.15
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:
## Forming the pool of models based on... ANN, ANA, AAN, Estimation progress: 40%50%60%70%80%90%100%... Done!
## Time elapsed: 2.03 seconds
## Model estimated: ETS(MAN)
## Persistence vector g:
## alpha beta
## 0.1165 0.0000
## Initial values were optimised.
##
## Loss function type: MSE; Loss function value: 0.1811
## Error standard deviation: 0.4331
## Sample size: 115
## Number of estimated parameters: 4
## Number of provided parameters: 1
## Number of degrees of freedom: 111
## Information criteria:
## AIC AICc BIC BICc
## 1999.144 1999.508 2010.124 2010.986
##
## 95% parametric prediction interval was constructed
## 50% of values are in the prediction interval
## Forecast errors:
## MPE: -188.2%; sCE: -2506.1%; Bias: -99.2%; MAPE: 188.3%
## MASE: 3.399; sMAE: 139.4%; sMSE: 244.1%; rMAE: 2.828; rRMSE: 2.307
This command has taken the data, split it into in-sample and holdout and produced the forecast of appropriate length to the holdout.