This vignette explains briefly how to use the function adam() and the related auto.adam() in smooth package. It does not aim at covering all aspects of the function, but focuses on the main ones.

ADAM is Augmented Dynamic Adaptive Model. It is a model that underlies ETS, ARIMA and regression, connecting them in a unified framework. The underlying model for ADAM is a Single Source of Error state space model, which is explained in detail separately in an online monograph.

The main philosophy of adam() function is to be agnostic of the provided data. This means that it will work with ts, msts, zoo, xts, data.frame, numeric and other classes of data. The specification of seasonality in the model is done using a separate parameter lags, so you are not obliged to transform the existing data to something specific, and can use it as is. If you provide a matrix, or a data.frame, or a data.table, or any other multivariate structure, then the function will use the first column for the response variable and the others for the explanatory ones. One thing that is currently assumed in the function is that the data is measured at a regular frequency. If this is not the case, you will need to introduce missing values manually.

In order to run the experiments in this vignette, we need to load the following packages:

require(greybox)
require(smooth)

ADAM ETS

First and foremost, ADAM implements ETS model, although in a more flexible way than (Hyndman et al. 2008): it supports different distributions for the error term, which are regulated via distribution parameter. By default, the additive error model relies on Normal distribution, while the multiplicative error one assumes Inverse Gaussian. If you want to reproduce the classical ETS, you would need to specify distribution="dnorm". Here is an example of ADAM ETS(MMM) with Normal distribution on AirPassengers data:

testModel <- adam(AirPassengers, "MMM", lags=c(1,12), distribution="dnorm",
                  h=12, holdout=TRUE)
summary(testModel)
#> 
#> Model estimated using adam() function: ETS(MMM)
#> Response variable: AirPassengers
#> Distribution used in the estimation: Normal
#> Loss function type: likelihood; Loss function value: 465.9969
#> Coefficients:
#>       Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha   0.7565     0.0947      0.569      0.9438 *
#> beta    0.0004     0.0191      0.000      0.0382  
#> gamma   0.0000     0.0522      0.000      0.1032  
#> 
#> Error standard deviation: 0.0346
#> Sample size: 132
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 128
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 939.9937 940.3087 951.5249 952.2939
plot(forecast(testModel,h=12,interval="prediction"))

You might notice that the summary contains more than what is reported by other smooth functions. This one also produces standard errors for the estimated parameters based on Fisher Information calculation. Note that this is computationally expensive, so if you have a model with more than 30 variables, the calculation of standard errors might take plenty of time. As for the default print() method, it will produce a shorter summary from the model, without the standard errors (similar to what es() does):

testModel
#> Time elapsed: 0.07 seconds
#> Model estimated using adam() function: ETS(MMM)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 465.9969
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.7565 0.0004 0.0000 
#> 
#> Sample size: 132
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 128
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 939.9937 940.3087 951.5249 952.2939 
#> 
#> Forecast errors:
#> ME: -9.668; MAE: 15.499; RMSE: 22.94
#> sCE: -44.196%; Asymmetry: -59.4%; sMAE: 5.904%; sMSE: 0.764%
#> MASE: 0.644; RMSSE: 0.732; rMAE: 0.204; rRMSE: 0.223

Also, note that the prediction interval in case of multiplicative error models are approximate. It is advisable to use simulations instead (which is slower, but more accurate):

plot(forecast(testModel,h=18,interval="simulated"))

If you want to do the residuals diagnostics, then it is recommended to use plot function, something like this (you can select, which of the plots to produce):

par(mfcol=c(3,4))
plot(testModel,which=c(1:11))
par(mfcol=c(1,1))
plot(testModel,which=12)

By default ADAM will estimate models via maximising likelihood function. But there is also a parameter loss, which allows selecting from a list of already implemented loss functions (again, see documentation for adam() for the full list) or using a function written by a user. Here is how to do the latter on the example of BJsales:

lossFunction <- function(actual, fitted, B){
  return(sum(abs(actual-fitted)^3))
}
testModel <- adam(BJsales, "AAN", silent=FALSE, loss=lossFunction,
                  h=12, holdout=TRUE)
testModel
#> Time elapsed: 0.02 seconds
#> Model estimated using adam() function: ETS(AAN)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: custom; Loss function value: 599.3072
#> Persistence vector g:
#> alpha  beta 
#> 1.000 0.227 
#> 
#> Sample size: 138
#> Number of estimated parameters: 2
#> Number of degrees of freedom: 136
#> Information criteria are unavailable for the chosen loss & distribution.
#> 
#> Forecast errors:
#> ME: 3.015; MAE: 3.129; RMSE: 3.866
#> sCE: 15.916%; Asymmetry: 91.7%; sMAE: 1.376%; sMSE: 0.029%
#> MASE: 2.626; RMSSE: 2.52; rMAE: 1.009; rRMSE: 1.009

Note that you need to have parameters actual, fitted and B in the function, which correspond to the vector of actual values, vector of fitted values on each iteration and a vector of the optimised parameters.

loss and distribution parameters are independent, so in the example above, we have assumed that the error term follows Normal distribution, but we have estimated its parameters using a non-conventional loss because we can. Some of distributions assume that there is an additional parameter, which can either be estimated or provided by user. These include Asymmetric Laplace (distribution="dalaplace") with alpha, Generalised Normal and Log-Generalised normal (distribution=c("gnorm","dlgnorm")) with shape and Student’s T (distribution="dt") with nu:

testModel <- adam(BJsales, "MMN", silent=FALSE, distribution="dgnorm", shape=3,
                  h=12, holdout=TRUE)

The model selection in ADAM ETS relies on information criteria and works correctly only for the loss="likelihood". There are several options, how to select the model, see them in the description of the function: ?adam(). The default one uses branch-and-bound algorithm, similar to the one used in es(), but only considers additive trend models (the multiplicative trend ones are less stable and need more attention from a forecaster):

testModel <- adam(AirPassengers, "ZXZ", lags=c(1,12), silent=FALSE,
                  h=12, holdout=TRUE)
#> Forming the pool of models based on... ANN , ANA , MNM , MAM , Estimation progress:    71 %86 %100 %... Done!
testModel
#> Time elapsed: 0.21 seconds
#> Model estimated using adam() function: ETS(MAM)
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 467.1158
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.7918 0.0000 0.0000 
#> 
#> Sample size: 132
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 128
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 942.2316 942.5465 953.7628 954.5317 
#> 
#> Forecast errors:
#> ME: 5.66; MAE: 18.165; RMSE: 24.481
#> sCE: 25.873%; Asymmetry: 52.1%; sMAE: 6.92%; sMSE: 0.87%
#> MASE: 0.754; RMSSE: 0.781; rMAE: 0.239; rRMSE: 0.238

Note that the function produces point forecasts if h>0, but it won’t generate prediction interval. This is why you need to use forecast() method (as shown in the first example in this vignette).

Similarly to es(), function supports combination of models, but it saves all the tested models in the output for a potential reuse. Here how it works:

testModel <- adam(AirPassengers, "CXC", lags=c(1,12),
                  h=12, holdout=TRUE)
testForecast <- forecast(testModel,h=18,interval="semiparametric", level=c(0.9,0.95))
testForecast
#>          Point forecast Lower bound (5%) Lower bound (2.5%) Upper bound (95%)
#> Jan 1960       415.4643         404.8979           402.9006          426.1449
#> Feb 1960       410.5010         385.7752           381.1895          435.8692
#> Mar 1960       473.1697         437.7122           431.1902          509.7810
#> Apr 1960       456.1741         415.7581           408.3819          498.1547
#> May 1960       456.8487         413.0935           405.1409          502.4412
#> Jun 1960       519.2177         468.4528           459.2369          572.1608
#> Jul 1960       577.1520         518.6865           508.0945          638.2209
#> Aug 1960       576.1572         516.4697           505.6710          638.5656
#> Sep 1960       503.0507         450.0809           440.5071          558.4768
#> Oct 1960       439.1892         392.6301           384.2186          487.9227
#> Nov 1960       383.5170         342.4956           335.0886          426.4725
#> Dec 1960       432.8311         386.4450           378.0704          481.4085
#> Jan 1961       443.3470         395.1627           386.4712          493.8415
#> Feb 1961       437.8972         389.1441           380.3637          489.0477
#> Mar 1961       504.5726         445.1967           434.5433          567.0434
#> Apr 1961       486.2818         426.3906           415.6800          549.4487
#> May 1961       486.8351         425.0177           413.9880          552.1445
#> Jun 1961       553.1119         484.1469           471.8250          625.8980
#>          Upper bound (97.5%)
#> Jan 1960            428.2182
#> Feb 1960            440.8828
#> Mar 1960            517.0713
#> Apr 1960            506.5729
#> May 1960            511.6172
#> Jun 1960            582.8270
#> Jul 1960            650.5464
#> Aug 1960            651.1763
#> Sep 1960            569.6863
#> Oct 1960            497.7822
#> Nov 1960            435.1673
#> Dec 1960            491.2423
#> Jan 1961            504.0713
#> Feb 1961            499.4243
#> Mar 1961            579.7574
#> Apr 1961            562.3403
#> May 1961            565.4992
#> Jun 1961            640.7643
plot(testForecast)

Yes, now we support vectors for the levels in case you want to produce several. In fact, we also support side for prediction interval, so you can extract specific quantiles without a hustle:

forecast(testModel,h=18,interval="semiparametric", level=c(0.9,0.95,0.99), side="upper")
#>          Point forecast Upper bound (90%) Upper bound (95%) Upper bound (99%)
#> Jan 1960       415.4643          423.7628          426.1449          430.6373
#> Feb 1960       410.5010          430.1360          435.8692          446.7600
#> Mar 1960       473.1697          501.4606          509.7810          525.6342
#> Apr 1960       456.1741          488.5647          498.1547          516.4783
#> May 1960       456.8487          491.9978          502.4412          522.4247
#> Jun 1960       519.2177          560.0244          572.1608          595.3930
#> Jul 1960       577.1520          624.2032          638.2209          665.0738
#> Aug 1960       576.1572          624.2280          638.5656          666.0442
#> Sep 1960       503.0507          545.7351          558.4768          582.9052
#> Oct 1960       439.1892          476.7165          487.9227          509.4103
#> Nov 1960       383.5170          416.5914          426.4725          445.4230
#> Dec 1960       432.8311          470.2333          481.4085          502.8418
#> Jan 1961       443.3470          482.2186          493.8415          516.1403
#> Feb 1961       437.8972          477.2621          489.0477          511.6707
#> Mar 1961       504.5726          552.6150          567.0434          594.7748
#> Apr 1961       486.2818          534.8294          549.4487          577.5781
#> May 1961       486.8351          537.0076          552.1445          581.2923
#> Jun 1961       553.1119          609.0428          625.8980          658.3396

A brand new thing in the function is the possibility to use several frequencies (double / triple / quadruple / … seasonal models). In order to show how it works, we will generate an artificial time series, inspired by half-hourly electricity demand using sim.gum() function:

set.seed(41)
ordersGUM <- c(1,1,1)
lagsGUM <- c(1,48,336)
initialGUM1 <- -25381.7
initialGUM2 <- c(23955.09, 24248.75, 24848.54, 25012.63, 24634.14, 24548.22, 24544.63, 24572.77,
                 24498.33, 24250.94, 24545.44, 25005.92, 26164.65, 27038.55, 28262.16, 28619.83,
                 28892.19, 28575.07, 28837.87, 28695.12, 28623.02, 28679.42, 28682.16, 28683.40,
                 28647.97, 28374.42, 28261.56, 28199.69, 28341.69, 28314.12, 28252.46, 28491.20,
                 28647.98, 28761.28, 28560.11, 28059.95, 27719.22, 27530.23, 27315.47, 27028.83,
                 26933.75, 26961.91, 27372.44, 27362.18, 27271.31, 26365.97, 25570.88, 25058.01)
initialGUM3 <- c(23920.16, 23026.43, 22812.23, 23169.52, 23332.56, 23129.27, 22941.20, 22692.40,
                 22607.53, 22427.79, 22227.64, 22580.72, 23871.99, 25758.34, 28092.21, 30220.46,
                 31786.51, 32699.80, 33225.72, 33788.82, 33892.25, 34112.97, 34231.06, 34449.53,
                 34423.61, 34333.93, 34085.28, 33948.46, 33791.81, 33736.17, 33536.61, 33633.48,
                 33798.09, 33918.13, 33871.41, 33403.75, 32706.46, 31929.96, 31400.48, 30798.24,
                 29958.04, 30020.36, 29822.62, 30414.88, 30100.74, 29833.49, 28302.29, 26906.72,
                 26378.64, 25382.11, 25108.30, 25407.07, 25469.06, 25291.89, 25054.11, 24802.21,
                 24681.89, 24366.97, 24134.74, 24304.08, 25253.99, 26950.23, 29080.48, 31076.33,
                 32453.20, 33232.81, 33661.61, 33991.21, 34017.02, 34164.47, 34398.01, 34655.21,
                 34746.83, 34596.60, 34396.54, 34236.31, 34153.32, 34102.62, 33970.92, 34016.13,
                 34237.27, 34430.08, 34379.39, 33944.06, 33154.67, 32418.62, 31781.90, 31208.69,
                 30662.59, 30230.67, 30062.80, 30421.11, 30710.54, 30239.27, 28949.56, 27506.96,
                 26891.75, 25946.24, 25599.88, 25921.47, 26023.51, 25826.29, 25548.72, 25405.78,
                 25210.45, 25046.38, 24759.76, 24957.54, 25815.10, 27568.98, 29765.24, 31728.25,
                 32987.51, 33633.74, 34021.09, 34407.19, 34464.65, 34540.67, 34644.56, 34756.59,
                 34743.81, 34630.05, 34506.39, 34319.61, 34110.96, 33961.19, 33876.04, 33969.95,
                 34220.96, 34444.66, 34474.57, 34018.83, 33307.40, 32718.90, 32115.27, 31663.53,
                 30903.82, 31013.83, 31025.04, 31106.81, 30681.74, 30245.70, 29055.49, 27582.68,
                 26974.67, 25993.83, 25701.93, 25940.87, 26098.63, 25771.85, 25468.41, 25315.74,
                 25131.87, 24913.15, 24641.53, 24807.15, 25760.85, 27386.39, 29570.03, 31634.00,
                 32911.26, 33603.94, 34020.90, 34297.65, 34308.37, 34504.71, 34586.78, 34725.81,
                 34765.47, 34619.92, 34478.54, 34285.00, 34071.90, 33986.48, 33756.85, 33799.37,
                 33987.95, 34047.32, 33924.48, 33580.82, 32905.87, 32293.86, 31670.02, 31092.57,
                 30639.73, 30245.42, 30281.61, 30484.33, 30349.51, 29889.23, 28570.31, 27185.55,
                 26521.85, 25543.84, 25187.82, 25371.59, 25410.07, 25077.67, 24741.93, 24554.62,
                 24427.19, 24127.21, 23887.55, 24028.40, 24981.34, 26652.32, 28808.00, 30847.09,
                 32304.13, 33059.02, 33562.51, 33878.96, 33976.68, 34172.61, 34274.50, 34328.71,
                 34370.12, 34095.69, 33797.46, 33522.96, 33169.94, 32883.32, 32586.24, 32380.84,
                 32425.30, 32532.69, 32444.24, 32132.49, 31582.39, 30926.58, 30347.73, 29518.04,
                 29070.95, 28586.20, 28416.94, 28598.76, 28529.75, 28424.68, 27588.76, 26604.13,
                 26101.63, 25003.82, 24576.66, 24634.66, 24586.21, 24224.92, 23858.42, 23577.32,
                 23272.28, 22772.00, 22215.13, 21987.29, 21948.95, 22310.79, 22853.79, 24226.06,
                 25772.55, 27266.27, 28045.65, 28606.14, 28793.51, 28755.83, 28613.74, 28376.47,
                 27900.76, 27682.75, 27089.10, 26481.80, 26062.94, 25717.46, 25500.27, 25171.05,
                 25223.12, 25634.63, 26306.31, 26822.46, 26787.57, 26571.18, 26405.21, 26148.41,
                 25704.47, 25473.10, 25265.97, 26006.94, 26408.68, 26592.04, 26224.64, 25407.27,
                 25090.35, 23930.21, 23534.13, 23585.75, 23556.93, 23230.25, 22880.24, 22525.52,
                 22236.71, 21715.08, 21051.17, 20689.40, 20099.18, 19939.71, 19722.69, 20421.58,
                 21542.03, 22962.69, 23848.69, 24958.84, 25938.72, 26316.56, 26742.61, 26990.79,
                 27116.94, 27168.78, 26464.41, 25703.23, 25103.56, 24891.27, 24715.27, 24436.51,
                 24327.31, 24473.02, 24893.89, 25304.13, 25591.77, 25653.00, 25897.55, 25859.32,
                 25918.32, 25984.63, 26232.01, 26810.86, 27209.70, 26863.50, 25734.54, 24456.96)
y <- sim.gum(orders=ordersGUM, lags=lagsGUM, nsim=1, frequency=336, obs=3360,
             measurement=rep(1,3), transition=diag(3), persistence=c(0.045,0.162,0.375),
             initial=rbind(initialGUM1,initialGUM2,initialGUM3))$data

We can then apply ADAM to this data:

testModel <- adam(y, "MMdM", lags=c(1,48,336), initial="backcasting",
                  silent=FALSE, h=336, holdout=TRUE)
testModel
#> Time elapsed: 1.37 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19566.52
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0213 0.0000 0.1834 0.2665 
#> Damping parameter: 0.7393
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39145.04 39145.07 39181.13 39181.24 
#> 
#> Forecast errors:
#> ME: 68.363; MAE: 141.934; RMSE: 182.802
#> sCE: 75.787%; Asymmetry: 49.6%; sMAE: 0.468%; sMSE: 0.004%
#> MASE: 0.191; RMSSE: 0.178; rMAE: 0.022; rRMSE: 0.023

Note that the more lags you have, the more initial seasonal components the function will need to estimate, which is a difficult task. This is why we used initial="backcasting" in the example above - this speeds up the estimation by reducing the number of parameters to estimate. Still, the optimiser might not get close to the optimal value, so we can help it. First, we can give more time for the calculation, increasing the number of iterations via maxeval (the default value is 40 iterations for each estimated parameter, e.g. $40 \times 5 = 200$ in our case):

testModel <- adam(y, "MMdM", lags=c(1,48,336), initial="backcasting",
                  silent=FALSE, h=336, holdout=TRUE, maxeval=10000)
testModel
#> Time elapsed: 1.68 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19566.51
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0214 0.0000 0.1825 0.2654 
#> Damping parameter: 0.7417
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39145.03 39145.06 39181.11 39181.23 
#> 
#> Forecast errors:
#> ME: 68.601; MAE: 142.026; RMSE: 182.928
#> sCE: 76.051%; Asymmetry: 49.8%; sMAE: 0.469%; sMSE: 0.004%
#> MASE: 0.191; RMSSE: 0.178; rMAE: 0.022; rRMSE: 0.023

This will take more time, but will typically lead to more refined parameters. You can control other parameters of the optimiser as well, such as algorithm, xtol_rel, print_level and others, which are explained in the documentation for nloptr function from nloptr package (run nloptr.print.options() for details). Second, we can give a different set of initial parameters for the optimiser, have a look at what the function saves:

testModel$B

and use this as a starting point for the reestimation (e.g. with a different algorithm):

testModel <- adam(y, "MMdM", lags=c(1,48,336), initial="backcasting",
                  silent=FALSE, h=336, holdout=TRUE, B=testModel$B)
testModel
#> Time elapsed: 0.68 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19566.51
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0214 0.0000 0.1824 0.2658 
#> Damping parameter: 0
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39145.02 39145.05 39181.11 39181.22 
#> 
#> Forecast errors:
#> ME: 68.533; MAE: 141.99; RMSE: 182.884
#> sCE: 75.976%; Asymmetry: 49.8%; sMAE: 0.468%; sMSE: 0.004%
#> MASE: 0.191; RMSSE: 0.178; rMAE: 0.022; rRMSE: 0.023

If you are ready to wait, you can change the initialisation to the initial="optimal", which in our case will take much more time because of the number of estimated parameters - 389 for the chosen model. The estimation process in this case might take 20 - 30 times more than in the example above.

In addition, you can specify some parts of the initial state vector or some parts of the persistence vector, here is an example:

testModel <- adam(y, "MMdM", lags=c(1,48,336), initial="backcasting",
                  silent=TRUE, h=336, holdout=TRUE, persistence=list(beta=0.1))
testModel
#> Time elapsed: 1.1 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 50157
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.1332 0.1000 0.3977 0.3539 
#> Damping parameter: 0.9305
#> Sample size: 3024
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 3019
#> Number of provided parameters: 1
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 100324.0 100324.0 100354.1 100354.1 
#> 
#> Forecast errors:
#> ME: -160302359550516; MAE: 160302359550620; RMSE: 2747785126744693
#> sCE: -177712397668157%; Asymmetry: -100%; sMAE: 528905945441.285%; sMSE: 8.21944710406262e+23%
#> MASE: 215869349742.776; RMSSE: 2672287825656.58; rMAE: 25055171305.002; rRMSE: 348578473096.1

The function also handles intermittent data (the data with zeroes) and the data with missing values. This is partially covered in the vignette on the oes() function. Here is a simple example:

testModel <- adam(rpois(120,0.5), "MNN", silent=FALSE, h=12, holdout=TRUE,
                  occurrence="odds-ratio")
testModel
#> Time elapsed: 0.02 seconds
#> Model estimated using adam() function: iETS(MNN)[O]
#> With backcasting initialisation
#> Occurrence model type: Odds ratio
#> Distribution assumed in the model: Mixture of Bernoulli and Gamma
#> Loss function type: likelihood; Loss function value: 23.6797
#> Persistence vector g:
#> alpha 
#>     0 
#> 
#> Sample size: 108
#> Number of estimated parameters: 3
#> Number of degrees of freedom: 105
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 215.9554 216.0697 224.0018 219.5872 
#> 
#> Forecast errors:
#> Asymmetry: -12.065%; sMSE: 27.894%; rRMSE: 0.791; sPIS: 540.224%; sCE: 5%

Finally, adam() is faster than es() function, because its code is more efficient and it uses a different optimisation algorithm with more finely tuned parameters by default. Let’s compare:

adamModel <- adam(AirPassengers, "CCC",
                  h=12, holdout=TRUE)
esModel <- es(AirPassengers, "CCC",
              h=12, holdout=TRUE)
"adam:"
#> [1] "adam:"
adamModel
#> Time elapsed: 0.84 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 4.4505
#> Average number of degrees of freedom: 127.5495
#> 
#> Forecast errors:
#> ME: -0.696; MAE: 15.606; RMSE: 22.305
#> sCE: -3.182%; sMAE: 5.945%; sMSE: 0.722%
#> MASE: 0.648; RMSSE: 0.712; rMAE: 0.205; rRMSE: 0.217
"es():"
#> [1] "es():"
esModel
#> Time elapsed: 0.83 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 4.1729
#> Average number of degrees of freedom: 127.8271
#> 
#> Forecast errors:
#> ME: -3.072; MAE: 15.417; RMSE: 22.031
#> sCE: -14.044%; sMAE: 5.873%; sMSE: 0.704%
#> MASE: 0.64; RMSSE: 0.703; rMAE: 0.203; rRMSE: 0.214

ADAM ARIMA

As mentioned above, ADAM does not only contain ETS, it also contains ARIMA model, which is regulated via orders parameter. If you want to have a pure ARIMA, you need to switch off ETS, which is done via model="NNN":

testModel <- adam(BJsales, "NNN", silent=FALSE, orders=c(0,2,2),
                  h=12, holdout=TRUE)
testModel
#> Time elapsed: 0.02 seconds
#> Model estimated using adam() function: ARIMA(0,2,2)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 240.5973
#> ARMA parameters of the model:
#>         Lag 1
#> MA(1) -0.7488
#> MA(2) -0.0175
#> 
#> Sample size: 138
#> Number of estimated parameters: 3
#> Number of degrees of freedom: 135
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 487.1947 487.3738 495.9764 496.4177 
#> 
#> Forecast errors:
#> ME: 2.963; MAE: 3.089; RMSE: 3.815
#> sCE: 15.642%; Asymmetry: 90.2%; sMAE: 1.359%; sMSE: 0.028%
#> MASE: 2.593; RMSSE: 2.487; rMAE: 0.996; rRMSE: 0.996

Given that both models are implemented in the same framework, they can be compared using information criteria.

The functionality of ADAM ARIMA is similar to the one of msarima function in smooth package, although there are several differences.

First, changing the distribution parameter will allow switching between additive / multiplicative models. For example, distribution="dlnorm" will create an ARIMA, equivalent to the one on logarithms of the data:

testModel <- adam(AirPassengers, "NNN", silent=FALSE, lags=c(1,12),
                  orders=list(ar=c(1,1),i=c(1,1),ma=c(2,2)), distribution="dlnorm",
                  h=12, holdout=TRUE)
testModel
#> Time elapsed: 0.14 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,2)[12]
#> With backcasting initialisation
#> Distribution assumed in the model: Log-Normal
#> Loss function type: likelihood; Loss function value: 483.4464
#> ARMA parameters of the model:
#>         Lag 1  Lag 12
#> AR(1) -0.8424 -0.7557
#>         Lag 1 Lag 12
#> MA(1)  0.6089 0.7222
#> MA(2) -0.0926 0.1492
#> 
#> Sample size: 132
#> Number of estimated parameters: 7
#> Number of degrees of freedom: 125
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  980.8928  981.7960 1001.0724 1003.2775 
#> 
#> Forecast errors:
#> ME: -27.035; MAE: 27.035; RMSE: 31.66
#> sCE: -123.594%; Asymmetry: -100%; sMAE: 10.3%; sMSE: 1.455%
#> MASE: 1.123; RMSSE: 1.01; rMAE: 0.356; rRMSE: 0.307

Second, if you want the model with intercept / drift, you can do it using constant parameter:

testModel <- adam(AirPassengers, "NNN", silent=FALSE, lags=c(1,12), constant=TRUE,
                  orders=list(ar=c(1,1),i=c(1,1),ma=c(2,2)), distribution="dnorm",
                  h=12, holdout=TRUE)
testModel
#> Time elapsed: 0.09 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,2)[12] with drift
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 483.6668
#> Intercept/Drift value: 1.5392
#> ARMA parameters of the model:
#>         Lag 1  Lag 12
#> AR(1) -0.5629 -0.9134
#>         Lag 1 Lag 12
#> MA(1)  0.2511 0.8799
#> MA(2) -0.0678 0.1041
#> 
#> Sample size: 132
#> Number of estimated parameters: 8
#> Number of degrees of freedom: 124
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  983.3335  984.5042 1006.3959 1009.2542 
#> 
#> Forecast errors:
#> ME: -12.771; MAE: 16.518; RMSE: 21.337
#> sCE: -58.384%; Asymmetry: -74.3%; sMAE: 6.293%; sMSE: 0.661%
#> MASE: 0.686; RMSSE: 0.681; rMAE: 0.217; rRMSE: 0.207

If the model contains non-zero differences, then the constant acts as a drift. Third, you can specify parameters of ARIMA via the arma parameter in the following manner:

testModel <- adam(AirPassengers, "NNN", silent=FALSE, lags=c(1,12),
                  orders=list(ar=c(1,1),i=c(1,1),ma=c(2,2)), distribution="dnorm",
                  arma=list(ar=c(0.1,0.1), ma=c(-0.96, 0.03, -0.12, 0.03)),
                  h=12, holdout=TRUE)
testModel
#> Time elapsed: 0 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,2)[12]
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 564.21
#> ARMA parameters of the model:
#>       Lag 1 Lag 12
#> AR(1)   0.1    0.1
#>       Lag 1 Lag 12
#> MA(1) -0.96  -0.12
#> MA(2)  0.03   0.03
#> 
#> Sample size: 132
#> Number of estimated parameters: 1
#> Number of degrees of freedom: 131
#> Number of provided parameters: 6
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1130.420 1130.451 1133.303 1133.378 
#> 
#> Forecast errors:
#> ME: 9.579; MAE: 17.084; RMSE: 19.15
#> sCE: 43.789%; Asymmetry: 61.9%; sMAE: 6.508%; sMSE: 0.532%
#> MASE: 0.709; RMSSE: 0.611; rMAE: 0.225; rRMSE: 0.186

Finally, the initials for the states can also be provided, although getting the correct ones might be a challenging task (you also need to know how many of them to provide; checking testModel$initial might help):

testModel <- adam(AirPassengers, "NNN", silent=FALSE, lags=c(1,12),
                  orders=list(ar=c(1,1),i=c(1,1),ma=c(2,0)), distribution="dnorm",
                  initial=list(arima=AirPassengers[1:24]),
                  h=12, holdout=TRUE)
testModel
#> Time elapsed: 0.05 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,0)[12]
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 494.9289
#> ARMA parameters of the model:
#>         Lag 1  Lag 12
#> AR(1) -0.4837 -0.0502
#>        Lag 1
#> MA(1) 0.3488
#> MA(2) 0.0964
#> 
#> Sample size: 132
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 127
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  999.8578 1000.3340 1014.2718 1015.4344 
#> 
#> Forecast errors:
#> ME: -18.06; MAE: 19.481; RMSE: 24.594
#> sCE: -82.564%; Asymmetry: -91.9%; sMAE: 7.422%; sMSE: 0.878%
#> MASE: 0.809; RMSSE: 0.785; rMAE: 0.256; rRMSE: 0.239

If you work with ADAM ARIMA model, then there is no such thing as “usual” bounds for the parameters, so the function will use the bounds="admissible", checking the AR / MA polynomials in order to make sure that the model is stationary and invertible (aka stable).

Similarly to ETS, you can use different distributions and losses for the estimation. Note that the order selection for ARIMA is done in auto.adam() function, not in the adam()! However, if you do orders=list(..., select=TRUE) in adam(), it will call auto.adam() and do the selection.

Finally, ARIMA is typically slower than ETS, mainly because its initial states are more difficult to estimate due to an increased complexity of the model. If you want to speed things up, use initial="backcasting" and reduce the number of iterations via maxeval parameter.

ADAM ETSX / ARIMAX / ETSX+ARIMA

Another important feature of ADAM is introduction of explanatory variables. Unlike in es(), adam() expects a matrix for data and can work with a formula. If the latter is not provided, then it will use all explanatory variables. Here is a brief example:

BJData <- cbind(BJsales,BJsales.lead)
testModel <- adam(BJData, "AAN", h=18, silent=FALSE)

If you work with data.frame or similar structures, then you can use them directly, ADAM will extract the response variable either assuming that it is in the first column or from the provided formula (if you specify one via formula parameter). Here is an example, where we create a matrix with lags and leads of an explanatory variable:

BJData <- cbind(as.data.frame(BJsales),as.data.frame(xregExpander(BJsales.lead,c(-7:7))))
colnames(BJData)[1] <- "y"
testModel <- adam(BJData, "ANN", h=18, silent=FALSE, holdout=TRUE, formula=y~xLag1+xLag2+xLag3)
testModel
#> Time elapsed: 0.08 seconds
#> Model estimated using adam() function: ETSX(ANN)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 197.2532
#> Persistence vector g (excluding xreg):
#> alpha 
#>     1 
#> 
#> Sample size: 132
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 127
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 404.5064 404.9826 418.9204 420.0830 
#> 
#> Forecast errors:
#> ME: 1.184; MAE: 1.62; RMSE: 2.25
#> sCE: 9.432%; Asymmetry: 50.8%; sMAE: 0.717%; sMSE: 0.01%
#> MASE: 1.328; RMSSE: 1.44; rMAE: 0.724; rRMSE: 0.897

Similarly to es(), there is a support for variables selection, but via the regressors parameter instead of xregDo, which will then use stepwise() function from greybox package on the residuals of the model:

testModel <- adam(BJData, "ANN", h=18, silent=FALSE, holdout=TRUE, regressors="select")

The same functionality is supported with ARIMA, so you can have, for example, ARIMAX(0,1,1), which is equivalent to ETSX(A,N,N):

testModel <- adam(BJData, "NNN", h=18, silent=FALSE, holdout=TRUE, regressors="select", orders=c(0,1,1))

The two models might differ because they have different initialisation in the optimiser and different bounds for parameters (ARIMA relies on invertibility condition, while ETS does the usual (0,1) bounds by default). It is possible to make them identical if the number of iterations is increased and the initial parameters are the same. Here is an example of what happens, when the two models have exactly the same parameters:

BJData <- BJData[,c("y",names(testModel$initial$xreg))];
testModel <- adam(BJData, "NNN", h=18, silent=TRUE, holdout=TRUE, orders=c(0,1,1),
                  initial=testModel$initial, arma=testModel$arma)
testModel
#> Time elapsed: 0 seconds
#> Model estimated using adam() function: ARIMAX(0,1,1)
#> With provided initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 426.7634
#> ARMA parameters of the model:
#>       Lag 1
#> MA(1) 0.259
#> 
#> Sample size: 132
#> Number of estimated parameters: 1
#> Number of degrees of freedom: 131
#> Number of provided parameters: 7
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 855.5268 855.5575 858.4096 858.4847 
#> 
#> Forecast errors:
#> ME: 0.642; MAE: 0.642; RMSE: 0.881
#> sCE: 5.111%; Asymmetry: 100%; sMAE: 0.284%; sMSE: 0.002%
#> MASE: 0.526; RMSSE: 0.564; rMAE: 0.287; rRMSE: 0.351
names(testModel$initial)[1] <- names(testModel$initial)[[1]] <- "level"
testModel2 <- adam(BJData, "ANN", h=18, silent=TRUE, holdout=TRUE,
                   initial=testModel$initial, persistence=testModel$arma$ma+1)
testModel2
#> Time elapsed: 0 seconds
#> Model estimated using adam() function: ETSX(ANN)
#> With provided initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 426.7634
#> Persistence vector g (excluding xreg):
#> alpha 
#> 1.259 
#> 
#> Sample size: 132
#> Number of estimated parameters: 1
#> Number of degrees of freedom: 131
#> Number of provided parameters: 7
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 855.5268 855.5575 858.4096 858.4847 
#> 
#> Forecast errors:
#> ME: 0.642; MAE: 0.642; RMSE: 0.881
#> sCE: 5.111%; Asymmetry: 100%; sMAE: 0.284%; sMSE: 0.002%
#> MASE: 0.526; RMSSE: 0.564; rMAE: 0.287; rRMSE: 0.351

Another feature of ADAM is the time varying parameters in the SSOE framework, which can be switched on via regressors="adapt":

testModel <- adam(BJData, "ANN", h=18, silent=FALSE, holdout=TRUE, regressors="adapt")
testModel$persistence
#>     alpha    delta1    delta2    delta3    delta4    delta5 
#> 0.2348972 0.1172567 0.2088164 0.2735241 0.1517711 0.2727661

Note that the default number of iterations might not be sufficient in order to get close to the optimum of the function, so setting maxeval to something bigger might help. If you want to explore, why the optimisation stopped, you can provide print_level=41 parameter to the function, and it will print out the report from the optimiser. In the end, the default parameters are tuned in order to give a reasonable solution, but given the complexity of the model, they might not guarantee to give the best one all the time.

Finally, you can produce a mixture of ETS, ARIMA and regression, by using the respective parameters, like this:

testModel <- adam(BJData, "AAN", h=18, silent=FALSE, holdout=TRUE, orders=c(1,0,0))
summary(testModel)
#> 
#> Model estimated using adam() function: ETSX(AAN)+ARIMA(1,0,0)
#> Response variable: y
#> Distribution used in the estimation: Normal
#> Loss function type: likelihood; Loss function value: 61.6429
#> Coefficients:
#>         Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha     0.1782     0.1658     0.0000      0.5060  
#> beta      0.0660     0.4057     0.0000      0.1782  
#> phi1[1]   1.0000     0.0383     0.9242      1.0757 *
#> xLag3     4.7620     2.8975    -0.9734     10.4902  
#> xLag7     0.5970     2.9093    -5.1617      6.3484  
#> xLag4     3.5532     2.7022    -1.7956      8.8953  
#> xLag6     1.5637     2.7116    -3.8038      6.9243  
#> xLag5     2.3473     2.4221    -2.4471      7.1357  
#> 
#> Error standard deviation: 0.3982
#> Sample size: 132
#> Number of estimated parameters: 9
#> Number of degrees of freedom: 123
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 141.2859 142.7613 167.2311 170.8331

This might be handy, when you explore a high frequency data, want to add calendar events, apply ETS and add AR/MA errors to it.

Finally, if you estimate ETSX or ARIMAX model and want to speed things up, it is recommended to use initial="backcasting", which will then initialise dynamic part of the model via backcasting and use optimisation for the parameters of the explanatory variables:

testModel <- adam(BJData, "AAN", h=18, silent=TRUE, holdout=TRUE, initial="backcasting")
summary(testModel)
#> 
#> Model estimated using adam() function: ETSX(AAN)
#> Response variable: y
#> Distribution used in the estimation: Normal
#> Loss function type: likelihood; Loss function value: 47.1377
#> Coefficients:
#>       Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha   0.7468     0.0906     0.5675      0.9259 *
#> beta    0.5235     0.2835     0.0000      0.7468  
#> xLag3   4.5690     2.3333    -0.0493      9.1821  
#> xLag7   0.4004     2.3452    -4.2415      5.0370  
#> xLag4   3.1373     2.0680    -0.9558      7.2257  
#> xLag6   1.0500     2.0720    -3.0511      5.1465  
#> xLag5   1.8081     1.9696    -2.0903      5.7022  
#> 
#> Error standard deviation: 0.3554
#> Sample size: 132
#> Number of estimated parameters: 8
#> Number of degrees of freedom: 124
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 110.2754 111.4462 133.3378 136.1961

Auto ADAM

While the original adam() function allows selecting ETS components and explanatory variables, it does not allow selecting the most suitable distribution and / or ARIMA components. This is what auto.adam() function is for.

In order to do the selection of the most appropriate distribution, you need to provide a vector of those that you want to check:

testModel <- auto.adam(BJsales, "XXX", silent=FALSE,
                       distribution=c("dnorm","dlaplace","ds"),
                       h=12, holdout=TRUE)
#> Evaluating models with different distributions... dnorm ,  Selecting ARIMA orders... 
#> Selecting differences... 
#> Selecting ARMA... |
#> The best ARIMA is selected. dlaplace ,  Selecting ARIMA orders... 
#> Selecting differences... 
#> Selecting ARMA... |
#> The best ARIMA is selected. ds ,  Selecting ARIMA orders... 
#> Selecting differences... 
#> Selecting ARMA... |-
#> The best ARIMA is selected. Done!
testModel
#> Time elapsed: 0.43 seconds
#> Model estimated using auto.adam() function: ETS(AAdN)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 237.6128
#> Persistence vector g:
#>  alpha   beta 
#> 0.9448 0.2979 
#> Damping parameter: 0.8789
#> Sample size: 138
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 134
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 483.2257 483.5264 494.9347 495.6756 
#> 
#> Forecast errors:
#> ME: 2.817; MAE: 2.967; RMSE: 3.654
#> sCE: 14.869%; Asymmetry: 88%; sMAE: 1.305%; sMSE: 0.026%
#> MASE: 2.491; RMSSE: 2.382; rMAE: 0.957; rRMSE: 0.954

This process can also be done in parallel on either the automatically selected number of cores (e.g. parallel=TRUE) or on the specified by user (e.g. parallel=4):

testModel <- auto.adam(BJsales, "ZZZ", silent=FALSE, parallel=TRUE,
                       h=12, holdout=TRUE)

If you want to add ARIMA or regression components, you can do it in the exactly the same way as for the adam() function. Here is an example of ETS+ARIMA:

testModel <- auto.adam(BJsales, "AAN", orders=list(ar=2,i=0,ma=0), silent=TRUE,
                       distribution=c("dnorm","dlaplace","ds","dgnorm"),
                       h=12, holdout=TRUE)
testModel
#> Time elapsed: 0.18 seconds
#> Model estimated using auto.adam() function: ETS(AAN)+ARIMA(2,0,0)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 240.9035
#> Persistence vector g:
#>  alpha   beta 
#> 0.3161 0.1483 
#> 
#> ARMA parameters of the model:
#>        Lag 1
#> AR(1) 0.7715
#> AR(2) 0.2285
#> 
#> Sample size: 138
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 133
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 491.8070 492.2616 506.4433 507.5631 
#> 
#> Forecast errors:
#> ME: 2.872; MAE: 3.027; RMSE: 3.731
#> sCE: 15.159%; Asymmetry: 87.9%; sMAE: 1.332%; sMSE: 0.027%
#> MASE: 2.541; RMSSE: 2.432; rMAE: 0.976; rRMSE: 0.974

However, this way the function will just use ARIMA(2,0,0) and fit it together with ETS(A,A,N). If you want it to select the most appropriate ARIMA orders from the provided (e.g. up to AR(2), I(1) and MA(2)), you need to add parameter select=TRUE to the list in orders:

testModel <- auto.adam(BJsales, "XXN", orders=list(ar=2,i=2,ma=2,select=TRUE),
                       distribution="default", silent=FALSE,
                       h=12, holdout=TRUE)
#> Evaluating models with different distributions... default ,  Selecting ARIMA orders... 
#> Selecting differences... 
#> Selecting ARMA... |
#> The best ARIMA is selected. Done!
testModel
#> Time elapsed: 0.12 seconds
#> Model estimated using auto.adam() function: ETS(AAdN)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 237.6128
#> Persistence vector g:
#>  alpha   beta 
#> 0.9448 0.2979 
#> Damping parameter: 0.8789
#> Sample size: 138
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 134
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 483.2257 483.5264 494.9347 495.6756 
#> 
#> Forecast errors:
#> ME: 2.817; MAE: 2.967; RMSE: 3.654
#> sCE: 14.869%; Asymmetry: 88%; sMAE: 1.305%; sMSE: 0.026%
#> MASE: 2.491; RMSSE: 2.382; rMAE: 0.957; rRMSE: 0.954

Knowing how to work with adam(), you can use similar principles, when dealing with auto.adam(). Just keep in mind that the provided persistence, phi, initial, arma and B won’t work, because this contradicts the idea of the model selection.

Finally, there is also the mechanism of automatic outliers detection, which extracts residuals from the best model, flags observations that lie outside the prediction interval of the width level in sample and then refits auto.adam() with the dummy variables for the outliers. Here how it works:

testModel <- auto.adam(AirPassengers, "PPP", silent=FALSE, outliers="use",
                       distribution="default",
                       h=12, holdout=TRUE)
#> Evaluating models with different distributions... default ,  Selecting ARIMA orders... 
#> Selecting differences... 
#> Selecting ARMA... |-
#> The best ARIMA is selected. 
#> Dealing with outliers...
testModel
#> Time elapsed: 4.51 seconds
#> Model estimated using auto.adam() function: ETSX(MMM)+SARIMA(3,0,0)[1](1,0,0)[12]
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 454.5344
#> Persistence vector g (excluding xreg):
#>  alpha   beta  gamma 
#> 0.0311 0.0000 0.0013 
#> 
#> ARMA parameters of the model:
#>         Lag 1 Lag 12
#> AR(1)  0.6621 0.3252
#> AR(2)  0.1822     NA
#> AR(3) -0.0421     NA
#> 
#> Sample size: 132
#> Number of estimated parameters: 9
#> Number of degrees of freedom: 123
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 927.0688 928.5442 953.0141 956.6161 
#> 
#> Forecast errors:
#> ME: -23.706; MAE: 23.706; RMSE: 29.31
#> sCE: -108.375%; Asymmetry: -100%; sMAE: 9.031%; sMSE: 1.247%
#> MASE: 0.984; RMSSE: 0.935; rMAE: 0.312; rRMSE: 0.285

If you specify outliers="select", the function will create leads and lags 1 of the outliers and then select the most appropriate ones via the regressors parameter of adam.

If you want to know more about ADAM, you are welcome to visit the online monograph.

Hyndman, Rob J, Anne B Koehler, J Keith Ord, and Ralph D Snyder. 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.