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: 465.9968
#> Coefficients:
#>       Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha   0.7565     0.0849     0.5883      0.9243 *
#> beta    0.0004     0.0187     0.0000      0.0375  
#> gamma   0.0000     0.0249     0.0000      0.0493  
#> 
#> Error standard deviation: 0.0365
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  965.9937  971.3621 1015.0013 1028.1078
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: 0.06 seconds
#> Model estimated using adam() function: ETS(MMM)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 465.9968
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.7565 0.0004 0.0000 
#> 
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  965.9937  971.3621 1015.0013 1028.1078 
#> 
#> Forecast errors:
#> ME: -9.684; MAE: 15.503; RMSE: 22.946
#> sCE: -44.272%; Asymmetry: -59.6%; sMAE: 5.906%; 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: 5
#> Number of degrees of freedom: 133
#> 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: 0.22 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: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  968.2316  973.6000 1017.2392 1030.3457 
#> 
#> 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       415.4836         392.4216           388.1332          439.0966
#> Feb 1960       410.5861         381.6388           376.3010          440.4178
#> Mar 1960       473.2776         433.4701           426.1861          514.5452
#> Apr 1960       456.3104         414.7785           407.2092          499.4957
#> May 1960       457.0062         414.3117           406.5413          501.4472
#> Jun 1960       519.4562         468.8172           459.6226          572.2611
#> Jul 1960       577.4786         519.5108           509.0033          638.0027
#> Aug 1960       576.5520         517.5935           506.9181          638.1614
#> Sep 1960       503.4345         452.1745           442.8907          556.9890
#> Oct 1960       439.5129         394.3119           386.1303          486.7585
#> Nov 1960       383.8169         344.1891           337.0180          425.2446
#> Dec 1960       433.1707         387.7127           379.4947          480.7290
#> Jan 1961       443.7121         395.1189           386.3578          494.6532
#> Feb 1961       438.3248         388.4275           379.4542          490.7334
#> Mar 1961       505.0727         444.2697           433.3779          569.1217
#> Apr 1961       486.7949         426.0409           415.1867          550.9184
#> May 1961       487.3681         426.4390           415.5549          551.6826
#> Jun 1961       553.7772         485.5434           473.3413          625.7443
#>          Upper bound (97.5%)
#> Jan 1960            443.7520
#> Feb 1960            446.3447
#> Mar 1960            522.8013
#> Apr 1960            508.1661
#> May 1960            510.3807
#> Jun 1960            582.8979
#> Jul 1960            650.2125
#> Aug 1960            650.6021
#> Sep 1960            567.8007
#> Oct 1960            496.3015
#> Nov 1960            433.6141
#> Dec 1960            490.3454
#> Jan 1961            504.9777
#> Feb 1961            501.3786
#> Mar 1961            582.1748
#> Apr 1961            564.0161
#> May 1961            564.8207
#> Jun 1961            640.4323
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       415.4836          433.7696          439.0966          449.2058
#> Feb 1960       410.5861          433.6496          440.4178          453.3020
#> Mar 1960       473.2776          505.1340          514.5452          532.5105
#> Apr 1960       456.3104          489.6214          499.4957          518.3716
#> May 1960       457.0062          491.2766          501.4472          520.8993
#> Jun 1960       519.4562          560.1578          572.2611          595.4286
#> Jul 1960       577.4786          624.1150          638.0027          664.6017
#> Aug 1960       576.5520          624.0146          638.1614          665.2669
#> Sep 1960       503.4345          544.6938          556.9890          580.5446
#> Oct 1960       439.5129          475.9075          486.7585          507.5516
#> Nov 1960       383.8169          415.7283          425.2446          443.4814
#> Dec 1960       433.1707          469.7975          480.7290          501.6851
#> Jan 1961       443.7121          482.9240          494.6532          517.1596
#> Feb 1961       438.3248          478.6466          490.7334          513.9462
#> Mar 1961       505.0727          554.3138          569.1217          597.5980
#> Apr 1961       486.7949          536.0686          550.9184          579.5008
#> May 1961       487.3681          536.7874          551.6826          580.3537
#> Jun 1961       553.7772          609.0880          625.7443          657.7935

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: 1.35 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 0.1825 0.2655 
#> Damping parameter: 0.929
#> Sample size: 3024
#> Number of estimated parameters: 343
#> Number of degrees of freedom: 2681
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39819.04 39907.09 41881.96 42234.80 
#> 
#> Forecast errors:
#> ME: 68.547; MAE: 142.003; RMSE: 182.897
#> sCE: 75.991%; Asymmetry: 49.8%; sMAE: 0.469%; 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: 1.44 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 0.1825 0.2655 
#> Damping parameter: 0.929
#> Sample size: 3024
#> Number of estimated parameters: 343
#> Number of degrees of freedom: 2681
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39819.04 39907.09 41881.96 42234.80 
#> 
#> Forecast errors:
#> ME: 68.547; MAE: 142.003; RMSE: 182.897
#> sCE: 75.991%; 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: 1.26 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 0.2658 
#> Damping parameter: 0
#> Sample size: 3024
#> Number of estimated parameters: 343
#> Number of degrees of freedom: 2681
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39819.02 39907.08 41881.94 42234.79 
#> 
#> Forecast errors:
#> ME: 68.53; MAE: 141.99; RMSE: 182.884
#> sCE: 75.972%; 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: 1.52 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: 54669.48
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.1030 0.1000 0.3052 0.3926 
#> Damping parameter: 0.9521
#> Sample size: 3024
#> Number of estimated parameters: 342
#> Number of degrees of freedom: 2682
#> Number of provided parameters: 1
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 110023.0 110110.5 112079.9 112430.5 
#> 
#> Forecast errors:
#> ME: -1.68600726586794e+27; MAE: 1.68600726586794e+27; RMSE: 1.50104309841494e+28
#> sCE: -1.86912029581763e+27%; Asymmetry: -100%; sMAE: 5.56285802326677e+24%; sMSE: 2.4528070854763e+49%
#> MASE: 2.27044251354004e+24; RMSSE: 1.4598008987814e+25; rMAE: 2.63522015435209e+23; rRMSE: 1.9041929669253e+24

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: 0.04 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: 14.6638
#> Persistence vector g:
#> alpha 
#>     0 
#> 
#> Sample size: 108
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 103
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 201.9237 202.1544 215.3343 206.5103 
#> 
#> 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: 0.79 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 17.3587
#> Average number of degrees of freedom: 114.6413
#> 
#> Forecast errors:
#> ME: 0.017; MAE: 15.691; RMSE: 22.444
#> sCE: 0.08%; sMAE: 5.978%; sMSE: 0.731%
#> MASE: 0.652; RMSSE: 0.716; rMAE: 0.206; rRMSE: 0.218
"es():"
#> [1] "es():"
esModel
#> Time elapsed: 0.66 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 17.1424
#> Average number of degrees of freedom: 114.8576
#> 
#> Forecast errors:
#> ME: -4.837; MAE: 15.417; RMSE: 22.04
#> sCE: -22.111%; sMAE: 5.873%; sMSE: 0.705%
#> 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: 240.5926
#> ARMA parameters of the model:
#>         Lag 1
#> MA(1) -0.7484
#> MA(2) -0.0149
#> 
#> Sample size: 138
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 133
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 491.1852 491.6397 505.8214 506.9413 
#> 
#> Forecast errors:
#> ME: 2.959; MAE: 3.085; RMSE: 3.81
#> sCE: 15.621%; Asymmetry: 90.2%; sMAE: 1.357%; sMSE: 0.028%
#> MASE: 2.59; RMSSE: 2.483; rMAE: 0.995; rRMSE: 0.994

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: 0.1 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: 458.6913
#> ARMA parameters of the model:
#>        Lag 1 Lag 12
#> AR(1) 0.2298 0.9995
#>         Lag 1  Lag 12
#> MA(1) -0.6584 -1.6001
#> MA(2) -0.0246  0.6505
#> 
#> Sample size: 132
#> Number of estimated parameters: 33
#> Number of degrees of freedom: 99
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  983.3826 1006.2806 1078.5151 1134.4182 
#> 
#> Forecast errors:
#> ME: -6.048; MAE: 12.125; RMSE: 15.222
#> sCE: -27.65%; Asymmetry: -48.6%; sMAE: 4.619%; sMSE: 0.336%
#> MASE: 0.503; RMSSE: 0.486; rMAE: 0.16; rRMSE: 0.148

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: 0.08 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: 469.7506
#> Intercept/Drift value: -0.0307
#> ARMA parameters of the model:
#>         Lag 1 Lag 12
#> AR(1) -0.6666 0.9999
#>         Lag 1  Lag 12
#> MA(1)  0.2238 -1.3737
#> MA(2) -0.2418  0.4423
#> 
#> Sample size: 132
#> Number of estimated parameters: 34
#> Number of degrees of freedom: 98
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1007.501 1032.037 1105.516 1165.419 
#> 
#> Forecast errors:
#> ME: -4.067; MAE: 10.469; RMSE: 14.357
#> sCE: -18.594%; Asymmetry: -35.4%; sMAE: 3.988%; sMSE: 0.299%
#> MASE: 0.435; RMSSE: 0.458; rMAE: 0.138; rRMSE: 0.139

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: 0.01 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: 534.8776
#> 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: 27
#> Number of degrees of freedom: 105
#> Number of provided parameters: 6
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1123.755 1138.294 1201.591 1237.085 
#> 
#> Forecast errors:
#> ME: 9.575; MAE: 17.082; RMSE: 19.148
#> sCE: 43.773%; 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: 488.888
#> ARMA parameters of the model:
#>         Lag 1  Lag 12
#> AR(1) -0.5068 -0.1066
#>        Lag 1
#> MA(1) 0.2985
#> MA(2) 0.0201
#> 
#> Sample size: 132
#> Number of estimated parameters: 31
#> Number of degrees of freedom: 101
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1039.776 1059.616 1129.143 1177.580 
#> 
#> Forecast errors:
#> ME: -13.557; MAE: 16.468; RMSE: 21.411
#> sCE: -61.975%; Asymmetry: -79.5%; sMAE: 6.274%; sMSE: 0.665%
#> MASE: 0.684; RMSSE: 0.683; rMAE: 0.217; rRMSE: 0.208

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: 0.09 seconds
#> Model estimated using adam() function: ETSX(ANN)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 197.1386
#> Persistence vector g (excluding xreg):
#> alpha 
#>     1 
#> 
#> Sample size: 132
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 126
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 406.2772 406.9492 423.5740 425.2146 
#> 
#> Forecast errors:
#> ME: 1.182; MAE: 1.618; RMSE: 2.247
#> sCE: 9.413%; Asymmetry: 50.7%; sMAE: 0.716%; sMSE: 0.01%
#> MASE: 1.326; RMSSE: 1.438; rMAE: 0.723; rRMSE: 0.896

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: 62.5902
#> ARMA parameters of the model:
#>        Lag 1
#> MA(1) 0.2401
#> 
#> 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 
#> 127.1804 127.2111 130.0632 130.1383 
#> 
#> Forecast errors:
#> ME: 0.636; MAE: 0.636; RMSE: 0.872
#> sCE: 5.063%; Asymmetry: 100%; sMAE: 0.281%; sMSE: 0.001%
#> MASE: 0.521; RMSSE: 0.558; rMAE: 0.284; rRMSE: 0.347
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: 62.5902
#> Persistence vector g (excluding xreg):
#>  alpha 
#> 1.2401 
#> 
#> 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 
#> 127.1804 127.2111 130.0632 130.1383 
#> 
#> Forecast errors:
#> ME: 0.636; MAE: 0.636; RMSE: 0.872
#> sCE: 5.063%; Asymmetry: 100%; sMAE: 0.281%; sMSE: 0.001%
#> MASE: 0.521; RMSSE: 0.558; rMAE: 0.284; rRMSE: 0.347

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 
#> 0.4868570 0.1609712 0.1191716 0.1594353 0.2062682 0.1044465

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: 57.9711
#> Coefficients:
#>         Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha     0.1746     0.1725     0.0000      0.5156  
#> beta      0.0743     0.4345     0.0000      0.1746  
#> phi1[1]   1.0000     0.0409     0.9190      1.0000 *
#> xLag3     4.7560     2.9209    -1.0272     10.5294  
#> xLag7     0.6129     2.9238    -5.1759      6.3920  
#> xLag4     3.5349     2.7253    -1.8610      8.9217  
#> xLag6     1.5622     2.7226    -3.8284      6.9436  
#> xLag5     2.3236     2.4342    -2.4960      7.1350  
#> 
#> Error standard deviation: 0.3921
#> Sample size: 132
#> Number of estimated parameters: 12
#> Number of degrees of freedom: 120
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 139.9421 142.5640 174.5358 180.9367

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: 46.091
#> Coefficients:
#>       Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha   0.7513     0.0918     0.5695      0.9328 *
#> beta    0.5038     0.2949     0.0000      0.7513  
#> xLag3   4.5727     2.3827    -0.1441      9.2829  
#> xLag7   0.4112     2.3897    -4.3194      5.1352  
#> xLag4   3.1462     2.1168    -1.0443      7.3308  
#> xLag6   1.0647     2.1153    -3.1227      5.2463  
#> xLag5   1.8214     2.0149    -2.1673      5.8044  
#> 
#> Error standard deviation: 0.3554
#> Sample size: 132
#> Number of estimated parameters: 10
#> Number of degrees of freedom: 122
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 112.1821 114.0003 141.0101 145.4490

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: 0.38 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: 237.6167
#> Persistence vector g:
#>  alpha   beta 
#> 0.9451 0.2974 
#> Damping parameter: 0.8791
#> Sample size: 138
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 132
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 487.2334 487.8746 504.7969 506.3766 
#> 
#> Forecast errors:
#> ME: 2.817; MAE: 2.968; RMSE: 3.654
#> sCE: 14.872%; 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: 0.16 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: 241.2479
#> Persistence vector g:
#> alpha  beta 
#>     0     0 
#> 
#> ARMA parameters of the model:
#>         Lag 1
#> AR(1)  1.3027
#> AR(2) -0.3186
#> 
#> Sample size: 138
#> Number of estimated parameters: 9
#> Number of degrees of freedom: 129
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 500.4957 501.9020 526.8410 530.3055 
#> 
#> Forecast errors:
#> ME: -0.159; MAE: 1.099; RMSE: 1.26
#> sCE: -0.842%; Asymmetry: -29%; sMAE: 0.484%; sMSE: 0.003%
#> MASE: 0.923; RMSSE: 0.821; rMAE: 0.355; rRMSE: 0.329

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: 0.11 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: 237.6167
#> Persistence vector g:
#>  alpha   beta 
#> 0.9451 0.2974 
#> Damping parameter: 0.8791
#> Sample size: 138
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 132
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 487.2334 487.8746 504.7969 506.3766 
#> 
#> Forecast errors:
#> ME: 2.817; MAE: 2.968; RMSE: 3.654
#> sCE: 14.872%; 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: 2.37 seconds
#> Model estimated using auto.adam() function: ETSX(MMM)
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 464.8552
#> Persistence vector g (excluding xreg):
#>  alpha   beta  gamma 
#> 0.7618 0.0000 0.0000 
#> 
#> Sample size: 132
#> Number of estimated parameters: 18
#> Number of degrees of freedom: 114
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  965.7104  971.7635 1017.6008 1032.3788 
#> 
#> Forecast errors:
#> ME: -9.845; MAE: 15.551; RMSE: 22.996
#> sCE: -45.008%; Asymmetry: -61.9%; sMAE: 5.924%; sMSE: 0.767%
#> MASE: 0.646; RMSSE: 0.734; rMAE: 0.205; rRMSE: 0.223

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.