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:

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: 473.8895
#> Coefficients:
#>       Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha   0.6172     0.0917     0.4358      0.7986 *
#> beta    0.0000     0.0186     0.0000      0.0368  
#> gamma   0.1905     0.0656     0.0607      0.3201 *
#> 
#> Error standard deviation: 0.0367
#> Sample size: 132
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 128
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 955.7791 956.0940 967.3103 968.0792
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: 0.05 seconds
#> Model estimated using adam() function: ETS(MMM)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 473.8895
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.6172 0.0000 0.1905 
#> 
#> Sample size: 132
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 128
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 955.7791 956.0940 967.3103 968.0792 
#> 
#> Forecast errors:
#> ME: -17.336; MAE: 18.16; RMSE: 25.138
#> sCE: -79.251%; Asymmetry: -90.7%; sMAE: 6.918%; sMSE: 0.917%
#> MASE: 0.754; RMSSE: 0.802; rMAE: 0.239; rRMSE: 0.244

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: 0.19 seconds
#> Model estimated using adam() function: ETS(MAM)
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 475.1973
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.6496 0.0000 0.2382 
#> 
#> Sample size: 132
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 128
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 958.3947 958.7096 969.9259 970.6948 
#> 
#> Forecast errors:
#> ME: -2.32; MAE: 16.219; RMSE: 21.671
#> sCE: -10.608%; Asymmetry: -6.2%; sMAE: 6.179%; sMSE: 0.682%
#> MASE: 0.673; RMSSE: 0.692; rMAE: 0.213; rRMSE: 0.21

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       421.3869         412.5346           410.8574          430.3180
#> Feb 1960       404.2151         378.8297           374.1284          430.2890
#> Mar 1960       478.2432         440.4960           433.5686          517.2865
#> Apr 1960       469.4753         426.7247           418.9335          513.9292
#> May 1960       483.3768         436.9943           428.5652          531.7112
#> Jun 1960       548.3281         495.2308           485.5863          603.6809
#> Jul 1960       605.2671         545.4208           534.5633          667.7113
#> Aug 1960       592.0438         532.2679           521.4364          654.4720
#> Sep 1960       492.9109         442.5413           433.4207          545.5438
#> Oct 1960       435.5918         390.9739           382.8960          482.2194
#> Nov 1960       383.7125         344.3754           337.2539          424.8229
#> Dec 1960       429.4814         385.5892           377.6416          475.3461
#> Jan 1961       446.7925         400.8209           392.5002          494.8448
#> Feb 1961       428.4600         381.0510           372.5092          478.1851
#> Mar 1961       506.7828         446.4769           435.6652          570.2693
#> Apr 1961       497.3506         435.1430           424.0310          563.0165
#> May 1961       511.9422         446.5961           434.9416          581.0002
#> Jun 1961       580.5781         508.4246           495.5299          656.7161
#>          Upper bound (97.5%)
#> Jan 1960            432.0478
#> Feb 1960            435.4488
#> Mar 1960            525.0770
#> Apr 1960            522.8546
#> May 1960            541.4400
#> Jun 1960            614.8273
#> Jul 1960            680.2989
#> Aug 1960            667.0696
#> Sep 1960            556.1714
#> Oct 1960            491.6356
#> Nov 1960            433.1253
#> Dec 1960            484.6072
#> Jan 1961            504.5510
#> Feb 1961            488.2690
#> Mar 1961            583.1988
#> Apr 1961            576.4312
#> May 1961            595.1262
#> Jun 1961            672.2639
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       421.3869          428.3295          430.3180          434.0648
#> Feb 1960       404.2151          424.3906          430.2890          441.4995
#> Mar 1960       478.2432          508.4000          517.2865          534.2323
#> Apr 1960       469.4753          503.7647          513.9292          533.3604
#> May 1960       483.3768          520.6389          531.7112          552.8988
#> Jun 1960       548.3281          590.9966          603.6809          627.9573
#> Jul 1960       605.2671          653.3911          667.7113          695.1304
#> Aug 1960       592.0438          640.1441          654.4720          681.9171
#> Sep 1960       492.9109          533.4585          545.5438          568.6990
#> Oct 1960       435.5918          471.5120          482.2194          502.7356
#> Nov 1960       383.7125          415.3822          424.8229          442.9125
#> Dec 1960       429.4814          464.8148          475.3461          495.5239
#> Jan 1961       446.7925          483.8083          494.8448          515.9936
#> Feb 1961       428.4600          466.7308          478.1851          500.1689
#> Mar 1961       506.7828          555.5991          570.2693          598.4731
#> Apr 1961       497.3506          547.8079          563.0165          592.2912
#> May 1961       511.9422          564.9905          581.0002          611.8328
#> Jun 1961       580.5781          639.0873          656.7161          690.6440

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: 0.97 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: 19575.11
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0186 0.0000 0.1810 0.2721 
#> Damping parameter: 0.8881
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39162.22 39162.25 39198.31 39198.42 
#> 
#> Forecast errors:
#> ME: 80.806; MAE: 144.186; RMSE: 185.09
#> sCE: 89.583%; Asymmetry: 57.6%; sMAE: 0.476%; sMSE: 0.004%
#> MASE: 0.194; RMSSE: 0.18; rMAE: 0.023; 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: 0.9 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: 19575.11
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0186 0.0000 0.1810 0.2721 
#> Damping parameter: 0.8881
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39162.22 39162.25 39198.31 39198.42 
#> 
#> Forecast errors:
#> ME: 80.806; MAE: 144.186; RMSE: 185.09
#> sCE: 89.583%; Asymmetry: 57.6%; sMAE: 0.476%; sMSE: 0.004%
#> MASE: 0.194; RMSSE: 0.18; rMAE: 0.023; 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: 1.23 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: 19574.32
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0185 0.0000 0.1813 0.2718 
#> Damping parameter: 0.9997
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39160.64 39160.66 39196.72 39196.83 
#> 
#> Forecast errors:
#> ME: 85.337; MAE: 146.164; RMSE: 187.409
#> sCE: 94.605%; Asymmetry: 60.2%; sMAE: 0.482%; sMSE: 0.004%
#> MASE: 0.197; RMSSE: 0.182; rMAE: 0.023; rRMSE: 0.024

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: 0.99 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: 55569.28
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.1023 0.1000 0.3078 0.5031 
#> Damping parameter: 0.9512
#> 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 
#> 111148.6 111148.6 111178.6 111178.7 
#> 
#> Forecast errors:
#> ME: -8.35383527144936e+39; MAE: 8.35383527144936e+39; RMSE: 1.52759937174104e+41
#> sCE: -9.26112441499184e+39%; Asymmetry: -100%; sMAE: 2.75628702827138e+37%; sMSE: 2.54036435427557e+75%
#> MASE: 1.12495972795497e+37; RMSSE: 1.48562752008939e+38; rMAE: 1.30569989341819e+36; rRMSE: 1.9378817190662e+37

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: 4
#> Number of degrees of freedom: 104
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 192.8464 192.9607 203.5749 194.4782 
#> 
#> Forecast errors:
#> Asymmetry: -1.833%; sMSE: 28.466%; rRMSE: 0.799; sPIS: -18.182%; sCE: 90.909%

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: 0.82 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 5.8562
#> Average number of degrees of freedom: 126.1438
#> 
#> Forecast errors:
#> ME: -17.536; MAE: 18.405; RMSE: 24.343
#> sCE: -80.169%; sMAE: 7.012%; sMSE: 0.86%
#> MASE: 0.764; RMSSE: 0.777; rMAE: 0.242; rRMSE: 0.236
"es():"
#> [1] "es():"
esModel
#> Time elapsed: 0.8 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 5.9289
#> Average number of degrees of freedom: 126.0711
#> 
#> Forecast errors:
#> ME: -14.557; MAE: 16.464; RMSE: 23.474
#> sCE: -66.548%; sMAE: 6.272%; sMSE: 0.8%
#> MASE: 0.684; RMSSE: 0.749; rMAE: 0.217; rRMSE: 0.228

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: 0.12 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: 483.6894
#> Intercept/Drift value: 1.5054
#> ARMA parameters of the model:
#>         Lag 1  Lag 12
#> AR(1) -0.5535 -0.9112
#>         Lag 1 Lag 12
#> MA(1)  0.2497 0.8703
#> MA(2) -0.0640 0.0992
#> 
#> Sample size: 132
#> Number of estimated parameters: 8
#> Number of degrees of freedom: 124
#> Information criteria:
#>       AIC      AICc       BIC      BICc 
#>  983.3788  984.5495 1006.4412 1009.2994 
#> 
#> Forecast errors:
#> ME: -12.561; MAE: 16.41; RMSE: 21.183
#> sCE: -57.422%; Asymmetry: -73.6%; sMAE: 6.252%; sMSE: 0.651%
#> MASE: 0.681; RMSSE: 0.676; rMAE: 0.216; rRMSE: 0.206

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: 0.06 seconds
#> Model estimated using adam() function: ETSX(ANN)
#> With backcasting initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 197.237
#> 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 
#> 404.4740 404.9502 418.8880 420.0506 
#> 
#> Forecast errors:
#> ME: 1.184; MAE: 1.619; RMSE: 2.249
#> sCE: 9.431%; Asymmetry: 50.8%; sMAE: 0.717%; sMSE: 0.01%
#> MASE: 1.328; RMSSE: 1.439; 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: 425.9002
#> ARMA parameters of the model:
#>        Lag 1
#> MA(1) 0.2556
#> 
#> 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 
#> 853.8004 853.8312 856.6832 856.7583 
#> 
#> Forecast errors:
#> ME: 0.64; MAE: 0.64; RMSE: 0.88
#> sCE: 5.102%; Asymmetry: 100%; sMAE: 0.283%; sMSE: 0.002%
#> MASE: 0.525; RMSSE: 0.563; rMAE: 0.286; 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: 425.9002
#> Persistence vector g (excluding xreg):
#>  alpha 
#> 1.2556 
#> 
#> 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 
#> 853.8004 853.8312 856.6832 856.7583 
#> 
#> Forecast errors:
#> ME: 0.64; MAE: 0.64; RMSE: 0.88
#> sCE: 5.102%; Asymmetry: 100%; sMAE: 0.283%; sMSE: 0.002%
#> MASE: 0.525; RMSSE: 0.563; rMAE: 0.286; 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 
#> 0.418436536 0.178429452 0.002808931 0.136392363 0.295716981 0.203650154

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: 60.753
#> Coefficients:
#>         Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha     0.2379     0.1651     0.0000      0.5643  
#> beta      0.0854     0.3845     0.0000      0.2379  
#> phi1[1]   0.9999     0.0474     0.9061      1.0936 *
#> xLag3     4.7444     3.0218    -1.2370     10.7182  
#> xLag7     0.5509     3.0319    -5.4506      6.5448  
#> xLag4     3.5020     2.8612    -2.1614      9.1583  
#> xLag6     1.4968     2.8703    -4.1848      7.1712  
#> xLag5     2.2980     2.5247    -2.6995      7.2892  
#> 
#> Error standard deviation: 0.3956
#> Sample size: 132
#> Number of estimated parameters: 9
#> Number of degrees of freedom: 123
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 139.5060 140.9814 165.4512 169.0533

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: 46.9657
#> Coefficients:
#>       Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha   0.7473     0.0908     0.5676      0.9268 *
#> beta    0.5206     0.2861     0.0000      0.7473  
#> xLag3   4.5697     2.4800    -0.3389      9.4727  
#> xLag7   0.4021     2.4920    -4.5302      5.3289  
#> xLag4   3.1388     2.1988    -1.2131      7.4859  
#> xLag6   1.0523     2.2024    -3.3068      5.4065  
#> xLag5   1.8102     2.0943    -2.3349      5.9507  
#> 
#> Error standard deviation: 0.3549
#> Sample size: 132
#> Number of estimated parameters: 8
#> Number of degrees of freedom: 124
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 109.9315 111.1022 132.9939 135.8521

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: 237.6128
#> 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: 0.17 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: 0.11 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: 237.6128
#> 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: 3.93 seconds
#> Model estimated using auto.adam() function: ETSX(MMM)+SARIMA(2,0,0)[12]
#> With backcasting initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 462.403
#> Persistence vector g (excluding xreg):
#>  alpha   beta  gamma 
#> 0.4120 0.0000 0.2028 
#> 
#> ARMA parameters of the model:
#>       Lag 12
#> AR(1) 0.2484
#> AR(2) 0.4580
#> 
#> Sample size: 132
#> Number of estimated parameters: 7
#> Number of degrees of freedom: 125
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 938.8059 939.7092 958.9855 961.1907 
#> 
#> Forecast errors:
#> ME: -21.218; MAE: 21.218; RMSE: 26.231
#> sCE: -97.001%; Asymmetry: -100%; sMAE: 8.083%; sMSE: 0.999%
#> MASE: 0.881; RMSSE: 0.837; rMAE: 0.279; rRMSE: 0.255

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.

Augmented Dynamic Adaptive Model

Ivan Svetunkov

2026-02-05

ADAM ETS

ADAM ARIMA

ADAM ETSX / ARIMAX / ETSX+ARIMA

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