Augmented Dynamic Adaptive Model

Ivan Svetunkov

2021-06-13

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 textbook.

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(Mcomp)
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 a N2568 data from M3 competition (if you provide an Mcomp object, adam() will automatically set the train and test sets, the forecast horizon and even the needed lags):

testModel <- adam(M3[[2568]], "MMM", lags=c(1,12), distribution="dnorm")
summary(testModel)
#> 
#> Model estimated using adam() function: ETS(MMM)
#> Response variable: M3..2568..
#> Distribution used in the estimation: Normal
#> Loss function type: likelihood; Loss function value: 869.9558
#> Coefficients:
#>              Estimate Std. Error Lower 2.5% Upper 97.5%  
#> alpha          0.0822     0.0225     0.0375      0.1267 *
#> beta           0.0298     0.0206     0.0000      0.0706  
#> gamma          0.0000     0.0555     0.0000      0.1097  
#> level       4553.2079    77.2964  4399.8351   4706.0971 *
#> trend          1.0039     0.0020     0.9999      1.0079 *
#> seasonal_1     1.1810     0.0208     1.1551      1.2302 *
#> seasonal_2     0.8152     0.0143     0.7893      0.8644 *
#> seasonal_3     0.8248     0.0145     0.7989      0.8740 *
#> seasonal_4     1.5787     0.0249     1.5528      1.6279 *
#> seasonal_5     0.7464     0.0131     0.7205      0.7956 *
#> seasonal_6     1.2653     0.0214     1.2394      1.3145 *
#> seasonal_7     0.8924     0.0155     0.8665      0.9416 *
#> seasonal_8     0.9106     0.0159     0.8847      0.9598 *
#> seasonal_9     1.2290     0.0227     1.2031      1.2782 *
#> seasonal_10    0.8835     0.0164     0.8575      0.9326 *
#> seasonal_11    0.8383     0.0155     0.8124      0.8875 *
#> 
#> Sample size: 116
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 99
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1773.912 1780.157 1820.723 1835.565
plot(forecast(testModel,h=18,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: 0.23 seconds
#> Model estimated using adam() function: ETS(MMM)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 869.9558
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.0822 0.0298 0.0000 
#> 
#> Sample size: 116
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 99
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1773.912 1780.157 1820.723 1835.565 
#> 
#> Forecast errors:
#> ME: 576.674; MAE: 798.134; RMSE: 996.154
#> sCE: 142.593%; sMAE: 10.964%; sMSE: 1.873%
#> MASE: 0.325; RMSSE: 0.314; rMAE: 0.352; rRMSE: 0.328

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 another M3 series:

lossFunction <- function(actual, fitted, B){
  return(sum(abs(actual-fitted)^3))
}
testModel <- adam(M3[[1234]], "AAN", silent=FALSE, loss=lossFunction)
testModel
#> Time elapsed: 0.02 seconds
#> Model estimated using adam() function: ETS(AAN)
#> Distribution assumed in the model: Normal
#> Loss function type: custom; Loss function value: 23993012
#> Persistence vector g:
#>  alpha   beta 
#> 0.6316 0.2494 
#> 
#> Sample size: 45
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 41
#> Information criteria are unavailable for the chosen loss & distribution.
#> 
#> Forecast errors:
#> ME: -346.9; MAE: 346.9; RMSE: 395.39
#> sCE: -34.086%; sMAE: 4.261%; sMSE: 0.236%
#> MASE: 4.8; RMSSE: 4.416; rMAE: 3.942; rRMSE: 3.567

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(M3[[1234]], "MMN", silent=FALSE, distribution="dgnorm", shape=3)

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(M3[[2568]], "ZXZ", lags=c(1,12), silent=FALSE)
#> Forming the pool of models based on... ANN , ANA , MNM , MAM , Estimation progress:    71 %86 %100 %... Done!
testModel
#> Time elapsed: 0.49 seconds
#> Model estimated using adam() function: ETS(MAM)
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 866.5561
#> Persistence vector g:
#>  alpha   beta  gamma 
#> 0.1036 0.0100 0.0000 
#> 
#> Sample size: 116
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 99
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1767.112 1773.357 1813.923 1828.766 
#> 
#> Forecast errors:
#> ME: 673.457; MAE: 829.876; RMSE: 1064.48
#> sCE: 166.524%; sMAE: 11.4%; sMSE: 2.138%
#> MASE: 0.338; RMSSE: 0.336; rMAE: 0.366; rRMSE: 0.351

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(M3[[2568]], "CXC", lags=c(1,12))
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%)
#> Sep 1992      10917.153         9323.220        9040.417174          12608.47
#> Oct 1992       7839.458         1553.164         474.124937          14695.43
#> Nov 1992       7454.090         1339.547         282.860624          14089.43
#> Dec 1992      10189.816         3303.161        2131.739857          17746.30
#> Jan 1993      10561.229         3651.255        2473.326107          18130.32
#> Feb 1993       7275.458         1535.470         526.340613          13422.69
#> Mar 1993       7386.608         1777.324         786.965597          13373.72
#> Apr 1993      14028.018         6961.201        5746.523605          21716.04
#> May 1993       6658.683         1842.404         975.971717          11724.79
#> Jun 1993      11401.590         5902.415        4930.808829          17262.85
#> Jul 1993       8024.271         3910.499        3166.931460          12333.62
#> Aug 1993       8227.646         4999.466        4412.558485          11593.28
#> Sep 1993      11141.207         9435.347        9134.370185          12958.76
#> Oct 1993       7999.688         1355.633         207.270405          15209.36
#> Nov 1993       7605.904         1120.112          -6.970025          14615.65
#> Dec 1993      10397.704         3134.145        1891.530272          18335.52
#> Jan 1994      10775.311         3482.336        2232.556823          18734.25
#> Feb 1994       7421.918         1309.147         230.370021          13950.02
#>          Upper bound (97.5%)
#> Sep 1992            12956.11
#> Oct 1992            16152.46
#> Nov 1992            15491.72
#> Dec 1992            19362.31
#> Jan 1993            19745.82
#> Feb 1993            14702.27
#> Mar 1993            14615.06
#> Apr 1993            23343.49
#> May 1993            12757.28
#> Jun 1993            18475.21
#> Jul 1993            13207.29
#> Aug 1993            12271.65
#> Sep 1993            13334.09
#> Oct 1993            16733.03
#> Nov 1993            16090.43
#> Dec 1993            20025.67
#> Jan 1994            20426.14
#> Feb 1994            15304.61
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%)
#> Sep 1992      10917.153          12214.99          12608.47          13367.78
#> Oct 1992       7839.458          13061.19          14695.43          17893.57
#> Nov 1992       7454.090          12514.31          14089.43          17165.07
#> Dec 1992      10189.816          15936.54          17746.30          21296.27
#> Jan 1993      10561.229          16320.18          18130.32          21678.16
#> Feb 1993       7275.458          11979.66          13422.69          16223.24
#> Mar 1993       7386.608          11972.34          13373.72          16089.04
#> Apr 1993      14028.018          19888.42          21716.04          25285.86
#> May 1993       6658.683          10553.87          11724.79          13977.79
#> Jun 1993      11401.590          15893.15          17262.85          19913.66
#> Jul 1993       8024.271          11341.39          12333.62          14238.52
#> Aug 1993       8227.646          10821.62          11593.28          13071.12
#> Sep 1993      11141.207          12534.44          12958.76          13779.08
#> Oct 1993       7999.688          13497.90          15209.36          18551.27
#> Nov 1993       7605.904          12957.18          14615.65          17848.32
#> Dec 1993      10397.704          16440.62          18335.52          22046.19
#> Jan 1994      10775.311          16836.57          18734.25          22447.86
#> Feb 1994       7421.918          12421.15          13950.02          16913.46

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:

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=cbind(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: 0.45 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 21761.94
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.9094 0.2123 0.0674 0.0659 
#> Damping parameter: 0.7132
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 43535.88 43535.91 43571.97 43572.08 
#> 
#> Forecast errors:
#> ME: 419.337; MAE: 846.283; RMSE: 1075.402
#> sCE: 464.259%; sMAE: 2.789%; sMSE: 0.126%
#> MASE: 1.13; RMSSE: 1.032; rMAE: 0.123; rRMSE: 0.127

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: 3.94 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19643.8
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0227 0.0226 0.1866 0.2329 
#> Damping parameter: 5e-04
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39299.60 39299.62 39335.68 39335.79 
#> 
#> Forecast errors:
#> ME: -30.121; MAE: 136.283; RMSE: 172.536
#> sCE: -33.347%; sMAE: 0.449%; sMSE: 0.003%
#> MASE: 0.182; RMSSE: 0.166; rMAE: 0.02; rRMSE: 0.02

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

testModel$B

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

testModel <- adam(y, "MMdM", lags=c(1,48,336), initial="backcasting",
                  silent=FALSE, h=336, holdout=TRUE, B=testModel$B)
testModel
#> Time elapsed: 0.65 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19643.8
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.0226 0.0199 0.1866 0.2325 
#> Damping parameter: 0.0062
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 39299.59 39299.62 39335.68 39335.79 
#> 
#> Forecast errors:
#> ME: -30.032; MAE: 136.283; RMSE: 172.534
#> sCE: -33.249%; sMAE: 0.449%; sMSE: 0.003%
#> MASE: 0.182; RMSSE: 0.166; rMAE: 0.02; rRMSE: 0.02

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: 0.37 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 21896.27
#> Persistence vector g:
#>  alpha   beta gamma1 gamma2 
#> 0.9332 0.1000 0.0518 0.0666 
#> Damping parameter: 0.9439
#> 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 
#> 43802.54 43802.56 43832.61 43832.69 
#> 
#> Forecast errors:
#> ME: 133.166; MAE: 793.664; RMSE: 1029.83
#> sCE: 147.432%; sMAE: 2.615%; sMSE: 0.115%
#> MASE: 1.06; RMSSE: 0.988; rMAE: 0.115; rRMSE: 0.122

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]
#> Occurrence model type: Odds ratio
#> Distribution assumed in the model: Mixture of Bernoulli and Gamma
#> Loss function type: likelihood; Loss function value: 59.3242
#> Persistence vector g:
#> alpha 
#> 7e-04 
#> 
#> Sample size: 108
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 103
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 276.8254 277.0562 290.2361 281.4121 
#> 
#> Forecast errors:
#> Bias: -100%; sMSE: 21.074%; rRMSE: 0.593; sPIS: 3580.727%; sCE: -550.881%

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(M3[[2568]], "CCC")
esModel <- es(M3[[2568]], "CCC")
"adam:"
#> [1] "adam:"
adamModel
#> Time elapsed: 1.98 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#> 
#> Number of models combined: 30
#> Sample size: 116
#> Average number of estimated parameters: 26.7162
#> Average number of degrees of freedom: 89.2838
#> 
#> Forecast errors:
#> ME: 640.352; MAE: 813.82; RMSE: 1033.638
#> sCE: 158.338%; sMAE: 11.18%; sMSE: 2.016%
#> MASE: 0.331; RMSSE: 0.326; rMAE: 0.359; rRMSE: 0.34
"es():"
#> [1] "es():"
esModel
#> Time elapsed: 3.95 seconds
#> Model estimated: ETS(CCC)
#> Initial values were optimised.
#> 
#> Loss function type: likelihood
#> Error standard deviation: 414.1228
#> Sample size: 116
#> Information criteria:
#> (combined values)
#>      AIC     AICc      BIC     BICc 
#> 1763.821 1769.594 1807.909 1820.587 
#> 
#> Forecast errors:
#> MPE: 2.9%; sCE: 91.1%; Bias: 49.3%; MAPE: 6.7%
#> MASE: 0.285; sMAE: 9.6%; sMSE: 1.4%; rMAE: 0.31; rRMSE: 0.281

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(M3[[1234]], "NNN", silent=FALSE, orders=c(0,2,2))
testModel
#> Time elapsed: 0.05 seconds
#> Model estimated using adam() function: ARIMA(0,2,2)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 255.293
#> ARMA parameters of the model:
#> MA:
#> theta1[1] theta2[1] 
#>   -1.0909    0.3210 
#> 
#> Sample size: 45
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 40
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 520.5861 522.1245 529.6194 532.5476 
#> 
#> Forecast errors:
#> ME: -348.345; MAE: 348.345; RMSE: 396.569
#> sCE: -34.228%; sMAE: 4.278%; sMSE: 0.237%
#> MASE: 4.82; RMSSE: 4.429; rMAE: 3.958; rRMSE: 3.578

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(M3[[2568]], "NNN", silent=FALSE, lags=c(1,12),
                  orders=list(ar=c(1,1),i=c(1,1),ma=c(2,2)), distribution="dlnorm")
testModel
#> Time elapsed: 0.92 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,2)[12]
#> Distribution assumed in the model: Log Normal
#> Loss function type: likelihood; Loss function value: 868.4544
#> ARMA parameters of the model:
#> AR:
#>  phi1[1] phi1[12] 
#>  -0.5461   0.0427 
#> MA:
#>  theta1[1]  theta2[1] theta1[12] theta2[12] 
#>    -0.6396    -0.1884    -0.3547    -0.1805 
#> 
#> Sample size: 116
#> Number of estimated parameters: 33
#> Number of degrees of freedom: 83
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1802.909 1830.275 1893.777 1958.820 
#> 
#> Forecast errors:
#> ME: 345.773; MAE: 592.327; RMSE: 731.203
#> sCE: 85.499%; sMAE: 8.137%; sMSE: 1.009%
#> MASE: 0.241; RMSSE: 0.231; rMAE: 0.262; rRMSE: 0.241

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

testModel <- adam(M3[[2568]], "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")
testModel
#> Time elapsed: 0.7 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,2)[12] with drift
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 896.6879
#> ARMA parameters of the model:
#> AR:
#>  phi1[1] phi1[12] 
#>  -0.4783   0.0481 
#> MA:
#>  theta1[1]  theta2[1] theta1[12] theta2[12] 
#>    -0.5178    -0.2590    -0.3023     0.0744 
#> 
#> Sample size: 116
#> Number of estimated parameters: 34
#> Number of degrees of freedom: 82
#> Information criteria:
#>      AIC     AICc      BIC     BICc 
#> 1861.376 1890.758 1954.998 2024.835 
#> 
#> Forecast errors:
#> ME: 219.85; MAE: 600.343; RMSE: 697.476
#> sCE: 54.362%; sMAE: 8.247%; sMSE: 0.918%
#> MASE: 0.244; RMSSE: 0.22; rMAE: 0.265; rRMSE: 0.23