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: 470.6091
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> alpha 0.6661 0.0819 0.5039 0.8278 *
#> beta 0.0038 0.0226 0.0000 0.0485
#> gamma 0.0298 0.0351 0.0000 0.0991
#> level 111.4423 5.5993 100.3511 122.5066 *
#> trend 1.0098 0.0028 1.0042 1.0154 *
#> seasonal_1 0.8973 0.0058 0.8859 0.9227 *
#> seasonal_2 0.8991 0.0096 0.8876 0.9245 *
#> seasonal_3 1.0297 0.0107 1.0182 1.0550 *
#> seasonal_4 0.9957 0.0090 0.9843 1.0211 *
#> seasonal_5 1.0021 0.0070 0.9907 1.0275 *
#> seasonal_6 1.1352 0.0093 1.1237 1.1605 *
#> seasonal_7 1.2382 0.0128 1.2268 1.2636 *
#> seasonal_8 1.2237 0.0112 1.2122 1.2490 *
#> seasonal_9 1.0642 0.0107 1.0527 1.0895 *
#> seasonal_10 0.9236 0.0096 0.9121 0.9489 *
#> seasonal_11 0.8004 0.0083 0.7889 0.8257 *
#>
#> Error standard deviation: 0.0379
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#> AIC AICc BIC BICc
#> 975.2182 980.5866 1024.2258 1037.3323
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: 0.12 seconds
#> Model estimated using adam() function: ETS(MMM)
#> With optimal initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 470.6091
#> Persistence vector g:
#> alpha beta gamma
#> 0.6661 0.0038 0.0298
#>
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#> AIC AICc BIC BICc
#> 975.2182 980.5866 1024.2258 1037.3323
#>
#> Forecast errors:
#> ME: -11.657; MAE: 16.154; RMSE: 22.504
#> sCE: -53.291%; Asymmetry: -70%; sMAE: 6.154%; sMSE: 0.735%
#> MASE: 0.671; RMSSE: 0.718; rMAE: 0.213; rRMSE: 0.219Also, 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):
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):
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 optimal initialisation
#> Distribution assumed in the model: Normal
#> Loss function type: custom; Loss function value: 599.2241
#> Persistence vector g:
#> alpha beta
#> 1.0000 0.2269
#>
#> Sample size: 138
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 134
#> Information criteria are unavailable for the chosen loss & distribution.
#>
#> Forecast errors:
#> ME: 3.015; MAE: 3.129; RMSE: 3.866
#> sCE: 15.918%; 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:
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: 0.55 seconds
#> Model estimated using adam() function: ETS(MAM)
#> With optimal initialisation
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 466.9086
#> Persistence vector g:
#> alpha beta gamma
#> 0.7807 0.0003 0.0002
#>
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#> AIC AICc BIC BICc
#> 967.8172 973.1857 1016.8249 1029.9313
#>
#> Forecast errors:
#> ME: 11.859; MAE: 22.322; RMSE: 26.996
#> sCE: 54.214%; Asymmetry: 68.4%; sMAE: 8.504%; sMSE: 1.058%
#> MASE: 0.927; RMSSE: 0.862; rMAE: 0.294; rRMSE: 0.262
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 411.7284 388.9060 384.6620 435.0953
#> Feb 1960 406.5590 377.9249 372.6446 436.0670
#> Mar 1960 466.6605 427.2715 420.0654 507.4993
#> Apr 1960 449.9171 408.8108 401.3205 492.6667
#> May 1960 450.5760 408.4541 400.7882 494.4225
#> Jun 1960 512.9144 462.9301 453.8543 565.0361
#> Jul 1960 569.9673 512.4185 501.9906 630.0703
#> Aug 1960 569.1866 510.7379 500.1577 630.2760
#> Sep 1960 498.4551 447.3865 438.1411 551.8259
#> Oct 1960 434.6231 389.7640 381.6464 481.5206
#> Nov 1960 378.8849 339.6652 332.5695 419.8925
#> Dec 1960 426.5414 381.4206 373.2682 473.7671
#> Jan 1961 433.3114 385.2139 376.5507 483.7699
#> Feb 1961 427.7775 377.6289 368.6300 480.5350
#> Mar 1961 490.9117 429.1080 418.0754 556.1836
#> Apr 1961 473.1945 410.6973 399.5832 539.3836
#> May 1961 473.7861 410.4375 399.1835 540.9272
#> Jun 1961 539.2199 466.5032 453.5946 616.3333
#> Upper bound (97.5%)
#> Jan 1960 439.7020
#> Feb 1960 441.9294
#> Mar 1960 515.6710
#> Apr 1960 501.2512
#> May 1960 503.2368
#> Jun 1960 575.5352
#> Jul 1960 642.1989
#> Aug 1960 642.6145
#> Sep 1960 562.6044
#> Oct 1960 490.9955
#> Nov 1960 428.1788
#> Dec 1960 483.3211
#> Jan 1961 494.0053
#> Feb 1961 491.2710
#> Mar 1961 569.5253
#> Apr 1961 552.9558
#> May 1961 554.7064
#> Jun 1961 632.1691
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 411.7284 429.8240 435.0953 445.0988
#> Feb 1960 406.5590 429.3724 436.0670 448.8109
#> Mar 1960 466.6605 498.1849 507.4993 525.2812
#> Apr 1960 449.9171 482.8908 492.6667 511.3560
#> May 1960 450.5760 484.3877 494.4225 513.6150
#> Jun 1960 512.9144 553.0895 565.0361 587.9036
#> Jul 1960 569.9673 616.2759 630.0703 656.4936
#> Aug 1960 569.1866 616.2460 630.2760 657.1600
#> Sep 1960 498.4551 539.5697 551.8259 575.3103
#> Oct 1960 434.6231 470.7477 481.5206 502.1658
#> Nov 1960 378.8849 410.4715 419.8925 437.9484
#> Dec 1960 426.5414 462.9081 473.7671 494.5886
#> Jan 1961 433.3114 472.1445 483.7699 506.0847
#> Feb 1961 427.7775 468.3510 480.5350 503.9517
#> Mar 1961 490.9117 541.0599 556.1836 585.3014
#> Apr 1961 473.1945 524.0113 539.3836 569.0174
#> May 1961 473.7861 525.3239 540.9272 571.0165
#> Jun 1961 539.2199 598.4040 616.3333 650.9165A 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))$dataWe 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: 0.81 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: 19869
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.1103 0.1068 0.4053 0.4211
#> Damping parameter: 0
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#> AIC AICc BIC BICc
#> 39750.01 39750.03 39786.09 39786.20
#>
#> Forecast errors:
#> ME: 60.796; MAE: 180.452; RMSE: 222.791
#> sCE: 66.587%; Asymmetry: 35.4%; sMAE: 0.588%; sMSE: 0.005%
#> MASE: 0.24; RMSSE: 0.214; rMAE: 0.027; rRMSE: 0.027
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: 2.48 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: 19642.89
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.0408 0.0408 0.1741 0.4820
#> Damping parameter: 0
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#> AIC AICc BIC BICc
#> 39297.78 39297.81 39333.86 39333.98
#>
#> Forecast errors:
#> ME: -124.828; MAE: 168.927; RMSE: 211.414
#> sCE: -136.718%; Asymmetry: -73.3%; sMAE: 0.551%; sMSE: 0.005%
#> MASE: 0.224; RMSSE: 0.203; rMAE: 0.025; rRMSE: 0.026
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:
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: 0.84 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: 19587.61
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.0234 0.0234 0.1969 0.2527
#> Damping parameter: 0.1706
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#> AIC AICc BIC BICc
#> 39187.23 39187.26 39223.31 39223.43
#>
#> Forecast errors:
#> ME: -144.41; MAE: 180.979; RMSE: 226.479
#> sCE: -158.165%; Asymmetry: -79%; sMAE: 0.59%; sMSE: 0.005%
#> MASE: 0.24; RMSSE: 0.218; rMAE: 0.027; rRMSE: 0.027
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: 0.64 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: 19991.1
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.1015 0.1000 0.1215 0.8448
#> Damping parameter: 0
#> Sample size: 3024
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 3019
#> Information criteria:
#> AIC AICc BIC BICc
#> 39992.20 39992.22 40022.27 40022.35
#>
#> Forecast errors:
#> ME: -55.633; MAE: 170.003; RMSE: 206.946
#> sCE: -60.932%; Asymmetry: -32.7%; sMAE: 0.554%; sMSE: 0.005%
#> MASE: 0.226; RMSSE: 0.199; rMAE: 0.025; rRMSE: 0.025The 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 optimal initialisation
#> Occurrence model type: Odds ratio
#> Distribution assumed in the model: Mixture of Bernoulli and Gamma
#> Loss function type: likelihood; Loss function value: 37.4787
#> 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
#> 229.2991 229.5299 242.7098 233.8857
#>
#> Forecast errors:
#> Asymmetry: -47.006%; sMSE: 26.461%; rRMSE: 0.901; sPIS: 1460.803%; sCE: -125.331%
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: 2.1 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#>
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 18.0026
#> Average number of degrees of freedom: 113.9974
#>
#> Forecast errors:
#> ME: 11.488; MAE: 22.026; RMSE: 26.785
#> sCE: 52.518%; sMAE: 8.391%; sMSE: 1.041%
#> MASE: 0.915; RMSSE: 0.855; rMAE: 0.29; rRMSE: 0.26
"es():"
#> [1] "es():"
esModel
#> Time elapsed: 2.04 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#>
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 18.0462
#> Average number of degrees of freedom: 113.9538
#>
#> Forecast errors:
#> ME: 1.919; MAE: 16.001; RMSE: 22.328
#> sCE: 8.773%; sMAE: 6.096%; sMSE: 0.724%
#> MASE: 0.664; RMSSE: 0.713; rMAE: 0.211; rRMSE: 0.217
