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: 475.7278
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> alpha 0.4143 0.0764 0.2630 0.5653 *
#> beta 0.0005 0.0044 0.0000 0.0092
#> gamma 0.0264 0.0394 0.0000 0.1041
#> level 108.5160 3.1894 102.1985 114.8183 *
#> trend 1.0114 0.0014 1.0087 1.0141 *
#> seasonal_1 0.9090 0.0083 0.8946 0.9348 *
#> seasonal_2 0.9031 0.0095 0.8887 0.9289 *
#> seasonal_3 1.0237 0.0108 1.0094 1.0495 *
#> seasonal_4 0.9963 0.0072 0.9819 1.0221 *
#> seasonal_5 0.9921 0.0083 0.9777 1.0179 *
#> seasonal_6 1.1169 0.0115 1.1026 1.1427 *
#> seasonal_7 1.2339 0.0131 1.2196 1.2598 *
#> seasonal_8 1.2267 0.0127 1.2123 1.2525 *
#> seasonal_9 1.0620 0.0105 1.0477 1.0879 *
#> seasonal_10 0.9221 0.0092 0.9077 0.9479 *
#> seasonal_11 0.8051 0.0085 0.7907 0.8309 *
#>
#> Error standard deviation: 0.0393
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#> AIC AICc BIC BICc
#> 985.4556 990.8240 1034.4632 1047.5697plot(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: 0.13 seconds
#> Model estimated using adam() function: ETS(MMM)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 475.7278
#> Persistence vector g:
#> alpha beta gamma
#> 0.4143 0.0005 0.0264
#>
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#> AIC AICc BIC BICc
#> 985.4556 990.8240 1034.4632 1047.5697
#>
#> Forecast errors:
#> ME: -14.479; MAE: 17.456; RMSE: 24.368
#> sCE: -66.193%; Asymmetry: -85%; sMAE: 6.65%; sMSE: 0.862%
#> MASE: 0.725; RMSSE: 0.778; rMAE: 0.23; rRMSE: 0.237Also, 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)
#> Distribution assumed in the model: Normal
#> Loss function type: custom; Loss function value: 599.224
#> Persistence vector g:
#> alpha beta
#> 1.0000 0.2265
#>
#> 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: 3.016; MAE: 3.13; RMSE: 3.867
#> sCE: 15.923%; Asymmetry: 91.7%; sMAE: 1.377%; sMSE: 0.029%
#> MASE: 2.627; RMSSE: 2.521; rMAE: 1.01; 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: 0.57 seconds
#> Model estimated using adam() function: ETS(MAM)
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 467.7828
#> Persistence vector g:
#> alpha beta gamma
#> 0.7487 0.0178 0.0006
#>
#> Sample size: 132
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 115
#> Information criteria:
#> AIC AICc BIC BICc
#> 969.5655 974.9339 1018.5731 1031.6796
#>
#> Forecast errors:
#> ME: 3.005; MAE: 16.312; RMSE: 22.838
#> sCE: 13.737%; Asymmetry: 37.5%; sMAE: 6.214%; sMSE: 0.757%
#> MASE: 0.677; RMSSE: 0.729; rMAE: 0.215; rRMSE: 0.222
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 414.8649 391.5810 387.2529 438.7114
#> Feb 1960 410.6283 381.2299 375.8122 440.9392
#> Mar 1960 473.2942 432.6534 425.2245 515.4579
#> Apr 1960 456.5806 414.0166 406.2690 500.8822
#> May 1960 459.5624 415.6970 407.7229 505.2633
#> Jun 1960 522.3406 470.1741 460.7155 576.7958
#> Jul 1960 581.7566 521.2214 510.2724 645.0648
#> Aug 1960 580.7394 518.9643 507.8064 645.4114
#> Sep 1960 508.5326 454.2424 444.4388 565.3792
#> Oct 1960 444.7669 396.6700 387.9920 495.1600
#> Nov 1960 388.7625 346.3430 338.6940 433.2272
#> Dec 1960 438.1822 389.0777 380.2390 489.7228
#> Jan 1961 448.1516 395.0905 385.5760 504.0040
#> Feb 1961 443.3565 387.4364 377.4563 502.4240
#> Mar 1961 510.7678 441.3690 429.0568 584.3951
#> Apr 1961 492.4934 421.8667 409.3953 567.6815
#> May 1961 495.4734 423.5192 410.8286 572.1420
#> Jun 1961 562.8933 480.1278 465.5488 651.1627
#> Upper bound (97.5%)
#> Jan 1960 443.4143
#> Feb 1960 446.9646
#> Mar 1960 523.9010
#> Apr 1960 509.7868
#> May 1960 514.4596
#> Jun 1960 587.7786
#> Jul 1960 657.8605
#> Aug 1960 658.4983
#> Sep 1960 576.8850
#> Oct 1960 505.3670
#> Nov 1960 442.2381
#> Dec 1960 500.1837
#> Jan 1961 515.3770
#> Feb 1961 514.4998
#> Mar 1961 599.5226
#> Apr 1961 583.1896
#> May 1961 587.9710
#> Jun 1961 669.4059plot(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 414.8649 433.3305 438.7114 448.9243
#> Feb 1960 410.6283 434.0594 440.9392 454.0387
#> Mar 1960 473.2942 505.8360 515.4579 533.8323
#> Apr 1960 456.5806 490.7443 500.8822 520.2711
#> May 1960 459.5624 494.7963 505.2633 525.2904
#> Jun 1960 522.3406 564.3029 576.7958 600.7209
#> Jul 1960 581.7566 630.5178 645.0648 672.9475
#> Aug 1960 580.7394 630.5379 645.4114 673.9335
#> Sep 1960 508.5326 552.3033 565.3792 590.4561
#> Oct 1960 444.7669 483.5624 495.1600 517.4083
#> Nov 1960 388.7625 422.9900 433.2272 452.8698
#> Dec 1960 438.1822 477.8430 489.7228 512.5310
#> Jan 1961 448.1516 491.0992 504.0040 528.8123
#> Feb 1961 443.3565 488.7360 502.4240 528.7796
#> Mar 1961 510.7678 567.2700 584.3951 617.4337
#> Apr 1961 492.4934 550.1430 567.6815 601.5694
#> May 1961 495.4734 554.2450 572.1420 606.7360
#> Jun 1961 562.8933 630.5419 651.1627 691.0383A 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: 1.11 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19500.06
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.0350 0.0000 0.1803 0.3057
#> Damping parameter: 0.9936
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#> AIC AICc BIC BICc
#> 39012.12 39012.14 39048.20 39048.31
#>
#> Forecast errors:
#> ME: 135.704; MAE: 186.674; RMSE: 229.875
#> sCE: 150.815%; Asymmetry: 74%; sMAE: 0.617%; sMSE: 0.006%
#> MASE: 0.252; RMSSE: 0.222; 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: 1.24 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19500.06
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.0351 0.0000 0.1797 0.3055
#> Damping parameter: 0.9935
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#> AIC AICc BIC BICc
#> 39012.11 39012.14 39048.20 39048.31
#>
#> Forecast errors:
#> ME: 135.758; MAE: 186.691; RMSE: 229.89
#> sCE: 150.876%; Asymmetry: 74.1%; sMAE: 0.618%; sMSE: 0.006%
#> MASE: 0.252; RMSSE: 0.222; rMAE: 0.027; rRMSE: 0.027
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$Band use this as a starting point for the reestimation (e.g. with a different algorithm):
testModel <- adam(y, "MMdM", lags=c(1,48,336), initial="backcasting",
silent=FALSE, h=336, holdout=TRUE, B=testModel$B)
testModel
#> Time elapsed: 0.45 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 19500.06
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.0351 0.0000 0.1799 0.3059
#> Damping parameter: 0.996
#> Sample size: 3024
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 3018
#> Information criteria:
#> AIC AICc BIC BICc
#> 39012.11 39012.14 39048.20 39048.31
#>
#> Forecast errors:
#> ME: 135.789; MAE: 186.716; RMSE: 229.915
#> sCE: 150.91%; Asymmetry: 74.1%; sMAE: 0.618%; sMSE: 0.006%
#> MASE: 0.252; RMSSE: 0.222; 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: 0.71 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Gamma
#> Loss function type: likelihood; Loss function value: 21804.83
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.9531 0.1000 0.0284 0.0468
#> Damping parameter: 0.7836
#> Sample size: 3024
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 3019
#> Information criteria:
#> AIC AICc BIC BICc
#> 43619.66 43619.68 43649.73 43649.81
#>
#> Forecast errors:
#> ME: 470.35; MAE: 889.006; RMSE: 1101.571
#> sCE: 522.727%; Asymmetry: 56.3%; sMAE: 2.94%; sMSE: 0.133%
#> MASE: 1.199; RMSSE: 1.063; rMAE: 0.129; rRMSE: 0.13The 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]
#> Occurrence model type: Odds ratio
#> Distribution assumed in the model: Mixture of Bernoulli and Gamma
#> Loss function type: likelihood; Loss function value: 68.1295
#> 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
#> 293.5997 293.8304 307.0103 298.1863
#>
#> Forecast errors:
#> Asymmetry: -19.201%; sMSE: 21.708%; rRMSE: 0.794; sPIS: 405.811%; sCE: -51.375%
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: 2.23 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#>
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 18.204
#> Average number of degrees of freedom: 113.796
#>
#> Forecast errors:
#> ME: 1.196; MAE: 15.085; RMSE: 21.536
#> sCE: 5.469%; sMAE: 5.747%; sMSE: 0.673%
#> MASE: 0.626; RMSSE: 0.687; rMAE: 0.198; rRMSE: 0.209"es():"
#> [1] "es():"esModel
#> Time elapsed: 2.14 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#>
#> Number of models combined: 30
#> Sample size: 132
#> Average number of estimated parameters: 18.0014
#> Average number of degrees of freedom: 113.9986
#>
#> Forecast errors:
#> ME: 5.112; MAE: 17.588; RMSE: 23.73
#> sCE: 23.368%; sMAE: 6.7%; sMSE: 0.817%
#> MASE: 0.73; RMSSE: 0.757; rMAE: 0.231; rRMSE: 0.23
