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:
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: 868.7509
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> alpha 0.1429 0.0557 0.0323 0.2530
#> beta 0.0121 0.0132 0.0000 0.0382
#> gamma 0.0100 0.0507 0.0000 0.1102
#> level 4407.1174 109.7941 4189.2622 4624.2858
#> trend 1.0064 0.0019 1.0026 1.0101
#> seasonal_1 1.1843 0.0217 1.1574 1.2402
#> seasonal_2 0.8172 0.0148 0.7903 0.8731
#> seasonal_3 0.8267 0.0149 0.7997 0.8826
#> seasonal_4 1.5608 0.0283 1.5338 1.6167
#> seasonal_5 0.7445 0.0136 0.7176 0.8004
#> seasonal_6 1.2706 0.0229 1.2436 1.3265
#> seasonal_7 0.8930 0.0161 0.8661 0.9489
#> seasonal_8 0.9137 0.0166 0.8868 0.9696
#> seasonal_9 1.2313 0.0234 1.2044 1.2872
#> seasonal_10 0.8835 0.0168 0.8565 0.9393
#> seasonal_11 0.8383 0.0159 0.8114 0.8942
#>
#> Sample size: 116
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 99
#> Information criteria:
#> AIC AICc BIC BICc
#> 1771.502 1777.747 1818.313 1833.156
plot(forecast(testModel,h=18,interval="parametric"))
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: 0.18 seconds
#> Model estimated using adam() function: ETS(MMM)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 868.7509
#> Persistence vector g:
#> alpha beta gamma
#> 0.1429 0.0121 0.0100
#>
#> Sample size: 116
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 99
#> Information criteria:
#> AIC AICc BIC BICc
#> 1771.502 1777.747 1818.313 1833.156
#>
#> Forecast errors:
#> ME: 645.912; MAE: 817.203; RMSE: 1043.544
#> sCE: 159.713%; sMAE: 11.226%; sMSE: 2.055%
#> MASE: 0.333; RMSSE: 0.329; rMAE: 0.361; rRMSE: 0.344
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):
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 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: 23993619
#> Persistence vector g:
#> alpha beta
#> 0.6348 0.2466
#>
#> 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: -347.014; MAE: 347.014; RMSE: 395.482
#> sCE: -34.097%; sMAE: 4.262%; sMSE: 0.236%
#> MASE: 4.801; RMSSE: 4.417; rMAE: 3.943; rRMSE: 3.568
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 beta
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(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: 0.74 seconds
#> Model estimated using adam() function: ETS(MAM)
#> Distribution assumed in the model: Inverse Gaussian
#> Loss function type: likelihood; Loss function value: 866.4522
#> Persistence vector g:
#> alpha beta gamma
#> 0.1096 0.0088 0.0000
#>
#> Sample size: 116
#> Number of estimated parameters: 17
#> Number of degrees of freedom: 99
#> Information criteria:
#> AIC AICc BIC BICc
#> 1766.904 1773.149 1813.715 1828.558
#>
#> Forecast errors:
#> ME: 635.957; MAE: 820.168; RMSE: 1044.133
#> sCE: 157.252%; sMAE: 11.267%; sMSE: 2.057%
#> MASE: 0.334; RMSSE: 0.33; rMAE: 0.362; rRMSE: 0.344
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 10886.130 9243.361 8966.660 12700.725
#> Oct 1992 7831.519 4292.762 3809.616 12391.888
#> Nov 1992 7437.760 4096.437 3635.120 11712.391
#> Dec 1992 10112.336 5856.079 5271.983 15567.949
#> Jan 1993 10478.405 6176.327 5580.797 15958.839
#> Feb 1993 7237.466 4226.174 3792.666 10978.934
#> Mar 1993 7354.970 4409.162 3979.430 10980.145
#> Apr 1993 14002.164 9150.384 8447.093 19980.513
#> May 1993 6647.976 4258.192 3892.750 9491.269
#> Jun 1993 11353.768 7881.474 7351.010 15478.587
#> Jul 1993 7973.697 5717.992 5359.163 10577.689
#> Aug 1993 8188.519 6298.729 5988.622 10319.498
#> Sep 1993 11052.078 9331.459 9043.073 12959.623
#> Oct 1993 7951.095 4342.503 3845.551 12576.735
#> Nov 1993 7567.014 4138.194 3661.816 11937.159
#> Dec 1993 10295.036 5938.505 5337.368 15860.975
#> Jan 1994 10653.875 6252.239 5639.939 16244.902
#> Feb 1994 7358.324 4257.516 3809.220 11201.412
#> Upper bound (97.5%)
#> Sep 1992 13091.87
#> Oct 1992 13556.76
#> Nov 1992 12796.42
#> Dec 1992 16952.12
#> Jan 1993 17340.31
#> Feb 1993 11899.32
#> Mar 1993 11862.79
#> Apr 1993 21434.68
#> May 1993 10158.92
#> Jun 1993 16443.81
#> Jul 1993 11168.53
#> Aug 1993 10790.26
#> Sep 1993 13372.51
#> Oct 1993 13752.32
#> Nov 1993 13041.58
#> Dec 1993 17268.87
#> Jan 1994 17650.49
#> Feb 1994 12144.65
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 10880.472 12257.295 12694.128 13554.02
#> Oct 1992 7836.843 11156.238 12399.946 15027.48
#> Nov 1992 7437.482 10552.958 11711.738 14152.77
#> Dec 1992 10120.871 14099.718 15580.822 18700.28
#> Jan 1993 10495.539 14502.427 15984.369 19097.21
#> Feb 1993 7238.434 9988.339 10980.548 13044.77
#> Mar 1993 7351.591 10021.508 10975.070 12950.42
#> Apr 1993 14003.358 18409.330 19982.218 23237.20
#> May 1993 6621.839 8729.270 9455.427 10937.37
#> Jun 1993 11371.633 14446.145 15502.513 17654.32
#> Jul 1993 7976.726 9930.462 10581.702 11891.71
#> Aug 1993 8177.270 9783.891 10305.729 11343.75
#> Sep 1993 11040.897 12486.089 12946.453 13854.20
#> Oct 1993 7962.470 11336.417 12594.089 15245.94
#> Nov 1993 7563.596 10750.698 11931.899 14416.88
#> Dec 1993 10284.532 14340.082 15844.983 19010.85
#> Jan 1994 10647.246 14729.539 16235.149 19394.44
#> Feb 1994 7356.377 10181.484 11198.437 13312.36
A brand new thing in the function is the possibility to use several frequencies (double / triple / quadruple / … seasonal models). Here is an example of what we can have in case of half-hourly data:
testModel <- adam(forecast::taylor, "MMdM", lags=c(1,48,336), silent=FALSE, h=336, holdout=TRUE)
testModel
#> Time elapsed: 42.53 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Inverse Gaussian
#> Loss function type: likelihood; Loss function value: 25478.85
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.9960 0.3651 0.0011 0.0040
#> Damping parameter: 0.75
#> Sample size: 3696
#> Number of estimated parameters: 390
#> Number of degrees of freedom: 3306
#> Information criteria:
#> AIC AICc BIC BICc
#> 51737.70 51829.97 54161.55 54540.58
#>
#> Forecast errors:
#> ME: 309.978; MAE: 829.421; RMSE: 1088.231
#> sCE: 351.993%; sMAE: 2.803%; sMSE: 0.135%
#> MASE: 1.276; RMSSE: 1.153; rMAE: 0.124; rRMSE: 0.133
Note that the more lags you have, the more initial seasonal components the function will need to estimate, which is a difficult task. 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 20 iterations for each optimised parameter. So, in case of the previous model it is 389*20=7780):
testModel <- adam(forecast::taylor, "MMdM", lags=c(1,48,336), silent=FALSE, h=336, holdout=TRUE,
maxeval=10000)
testModel
#> Time elapsed: 27.16 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Inverse Gaussian
#> Loss function type: likelihood; Loss function value: 25623.23
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.9959 0.3651 0.0011 0.0040
#> Damping parameter: 0.7226
#> Sample size: 3696
#> Number of estimated parameters: 390
#> Number of degrees of freedom: 3306
#> Information criteria:
#> AIC AICc BIC BICc
#> 52026.47 52118.75 54450.32 54829.35
#>
#> Forecast errors:
#> ME: 349.11; MAE: 849.097; RMSE: 1109.359
#> sCE: 396.429%; sMAE: 2.87%; sMSE: 0.141%
#> MASE: 1.306; RMSSE: 1.175; rMAE: 0.127; rRMSE: 0.135
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 (e.g. with a different algorithm):
testModel <- adam(forecast::taylor, "MMdM", lags=c(1,48,336), silent=FALSE, h=336, holdout=TRUE,
B=testModel$B)
testModel
#> Time elapsed: 41.02 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Inverse Gaussian
#> Loss function type: likelihood; Loss function value: 25123.54
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.9964 0.9964 0.0034 0.0000
#> Damping parameter: 0.6517
#> Sample size: 3696
#> Number of estimated parameters: 390
#> Number of degrees of freedom: 3306
#> Information criteria:
#> AIC AICc BIC BICc
#> 51027.08 51119.36 53450.93 53829.97
#>
#> Forecast errors:
#> ME: -9.782; MAE: 755.618; RMSE: 1041.161
#> sCE: -11.108%; sMAE: 2.554%; sMSE: 0.124%
#> MASE: 1.162; RMSSE: 1.103; rMAE: 0.113; rRMSE: 0.127
Finally, we can speed up the process by using a different initialisation of the state vector, such as backcasting:
testModel <- adam(forecast::taylor, "MMdM", lags=c(1,48,336), silent=FALSE, h=336, holdout=TRUE,
initial="b")
The result might be less accurate than in case of the optimisation, but it should be faster.
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(forecast::taylor, "MMdM", lags=c(1,48,336), silent=TRUE, h=336, holdout=TRUE,
initial=list(level=30000, trend=1), persistence=list(beta=0.1))
testModel
#> Time elapsed: 41.44 seconds
#> Model estimated using adam() function: ETS(MMdM)[48, 336]
#> Distribution assumed in the model: Inverse Gaussian
#> Loss function type: likelihood; Loss function value: 25738.4
#> Persistence vector g:
#> alpha beta gamma1 gamma2
#> 0.9670 0.1000 0.0001 0.0330
#> Damping parameter: 0.7634
#> Sample size: 3696
#> Number of estimated parameters: 387
#> Number of degrees of freedom: 3309
#> Number of provided parameters: 3
#> Information criteria:
#> AIC AICc BIC BICc
#> 52250.81 52341.59 54656.01 55028.91
#>
#> Forecast errors:
#> ME: 171.838; MAE: 764.081; RMSE: 1039.778
#> sCE: 195.129%; sMAE: 2.582%; sMSE: 0.123%
#> MASE: 1.175; RMSSE: 1.102; rMAE: 0.114; rRMSE: 0.127
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)
#> Occurrence model type: Odds ratio
#> Distribution assumed in the model: Mixture of Bernoulli and Inverse Gaussian
#> Loss function type: likelihood; Loss function value: -24.4472
#> 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
#> 109.8981 110.1289 123.3087 114.4847
#>
#> Forecast errors:
#> Bias: -47.26%; sMSE: 17.771%; rRMSE: 0.909; sPIS: 1284.304%; sCE: -222.302%
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: 1.97 seconds
#> Model estimated: ETS(CCC)
#> Loss function type: likelihood
#>
#> Number of models combined: 30
#> Sample size: 116
#> Average number of estimated parameters: 22.496
#> Average number of degrees of freedom: 93.504
#>
#> Forecast errors:
#> ME: 626.704; MAE: 810.672; RMSE: 1029.509
#> sCE: 154.964%; sMAE: 11.136%; sMSE: 2%
#> MASE: 0.33; RMSSE: 0.325; rMAE: 0.358; rRMSE: 0.339
"es():"
#> [1] "es():"
esModel
#> Time elapsed: 4.02 seconds
#> Model estimated: ETS(CCC)
#> Initial values were optimised.
#>
#> Loss function type: MSE
#> Error standard deviation: 415.2322
#> Sample size: 116
#> Information criteria:
#> (combined values)
#> AIC AICc BIC BICc
#> 1763.958 1769.656 1807.938 1820.633
#>
#> Forecast errors:
#> MPE: 2.6%; sCE: 83.7%; Bias: 46%; MAPE: 6.7%
#> MASE: 0.284; sMAE: 9.6%; sMSE: 1.3%; rMAE: 0.308; rRMSE: 0.276
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: 0.18 seconds
#> Model estimated using adam() function: ARIMA(0,2,2)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 253.4134
#> ARMA parameters of the model:
#> MA:
#> theta1[1] theta2[1]
#> -0.9930 -0.0784
#>
#> Sample size: 45
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 40
#> Information criteria:
#> AIC AICc BIC BICc
#> 516.8269 518.3654 525.8602 528.7884
#>
#> Forecast errors:
#> ME: -360.822; MAE: 360.822; RMSE: 408.083
#> sCE: -35.454%; sMAE: 4.432%; sMSE: 0.251%
#> MASE: 4.992; RMSSE: 4.558; rMAE: 4.1; rRMSE: 3.682
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: 0.43 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: 921.5619
#> ARMA parameters of the model:
#> AR:
#> phi1[1] phi1[12]
#> 0.0926 0.1027
#> MA:
#> theta1[1] theta2[1] theta1[12] theta2[12]
#> -0.9155 0.2220 -0.1598 -0.0181
#>
#> Sample size: 116
#> Number of estimated parameters: 33
#> Number of degrees of freedom: 83
#> Information criteria:
#> AIC AICc BIC BICc
#> 1909.124 1936.490 1999.992 2065.035
#>
#> Forecast errors:
#> ME: 359.234; MAE: 611.446; RMSE: 716.375
#> sCE: 88.827%; sMAE: 8.399%; sMSE: 0.968%
#> MASE: 0.249; RMSSE: 0.226; rMAE: 0.27; rRMSE: 0.236
Second, it does not have intercept. If you want to have one, you can do this reintroducing ETS component and imposing some restrictions:
testModel <- adam(M3[[2568]], "ANN", silent=FALSE, lags=c(1,12), persistence=0,
orders=list(ar=c(1,1),i=c(1,1),ma=c(2,2)), distribution="dnorm")
testModel
#> Time elapsed: 0.38 seconds
#> Model estimated using adam() function: ETS(ANN)+SARIMA(1,1,2)[1](1,1,2)[12]
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 920.849
#> Persistence vector g:
#> alpha
#> 0
#>
#> ARMA parameters of the model:
#> AR:
#> phi1[1] phi1[12]
#> 0.1597 0.0800
#> MA:
#> theta1[1] theta2[1] theta1[12] theta2[12]
#> -0.9542 0.0936 -0.1334 0.0385
#>
#> Sample size: 116
#> Number of estimated parameters: 34
#> Number of degrees of freedom: 82
#> Number of provided parameters: 1
#> Information criteria:
#> AIC AICc BIC BICc
#> 1909.698 1939.081 2003.320 2073.157
#>
#> Forecast errors:
#> ME: 440.603; MAE: 667.885; RMSE: 787.571
#> sCE: 108.947%; sMAE: 9.175%; sMSE: 1.17%
#> MASE: 0.272; RMSSE: 0.249; rMAE: 0.295; rRMSE: 0.259
This way we get the global level, which acts as an intercept. The drift is not supported in the model either.
Third, you can specify parameters of ARIMA via the arma
parameter in the following manner:
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="dnorm",
arma=list(ar=c(0.1,0.1), ma=c(-0.96, 0.03, -0.12, 0.03)))
testModel
#> Time elapsed: 0.42 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,2)[12]
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 920.0852
#> ARMA parameters of the model:
#> AR:
#> phi1[1] phi1[12]
#> 0.1 0.1
#> MA:
#> theta1[1] theta2[1] theta1[12] theta2[12]
#> -0.96 0.03 -0.12 0.03
#>
#> Sample size: 116
#> Number of estimated parameters: 27
#> Number of degrees of freedom: 89
#> Number of provided parameters: 6
#> Information criteria:
#> AIC AICc BIC BICc
#> 1894.170 1911.352 1968.517 2009.355
#>
#> Forecast errors:
#> ME: 435.692; MAE: 661.24; RMSE: 779.401
#> sCE: 107.733%; sMAE: 9.084%; sMSE: 1.146%
#> MASE: 0.269; RMSSE: 0.246; rMAE: 0.292; rRMSE: 0.257
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(M3[[2568]], "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=M3[[2568]]$x[1:24]))
testModel
#> Time elapsed: 0.3 seconds
#> Model estimated using adam() function: SARIMA(1,1,2)[1](1,1,0)[12]
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 919.7511
#> ARMA parameters of the model:
#> AR:
#> phi1[1] phi1[12]
#> 0.0817 0.0966
#> MA:
#> theta1[1] theta2[1]
#> -0.9937 0.0823
#>
#> Sample size: 116
#> Number of estimated parameters: 31
#> Number of degrees of freedom: 85
#> Information criteria:
#> AIC AICc BIC BICc
#> 1901.502 1925.121 1986.863 2043.001
#>
#> Forecast errors:
#> ME: 428.915; MAE: 641.698; RMSE: 760.638
#> sCE: 106.057%; sMAE: 8.815%; sMSE: 1.092%
#> MASE: 0.261; RMSSE: 0.24; rMAE: 0.283; rRMSE: 0.251
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()
!
Finally, ARIMA is typically slower than ETS, mainly because the maxeval
is set by default to be at least 1000. But this is inevitable due to an increased complexity of the model - otherwise it won’t be estimated properly. If you want to speed things up, use initial="backcasting"
and reduce the number of iterations.
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:
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)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 206.7573
#> Persistence vector g (excluding xreg):
#> alpha
#> 0.9993
#>
#> Sample size: 132
#> Number of estimated parameters: 6
#> Number of degrees of freedom: 126
#> Information criteria:
#> AIC AICc BIC BICc
#> 425.5146 426.1866 442.8114 444.4520
#>
#> Forecast errors:
#> ME: 0.644; MAE: 1.418; RMSE: 1.827
#> sCE: 5.13%; sMAE: 0.628%; sMSE: 0.007%
#> MASE: 1.163; RMSSE: 1.169; rMAE: 0.633; rRMSE: 0.728
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:
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):
The two models might differ because they have different initialisation in the optimiser. 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$arma)
testModel
#> Time elapsed: 0 seconds
#> Model estimated using adam() function: ARIMAX(0,1,1)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 71.0402
#> ARMA parameters of the model:
#> MA:
#> theta1[1]
#> 0.2447
#>
#> Sample size: 132
#> Number of estimated parameters: 1
#> Number of degrees of freedom: 131
#> Number of provided parameters: 7
#> Information criteria:
#> AIC AICc BIC BICc
#> 144.0804 144.1112 146.9632 147.0384
#>
#> Forecast errors:
#> ME: 0.543; MAE: 0.579; RMSE: 0.753
#> sCE: 4.325%; sMAE: 0.256%; sMSE: 0.001%
#> MASE: 0.475; RMSSE: 0.482; rMAE: 0.259; rRMSE: 0.3
names(testModel$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)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 1e+300
#> Persistence vector g (excluding xreg):
#> alpha
#> 1.2447
#>
#> 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
#> 144.0804 144.1112 146.9632 147.0384
#>
#> Forecast errors:
#> ME: 0.543; MAE: 0.579; RMSE: 0.753
#> sCE: 4.325%; sMAE: 0.256%; sMSE: 0.001%
#> MASE: 0.475; RMSSE: 0.482; rMAE: 0.259; rRMSE: 0.3
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
#> 7.658422e-01 3.055245e-04 7.349038e-06 1.069980e-01 1.115520e-01 3.778273e-02
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(1,0,1))
summary(testModel)
#> Model estimated using adam() function: ETSX(AAN)+ARIMA(1,0,1)
#> Response variable: y
#> Distribution used in the estimation: Normal
#> Loss function type: likelihood; Loss function value: 97.2232
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> alpha 0.9995 0.1509 0.7008 1.0000
#> beta 0.0000 0.0153 0.0000 0.0303
#> phi1[1] 0.3262 0.1018 0.1247 0.5275
#> theta1[1] -0.1739 0.1669 -0.3362 0.1559
#> level 32.1939 13.5465 5.3706 58.9680
#> trend 0.0055 0.0581 -0.1096 0.1203
#> ARIMAState1 0.0103 1.6070 -3.1718 3.1865
#> xLag3 5.0720 0.2038 4.6685 5.4748
#> xLag7 1.5185 0.2714 0.9810 2.0550
#> xLag4 4.4649 0.3578 3.7565 5.1720
#> xLag6 2.6176 0.3969 1.8316 3.4021
#> xLag5 3.2385 0.3853 2.4756 3.9999
#>
#> Sample size: 132
#> Number of estimated parameters: 13
#> Number of degrees of freedom: 119
#> Information criteria:
#> AIC AICc BIC BICc
#> 220.4464 223.5312 257.9228 265.4539
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.
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(M3[[1234]], "XXX", silent=FALSE,
distribution=c("dnorm","dlaplace","ds"))
#> Evaluating models with different distributions... dnorm , dlaplace , ds , Done!
testModel
#> Time elapsed: 0.28 seconds
#> Model estimated using adam() function: ETS(AAN)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 255.2972
#> Persistence vector g:
#> alpha beta
#> 0.6828 0.2276
#>
#> Sample size: 45
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 40
#> Information criteria:
#> AIC AICc BIC BICc
#> 520.5943 522.1328 529.6277 532.5559
#>
#> Forecast errors:
#> ME: -348.216; MAE: 348.216; RMSE: 396.392
#> sCE: -34.215%; sMAE: 4.277%; sMSE: 0.237%
#> MASE: 4.818; RMSSE: 4.427; rMAE: 3.957; rRMSE: 3.576
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
):
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(M3[[1234]], "AAN", orders=list(ar=2,i=2,ma=2), silent=TRUE,
distribution=c("dnorm","dlaplace","ds","dgnorm"))
testModel
#> Time elapsed: 0.43 seconds
#> Model estimated using adam() function: ETS(AAN)+ARIMA(2,2,2)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 255.3152
#> Persistence vector g:
#> alpha beta
#> 2e-04 0e+00
#>
#> ARMA parameters of the model:
#> AR:
#> phi1[1] phi2[1]
#> -0.6085 0.1040
#> MA:
#> theta1[1] theta2[1]
#> -0.1176 -0.9393
#>
#> Sample size: 45
#> Number of estimated parameters: 13
#> Number of degrees of freedom: 32
#> Information criteria:
#> AIC AICc BIC BICc
#> 536.6304 548.3724 560.1170 582.4658
#>
#> Forecast errors:
#> ME: -312.957; MAE: 312.957; RMSE: 359.974
#> sCE: -30.75%; sMAE: 3.844%; sMSE: 0.195%
#> MASE: 4.33; RMSSE: 4.02; rMAE: 3.556; rRMSE: 3.248
However, this way the function will just use ARIMA(2,2,2) and fit it together with ETS. 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(M3[[1234]], "XXN", orders=list(ar=2,i=2,ma=2,select=TRUE),
distribution="default", silent=FALSE)
#> Evaluating models with different distributions... default , Selecting ARIMA orders... 5 %38 %71 % 100 %. The best ARIMA is selected.
#> Done!
testModel
#> Time elapsed: 0.11 seconds
#> Model estimated using adam() function: ETS(AAN)
#> Distribution assumed in the model: Normal
#> Loss function type: likelihood; Loss function value: 255.2972
#> Persistence vector g:
#> alpha beta
#> 0.6828 0.2276
#>
#> Sample size: 45
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 40
#> Information criteria:
#> AIC AICc BIC BICc
#> 520.5943 522.1328 529.6277 532.5559
#>
#> Forecast errors:
#> ME: -348.216; MAE: 348.216; RMSE: 396.392
#> sCE: -34.215%; sMAE: 4.277%; sMSE: 0.237%
#> MASE: 4.818; RMSSE: 4.427; rMAE: 3.957; rRMSE: 3.576
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 thw width level
in sample and then refits auto.adam()
with the dummy variables for the outliers. Here how it works:
testModel <- auto.adam(Mcomp::M3[[2568]], "PPP", silent=FALSE, outliers="use",
distribution="default")
#> Evaluating models with different distributions... default ,
#> Dealing with outliers...
testModel
#> Time elapsed: 1.01 seconds
#> Model estimated using adam() function: ETSX(MMdM)
#> Distribution assumed in the model: Inverse Gaussian
#> Loss function type: likelihood; Loss function value: 854.3258
#> Persistence vector g (excluding xreg):
#> alpha beta gamma
#> 0.0233 0.0233 0.0196
#> Damping parameter: 0.9529
#> Sample size: 116
#> Number of estimated parameters: 22
#> Number of degrees of freedom: 94
#> Information criteria:
#> AIC AICc BIC BICc
#> 1752.652 1763.533 1813.231 1839.094
#>
#> Forecast errors:
#> ME: 752.6; MAE: 867.837; RMSE: 1109.625
#> sCE: 186.094%; sMAE: 11.922%; sMSE: 2.323%
#> MASE: 0.353; RMSSE: 0.35; rMAE: 0.383; rRMSE: 0.365
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 textbook (this is a work in progress at the moment).
Hyndman, Rob J, Anne B Koehler, J Keith Ord, and Ralph D Snyder. 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.