Complex Exponential Smoothing
Function estimates CES in state space form with information potential equal to errors and returns several variables.
ces(data, seasonality = c("none", "simple", "partial", "full"), lags = c(frequency(data)), formula = NULL, regressors = c("use", "select", "adapt"), initial = c("backcasting", "optimal", "complete"), a = NULL, b = NULL, loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE", "MSCE"), h = 0, holdout = FALSE, bounds = c("admissible", "none"), silent = TRUE, model = NULL, ...) auto.ces(data, seasonality = c("none", "simple", "partial", "full"), lags = c(frequency(data)), formula = NULL, regressors = c("use", "select", "adapt"), initial = c("backcasting", "optimal", "complete"), ic = c("AICc", "AIC", "BIC", "BICc"), loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE", "MSCE"), h = 0, holdout = FALSE, bounds = c("admissible", "none"), silent = TRUE, ...) ces_old(data, seasonality = c("none", "simple", "partial", "full"), initial = c("backcasting", "optimal"), a = NULL, b = NULL, ic = c("AICc", "AIC", "BIC", "BICc"), loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE", "MSCE"), h = 10, holdout = FALSE, bounds = c("admissible", "none"), silent = c("all", "graph", "legend", "output", "none"), ...)
data: Vector, containing data needed to be forecasted. If a matrix (or data.frame / data.table) is provided, then the first column is used as a response variable, while the rest of the matrix is used as a set of explanatory variables. formula can be used in the latter case in order to define what relation to have.
seasonality: The type of seasonality used in CES. Can be: none
simple - Simple seasonality, using lagged CES (based on t-m observation, where m is the seasonality lag); partial - Partial seasonality with the real seasonal component (equivalent to additive seasonality); full - Full seasonality with complex seasonal component (can do both multiplicative and additive seasonality, depending on the data). First letter can be used instead of full words.In case of the auto.ces() function, this parameter defines which models to try.
lags: Vector of lags to use in the model. Allows defining multiple seasonal models.
formula: Formula to use in case of explanatory variables. If NULL, then all the variables are used as is. Can also include trend, which would add the global trend. Only needed if data is a matrix or if trend is provided.
regressors: The variable defines what to do with the provided explanatory variables: "use" means that all of the data should be used, while "select" means that a selection using ic should be done, "adapt" will trigger the mechanism of time varying parameters for the explanatory variables.
initial: Should be a character, which can be "optimal", meaning that all initial states are optimised, or "backcasting", meaning that the initials of dynamic part of the model are produced using backcasting procedure (advised for data with high frequency). In the latter case, the parameters of the explanatory variables are optimised. This is recommended for CESX. Alternatively, you can set initial="complete"
backcasting, which means that all states (including explanatory variables) are initialised via backcasting.
a: First complex smoothing parameter. Should be a complex number.
NOTE! CES is very sensitive to a and b values so it is advised either to leave them alone, or to use values from previously estimated model.
b: Second complex smoothing parameter. Can be real if seasonality="partial". In case of seasonality="full" must be complex number.
loss: The type of Loss Function used in optimization. loss can be:
likelihood - the model is estimated via the maximisation of the likelihood of the function specified in distribution;MSE (Mean Squared Error),MAE (Mean Absolute Error),HAM (Half Absolute Moment),LASSO - use LASSO to shrink the parameters of the model;RIDGE - use RIDGE to shrink the parameters of the model;TMSE - Trace Mean Squared Error,GTMSE - Geometric Trace Mean Squared Error,MSEh - optimisation using only h-steps ahead error,MSCE - Mean Squared Cumulative Error.In case of LASSO / RIDGE, the variables are not normalised prior to the estimation, but the parameters are divided by the mean values of explanatory variables.
Note that model selection and combination works properly only for the default loss="likelihood".
Furthermore, just for fun the absolute and half analogues of multistep estimators are available: MAEh, TMAE, GTMAE, MACE, HAMh, THAM, GTHAM, CHAM.
Last but not least, user can provide their own function here as well, making sure that it accepts parameters actual, fitted and B. Here is an example:
lossFunction <- function(actual, fitted, B)return(mean(abs(actual-fitted)))
loss=lossFunction
h: The forecast horizon. Mainly needed for the multistep loss functions.
holdout: Logical. If TRUE, then the holdout of the size h
is taken from the data (can be used for the model testing purposes).
bounds: The type of bounds for the persistence to use in the model estimation. Can be either admissible - guaranteeing the stability of the model, or none - no restrictions (potentially dangerous).
silent: Specifies, whether to provide the progress of the function or not. If TRUE, then the function will print what it does and how much it has already done.
model: A previously estimated GUM model, if provided, the function will not estimate anything and will use all its parameters.
...: Other non-documented parameters. See adam for details. However, there are several unique parameters passed to the optimiser in comparison with adam: 1. algorithm0, which defines what algorithm to use in nloptr for the initial optimisation. By default, this is "NLOPT_LN_BOBYQA". 2. algorithm determines the second optimiser. By default this is "NLOPT_LN_NELDERMEAD". 3. maxeval0 and maxeval, that determine the number of iterations for the two optimisers. By default, maxeval0=1000, maxeval=40*k, where k is the number of estimated parameters. 4. xtol_rel0 and xtol_rel, which are 1e-8 and 1e-6 respectively. There are also ftol_rel0, ftol_rel, ftol_abs0 and ftol_abs, which by default are set to values explained in the nloptr.print.options() function.
ic: The information criterion to use in the model selection.
Object of class "adam" is returned with similar elements to the adam function.
The function estimates Complex Exponential Smoothing in the state space form described in Svetunkov et al. (2022) with the information potential equal to the approximation error.
The auto.ces() function implements the automatic seasonal component selection based on information criteria.
ces_old() is the old implementation of the model and will be discontinued starting from smooth v4.5.0.
ces() uses two optimisers to get good estimates of parameters. By default these are BOBYQA and then Nelder-Mead. This can be regulated via ... - see details below.
For some more information about the model and its implementation, see the vignette: vignette("ces","smooth")
y <- rnorm(100,10,3) ces(y, h=20, holdout=FALSE) y <- 500 - c(1:100)*0.5 + rnorm(100,10,3) ces(y, h=20, holdout=TRUE) ces(BJsales, h=8, holdout=TRUE) ces(AirPassengers, h=18, holdout=TRUE, seasonality="s") ces(AirPassengers, h=18, holdout=TRUE, seasonality="p") ces(AirPassengers, h=18, holdout=TRUE, seasonality="f") y <- ts(rnorm(100,10,3),frequency=12) # CES with and without holdout auto.ces(y,h=20,holdout=TRUE) auto.ces(y,h=20,holdout=FALSE) # Selection between "none" and "full" seasonalities auto.ces(AirPassengers, h=12, holdout=TRUE, seasonality=c("n","f"), ic="AIC") ourModel <- auto.ces(AirPassengers) summary(ourModel) forecast(ourModel, h=12)
Svetunkov I. (2023) Smooth forecasting with the smooth package in R. arXiv:2301.01790. tools:::Rd_expr_doi("10.48550/arXiv.2301.01790") .
Svetunkov I. (2015 - Inf) "smooth" package for R - series of posts about the underlying models and how to use them: https://openforecast.org/category/r-en/smooth/.
Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM) (1st ed.). Chapman and Hall/CRC. tools:::Rd_expr_doi("10.1201/9781003452652") , online version: https://openforecast.org/adam/.
Snyder, R. D., 1985. Recursive Estimation of Dynamic Linear Models. Journal of the Royal Statistical Society, Series B (Methodological) 47 (2), 272-276.
Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. tools:::Rd_expr_doi("10.1007/978-3-540-71918-2") .
Svetunkov, I., Kourentzes, N., & Ord, J. K. (2022). Complex exponential smoothing. Naval Research Logistics, 69(8), 1108–1123. https://doi.org/10.1002/nav.22074
adam, es
Ivan Svetunkov, ivan@svetunkov.com