smooth: forecasting using state-space models

Ivan Svetunkov

2021-06-13

This vignette explains how to use functions in smooth package, what they produce, what each field in outputs and what returned values mean. Underlying statistical models are not discussed here, but if you want to know more about them, then there is a document “Statistical models underlying functions of ‘smooth’ package for R”. Some of the features of the package are also explained in my blog.

The package includes the following functions:

  1. adam() - ADAM - Advanced Dynamic Adaptive Model, the model that encompasses both ETS and ARIMA;
  2. es() - Exponential Smoothing;
  3. ssarima() - State-Space ARIMA, also known as Several Seasonalities ARIMA and msarima(), aka Multiple Seasonal ARIMA;
  4. ces() - Complex Exponential Smoothing;
  5. gum() - Generalised Univariate Model;
  6. sma() - Simple Moving Average in state-space form;
  7. Simulate functions of the package.
  8. smoothCombine() - function that combines forecasts of the main univariate functions of smooth package based on information criteria.
  9. oes() - Occurrence part of iETS model – function that estimates probability of occurrence of variable using one of the following model types: 1. Fixed probability; 2. Odds ratio probability; 3. Inverse odds ratio probability; 4. Direct probability; 5. General. It can also select the most appropriate model among these five. The model produced by oes() can then be used in any forecasting function as input variable for occurrence parameter. This is the new function introduced in smooth v2.5.0, substituting the old iss() function.

The functions (1) - (4) and (6) return object of class smooth, (5) returns the object of class vsmooth, (7) returns smooth.sim class and finally (8) returns oes or viss (depending on the function used). There are several methods for these classes in the package.

Some other functions, which are not considered as core and important:

  1. cma() - Centred Moving Average based on sma() and msarima()
  2. msdecompose - Multiple Seasonal Decomposition based on centred moving averages. Useful if a series with several frequencies need to be decomposed. The frequencies are specified separately in the lags parameter. Given that this function does not rely on any state space model, it might be moved to a different package at some point.

Methods for the class smooth

There are several methods that can be used together with the forecasting functions of the package. When a model is saved to some object ourModel, these function will do some magic. Here’s the list of all the available methods with brief explanations:

  1. print(ourModel) – function prints brief output with explanation of what was fitted, with what parameters and errors;
  2. summary(ourModel) – returns an output with parameters, standard errors and confidence intervals in case of adam() function. For the other functions it is equivalent to print(ourModel);
  3. actuals(ourModel) – returns actual values;
  4. fitted(ourModel) – fitted values of the model;
  5. residuals(ourModel) – residuals of constructed model;
  6. rstandard(ourModel) – standardised residuals of the model;
  7. rstudent(ourModel) – studentised residuals of the model;
  8. rmultistep(ourModel) – returns in sample multiple steps ahead forecast errors;
  9. AIC(ourModel), BIC(ourModel), AICc(ourModel) and BICc(ourModel) – information criteria of the constructed model. AICc() and BICc() functions are not standard stats functions and are imported from greybox package and modified in smooth for the specific models;
  10. plot(ourModel) – produces plots for the diagnostics of the constructed model. There are 9 options of what to produce, see ?plot.smooth() for more details. Note that if the number of states is higher than 10, then several graphs are produced for the option which=9.
  11. forecast(ourModel) – point and interval forecasts. Returns object of class smooth.forecast;
  12. plot(forecast(ourModel)) – produces graph with actuals, forecast, fitted and prediction interval using graphmaker() function from greybox package.
  13. simulate(ourModel) – produces data simulated from provided model (not supported by adam() yet);
  14. logLik(ourModel) – returns log-likelihood of the model;
  15. nobs(ourModel) – returns number of observations in-sample we had;
  16. nparam(ourModel) – number of estimated parameters (originally from greybox package);
  17. pointLik(ourModel) – likelihood values for each separate observation;
  18. sigma(ourModel) – variance of the residuals of the model;
  19. lags(ourModel) – lags of the model (used with ssarima() and gum());
  20. orders(ourModel) – orders of the model (can be used with ssarima(), gum() and sma());
  21. modelType(ourModel) – returns the type of the model. Returns something like “MMM” for ETS(MMM). Can be used with es(), ces() and ets();
  22. errorType(ourModel) – the type of the error of a model (additive or multiplicative);
  23. coef(ourModel) – returns the vector of all the estimated coefficients of the model;
  24. confint(ourModel) – returns confidence intervals for parameters of the model. Only works with adam();
  25. vcov(ourModel) – returns the covariance matrix of parameters. Only works with adam();
  26. coefbootstrap(ourModel) – returns the bootstrapped coefficients of model (imported from greybox package). Only works with adam();
  27. formula(ourModel) – returns the formula for the measurement equation. This is a proper formula in case of adam() and only decorative one for all the other functions;
  28. multicov(ourModel) – covariance matrix of multiple steps ahead forecast errors;
  29. refit(ourModel) – produces bootstrapped trajectories from adam() model. Only works with adam();
  30. reforecast(ourModel) – produces forecasts based on bootstrapped trajectories from a model. Only works with adam();