Model time series using mixture autoregressive (MAR) models. Implemented are frequentist (EM) and Bayesian methods for estimation, prediction and model evaluation. See Wong and Li (2002) <doi:10.1111/1467-9868.00222>, Boshnakov (2009) <doi:10.1016/j.spl.2009.04.009>), and the extensive references in the documentation.

## Details

Package mixAR provides functions for modelling with mixture autoregressive (MixAR) models. The S4 class "MixARGaussian" can be used when the error distributions of the components are standard Gaussian. The class "MixARgen" admits arbitrary (well, within reason) distributions for the error components. Both classes inherit from the virtual class "MixAR".

Estimation can be done with fit_mixAR. Currently, the EM algorithm is used for estimation.

For "MixARGaussian" the M-step of the EM algorithm reduces to a system of linear equations. For "MixARgen" the problem is substantially non-linear. The implementation is fairly general but currently not optimised for efficiency. The specification of the error distributions went through several stages and may still be reviewed. However, backward compatibility will be kept.

## Author

Georgi N. Boshnakov [aut, cre], Davide Ravagli [aut]

Maintainer: Georgi N. Boshnakov <georgi.boshnakov@manchester.ac.uk>

## References

Akinyemi MI (2013). Mixture autoregressive models: asymptotic properties and application to financial risk. Ph.D. thesis, Probability and Statistics Group, School of Mathematics, University of Manchester.

Boshnakov GN (2009). “Analytic expressions for predictive distributions in mixture autoregressive models.” Stat. Probab. Lett. , 79(15), 1704--1709. doi: 10.1016/j.spl.2009.04.009 , https://doi.org/10.1016/j.spl.2009.04.009.

Boshnakov GN (2011). “On First and Second Order Stationarity of Random Coefficient Models.” Linear Algebra Appl., 434(2), 415--423. doi: 10.1016/j.laa.2010.09.023 , https://doi.org/10.1016/j.laa.2010.09.023.

Fong PW, Li WK, Yau CW, Wong CS (2007). “On a mixture vector autoregressive model.” Can. J. Stat. , 35(1), 135--150.

Ravagli D, Boshnakov GN (2020). “Bayesian analysis of mixture autoregressive models covering the complete parameter space.” https://arxiv.org/abs/2006.11041.

Ravagli D, Boshnakov GN (2020). “Portfolio optimization with mixture vector autoregressive models.” https://arxiv.org/abs/2005.13396.

Hossain AS (2012). Complete Bayesian analysis of some mixture time series models. Ph.D. thesis, Probability and Statistics Group, School of Mathematics, University of Manchester.

Wong CS (1998). Statistical inference for some nonlinear time series models. Ph.D. thesis, University of Hong Kong, Hong Kong .

Wong CS, Li WK (2000). “On a mixture autoregressive model.” J. R. Stat. Soc., Ser. B, Stat. Methodol. , 62(1), 95--115.

Wong CS, Li WK (2001). “On a logistic mixture autoregressive model.” Biometrika , 88(3), 833--846. doi: 10.1093/biomet/88.3.833 , https://doi.org/10.1093/biomet/88.3.833.

Wong CS, Li WK (2001). “On a mixture autoregressive conditional heteroscedastic model.” J. Am. Stat. Assoc. , 96(455), 982--995. doi: 10.1198/016214501753208645 , https://doi.org/10.1198/016214501753208645.

## Examples

## object 'exampleModels' contains a number of models for examples and testing
names(exampleModels)
#>  [1] "WL_ibm"     "WL_A"       "WL_B"       "WL_I"       "WL_II"
#>  [6] "WL_ibm_gen" "WL_ibm_t3v" "WL_ibm_tf"  "WL_At"      "WL_Bt_1"
#> [11] "WL_Bt_2"    "WL_Bt_3"    "WL_Ct_1"    "WL_Ct_2"    "WL_Ct_3"   exampleModels$WL_ibm #> (To see the internal structure of the object, use function 'str'.) #> #> An object of class "MixARGaussian" #> Number of components: 3 #> prob shift scale order ar_1 ar_2 #> Comp_1 0.5439 0 4.8227 2 0.6792 0.3208 #> Comp_2 0.4176 0 6.0082 2 1.6711 -0.6711 #> Comp_3 0.0385 0 18.1716 1 1.0000 #> #> Distributions of the error components: #> standard Gaussian #> ## some of the models below are available in object 'exampleModels'; ## the examples here show how to create them from scratch mo_WLprob <- c(0.5439, 0.4176, 0.0385) # model coefficients from Wong&Li mo_WLsigma <- c(4.8227, 6.0082, 18.1716) mo_WLar <- list(c(0.6792, 0.3208), c(1.6711, -0.6711), 1) mo_WL <- new("MixARGaussian", prob = mo_WLprob, scale = mo_WLsigma, arcoef = mo_WLar) mo_WL_A <- new("MixARGaussian" # WongLi, model A , prob = c(0.5, 0.5) , scale = c(5, 1) , shift = c(0, 0) , arcoef = list(c(0.5), c(1.1)) ) mo_WL_B <- new("MixARGaussian" # WongLi, model B , prob = c(0.75, 0.25) , scale = c(5, 1) , shift = c(0, 0) , arcoef = list(c(0.5), c(1.4)) ) mo_WL_I <- new("MixARGaussian" # WongLi, model I , prob = c(0.4, 0.3, 0.3) , scale = c(1, 1, 5) , shift = c(0, 0, -5) , arcoef = list(c(0.9, -0.6), c(-0.5), c(1.50, -0.74, 0.12)) ) mo_WL_II <- new("MixARGaussian" # WongLi, model II , prob = c(0.4, 0.3, 0.3) , scale = c(1, 1, 5) , shift = c(5, 0, -5) , arcoef = list(c(0.9, -0.6), c(-0.7, 0), c( 0, 0.80)) ) ## MixAR models with arbitrary dist. of the components ## (user interface not finalized) ## Gaussian mo_WLgen <- new("MixARgen", prob = mo_WLprob, scale = mo_WLsigma, arcoef = mo_WLar, dist = list(dist_norm)) ## t_3 mo_WLt3v <- new("MixARgen", prob = mo_WLprob, scale = mo_WLsigma, arcoef = mo_WLar, dist = list(fdist_stdt(3, fixed = FALSE))) ## t_20, t_30, t_40 (can be used to start estimation) mo_WLtf <- new("MixARgen", prob = mo_WLprob, scale = mo_WLsigma, arcoef = mo_WLar, dist = list(generator = function(par) fn_stdt(par, fixed = FALSE), param = c(20, 30, 40))) ## data(ibmclose, package = "fma") # for ibmclose' ## The examples below are quick but some of them are marked as 'not run' ## to avoid cumulative time of more than 5s on CRAN. ## fit a MAR(2,2,1) model a0a <- fit_mixAR(as.numeric(fma::ibmclose), c(2, 2, 1), crit = 1e-4) ## same with 2 sets of automatically generated initial values. # \donttest{ a0b <- fit_mixAR(as.numeric(fma::ibmclose), c(2, 2, 1), 2, crit = 1e-4) # } ## fix the shift parameters: a1a <- fit_mixAR(as.numeric(fma::ibmclose), c(2, 2, 1), fix = "shift", crit = 1e-4) ## ... with 3 sets of automatically generated initial values. # \donttest{ a1b <- fit_mixAR(as.numeric(fma::ibmclose), c(2, 2, 1), 3, fix = "shift", crit = 1e-4) # } # \donttest{ ## specify the model using a MixAR model object a1c <- fit_mixAR(as.numeric(fma::ibmclose), a1a$model, init = a0a$model, fix = "shift", crit = 1e-4) ## fit a model like mo_WL using as initial values 2 automatically generated sets. a2 <- fit_mixAR(as.numeric(fma::ibmclose), mo_WL, 2, fix = "shift", permute = TRUE, crit = 1e-4) # } moT_B3 <- new("MixARgen" , prob = c(0.3, 0.3, 0.4) , scale = c(2, 1, 0.5) , shift = c(5, -5, 0) , arcoef = list(c(0.5, 0.24), c(-0.9), c(1.5, -0.74, 0.12)) # t4, t4, t10 , dist = distlist("stdt", c(4,10), fixed = c(FALSE, TRUE), tr = c(1, 1, 2)) ) moT_C1 <- new("MixARgen" , prob = c(0.3, 0.3, 0.4) , scale = c(2, 1, 0.5) , shift = c(5, -5, 0) , arcoef = list(c(0.5, 0.24), c(-0.9), c(1.5, -0.74, 0.12)) # t4, t7, N(0,1) , dist = distlist(c("stdt", "stdt", "stdnorm"), c(4,7)) ) ## demonstrate reuse of existing models exampleModels$WL_Bt_1
#> (To see the internal structure of the object, use function 'str'.)
#>
#> An object of class "MixARgen"
#> Number of components: 3
#>        prob shift scale order ar_1 ar_2  ar_3
#> Comp_1 0.3    5   2.0     2    0.5  0.24
#> Comp_2 0.3   -5   1.0     1   -0.9
#> Comp_3 0.4    0   0.5     3    1.5 -0.74 0.12
#>
#> Distributions of the error components:
#> 	Component 1: Student t with 4 df
#> 	Component 2: Student t with 6 df
#> 	Component 3: Student t with 10 df
#> moT_C2 <- new("MixARgen"
, model = exampleModels$WL_Bt_1 , dist = distlist(c("stdt", "stdt", "stdnorm"), c(4,7)) # t4, t7, N(0,1) ) moT_C3 <- new("MixARGaussian", model = exampleModels$WL_Bt_1 )
`