| Title: | Mixture Autoregressive Models |
|---|---|
| Description: | 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. |
| Authors: | Georgi N. Boshnakov [aut, cre], Davide Ravagli [aut] |
| Maintainer: | Georgi N. Boshnakov <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.22.7.9000 |
| Built: | 2024-09-13 02:54:15 UTC |
| Source: | https://github.com/geobosh/mixar |
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.
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.
Georgi N. Boshnakov [aut, cre], Davide Ravagli [aut]
Maintainer: Georgi N. Boshnakov <[email protected]>
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.
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.
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.” 2006.11041, https://arxiv.org/abs/2006.11041.
Ravagli D, Boshnakov GN (2020). “Portfolio optimization with mixture vector autoregressive models.” 2005.13396, 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.
Wong CS, Li WK (2001). “On a mixture autoregressive conditional heteroscedastic model.” J. Am. Stat. Assoc., 96(455), 982-995. doi:10.1198/016214501753208645.
## object 'exampleModels' contains a number of models for examples and testing names(exampleModels) exampleModels$WL_ibm ## 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. 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. a1b <- fit_mixAR(as.numeric(fma::ibmclose), c(2, 2, 1), 3, fix = "shift", crit = 1e-4) ## 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 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 )## object 'exampleModels' contains a number of models for examples and testing names(exampleModels) exampleModels$WL_ibm ## 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. 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. a1b <- fit_mixAR(as.numeric(fma::ibmclose), c(2, 2, 1), 3, fix = "shift", crit = 1e-4) ## 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 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 )
Samples parameters of a mixture autoregressive model from respective posterior distributions.
bayes_mixAR(y, model, fix_shift = FALSE, a = .2, c = 2, tau, nsim, burnin)bayes_mixAR(y, model, fix_shift = FALSE, a = .2, c = 2, tau, nsim, burnin)
y |
a time series (currently a numeric vector). |
model |
an object of class |
fix_shift |
should |
a, c
|
numeric hyperparameters, default values are from Richardson and Green (1997). |
tau |
|
nsim |
|
burnin |
|
For details see Ravagli and Boshnakov (2020).
a list with following elements:
mix_weights |
a |
scale |
a |
precision |
a |
shift |
a |
mu |
a |
ARcoeff |
a list which elements are matrices, one for each AR component in the mixture. |
acc_rate |
|
n_samp |
the sample size, calculated as |
LatentZ |
the latest Z variables drawn (for utility only). |
n_comp |
the number of components in the mixture. |
fix_shift |
same as input, whether the shift parameter was kept fixed or not. |
Davide Ravagli
Richardson S, Green PJ (1997). “On Bayesian Analysis of Mixtures with an Unknown Number of Components.” J. R. Stat. Soc., Ser. B, Stat. Methodol., 59(4), 731-792.
Ravagli D, Boshnakov GN (2020). “Bayesian analysis of mixture autoregressive models covering the complete parameter space.” 2006.11041, https://arxiv.org/abs/2006.11041.
prob <- c(0.5, 0.5) sigma <- c(1, 2) ar <- list(-0.5, 1) model <- new("MixARGaussian", prob = prob, scale = sigma, arcoef = ar) ## MAR(1,1) model y <- mixAR_sim(model, 300, rep(0, max(model@order))) bayes_mixAR(y, model, fix_shift = FALSE, tau = c(.15,.25), nsim = 20, burnin = 10)prob <- c(0.5, 0.5) sigma <- c(1, 2) ar <- list(-0.5, 1) model <- new("MixARGaussian", prob = prob, scale = sigma, arcoef = ar) ## MAR(1,1) model y <- mixAR_sim(model, 300, rep(0, max(model@order))) bayes_mixAR(y, model, fix_shift = FALSE, tau = c(.15,.25), nsim = 20, burnin = 10)
Reversible Jump MCMC algorithm to choose the optimal autoregressive order of each component of a mixture autoregressive model.
Choose_pk(y, model, fix_shift = FALSE, tau, pmax, method, par = NULL, nsim)Choose_pk(y, model, fix_shift = FALSE, tau, pmax, method, par = NULL, nsim)
y |
a time series. Currently a |
model |
an object inheriting from class |
fix_shift |
whether the shift/mean parameter should be kept fixed to its
starting value or not. Default is |
tau |
tuning parameters for Metropolis-Hastings algorithm in sampling AR coefficients. |
pmax |
the largest autoregressive order allowed for each component. |
method |
character vector of length 1. Method for calculating probability of
new AR order to be increased/decreased by 1 unit. Currently
available |
par |
|
nsim |
|
out |
a dataframe with |
fix_shift |
the choice made for the shift/mean parameters. |
method |
the method used to increase/decrease AR orders. |
Choose_pk currenlty supports class "MixARGaussian"
only.
Davide Ravagli
Ravagli D, Boshnakov GN (2020). “Bayesian analysis of mixture autoregressive models covering the complete parameter space.” 2006.11041, https://arxiv.org/abs/2006.11041.
bx_dx for more details on the method
model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = list(-0.5, 1) ) # note: arcoef != list(-0.5, 1.1) == exampleModels$WL_At@arcoef@a set.seed(1234) n <- 50 # 200 y <- mixAR_sim(model, n, rep(0, max(model@order)), nskip = 100) nsim <- 25 # 100 pk_star <- Choose_pk(y, model, tau = c(.15, .25), pmax = 5, method = "NULL", nsim = nsim)model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = list(-0.5, 1) ) # note: arcoef != list(-0.5, 1.1) == exampleModels$WL_At@arcoef@a set.seed(1234) n <- 50 # 200 y <- mixAR_sim(model, n, rep(0, max(model@order)), nskip = 100) nsim <- 25 # 100 pk_star <- Choose_pk(y, model, tau = c(.15, .25), pmax = 5, method = "NULL", nsim = nsim)
Compute the log-likelihood of a MixAR model for a univariate time series.
cond_loglik(model, x, index) cond_loglikS(model, x, index)cond_loglik(model, x, index) cond_loglikS(model, x, index)
model |
a MixAR model. |
x |
a time series or numeric vector. |
index |
a vector of integers giving the indices in |
cond_loglik computes the conditional log-likelihood of a MixAR
model. Conditional here means conditional on the first p values
being fixed, where p is the maximum AR order of the components
of the model.
Argument index can be used to compute the sum over a subset of
time points.
cond_loglikS is a variant of cond_loglik for the case
when the input model contains seasonal AR coefficients.
the log-likelihood, a numeric value
Georgi N. Boshnakov and Davide Ravagli
## data(ibmclose, package = "fma") # doesn't work with fma v2.4, using '::' cond_loglik(exampleModels$WL_ibm, as.numeric(fma::ibmclose)) cond_loglik(exampleModels$WL_ibm_gen, as.numeric(fma::ibmclose)) data(lynx) # for 'lynx' data sar <- new("raggedCoefS", a = list(c(1.1022, -0.2835), c(1.5279, -0.8871)), as = list(c(0, 0), 0), s = 10) ## SMAR(2; 2, 2)(2, 1)_10 model_s10 <- new("MixARGaussian", prob = c(.3, .7), scale = c(.08, .202), arcoef = sar, shift = c(.7,1)) cond_loglikS(model_s10, log(lynx)) cond_loglikS(model_s10, log(lynx), index = 45:114) # on reduced dataset## data(ibmclose, package = "fma") # doesn't work with fma v2.4, using '::' cond_loglik(exampleModels$WL_ibm, as.numeric(fma::ibmclose)) cond_loglik(exampleModels$WL_ibm_gen, as.numeric(fma::ibmclose)) data(lynx) # for 'lynx' data sar <- new("raggedCoefS", a = list(c(1.1022, -0.2835), c(1.5279, -0.8871)), as = list(c(0, 0), 0), s = 10) ## SMAR(2; 2, 2)(2, 1)_10 model_s10 <- new("MixARGaussian", prob = c(.3, .7), scale = c(.08, .202), arcoef = sar, shift = c(.7,1)) cond_loglikS(model_s10, log(lynx)) cond_loglikS(model_s10, log(lynx), index = 45:114) # on reduced dataset
The noise distributions are specified by a list of functions for the density, quantiles, etc. This object demonstrates this for the standard normal distribution.
dist_normdist_norm
This is a list of functions or names of functions for calculations related to the standard normal distribution. Currently it has elements with the following names: "pdf", "cdf", "rand", "logpdf", "Fscore", "xFscore", "Parscore", "get_param", "set_param", "any_param", "show".
dist_norm may be used to specify the noise distribution for
MixAR models. It can be used as a template if other
distributions are needed, see also fdist_stdnorm.
dist_norm dist_norm$pdf dist_norm$cdfdist_norm dist_norm$pdf dist_norm$cdf
Optimise the scale parameters in MixAR models from class MixARgen. Internal function.
em_est_dist(tau, etk, parscore, sigma, nu, logpdf)em_est_dist(tau, etk, parscore, sigma, nu, logpdf)
tau |
conditional probabilities, an object of class "MixComp", see 'Details'. |
etk |
component residuals, see 'Details'. |
parscore |
the score function(s), see 'Details'. |
sigma |
current values of the scale parameters, a numeric vector. |
nu |
current values of the parameters. w.r.t. which optimisation is done. |
logpdf |
the log of pdf as a function of the parameters. |
One or more of the error distributions of a MixAR model may have
parameters that are considered unknown. In that case
em_est_dist can be used to optimise with respect to them.
The representation of the error distributions in "MixARgen" models
carries all the necessary information about
parameters. em_est_dist works by extracting their current
values from logpdf, passes them to the optimisation function
(or equation solver) and stores the result back into logpdf.
em_est_dist is quite general, as long as logpdf is
prepared according to the conventions it expects (this is so if they
are valid elements of the dist slot of "MixARgen" objects).
the new values of the parameters
Calculates estimates of scale parameters of MixAR models from conditional probabilities and mixture ‘residuals’. Used in EM algorithm.
tauetk2sigmahat(tau, etk) em_est_sigma(tau, etk, Fscore, sigma, dontfix = rep(TRUE, length(sigma)), compwise = FALSE)tauetk2sigmahat(tau, etk) em_est_sigma(tau, etk, Fscore, sigma, dontfix = rep(TRUE, length(sigma)), compwise = FALSE)
tau |
the conditional probabilities for the groups, a |
etk |
component "residuals", MixComp object(?). |
Fscore |
the score function(s) of the noise distributions. |
sigma |
current values of the scale parameters. |
compwise |
if |
dontfix |
a logical vector containig |
tauetk2sigmahat calculates estimates of the scale parameters
for a MixAR time series with Gaussian components. There is an explicit
formula in that case.
em_est_sigma calculates estimates of the scale parameters in
the general case. The non-linear equations are solved using functions
from package BB. The equations for the components can often be
solved independently. When that is the case, compwise may speed
things up a little.
The new values of the scale parameters, a numeric vector
Gaussian EM-step with random initialisation.
em_rinit(y, order, partempl) etk2tau(etk)em_rinit(y, order, partempl) etk2tau(etk)
y |
time series. |
order |
MixAR order, vector of length the number of components. |
partempl |
parameter template, a list containing one element for each mixture
component, see |
etk |
MixAR component residuals, a matrix. |
em_rinit generates random MAR residuals, performs a non-distributional
E-step, and a Gaussian M-step.
etk2tau estimates tau from component residuals
only. Note that this is unlike em_tau, which also needs
the noise pdf's, as well as estimates of the mixture probabilities.
em_rinit uses etk2tau to start the EM algorithm.
for em_rinit, an object from class "MixARGaussian"
for etk2tau, a matrix representing tau (i-th row
contains probabilities corresponding to the i-th observation)
Georgi N. Boshnakov
Create estimation templates from MixAR model objects.
est_templ(model, shift = TRUE, ...)est_templ(model, shift = TRUE, ...)
model |
a |
shift |
logical, see Details. |
... |
currently not used. |
Argument model is used as a template to specify values of
parameters and/or which parameters to estimate or fix. In general,
If a value of a parameter in model is NA, then it is to
be estimated. Otherwise the parameter is taken as is.
The current implementation is incomplete. In particular, the AR parameters are always designated for estimation.
Argument shift can be used to overwrite some or values
component shift in model. If shift has length
one, it is replicated to the number of MixAR components. If
shift[k] is TRUE, then the shift coefficient for the
k-th component is set to NA to request its
estimation. Otherwise, the value of the shift for the k-th component
in model is taken.
Argument shift has a default of TRUE which causes the
shift coefficients to be estimated irrespectively of their values in
model.
est_templ returns a list with as many components as there are
MixAR components in the model. The k-th component of the list is itself a list
specifing which parameters of the i-th MixAR component to estimate or fix.
a list, as described in Details.
exampleModels$WL_A est_templ(exampleModels$WL_A) est_templ(exampleModels$WL_A, shift = FALSE) exampleModels$WL_I est_templ(exampleModels$WL_I)exampleModels$WL_A est_templ(exampleModels$WL_A) est_templ(exampleModels$WL_A, shift = FALSE) exampleModels$WL_I est_templ(exampleModels$WL_I)
MixAR models for examples and testing.
exampleModelsexampleModels
Coefficients of models from the examples in
Wong and Li (2000). Variations on these with different
noise distributions are used throughout the examples in mixAR.
The models are from classes inheriting from class "MixAR".
exampleModels is a list with the following components:
|
WL_ibm WL_A WL_B WL_I WL_II WL_ibm_gen WL_ibm_t3v WL_ibm_tf WL_At WL_Bt_1 WL_Bt_2 WL_Bt_3 WL_Ct_1 WL_Ct_2 WL_Ct_3 |
Each component is a MixAR model, i.e. an object inheriting from class
"MixAR".
Wong CS, Li WK (2000). “On a mixture autoregressive model.” J. R. Stat. Soc., Ser. B, Stat. Methodol., 62(1), 95-115.
## use these instead of moWL, moWL_A, moWL_B, etc. exampleModels$WL_ibm exampleModels$WL_A exampleModels$WL_B # what is the difference between A and B? show_diff(exampleModels$WL_A, exampleModels$WL_B) exampleModels$WL_I exampleModels$WL_II #show_diff(exampleModels$WL_I, exampleModels$WL_II) exampleModels$WL_ibm_gen exampleModels$WL_ibm_t3v exampleModels$WL_ibm_tf #show_diff(exampleModels$WL_ibm_gen, exampleModels$WL_ibm_t3v) exampleModels$WL_At exampleModels$WL_Bt_1 exampleModels$WL_Bt_2 exampleModels$WL_Bt_3 ## what is different between Bt_2 and Bt_1? (df of component 2) show_diff(exampleModels$WL_Bt_2, exampleModels$WL_Bt_1) exampleModels$WL_Ct_1 exampleModels$WL_Ct_2 exampleModels$WL_Ct_3 ## The models were created with something like: moWLprob <- c(0.5439,0.4176,0.0385) moWLsigma <- c(4.8227,6.0082,18.1716) moWLar <- list(c(0.6792,0.3208), c(1.6711,-0.6711), 1) moWL <- new("MixARGaussian", prob = moWLprob, scale = moWLsigma, arcoef = moWLar) moWLgen <- new("MixARgen", prob = moWLprob, scale = moWLsigma, arcoef = moWLar, dist = list(dist_norm)) ## clean up a bit rm(moWLprob, moWLsigma, moWLar, moWL, moWLgen)## use these instead of moWL, moWL_A, moWL_B, etc. exampleModels$WL_ibm exampleModels$WL_A exampleModels$WL_B # what is the difference between A and B? show_diff(exampleModels$WL_A, exampleModels$WL_B) exampleModels$WL_I exampleModels$WL_II #show_diff(exampleModels$WL_I, exampleModels$WL_II) exampleModels$WL_ibm_gen exampleModels$WL_ibm_t3v exampleModels$WL_ibm_tf #show_diff(exampleModels$WL_ibm_gen, exampleModels$WL_ibm_t3v) exampleModels$WL_At exampleModels$WL_Bt_1 exampleModels$WL_Bt_2 exampleModels$WL_Bt_3 ## what is different between Bt_2 and Bt_1? (df of component 2) show_diff(exampleModels$WL_Bt_2, exampleModels$WL_Bt_1) exampleModels$WL_Ct_1 exampleModels$WL_Ct_2 exampleModels$WL_Ct_3 ## The models were created with something like: moWLprob <- c(0.5439,0.4176,0.0385) moWLsigma <- c(4.8227,6.0082,18.1716) moWLar <- list(c(0.6792,0.3208), c(1.6711,-0.6711), 1) moWL <- new("MixARGaussian", prob = moWLprob, scale = moWLsigma, arcoef = moWLar) moWLgen <- new("MixARgen", prob = moWLprob, scale = moWLsigma, arcoef = moWLar, dist = list(dist_norm)) ## clean up a bit rm(moWLprob, moWLsigma, moWLar, moWL, moWLgen)
Estimate a MixAR model for a time series. This is a generic function. The methods defined in package mixAR are described here.
fit_mixAR(x, model, init, fix, ...)fit_mixAR(x, model, init, fix, ...)
x |
a time series. |
model |
model, object inheriting from MixAR class. |
init |
what initializations to do, see Details. |
fix |
which parameters to fix, see Details. |
... |
additional arguments for the methods. |
Method dispatch is done on the first three arguments:
x, model and init.
model specifies the model to fit. If model inherits from
"MixAR", it is used as a template. If init is missing,
the parameters of model are also used as initial values.
model can also be a numeric vector specifying the order of a
MixAR model with Gaussian components.
Argument init can be used to give initial values in variety of
ways. If it is a MixAR object it doesn't need to be of the same class
as model, to allow using as initial values common parameters
of different MixAR models. A positive integer value of init
asks to run the fitting procedure init times, each time
generating random initial values.
init can also be a list. In that case, each component of the
list should itself be an acceptable value for init and the
fitting procedure is run with each component of init.
Argument fix can be given in a number of ways. Note however
that currently there is no method dispatch on it.
Currently the default method for fit_mixAR just throws error,
since there seems no suitable default task to do.
See individual methods for further details.
a MixAR model or a list of MixAR models, depending on the arguments.
signature(x = "ANY", model = "ANY", init = "ANY")The default method throws error.
signature(x = "ANY", model = "MixAR", init = "missing")This is equivalent to setting init = model.
signature(x = "ANY", model = "MixAR", init = "MixAR")model is a template for the result, init specifies
initial values for the parameters. In principle, model and
init may be from different classes, to allow for example
using AR coefficients from a Gaussian fit for other distributions.
signature(x = "ANY", model = "MixAR", init = "numeric")init must be a single positive integer here. The model is
fitted init times, each time starting with a new set of
randomly generated initial values. If select is TRUE,
the default, the model with the largest likelihood is returned,
otherwise a list containing the init fitted models is
returned.
signature(x = "ANY", model = "MixAR", init = "list")Each element of the list init should be an acceptable value
for init. The model is fitted with the initial value set to each
element of init. A list containing the fitted models is
returned.
signature(x = "ANY", model = "MixARGaussian", init = "MixAR")signature(x = "ANY", model = "numeric", init = "missing")This is equivalent to setting init = 1.
signature(x = "ANY", model = "numeric", init = "numeric")A numeric model should be a vector of non-negative integers
specifying the order of the MixAR model. The distribution of the
components is assumed Gaussian.
## model coefficients from Wong&Li (IBM fit) prob <- exampleModels$WL_ibm@prob # c(0.5439, 0.4176, 0.0385) sigma <- exampleModels$WL_ibm@scale # c(4.8227, 6.0082, 18.1716) ar <- exampleModels$WL_ibm@arcoef@a # list(c(0.6792, 0.3208), c(1.6711, -0.6711), 1) ## data(ibmclose, package = "fma") # `ibmclose' mot30 <- new("MixARgen", prob = prob, scale = sigma, arcoef = ar, dist = distlist("stdt", c(30, 30, 30))) mot20_30_40 <- new("MixARgen", prob = prob, scale = sigma, arcoef = ar, dist = distlist("stdt", c(20, 30, 40))) mo_t20_t30_norm <- new("MixARgen", prob = prob, scale = sigma, arcoef = ar, dist = distlist(c("stdt", "stdt", "stdnorm"), c(20, 30))) ## Gaussian components fi0 <- fit_mixAR(fma::ibmclose, exampleModels$WL_ibm, fix = "shift", crit = 1e-4) fi0$model if(FALSE){ # don't run on CRAN to save a couple of seconds ## remove minniter/maxniter below for realistic results. ## std-t components fi1 <- fit_mixAR(fma::ibmclose, mot30, fix = "shift", crit = 1e-4, minniter = 1, maxniter = 3) fi1$model ## 1st and 2nd components std-t, 3rd Gaussian fi2 <- fit_mixAR(fma::ibmclose, mo_t20_t30_norm, fix = "shift", crit = 1e-4, minniter = 1, maxniter = 3) fi2$model }## model coefficients from Wong&Li (IBM fit) prob <- exampleModels$WL_ibm@prob # c(0.5439, 0.4176, 0.0385) sigma <- exampleModels$WL_ibm@scale # c(4.8227, 6.0082, 18.1716) ar <- exampleModels$WL_ibm@arcoef@a # list(c(0.6792, 0.3208), c(1.6711, -0.6711), 1) ## data(ibmclose, package = "fma") # `ibmclose' mot30 <- new("MixARgen", prob = prob, scale = sigma, arcoef = ar, dist = distlist("stdt", c(30, 30, 30))) mot20_30_40 <- new("MixARgen", prob = prob, scale = sigma, arcoef = ar, dist = distlist("stdt", c(20, 30, 40))) mo_t20_t30_norm <- new("MixARgen", prob = prob, scale = sigma, arcoef = ar, dist = distlist(c("stdt", "stdt", "stdnorm"), c(20, 30))) ## Gaussian components fi0 <- fit_mixAR(fma::ibmclose, exampleModels$WL_ibm, fix = "shift", crit = 1e-4) fi0$model if(FALSE){ # don't run on CRAN to save a couple of seconds ## remove minniter/maxniter below for realistic results. ## std-t components fi1 <- fit_mixAR(fma::ibmclose, mot30, fix = "shift", crit = 1e-4, minniter = 1, maxniter = 3) fi1$model ## 1st and 2nd components std-t, 3rd Gaussian fi2 <- fit_mixAR(fma::ibmclose, mo_t20_t30_norm, fix = "shift", crit = 1e-4, minniter = 1, maxniter = 3) fi2$model }
Estimate a linear regression model for time series with residuals from a mixture autoregressive process.
fit_mixARreg(x, y, mixARmodel, EMinit, ...) mixARreg(x, y, mixARmodel, tol = 1e-6, niter = 200)fit_mixARreg(x, y, mixARmodel, EMinit, ...) mixARreg(x, y, mixARmodel, tol = 1e-6, niter = 200)
x |
the response time series (currently a numeric vector). |
y |
|
mixARmodel |
An object inheriting from class |
EMinit |
starting values for EM estimation of MixAR parameters. If present,
must be a named list, containing at least |
tol |
threshold for convergence criterion. |
... |
passed on to |
niter |
maximal number of iterations. |
fit_mixARreg is a generic function.
Currently there is no default method for fit_mixARreg.
Arguments y, mixARmodel, EMinit can be given in a
number of ways, see individual methods for details.
Argument mixARmodel gives the details of the the MixAR part of
the model and initial values for the parameters. For
fit_mixARreg this can alternatively be done with a list using
argument EMinit. Currently, at least one of the two must be
supplied, and if both are present EMinit is ignored.
mixARreg performs a two-step estimation of a linear regression
model with mixture autoregressive residuals. It is the workhorse for
fit_mixARreg which calls it to do the computations.
reg |
The summary output of the regression part of the model. |
mixARmodel |
Estimates of the mixture autoregressive part of the model. |
niter |
The number of iterations until convergence. |
signature(x = "ANY", y = "data.frame", mixARmodel =
"MixAR", EMinit = "missing")
Covariates y are supplied as data.frame: each column
corresponds to one covariate. Initialization of MixAR paramters is
done using input mixARmodel
signature(x = "ANY", y = "matrix",
mixARmodel = "MixAR", EMinit = "missing")
Covariates y are supplied as matrix: each column
corresponds to one covariate. Initialization of MixAR paramters is
done using input mixARmodel
signature(x = "ANY", y = "numeric", mixARmodel = "MixAR", EMinit = "missing")
Covariates y is supplied as numeric: this method handles the
simple regression case with a single covairate.
Initialization of MixAR paramters is done using input mixARmodel
signature(x = "ANY", y = "ANY", mixARmodel = "missing", EMinit = "list")EMinit must be a named list (see 'Arguments').
signature(x = "ANY", y = "ANY", mixARmodel = "MixAR", EMinit = "list")When both mixARmodel and EMinit are supplied, the second is ignored.
Estimation is done using the function mixARreg within each
method.
Davide Ravagli and Georgi N. Boshnakov
## Simulate covariates set.seed(1234) n <- 50 # for CRAN y <- data.frame(rnorm(n, 7, 1), rt(n, 3), rnorm(n, 3, 2)) ## Build mixAR part model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = exampleModels$WL_At@arcoef@a ) # list(-0.5, 1.1) ## Simulate from MixAR part u <- mixAR_sim(model, n, 0) x <- 10 + y[, 1] + 3 * y[, 2] + 2 * y[, 3] + u ## Fit model ## Using MixARGaussian fit_mixARreg(x = x, y = y, mixARmodel = model, niter = 3) ## Using EMinit EMinit <- list(prob = exampleModels$WL_At@prob, scale = exampleModels$WL_At@scale, arcoef = exampleModels$WL_At@arcoef@a) fit_mixARreg(x = x, y = y, EMinit = EMinit, niter = 3)## Simulate covariates set.seed(1234) n <- 50 # for CRAN y <- data.frame(rnorm(n, 7, 1), rt(n, 3), rnorm(n, 3, 2)) ## Build mixAR part model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = exampleModels$WL_At@arcoef@a ) # list(-0.5, 1.1) ## Simulate from MixAR part u <- mixAR_sim(model, n, 0) x <- 10 + y[, 1] + 3 * y[, 2] + 2 * y[, 3] + u ## Fit model ## Using MixARGaussian fit_mixARreg(x = x, y = y, mixARmodel = model, niter = 3) ## Using EMinit EMinit <- list(prob = exampleModels$WL_At@prob, scale = exampleModels$WL_At@scale, arcoef = exampleModels$WL_At@arcoef@a) fit_mixARreg(x = x, y = y, EMinit = EMinit, niter = 3)
Estimate a MixVAR model for a multivariate time series. This is a generic function. The methods defined in package MixAR are described here.
fit_mixVAR(x, model, fix, ...)fit_mixVAR(x, model, fix, ...)
x |
a multivariate time series (currently a numeric matrix). |
model |
model, object inheriting from MixVAR class. |
fix |
if TRUE, fix the shift parameters. |
... |
additional arguments for the methods (not currently used). |
model specifies the model to fit. If model inherits from
"MixVAR", it is used as a template. Estimation is done via
EM-Algorithm, using the function mixVARfit.
Currently the default method for fit_mixAR just throws error,
since there seems no suitable default task to do.
a MixVAR model.
signature(x = "ANY", model = "MixVAR")signature(x = "ANY", model = "ANY")AR <- list() AR[[1]] <- array(c(0.5, -0.3, -0.6, 0, 0, 0.5, 0.4, 0.5, -0.3), dim = c(3, 3, 1)) AR[[2]] <- array(c(-0.5, 0.3, 0, 1, 0, -0.5, -0.4, -0.2, 0.5), dim = c(3, 3, 1)) prob <- c(0.75, 0.25) shift <- cbind(c(0, 0, 0), c(0, 0, 0)) Sigma1 <- cbind(c(1, 0.5, -0.4), c(0.5, 2, 0.8), c(-0.4, 0.8, 4)) Sigma2 <- cbind(c(1, 0.2, 0), c(0.2, 2, -0.15), c(0, -0.15, 4)) Sigma <- array(c(Sigma1, Sigma2), dim = c(3, 3, 2)) m <- new("MixVARGaussian", prob = prob, vcov = Sigma, arcoef = AR, shift = shift) set.seed(1234) y <- mixVAR_sim(m, n = 100, init = matrix(0, ncol = 3), nskip = 50, flag = FALSE) fit_mixVAR(y, m, tol = 1e-3) mixVARfit(y, m, tol = 1e-3)AR <- list() AR[[1]] <- array(c(0.5, -0.3, -0.6, 0, 0, 0.5, 0.4, 0.5, -0.3), dim = c(3, 3, 1)) AR[[2]] <- array(c(-0.5, 0.3, 0, 1, 0, -0.5, -0.4, -0.2, 0.5), dim = c(3, 3, 1)) prob <- c(0.75, 0.25) shift <- cbind(c(0, 0, 0), c(0, 0, 0)) Sigma1 <- cbind(c(1, 0.5, -0.4), c(0.5, 2, 0.8), c(-0.4, 0.8, 4)) Sigma2 <- cbind(c(1, 0.2, 0), c(0.2, 2, -0.15), c(0, -0.15, 4)) Sigma <- array(c(Sigma1, Sigma2), dim = c(3, 3, 2)) m <- new("MixVARGaussian", prob = prob, vcov = Sigma, arcoef = AR, shift = shift) set.seed(1234) y <- mixVAR_sim(m, n = 100, init = matrix(0, ncol = 3), nskip = 50, flag = FALSE) fit_mixVAR(y, m, tol = 1e-3) mixVARfit(y, m, tol = 1e-3)
These functions and objects are mostly internal and should not be needed for routine use. Generate noise distribution, currently standard normal and standardised t-distributions. These functions can be used as templates for new distributions.
fdist_stdnorm() fdist_stdt(df, fixed = TRUE) fn_stdt(df, fixed = TRUE) b_show(x) distlist(type, param, ncomp = NULL, fixed = FALSE, tr = NULL, ...) ed_nparam ed_parse(s) ed_skeleton(df, fixed = FALSE, n = length(df), tr = NULL) ed_src ed_stdnorm ed_stdt ed_stdt0 ed_stdt1 ft_stdtfdist_stdnorm() fdist_stdt(df, fixed = TRUE) fn_stdt(df, fixed = TRUE) b_show(x) distlist(type, param, ncomp = NULL, fixed = FALSE, tr = NULL, ...) ed_nparam ed_parse(s) ed_skeleton(df, fixed = FALSE, n = length(df), tr = NULL) ed_src ed_stdnorm ed_stdt ed_stdt0 ed_stdt1 ft_stdt
df |
degrees of freedom |
fixed |
if TRUE, the parameters are fixed, otherwise they are variable, see Details. |
x |
a fitted object. |
type |
list of distributions. |
param |
parameters. |
ncomp |
number of components. |
tr |
transformation. |
... |
not used. |
s |
named vector. |
n |
number of different degrees of freedom. |
If argument fixed is TRUE, estimation functions assume that the
parameter(s) are fixed, otherwise they estimate it. The support is
incomplete, see below.
fdist_stdnorm is for the standard normal distribution. For
example dist_norm is generated by it.
fdist_stdt is for the t-distribution with df degrees of
freedom.
fn_stdt is also for the t-distribution but the degrees of
freedom, df, may be a vector. The value is a list of
distributions. Although the list can be obtained by repeated calls of
fdist_stdt
The support is incomplete. In particular, if parameter fixed
is TRUE, changes to the parameter(s) should probably not be allowed
(this can be achieved by simply dropping the corresponding function
from the list). However, a thorough rethinking is necessary, as I
introduced it on the fly while developing estimation functions and
forbidding changes may necessitate changes in the code. Changes are
useful for estimation for convenience but also to avoid recreating the
whole distributions again and again.
However, there is a major drawback, which in the final version needs to be addressed satisfactorily. Since parameters are held in local environments, changes to the parameters are reflected in copies of the objects. For example, an estimation function (or the user) may call another function with a model containing an object generated by the above functions and assign the result to a new object. However if the parameters of the noise distribution are changed in the process this will be reflected in the original model.
Note that the above effect is valid only if an object generated by the above functions is reused. Objects created by different calls have different environments, so the problem does not arise for them.
stdt3 <- fdist_stdt(3) stdt3v <- fdist_stdt(3, fixed = FALSE) fn_stdt(c(20, 30, 40), fixed = FALSE) mo_tf <- new("MixARgen", prob = exampleModels$WL_ibm@prob, scale = exampleModels$WL_ibm@scale, arcoef = exampleModels$WL_ibm@arcoef@a, dist = list(generator = function(par) fn_stdt(par, fixed = FALSE), param = c(20, 30, 40))) mo_tf str(mo_tf) noise_dist(mo_tf, "pdf") parameters(mo_tf) parameters(mo_tf, names = TRUE) get_edist(mo_tf) noise_params(mo_tf)stdt3 <- fdist_stdt(3) stdt3v <- fdist_stdt(3, fixed = FALSE) fn_stdt(c(20, 30, 40), fixed = FALSE) mo_tf <- new("MixARgen", prob = exampleModels$WL_ibm@prob, scale = exampleModels$WL_ibm@scale, arcoef = exampleModels$WL_ibm@arcoef@a, dist = list(generator = function(par) fn_stdt(par, fixed = FALSE), param = c(20, 30, 40))) mo_tf str(mo_tf) noise_dist(mo_tf, "pdf") parameters(mo_tf) parameters(mo_tf, names = TRUE) get_edist(mo_tf) noise_params(mo_tf)
get_edist in package mixAR Methods for function get_edist in package mixAR
get_edist gives the error (or noise) distribution of MixAR
objects.
Currently the distribution is returned as a list of functions. The list contains one element for each component. If the error distributions of all components are the same, then the list may contain a single element representing the common error distribution.
Note that the distribution is not necessarily stored in slot
dist in this format, see the description of this slot in class
"MixARgen".
Such a slot may even not exist if the distribution of the error
components is fixed as is the case for class MixARGaussian.
Each subclass of MixAR needs to define a method for
get_edist.
signature(model = "MixAR")Issue an error message and stop.
This method is invoked for subclasses of MixAR that have
not defined their own method for get_edist. This is an
error.
signature(model = "MixARGaussian")Return an object representing the fact that the error
distributions of the components of MixARGaussian objects
are standard normal.
signature(model = "MixARgen")Return an object representing the error distributions of the
components of MixARgen objects. The distributions are not
necessarilly the same for such objects.
"MixComp"
Generalised inner product and methods for class MixComp. The methods for MixComp provide for very convenient computing with MixAR models.
inner(x, y, star = "*", plus = .mplus)inner(x, y, star = "*", plus = .mplus)
x |
the first argument. |
y |
the second argument. |
star |
function to apply to pairs of elements from |
plus |
function to apply to combine the results from the pairs, default is addition, as for the usual inner product. |
inner computes a generalised inner product x . y, where
multiplication and summation can be replaced by other functions.
The default method of inner applies star to the
corresponding pairs of elements and combines them with plus.
There is no recycling, if x and y have different
lengths, an error is raised. The elements of x and y
are accessed with "[[". plus should be an n-ary
operation.
the inner product, the type of the result depends on the arguments
Methods for inner product between a "MixComp" object and a
vector are similar to a product between a matrix and a vector but
comply with the conventions of class "MixComp". For this reason
they are described in the help page for class
"MixComp", along with methods for other functions
and operators applied to "MixComp" objects.
signature(x = "ANY", y = "ANY", star = "ANY", plus = "ANY")This is the default method, see section Details.
signature(x = "MixComp", y = "missing", star = "missing", plus = "missing")see "MixComp".
signature(x = "MixComp", y = "numeric", star = "missing", plus = "missing")see "MixComp".
signature(x = "numeric", y = "MixComp", star = "missing", plus = "missing")see "MixComp".
signature(x = "MixComp", y = "numeric", star = "ANY", plus = "ANY")see "MixComp".
signature(x = "MixComp", y = "numeric", star = "ANY", plus = "missing")see "MixComp".
"MixComp"
inner(1:3, 2:4) # [1] 20 class(inner(1:3, 2:4)) # [1] "integer" ## compare to: 1:3 %*% 2:4 # 20, but (1,1)-matrix class(1:3 %*% 2:4) # matrixinner(1:3, 2:4) # [1] 20 class(inner(1:3, 2:4)) # [1] "integer" ## compare to: 1:3 %*% 2:4 # 20, but (1,1)-matrix class(1:3 %*% 2:4) # matrix
Checks if a MixAR model is stable. This is also the second order stationarity condition.
isStable(x)isStable(x)
x |
the model |
If each component of a MixAR model corresponds to a stable autoregression model, then the MixAR model is also stable. However, the MixAR model may be stable also when some of its components correspond to integrated or explosive AR models, see the references.
True if the model is stable (second order stationary), FALSE otherwise.
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.
Wong CS, Li WK (2000). “On a mixture autoregressive model.” J. R. Stat. Soc., Ser. B, Stat. Methodol., 62(1), 95-115.
isStable(exampleModels$WL_I) isStable(exampleModels$WL_II)isStable(exampleModels$WL_I) isStable(exampleModels$WL_II)
Takes the output from a MCMC simulation of parameters of a mixture, and detects whether labels switch has occured while sampling, using the method by Celeux (2000).
label_switch(x, m)label_switch(x, m)
x |
output from an MCMC sampling of a mixture. A |
m |
the number of observations in the sample that will be used to
initialise the algorithm. |
Function can be directly executed when x is one of
mix_weights, scale, precision, shift or
mu from bayes_mixAR output. ARcoeff cannot be input
as it is, but element from the list may be used.
A list of 2:
x |
The input matrix, with adjusted labels |
true_perm |
The "true" permutation at each iteration. |
There is no absolute choice on what x should be to obtain the
"true" permutation at any given point. User is subject to make the
most suitable choice, given output of their MCMC.
Davide Ravagli
Celeux G (2000). Bayesian Inference of Mixture: The Label Switching Problem.. Payne R., Green P. (eds) COMPSTAT. Physica, Heidelberg.
model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = exampleModels$WL_At@arcoef@a ) # list(-0.5, 1.1) y <- mixAR_sim(model, n = 300, init = rep(0, which.max(model@order))) ## just examples, use larger numbers in practice nsim <- 30 # 200 burnin <- 10 # 100 x <- bayes_mixAR(y, model, fix_shift = FALSE, tau = c(.15, .25), nsim = nsim, burnin = burnin) label_switch(x$mix_weights, m = 5)model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = exampleModels$WL_At@arcoef@a ) # list(-0.5, 1.1) y <- mixAR_sim(model, n = 300, init = rep(0, which.max(model@order))) ## just examples, use larger numbers in practice nsim <- 30 # 200 burnin <- 10 # 100 x <- bayes_mixAR(y, model, fix_shift = FALSE, tau = c(.15, .25), nsim = nsim, burnin = burnin) label_switch(x$mix_weights, m = 5)
Give a numeric vector containing non-redundant parameters of a MixAR model in a form suitable for use by optimisation routines. The methods defined in package mixAR for this generic function are described here.
lik_params(model)lik_params(model)
model |
a MixAR model. |
lik_params gives the parameters of a MixAR model as a numeric
vector.
This is a generic function. Parameters common to all MixAR models are arranged as described below. There are no other parameters when the error distributions do not contain parameters of their own. Methods for sub-classes with additional parameters should append them after the common parameters.
If is the number of components and is the
probability associated with the th component, then the
parameters are put in a vector as follows:
component probabilities, ,
(note: is not included)
scales, ,
shifts, ,
AR coefficients of the 1st component,
AR coefficients of the 2nd component,
...
AR coefficients of the th component.
A numeric vector containing all parameters except the probability associated with the last component.
signature(model = "MixAR")signature(model = "MixARgen")The probability associated with the th component is omitted
as it is redundant. This makes it possible to try unconstrained
optimisation though it is not likely to give useful results since
there are other restrictions on the probabilities.
Georgi N. Boshnakov
Create a function for the computation of the conditional likelihood of MixAR models for a given time series. The methods for this generic function defined in package mixAR are described here.
make_fcond_lik(model, ts)make_fcond_lik(model, ts)
model |
a |
ts |
the time series |
The returned value is a function, say f(x), whose only argument
is a numeric vector of parameters with the arrangement of
lik_params, for which it computes the conditional
loglikelihood.
f can be given to optimisation routines.
Argument model is an object inheriting from MixAR and
determines the structure of the MixAR model for the function,
f, that it creates. So, properties of the model,
such as number of components, AR order, and distribution of the noise
components are fixed when f is created and only the numeric
values of the parameters are changed by calls to it.
a function of one argument, the parameters of a MixAR model as a
numeric vector with the arrangement of lik_params, for which
it computes the conditional loglikelihood
The environment of the returned function contains the time series and
the model object (initially argument model, later the model
used in the last call to f). So, these things can be extracted
from f.
Is it necessary to create convenience functions?
signature(model = "MixAR", ts = "numeric")The function implements the method by Chib (1995) and Chib and Jeliazkov (2001) for calculation of the marginal loglikelihood of a mixture autoregressive model. It automatically finds high density values for model parameters, and evaluates the likelihood at such points.
marg_loglik(y, model, tau, nsim, prob_mod)marg_loglik(y, model, tau, nsim, prob_mod)
y |
a time series (currently a numeric vector). |
model |
object of formal class |
tau |
tuning parameter for Metropolis-Hasting move to update autoregressive parameters. |
nsim |
sample size on which to evaluate highest density values. |
prob_mod |
this is currently the output from |
nsim is the sample size on which to evaluate highest density
values for each set of parameters. For example, choosing
nsim=1000 results in 1000*(g+3) (1000 iterations for
each autoregressive component, plus 1000 for mean and scale parameters
and mixing weights).
A list containing the following elements:
marg_loglik |
value of the marginal loglikelihood. |
phi_hd |
set of highest density autoregressive parameters. |
prec_hd |
set of highest density precision parameters. |
mu_hd |
set of highest density mean parameters. |
weig_hd |
set of highest density mixing weights. |
Davide Ravagli
Chib S (1995). “Marginal likelihood from the Gibbs output.” J. A. Stat. Ass., 90(432), 1313-1321.
Chib S, Jeliazkov I (2001). “Marginal likelihood from the Metropolis-Hastings output.” J. A. Stat. Ass., 96(453), 270-281.
prob <- c(0.5, 0.5) sigma <- c(1, 2) arco <- list(-0.5, 1) model <- new("MixARGaussian", prob = prob, scale = sigma, arcoef = arco) set.seed(1234) y <- mixAR_sim(model, 250, rep(0, max(model@order)), nskip = 100) # data nsim <- 10 # 50 marg_loglik(y, model, tau = c(.15, .25), nsim = nsim, 0.5)prob <- c(0.5, 0.5) sigma <- c(1, 2) arco <- list(-0.5, 1) model <- new("MixARGaussian", prob = prob, scale = sigma, arcoef = arco) set.seed(1234) y <- mixAR_sim(model, 250, rep(0, max(model@order)), nskip = 100) # data nsim <- 10 # 50 marg_loglik(y, model, tau = c(.15, .25), nsim = nsim, 0.5)
Compute component residuals for MixAR models.
mix_ek(model, x, index, xcond, scale)mix_ek(model, x, index, xcond, scale)
model |
a model. |
x |
time series. |
index |
a vector of positive integer specifying the indices for which to compute the residuls, has a natural default. |
xcond |
the past values needed for the conditional distribution, a numeric vector of length at least the maximal AR order of the components. |
scale |
logical or missing, if |
mix_ek computes component residuals from MixAR models.
It is highly desirable to use it along with mix_hatk and
the underlying function mixFilter. Doing this ensures
transparent code and easy maintenance. Also, more efficient
implementation can be introduced without changing other code.
signature(model = "MixAR", x = "numeric", index = "missing",
xcond = "numeric", scale = "logical")signature(model = "MixAR", x = "numeric", index = "missing",
xcond = "numeric", scale = "missing")signature(model = "MixAR", x = "numeric", index = "numeric",
xcond = "missing", scale = "logical")signature(model = "MixAR", x = "numeric", index = "numeric",
xcond = "missing", scale = "missing")Georgi N. Boshnakov
mixFilter which is used by mix_ek to do the job,
MixComp-class for easy manipulation of the returned
object.
class "MixAR"
Function and methods to compute component predictions for MixAR models
mix_hatk(model, x, index, xcond)mix_hatk(model, x, index, xcond)
model |
a model. |
x |
time series. |
index |
a vector of positive integers specifying the indices for which to compute the residuals, has a natural default. |
xcond |
the past values needed for the conditional distribution, a numeric vector of length at least the maximal AR order of the components. |
signature(model = "MixAR", x = "numeric", index = "numeric",
xcond = "missing")Georgi N. Boshnakov
class "MixAR"
Conditional moments of MixAR models.
mix_location(model, x, index, xcond) mix_variance(model, x, index, xcond) mix_central_moment(model, x, index, xcond, k) mix_moment(model, x, index, xcond, k) mix_kurtosis(...) mix_ekurtosis(...)mix_location(model, x, index, xcond) mix_variance(model, x, index, xcond) mix_central_moment(model, x, index, xcond, k) mix_moment(model, x, index, xcond, k) mix_kurtosis(...) mix_ekurtosis(...)
model |
a MixAR object. |
x |
a time series. |
index |
a vector of indices in |
xcond |
a time series, the point prediction is computed for the
first value after the end of the time series. Only the last
|
k |
a positive integer specifying the moment to compute. |
... |
passed on to |
These functions compute conditional moments and related quantities.
kurtosis and ekurtosis compute conditional kurtosis and
excess kurtosis, respectively. Effectively, they have the same
parameters as mix_central_moment, since they pass "..."
to it along with k = 4. It is an error to supply argument
k to the kurtosis functions.
when called with one argument (model), a function with argument xcond;
otherwise if xcond is not missing, a single numeric value;
otherwise a vector of length length(index).
I wrote the above description recently from reading six years old code, it may need further verification.
Georgi N. Boshnakov
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.
mix_pdf, mix_cdf, mix_qf
for the predictive distributions (pdf, cdf, quantiles);
## data(ibmclose, package = "fma") # `ibmclose' ibmclose <- as.numeric(fma::ibmclose) length(ibmclose) # 369 max(exampleModels$WL_ibm@order) # 2 ## compute point predictions for t = 3,...,369 pred <- mix_location(exampleModels$WL_ibm, ibmclose) plot(pred) ## compute one-step point predictions for t = 360,...369 mix_location(exampleModels$WL_ibm, ibmclose, index = 369 - 9:0 ) f <- mix_location(exampleModels$WL_ibm) # a function ## predict the value after the last f(ibmclose) ## a different way to compute one-step point predictions for t = 360,...369 sapply(369 - 10:1, function(k) f(ibmclose[1:k])) ## the results are the same, but notice that xcond gives past values ## while index above specifies the times for which to compute the predictions. identical(sapply(369 - 10:1, function(k) f(ibmclose[1:k])), mix_location(exampleModels$WL_ibm, ibmclose, index = 369 - 9:0 )) ## conditional variance f <- mix_variance(exampleModels$WL_ibm) # a function ## predict the value after the last f(ibmclose) ## a different way to compute one-step point predictions for t = 360,...369 sapply(369 - 10:1, function(k) f(ibmclose[1:k])) ## the results are the same, but notice that xcond gives past values ## while index above specifies the times for which to compute the predictions. identical(sapply(369 - 10:1, function(k) f(ibmclose[1:k])), mix_variance(exampleModels$WL_ibm, ibmclose, index = 369 - 9:0 )) # interesting example # bimodal distribution, low kurtosis, 4th moment not much larger than 2nd moWL <- exampleModels$WL_ibm mix_location(moWL,xcond = c(500,450)) mix_kurtosis(moWL,xcond = c(500,450)) f1pdf <- mix_pdf(moWL,xcond = c(500,450)) f1cdf <- mix_cdf(moWL,xcond = c(500,450)) gbutils::plotpdf(f1pdf,cdf=f1cdf) gbutils::plotpdf(f1cdf,cdf=f1cdf) f1cdf(c(400,480)) mix_variance(moWL,xcond = c(500,450)) mix_central_moment(moWL,xcond = c(500,450), k=2) sqrt(mix_variance(moWL,xcond = c(500,450))) sqrt(mix_central_moment(moWL,xcond = c(500,450), k=2))## data(ibmclose, package = "fma") # `ibmclose' ibmclose <- as.numeric(fma::ibmclose) length(ibmclose) # 369 max(exampleModels$WL_ibm@order) # 2 ## compute point predictions for t = 3,...,369 pred <- mix_location(exampleModels$WL_ibm, ibmclose) plot(pred) ## compute one-step point predictions for t = 360,...369 mix_location(exampleModels$WL_ibm, ibmclose, index = 369 - 9:0 ) f <- mix_location(exampleModels$WL_ibm) # a function ## predict the value after the last f(ibmclose) ## a different way to compute one-step point predictions for t = 360,...369 sapply(369 - 10:1, function(k) f(ibmclose[1:k])) ## the results are the same, but notice that xcond gives past values ## while index above specifies the times for which to compute the predictions. identical(sapply(369 - 10:1, function(k) f(ibmclose[1:k])), mix_location(exampleModels$WL_ibm, ibmclose, index = 369 - 9:0 )) ## conditional variance f <- mix_variance(exampleModels$WL_ibm) # a function ## predict the value after the last f(ibmclose) ## a different way to compute one-step point predictions for t = 360,...369 sapply(369 - 10:1, function(k) f(ibmclose[1:k])) ## the results are the same, but notice that xcond gives past values ## while index above specifies the times for which to compute the predictions. identical(sapply(369 - 10:1, function(k) f(ibmclose[1:k])), mix_variance(exampleModels$WL_ibm, ibmclose, index = 369 - 9:0 )) # interesting example # bimodal distribution, low kurtosis, 4th moment not much larger than 2nd moWL <- exampleModels$WL_ibm mix_location(moWL,xcond = c(500,450)) mix_kurtosis(moWL,xcond = c(500,450)) f1pdf <- mix_pdf(moWL,xcond = c(500,450)) f1cdf <- mix_cdf(moWL,xcond = c(500,450)) gbutils::plotpdf(f1pdf,cdf=f1cdf) gbutils::plotpdf(f1cdf,cdf=f1cdf) f1cdf(c(400,480)) mix_variance(moWL,xcond = c(500,450)) mix_central_moment(moWL,xcond = c(500,450), k=2) sqrt(mix_variance(moWL,xcond = c(500,450))) sqrt(mix_central_moment(moWL,xcond = c(500,450), k=2))
Function and methods to get the number of component in a mixture
object. For "MixComp" objects this is equivalent to
ncol.
mix_ncomp(x)mix_ncomp(x)
x |
an object, such as |
a number
signature(x = "MixAR")signature(x = "MixComp")Georgi N. Boshnakov
Gives conditional probability densities and distribution functions of mixture autoregressive models.
mix_pdf gives a probability density, mix_cdf a
distribution function. If argument x is supplied, the functions
are evaluated for the specified values of x, otherwise function
objects are returned and can be used for further computations, eg for
graphs.
mix_pdf and mix_cdf have methods with the following
signatures.
signature(model = "MixARGaussian", x = "missing", index = "missing", xcond = "numeric")Return (as a function of one argument) the conditional density
(respectively cdf), , of given
the past values xcond. The values in xcond are in
natural time order, e.g. the last value in xcond is
. xcond must contain enough values for the
computation of the conditional density (cdf) but if more are given,
only the necessary ones are used.
signature(model = "MixARGaussian", x = "numeric", index = "missing", xcond = "numeric")Compute the conditional density (respectively cdf) at the values given
by x.
signature(model = "MixARGaussian", x = "numeric", index = "numeric", xcond = "missing")Compute conditional densities (respectively cdf) for times
specified in index. For each index the
past values needed for the computation of the pdf (cdf) are
...,x[t-2],x[t-1].
signature(model = "MixARgen", x = "missing", index = "missing",
xcond = "numeric")signature(model = "MixARgen", x = "numeric", index = "missing",
xcond = "numeric")signature(model = "MixARgen", x = "numeric", index = "numeric",
xcond = "missing")Georgi N. Boshnakov
mix_moment for examples and computation of summary statistics of the
predictive distributions
mix_qf for computation of quantiles.
Gives conditional quantile functions of mixture autoregressive models.
mix_qf(model, p, x, index, xcond)mix_qf(model, p, x, index, xcond)
model |
mixAR model. |
p |
vector of probabilities. |
x |
time series. |
index |
vector of positive integers. |
xcond |
the past values needed for the conditional distribution, a numeric vector of length at least the maximal AR order of the components. |
depending on the arguments, a function for computing quantiles or a numeric vector representing quantiles, see sections 'Details' and 'Methods
signature(model = "MixARGaussian", p = "missing", x = "missing", index = "missing", xcond = "numeric")signature(model = "MixARGaussian", p = "numeric", x = "missing", index = "missing", xcond = "numeric")signature(model = "MixARGaussian", p = "numeric", x = "numeric", index = "numeric", xcond = "missing")Georgi N. Boshnakov
mix_moment for examples
Compute standard errors of estimates of MixAR models.
mix_se(x, model, fix_shift)mix_se(x, model, fix_shift)
x |
time series. |
model |
MixAR model, an object inheriting from class “MixAR”. |
fix_shift |
|
For formulas used in the computation, see Wong (1998).
a list with components:
standard_errors |
Standard error of parameter estimates, |
covariance_matrix |
The covariance matrix, obtained as inverse of the information matrix, |
Complete_Information |
Complete information matrix, |
Missing_Information |
Missing information matrix. |
signature(x = "ANY", model = "list")signature(x = "ANY", model = "MixAR")signature(x = "ANY", model = "MixARGaussian")Davide Ravagli
Wong CS (1998). Statistical inference for some nonlinear time series models. Ph.D. thesis, University of Hong Kong, Hong Kong.
## Example with IBM data ## data(ibmclose, package = "fma") moWLprob <- exampleModels$WL_ibm@prob # 2019-12-15; was: c(0.5339,0.4176,0.0385) moWLsigma <- exampleModels$WL_ibm@scale # c(4.8227,6.0082,18.1716) moWLar <- list(-0.3208, 0.6711,0) # @Davide - is this from some model? moWLibm <- new("MixARGaussian", prob = moWLprob, scale = moWLsigma, arcoef = moWLar) IBM <- diff(fma::ibmclose) mix_se(as.numeric(IBM), moWLibm, fix_shift = TRUE)$'standard_errors'## Example with IBM data ## data(ibmclose, package = "fma") moWLprob <- exampleModels$WL_ibm@prob # 2019-12-15; was: c(0.5339,0.4176,0.0385) moWLsigma <- exampleModels$WL_ibm@scale # c(4.8227,6.0082,18.1716) moWLar <- list(-0.3208, 0.6711,0) # @Davide - is this from some model? moWLibm <- new("MixARGaussian", prob = moWLprob, scale = moWLsigma, arcoef = moWLar) IBM <- diff(fma::ibmclose) mix_se(as.numeric(IBM), moWLibm, fix_shift = TRUE)$'standard_errors'
BIC calculations for mixture autoregressive models.
mixAR_BIC(y, model, fix = NULL, comp_loglik = TRUE, index) BIC_comp(x, y)mixAR_BIC(y, model, fix = NULL, comp_loglik = TRUE, index) BIC_comp(x, y)
y |
a time series. |
model |
the model for which to calculate BIC, an object inheriting from
class |
fix |
If |
comp_loglik |
Should the loglikelihood be calculated? Default is
|
index |
Discard the first |
x |
a list containing a combination of |
mixAR_BIC calculates the BIC criterion of a given MixAR
object with respect to a specified time series.
If index is specified, it has to be at least equal to the
largest autoregressive order. The function calculates BIC on the last
(index + 1):n data points.
BIC_comp calculates the value of BIC for the models listed in
x with respect to the specified time series y.
If the distributions of the components contain estimated parameters, then their number is included in the number of parameters for the calculation of BIC.
If comp_loglik = TRUE, the function calculates BIC based on the
given model, data and index.
If comp_loglik = FALSE and model is output from
fit_mixAR, it returns object vallogf from that list.
Davide Ravagli
model1 <- new("MixARGaussian", prob = c(0.5, 0.5), scale = c(1, 2), arcoef = list(-0.5, 1.1)) model2 <- new("MixARGaussian", prob = c(0.5, 0.3, 0.2), scale = c(1, 3, 8), arcoef = list(c(-0.5, 0.5), 1, 0.4)) set.seed(123) y <- mixAR_sim(model1, 400, c(0, 0, 0), nskip = 100) mixAR_BIC(y, model1) model_fit1 <- fit_mixAR(y, model1) model_fit2 <- fit_mixAR(y, model2, crit = 1e-4) mixAR_BIC(y, model_fit1) mixAR_BIC(y, model_fit2) BIC_comp(list(model1, model2, model_fit1, model_fit2), y) mixAR_BIC(y, model_fit1, index = 20) mixAR_BIC(y, model_fit2, index = 20)model1 <- new("MixARGaussian", prob = c(0.5, 0.5), scale = c(1, 2), arcoef = list(-0.5, 1.1)) model2 <- new("MixARGaussian", prob = c(0.5, 0.3, 0.2), scale = c(1, 3, 8), arcoef = list(c(-0.5, 0.5), 1, 0.4)) set.seed(123) y <- mixAR_sim(model1, 400, c(0, 0, 0), nskip = 100) mixAR_BIC(y, model1) model_fit1 <- fit_mixAR(y, model1) model_fit2 <- fit_mixAR(y, model2, crit = 1e-4) mixAR_BIC(y, model_fit1) mixAR_BIC(y, model_fit2) BIC_comp(list(model1, model2, model_fit1, model_fit2), y) mixAR_BIC(y, model_fit1, index = 20) mixAR_BIC(y, model_fit2, index = 20)
Compute conditional probabilities for the E-step of the EM algorithm for MixAR models. Internal function.
mixAR_cond_probs(model, y, indx = NULL)mixAR_cond_probs(model, y, indx = NULL)
model |
an object from a sub-class of "MixAR". |
y |
the time series, a numeric vector. |
indx |
indices of elements for which to compute residuals. |
This is essentially the E-step for the MixAR models.
the conditional probabilities, an object from class "MixComp".
Carry out diagnostic checks and tests on fitted mixAR models.
## S3 method for class 'MixAR' tsdiag(object, gof.lag = NULL, y, ask = interactive(), ..., plot = interactive(), std.resid = FALSE) mixAR_diag(model, y, ...)## S3 method for class 'MixAR' tsdiag(object, gof.lag = NULL, y, ask = interactive(), ..., plot = interactive(), std.resid = FALSE) mixAR_diag(model, y, ...)
model, object
|
the model on which to perform the checks, an object from class
|
gof.lag |
Goodness of fit lag(s) for the Ljung-Box tests. Vector containing
one or more positive integers. how many lags to compute for acf and pacf? The default is as that of
|
y |
a time series, currently a |
ask |
if |
plot |
if |
std.resid |
if |
... |
for |
It is recommended to use tsdiag. mixAR_diag is
essentially deprecated and is still here for compatibility with old
code. Moreover, the tsdiag method is more flexible. The only
advantage of mixAR_diag is that it accepts also a list for
argument model but this is equivalent to calling tsdiag
with object = model$model.
The function calculates several types of residuals, provides diagnostic plots for each of them, and returns numerical results. The following choices are currently available:
ACF/PACF of residuals,
ACF/PACF of U_residuals,
ACF/PACF of tau_residuals,
ACF/Histogram of tau_residuals.
In interactive sessions the user is presented with a menu to select
plot(s) from the available ones. The choice can be restricted to a
subset of them by giving argument plot a vector of integers.
This is most useful to select a particular plot, with somethinng like
plot = 2 in the call to tsdiag. plot is used as
an index vector, so plot = -1 would remove the first item
listed above from the offered alternatives.
Transformations on the data are performed, as described in Smith (1985).
Four types of residuals are computed:
difference (possibly scaled) between observed values and point predictions.
probability integral transform of the data using the CDF of the conditional distributions implied by the fitted model. For a good model these should resemble an IID sequence uniformly distributed on (0,1).
set of transformed U_residuals with the quantile function
of the standard normal distribution (qnorm). For a good
model these should resemble an IID sequence from N(0,1).
These residuals are calculated as the component specific residual
e_tk divided by its corresponding scale sigma_k,
according to under which component y_t has largest density. Under
correct model specification, these should be jointly
Normal. Shapiro-Wilk test is performed on this set of residual to
assess the hypothesis.
For all types of residual results for the Ljung-Box test are provided. This test is particularly relevant for the V- and tau-residuals.
Kolmogorov-Smirnov test is carried out for the U_residuals to assess the hypothesis of uniform distribution.
Shapiro-Wilk test of normality is applied to V- and tau-residuals.
returns invisibly a list with class "tsdiagMixAR", currently
containing the following components:
residuals |
ordinary residuals, |
U_residuals |
see Details, |
V_residuals |
see Details, |
tau_residuals |
see Details, |
BIC |
the value of the BIC criterion, a number. |
Each component, except BIC, is a list containing the residuals
in component value, Ljung-Box test in "Ljung-Box" and
possibly other tests suitable for the corresponding type of
residuals.
This function should be used for diagnostic checking of MixARGaussian
objects only.
Davide Ravagli and Georgi N. Boshnakov
Smith JQ (1985). “Diagnostic checks of non-standard time series models.” Journal of Forecasting, 4(3), 283-291. doi:10.1002/for.3980040305, https://onlinelibrary.wiley.com/doi/pdf/10.1002/for.3980040305, https://onlinelibrary.wiley.com/doi/abs/10.1002/for.3980040305.
Wong CS, Li WK (2000). “On a mixture autoregressive model.” J. R. Stat. Soc., Ser. B, Stat. Methodol., 62(1), 95-115.
model1 <- new("MixARGaussian", prob = c(0.5, 0.5), scale = c(1, 2), arcoef = list(-0.5, 1.1)) set.seed(123) y <- mixAR_sim(model1, 400, c(0,0,0), nskip = 100) fit1 <- fit_mixAR(y, model1) d <- tsdiag(fit1$model, c(10, 20, 50), y) d ## This will put each plot in a separate file (mydiag01.pdf, ..., mydiag04.pdf) ## pdf("mydiag%02d.pdf", onefile = FALSE) ## d <- tsdiag(fit1$model, c(10, 20, 50), y, ask = FALSE) ## dev.off()model1 <- new("MixARGaussian", prob = c(0.5, 0.5), scale = c(1, 2), arcoef = list(-0.5, 1.1)) set.seed(123) y <- mixAR_sim(model1, 400, c(0,0,0), nskip = 100) fit1 <- fit_mixAR(y, model1) d <- tsdiag(fit1$model, c(10, 20, 50), y) d ## This will put each plot in a separate file (mydiag01.pdf, ..., mydiag04.pdf) ## pdf("mydiag%02d.pdf", onefile = FALSE) ## d <- tsdiag(fit1$model, c(10, 20, 50), y, ask = FALSE) ## dev.off()
Simulate from MixAR models
mixAR_sim(model, n, init, nskip = 100, flag = FALSE) mixAny_sim(model, n, init, nskip=100, flag = FALSE, theta, galpha0, galpha, gbeta)mixAR_sim(model, n, init, nskip = 100, flag = FALSE) mixAny_sim(model, n, init, nskip=100, flag = FALSE, theta, galpha0, galpha, gbeta)
model |
model from which to simulate, an object inheriting from class |
init |
initial values, numeric vector. |
n |
size of the simulated series. |
nskip |
number of burn-in values, see Details. |
flag |
if |
theta |
ma coef, a list. |
galpha0 |
alpha0[k], k=1,...,g. |
galpha |
garch alpha. |
gbeta |
garch beta. |
mixAR_sim simulates a series of length nskip+n and
returns the last n values.
mixAny_sim simulates from a MixAR model with GARCH
innovations. mixAny_sim was a quick fix for Shahadat and needs
consolidation.
The vector init provides the initial values for
. Its length must be at least equal to the maximal AR
order. If it is longer, only the last max(model@order) elements
are used.
a numeric vector of length n. If flag = TRUE it has
attribute regimes containing z.
exampleModels$WL_ibm ## simulate a continuation of BJ ibm data ts1 <- mixAR_sim(exampleModels$WL_ibm, n = 30, init = c(346, 352, 357), nskip = 0) # a simulation based estimate of the 1-step predictive distribution # for the first date after the data. s1 <- replicate(1000, mixAR_sim(exampleModels$WL_ibm, n = 1, init = c(346, 352, 357), nskip = 0)) plot(density(s1)) # load ibm data from BJ ## data(ibmclose, package = "fma") # overlay the 'true' predictive density. pdf1 <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) curve(pdf1, add = TRUE, col = 'blue') # estimate of 5% quantile of predictive distribution quantile(s1, 0.05) # Monte Carlo estimate of "expected shortfall" # (but the data has not been converted into returns...) mean(s1[ s1 <= quantile(s1, 0.05) ])exampleModels$WL_ibm ## simulate a continuation of BJ ibm data ts1 <- mixAR_sim(exampleModels$WL_ibm, n = 30, init = c(346, 352, 357), nskip = 0) # a simulation based estimate of the 1-step predictive distribution # for the first date after the data. s1 <- replicate(1000, mixAR_sim(exampleModels$WL_ibm, n = 1, init = c(346, 352, 357), nskip = 0)) plot(density(s1)) # load ibm data from BJ ## data(ibmclose, package = "fma") # overlay the 'true' predictive density. pdf1 <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) curve(pdf1, add = TRUE, col = 'blue') # estimate of 5% quantile of predictive distribution quantile(s1, 0.05) # Monte Carlo estimate of "expected shortfall" # (but the data has not been converted into returns...) mean(s1[ s1 <= quantile(s1, 0.05) ])
Relabel the components of a MixAR model.
mixAR_switch(model, perm) mixAR_permute(model, perm)mixAR_switch(model, perm) mixAR_permute(model, perm)
model |
a MixAR model |
perm |
a permutation for relabeling |
If the permutation is the identity permutation the model is returned
as is. Otherwise the order of the components is changed according to
perm. Basically, perm is used as index,
e.g. prob[perm], etc.
Currently the function only reorders the "usual" components. Subclasses of "MixAR" may contain other parameters (e.g. different error distributions). So this function may not be appropriate for them.
"MixAR" — mixture autoregressive models Mixture autoregressive models
A virtual Class: no objects can be created from it.
Derived classes add distribution properties, e.g. use class
"MixARGaussian" for MixAR models with Gaussian
error components.
prob:the mixing probabilities, "numeric".
order:the AR orders, "numeric".
shift:intercept terms, "numeric".
scale:scaling factor, "numeric".
arcoef:autoregressive coefficients, an object from class
"raggedCoef" containing one row for each mixture component.
signature(x = "ANY", model = "MixAR", init = "list"): ...
signature(x = "ANY", model = "MixAR", init = "missing"): ...
signature(x = "ANY", model = "MixAR", init = "MixAR"): ...
signature(x = "ANY", model = "MixAR", init = "numeric"): ...
signature(x = "ANY", model = "MixARGaussian", init = "MixAR"): ...
signature(model = "MixAR"): ...
signature(.Object = "MixAR"): ...
signature(model = "MixAR"): ...
signature(model = "MixAR", ts = "numeric"): ...
signature(model = "MixAR", x = "numeric", index = "numeric", xcond = "missing", scale = "missing"): ...
signature(model = "MixAR", x = "numeric", index = "numeric", xcond = "missing", scale = "logical"): ...
signature(model = "MixAR", x = "numeric", index = "missing", xcond = "numeric", scale = "missing"): ...
signature(model = "MixAR", x = "numeric", index = "missing", xcond = "numeric", scale = "logical"): ...
signature(model = "MixAR", x = "numeric", index = "numeric", xcond = "missing"): ...
signature(x = "MixAR"): ...
signature(template = "MixAR"): ...
signature(model = "MixAR"): ...
signature(model = "MixAR"): ...
signature(model = "MixAR"): ...
signature(model = "MixAR"): ...
signature(x = "MixAR"): ...
Georgi N. Boshnakov
codemixAR,
classes "MixARGaussian",
"MixARgen"
## some models from subclasses of (virtual) class "MixAR" names(exampleModels) exampleModels$WL_A exampleModels$WL_At ## modify an existing model, here change the mixture weights mixAR(exampleModels$WL_A, coef = list((prob = c(0.4, 0.6))))## some models from subclasses of (virtual) class "MixAR" names(exampleModels) exampleModels$WL_A exampleModels$WL_At ## modify an existing model, here change the mixture weights mixAR(exampleModels$WL_A, coef = list((prob = c(0.4, 0.6))))
Generic function with methods for creating MixAR objects.
mixAR(template, coef, ..., filler = NA_real_)mixAR(template, coef, ..., filler = NA_real_)
template |
an object to be used as a template for the new object, typically
inheriting from |
coef |
parameters for the new object a list with components
|
... |
further arguments for methods. |
filler |
value for unspecified parameters, default is |
mixAR provides an alternative to the function new for
specifying MixAR models.
If template is numeric vector, it is taken to specify the AR order
of the model and the number of mixture components. A Gaussian MixAR
model is created with parameters filled initially with NA's and then
updated with values given by coef. coef does not need to
have values for all parameters and may be missing altogether. If NA's
are not suitable for initialisation, a suitable value can be specified
with filler.
If template is a MixAR object, then the new object will
have the class of template. The new object is set
initially to a copy of template and then updated with
parameters specified by coef (and maybe others for some
methods).
In principle, the numeric parameters are vectors of length the number of components of the MixAR model. For convenience, single values are replicated to the number of components. For this to work, at least one component must be specified completely, for example the order. It is an error for the parameters to imply conflicting number of components.
signature(template = "ANY")signature(template = "MixAR")class "MixARGaussian",
class "MixARgen"
mixAR(coef = list(prob = c(.5,.5), scale = c(1,2), arcoef = list(.5, 1.1), shift = c(0,0), order = c(1,1))) mixAR(template = c(1,1)) mixAR(coef = list(order = c(1,1))) # same m2 <- new("MixARGaussian", order = c(3, 2, 1), arcoef = matrix(c(1:3, c(1:2, 0), c(1, 0, 0)), nrow = 3, byrow = TRUE)) m2a <- mixAR(m2, list(prob = c(0.5, 0.25, 0.25))) show_diff(m2, m2a)mixAR(coef = list(prob = c(.5,.5), scale = c(1,2), arcoef = list(.5, 1.1), shift = c(0,0), order = c(1,1))) mixAR(template = c(1,1)) mixAR(coef = list(order = c(1,1))) # same m2 <- new("MixARGaussian", order = c(3, 2, 1), arcoef = matrix(c(1:3, c(1:2, 0), c(1, 0, 0)), nrow = 3, byrow = TRUE)) m2a <- mixAR(m2, list(prob = c(0.5, 0.25, 0.25))) show_diff(m2, m2a)
Fit a mixture autoregressive model to a univariate time series using the EM algorithm.
mixARemFixedPoint(y, model, est_shift = TRUE, crit = 1e-14, maxniter = 200, minniter = 10, verbose = FALSE) mixARgenemFixedPoint(y, model, crit = 1e-14, maxniter = 200, minniter = 10, verbose = FALSE, ...)mixARemFixedPoint(y, model, est_shift = TRUE, crit = 1e-14, maxniter = 200, minniter = 10, verbose = FALSE) mixARgenemFixedPoint(y, model, crit = 1e-14, maxniter = 200, minniter = 10, verbose = FALSE, ...)
y |
a univariate time series. |
model |
an object of class MixAR, a mixture autoregressive model providing the model specifications and initial values for the parameters. |
est_shift |
if TRUE optimise also w.r.t. the shift (constant) terms of the AR components, if FALSE keep the shift terms fixed. |
crit |
stop iterations when the relative change in the log-likelihood becomes smaller than this value. |
maxniter |
maximum number of iterations. |
minniter |
minimum number of iterations, do at leat that many iterations. |
... |
further arguments to be passed on to the M-step optimiser. |
verbose |
print more details during optimisation. |
mixARemFixedPoint and mixARgenemFixedPoint estimate
MixAR models with the EM algorithm. For mixARemFixedPoint, the
distribution of the components are fixed to be Gaussian. For
mixARgenemFixedPoint, the distributions can, in principle be
arbitrary (well, to a point).
Starting with model, the expectation and maximisation steps of
the EM algorithm are repeated until convergence is detected or the
maximum number of iterations, maxniter is exceeded.
Currently the convergence check is very basic—the iterations stop
when the relative change in the log-likelihood in the last two
iterations is smaller than the threshold value specified by
crit and at least minniter iterations have been done.
The EM algorithm may converge very slowly. To do additional iterations use the returned value in another call of this function.
the fitted model as an object inheriting from "MixAR".
This function was not intended to be called directly by the user (hence the inconvenient name).
Georgi N. Boshnakov
fit_mixAR which uses these functions for estimation,
classes
"MixARGaussian",
"MixARgen"
## data(ibmclose, package = "fma") # ibm data from BJ m0 <- exampleModels$WL_ibm m1 <- mixARemFixedPoint(fma::ibmclose, m0) m1a <- mixARemFixedPoint(fma::ibmclose, m1$model) show_diff(m1$model, m1a$model) mixARemFixedPoint(fma::ibmclose, m0, est_shift = FALSE) ## simulate a continuation of ibmclose, assuming m0 ts1 <- mixAR_sim(m0, n = 50, init = c(346, 352, 357), nskip = 0) m2a <- mixARemFixedPoint(ts1, m0, est_shift = FALSE)$model m2b <- mixARemFixedPoint(diff(ts1), m0, est_shift = FALSE)$model## data(ibmclose, package = "fma") # ibm data from BJ m0 <- exampleModels$WL_ibm m1 <- mixARemFixedPoint(fma::ibmclose, m0) m1a <- mixARemFixedPoint(fma::ibmclose, m1$model) show_diff(m1$model, m1a$model) mixARemFixedPoint(fma::ibmclose, m0, est_shift = FALSE) ## simulate a continuation of ibmclose, assuming m0 ts1 <- mixAR_sim(m0, n = 50, init = c(346, 352, 357), nskip = 0) m2a <- mixARemFixedPoint(ts1, m0, est_shift = FALSE)$model m2b <- mixARemFixedPoint(diff(ts1), m0, est_shift = FALSE)$model
Class "MixARGaussian" represents MixAR models with Gaussian
noise components.
Objects can be created by calls of the form new("MixARGaussian",
...), giving the elements of the model as named arguments, see the
examples below. All elements of the model, except arcoef, are
simple numeric vectors. From version 0.19-15 of package MixAR it is
possible to create objects using MixARGaussian(...). The two
forms are completely equivalent.
arcoef contains the AR coefficients, one numeric vector for
each mixture component. It can be given as a
"raggedCoef" object or as a list of numeric
vectors.
To input a model with seasonal AR coefficients, argument passed to arcoef can be passed as a raggedCoefS object, or as a list of three elements.
For the latter, seasonality s must be explicitly indicated.
AR coefficients can be given as list or matrix within the main list (one for main AR coefficients, named a, and one for seasonal AR coefficients, as). Each row of a input matrix/element of the list denotes one component of the mixture.
If not named, initialisation takes the first passed element to be a and the second to be as.
The AR order of the model is inferred from arcoef
argument. If argument order is given, it is checked for
consistency with arcoef. The shift slot defaults to a
vector of zeroes and the scale slot to a vector of ones.
The distribution of the noise components is standard Gaussian, N(0,1).
All slots except arcoef are numeric vectors of length
equal to the number of components in the model.
prob:probabilities of the mixture components
order:AR orders of the components
shift:the shift (intercept) terms of the AR components
scale:the standard deviations of the noise terms of the AR components
arcoef:The AR components, object of class "raggedCoef"
Class "MixAR", directly.
signature(model = "MixARGaussian", x = "numeric", index = "numeric", xcond = "missing"):
...
signature(model = "MixARGaussian", x = "numeric", index = "missing", xcond = "numeric"):
...
signature(x = "ANY", model = "MixARGaussian", init = "MixAR"): ...
signature(model = "MixARGaussian"): ...
signature(model = "MixARGaussian", x = "missing", index = "missing", xcond = "numeric"): ...
signature(model = "MixARGaussian", x = "missing", index = "missing", xcond = "numeric"): ...
signature(model = "MixARGaussian", x = "numeric", index = "missing", xcond = "numeric"): ...
signature(model = "MixARGaussian", x = "numeric", index = "numeric", xcond = "missing"): ...
signature(model = "MixARGaussian"): ...
signature(model = "MixARGaussian"): ...
Georgi N. Boshnakov
showClass("MixARGaussian") ## load ibm data from BJ ## data(ibmclose, package = "fma") ## compute a predictive density, assuming exampleModels$WL_ibm model ## for the first date after the end of the data pdf1 <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) ## plot the predictive density ## (cdf is used to determine limits on the x-axis) cdf1 <- mix_cdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) gbutils::plotpdf(pdf1, cdf = cdf1, lq = 0.001, uq = 0.999) ## compute lower 5% quantile of cdf1 gbutils::cdf2quantile(0.05, cdf = cdf1)showClass("MixARGaussian") ## load ibm data from BJ ## data(ibmclose, package = "fma") ## compute a predictive density, assuming exampleModels$WL_ibm model ## for the first date after the end of the data pdf1 <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) ## plot the predictive density ## (cdf is used to determine limits on the x-axis) cdf1 <- mix_cdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) gbutils::plotpdf(pdf1, cdf = cdf1, lq = 0.001, uq = 0.999) ## compute lower 5% quantile of cdf1 gbutils::cdf2quantile(0.05, cdf = cdf1)
"MixARgen"
A class for MixAR models with arbitrary noise distributions. "MixARgen"
inherits from "MixAR".
Objects can be created by calls of the form new("MixARgen",
dist, ...) or mixARgen(...). The two forms are completely
equivalent. The latter is available from version 0.19-15 of package
MixAR.
Most slots are inherited from class "MixAR".
prob:the mixing probabilities, "numeric".
order:the AR orders, "numeric".
shift:intercept terms, "numeric".
scale:scaling factor, "numeric".
arcoef:autoregressive coefficients, an object from class
"raggedCoef" containing one row for each mixture component.
dist:Object of class "list", representing the
noise distributions. The list contains one element for each
component of the MixAR model or a single element if the
noise distribution is the same for all components.
If the distributions do not contain parameters (e.g. Gaussian or
) it is sufficient to give the list of functions in the
element dist of the list.
If the distributions do contain parameters the recommended
arrangement is to give a list with components generator and
param, such that a call generator(param) should
produce the required list of distributions.
This is not finalised but if changed, backward compatibility with existing objects will be maintained.
Class "MixAR", directly.
signature(model = "MixARgen"): ...
signature(.Object = "MixARgen"): ...
signature(model = "MixARgen"): ...
signature(model = "MixARgen", x = "missing", index = "missing", xcond = "numeric"): ...
signature(model = "MixARgen", x = "numeric", index = "missing", xcond = "numeric"): ...
signature(model = "MixARgen", x = "numeric", index = "numeric", xcond = "missing"): ...
signature(model = "MixARgen", x = "missing", index = "missing", xcond = "numeric"): ...
signature(model = "MixARgen", x = "numeric", index = "missing", xcond = "numeric"): ...
signature(model = "MixARgen", x = "numeric", index = "numeric", xcond = "missing"): ...
signature(model = "MixARgen"): ...
signature(model = "MixARgen"): ...
signature(model = "MixARgen"): ...
showClass("MixARgen") exampleModels$WL_ibm_gen@dist noise_dist(exampleModels$WL_ibm_gen, "cdf") noise_dist(exampleModels$WL_ibm_gen, "pdf") noise_dist(exampleModels$WL_ibm_gen, "pdf", expand = TRUE) noise_dist(exampleModels$WL_ibm_gen, "cdf", expand = TRUE) ## data(ibmclose, package = "fma") # for `ibmclose' pdf1 <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) cdf1 <- mix_cdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) gbutils::plotpdf(pdf1, cdf = cdf1, lq = 0.001, uq = 0.999) pdf1gen <- mix_pdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)) cdf1gen <- mix_cdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)) gbutils::plotpdf(pdf1gen, cdf = cdf1gen, lq = 0.001, uq = 0.999) length(fma::ibmclose) cdf1gena <- mix_cdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)[-(369:369)]) pdf1gena <- mix_pdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)[-(369:369)]) gbutils::plotpdf(pdf1gena, cdf = cdf1gena, lq = 0.001, uq = 0.999) pdf1a <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)[-(369:369)]) cdf1a <- mix_cdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)[-(369:369)]) gbutils::plotpdf(pdf1a, cdf = cdf1a, lq = 0.001, uq = 0.999) cdf1gena <- mix_cdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)[-(369:369)]) cond_loglik(exampleModels$WL_ibm, as.numeric(fma::ibmclose)) cond_loglik(exampleModels$WL_ibm_gen, as.numeric(fma::ibmclose)) ts1gen <- mixAR_sim(exampleModels$WL_ibm_gen, n = 30, init = c(346, 352, 357), nskip = 0) plot(ts1gen) plot(mixAR_sim(exampleModels$WL_ibm_gen, n = 100, init = c(346, 352, 357), nskip = 0), type = "l") plot(diff(mixAR_sim(exampleModels$WL_ibm_gen, n = 100, init = c(346, 352, 357), nskip = 0)), type = "l") noise_dist(exampleModels$WL_ibm_gen, "Fscore") prob <- exampleModels$WL_ibm@prob scale <- exampleModels$WL_ibm@scale arcoef <- exampleModels$WL_ibm@arcoef@a mo_WLt3 <- new("MixARgen", prob = prob, scale = scale, arcoef = arcoef, dist = list(fdist_stdt(3))) mo_WLt30 <- new("MixARgen", prob = prob, scale = scale, arcoef = arcoef, dist = list(fdist_stdt(30)))showClass("MixARgen") exampleModels$WL_ibm_gen@dist noise_dist(exampleModels$WL_ibm_gen, "cdf") noise_dist(exampleModels$WL_ibm_gen, "pdf") noise_dist(exampleModels$WL_ibm_gen, "pdf", expand = TRUE) noise_dist(exampleModels$WL_ibm_gen, "cdf", expand = TRUE) ## data(ibmclose, package = "fma") # for `ibmclose' pdf1 <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) cdf1 <- mix_cdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)) gbutils::plotpdf(pdf1, cdf = cdf1, lq = 0.001, uq = 0.999) pdf1gen <- mix_pdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)) cdf1gen <- mix_cdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)) gbutils::plotpdf(pdf1gen, cdf = cdf1gen, lq = 0.001, uq = 0.999) length(fma::ibmclose) cdf1gena <- mix_cdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)[-(369:369)]) pdf1gena <- mix_pdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)[-(369:369)]) gbutils::plotpdf(pdf1gena, cdf = cdf1gena, lq = 0.001, uq = 0.999) pdf1a <- mix_pdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)[-(369:369)]) cdf1a <- mix_cdf(exampleModels$WL_ibm, xcond = as.numeric(fma::ibmclose)[-(369:369)]) gbutils::plotpdf(pdf1a, cdf = cdf1a, lq = 0.001, uq = 0.999) cdf1gena <- mix_cdf(exampleModels$WL_ibm_gen, xcond = as.numeric(fma::ibmclose)[-(369:369)]) cond_loglik(exampleModels$WL_ibm, as.numeric(fma::ibmclose)) cond_loglik(exampleModels$WL_ibm_gen, as.numeric(fma::ibmclose)) ts1gen <- mixAR_sim(exampleModels$WL_ibm_gen, n = 30, init = c(346, 352, 357), nskip = 0) plot(ts1gen) plot(mixAR_sim(exampleModels$WL_ibm_gen, n = 100, init = c(346, 352, 357), nskip = 0), type = "l") plot(diff(mixAR_sim(exampleModels$WL_ibm_gen, n = 100, init = c(346, 352, 357), nskip = 0)), type = "l") noise_dist(exampleModels$WL_ibm_gen, "Fscore") prob <- exampleModels$WL_ibm@prob scale <- exampleModels$WL_ibm@scale arcoef <- exampleModels$WL_ibm@arcoef@a mo_WLt3 <- new("MixARgen", prob = prob, scale = scale, arcoef = arcoef, dist = list(fdist_stdt(3))) mo_WLt30 <- new("MixARgen", prob = prob, scale = scale, arcoef = arcoef, dist = list(fdist_stdt(30)))
Simulate white noise series from a list of functions and vector of regimes. This function is used internally for simulation from MixAR models.
mixARnoise_sim(rdist, z)mixARnoise_sim(rdist, z)
rdist |
a list of functions for random number generation, see ‘Details’. |
z |
a vector of positive integers specifying the 'regimes'. |
If the length of the list rdist is max(z), then
z[[i]] is the random number generator for regime i.
Alternatively, if rdist is of length one, then the same
generator will be used for all regimes.
mixARnoise_sim returns a vector, say y, of the same
length as z, such that y[i] is generated by
z[[i]].
a numeric vector
## MixAR with 2 components: N(0,1) and t_5 set.seed = 1234 z <- sample(2, size = 5, replace = TRUE) mixARnoise_sim(list(rnorm, function(n) rt(n, 5)), z)## MixAR with 2 components: N(0,1) and t_5 set.seed = 1234 z <- sample(2, size = 5, replace = TRUE) mixARnoise_sim(list(rnorm, function(n) rt(n, 5)), z)
"MixComp" — manipulation of MixAR time seriesClass "MixComp" represents components of mixture autoregressive
time series and their transformations obtained by arithmetic and
related operations. Methods are provided to allow convenient
computation with such time series.
Objects can be created by calls of the form new("MixComp", ...).
It is more usual however to obtain such objects initially
from functions such as mix_ek. Methods are defined to allow for
convenient and intuitive further manipulation of such objects.
Internally, an object of class MixComp is a matrix with one column
for each component. However, methods for arithmetic operations
involving MixComp objects are defined to perform natural
operations for mixture objects. For example, multiplication by
vectors is commutative and “does the right thing”.
m:Object of class "matrix" with one column correponding to
each component of the mixture AR model.
Arithmetic operations involving MixComp objects are defined
to allow for convenient execution of computations for mixture
autoregressive models, see class "MixComp".
signature(e1 = "MixComp", e2 = "missing"):
unary minus for "MixComp" objects.
signature(e1 = "numeric", e2 = "MixComp"):
If e2 is thought of as a matrix, , then the number
of elements of e1 must be the same as the number of rows
of and each column of is subtracted from
e1, see also "mix_ek", "mix_hatk".
As a special case, if has only one row, then it is
subtracted from each element of e1, i.e. that row is
replicated to obtain a matrix with as many rows as the length of
e1 and then subtracted from e1 as above.
The result is a MixComp object.
signature(e1 = "MixComp", e2 = "numeric"):
This is analogous to the above method. (FIXME: the code
of this function does not deal with the special case as in the
above method. Is this an omission or I have done it on purpose?)
signature(e1 = "function", e2 = "MixComp"):
This applies the function e1 to each element of
e2. Together with the arithmetic operations this allows for
easy computation with MixComp objects (e.g. pdfs, likelihoods).
signature(e1 = "character", e2 = "MixComp"):
signature(e1 = "list", e2 = "MixComp"):
If e1 is of length one it specifies a function to be
applied to each element of e2, otherwise it is a list of
functions, such that the th function is applied to the
th column of e2@m.
signature(e1 = "MixComp", e2 = "MixComp"): ...
signature(e1 = "MixComp", e2 = "numeric"):
see the following.
signature(e1 = "numeric", e2 = "MixComp"):
“Column” of the MixComp object is multiplied by
the th element of the numeric vector, i.e. each “row”
of the MixComp object is multiplied by the vector (or, the
vector is replicated to a matrix to be multiplied by the
MixComp object).
signature(e1 = "function", e2 = "MixComp"):
Multiplying a function by a MixComp object actually
applies the function to each element of the object.
This is a misuse of methods, prefer operator
%of% which does the same.
signature(e1 = "character", e2 = "MixComp"):
The first argument is a name of a function which is
applied to each element of the MixComp object. This is a
misuse of methods, see operator %of%
which does the same.
signature(e1 = "MixComp", e2 = "numeric"):
signature(e1 = "numeric", e2 = "MixComp"):
Division works analogously to "*".
signature(e1 = "MixComp", e2 = "numeric"):
If k is a scalar, raise each element of e1@m to
power k.
(For consistency this operation should have the semantics of "*" and "/" but this operator probably makes sense only for scalar 'e2', where the semantics doesn't matter. So, don't bother for now.)
signature(e1 = "numeric", e2 = "MixComp"):
signature(e1 = "MixComp", e2 = "numeric"):
Addition involving MixComp objects works analogously to subtraction.
signature(x = "MixComp", y = "missing", star = "missing", plus = "missing"):
With one argument inner computes the sum of the columns of
the argument. This is conceptually equivalent to y being a
vector of ones.
signature(x = "MixComp", y = "numeric", star = "missing", plus = "missing"):
signature(x = "numeric", y = "MixComp", star = "missing", plus = "missing"):
The number of elements of the numeric argument should be equal to
the number of rows of the MixComp object. Effectively,
computes the inner product of the two arguments. The order of the
arguments does not matter.
Returns a numeric vector.
signature(x = "MixComp", y = "numeric", star = "ANY", plus = "ANY"):
Computes a generalised inner product of x with y using the
specified functions in place of the usual "*" and "+"
operations. The defaults for star and + are
equivalent to multiplication and addition, respectively.
Note that "+" is a binary operation (not -ary) in R. So
technically the correct way to specify the default operation here
is "sum" or sum. Since it is easy to make this mistake, if
plus == "+", it is replaced by "sum".
(In fact, plus is given a single argument, the vector of
values to work on. Since "+" works as a unary operator on one
argument, it would give surprising results if left as is.)
signature(x = "MixComp", y = "numeric", star = "ANY", plus = "missing"):
This is a more efficient implementation for the case when
plus = sum.
signature(x = "MixComp"):
Number of components.
signature(x = "MixComp")A "MixComp" object is essentially a matrix. This method
gives the dimension of the underlying matrix. This method
indirectly ensures that nrow() and ncol() work
naturally for "MixComp" objects.
Georgi N. Boshnakov
## dim, nrow, ncol a <- new("MixComp", m = matrix(c(1:7, 11:17, 21:27), ncol = 3)) a dim(a) nrow(a) ncol(a) mix_ncomp(a) -a a - 1:7 1:7 + a 2*a b <- new("MixComp", m = matrix(rnorm(18), ncol = 3)) ## apply a function to the columns of a MixComp object pnorm %of% b ## apply a separate function to to each column flist <- list(function(x) pnorm(x), function(x) pt(x, df = 5), function(x) pt(x, df = 4) ) flist %of% b## dim, nrow, ncol a <- new("MixComp", m = matrix(c(1:7, 11:17, 21:27), ncol = 3)) a dim(a) nrow(a) ncol(a) mix_ncomp(a) -a a - 1:7 1:7 + a 2*a b <- new("MixComp", m = matrix(rnorm(18), ncol = 3)) ## apply a function to the columns of a MixComp object pnorm %of% b ## apply a separate function to to each column flist <- list(function(x) pnorm(x), function(x) pt(x, df = 5), function(x) pt(x, df = 4) ) flist %of% b
Filter time series with MixAR filters, a generic function with no default method (currently).
mixFilter(x, coef, index, shift = 0, residual = FALSE, scale = 1)mixFilter(x, coef, index, shift = 0, residual = FALSE, scale = 1)
x |
time series |
coef |
the filter coefficients |
index |
indices for which to calculate the filtered values. |
shift |
optional shifts (intercept) terms. |
residual |
If FALSE (default) calculate “predictions”, if TRUE calculate “residuals”. |
scale |
optional scale factor(s), makes sense only when
|
a MixComp object
signature(x = "ANY", coef = "ANY", index = "ANY")This method simply prints an error message and stops.
signature(x = "numeric", coef = "raggedCoef", index = "numeric")Georgi N. Boshnakov
M-step for models from class MixARgen. This function is for use by other functions.
mixgenMstep(y, tau, model, index, fix = NULL, comp_sigma = FALSE, method = "BBsolve", maxit = 100, trace = FALSE, lessverbose = TRUE, ...)mixgenMstep(y, tau, model, index, fix = NULL, comp_sigma = FALSE, method = "BBsolve", maxit = 100, trace = FALSE, lessverbose = TRUE, ...)
y |
time series, a numeric vector. |
tau |
conditional probabilities, an object of class "MixComp". |
model |
the current model, an object from a subclass of class "MixAR". |
index |
indices of observations for which to compute residuals, a vector of positive integers, see 'Details'. |
method |
optimisation or equation solving method for package BB |
... |
arguments to pass on to optimisation functions, not thought over yet. Do not use until this notice is removed. |
comp_sigma |
If TRUE optimise the scale parameters using univariate optimisation. (note: does not work with argument 'fix' yet.) |
fix |
specify parameters to be held fixed during optimisation, see 'Details'. |
maxit |
maximal number of iterations for BB optimisers and solvers. Meant mainly for testing. |
trace |
if TRUE, BB optimisers and solvers will print information about their proceedings. Meant mainly for testing. |
lessverbose |
if TRUE, print a dot instead of more verbose information. |
mixgenMstep is an implementation of the M-step of the EM
algorithm for mixture autoregressive models specified by objects of
class "MixARgen". The function was build and modified incrementally
with the main goal of providing flexibility. Speed will be addressed
later.
By default optimisation is done with respect to all parameters.
Argument fix may be a list with elements "prob", "shift",
"scale" and "arcoef". These elements should be logical vectors
containing TRUE in the positions of the fixed parameters.
Elements with no fixed parameters may be omitted. (Currently the
"prob" element is ignored, i.e. it is not possible to fix any of the
component probabilities.)
If fix = "shift" the shift parameters are kept fixed. This is
equivalent to fix = list(shift = rep(TRUE,g)).
The parameters (if any) of the distributions of the error components are estimated by default. Currently the above method cannot be used to fix some of them. This can be achieved however by modifying the distribution part of the model since that incorporates information about the parameters and whether they are fixed or not.
fit_mixAR and
mixARgenemFixedPoint
which are meant to be called by users.
Internal functions for EM estimation of MixAR models with Gaussian components: sums of products and crossproducts; M-step for MixAR estimation; estimation of autoregressive part of the model.
tauCorrelate(y, tau, order) tau2arcoef(y, tau, order, est_shift = TRUE) mixMstep(y, tau, order, index, est_shift = TRUE)tauCorrelate(y, tau, order) tau2arcoef(y, tau, order, est_shift = TRUE) mixMstep(y, tau, order, index, est_shift = TRUE)
y |
time series. |
tau |
conditional probabilties for the observations to belong to each of
the components, a |
order |
order of the MixAR model, numeric vector of length the number of mixture components. |
index |
indices of the observations to include in the likelihood
calculations, typically |
est_shift |
if TRUE include shifts (intercepts) in the AR components, otherwise set them to zero. |
mixMstep performs an M-step for estimation of MixAR models with
Gaussian components.
tauCorrelate computes crossproducts needed for EM estimation
of MixAR models with Gaussian components.
tau2arcoef computes the AR coefficients by solving
Yule-Walker-type equations for each component.
For mixMstep, a MixAR model, an object of class MixARGaussian.
For tauCorrelate, a named list with the following components:
Stau |
|
Stauy |
|
Stauyy |
For tau2arcoef, a list with two components:
shift |
the shift (intercept) terms, a numeric vector |
arcoef |
the AR coefficients as a list, whose i-th component contains the coefficients for component i (as a numeric vector) |
Provides estimation via EM-Algorithm for mixture autoregressive models including seasonal AR parameters.
mixSARfit(y, model, est_shift = FALSE, tol = 10^-14)mixSARfit(y, model, est_shift = FALSE, tol = 10^-14)
y |
a time series (currently a numeric vector). |
model |
an object of class |
est_shift |
if missing or |
tol |
threshold for stopping criterion. |
This function only works for "MixAR" objects in which slot
arcoef is of class "raggedCoefS".
A list of 2:
model |
an object of class |
vallogf |
the value of the loglikelihood function for the returned model. |
Davide Ravagli and Georgi N. Boshnakov
ar1 <- list(c(0.5, -0.5), c(1.1, 0, -0.5)) ar12 <- list(0, c(-0.3, 0.1)) s = 12 rag <- new("raggedCoefS", a = ar1, as = ar12, s = s) model <- new("MixARGaussian", prob = exampleModels$WL_A@prob, # c(0.5, 0.5) scale = exampleModels$WL_A@scale, # c(5, 1) arcoef = rag) set.seed(1234) y <- mixAR_sim(model, n = 100, init = rep(0, 24)) mixSARfit(y, model) ## fix the intercepts to zero mixSARfit(y, model, est_shift = FALSE, tol = 10e-4)ar1 <- list(c(0.5, -0.5), c(1.1, 0, -0.5)) ar12 <- list(0, c(-0.3, 0.1)) s = 12 rag <- new("raggedCoefS", a = ar1, as = ar12, s = s) model <- new("MixARGaussian", prob = exampleModels$WL_A@prob, # c(0.5, 0.5) scale = exampleModels$WL_A@scale, # c(5, 1) arcoef = rag) set.seed(1234) y <- mixAR_sim(model, n = 100, init = rep(0, 24)) mixSARfit(y, model) ## fix the intercepts to zero mixSARfit(y, model, est_shift = FALSE, tol = 10e-4)
Simulate data from multivariate MixAR models under the assumptions of multivariate Gaussian innovarion
mixVAR_sim(model, n, init, nskip = 100, flag = FALSE)mixVAR_sim(model, n, init, nskip = 100, flag = FALSE)
model |
model from which to simulate, an object inheriting from class
|
n |
size of simulated multivariate series. |
init |
initial values, a numeric matrix. If |
nskip |
number of burn-in values. |
flag |
if |
mixVAR_sim simulates a series of length nskip + n and returns
the last n values. init provides initial values for the
algorithm. Each row is considered as a time point. The number of rows
must be at least equal to the maximal AR order.
a numeric matrix with n rows.
Davide Ravagli
AR <- list() AR[[1]] <- array(c(0.5,-0.3,-0.6,0,0,0.5,0.4,0.5,-0.3), dim = c(3,3,1)) AR[[2]] <- array(c(-0.5,0.3,0,1,0,-0.5,-0.4,-0.2, 0.5), dim = c(3,3,1)) prob <- c(0.75, 0.25) shift <- cbind(c(0,0,0), c(0,0,0)) Sigma1 <- cbind(c(1, 0.5, -0.4), c(0.5, 2, 0.8), c(-0.4, 0.8, 4)) Sigma2 <- cbind(c(1,0.2, 0), c(0.2, 2, -0.15), c(0, -0.15, 4)) Sigma <- array(c(Sigma1, Sigma2), dim = c(3,3,2)) m <- new("MixVARGaussian", prob=prob, vcov=Sigma, arcoef=AR, shift=shift) mixVAR_sim(m, n=500, init=matrix(rep(0,3), ncol=3), nskip=100, flag=FALSE)AR <- list() AR[[1]] <- array(c(0.5,-0.3,-0.6,0,0,0.5,0.4,0.5,-0.3), dim = c(3,3,1)) AR[[2]] <- array(c(-0.5,0.3,0,1,0,-0.5,-0.4,-0.2, 0.5), dim = c(3,3,1)) prob <- c(0.75, 0.25) shift <- cbind(c(0,0,0), c(0,0,0)) Sigma1 <- cbind(c(1, 0.5, -0.4), c(0.5, 2, 0.8), c(-0.4, 0.8, 4)) Sigma2 <- cbind(c(1,0.2, 0), c(0.2, 2, -0.15), c(0, -0.15, 4)) Sigma <- array(c(Sigma1, Sigma2), dim = c(3,3,2)) m <- new("MixVARGaussian", prob=prob, vcov=Sigma, arcoef=AR, shift=shift) mixVAR_sim(m, n=500, init=matrix(rep(0,3), ncol=3), nskip=100, flag=FALSE)
"MixVAR" — mixture vector autoregressive models Mixture vector autoregressive models
A virtual Class: No objects may be created from it.
Derived classes add distribution properties, e.g. use class
"MixVARGaussian" for MixVAR models with Gaussian
error components.
prob:the mixing probabilities,
an object of class "numeric"
order:Object of class "numeric" ~~
shift:Object of class "matrix" ~~
vcov:Object of class "array" ~~
arcoef:Object of class "raggedCoefV" ~~
signature(x = "ANY", model = "MixAR"): ...
Davide Ravagli
class "MixVARGaussian"
Provides EM-estimation of mixture autoregressive models for multivariate time series
mixVARfit(y, model, fix = FALSE, tol = 10^-6, verbose = FALSE)mixVARfit(y, model, fix = FALSE, tol = 10^-6, verbose = FALSE)
y |
a data matrix. |
model |
an object of class |
tol |
Threshold for convergence criterion. |
fix |
if TRUE, fix the shift parameters. |
verbose |
if |
Estimation is done under the assumption of multivariate Gaussian innovations.
An object of class MixVARGaussian with EM estimates of model
parameters.
Davide Ravagli
Fong PW, Li WK, Yau CW, Wong CS (2007). “On a Mixture Vector Autoregressive Model.” The Canadian Journal of Statistics / La Revue Canadienne de Statistique, 35(1), 135–150. ISSN 03195724, doi:10.1002/cjs.5550350112.
fit_mixVAR-methods for examples
Class MixVARGaussian represents MixAR models with multivariate Gaussian noise components.
Objects can be created by calls of the form
new("MixVARGaussian", ...), giving the elements of the model as
named arguments, see the examples below.
arcoef contains the AR coefficients, one numeric array for
each mixture component. It can be given as a
"raggedCoefV" object or as a list of numeric
arrays.
The AR order of the model is inferred from arcoef
argument. If argument order is given, it is checked for
consistency with arcoef. The shift slot defaults to a
matrix of zeroes and the vcov slot to an array of
identity matrices, one for each component.
The distribution of the noise components is standard multivariate Gaussian, N(0,1).
All slots except arcoef are numeric vectors of length
equal to the number of components in the model.
prob:probabilities of the mixture components,
order:AR orders of the components,
shift:the shift (intercept) terms of the AR components,
vcov:covariance matrices of the noise terms of the AR components,
arcoef:The AR components, object of class "raggedCoefV".
Class "MixAR", directly.
signature(x = "ANY", model = "MixARGaussian"):
...
Davide Ravagli
class "MixAR"
showClass("MixVARGaussian") ## Create array of covariance matrices Sigma1 <- cbind(c(0.0013, 0.0011), c(0.0011, 0.0012)) Sigma2 <- cbind(c(0.0072, 0.0047), c(0.0047, 0.0039)) Sigma <- array(c(Sigma1, Sigma2), dim=c(2,2,2)) ## Create list of AR coefficients AR <- list() AR[[1]] <- array(c(0.0973, -0.0499, 0.2927, 0.4256, ## VAR(2;4) -0.0429, 0.0229, -0.1515, -0.1795, -0.0837, -0.1060, -0.1530, 0.1947, -0.1690, -0.0903, 0.1959, 0.0955), dim=c(2,2,4)) AR[[2]] <- array(c(0.3243, 0.2648, 0.4956, 0.2870, ## VAR(2;3) -0.1488, 0.0454, -0.0593, -0.3629, 0.1314, 0.0274, 0.0637, 0.0485), dim=c(2,2,3)) ## Create vector of mixing weights prob <- c(0.6376, 0.3624) ## Create matrix of shift parameters shift <- cbind(c(0.0044, 0.0020), c(-0.0039, -0.0014)) ## Build "MixVARGaussian" model new("MixVARGaussian", prob=prob, vcov=Sigma, arcoef=AR, shift=shift)showClass("MixVARGaussian") ## Create array of covariance matrices Sigma1 <- cbind(c(0.0013, 0.0011), c(0.0011, 0.0012)) Sigma2 <- cbind(c(0.0072, 0.0047), c(0.0047, 0.0039)) Sigma <- array(c(Sigma1, Sigma2), dim=c(2,2,2)) ## Create list of AR coefficients AR <- list() AR[[1]] <- array(c(0.0973, -0.0499, 0.2927, 0.4256, ## VAR(2;4) -0.0429, 0.0229, -0.1515, -0.1795, -0.0837, -0.1060, -0.1530, 0.1947, -0.1690, -0.0903, 0.1959, 0.0955), dim=c(2,2,4)) AR[[2]] <- array(c(0.3243, 0.2648, 0.4956, 0.2870, ## VAR(2;3) -0.1488, 0.0454, -0.0593, -0.3629, 0.1314, 0.0274, 0.0637, 0.0485), dim=c(2,2,3)) ## Create vector of mixing weights prob <- c(0.6376, 0.3624) ## Create matrix of shift parameters shift <- cbind(c(0.0044, 0.0020), c(-0.0039, -0.0014)) ## Build "MixVARGaussian" model new("MixVARGaussian", prob=prob, vcov=Sigma, arcoef=AR, shift=shift)
Multi-step predictions for MixAR models.
multiStep_dist(model, maxh, N, xcond, ...)multiStep_dist(model, maxh, N, xcond, ...)
model |
a MixAR model. |
maxh |
maximal horizon, a positive integer. |
N |
an integer specifying the number of simulation samples to use, see 'Details. This argument is used only by simulation based methods. |
xcond |
the past values needed for the conditional distribution, a numeric vector of length at least the maximal AR order of the components. |
... |
used only in some methods, see the details for the individual methods. |
The function currently implements two methods: the exact method due to Boshnakov (2009) and a simulation method described by (Wong and Li 2000) for Gaussian MixAR models but valid more generally.
The simulation method is available for any MixAR model, while the exact method is currently implemented only for models with Gaussian components ("MixARGaussian" class).
multiStep_dist returns a function which can be used to obtain
various properties of the predictive distribution for lags up to
maxh.
If argument N is missing the exact method is tried. Currently
an error will result if the exact method is not implemented for
model.
If argument N is given it must be a scalar numeric value, the
number of simulations to be performed to construct an approximation
for the predictive distributions.
The simulation is done by multiStep_dist. The properties
obtained later from the function returned by multiStep_dist use
the samples generated by the call to multiStep_dist.
To do a simulation with different parameters (e.g., with larger
N) call multiStep_dist again.
If xcond is missing multiStep_dist returns a function
with arguments h, what and xcond.
If xcond is supplied, then it is fixed to that value and the
arguments of the returned function are h, what and
'...'. The dots argument is currently used in the case of the
simulation method, see below.
Let f be the function returned by multiStep_dist.
Argument h is the required prediction horizon and can be a
number in the interval . Argument what is the
required property of the predictive distribution for lag
h. If what is a function, it is applied to the simulated
sample for the requested horizon (currently available only for
the simulation method). If what is a character string, the
corresponding property of the predictive distribution for horizon
h is returned.
Currently possible values for what are:
the probability density function.
the cumulative distribution function.
the location (conditional mean).
the conditional variance, a.k.a (squared) volatility.
the conditional standard deviation, a.k.a volatility.
the conditional skewness.
the conditional kurtosis.
Note that what = "pdf" and what = "cdf" return functions
even in the simulation case. For "pdf" the function is constructed
using density and the "..." arguments passed to f will
be passed on to density if finer control is needed.
If what is none of the above, the raw object is returned
currently (but this may change).
a function as described in sections ‘Details’ and ‘Methods’
The Details section gives a rather detailed description of the function, so the descriptions below are brief.
signature(model = "MixAR", maxh = "numeric", N = "numeric",
xcond = "numeric")Non-missing N requests the simulation method. The
predictive distribution is approximated by simulating N
of future paths up to horizon maxh and using a
non-parametric estimate. Arguments "..." are passed to
density to allow finer control.
signature(model = "MixARGaussian", maxh = "numeric", N = "missing",
xcond = "missing")Computes the predictive distribution using the exact method.
Returns a function with arguments h, what and xcond.
signature(model = "MixARGaussian", maxh = "numeric", N = "missing",
xcond = "ANY")Computes the predictive distribution using the exact method.
Returns a function with arguments h and what.
(i.e., xcond is fixed to the supplied argument xcond).
Georgi N. Boshnakov
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.
Wong CS, Li WK (2000).
“On a mixture autoregressive model.”
J. R. Stat. Soc., Ser. B, Stat. Methodol., 62(1), 95-115.
## exact method, without xcond dist <- multiStep_dist(exampleModels$WL_ibm, maxh = 3) tfpdf <- dist(3, "pdf", xcond = c(560, 600)) # xcond is argument to 'dist' here tfcdf <- dist(3, "cdf", xcond = c(560, 600)) ## plot the pdf (gbutils::plotpdf determines suitable range automatically) gbutils::plotpdf(tfpdf, cdf = tfcdf) args(dist(3, "pdf", xcond = c(500, 600))) # x ## use a simulation method with N = 1000 tf <- multiStep_dist(exampleModels$WL_ibm, maxh = 3, N = 1000, xcond = c(560, 600)) args(tf) # (h, what, ...) ## the exact method may also be used with fixed xcond: tfe <- multiStep_dist(exampleModels$WL_ibm, maxh = 3, xcond = c(560, 600)) ## get pdf and cdf for horizon 3 tfepdf <- tfe(3, "pdf") tfecdf <- tfe(3, "cdf") ## plot the pdf gbutils::plotpdf(tfepdf, cdf = tfecdf) tf(3, "location") tf(1, "location") mix_location(exampleModels$WL_ibm, xcond = c(560, 600)) ## larger simulation gives better approximation, in general tf <- multiStep_dist(exampleModels$WL_ibm, maxh = 3, N = 10000, xcond = c(560, 600)) tf(1, "location") tf1000pdf <- tf(3, "pdf") tf1000cdf <- tf(3, "cdf") gbutils::plotpdf(tf1000pdf, cdf = tf1000cdf) ## plot the exact and simulated pdf's together for comparison gbutils::plotpdf(tfepdf, cdf = tfecdf) curve(tf1000pdf, add = TRUE, col = "red") ## get the raw data tfs <- tf(1, "sampled") apply(tfs, 2, mean) # location for lags from 1 to maxh (here 3) tf(1, "location") tf(1, "variance") tf(1, "sd") mix_variance(exampleModels$WL_ibm, xcond = c(560, 600)) sqrt(mix_variance(exampleModels$WL_ibm, xcond = c(560, 600))) mix_kurtosis(exampleModels$WL_ibm, xcond = c(359, 200)) mix_kurtosis(exampleModels$WL_ibm, xcond = c(359, 400))## exact method, without xcond dist <- multiStep_dist(exampleModels$WL_ibm, maxh = 3) tfpdf <- dist(3, "pdf", xcond = c(560, 600)) # xcond is argument to 'dist' here tfcdf <- dist(3, "cdf", xcond = c(560, 600)) ## plot the pdf (gbutils::plotpdf determines suitable range automatically) gbutils::plotpdf(tfpdf, cdf = tfcdf) args(dist(3, "pdf", xcond = c(500, 600))) # x ## use a simulation method with N = 1000 tf <- multiStep_dist(exampleModels$WL_ibm, maxh = 3, N = 1000, xcond = c(560, 600)) args(tf) # (h, what, ...) ## the exact method may also be used with fixed xcond: tfe <- multiStep_dist(exampleModels$WL_ibm, maxh = 3, xcond = c(560, 600)) ## get pdf and cdf for horizon 3 tfepdf <- tfe(3, "pdf") tfecdf <- tfe(3, "cdf") ## plot the pdf gbutils::plotpdf(tfepdf, cdf = tfecdf) tf(3, "location") tf(1, "location") mix_location(exampleModels$WL_ibm, xcond = c(560, 600)) ## larger simulation gives better approximation, in general tf <- multiStep_dist(exampleModels$WL_ibm, maxh = 3, N = 10000, xcond = c(560, 600)) tf(1, "location") tf1000pdf <- tf(3, "pdf") tf1000cdf <- tf(3, "cdf") gbutils::plotpdf(tf1000pdf, cdf = tf1000cdf) ## plot the exact and simulated pdf's together for comparison gbutils::plotpdf(tfepdf, cdf = tfecdf) curve(tf1000pdf, add = TRUE, col = "red") ## get the raw data tfs <- tf(1, "sampled") apply(tfs, 2, mean) # location for lags from 1 to maxh (here 3) tf(1, "location") tf(1, "variance") tf(1, "sd") mix_variance(exampleModels$WL_ibm, xcond = c(560, 600)) sqrt(mix_variance(exampleModels$WL_ibm, xcond = c(560, 600))) mix_kurtosis(exampleModels$WL_ibm, xcond = c(359, 200)) mix_kurtosis(exampleModels$WL_ibm, xcond = c(359, 400))
Functions for the distributions of the components of MixAR models
get_edist(model) noise_dist(model, what, expand = FALSE) noise_rand(model, expand = FALSE) noise_params(model) set_noise_params(model, nu)get_edist(model) noise_dist(model, what, expand = FALSE) noise_rand(model, expand = FALSE) noise_params(model) set_noise_params(model, nu)
model |
a model. |
what |
the property, a character string. |
expand |
if TRUE, expand the list to length equal to the number of components, see Details. |
nu |
degrees of freedom. |
get_edist gives the distributions of the noise components of
model.
noise_dist gives property what of the noise
distribution.
noise_rand gives a list of functions for simulation from the
component distributions.
In each case, the list contains one element for each component but if
it is of length one, then the only element is common for all
components. To force a complete list even in this case, use
expand = TRUE.
noise_params gives the parameters of the model as a numeric
vector.
The distribution is specified as a list. Element "dist" contain the distribution. Element "generator" is a function that generates a distribution like the one specified. If "dist" is absent or NULL, the generator is called to generate a distribution object.
Initially the distribution itself was used for slot dist. For
compatibility with old code using that format, this is still
supported.
fdist_stdnorm,
fdist_stdt,
fn_stdt.
noise_dist in package mixAR Methods for function noise_dist in package mixAR
signature(model = "MixAR")signature(model = "MixARGaussian")signature(model = "MixARgen")noise_params in package mixAR Methods for function noise_params in package mixAR
signature(model = "MixAR")signature(model = "MixARgen")noise_rand in package mixAR Methods for function noise_rand in package mixAR
signature(model = "MixAR")signature(model = "MixARGaussian")signature(model = "MixARgen")Set or extract the parameters of MixAR objects.
parameters(model, namesflag = FALSE, drop = character(0)) parameters(model) <- value ## S4 replacement method for signature 'MixAR' parameters(model) <- value ## S4 replacement method for signature 'ANY' parameters(model) <- valueparameters(model, namesflag = FALSE, drop = character(0)) parameters(model) <- value ## S4 replacement method for signature 'MixAR' parameters(model) <- value ## S4 replacement method for signature 'ANY' parameters(model) <- value
model |
a model. |
namesflag |
if TRUE, generate names. |
drop |
names of parameters not to include in the returned value, a character vector. The default is to return all parameters, see Details. |
value |
values of the parameters, numeric. |
This is a generic function. The dispatch is on argument model.
The default calls coef.
parameters extracts the parameters of a MixAR object. It
returns a numeric vector. If namesflag is TRUE the
returned vector is named, so that the parameters can be referred to by
names. Argument drop is a character vector giving names of
parameters not to be included in the returned value.
This function can be useful for setting parameters from optimisation routines.
set_parameters is deprecated,
use parameters(model) <- value instead.
a vector of parameters, maybe with names.
signature(model = "ANY")signature(model = "MixAR")parameters(exampleModels$WL_ibm) parameters(exampleModels$WL_ibm, namesflag = TRUE) ## drop orders parameters(exampleModels$WL_ibm, namesflag = TRUE, drop = "order") ## drop orders and mixing weights parameters(exampleModels$WL_ibm, namesflag = TRUE, drop = c("order", "prob")) parameters(exampleModels$WL_I, namesflag = TRUE) parameters(exampleModels$WL_II, namesflag = TRUE)parameters(exampleModels$WL_ibm) parameters(exampleModels$WL_ibm, namesflag = TRUE) ## drop orders parameters(exampleModels$WL_ibm, namesflag = TRUE, drop = "order") ## drop orders and mixing weights parameters(exampleModels$WL_ibm, namesflag = TRUE, drop = c("order", "prob")) parameters(exampleModels$WL_I, namesflag = TRUE) parameters(exampleModels$WL_II, namesflag = TRUE)
The infix operator %of% is a generic function which applies
functions to objects. This page describes the function and the
methods defined in package mixAR.
"%of%"(e1, e2) e1 %of% e2"%of%"(e1, e2) e1 %of% e2
e1 |
usually a function, the name of a function, a character vector, or a list of functions, see Details. |
e2 |
an object, usually matrix-like. |
%of% is a generic function with dispatch on both arguments.
It is intended to be used mainly in infix form.
%of% transforms each “column” of a matrix-like object by a
function. If e1 specifies a single function, that is applied to all
columns. Otherwise length(e1) should equal the number of
“columns” of e2 and e1[[i]] is applied to the
i-th “column” of e2.
The mental model is that the first argument, e1, is (converted
to) a list of functions containing one function for each column of
e2. The i-th function is applied to each element of the i-th
column.
The methods for "MixComp" objects allow for very transparent
and convenient computing with "MixAR" objects.
for the default method, a matrix;
for methods with e2 from class MixComp, a MixComp
object with its slot m replaced by the result of applying
e1 to its elements, see the descriptions of the individual
methods for details;
Below are the descriptions of the methods for %of% defined by
package mixAR.
signature(e1 = "ANY", e2 = "ANY")This is the default method. It uses apply() to evaluate
e1 for each element of the matrix e2, without
checking the arguments. If the arguments are not suitable for
apply(), any error messages will come from it. So, for this
method e1 is a function (or the name of a function) and
e2 is a matrix or array.
signature(e1 = "function", e2 = "MixComp")Create (and return) a MixComp object with its slot m
replaced by the result of applying the function e1 to each
element of the MixComp object e2, see class
"MixComp".
signature(e1 = "character", e2 = "MixComp")Here e1 contains the names of one or more functions. If
length(e1) = 1, this is equivalent to the method for
e1 of class "function".
If length(e1) > 1, then for each i the function
specified by e1[i] is applied to the ith column of
e2@m. In this case there is no recycling: e1 must
have ncol(e2@m) elements.
signature(e1 = "list", e2 = "MixComp")Here each element of e1 is a function or the name of a
function. It works analogously to the method with e1 from
class "character". If length(e1) = 1, then
e1[[1]] is applied to each element of
e1@m. Otherwise, if length(e1) > 1, then
e1[[i]] is applied to the ith column of e2@m.
The code is rather inefficient for some of the methods.
Maybe should require that the functions in the first argument are vectorised. (Some methods effectively assume it.)
Georgi N. Boshnakov
class "MixComp"
m <- matrix(rnorm(18), ncol = 3) ## defult method pm1 <- pnorm %of% m f3 <- list(pnorm, function(x, ...) pnorm(x, mean = 0.1), function(x, ...) pnorm(x, mean = -0.1) ) ## no method for f from "list" yet: ## pm2 <- f3 %of% m mc <- new("MixComp", m = m) pnorm %of% mc pmc3 <- f3 %of% mc ## result is equivalent to applying f3[[i] to m[ , i]: all.equal(pmc3@m, cbind(f3[[1]](m[ , 1]), f3[[2]](m[ , 2]), f3[[3]](m[ , 3])))m <- matrix(rnorm(18), ncol = 3) ## defult method pm1 <- pnorm %of% m f3 <- list(pnorm, function(x, ...) pnorm(x, mean = 0.1), function(x, ...) pnorm(x, mean = -0.1) ) ## no method for f from "list" yet: ## pm2 <- f3 %of% m mc <- new("MixComp", m = m) pnorm %of% mc pmc3 <- f3 %of% mc ## result is equivalent to applying f3[[i] to m[ , i]: all.equal(pmc3@m, cbind(f3[[1]](m[ , 1]), f3[[2]](m[ , 2]), f3[[3]](m[ , 3])))
All permutations of the columns of a matrix
permn_cols(m)permn_cols(m)
m |
a matrix |
This function is a wrapper for permn from package ‘combinat’.
a list with one element for each permutation of the columns. Each element of the list is an unnamed list with two components:
the permutation, a vector of positive integers,
a matrix obtained by permuting the columns of m.
Georgi N. Boshnakov
m <- matrix(c(11:14,21:24,31:34), ncol=3) pm <- permn_cols(m) pm[[2]]m <- matrix(c(11:14,21:24,31:34), ncol=3) pm <- permn_cols(m) pm[[2]]
Closing prices of four stocks.
data("PortfolioData1")data("PortfolioData1")
A data frame with 867 observations on the following 4 variables.
DELLnumeric, Dell Technologies Inc.
MSFTnumeric, Microsoft Corporation.
INTCnumeric, Intel Corporation.
IBMnumericr, International Business Machine Corporation.
Time series of daily adjusted close prices of the above stocks from 2 January 2016 to 29 January 2020 (867 observations).
‘https://finance.yahoo.com/’
data(PortfolioData1) dim(PortfolioData1) head(PortfolioData1)data(PortfolioData1) dim(PortfolioData1) head(PortfolioData1)
Exact predictive parameters for multi-step MixAR prediction.
predict_coef(model, maxh)predict_coef(model, maxh)
model |
a MixAR model. |
maxh |
maximal horizon. |
predict_coef() implements the method of
Boshnakov (2009) for the h-step prediction
of MixAR processes. The h-step predictive distribution has a MixAR
distribution with components and this function computes its
parameters.
predict_coef() implements the results by
Boshnakov (2009) to compute the parameters
of the predictive distributions. predict_coef() is mostly a
helper function, use multiStep_dist for
prediction/forecasting (the exact method for multiStep_dist
uses predict_coef() to do the main work).
predict_coef() returns a list of lists containing the
quantities needed for each horizon , see section Value.
Alternatiely, the parameters can be obtained as MixAR models
by calling the function generated by the exact method of
multiStep_dist with argument what = "MixAR".
a list with components:
arcoefs |
a list, |
sigmas |
a list, |
probs |
a list, |
sStable |
a list, |
Georgi N. Boshnakov
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.
Small utilities for ragged objects. Modify the elements of raggedCoef objects, extract them as a vector.
rag_modify(rag, v) ragged2vec(x)rag_modify(rag, v) ragged2vec(x)
rag |
the |
v |
vector of values to replace the old ones. |
x |
a |
An error will occur if the length of v is not equal to
sum(rag@p).
rag_modify is, in a sense, the inverse of ragged2vec.
for rag_modify, a raggedCoef object of the same order as
rag but with coefficients replaced by the new values.
for ragged2vec, a numeric vector.
rag1 <- new("raggedCoef", list(1, 2:3, 4:6)) a1 <- (1:6)^2 rag1a <- rag_modify(rag1, a1) rag2 <- new("raggedCoef", list(1, numeric(0), 4:6)) # a zero-length ccomponent a2 <- (1:4)^2 rag2a <- rag_modify(rag2, a2)rag1 <- new("raggedCoef", list(1, 2:3, 4:6)) a1 <- (1:6)^2 rag1a <- rag_modify(rag1, a1) rag2 <- new("raggedCoef", list(1, numeric(0), 4:6)) # a zero-length ccomponent a2 <- (1:4)^2 rag2a <- rag_modify(rag2, a2)
"raggedCoef" — ragged list objects
Some models have several several vectors of parameters, possibly of
different lengths, such that in some circumstances they are thought of
as lists, in others as matrices after suitable padding with zeroes.
Class "raggedCoef" represents such ragged lists. In package
"MixAR" it is used to hold the autoregressive coefficients of
MixAR models.
raggedCoef(p, value = NA_real_)raggedCoef(p, value = NA_real_)
p |
orders, vector of integers. |
value |
typically, a list, but see Details. |
Class "raggedCoef" is for objects that can be considered as
both, lists and matrices. The elements of the list are vectors,
possibly of different lengths. When the object is viewed as a matrix,
each element of the list (suitably padded with zeroes or NAs)
represents a row of a matrix.
The recommended way to create objects from class "raggedCoef"
is with the function raggedCoef.
If value is a "raggedCoef" object it is returned.
If value is a list, it is converted to "raggedCoef" using
new().
If argument p is missing, it is inferred from the
lengths of the elements of the list.
If argument p is not missing, a consistency check
is made to ensure that the order of the object is as specified by
p.
Otherwise, if value is of length one, it is replicated to form
a ragged list with i-th element a vector of length
p[i]. Although not checked, the intention here is that
value is from some atomic class. The default for value
is NA_real_ to give a convenient way to create a ragged list.
Finally, if none of the above applies, value is effectively assumed to
be a vector of length sum(p), although other cases are
admissible (but I don't remember if this was intended). In this case,
value is reshaped into a ragged list to match p. This is
convenient when, for example, the elements of a ragged array are
obtained from an optimisation routine which expects plain vector.
Below we describe the "initialize" method that underlies
new("raggedCoef", ...). The recommended way to create
"raggedCoef" objects is with the function raggedCoef,
see section Details.
Objects can also be created by calls of the form
new("raggedCoef", v), where v is a list whose elements
are numeric vectors, or new("raggedCoef", v1, v2, ...), where
v1, v2, ... are numeric vectors. The two forms are equivalent
if v = list(v1, v2, ...).
The elements of the list v may be named.
Similarly, named arguments can be used in the second form, say
new("raggedCoef", name1 = v1, name2 = v2, ...).
In both cases the names are preserved in the internal representaion,
but not used.
If the arguments are not as specified above the result should be
considered undefined.
Currently, if there are other arguments after the list v, they
are ignored with a warning. If the first argument is not a list then
all arguments must be numeric and an error is raised if this is
not the case. For completeness, we mention that exactly two arguments named
a, and p are also accepted by new(), eg
new("raggedCoef", p = c(1, 2), a = list(3, 4:5)), but these
are assigned to the slots without any checking. so it is
most flexible (and recommended) to use raggedCoef() instead.
a:Object of class "list" containing the values.
p:Object of class "numeric" containing the
lengths of the components of a.
Indexing with "[" treats a raggedCoef object as a matrix, while
"[[" treats the object as list (it works on slot a).
Note that there is a difference between x[2,] (or the
equivalent x[2]) and x[[2]]—the former gives a vector
of length max(p), so potentially padded with zeroes, while the
latter gives the component with its “natural” length.
The replacement variants of "[" and "[[" do not change the structure of the object and issue errors if the replacement value would result in that. In situations where the checks are deemed redundant, direct assignments to the corresponding slots may be used.
signature(x = "raggedCoef", i = "missing", j = "missing", drop = "ANY"):
signature(x = "raggedCoef", i = "missing", j = "numeric", drop = "ANY"):
signature(x = "raggedCoef", i = "numeric", j = "missing", drop = "ANY"):
signature(x = "raggedCoef", i = "numeric", j = "numeric", drop = "ANY"):
Indexing with "[" treats a raggedCoef object as a matrix
with one row for each component and number of columns equal to
max(p). However, x[2] is equivalent to x[2,]
which is different from the treatment of matrix objects in
base R.
signature(x = "raggedCoef", i = "ANY", j = "missing"):
signature(x = "raggedCoef", i = "ANY", j = "ANY"):
"[[" extracts the corresponding element of slot a.
signature(x = "raggedCoef", i = "ANY", j = "ANY", value = "numeric"):
Replace the j-th element of i-th row with value.
All arguments must be scalars.
signature(x = "raggedCoef", i = "ANY", j = "missing", value = "numeric"):
signature(x = "raggedCoef", i = "ANY", j = "missing", value = "numeric"):
Replace the i-th row with value. Argument i must be
a scalar while the length of value must be the same as that
of x@a[[i]]. The methods for "[" and "[[" with this
signature coinside.
signature(x = "raggedCoef", i = "ANY", j = "missing", value = "list"):
The elements of value must have the same lengths as the
elements they are replacing.
signature(x = "raggedCoef", i = "ANY", j = "missing", value = "matrix"):
This is essentially the reverse od the corresponding
non-replacement operator. value must have at least as many
columns as the longest element of x that is replaced.
signature(x = "raggedCoef", i = "ANY", j = "ANY", value = "numeric"): ...
signature(x = "raggedCoef", i = "missing", j = "missing", value = "list"): ...
signature(x = "raggedCoef", i = "missing", j = "missing", value = "matrix"): ...
signature(x = "raggedCoef", i = "missing", j = "missing", value = "numeric"): ...
signature(.Object = "raggedCoef"):
Creates objects of class raggedCoef. This method is used
internally by new(). Users should use new() for
creation of objects from this class, see the examples.
signature(object = "raggedCoef"): ...
signature(x = "numeric", coef = "raggedCoef", index = "numeric"):
Apply a mixture filter to a time series.
signature(x = "raggedCoef"):
Gives x@p, which is the same as lengths(x@a).
signature(x = "raggedCoef"):
Gives the total number of coefficients (sum(x@p)).
signature(x = "raggedCoef"):
Are there NA's in x@a?
signature(x = "raggedCoef"):
The dimension of the object, when viewed as a matrix.
The presence of this method also ensures that nrow()
and related functions give the expected result.
Slot p is redundant but convenient.
Georgi N. Boshnakov
class "MixARGaussian"
ragged1 <- list(1, 2:3, 4:6) ragged2 <- list(a = 1, b = 2:3, c = 4:6) raggedCoef(1:3) # only order given, fill with NA's raggedCoef(1:3, 0) # fill with a number (zero in this case) ## init with a list raggedCoef(ragged1) raggedCoef(value = ragged1) ## error, since the shape of ragged1 is not c(2, 2, 3): ## raggedCoef(c(2, 2, 3), value = ragged1) ## init with a flattened list raggedCoef(p = 1:3, value = 1:6) ## specify each component separately ragA <- new("raggedCoef", 1, 2:3, 4:6) ragB <- new("raggedCoef", list(1, 2:3, 4:6)) # same identical(ragA, ragB) #TRUE ## extract as a matrix ragA[] ## extract the 2nd component ragA[2] # c(2, 3, 0) ("[" pads with 0's) ragA[[2]] # c(2, 3) ("[[" does not pad) ## get the 2nd and 3rd components as a matrix ragA[2:3, ] # "[" treats object (almost) as matrix ragA[2:3] # same (though not as for "matrix") ## names are kept in the list but currently not used ragC <- new("raggedCoef", list(a = 1, b = 2:3, c = 4:6)) ragC1 <- new("raggedCoef", a = 1, b = 2:3, c = 4:6) identical(ragC, ragC1) # TRUE names(ragC@a) # [1] "a" "b" "c" length(ragA) dim(ragA) c(nrow(ragA), ncol(ragA)) c(NROW(ragA), NCOL(ragA))ragged1 <- list(1, 2:3, 4:6) ragged2 <- list(a = 1, b = 2:3, c = 4:6) raggedCoef(1:3) # only order given, fill with NA's raggedCoef(1:3, 0) # fill with a number (zero in this case) ## init with a list raggedCoef(ragged1) raggedCoef(value = ragged1) ## error, since the shape of ragged1 is not c(2, 2, 3): ## raggedCoef(c(2, 2, 3), value = ragged1) ## init with a flattened list raggedCoef(p = 1:3, value = 1:6) ## specify each component separately ragA <- new("raggedCoef", 1, 2:3, 4:6) ragB <- new("raggedCoef", list(1, 2:3, 4:6)) # same identical(ragA, ragB) #TRUE ## extract as a matrix ragA[] ## extract the 2nd component ragA[2] # c(2, 3, 0) ("[" pads with 0's) ragA[[2]] # c(2, 3) ("[[" does not pad) ## get the 2nd and 3rd components as a matrix ragA[2:3, ] # "[" treats object (almost) as matrix ragA[2:3] # same (though not as for "matrix") ## names are kept in the list but currently not used ragC <- new("raggedCoef", list(a = 1, b = 2:3, c = 4:6)) ragC1 <- new("raggedCoef", a = 1, b = 2:3, c = 4:6) identical(ragC, ragC1) # TRUE names(ragC@a) # [1] "a" "b" "c" length(ragA) dim(ragA) c(nrow(ragA), ncol(ragA)) c(NROW(ragA), NCOL(ragA))
"raggedCoefS" — ragged list
Ragged list used to hold coefficients of MixAR models with seasonal AR parameters.
Objects are created by calls of the form
new("raggedCoefS", a = list(v1, v2 , ...), as = list(vs1, vs2, ...), s).
If orders p and ps are specified, a consistency check is made.
Object of class "list" containing AR values. Each element of the list must be "numeric"
Object of class "numeric" containing the lengths of components in a. If missing, it is generated based on lengths of elements of a.
Object of class "list" containing seasonal AR values. Each element of the list must be "numeric"
Object of class "numeric" containing the lengths of elements of as. If missing, it is generated based on lengths of elements of as.
A single element "numeric" vector
determining the seasonality in the model(monthly, quarterly, etc..).
Indexing with "[" treats a raggedCoef object as a matrix
(one row for each component), while
"[[" treats the object as list (it works on slot a). Specifically,
"[[1]]" picks the systematic AR parameters, "[[2]]" picks seasonal AR
parameters.
The replacement variants of "[" and "[[" do not change the structure of the object.
Replacement methods only work for subsets
x[[i]], x[[i]][[j]], x[[i]][[j]][k] for suitable i,
j and k.
i must be equal to 1 for x@a and 2 for x@as.
signature(x = "raggedCoefS", i = "missing", j = "missing"):
returns the complete matrix of coefficients, one row corresponding to one component, with '0's to match different orders
signature(x = "raggedCoefS", i = "missing", j =
"missing"):
signature(x = "raggedCoefS", i = "numeric", j =
"missing"):
signature(x = "raggedCoefS", i = "numeric", j = "numeric"):
Indexing with "[" treats a raggedCoef object as a matrix
with one row for each component and number of columns equal to
max(p) + max(ps) in increasing lag.
However, x[2] is equivalent to x[2,]
which is different from the treatment of matrix objects in
base R.
signature(x = "raggedCoefS"), i = "numeric":
if i=1 selects the list of systematic AR parameters; if i=2 selects the list of seasonal AR parameters.
signature(x = "raggedCoefS"), i = "numeric", j = "numeric":
signature(x = "raggedCoefS"), i = "numeric", j = "numeric", k = "numeric":
j and k are used to select specific elements from the listt of interest.
Davide Ravagli
class "raggedCoef"
showClass("raggedCoefS") ragA <- new("raggedCoefS", a = list( c(0.5, -0.5), 1), as = list(0, c(0.3, -0.1) ), s = 12) ragB <- new("raggedCoefS", a = list( c(0.5, -0.5), 1), p = c(2, 1), as = list(0, c(0.3, -0.1) ), ps = c(1, 2), s = 12) # same ## Elements selection examples ragA[] ## matrix of coefficients ragA[1]; ragA[1, ] ## vector of coefficients from first component ragA[[2]] ## list of seasonal AR parameters ragA[[2]][[1]] ## vector of seasonal AR parameters from first component ## Replacement of values in 'raggedCoefS' objects ragB[[2]] <- list(1, c(-0.5,0.5)) ragB[[2]][[2]] <- c(20, 22) ragB[[1]][[1]][1] <- 0showClass("raggedCoefS") ragA <- new("raggedCoefS", a = list( c(0.5, -0.5), 1), as = list(0, c(0.3, -0.1) ), s = 12) ragB <- new("raggedCoefS", a = list( c(0.5, -0.5), 1), p = c(2, 1), as = list(0, c(0.3, -0.1) ), ps = c(1, 2), s = 12) # same ## Elements selection examples ragA[] ## matrix of coefficients ragA[1]; ragA[1, ] ## vector of coefficients from first component ragA[[2]] ## list of seasonal AR parameters ragA[[2]][[1]] ## vector of seasonal AR parameters from first component ## Replacement of values in 'raggedCoefS' objects ragB[[2]] <- list(1, c(-0.5,0.5)) ragB[[2]][[2]] <- c(20, 22) ragB[[1]][[1]][1] <- 0
"raggedCoefV" — ragged list
Ragged list used to hold coefficients of MixVAR models.
Objects are created by calls of the form
new("raggedCoefV", a = list(v1, v2 ,...).
a:Object of class "list" containing AR values. Each element of
the list must be "array".
p:Object of class "numeric" containing the length of arrays in
a (AR orders). If missing, it is generated based on lengths
of elements of a.
Indexing with "[" and "[[" works on slot a.
"[" and "[[" can be use alternatively. Specifically, "[]" and "[[]]"
produce the same result, the complete list of AR coefficients.
Similarly, [i,], [i] and [[i]] all return the
i^th element of the list, the array for i^th component. [,j]
returns an array with j^th lag autoregressive parameters for each
component.
signature(x = "raggedCoefV", i = "missing", j = "ANY", drop = "ANY"): ...
signature(x = "raggedCoefV", i = "missing", j = "numeric", drop = "ANY"): ...
signature(x = "raggedCoefV", i = "numeric", j = "missing", drop = "ANY"): ...
signature(x = "raggedCoefV", i = "numeric", j = "numeric", drop = "ANY"): ...
signature(x = "raggedCoefV", i = "missing", j = "ANY"): ...
signature(x = "raggedCoefV", i = "numeric", j = "ANY"): ...
signature(.Object = "raggedCoefV"): ...
signature(object = "raggedCoefV"): ...
Davide Ravagli
class "MixVAR"
showClass("raggedCoefV") AR <- list() AR[[1]] <- array(c(0.0973, -0.0499, 0.2927, 0.4256, ## VAR(2;4) -0.0429, 0.0229, -0.1515, -0.1795, -0.0837, -0.1060, -0.1530, 0.1947, -0.1690, -0.0903, 0.1959, 0.0955), dim=c(2,2,4)) AR[[2]] <- array(c(0.3243, 0.2648, 0.4956, 0.2870, ## VAR(2;3) -0.1488, 0.0454, -0.0593, -0.3629, 0.1314, 0.0274, 0.0637, 0.0485), dim=c(2,2,3)) new("raggedCoefV", AR) new("raggedCoefV", a=AR, p=c(4,3))showClass("raggedCoefV") AR <- list() AR[[1]] <- array(c(0.0973, -0.0499, 0.2927, 0.4256, ## VAR(2;4) -0.0429, 0.0229, -0.1515, -0.1795, -0.0837, -0.1060, -0.1530, 0.1947, -0.1690, -0.0903, 0.1959, 0.0955), dim=c(2,2,4)) AR[[2]] <- array(c(0.3243, 0.2648, 0.4956, 0.2870, ## VAR(2;3) -0.1488, 0.0454, -0.0593, -0.3629, 0.1314, 0.0274, 0.0637, 0.0485), dim=c(2,2,3)) new("raggedCoefV", AR) new("raggedCoefV", a=AR, p=c(4,3))
Filter a time series with options to shift and scale. This function is used by mixFilter.
raghat1(filter, x, index, shift = 0, residual = FALSE, scale = 1)raghat1(filter, x, index, shift = 0, residual = FALSE, scale = 1)
filter |
The coefficients of the filter, numeric, see Details. |
x |
time series, numeric. |
index |
indices for which to compute the filtered values, numeric. |
shift |
a constant to be added to each filtered element, a number. |
residual |
if TRUE calculate a ‘residual’, otherwise calculate a ‘hat’ value. |
scale |
if |
This function is used by mixFilter. Applies an autoregressive
filter to a time series for indices specified by index.
Note that ‘filter’ here is equivalent to calculating one-step
predictions (or residuals if residual=TRUE) from
autoregressions.
index should not specify indices smaller than
length(filter)+1 or larger than length(x)+1. The value
length(x)+1 can legitimately be used to calculate a prediction
(but not a residual of course) for the first value after the end of the
series.
A numeric vector of length equal to length(index).
This should probably use filter but for the purposes of this
package filter is usually short and the calculation is
vectorised w.r.t. index, so should not be terribly slow.
Translations of functions from my Mathematica sources. Not used currently?
randomArCoefficients(ts, wv, pk, pmax, partempl, sub_size = 10, condthr = 10, nattempt = 10, startfrom = pmax + 1) randomMarParametersKernel(ts, ww, pk, pmax, partempl, ...) randomMarResiduals(ts, p, partempl) tsDesignMatrixExtended(ts, p, ind, partempl)randomArCoefficients(ts, wv, pk, pmax, partempl, sub_size = 10, condthr = 10, nattempt = 10, startfrom = pmax + 1) randomMarParametersKernel(ts, ww, pk, pmax, partempl, ...) randomMarResiduals(ts, p, partempl) tsDesignMatrixExtended(ts, p, ind, partempl)
ts |
time series. |
wv, ww
|
a vector of weights (?). |
pk |
the AR order of the requested component. |
pmax |
the maximal AR order in the model. Needed since it cannot be determined by functions working on a single component. |
partempl |
parameter template, a list containing one element for each mixture component, see Details. |
sub_size |
the size of the subsample to use, default is 10. |
condthr |
threshold for the condition number. |
nattempt |
if |
startfrom |
the starting index (in |
... |
arguments to pass on to |
p |
a vector of non-negative integers, the MixAR order. |
ind |
a vector of positive integers specifying the indices of the observations to use for the “response” variable. |
randomArCoefficients tries small subsamples (not necessarilly
contiguous) from the observations in search of a cluster hopefully
belonging to one mixture component and estimates the corresponding
shift and AR parameters.
randomMarResiduals selects random parameters for each mixture
component and returns the corresponding residuals.
randomMarParametersKernel is a helper function which does the
computation for one component.
tsDesignMatrixExtended forms the extended design matrix
corresponding to a subsample. This is used for least square estimation
of the parameters.
Georgi N. Boshnakov
row_lengths in package mixAR
Determine the lengths of the ‘rows’ of a ragged object.
Some objects in this package contain (effectively) lists of vectors. These vectors are considered ‘rows’ and this function returns their lengths (as a vector).
signature(x = "ANY")The default method. Applies length to the elements of the
argument (2020-03-28: now using lengths(x)).
signature(x = "raggedCoef")Returns the lengths of the rows of the components, a numeric vector.
signature(x = "MixAR")Returns the AR orders of the model components, a numeric vector.
Sampling functions for Bayesian analysis of mixture autoregressive
models. Draws observations from posterior distributions of the latent
variables Z_ts and the parameters of mixture autoregressive
models.
sampZpi(y, pk, prob, mu, AR, sigma, nsim, d) sampMuShift(y, pk, prec, nk, shift, z, AR, nsim) sampSigmaTau(y, pk, prec, nk, AR, mu, z, a, c, nsim)sampZpi(y, pk, prob, mu, AR, sigma, nsim, d) sampMuShift(y, pk, prec, nk, shift, z, AR, nsim) sampSigmaTau(y, pk, prec, nk, AR, mu, z, a, c, nsim)
y |
a time series (currently a |
pk |
|
prob |
|
mu |
|
shift |
|
AR |
|
sigma |
|
nsim |
dessired sample size. |
d |
|
prec |
|
nk |
output from |
z |
output |
a, c
|
hyperparameters. |
sampZpi draws observations from posterior distributions of the
latent variables Z_ts and mixing weights of a Mixture
autoregressive model.
sampSigmaTau draws observations from posterior distributions of
the precisions tau_k of a Mixture autoregressive model, and
obtains scales sigma_k by transformation.
sampMuShift Draws observations from posterior distributions of
the means mu_k of a Mixture autoregressive model, and obtains
shifts phi_k0 by transformation.
for sampZpi, a list containing the following elements:
mix_weights |
|
latentZ |
|
nk |
Vector of length |
for sampMuShift, a list containing the following elements:
shift |
|
mu |
|
for sampSigmaTau, a list containing the following elements:
scale |
|
precision |
|
lambda |
|
Davide Ravagli
model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = exampleModels$WL_At@arcoef@a ) # list(-0.5, 1.1) prob <- model@prob sigma <- model@scale prec <- 1 / sigma ^ 2 g <- length(model@prob) d <- rep(1, g) pk <- model@arcoef@p p <- max(pk) shift <- mu <- model@shift AR <- model@arcoef@a model set.seed(1234) n <- 50 # 500 nsim <- 50 y <- mixAR_sim(model, n = n, init = 0) x <- sampZpi(y, pk, prob, shift, AR, sigma, nsim = nsim, d) x1 <- sampMuShift(y, pk, prec, nk = x$nk, shift, z = x$latentZ, AR, nsim = nsim) x2 <- sampSigmaTau(y, pk, prec, nk = x$nk, AR, mu = x1$mu, z = x$latentZ, a = 0.2, c = 2, nsim = nsim)model <- new("MixARGaussian", prob = exampleModels$WL_At@prob, # c(0.5, 0.5) scale = exampleModels$WL_At@scale, # c(1, 2) arcoef = exampleModels$WL_At@arcoef@a ) # list(-0.5, 1.1) prob <- model@prob sigma <- model@scale prec <- 1 / sigma ^ 2 g <- length(model@prob) d <- rep(1, g) pk <- model@arcoef@p p <- max(pk) shift <- mu <- model@shift AR <- model@arcoef@a model set.seed(1234) n <- 50 # 500 nsim <- 50 y <- mixAR_sim(model, n = n, init = 0) x <- sampZpi(y, pk, prob, shift, AR, sigma, nsim = nsim, d) x1 <- sampMuShift(y, pk, prec, nk = x$nk, shift, z = x$latentZ, AR, nsim = nsim) x2 <- sampSigmaTau(y, pk, prec, nk = x$nk, AR, mu = x1$mu, z = x$latentZ, a = 0.2, c = 2, nsim = nsim)
Show differences between two MixAR models in a way that enables quick
comparison between them. This is a generic function, package
mixAR defines methods for MixAR models.
show_diff(model1, model2)show_diff(model1, model2)
model1, model2
|
the MixAR models to be compared. |
show_diff() is a generic function with dispatch on both
arguments.
show_diff() prints the differences between two
models in convenient form for comparison. The methods for MixAR models
allow to see differences between similar models at a glance.
The function is called for the side effect of printing the differences between the two models and has no useful return value.
signature(model1 = "MixAR", model2 = "MixAR")signature(model1 = "MixARGaussian", model2 = "MixARgen")signature(model1 = "MixARgen", model2 = "MixARGaussian")signature(model1 = "MixARgen", model2 = "MixARgen")Georgi N. Boshnakov
## the examples reveal that the models below ## differ only in the noise distributions show_diff(exampleModels$WL_Ct_3, exampleModels$WL_Bt_1) show_diff(exampleModels$WL_Bt_1, exampleModels$WL_Ct_3) show_diff(exampleModels$WL_Ct_2, exampleModels$WL_Bt_3)## the examples reveal that the models below ## differ only in the noise distributions show_diff(exampleModels$WL_Ct_3, exampleModels$WL_Bt_1) show_diff(exampleModels$WL_Bt_1, exampleModels$WL_Ct_3) show_diff(exampleModels$WL_Ct_2, exampleModels$WL_Bt_3)
Perform simulation experiments
simuExperiment(model, simu, est, N = 100, use_true = FALSE, raw = FALSE, init_name = "init", keep = identity, summary_fun = .fsummary, ...)simuExperiment(model, simu, est, N = 100, use_true = FALSE, raw = FALSE, init_name = "init", keep = identity, summary_fun = .fsummary, ...)
model |
the model, see 'Details'. |
simu |
arguments for the simulation function, a list, see 'Details'. |
est |
arguments for the estimation function, a list, see 'Details'. |
N |
number of simulations. |
use_true |
if TRUE, use also the "true" coefficients as initial values, see 'Details'. |
raw |
if TRUE, include the list of estimated models in the returned value. |
init_name |
name of the argument of the estimation function which specifies the initial values for estimation, not always used, see ‘Details’. |
keep |
what values to keep from each simulation run, a function, see 'Details'. |
summary_fun |
A function to apply at the end of the experiment to obtain a summary, see 'Details'. |
... |
additional arguments to pass on to the summary function. NOTE: this may change. |
Argument model specifies the underlying model and is not always
needed, see the examples.
Argument simu specifies how to simulate the data.
Argument est specifies the estimation procedure.
Argument N specifies the number of simulation runs.
The remaining arguments control details of the simulations, mostly
what is returned.
Basically, simuExperiment does N simulation-estimation
runs.
The keep function is applied to the value obtained from each
run.
The results from keep are assembled in a list (these are the
'raw' results).
Finally, the summary function (argument summary_fun) is applied
to the raw list.
simu and est are lists with two elements: fun and
args. fun is a function or the name of a
function. args is a list of arguments to that function. The
first argument of the estimation function, est$fun, is the
simulated data. This argument is inserted by simuExperiment and
should not be put in est$args.
The value returned by the summary function is the main part of the
result. If raw = TRUE, then the raw list is returned, as well.
Further fields may be made possible through additional arguments but
'Summary' and 'Raw' are guaranteed to be as described here.
simuExperiment uses init_name only if use_true is
TRUE to arrange a call of the estimation function with initial value
model. Obviously, simuExperiment does not know how (or
if) the estimation function does with its arguments.
The function specified by argument keep is called with one
argument when use_true is FALSE and two arguments otherwise.
A list with one or more elements, depending on the arguments.
Summary |
a summary of the experiment, by default sample means and standard deviations of the estimates. |
Raw |
A list of the estimated models. |
Georgi N. Boshnakov
## explore dist. of the mean of a random sample of length 5. ## (only illustration, such simple cases hardly need simuExperiment) sim1 <- list(fun="rnorm", args = list(n=5, mean=3, sd = 2)) est1 <- list(fun=mean, args = list()) # a basic report function fsum1 <- function(x){ wrk <- do.call("c",x) c(n = length(wrk), mean = mean(wrk), sd = sd(wrk))} a1 <- simuExperiment(TRUE, simu = sim1, est = est1, N = 1000, summary_fun = fsum1) # explore also the dist. of the sample s.d. est2 <- est1 est2$fun <- function(x) c(xbar = mean(x), s = sd(x)) a2 <- simuExperiment(TRUE, simu = sim1, est = est2, N = 1000) # keep the raw sample means and s.d.'s for further use a2a <- simuExperiment(TRUE, simu = sim1, est = est2, N = 1000, raw = TRUE) a2a$Summary # replicate a2a$Summary s5 <- sapply(a2a$Raw, identity) apply(s5, 1, mean) apply(s5, 1, sd) hist(s5[1,], prob=TRUE) lines(density(s5[1,])) curve(dnorm(x, mean(s5[1,]), sd(s5[1,])), add = TRUE, col = "red") mixAR:::.fsummary(a2a$Raw) mixAR:::.fsummary(a2a$Raw, merge = TRUE)## explore dist. of the mean of a random sample of length 5. ## (only illustration, such simple cases hardly need simuExperiment) sim1 <- list(fun="rnorm", args = list(n=5, mean=3, sd = 2)) est1 <- list(fun=mean, args = list()) # a basic report function fsum1 <- function(x){ wrk <- do.call("c",x) c(n = length(wrk), mean = mean(wrk), sd = sd(wrk))} a1 <- simuExperiment(TRUE, simu = sim1, est = est1, N = 1000, summary_fun = fsum1) # explore also the dist. of the sample s.d. est2 <- est1 est2$fun <- function(x) c(xbar = mean(x), s = sd(x)) a2 <- simuExperiment(TRUE, simu = sim1, est = est2, N = 1000) # keep the raw sample means and s.d.'s for further use a2a <- simuExperiment(TRUE, simu = sim1, est = est2, N = 1000, raw = TRUE) a2a$Summary # replicate a2a$Summary s5 <- sapply(a2a$Raw, identity) apply(s5, 1, mean) apply(s5, 1, sd) hist(s5[1,], prob=TRUE) lines(density(s5[1,])) curve(dnorm(x, mean(s5[1,]), sd(s5[1,])), add = TRUE, col = "red") mixAR:::.fsummary(a2a$Raw) mixAR:::.fsummary(a2a$Raw, merge = TRUE)
Compute moments and absolute moments of standardised-t, t and normal distributions.
stdnormmoment(k) stdnormabsmoment(k) stdtmoment(nu, k) stdtabsmoment(nu, k) tabsmoment(nu, k)stdnormmoment(k) stdnormabsmoment(k) stdtmoment(nu, k) stdtabsmoment(nu, k) tabsmoment(nu, k)
k |
numeric vector, moments to compute. |
nu |
a number, degrees of freedom. |
These functions compute moments of standardised-t and standard normal distibutions. These distributions have mean zero and variance 1. Standardised-t is often prefferred over Student-t for innovation distributions, since its variance doesn't depend on its parameter (degrees of freedom). The absolute moments of the usual t-distributions are provided, as well.
The names of the functions start with an abbreviated name of the
distribution concerned: stdnorm (N(0,1)), stdt
(standardised-t), t (Student-t).
The functions with names ending in absmoment()
(stdnormabsmoment(), stdtabsmoment() and tabsmoment())
compute absolute moments, The rest (stdnormmoment() and
stdtmoment()) compute ordinary moments.
The absolute moments are valid for (at least) k >= 0, not
necessarily integer. The ordinary moments are currently intended only
for integer moments and return NaN's for fractional ones, with
warnings.
Note that the Student-t and standardised-t with degrees
of freedom have finite (absolute) moments only for .
As a consequence, standardised-t is defined only for
(otherwise the variance is infinite).
stdtabsmoment returns Inf for any .
stdtmoment returns Inf for even integer k's, such
that . However, for odd integers it returns
zero and for non-integer moments it returns NaN.
Here is an example, where the first two k's are smaller than
nu, while the others are not:
stdtabsmoment(nu = 5, k = c(4, 4.5, 5, 5.5)) ##: [1] 9.00000 29.31405 Inf Inf stdtmoment(nu = 5, k = c(4, 4.5, 5, 5.5)) ##: [1] 9 NaN 0 NaN
These functions are designed to work with scalar nu but this
is not enforced.
numeric vector of the same length as k.
Georgi N. Boshnakov
Würtz D, Chalabi Y, Luksan L (2006). “Parameter Estimation of ARMA Models with GARCH / APARCH Errors An R and SPlus Software Implementation.” http://www-stat.wharton.upenn.edu/%7Esteele/Courses/956/RResources/GarchAndR/WurtzEtAlGarch.pdf.
## some familiar positive integer moments stdnormmoment(1:6) ## fractional moments of N(0,1) currently give NaN stdnormmoment(seq(1, 6, by = 0.5)) ## abs moments don't need to be integer curve(stdnormabsmoment, from = 0, to = 6, type = "l", col = "blue") ## standardised-t stdtmoment(5, 1:6) stdtabsmoment(5, 1:6) stdtabsmoment(5, 1:6) ## Student-t tabsmoment(5, 1:6)## some familiar positive integer moments stdnormmoment(1:6) ## fractional moments of N(0,1) currently give NaN stdnormmoment(seq(1, 6, by = 0.5)) ## abs moments don't need to be integer curve(stdnormabsmoment, from = 0, to = 6, type = "l", col = "blue") ## standardised-t stdtmoment(5, 1:6) stdtabsmoment(5, 1:6) stdtabsmoment(5, 1:6) ## Student-t tabsmoment(5, 1:6)
Translations of some of my MixAR Mathematica functions. Not sure if these are still used.
tomarparambyComp(params) tomarparambyType(params) permuteArpar(params)tomarparambyComp(params) tomarparambyType(params) permuteArpar(params)
params |
the parameters of the MixAR model, a list, see Details. |
tomarparambyComp is for completeness, my Mathematica programs
do not have this currently.
The arrangement of the parameters of MixAR models in package
"MixAR" is “by type”: slot prob contains the mixture
probabilities (weights), shift contains intercepts, and so on.
An alternative representation is “by component”: a list whose k-th elements contains all parameters associated with the k-th mixture component. The functions described here use the following order for the parameter of the k-th component: prob_k, shift_k, arcoeff_k, sigma2_k.
tomarparambyType takes an argument, params, arranged
“by component” and converts it to “by type”.
tomarparambyComp does the inverse operation, from “by type”
to “by component”.
permuteArpar creates all permutaions of the components of a
MixAR model. It takes a “by component” argument. The autoregressive
orders are not permuted, in that if the input model has AR orders
c(2, 1, 3), all permuted models are also c(2, 1, 3).
The AR coefficients of shorter or longer components are padded with
zeroes or truncated, respectively, see the unexported
adjustLengths().
For tomarparambyComp, a list containing the parameters of
the model arranged “by component”, see Details.
For tomarparambyType, a list containing the parameters of
the model arranged “by type”. It contains the following elements.
prob |
mixture probabilities, a numeric vector, |
shift |
shifts, a numeric vector, |
arcoef |
autoregressive coefficients, |
s2 |
noise variances, a numeric vector. |
For permuteArpar, a list with one element (arranged “by
type”) for each possible permutation of the AR parameters.
Georgi N. Boshnakov
bycomp <- list(list(0.1, 10, 0.11, 1), list(0.2, 20, c(0.11, 0.22), 2), list(0.3, 30, c(0.11, 0.22, 0.33), 3) ) bytype <- tomarparambyType(bycomp) identical(bycomp, tomarparambyComp(bytype)) # TRUE permuteArpar(bycomp)bycomp <- list(list(0.1, 10, 0.11, 1), list(0.2, 20, c(0.11, 0.22), 2), list(0.3, 30, c(0.11, 0.22, 0.33), 3) ) bytype <- tomarparambyType(bycomp) identical(bycomp, tomarparambyComp(bytype)) # TRUE permuteArpar(bycomp)