# beyondWhittle v1.1.1

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## Bayesian Spectral Inference for Stationary Time Series

Implementations of Bayesian parametric, nonparametric and semiparametric procedures for univariate and multivariate time series. The package is based on the methods presented in C. Kirch et al (2018) <doi:10.1214/18-BA1126> and A. Meier (2018) <https://opendata.uni-halle.de//handle/1981185920/13470>. It was supported by DFG grant KI 1443/3-1.

## Functions in beyondWhittle

 Name Description arma_conditional Negative ARMA(p, q) log likelihood beyondWhittle-package Bayesian spectral inference for stationary time series betaBasis_k Construct Bernstein polynomial basis of degree k on omega center mean center a numerical vector fast_ift Fast Inverse Fourier Transform VARMAcov_muted This is a nearly exact copy of the MTS::VARMAcov function, where the output commands at the end are removed. This has to be done because the function is called repeatedly within the MCMC algorithm. For future versions of the package, a better solution is intended. fast_ft Fast Fourier Transform Adj adjoint of complex matrix gibbs_VAR_nuisance_intern Gibbs sampling algorithm for VAR model gibbs_ar Gibbs sampler for an autoregressive model with PACF parametrization. cx_cube_from_ComplexVector I/O: Only used within Rcpp gibbs_multivariate_nuisance Gibbs sampler for corrected parametric likelihood + Bernstein-Dirichlet mixture, including possibility of using time series as mere nuisance parameter Multivariate case gibbs_multivariate_nuisance_cpp Gibbs sampler in Cpp dbList Construct Bernstein polynomial basises of degree up to kmax on omega gibbs_vnp Gibbs sampler for multivaiate Bayesian nonparametric inference with Whittle likelihood llike Log corrected parametric AR likelihood (Gaussian) acvBlockMatrix Build an nd times nd Block Toeplitz matrix from the (d times d) autocovariances gamma(0),...,gamma(n-1) hasEigenValueSmallerZero Does a matrix have an eigenvalue smaller than 0? acvMatrix Build an n times n Toeplitz matrix from the autocovariance values gamma(0),...,gamma(n-1) llike_AR Time domain AR(p) likelihood for nuisance/noise time series is_hpd Check if a matrix is Hermitian positive definite is_quadratic Is l quadratic? complexValuedPsd Inverse function to realValuedPsd cube_from_NumericVector I/O: Only used within Rcpp Note: Same workaround as cx_cube_from_ComplexVector genEpsARMAC Get epsilon process (i.e. model residuals) for ARMA(p,q) unit_trace_I_l Range intervals I_l, see (63) in Mittelbach et al. plot.gibbs_psd Plot method for gibbs_psd class mpdgrm Compute Periodgram matrix from (complex-valued) Fourier coefficients uniformmax_multi Helping function for uci_matrix mixtureWeight Get mixture weights of Bernstein-Dirchlet-Mixtures phiFromBeta_normalInverseWishart Convert vector parametrization (beta) to matrix-parametrization (phi), the latter as e.g. used in MTS::VAR()\$ar coarsened_bernstein Construct coarsened Bernstein polynomial basis of degree l on omega unit_trace_log_f_l Get log(f_l), see (66) in Mittelbach et al. Helping function for unit_trace_runif gibbs_nuisance Gibbs sampler for corrected parametric likelihood + Bernstein-Dirichlet mixture, including possibility of using time series as mere nuisance parameter get_U_cpp Get U from phi, vectorized, cpp internal only coarsened_bernstein_i Helping function for coarsened_bernstein unit_trace_mu Get mu vector, see (36) in Mittelbach et al. Helping function for unit_trace_runif fast_mean Help function to compute the mean. my_rdirichlet Generate a random samples from a Dirichlet distribution logfuller Fuller Logarithm gibbs_var Gibbs sampler for vector autoregressive model. lpost Log posterior = log prior + log corrected parametric likelihood get_f_matrix Construct psd mixture densityMixture Construct a density mixture from mixture weights and density functions. epsilon_var epsilon process (residuals) of VAR model gibbs_AR_nuisance_intern Gibbs sampler for Bayesian AR model in PACF parametrization, including support for TS to be a nuisance parameter lpost_AR Log Posterior = Log Prior + (conditional) Log Likelihood mdft Multivariate discrete (fast) Fourier Transform psd_arma ARMA(p,q) spectral density function qpsd Compute normalized PSD in the Bernstein-Dirichlet parametrization. lprior_AR Log prior for PACF (~Beta) and sigma2 (~InverseGamma), unnormalized print_warn Help function to print debugging messages print.gibbs_psd Print method for gibbs_psd class nll_norm Negative log likelihood of iid standard normal observations [unit variance] Note: deprecated plotMPsd Visualization of multivariate PSDs Used in plot.gibbs_psd realValuedPsd Store imaginary parts above and real parts below the diagonal lprior Log prior of Bernstein-Dirichlet mixture and parametric working model -- all unnormalized is_spd Check if a matrix is symmetric positive definite llike_var_partial VAR(p) partial likelihood (unnormalized) Note: Fine for fixed p, but not suited for model comparison lik_ar Likelihood of an autoregressive time series model with i.i.d. normal innovations midft Multivariate inverse discrete (fast) Fourier Transform print_mcmc_state Help function to print MCMC state sim_varma Simulate from a VARMA model missingValues_str_help Get string representation for missing values position from vector index print_summary_gibbs_psd_help Helping function for print and summary (both are quite similar) fourier_freq Fourier frequencies logDet_stickBreaking Log determinant of stick breaking transformation V -> p unit_trace_nu Get log(nu) vector, see (38) in Mittelbach et al. Helping function for unit_trace_runif gibbs_npc Gibbs sampler for Bayesian semiparametric inference with the corrected AR likelihood gibbs_np Gibbs sampler for Bayesian nonparametric inference with Whittle likelihood unit_trace_p Get p vector, see (67) in Mittelbach et al. pFromV Get p from v in Stick Breaking DP representation transfer_polynomial VARMA transfer polynomials psd_dummy_model Time series model X_t=e_t, E[e_t]=0 omegaFreq Fourier frequencies rescaled on the unit interval summary.gibbs_psd Summary method for gibbs_psd class psd_array Convert psd vector to array (compatibility: to use plotMPsd for univariate functions as well) pacf2AR C++ function for computing AR coefficients from PACF. See Section III in Barndorff-Nielsen and Schou (1973) llike_var VAR(p) likelihood llike_var_full VAR(p) full likelihood psd_varma VARMA(p,q) spectral density function sldmvnorm sum of multivariate normal log densities with mean 0 and covariance Sigma, unnormalized unit_trace_runif_single Obtain one uniform draw from d times d Hpd matrices with unit trace See Algorithm 2 in Mittelbach et al. (adjusted to complex case) uniformmax_help Helping function for uci_matrix uniformmax Uniform maximum, as needed for uniform credible intervals unit_trace_log_d Get log(d) vector, see (39) in Mittelbach et al, adjusted to complex case Helping function for unit_trace_runif unit_trace_log_c Get log(c) vector, see (70) in Mittelbach et al. Helping function for unit_trace_runif scree_type_ar Negative log AR likelihood values for scree-type plots rmvnorm Simulate from a Multivariate Normal Distribution pacf_to_ar Convert partial autocorrelation coefficients to AR coefficients. uci_help Helping function for uci_matrix uci_matrix Uniform credible intervals in matrix-valued case unit_trace_x_from_phi Get x from phi, see (62) in Mittelbach et al. unrollPsd Redundantly roll out a PSD from length N=floor(n/2) to length n unit_trace_sigma2 Get sigma2 vector, see (70) in Mittelbach et al. Helping function for unit_trace_runif vFromP Get v from p (DP inverse stick breaking) Note: p is assumed to have length L, i.e. it does NOT contain p_0 unit_trace_runif Draw uniformly from Hpd matrices with unit trace unit_trace_q Get q vector, see (68) in Mittelbach et al. unit_trace_L_from_x Get L (lower triangular Cholesky) from x Called U^* in Mittelbach et al, see (60) there unit_trace_U_from_phi Get U (Hpd with unit trace) matrix from phi (hyperspherical coordinates) vector. psd_varma_help helping function for psd_varma varma_transfer2psd Get VARMA PSD from transfer polynomials Helping function for psd_varma acceptanceRate Computing acceptance rate based on trace Note: Only use for traces from continous distributions! VAR_regressor_matrix VAR regressor matrix, see Section 2.2.3 in Koop and Korobilis (2010) No Results!