exdqlm: Extended Dynamic Quantile Linear Models

Description

Routines for Bayesian estimation and analysis of dynamic quantile linear models utilizing the extended asymmetric Laplace error distribution, also known as extended dynamic quantile linear models (exDQLM) described in Barata et al (2020) doi:10.1214/21-AOAS1497.

Author(s)

Maintainer: Raquel Barata raquel.a.barata@gmail.com

Authors:

• Antonio Aguirre

Other contributors:

• Raquel Prado [thesis advisor]

• Bruno Sanso [thesis advisor]

Addition for exdqlm objects

Description

Combines two state space blocks into a single state space model for an exDQLM.

Usage

## S3 method for class 'exdqlm'
m1 + m2

Arguments

Value

A object of class "exdqlm" containing the new combined state space model components:

• FF - Observational vector.

• GG - Evolution matrix.

• m0 - Prior mean of the state vector.

• C0 - Prior covariance of the state vector.

Examples

trend.comp = polytrendMod(2,rep(0,2),10*diag(2))
seas.comp = seasMod(365,c(1,2,4),C0=10*diag(6))
model = trend.comp + seas.comp

Monthly time-series of water flow at Big Tree water gauge.

Description

Average monthly natural water flow (cubic feet per second) at the Big Tree gauge of the San Lorenzo river in Santa Cruz County, CA from 1937 through 2014.

Usage

BTflow

Format

A time series of length 936.

Source

https://waterdata.usgs.gov/nwis/

References

U.S. Geological Survey (2016). National Water Information System data available on the World Wide Web (USGS Water Data for the Nation). https://waterdata.usgs.gov/nwis/.

Daily time-series of ELI anomalies.

Description

ELI anomalies on the daily time-scale from January 1, 1979 to December 31, 2019 with all February 29ths omitted.

Usage

ELIanoms

Format

A time series of length 14965.

Source

https://portal.nersc.gov/archive/home/projects/cascade/www/ELI

References

Patricola, C.M., O’Brien, J.P., Risser, M.D. et al. Maximizing ENSO as a source of western US hydroclimate predictability. Clim Dyn 54, 351–372 (2020). doi:10.1007/s00382-019-05004-8

exdqlm objects

Description

as.exdqlm attempts to turn a list into an exdqlm object. Works for time-invariant dlm objects created using the dlm package.

Usage

as.exdqlm(m)

Arguments

Value

A object of class "exdqlm" containing the state space model components:

• FF - Observational vector.

• GG - Evolution matrix.

• m0 - Prior mean of the state vector.

• C0 - Prior covariance of the state vector.

Plot a component of an exDQLM

Description

The function plots the dynamic MAP estimates and 95% credible intervals (CrIs) of a specified component of an exDQLM. Alternatively, if just.theta=TRUE the MAP estimates and 95% credible intervals (CrIs) of a single element of the dynamic state vector are plotted.

Usage

compPlot(
  m1,
  index,
  add = FALSE,
  col = "purple",
  just.theta = FALSE,
  cr.percent = 0.95
)

Arguments

Value

A list of the following is returned:

• map.comp - MAP estimate of the dynamic component (or element of the state vector).

• lb.comp - Lower bound of the 95% CrIs of the dynamic component (or element of the state vector).

• ub.comp - Upper bound of the 95% CrIs of the dynamic component (or element of the state vector).

Examples

y = scIVTmag[1:365]
trend.comp = polytrendMod(2,rep(0,2),10*diag(2))
seas.comp = seasMod(365,c(1,2,4),C0=10*diag(6))
model = trend.comp + seas.comp
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.98,1),dim.df = c(2,6),
                   gam.init=-3.5,sig.init=15,tol=0.05)
# plot first harmonic component
compPlot(M0,index=c(3,4),col="blue")

Density Function for the Extended Asymmetric Laplace (exAL) Distribution

Description

Computes the PDF of the Extended Asymmetric Laplace (exAL) distribution. Vectorized over x.

Usage

dexal(x, p0 = 0.5, mu = 0, sigma = 1, gamma = 0, log = FALSE)

Arguments

Value

Numeric vector of densities (same length as x).

Examples

dexal(0)
dexal(1, p0 = 0.75, mu = 0, sigma = 2, gamma = 0.25)
dexal(seq(-3, 3, by = 0.1), p0 = 0.3, mu = 0, sigma = 1, gamma = -0.45)

exDQLM Diagnostics

Description

The function computes the following for the model(s) provided: the posterior predictive loss criterion based off the check loss, the one-step-ahead distribution sequence and its KL divergence from normality. The function also plots the following: the qq-plot and ACF plot corresponding to the one-step-ahead distribution sequence, and a time series plot of the MAP standard forecast errors.

Usage

exdqlmDiagnostics(
  m1,
  m2 = NULL,
  plot = TRUE,
  cols = c("red", "blue"),
  ref = NULL
)

Arguments

Value

A object of class "exdqlmDiagnostics" containing the following:

• m1.uts - The one-step-ahead distribution sequence of m1.

• m1.KL - The KL divergence of m1.uts and a standard normal.

• m1.pplc - The posterior predictive loss criterion of m1 based off the check loss function.

• m1.qq - The ordered pairs of the qq-plot comparing m1.uts with a standard normal distribution.

• m1.acf - The autocorrelations of m1.uts by lag.

• m1.rt - Run-time of the original model m1 in seconds.

• m1.msfe - MAP standardized one-step-ahead forecast errors from the original model m1.

• y - The original time-series used to fit m1.

If m2 is provided, analogous results for m2 are also included in the list.

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,mean(y),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.95),dim.df = c(1),
                  gam.init=-3.5,sig.init=15)
M0.diags = exdqlmDiagnostics(M0,plot=FALSE)

k-step-ahead quantile forecasts

Description

Computes filtered and k-step-ahead forecast quantiles from a fitted dynamic quantile model and optionally adds them to an existing plot.

Usage

exdqlmForecast(
  start.t,
  k,
  m1,
  fFF = NULL,
  fGG = NULL,
  plot = TRUE,
  add = FALSE,
  cols = c("purple", "magenta"),
  cr.percent = 0.95
)

Arguments

Value

A object of class "exdqlmForecast" containing the following:

• start.t Integer index at which forecasts start (within the span of the fitted model in m1).

• k Integer; number of steps ahead forecasted.

• m1 A fitted exDQLM model object, typically returned by exdqlmISVB() or exdqlmMCMC().

• cr.percent Numeric in (0, 1); the probability mass for the credible intervals (e.g., 0.95).

• fa Forecast state mean vectors (p \times k matrix).

• fR Forecast state covariance matrices (p \times p \times k array).

• ff Forecast quantile means (length-k numeric).

• fQ Forecast quantile variances (length-k numeric).

Examples

 # Toy example; keep small and fast
 y <- scIVTmag[1:100]
 model <- polytrendMod(1, stats::quantile(y, 0.85), 10)
 M0 <- exdqlmISVB(y, p0 = 0.85, model, df = c(0.98), dim.df = c(1),
                  gam.init = -3.5, sig.init = 15)
 exdqlmForecast(start.t = 90, k = 10, m1 = M0)

exDQLM - ISVB algorithm

Description

The function applies an Importance Sampling Variational Bayes (ISVB) algorithm to estimate the posterior of an exDQLM.

Usage

exdqlmISVB(
  y,
  p0,
  model,
  df,
  dim.df,
  fix.gamma = FALSE,
  gam.init = NA,
  fix.sigma = TRUE,
  sig.init = NA,
  dqlm.ind = FALSE,
  exps0,
  tol = 0.1,
  n.IS = 500,
  n.samp = 200,
  PriorSigma = NULL,
  PriorGamma = NULL,
  verbose = TRUE
)

Arguments

Value

A object of class "exdqlmISVB" containing the following:

• y - Time-series data used to fit the model.

• run.time - Algorithm run time in seconds.

• iter - Number of iterations until convergence was reached.

• dqlm.ind - Logical value indicating whether gamma was fixed at 0, reducing the exDQLM to the special case of the DQLM.

• model - List of the state-space model including GG, FF, prior parameters m0 and C0.

• p0 - The quantile which was estimated.

• df - Discount factors used for each block.

• dim.df - Dimension used for each block of discount factors.

• sig.init - Initial value for sigma, or value at which sigma was fixed if fix.sigma=TRUE.

• seq.sigma - Sequence of sigma estimated by the algorithm until convergence.

• samp.theta - Posterior sample of the state vector variational distribution.

• samp.post.pred - Sample of the posterior predictive distributions.

• map.standard.forecast.errors - MAP standardized one-step-ahead forecast errors.

• samp.sigma - Posterior sample of scale parameter sigma variational distribution.

• samp.vts - Posterior sample of latent parameters, v_t, variational distributions.

• theta.out - List containing the variational distribution of the state vector including filtered distribution parameters (fm and fC) and smoothed distribution parameters (sm and sC).

• vts.out - List containing the variational distributions of latent parameters v_t.

• fix.sigma Logical value indicating whether sigma was fixed at sig.init.

If dqlm.ind=FALSE, the object also contains:

• gam.init - Initial value for gamma, or value at which gamma was fixed if fix.gamma=TRUE.

• seq.gamma - Sequence of gamma estimated by the algorithm until convergence.

• samp.gamma - Posterior sample of skewness parameter gamma variational distribution.

• samp.sts - Posterior sample of latent parameters, s_t, variational distributions.

• gammasig.out - List containing the IS estimate of the variational distribution of sigma and gamma.

• sts.out - List containing the variational distributions of latent parameters s_t.

• fix.gamma Logical value indicating whether gamma was fixed at gam.init.

Or if dqlm.ind=TRUE, the object also contains:

• sig.out - List containing the IS estimate of the variational distribution of sigma.

Examples

y = scIVTmag[1:1095]
trend.comp = polytrendMod(1,mean(y),10)
seas.comp = seasMod(365,c(1,2,4),C0=10*diag(6))
model = trend.comp + seas.comp
M0 = exdqlmISVB(y,p0=0.85,model,df=c(1,1),dim.df = c(1,6),
                 gam.init=-3.5,sig.init=15,tol=0.05)

exDQLM - MCMC algorithm

Description

The function applies a Markov chain Monte Carlo (MCMC) algorithm to sample the posterior of an exDQLM.

Usage

exdqlmMCMC(
  y,
  p0,
  model,
  df,
  dim.df,
  fix.gamma = FALSE,
  gam.init = NA,
  fix.sigma = FALSE,
  sig.init = NA,
  dqlm.ind = FALSE,
  Sig.mh,
  joint.sample = FALSE,
  n.burn = 2000,
  n.mcmc = 1500,
  init.from.isvb = TRUE,
  PriorSigma = NULL,
  PriorGamma = NULL,
  verbose = TRUE
)

Arguments

Value

A object of class "exdqlmMCMC" containing the following:

• y - Time-series data used to fit the model.

• run.time - Algorithm run time in seconds.

• model - List of the state-space model including GG, FF, prior parameters m0 and C0.

• p0 - The quantile which was estimated.

• df - Discount factors used for each block.

• dim.df - Dimension used for each block of discount factors.

• samp.theta - Posterior sample of the state vector.

• samp.post.pred - Sample of the posterior predictive distributions.

• map.standard.forecast.errors - MAP standardized one-step-ahead forecast errors.

• samp.sigma - Posterior sample of scale parameter sigma.

• samp.vts - Posterior sample of latent parameters, v_t.

• theta.out - List containing the distributions of the state vector including filtered distribution parameters (fm and fC) and smoothed distribution parameters (sm and sC).

• n.burn Number of MCMC iterations that were burned.

• n.mcmc Number of MCMC iterations that were sampled.

If dqlm.ind=FALSE, the object also contains the following:

• samp.gamma - Posterior sample of skewness parameter gamma.

• samp.sts - Posterior sample of latent parameters, s_t.

• init.log.sigma - Burned samples of log sigma from the random walk MH joint sampling of sigma and gamma.

• init.logit.gamma - Burned samples of logit gamma from the random walk MH joint sampling of sigma and gamma.

• accept.rate - Acceptance rate of the MH step.

• Sig.mh - Covariance matrix used in MH step to jointly sample sigma and gamma.

Examples

y = scIVTmag[1:100]
trend.comp = polytrendMod(1,mean(y),10)
seas.comp = seasMod(365,c(1,2,4),C0=10*diag(6))
model = trend.comp + seas.comp
M2 = exdqlmMCMC(y,p0=0.85,model,df=c(1,1),dim.df = c(1,6),
                gam.init=-3.5,sig.init=15,
                n.burn=100,n.mcmc=150)

Plot exDQLM

Description

The function plots the MAP estimates and 95% credible intervals (CrIs) of the dynamic quantile of an exDQLM.

Usage

exdqlmPlot(m1, add = FALSE, col = "purple", cr.percent = 0.95)

Arguments

Value

A list of the following is returned:

• map.quant - MAP estimate of the dynamic quantile.

• lb.quant - Lower bound of the 95% CrIs of the dynamic quantile.

• ub.quant - Upper bound of the 95% CrIs of the dynamic quantile.

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                   gam.init=-3.5,sig.init=15)
exdqlmPlot(M0,col="blue")

exdqlm objects

Description

is.exdqlm tests if its argument is a exdqlm object.

Usage

is.exdqlm(m)

Arguments

exdqlmDiagnostic objects

Description

is.exdqlmDiagnostic tests if its argument is a exdqlmDiagnostic object.

Usage

is.exdqlmDiagnostic(x)

Arguments

exdqlmForecast objects

Description

is.exdqlmForecast tests if its argument is a exdqlmForecast object.

Usage

is.exdqlmForecast(x)

Arguments

exdqlmISVB objects

Description

is.exdqlmISVB tests if its argument is a exdqlmISVB object.

Usage

is.exdqlmISVB(m)

Arguments

exdqlmMCMC objects

Description

is.exdqlmMCMC tests if its argument is a exdqlmMCMC object.

Usage

is.exdqlmMCMC(m)

Arguments

Monthly Niño 3.4 Index.

Description

Monthly Niño 3.4 sea surface temperature (SST) Index time series.

Usage

nino34

Format

A time series of length 936.

Source

https://psl.noaa.gov/gcos_wgsp/Timeseries/Nino34/

References

Rayner N. A., D. E. Parker, E. B. Horton, C. K. Folland, L. V. Alexander, D. P. Rowell, E. C. Kent, A. Kaplan. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. J. Geophys. Res., 108 (D14), 4407 (2003). doi:10.1029/2002JD002670

Cumulative Distribution Function (CDF) for the exAL Distribution

Description

Vectorized over q.

Usage

pexal(
  q,
  p0 = 0.5,
  mu = 0,
  sigma = 1,
  gamma = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

Value

Numeric vector of CDF values (same length as q).

Examples

pexal(0)
pexal(c(-1, 0, 1), p0 = 0.5, mu = 0, sigma = 1, gamma = 0.1)

Plot Method for exdqlmDiagnostic Objects

Description

Plot Method for exdqlmDiagnostic Objects

Usage

## S3 method for class 'exdqlmDiagnostic'
plot(x, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,mean(y),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.95),dim.df = c(1),
                  gam.init=-3.5,sig.init=15)
M0.diags = exdqlmDiagnostics(M0,plot=FALSE)
plot(M0.diags)

Plot Method for exdqlmForecast Objects

Description

Plot Method for exdqlmForecast Objects

Usage

## S3 method for class 'exdqlmForecast'
plot(x, ...)

Arguments

Examples

 y <- scIVTmag[1:100]
 model <- polytrendMod(1, stats::quantile(y, 0.85), 10)
 M0 <- exdqlmISVB(y, p0 = 0.85, model, df = c(0.98), dim.df = c(1),
                  gam.init = -3.5, sig.init = 15)
 M0.forecast = exdqlmForecast(start.t = 90, k = 10, m1 = M0)
 plot(M0.forecast)

Plot Method for exdqlmISVB Objects

Description

Plot Method for exdqlmISVB Objects

Usage

## S3 method for class 'exdqlmISVB'
plot(x, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                   gam.init=-3.5,sig.init=15)
plot(M0)

Plot Method for exdqlmMCMC Objects

Description

Plot Method for exdqlmMCMC Objects

Usage

## S3 method for class 'exdqlmMCMC'
plot(x, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M2 = exdqlmMCMC(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                gam.init=-3.5,sig.init=15,
                n.burn=100,n.mcmc=150)
plot(M2)                

Create an n-th order polynomial exDQLM component

Description

The function creates an n-th order polynomial exDQLM component.

Usage

polytrendMod(order, m0, C0)

Arguments

Value

A object of class "exdqlm" containing the following:

• FF - Observational vector.

• GG - Evolution matrix.

• m0 - Prior mean of the state vector.

• C0 - Prior covariance of the state vector.

Examples

# create a second order polynomial component
trend.comp = polytrendMod(2,rep(0,2),10*diag(2))

Print exDQLM model details

Description

Print the details of the exDQLM model.

Usage

## S3 method for class 'exdqlm'
print(x, ...)

Arguments

Print Method for exdqlmDiagnostic Objects

Description

Print Method for exdqlmDiagnostic Objects

Usage

## S3 method for class 'exdqlmDiagnostic'
print(x, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,mean(y),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.95),dim.df = c(1),
                  gam.init=-3.5,sig.init=15)
M0.diags = exdqlmDiagnostics(M0,plot=FALSE)
print(M0.diags)

Print Method for exdqlmForecast Objects

Description

Print Method for exdqlmForecast Objects

Usage

## S3 method for class 'exdqlmForecast'
print(x, ...)

Arguments

Examples

 y <- scIVTmag[1:100]
 model <- polytrendMod(1, stats::quantile(y, 0.85), 10)
 M0 <- exdqlmISVB(y, p0 = 0.85, model, df = c(0.98), dim.df = c(1),
                  gam.init = -3.5, sig.init = 15)
 M0.forecast = exdqlmForecast(start.t = 90, k = 10, m1 = M0)
 print(M0.forecast)

Print Method for exdqlmISVB Objects

Description

Print Method for exdqlmISVB Objects

Usage

## S3 method for class 'exdqlmISVB'
print(x, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                   gam.init=-3.5,sig.init=15)
print(M0)

Print Method for exdqlmMCMC Objects

Description

Print Method for exdqlmMCMC Objects

Usage

## S3 method for class 'exdqlmMCMC'
print(x, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M2 = exdqlmMCMC(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                gam.init=-3.5,sig.init=15,
                n.burn=100,n.mcmc=150)
print(M2)                

Quantile Function for the exAL Distribution

Description

Vectorized over p.

Usage

qexal(
  p,
  p0 = 0.5,
  mu = 0,
  sigma = 1,
  gamma = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

Value

Numeric vector of quantiles (same length as p).

Examples

p <- seq(0.1, 0.9, by = 0.2)
q <- qexal(p, p0 = 0.5, mu = 0, sigma = 1, gamma = 0)
all.equal(p, pexal(q, p0 = 0.5, mu = 0, sigma = 1, gamma = 0), tol = 1e-4)

Random Sample Generation for the exAL Distribution

Description

Random Sample Generation for the exAL Distribution

Usage

rexal(n, p0 = 0.5, mu = 0, sigma = 1, gamma = 0)

Arguments

Value

Numeric vector of length n.

Examples

set.seed(1); length(rexal(10))
rexal(3, p0 = 0.5, mu = c(-1, 0, 1))

Time series of daily average magnitude IVT in Santa Cruz, CA.

Description

ECMWF Re-Analysis 5 (ERA5) daily average magnitude IVT in Santa Cruz, CA (approximately 22 N, 122 W) from January 1, 1979 to December 31, 2019 with all February 29ths omitted.

Usage

scIVTmag

Format

A time series of length 14965.

Source

https://cds.climate.copernicus.eu

References

Hersbach, H, Bell, B, Berrisford, P, et al. The ERA5 global reanalysis. Q J R Meteorol Soc. 2020; 146: 1999– 2049. doi:10.1002/qj.3803

Create Fourier representation of a periodic exDQLM component

Description

The function creates a Fourier form periodic component for given period and harmonics.

Usage

seasMod(p, h, m0, C0)

Arguments

Value

A object of class "exdqlm" containing the following:

• FF - Observational vector.

• GG - Evolution matrix.

• m0 - Prior mean of the state vector.

• C0 - Prior covariance of the state vector.

Examples

# create a seasonal component with first, second and fourth harmonics of a period of 365
seas.comp = seasMod(365,c(1,2,4),C0=10*diag(6))

Summary exDQLM model details

Description

Print the details of the exDQLM model.

Usage

## S3 method for class 'exdqlm'
summary(object, ...)

Arguments

Summary Method for exdqlmDiagnostic Objects

Description

Summary Method for exdqlmDiagnostic Objects

Usage

## S3 method for class 'exdqlmDiagnostic'
summary(object, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,mean(y),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.95),dim.df = c(1),
                  gam.init=-3.5,sig.init=15)
M0.diags = exdqlmDiagnostics(M0,plot=FALSE)
summary(M0.diags)

Summary Method for exdqlmForecast Objects

Description

Summary Method for exdqlmForecast Objects

Usage

## S3 method for class 'exdqlmForecast'
summary(object, ...)

Arguments

Examples

 y <- scIVTmag[1:100]
 model <- polytrendMod(1, stats::quantile(y, 0.85), 10)
 M0 <- exdqlmISVB(y, p0 = 0.85, model, df = c(0.98), dim.df = c(1),
                  gam.init = -3.5, sig.init = 15)
 M0.forecast = exdqlmForecast(start.t = 90, k = 10, m1 = M0)
 summary(M0.forecast)

Summary Method for exdqlmISVB Objects

Description

Summary Method for exdqlmISVB Objects

Usage

## S3 method for class 'exdqlmISVB'
summary(object, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M0 = exdqlmISVB(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                   gam.init=-3.5,sig.init=15)
summary(M0)

Summary Method for exdqlmMCMC Objects

Description

Summary Method for exdqlmMCMC Objects

Usage

## S3 method for class 'exdqlmMCMC'
summary(object, ...)

Arguments

Examples

y = scIVTmag[1:100]
model = polytrendMod(1,quantile(y,0.85),10)
M2 = exdqlmMCMC(y,p0=0.85,model,df=c(0.98),dim.df = c(1),
                gam.init=-3.5,sig.init=15,
                n.burn=100,n.mcmc=150)
summary(M2)                

Transfer Function exDQLM - ISVB algorithm

Description

The function applies an Importance Sampling Variational Bayes (ISVB) algorithm to estimate the posterior of an exDQLM with exponential decay transfer function component.

Usage

transfn_exdqlmISVB(
  y,
  p0,
  model,
  X,
  df,
  dim.df,
  lam,
  tf.df,
  fix.gamma = FALSE,
  gam.init = NA,
  fix.sigma = TRUE,
  sig.init = NA,
  dqlm.ind = FALSE,
  exps0,
  tol = 0.1,
  n.IS = 500,
  n.samp = 200,
  PriorSigma = NULL,
  PriorGamma = NULL,
  tf.m0 = rep(0, 2),
  tf.C0 = diag(1, 2),
  verbose = TRUE
)

Arguments

Value

A object of class "exdqlmISVB" containing the following:

• run.time - Algorithm run time in seconds.

• iter - Number of iterations until convergence was reached.

• dqlm.ind - Logical value indicating whether gamma was fixed at 0, reducing the exDQLM to the special case of the DQLM.

• model - List of the augmented state-space model including GG, FF, prior parameters m0 and C0.

• p0 - The quantile which was estimated.

• df - Discount factors used for each block, including transfer function component.

• dim.df - Dimension used for each block of discount factors, including transfer function component.

• lam - Transfer function rate parameter lambda.

• sig.init - Initial value for sigma, or value at which sigma was fixed if fix.sigma=TRUE.

• seq.sigma - Sequence of sigma estimated by the algorithm until convergence.

• samp.theta - Posterior sample of the state vector variational distribution.

• samp.post.pred - Sample of the posterior predictive distributions.

• map.standard.forecast.errors - MAP standardized one-step-ahead forecast errors.

• samp.sigma - Posterior sample of scale parameter sigma variational distribution.

• samp.vts - Posterior sample of latent parameters, v_t, variational distributions.

• theta.out - List containing the variational distribution of the state vector including filtered distribution parameters (fm and fC) and smoothed distribution parameters (sm and sC).

• vts.out - List containing the variational distributions of latent parameters v_t.

• median.kt - Median number of time steps until the effect of X_t is less than or equal to 1e-3.

If dqlm.ind=FALSE, the object also contains:

• gam.init - Initial value for gamma, or value at which gamma was fixed if fix.gamma=TRUE.

• seq.gamma - Sequence of gamma estimated by the algorithm until convergence.

• samp.gamma - Posterior sample of skewness parameter gamma variational distribution.

• samp.sts - Posterior sample of latent parameters, s_t, variational distributions.

• gammasig.out - List containing the IS estimate of the variational distribution of sigma and gamma.

• sts.out - List containing the variational distributions of latent parameters s_t.

Or if dqlm.ind=TRUE, the object also contains:

• sig.out - List containing the IS estimate of the variational distribution of sigma.

Examples

y = scIVTmag[1:1095]
X = ELIanoms[1:1095]
trend.comp = polytrendMod(1,mean(y),10)
seas.comp = seasMod(365,c(1,2,4),C0=10*diag(6))
model = trend.comp + seas.comp
M1 = transfn_exdqlmISVB(y,p0=0.85,model=model,
                          X,df=c(1,1),dim.df = c(1,6),
                          gam.init=-3.5,sig.init=15,
                          lam=0.38,tf.df=c(0.97,0.97))