Package 'arfima'

Title: Fractional ARIMA (and Other Long Memory) Time Series Modeling
Description: Simulates, fits, and predicts long-memory and anti-persistent time series, possibly mixed with ARMA, regression, transfer-function components. Exact methods (MLE, forecasting, simulation) are used. Bug reports should be done via GitHub (at <https://github.com/JQVeenstra/arfima>), where the development version of this package lives; it can be installed using devtools.
Authors: JQ (Justin) Veenstra [aut, cre], A.I. McLeod [aut]
Maintainer: JQ Veenstra <[email protected]>
License: MIT + file LICENSE
Version: 1.8-1
Built: 2024-11-06 06:14:45 UTC
Source: CRAN

Help Index

Simulates, fits, and predicts persistent and anti-persistent time series. arfima

Description

Simulates with arfima.sim, fits with arfima, and predicts with a method for the generic function. Plots predictions and the original time series. Has the capability to fit regressions with ARFIMA/ARIMA-FGN/ARIMA-PLA errors, as well as transfer functions/dynamic regression.

Details

Package: arfima
Type: Package
Version: 1.4-0
Date: 2017-06-20
License: MIT

A list of functions:

arfima.sim - Simulates an ARFIMA, ARIMA-FGN, or ARIMA-PLA (three classes of mixed ARIMA hyperbolic decay processes) process, with possible seasonal components.

arfima - Fits an ARIMA-HD (default single-start) model to a series, with options for regression with ARIMA-HD errors and dynamic regression (transfer functions). Allows for fixed parameters as well as choices for the optimizer to be used.

arfima0 - Simplified version of arfima

weed - Weeds out modes too close to each other in the same fit. The modes with the highest log-likelihoods are kept

print.arfima - Prints the relevant output of an arfima fitted object, such as parameter estimates, standard errors, etc.

summary.arfima - A much more detailed version of print.arfima

coef.arfima - Extracts the coefficients from a arfima object

vcov.arfima - Theoretical and observed covariance matrices of the coefficients

residuals.arfima - Extracts the residuals or regression residuals from a arfima object

fitted.arfima - Extracts the fitted values from a arfima object

tacvfARFIMA - Computes the theoretical autocovariance function of a supplied model. The model is checked for stationarity and invertibility.

iARFIMA - Computes the Fisher information matrix of all non-FGN components of the given model. Can be computed (almost) exactly or through a psi-weights approximation. The approximation takes more time.

IdentInvertQ - Checks whether the model is identifiable, stationary, and invertible. Identifiability is checked through the information matrix of all non-FGN components, as well as whether both types of fractional noise are present, both seasonally and non-seasonally.

lARFIMA and lARFIMAwTF - Computes the log-likelihood of a given model with a given series. The second admits transfer function data.

predict.arfima - Predicts from an arfima object. Capable of exact minimum mean squared error predictions even with integer d > 0 and/or integer dseas > 0. Does not include transfer function/leading indicators as of yet. Returns a predarfima object, which is composed of: predictions, and standard errors (exact and, if possible, limiting).

print.predarfima - Prints the relevant output from a predarfima object: the predictions and their standard deviations.

plot.predarfima - Plots a predarfima object. This includes the original time series, the forecasts and as default the standard 95% prediction intervals (exact and, if available, limiting).

logLik.arfima, AIC.arfima, BIC.arfima - Extracts the requested values from an arfima object

distance - Calculates the distances between the modes

removeMode - Removes a mode from a fit

tacvf - Calculates the theoretical autocovariance functions (tacvfs) from a fitted arfima object

plot.tacvf - Plots the tacvfs

print.tacvf - Prints the tacvfs

tacfplot - Plots the theoretical autocorrelation functions (tacfs) of different models on the same data

SeriesJ, tmpyr - Two datasets included with the package

Author(s)

JQ (Justin) Veenstra, A. I. McLeod

Maintainer: JQ (Justin) Veenstra <[email protected]>

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

Examples

set.seed(8564)
sim <- arfima.sim(1000, model = list(phi = c(0.2, 0.1), dfrac = 0.4, theta = 0.9))
fit <- arfima(sim, order = c(2, 0, 1), back=TRUE)

fit

data(tmpyr)

fit1 <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3), dmean = FALSE)
fit1

plot(tacvf(fit1), maxlag = 30, tacf = TRUE)

fit2 <- arfima(tmpyr, order = c(1, 0, 0), numeach = c(3, 3), autoweed = FALSE,
dmean = FALSE)

fit2

fit2 <- weed(fit2)

fit2

tacfplot(fits = list(fit1, fit2))

fit3 <- removeMode(fit2, 2)

fit3

coef(fit2)
vcov(fit2)

fit1fgn <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3),
dmean = FALSE, lmodel = "g")
fit1fgn

fit1hd <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3),
dmean = FALSE, lmodel = "h")
fit1hd

data(SeriesJ)
attach(SeriesJ)

fitTF <- arfima(YJ, order= c(2, 0, 0), xreg = XJ, reglist =
list(regpar = c(1, 2, 3)), lmodel = "n", dmean = FALSE)
fitTF

detach(SeriesJ)

set.seed(4567)

sim <- arfima.sim(1000, model = list(phi = 0.3, dfrac = 0.4, dint = 1),
sigma2 = 9)

X <- matrix(rnorm(2000), ncol = 2)

simreg <- sim + crossprod(t(X), c(2, 3))

fitreg <- arfima(simreg, order = c(1, 1, 0), xreg = X)

fitreg

plot(sim)

lines(residuals(fitreg, reg = TRUE)[[1]], col = "blue")
##pretty much a perfect match.

Information criteria for arfima objects

Description

Computes information criteria for arfima objects. See AIC for more details.

Usage

## S3 method for class 'arfima'
AIC(object, ..., k = 2)

Arguments

object

An object of class "arfima". Note these functions can only be called on one object at a time because of possible multimodality.
...

Other models fit to data for which to extract the AIC/BIC. Not recommended, as an arfima object can be multimodal.
k

The penalty term to be used. See AIC.

Value

The information criteria for each mode in a vector.

Author(s)

JQ (Justin) Veenstra

Examples

set.seed(34577)
sim <- arfima.sim(500, model = list(theta = 0.9, phi = 0.5, dfrac = 0.4))
fit1 <- arfima(sim, order = c(1, 0, 1), cpus = 2, back=TRUE)
fit2 <- arfima(sim, order = c(1, 0, 1), cpus = 2, lmodel = "g", back=TRUE)
fit3 <- arfima(sim, order = c(1, 0, 1), cpus = 2, lmodel = "h", back=TRUE)

AIC(fit1)
AIC(fit2)
AIC(fit3)

Fit ARFIMA, ARIMA-FGN, and ARIMA-PLA (multi-start) models Fits ARFIMA/ARIMA-FGN/ARIMA-PLA multi-start models to times series data. Options include fixing parameters, whether or not to fit fractional noise, what type of fractional noise (fractional Gaussian noise (FGN), fractionally differenced white noise (FDWN), or the newly introduced power-law autocovariance noise (PLA)), etc. This function can fit regressions with ARFIMA/ARIMA-FGN/ARIMA-PLA errors via the xreg argument, including dynamic regression (transfer functions).

Description

Fits by direct optimization using optim. The optimizer choices are: 0 - BFGS; 1 - Nealder-Mead; 2 - SANN; otherwise CG.

Usage

arfima(
  z,
  order = c(0, 0, 0),
  numeach = c(1, 1),
  dmean = TRUE,
  whichopt = 0,
  itmean = FALSE,
  fixed = list(phi = NA, theta = NA, frac = NA, seasonal = list(phi = NA, theta = NA,
    frac = NA), reg = NA),
  lmodel = c("d", "g", "h", "n"),
  seasonal = list(order = c(0, 0, 0), period = NA, lmodel = c("d", "g", "h", "n"),
    numeach = c(1, 1)),
  useC = 3,
  cpus = 1,
  rand = FALSE,
  numrand = NULL,
  seed = NA,
  eps3 = 0.01,
  xreg = NULL,
  reglist = list(regpar = NA, minn = -10, maxx = 10, numeach = 1),
  check = F,
  autoweed = TRUE,
  weedeps = 0.01,
  adapt = TRUE,
  weedtype = c("A", "P", "B"),
  weedp = 2,
  quiet = FALSE,
  startfit = NULL,
  back = FALSE
)

Arguments

z

The data set (time series)
order

The order of the ARIMA model to be fit: c(p, d, q). We have that p is the number of AR parameters (phi), d is the amount of integer differencing, and q is the number of MA parameters (theta). Note we use the Box-Jenkins convention for the MA parameters, in that they are the negative of arima: see "Details".
numeach

The number of starts to fit for each parameter. The first argument in the vector is the number of starts for each AR/MA parameter, while the second is the number of starts for the fractional parameter. When this is set to 0, no fractional noise is fit. Note that the number of starts in total is multiplicative: if we are fitting an ARFIMA(2, d, 2), and use the older number of starts (c(2, 2)), we will have 2^2 * 2 * 2^2 = 32 starting values for the fits. Note that the default has changed from c(2, 2) to c(1, 1) since package version 1.4-0
dmean

Whether the mean should be fit dynamically with the optimizer. Note that the likelihood surface will change if this is TRUE, but this is usually not worrisome. See the referenced thesis for details.
whichopt

Which optimizer to use in the optimization: see "Details".
itmean

This option is under investigation, and will be set to FALSE automatically until it has been decided what to do.

Whether the mean should be fit iteratively using the function TrenchMean. Currently itmean, if set to TRUE, has higher priority that dmean: if both are TRUE, dmean will be set to FALSE, with a warning.
fixed

A list of parameters to be fixed. If we are to fix certain elements of the AR process, for example, fixed$phi must have length equal to p. Any numeric value will fix the parameter at that value; for example, if we are modelling an AR(2) process, and we wish to fix only the first autoregressive parameter to 0, we would have fixed = list(phi = c(0, NA)). NA corresponds to that parameter being allowed to change in the optimization process. We can fix the fractional parameters, and unlike arima, can fix the seasonal parameters as well. Currently, fixing regression/transfer function parameters is disabled.
lmodel

The long memory model (noise type) to be used: "d" for FDWN, "g" for FGN, "h" for PLA, and "n" for none (i.e. ARMA short memory models). Default is "d".
seasonal

The seasonal components of the model we wish to fit, with the same components as above. The period must be supplied.
useC

How much interfaced C code to use: an integer between 0 and 3. The value 3 is strongly recommended. See "Details".
cpus

The number of CPUs used to perform the multi-start fits. A small number of fits and a high number of cpus (say both equal 4) with n not large can actually be slower than when cpus = 1. The number of CPUs should not exceed the number of threads available to R.
rand

Whether random starts are used in the multistart method. Defaults to FALSE.
numrand

The number of random starts to use.
seed

The seed for the random starts.
eps3

How far to start from the boundaries when using a grid for the multi-starts (i.e. when rand is FALSE.)
xreg

A matrix, data frame, or vector of regressors for regression or transfer functions.
reglist

A list with the following elements:

  • regpar - either NA or a list, matrix, data frame, or vector with 3 columns. If regpar is a vector, the matrix xreg must have one row or column only. In order, the elements of regpar are: r, s, and b. The values of r are the the orders of the delta parameters as in Box, Jenkins and Reinsel, the values of s are the orders of omega parameters, and the values of b are the backshifting to be done.

  • minn - the minimum value for the starting value of the search, if reglist$numeach > 1.

  • maxx - the maximum value for the starting value of the search, if reglist$numeach > 1.

  • numeach - the number of starts to try for each regression parameter.

check

If TRUE, checks at each optim iteration whether the model is identifiable. This makes the optimization much slower.
autoweed

Whether to automatically (before the fit is returned) weed out modes found that are found that are close together (usually the same point.)
weedeps

The maximum distance between modes that are close together for the mode with the lower log-likelihood to be weeded out. If adapt is TRUE (default) this value changes.
adapt

If TRUE, if dim is the dimensionality of the search, weedeps is changed to (1+weedeps)dim−1(1 + weedeps)^{dim} - 1.
weedtype

The type of weeding to be done. See weed.
weedp

The p in the p-norm to be used in the weeding. p = 2 (default) is Euclidean distance.
quiet

If TRUE, no auxiliary output is generated. The default (FALSE) has information of fits being proformed.
startfit

Meant primarily for debugging (for now), allows starting places for the fitting process. Overrides numeach.
back

Setting this to true will restore the defaults in numeach.

Details

A word of warning: it is generally better to use the default, and only use Nelder-Mead to check for spurious modes. SANN takes a long time (and may only find one mode), and CG may not be stable.

If using Nelder-Mead, it must be stressed that Nelder-Mead can take out non-spurious modes or add spurious modes: we have checked visually where we could. Therefore it is wise to use BFGS as the default and if there are modes close to the boundaries, check using Nelder-Mead.

The moving average parameters are in the Box-Jenkins convention: they are the negative of the parameters given by arima. That is, the model to be fit is, in the case of a non-seasonal ARIMA model, phi(B) (1-B)^d z[t] = theta(B) a[t], where phi(B) = 1 - phi(1) B - ... - phi(p) B^p and theta(B) = 1 - theta(1) B - ... - theta(q) B^q.

For the useC parameter, a "0" means no C is used; a "1" means C is only used to compute the log-likelihood, but not the theoretical autocovariance function (tacvf); a "2" means that C is used to compute the tacvf and not the log-likelihood; and a "3" means C is used to compute everything.

Value

An object of class "arfima". In it, full information on the fit is given, though not printed under the print.arfima method. The phis are the AR parameters, and the thetas are the MA parameters. Residuals, regression residuals, etc., are all available, along with the parameter values and standard errors. Note that the muHat returned in the arfima object is of the differenced series, if differencing is applied.

Note that if multiple modes are found, they are listed in order of log-likelihood value.

Author(s)

JQ (Justin) Veenstra

References

McLeod, A. I., Yu, H. and Krougly, Z. L. (2007) Algorithms for Linear Time Series Analysis: With R Package Journal of Statistical Software, Vol. 23, Issue 5

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

P. Borwein (1995) An efficient algorithm for Riemann Zeta function Canadian Math. Soc. Conf. Proc., 27, pp. 29-34.

See Also

arfima.sim, SeriesJ, arfima-package

Examples

set.seed(8564)
sim <- arfima.sim(1000, model = list(phi = c(0.2, 0.1),
dfrac = 0.4, theta = 0.9))
fit <- arfima(sim, order = c(2, 0, 1), back=TRUE)

fit

data(tmpyr)

fit <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3))
fit

plot(tacvf(fit), maxlag = 30, tacf = TRUE)

data(SeriesJ)
attach(SeriesJ)

fitTF <- arfima(YJ, order= c(2, 0, 0), xreg = XJ, reglist =
list(regpar = c(2, 2, 3)), lmodel = "n")
fitTF

detach(SeriesJ)

Simulate an ARFIMA time series.

Description

This function simulates an long memory ARIMA time series, with one of fractionally differenced white noise (FDWN), fractional Gaussian noise (FGN), power-law autocovariance (PLA) noise, or short memory noise and possibly seasonal effects.

Usage

arfima.sim(
  n,
  model = list(phi = numeric(0), theta = numeric(0), dint = 0, dfrac = numeric(0), H =
    numeric(0), alpha = numeric(0), seasonal = list(phi = numeric(0), theta = numeric(0),
    dint = 0, period = numeric(0), dfrac = numeric(0), H = numeric(0), alpha =
    numeric(0))),
  useC = 3,
  sigma2 = 1,
  rand.gen = rnorm,
  muHat = 0,
  zinit = NULL,
  innov = NULL,
  ...
)

Arguments

n

The number of points to be generated.
model

The model to be simulated from. The phi and theta arguments should be vectors with the values of the AR and MA parameters. Note that Box-Jenkins notation is used for the MA parameters: see the "Details" section of arfima. The dint argument indicates how much differencing should be required to make the process stationary. The dfrac, H, and alpha arguments are FDWN, FGN and PLA values respectively; note that only one (or none) of these can have a value, or an error is returned. The seasonal argument is a list, with the same parameters, and a period, as the model argument. Note that with a seasonal model, we can have mixing of FDWN/FGN/HD noise: one in the non-seasonal part, and the other in the seasonal part.
useC

How much interfaced C code to use: an integer between 0 and 3. The value 3 is strongly recommended. See the "Details" section of arfima.
sigma2

The desired variance for the innovations of the series.
rand.gen

The distribution of the innovations. Any distribution recognized by R is possible
muHat

The theoretical mean of the series before integration (if integer integration is done)
zinit

Used for prediction; not meant to be used directly. This allows a start of a time series to be specified before inverse differencing (integration) is applied.
innov

Used for prediction; not meant to be used directly. This allows for the use of given innovations instead of ones provided by rand.gen.
...

Other parameters passed to the random variate generator; currently not used.

Details

A suitably defined stationary series is generated, and if either of the dints (non-seasonal or seasonal) are greater than zero, the series is integrated (inverse-differenced) with zinit equalling a suitable amount of 0s if not supplied. Then a suitable amount of points are taken out of the beginning of the series (i.e. dint + period * seasonal dint = the length of zinit) to obtain a series of length n. The stationary series is generated by calculating the theoretical autovariance function and using it, along with the innovations to generate a series as in McLeod et. al. (2007). Note: if you would like to fit a function from a fitted arfima model, the function sim_from_fitted can be used.

Value

A sample from a multivariate normal distribution that has a covariance structure defined by the autocovariances generated for given parameters. The sample acts like a time series with the given parameters.

Author(s)

JQ (Justin) Veenstra

References

McLeod, A. I., Yu, H. and Krougly, Z. L. (2007) Algorithms for Linear Time Series Analysis: With R Package Journal of Statistical Software, Vol. 23, Issue 5

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

P. Borwein (1995) An efficient algorithm for Riemann Zeta function Canadian Math. Soc. Conf. Proc., 27, pp. 29-34.

See Also

arfima, sim_from_fitted

Examples

set.seed(6533)
sim <- arfima.sim(1000, model = list(phi = .2, dfrac = .3, dint = 2))

fit <- arfima(sim, order = c(1, 2, 0))
fit

Exact MLE for ARFIMA The time series is corrected for the sample mean and then exact MLE is used for the other parameters. This is a simplified version of the arfima() function that may be useful in simulations and bootstrapping.

Description

The sample mean is asymptotically efficient.

Usage

arfima0(z, order = c(0, 0, 0), lmodel = c("FD", "FGN", "PLA", "NONE"))

Arguments

z

time series
order

(p,d,q) where p=order AR, d=regular difference, q=order MA
lmodel

type of long-memory component: FD, FGN, PLA or NONE

Value

list with components:

bHat

transformed optimal parameters
alphaHat

estimate of alpha
HHat

estimate of H
dHat

estimate of d
phiHat

estimate of phi
thetaHat

estimate of theta
wLL

optimized value of Whittle approximate log-likelihood
LL

corresponding exact log-likelihood
convergence

convergence indicator

Author(s)

JQ (Justin) Veenstra and A. I. McLeod

Examples

z <- rnorm(100)
arfima0(z, lmodel="FGN")

Prints changes to the package since the last update. Started in 1.4-0

Description

Prints changes to the package since the last update. Started in 1.4-0

Usage

arfimachanges()

Converts AR/MA coefficients from operator space to the PACF space

Description

Converts AR/MA coefficients from operator space to the PACF box-space; usually for internal use

Usage

ARToPacf(phi)

Arguments

phi

The AR/MA coefficients in operator space

Value

The AR/MA coefficients in the PACF space

Author(s)

A. I. McLeod

References

Barndorff-Nielsen O. E., Schou G. (1973). "On the parametrization of autoregressive models by partial autocorrelations." Journal of Multivariate Analysis, 3, 408-419

McLeod A. I., Zhang Y (2006). "Partial autocorrelation parameterization for subset autore- gression." Journal of Time Series Analysis, 27(4), 599-612

Finds the best modes of an arfima fit.

Description

Finds the best modes of an arfima fit with respect to log-likelihood.

Usage

bestModes(object, bestn)

Arguments

object

An object of class "arfima".
bestn

The top number of modes to keep with respect to the log-likelihood.

Details

This is the easiest way to remove modes with lower log-likelihoods.

Value

The bestn "best" modes.

Author(s)

JQ (Justin) Veenstra

See Also

arfima

Examples

set.seed(8765)
sim <- arfima.sim(1000, model = list(phi = 0.4, theta = 0.9, dfrac = 0.4))
fit <- arfima(sim, order = c(1, 0, 1), back=TRUE)
fit
fit <- bestModes(fit, 2)
fit

Extract Model Coefficients

Description

Extracts the coefficients from a arfima fit.

Usage

## S3 method for class 'arfima'
coef(object, tpacf = FALSE, digits = max(4, getOption("digits") - 3), ...)

Arguments

object

A fitted arfima object.
tpacf

If TRUE, the (ARMA) coefficients are in the transformed PACF space.
digits

The number of digits to print
...

Other optional arguments. Currently not used.

Value

A matrix of coefficients. The rows are for the modes, and the columns are for the model variables.

Author(s)

JQ (Justin) Veenstra

Examples

set.seed(8564)
sim <- arfima.sim(1000, model = list(phi = c(0.2, 0.1), dfrac = 0.4, theta = 0.9))
fit <- arfima(sim, order = c(2, 0, 1), back=TRUE)

fit
coef(fit)

The distance between modes of an arfima fit.

Description

The distance between modes of an arfima fit.

Usage

distance(ans, p = 2, digits = 4)

Arguments

ans

An object of class "arfima".
p

The p in the p-norm to be used.
digits

The number of digits to print.

Value

A list of two data frames: one with distances in operator space, the second with distances in the transformed (PACF) space.

Author(s)

JQ (Justin) Veensta

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

Examples

set.seed(8564)
sim <- arfima.sim(1000, model = list(phi = c(0.2, 0.1), dfrac = 0.4, theta = 0.9))
fit <- arfima(sim, order = c(2, 0, 1), back=TRUE)

fit

distance(fit)

Extract Model Fitted Values

Description

Extract fitted values from an arfima object.

Usage

## S3 method for class 'arfima'
fitted(object, ...)

Arguments

object

A arfima object.
...

Optional parameters. Currently not used.

Value

A list of vectors of fitted values, one for each mode.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

arfima, resid.arfima

Examples

set.seed(8564)
sim <- arfima.sim(1000, model = list(phi = c(0.2, 0.1), dfrac = 0.4, theta = 0.9))
fit <- arfima(sim, order = c(2, 0, 1), back=TRUE)

fit

resid <- resid(fit)
par(mfrow = c(1, 3))
fitted <- fitted(fit)
plot(fitted[[1]], resid[[1]])
plot(fitted[[2]], resid[[2]])
plot(fitted[[3]], resid[[3]])
par(mfrow = c(1, 1))

The Fisher information matrix of an ARFIMA process

Description

Computes the approximate or (almost) exact Fisher information matrix of an ARFIMA process

Usage

iARFIMA(
  phi = numeric(0),
  theta = numeric(0),
  phiseas = numeric(0),
  thetaseas = numeric(0),
  period = 0,
  dfrac = TRUE,
  dfs = FALSE,
  exact = TRUE
)

Arguments

phi

The autoregressive parameters in vector form.
theta

The moving average parameters in vector form. See Details for differences from arima.
phiseas

The seasonal autoregressive parameters in vector form.
thetaseas

The seasonal moving average parameters in vector form. See Details for differences from arima.
period

The periodicity of the seasonal components. Must be >= 2.
dfrac

TRUE if we include the fractional d parameter, FALSE otherwise
dfs

TRUE if we include the seasonal fractional d parameter, FALSE otherwise
exact

If FALSE, calculate the approximate information matrix via psi-weights. Otherwise the (almost) exact information matrix will be calculated. See "Details".

Details

The matrices are calculated as outlined in Veenstra and McLeod (2012), which draws on many references. The psi-weights approximation has a fixed maximum lag for the weights as 2048 (to be changed to be adaptable.) The fractional difference(s) by AR/MA components have a fixed maximum lag of 256, also to be changed. Thus the exact matrix has some approximation to it. Also note that the approximate method takes much longer than the "exact" one.

The moving average parameters are in the Box-Jenkins convention: they are the negative of the parameters given by arima.

Value

The information matrix of the model.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

IdentInvertQ

Examples

tick <- proc.time()
exactI <- iARFIMA(phi = c(.4, -.2), theta = c(.7), phiseas = c(.8, -.4),
	d = TRUE, dfs = TRUE, period = 12)
proc.time() - tick
tick <- proc.time()
approxI <- iARFIMA(phi = c(.4, -.2), theta = c(.7), phiseas = c(.8, -.4), 
	d = TRUE, dfs = TRUE, period = 12, exact = FALSE)
proc.time() - tick
exactI
max(abs(exactI - approxI))

Checks invertibility, stationarity, and identifiability of a given set of parameters

Description

Computes whether a given long memory model is invertible, stationary, and identifiable.

Usage

IdentInvertQ(
  phi = numeric(0),
  theta = numeric(0),
  phiseas = numeric(0),
  thetaseas = numeric(0),
  dfrac = numeric(0),
  dfs = numeric(0),
  H = numeric(0),
  Hs = numeric(0),
  alpha = numeric(0),
  alphas = numeric(0),
  delta = numeric(0),
  period = 0,
  debug = FALSE,
  ident = TRUE
)

Arguments

phi

The autoregressive parameters in vector form.
theta

The moving average parameters in vector form. See Details for differences from arima.
phiseas

The seasonal autoregressive parameters in vector form.
thetaseas

The seasonal moving average parameters in vector form. See Details for differences from arima.
dfrac

The fractional differencing parameter.
dfs

The seasonal fractional differencing parameter.
H

The Hurst parameter for fractional Gaussian noise (FGN). Should not be mixed with dfrac or alpha: see "Details".
Hs

The Hurst parameter for seasonal fractional Gaussian noise (FGN). Should not be mixed with dfs or alphas: see "Details".
alpha

The decay parameter for power-law autocovariance (PLA) noise. Should not be mixed with dfrac or H: see "Details".
alphas

The decay parameter for seasonal power-law autocovariance (PLA) noise. Should not be mixed with dfs or Hs: see "Details".
delta

The delta parameters for transfer functions.
period

The periodicity of the seasonal components. Must be >= 2.
debug

When TRUE and model is not stationary/invertible or identifiable, prints some helpful output.
ident

Whether to test for identifiability.

Details

This function tests for identifiability via the information matrix of the ARFIMA process. Whether the process is stationary or invertible amounts to checking whether all the variables fall in correct ranges.

The moving average parameters are in the Box-Jenkins convention: they are the negative of the parameters given by arima.

If dfrac/H/alpha are mixed and/or dfs/Hs/alphas are mixed, an error will not be thrown, even though only one of these can drive the process at either level. Note also that the FGN or PLA have no impact on the identifiability of the model, as information matrices containing these parameters currently do not have known closed form. These two parameters must be within their correct ranges (0<H<1 for FGN and 0 < alpha < 3 for PLA.)

Value

TRUE if the model is stationary, invertible and identifiable. FALSE otherwise.

Author(s)

Justin Veenstra

References

McLeod, A.I. (1999) Necessary and sufficient condition for nonsingular Fisher information matrix in ARMA and fractional ARMA models The American Statistician 53, 71-72.

Veenstra, J. and McLeod, A. I. (2012, Submitted) Improved Algorithms for Fitting Long Memory Models: With R Package

See Also

iARFIMA

Examples

IdentInvertQ(phi = 0.3, theta = 0.3)
IdentInvertQ(phi = 1.2)

Exact log-likelihood of a long memory model

Description

Computes the exact log-likelihood of a long memory model with respect to a given time series.

Usage

lARFIMA(
  z,
  phi = numeric(0),
  theta = numeric(0),
  dfrac = numeric(0),
  phiseas = numeric(0),
  thetaseas = numeric(0),
  dfs = numeric(0),
  H = numeric(0),
  Hs = numeric(0),
  alpha = numeric(0),
  alphas = numeric(0),
  period = 0,
  useC = 3
)

Arguments

z

A vector or (univariate) time series object, assumed to be (weakly) stationary.
phi

The autoregressive parameters in vector form.
theta

The moving average parameters in vector form. See Details for differences from arima.
dfrac

The fractional differencing parameter.
phiseas

The seasonal autoregressive parameters in vector form.
thetaseas

The seasonal moving average parameters in vector form. See Details for differences from arima.
dfs

The seasonal fractional differencing parameter.
H

The Hurst parameter for fractional Gaussian noise (FGN). Should not be mixed with dfrac or alpha: see "Details".
Hs

The Hurst parameter for seasonal fractional Gaussian noise (FGN). Should not be mixed with dfs or alphas: see "Details".
alpha

The decay parameter for power-law autocovariance (PLA) noise. Should not be mixed with dfrac or H: see "Details".
alphas

The decay parameter for seasonal power-law autocovariance (PLA) noise. Should not be mixed with dfs or Hs: see "Details".
period

The periodicity of the seasonal components. Must be >= 2.
useC

How much interfaced C code to use: an integer between 0 and 3. The value 3 is strongly recommended. See "Details".

Details

The log-likelihood is computed for the given series z and the parameters. If two or more of dfrac, H or alpha are present and/or two or more of dfs, Hs or alphas are present, an error will be thrown, as otherwise there is redundancy in the model. Note that non-seasonal and seasonal components can be of different types: for example, there can be seasonal FGN with FDWN at the non-seasonal level.

The moving average parameters are in the Box-Jenkins convention: they are the negative of the parameters given by arima.

For the useC parameter, a "0" means no C is used; a "1" means C is only used to compute the log-likelihood, but not the theoretical autocovariance function (tacvf); a "2" means that C is used to compute the tacvf and not the log-likelihood; and a "3" means C is used to compute everything.

Note that the time series is assumed to be stationary: this function does not do any differencing.

Value

The exact log-likelihood of the model given with respect to z, up to an additive constant.

Author(s)

Justin Veenstra

References

Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (2008) Time Series Analysis: Forecasting and Control. 4th Edition. John Wiley and Sons, Inc., New Jersey.

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

arfima

lARFIMAwTF

tacvfARFIMA

Examples

set.seed(3452)
sim <- arfima.sim(1000, model = list(phi = c(0.3, -0.1)))
lARFIMA(sim, phi = c(0.3, -0.1))

Exact log-likelihood of a long memory model with a transfer function model and series included Computes the exact log-likelihood of a long memory model with respect to a given time series as well as a transfer fucntion model and series. This function is not meant to be used directly.

Description

Once again, this function should not be used externally.

Usage

lARFIMAwTF(
  z,
  phi = numeric(0),
  theta = numeric(0),
  dfrac = numeric(0),
  phiseas = numeric(0),
  thetaseas = numeric(0),
  dfs = numeric(0),
  H = numeric(0),
  Hs = numeric(0),
  alpha = numeric(0),
  alphas = numeric(0),
  xr = numeric(0),
  r = numeric(0),
  s = numeric(0),
  b = numeric(0),
  delta = numeric(0),
  omega = numeric(0),
  period = 0,
  useC = 3,
  meanval = 0
)

Arguments

z

A vector or (univariate) time series object, assumed to be (weakly) stationary.
phi

The autoregressive parameters in vector form.
theta

The moving average parameters in vector form. See Details for differences from arima.
dfrac

The fractional differencing parameter.
phiseas

The seasonal autoregressive parameters in vector form.
thetaseas

The seasonal moving average parameters in vector form. See Details for differences from arima.
dfs

The seasonal fractional differencing parameter.
H

The Hurst parameter for fractional Gaussian noise (FGN). Should not be mixed with dfrac or alpha: see "Details".
Hs

The Hurst parameter for seasonal fractional Gaussian noise (FGN). Should not be mixed with dfs or alphas: see "Details".
alpha

The decay parameter for power-law autocovariance (PLA) noise. Should not be mixed with dfrac or H: see "Details".
alphas

The decay parameter for seasonal power-law autocovariance (PLA) noise. Should not be mixed with dfs or Hs: see "Details".
xr

The regressors in vector form
r

The order of the delta(s)
s

The order of the omegas(s)
b

The backshifting to be done
delta

Transfer function parameters as in Box, Jenkins, and Reinsel. Corresponds to the "autoregressive" part of the dynamic regression.
omega

Transfer function parameters as in Box, Jenkins, and Reinsel. Corresponds to the "moving average" part of the dynamic regression: note that omega_0 is not restricted to 1. See "Details" for issues.
period

The periodicity of the seasonal components. Must be >= 2.
useC

How much interfaced C code to use: an integer between 0 and 3. The value 3 is strongly recommended. See "Details".
meanval

If the mean is to be estimated dynamically, the mean.

Value

A log-likelihood value

Author(s)

Justin Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

Extract Log-Likelihood Values

Description

Extracts log-likelihood values from a arfima fit.

Usage

## S3 method for class 'arfima'
logLik(object, ...)

Arguments

object

A fitted arfima object
...

Optional arguments not currently used.

Details

Uses the function DLLoglikelihood from the package ltsa. The log-likelihoods returned are exact up to an additive constant.

Value

A vector of log-likelihoods, one for each mode, is returned, along with the degrees of freedom.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

AIC.arfima

Converts AR/MA coefficients from the PACF space to operator space

Description

Converts AR/MA coefficients from PACF box-space to operator space; usually for internal use

Usage

PacfToAR(pi)

Arguments

pi

The AR/MA coefficients in PACF box-space

Value

The AR/MA coefficients in operator space.

Author(s)

A. I. McLeod

References

Barndorff-Nielsen O. E., Schou G. (1973). "On the parametrization of autoregressive models by partial autocorrelations." Journal of Multivariate Analysis, 3, 408-419

McLeod A. I. , Zhang Y (2006). "Partial autocorrelation parameterization for subset autore- gression." Journal of Time Series Analysis, 27(4), 599-612

Plots the original time series, the predictions, and the prediction intervals for a predarfima object.

Description

This function takes a predarfima object generated by predict.arfima and plots all of the information contained in it. The colour code is as follows:

Usage

## S3 method for class 'predarfima'
plot(
  x,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ylim = NULL,
  numback = 5,
  xlim = NULL,
  ...
)

Arguments

x

A predarfima object
xlab

Optional
ylab

Optional
main

Optional
ylim

Optional
numback

The number of last values of the original series to plot defined by the user. The default is five
xlim

Optional
...

Currently not used

Details

grey: exact prediction red: exact prediction intervals (PIs) orange: limiting PIs

See predict.arfima.

Value

None. Generates a plot

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

predict.arfima, print.predarfima

Examples

set.seed(82365)
sim <- arfima.sim(1000, model = list(dfrac = 0.4, theta=0.9, dint = 1))
fit <- arfima(sim, order = c(0, 1, 1), back=TRUE)
fit
pred <- predict(fit, n.ahead = 5)
pred
plot(pred)
#Let's look at more context
plot(pred, numback = 50)

Plots the output from a call to tacvf

Description

Plots the theoretical autocovariance functions of the modes for a fitted arfima object

Usage

## S3 method for class 'tacvf'
plot(
  x,
  type = "o",
  pch = 20,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  xlim = NULL,
  ylim = NULL,
  tacf = FALSE,
  maxlag = NULL,
  lag0 = !tacf,
  ...
)

Arguments

x

A tacvf object from a call to said function
type

See plot. The default is recommended for short maxlag
pch

See plot
xlab

See plot
ylab

See plot
main

See plot
xlim

See plot
ylim

See plot
tacf

If TRUE, plots the theoretical autocorellations instead
maxlag

The maximum lag for the plot
lag0

Whether or not to plot lag 0 of the tacvfs/tacfs. Default !tacf. Used by tacfplot.
...

Currently not used

Details

Only plots up to nine tacvfs. It is highly recommended that the arfima object be weeded before calling tacvf

Value

None. There is a plot as output.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

tacvf

Examples

set.seed(1234)
sim <- arfima.sim(1000, model = list(theta = 0.99, dfrac = 0.49))
fit <- arfima(sim, order = c(0, 0, 1))
plot(tacvf(fit))
plot(tacvf(fit), tacf = TRUE)

Predicts from a fitted object.

Description

Performs prediction of a fitted arfima object. Includes prediction for each mode and exact and limiting prediction error standard deviations. NOTE: the standard errors in beta are currently not taken into account in the prediction intervals shown. This will be updated as soon as possible.

Usage

## S3 method for class 'arfima'
predict(
  object,
  n.ahead = 1,
  prop.use = "default",
  newxreg = NULL,
  predint = 0.95,
  exact = c("default", T, F),
  setmuhat0 = FALSE,
  cpus = 1,
  trend = NULL,
  n.use = NULL,
  xreg = NULL,
  ...
)

Arguments

object

A fitted arfima object
n.ahead

The number of steps ahead to predict
prop.use

The proportion (between 0 and 1) or percentage (between >1 and 100) of data points to use for prediction. Defaults to the string "default", which sets the number of data points n.use to the minimum of the series length and 1000. Overriden by n.use.
newxreg

If a regression fit, the new regressors
predint

The percentile to use for prediction intervals assuming normal deviations.
exact

Controls whether exact (based on the theoretical autocovariance matrix) prediction variances are calculated (which is recommended), as well as whether the exact prediction formula is used when the process is differenced (which can take a fair amount of time if the length of the series used to predict is large). Defaults to the string "default", which is TRUE for the first and FALSE for the second. A Boolean value (TRUE or FALSE) will set both to this value.
setmuhat0

Experimental. Sets muhat equal to zero
cpus

The number of CPUs to use for prediction. Currently not implemented
trend

An optional vector the length of n.ahead or longer to add to the predictions
n.use

Directly set the number mentioned in prop.use.
xreg

Alias for newxreg
...

Optional arguments. Currently not used

Value

A list of lists, ceiling(prop.use * n)one for each mode with relavent details about the prediction

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

arfima, plot.predarfima, print.predarfima

Examples

set.seed(82365)
sim <- arfima.sim(1000, model = list(dfrac = 0.4, theta=0.9, dint = 1))
fit <- arfima(sim, order = c(0, 1, 1), back=TRUE)
fit
pred <- predict(fit, n.ahead = 5)
pred
plot(pred, numback=50)
#Predictions aren't really different due to the
#series.  Let's see what happens when we regress!

set.seed(23524)
#Forecast 5 ahead as before
#Note that we need to integrate the regressors, since time series regression
#usually assumes that regressors are of the same order as the series.
n.fore <- 5
X <- matrix(rnorm(3000+3*n.fore), ncol = 3)
X <- apply(X, 2, cumsum)
Xnew <- X[1001:1005,]
X <- X[1:1000,]
beta <- matrix(c(2, -.4, 6), ncol = 1)
simX <- sim + as.vector(X%*%beta)
fitX <- arfima(simX, order = c(0, 1, 1), xreg = X, back=TRUE)
fitX
#Let's compare predictions.
predX <- predict(fitX, n.ahead = n.fore, xreg = Xnew)
predX
plot(predX, numback = 50)
#With the mode we know is really there, it looks better.
fitX <- removeMode(fitX, 2)
predXnew <- predict(fitX, n.ahead = n.fore, xreg = Xnew)
predXnew
plot(predXnew, numback=50)

Prints a Fitted Object

Description

Prints a fitted arfima object's relevant details

Usage

## S3 method for class 'arfima'
print(x, digits = max(6, getOption("digits") - 3), ...)

Arguments

x

A fitted arfima object
digits

The number of digits to print
...

Optional arguments. See print.

Value

The object is returned invisibly

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

Prints predictions and prediction intervals

Description

Prints the output of predict on an arfima object

Usage

## S3 method for class 'predarfima'
print(x, digits = max(6, getOption("digits") - 3), ...)

Arguments

x

An object of class "predarfima"
digits

The number of digits to print
...

Currently not used

Details

Prints all the relavent output of the prediction function of the arfima package

Value

x is returned invisibly

Author(s)

JQ (Justin) Veenstra

See Also

arfima, predict.arfima, predict, plot.predarfima

Examples

set.seed(82365)
sim <- arfima.sim(1000, model = list(dfrac = 0.4, theta=0.9, dint = 1))
fit <- arfima(sim, order = c(0, 1, 1), back=TRUE)
fit
pred <- predict(fit, n.ahead = 5)
pred
plot(pred)

Prints the output of a call to summary on an arfima object

Description

Prints the output of a call to summary on an arfima object

Usage

## S3 method for class 'summary.arfima'
print(
  x,
  digits = max(6, getOption("digits") - 3),
  signif.stars = getOption("show.signif.stars"),
  ...
)

Arguments

x

A summary.arfima object
digits

The number of digits to print
signif.stars

Whether to print stars on significant output
...

Currently not used

Value

Returns the object x invisibly

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

arfima, print.arfima, summary.arfima, print

Examples

set.seed(54678)
sim <- arfima.sim(1000, model = list(phi = 0.9, H = 0.3))
fit <- arfima(sim, order = c(1, 0, 0), lmodel = "g", back=TRUE)
summary(fit)

Prints a tacvf object.

Description

Prints the output of a call to tacvf on an arfima object

Usage

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

Arguments

x

The tacvf object.
...

Optional arguments. See print.

Value

The object is returned invisibly

Author(s)

JQ (Justin) Veenstra

See Also

tacvf, plot.tacvf

Removes a mode from an arfima fit.

Description

This function is useful if one suspects a mode is spurious and does not want to call the weed function.

Usage

removeMode(object, num)

Arguments

object

An object of class "arfima".
num

The number of the mode as in the printed value of the object.

Value

The original object with the mode removed.

Author(s)

JQ (Justin) Veenstra

See Also

arfima

Examples

set.seed(8765)
sim <- arfima.sim(1000, model = list(phi = 0.4, theta = 0.9, dfrac = 0.4))
fit <- arfima(sim, order = c(1, 0, 1), back=TRUE)
fit
fit <- removeMode(fit, 3)
fit

Extract the Residuals of a Fitted Object

Description

Extracts the residuals or regression residuals from a fitted arfima object

Usage

## S3 method for class 'arfima'
residuals(object, reg = FALSE, ...)

Arguments

object

A fitted arfima object
reg

Whether to extract the regression residuals instead. If TRUE, throws an error if no regression was done.
...

Optional parameters. Currently not used.

Value

A list of vectors of residuals, one for each mode.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

arfima, fitted.arfima

Examples

set.seed(8564)
sim <- arfima.sim(1000, model = list(phi = c(0.2, 0.1), dfrac = 0.4, theta = 0.9))
fit <- arfima(sim, order = c(2, 0, 1), back=TRUE)

fit

resid <- resid(fit)
par(mfrow = c(1, 3))
plot(resid[[1]])
plot(resid[[2]])
plot(resid[[3]])
fitted <- fitted(fit)
plot(fitted[[1]], resid[[1]])
plot(fitted[[2]], resid[[2]])
plot(fitted[[3]], resid[[3]])
par(mfrow = c(1, 1))

Series J, Gas Furnace Data

Description

Gas furnace data, sampling interval 9 seconds; observations for 296 pairs of data points.

Format

List with ts objects XJ and YJ.

Details

XJ is input gas rate in cubic feet per minute, YJ is percentage carbon dioxide (CO2) in outlet gas. X is the regressor.

Box, Jenkins, and Reinsel (2008) fit an AR(2) to YJ, with transfer function specifications r = 2, s = 2, and b = 3, regressing on XJ. Our package agrees with their results.

Source

Box, Jenkins and Reinsel(2008). Time Series Analysis: Forecasting and Control.

References

Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (2008) Time Series Analysis: Forecasting and Control. 4th Edition. John Wiley and Sons, Inc., New Jersey.

Veenstra, J. and McLeod, A. I. (Working Paper). The arfima R package: Exact Methods for Hyperbolic Decay Time Series

Examples

data(SeriesJ)
attach(SeriesJ)

fitTF <- arfima(YJ, order= c(2, 0, 0), xreg = XJ, reglist =
list(regpar = c(2, 2, 3)), lmodel = "n")
fitTF ## agrees fairly closely with Box et. al.


detach(SeriesJ)

Simulate an ARFIMA time series from a fitted arfima object.

Description

This function simulates an long memory ARIMA time series, with one of fractionally differenced white noise (FDWN), fractional Gaussian noise (FGN), power-law autocovariance (PLA) noise, or short memory noise and possibly seasonal effects.

Usage

sim_from_fitted(n, model, X = NULL, seed = NULL)

Arguments

n

The number of points to be generated.
model

The model to be simulated from. The phi and theta arguments should be vectors with the values of the AR and MA parameters. Note that Box-Jenkins notation is used for the MA parameters: see the "Details" section of arfima.
X

The xreg matrix to add to the series, required if there is an xreg argument in model. An error will be thrown if there is a mismatch between this argument and whether model was called with a external regressor
seed

An optional seed that will be set before the simulation. If model is multimodal, a seed will be chosen randomly if not provided, and all modes will simulate a time series with said seed set.

Details

A suitably defined stationary series is generated, and if either of the dints (non-seasonal or seasonal) are greater than zero, the series is integrated (inverse-differenced) with zinit equalling a suitable amount of 0s if not supplied. Then a suitable amount of points are taken out of the beginning of the series (i.e. dint + period * seasonal dint = the length of zinit) to obtain a series of length n. The stationary series is generated by calculating the theoretical autovariance function and using it, along with the innovations to generate a series as in McLeod et. al. (2007). Note: if you would like to fit from parameters, use the funtion, arfima.sim.

Value

A sample (or list of samples) from a multivariate normal distribution that has a covariance structure defined by the autocovariances generated for given parameters. The sample acts like a time series with the given parameters. The returned value will be a list if the fit is multimodal.

Author(s)

JQ (Justin) Veenstra

References

McLeod, A. I., Yu, H. and Krougly, Z. L. (2007) Algorithms for Linear Time Series Analysis: With R Package Journal of Statistical Software, Vol. 23, Issue 5

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

P. Borwein (1995) An efficient algorithm for Riemann Zeta function Canadian Math. Soc. Conf. Proc., 27, pp. 29-34.

See Also

arfima, arfima.sim

Examples

set.seed(6533)
sim <- arfima.sim(1000, model = list(phi = .2, dfrac = .3, dint = 2))

fit <- arfima(sim, order = c(1, 2, 0))
fit

sim2 <- sim_from_fitted(100, fit)

fit2 <- arfima(sim2, order = c(1, 2, 0))
fit2

set.seed(2266)
#Fairly pathological series to fit for this package
series = arfima.sim(500, model=list(phi = 0.98, dfrac = 0.46))

X = matrix(rnorm(1000), ncol = 2)
colnames(X) <- c('c1', 'c2')
series_added <- series + X%*%c(2, 5)

fit <- arfima(series, order = c(1, 0, 0), numeach = c(2, 2))
fit_X <- arfima(series_added, order=c(1, 0, 0), xreg=X, numeach = c(2, 2))

from_series <- sim_from_fitted(1000, fit)
 
fit1a <- arfima(from_series[[1]], order = c(1, 0, 0), numeach = c(2, 2))
fit1a
fit1 <- arfima(from_series[[1]], order = c(1, 0, 0))
fit1
fit2 <- arfima(from_series[[1]], order = c(1, 0, 0))
fit2
fit3 <- arfima(from_series[[1]], order = c(1, 0, 0))
fit3
fit4 <- arfima(from_series[[1]], order = c(1, 0, 0))
fit4

Xnew = matrix(rnorm(2000), ncol = 2)
from_series_X <- sim_from_fitted(1000, fit_X, X=Xnew)

fit_X1a <- arfima(from_series_X[[1]], order=c(1, 0, 0), xreg=Xnew, numeach = c(2, 2))
fit_X1a
fit_X1 <- arfima(from_series_X[[1]], order=c(1, 0, 0), xreg=Xnew)
fit_X1
fit_X2 <- arfima(from_series_X[[2]], order=c(1, 0, 0), xreg=Xnew)
fit_X2
fit_X3 <- arfima(from_series_X[[3]], order=c(1, 0, 0), xreg=Xnew)
fit_X3
fit_X4 <- arfima(from_series_X[[4]], order=c(1, 0, 0), xreg=Xnew)
fit_X4

Extensive Summary of an Object

Description

Provides a very comprehensive summary of a fitted arfima object. Includes correlation and covariance matrices (observed and expected), the Fisher Information matrix of those parameters for which it is defined, and more, for each mode.

Usage

## S3 method for class 'arfima'
summary(object, digits = max(4, getOption("digits") - 3), ...)

Arguments

object

A fitted arfima object
digits

The number of digits to print
...

Optional arguments, currently not used.

Value

A list of lists (one for each mode) of all relevant information about the fit that can be passed to print.summary.arfima.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

arfima, iARFIMA, vcov.arfima

Examples

data(tmpyr)

fit <- arfima(tmpyr, order = c(1, 0, 1), back=TRUE)
fit

summary(fit)

Plots the theoretical autocorralation functions (tacfs) of one or more fits.

Description

Plots the theoretical autocorralation functions (tacfs) of one or more fits.

Usage

tacfplot(
  fits = list(),
  modes = "all",
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  xlim = NULL,
  ylim = NULL,
  maxlag = 20,
  lag0 = FALSE,
  ...
)

Arguments

fits

A list of objects of class "arfima".
modes

Either "all" or a vector of the same length as fits for which the tacfs will be ploted.
xlab

Optional. Usually better to be generated by the function.
ylab

Optional. Usually better to be generated by the function.
main

Optional. Usually better to be generated by the function.
xlim

Optional. Usually better to be generated by the function.
ylim

Optional. Usually better to be generated by the function.
maxlag

Optional. Used to limit the length of tacfs. Highly recommended to be a value from 20 - 50.
lag0

Whether or not the lag 0 tacf should be printed. Since this is always 1 for all tacfs, recommended to be TRUE. It is easier to see the shape of the tacfs.
...

Optional. Currently not used.

Value

NULL. However, there is a plot output.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

tacvf, plot.tacvf

Examples

set.seed(34577)
sim <- arfima.sim(500, model = list(theta = 0.9, phi = 0.5, dfrac = 0.4))
fit1 <- arfima(sim, order = c(1, 0, 1), cpus = 2, back=TRUE)
fit2 <- arfima(sim, order = c(1, 0, 1), cpus = 2, lmodel = "g", back=TRUE)
fit3 <- arfima(sim, order = c(1, 0, 1), cpus = 2, lmodel = "h", back=TRUE)
fit1
fit2
fit3
tacfplot(fits = list(fit1, fit2, fit3), maxlag = 30)

Extracts the tacvfs of a fitted object

Description

Extracts the theoretical autocovariance functions (tacvfs) from a fitted arfima or one of its modes (an ARFIMA) object.

Usage

tacvf(obj, xmaxlag = 0, forPred = FALSE, n.ahead = 0, nuse = -1, ...)

Arguments

obj

An object of class "arfima" or "ARFIMA". The latter class is a mode of the former.
xmaxlag

The number of extra points to be added on to the end. That is, if the original series has length 300, and xmaxlag = 5, the tacvfs will go from lag 0 to lag 304.
forPred

Should only be TRUE from a call to predict.arfima.
n.ahead

Only used internally.
nuse

Only used internally.
...

Optional arguments, currently not used.

Value

A list of tacvfs, one for each mode, the length of the time series.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

plot.tacvf, print.tacvf, tacfplot, arfima

The theoretical autocovariance function of a long memory process.

Description

Calculates the tacvf of a mixed long memory-ARMA (with posible seasonal components). Combines long memory and ARMA (and non-seasonal and seasonal) parts via convolution.

Usage

tacvfARFIMA(
  phi = numeric(0),
  theta = numeric(0),
  dfrac = numeric(0),
  phiseas = numeric(0),
  thetaseas = numeric(0),
  dfs = numeric(0),
  H = numeric(0),
  Hs = numeric(0),
  alpha = numeric(0),
  alphas = numeric(0),
  period = 0,
  maxlag,
  useCt = T,
  sigma2 = 1
)

Arguments

phi

The autoregressive parameters in vector form.
theta

The moving average parameters in vector form. See Details for differences from arima.
dfrac

The fractional differencing parameter.
phiseas

The seasonal autoregressive parameters in vector form.
thetaseas

The seasonal moving average parameters in vector form. See Details for differences from arima.
dfs

The seasonal fractional differencing parameter.
H

The Hurst parameter for fractional Gaussian noise (FGN). Should not be mixed with dfrac or alpha: see "Details".
Hs

The Hurst parameter for seasonal fractional Gaussian noise (FGN). Should not be mixed with dfs or alphas: see "Details".
alpha

The decay parameter for power-law autocovariance (PLA) noise. Should not be mixed with dfrac or H: see "Details".
alphas

The decay parameter for seasonal power-law autocovariance (PLA) noise. Should not be mixed with dfs or Hs: see "Details".
period

The periodicity of the seasonal components. Must be >= 2.
maxlag

The number of terms to compute: technically the output sequence is from lags 0 to maxlag, so there are maxlag + 1 terms.
useCt

Whether or not to use C to compute the (parts of the) tacvf.
sigma2

Used in arfima.sim: determines the value of the innovation variance. The tacvf sequence is multiplied by this value.

Details

The log-likelihood is computed for the given series z and the parameters. If two or more of dfrac, H or alpha are present and/or two or more of dfs, Hs or alphas are present, an error will be thrown, as otherwise there is redundancy in the model. Note that non-seasonal and seasonal components can be of different types: for example, there can be seasonal FGN with FDWN at the non-seasonal level.

The moving average parameters are in the Box-Jenkins convention: they are the negative of the parameters given by arima.

Value

A sequence of length maxlag + 1 (lags 0 to maxlag) of the tacvf of the given process.

Author(s)

JQ (Justin) Veenstra and A. I. McLeod

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

P. Borwein (1995) An efficient algorithm for Riemann Zeta function Canadian Math. Soc. Conf. Proc., 27, pp. 29-34.

Examples

t1 <- tacvfARFIMA(phi = c(0.2, 0.1), theta = 0.4, dfrac = 0.3, maxlag = 30)
t2 <- tacvfARFIMA(phi = c(0.2, 0.1), theta = 0.4, H = 0.8, maxlag = 30)
t3 <- tacvfARFIMA(phi = c(0.2, 0.1), theta = 0.4, alpha = 0.4, maxlag = 30)
plot(t1, type = "o", col = "blue", pch = 20)
lines(t2, type = "o", col = "red", pch = 20)
lines(t3, type = "o", col = "purple", pch = 20)  #they decay at about the same rate

Temperature Data

Description

Central England mean yearly temperatures from 1659 to 1976

Format

A ts tmpyr

Details

Hosking notes that while the ARFIMA(1, d, 1) has a lower AIC, it is not much lower than the AIC of the ARFIMA(1, d, 0).

Bhansali and Kobozka find: muHat = 9.14, d = 0.28, phi = -0.77, and theta = -0.66 for the ARFIMA(1, d, 1), which is close to our result, although our result reveals trimodality if numeach is large enough. The third mode is close to Hosking's fit of an ARMA(1, 1) to these data, while the second is very antipersistent.

Our package gives a very close result to Hosking for the ARFIMA(1, d, 0) case, although there is also a second mode. Given how close it is to the boundary, it may or may not be spurious. A check with dmean = FALSE shows that it is not the optimized mean giving a spurious mode.

If, however, we use whichopt = 1, we only have one mode. Note that Nelder-Mead sometimes does take out non-spurious modes, or add spurious modes to the surface.

Source

https://hadleyserver.metoffice.gov.uk/hadobs/hadcet/

References

Parker, D.E., Legg, T.P., and Folland, C.K. (1992). A new daily Central England Temperature Series, 1772-1991. Int. J. Clim., Vol 12, pp 317-342

Manley,G. (1974). Central England Temperatures: monthly means 1659 to 1973. Q.J.R. Meteorol. Soc., Vol 100, pp 389-405.

Hosking, J. R. M. (1984). Modeling persistence in hydrological time series using fractional differencing, Water Resour. Res., 20(12)

Bhansali, R. J. and Koboszka, P. S. (2003) Prediction of Long-Memory Time Series In Doukhan, P., Oppenheim, G. and Taqqu, M. S. (Eds) Theory and Applications of Long-Range Dependence (pp355-368) Birkhauser Boston Inc.

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

Examples

data(tmpyr)

fit <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3), dmean = TRUE, back=TRUE)
fit
##suspect that fourth mode may be spurious, even though not close to a boundary
##may be an induced mode from the optimization of the mean

fit <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3), dmean = FALSE, back=TRUE)
fit

##perhaps so


plot(tacvf(fit), maxlag = 30, tacf = TRUE)

fit1 <- arfima(tmpyr, order = c(1, 0, 0), dmean = TRUE, back=TRUE)
fit1

fit2 <- arfima(tmpyr, order = c(1, 0, 0), dmean = FALSE, back=TRUE)
fit2  ##still bimodal.  Second mode may or may not be spurious.

fit3 <- arfima(tmpyr, order = c(1, 0, 0), dmean = FALSE, whichopt = 1, numeach = c(3, 3))
fit3  ##Unimodal.  So the second mode was likely spurious.

plot(tacvf(fit2), maxlag = 30, tacf = TRUE)
##maybe not spurious.  Hard to tell without visualizing the surface.

##compare to plotted tacf of fit1:  looks alike
plot(tacvf(fit1), maxlag = 30, tacf = TRUE)

tacfplot(list(fit1, fit2))

Extracts the Variance-Covariance Matrix

Description

Extracts the variance-covariance matrices (one or two for each mode) from a fitted arfima object.

Usage

## S3 method for class 'arfima'
vcov(
  object,
  type = c("b", "o", "e"),
  cor = FALSE,
  digits = max(4, getOption("digits") - 3),
  tapprox = FALSE,
  summ = FALSE,
  ...
)

Arguments

object

A fitted arfima object
type

Which type of covariance matrix to return: "o" is the observed matrix (from solving the Hessian), "e" is the expected matrix (from solving the information matrix), and "b" is both.
cor

Whether or not the correlation matrix should be returned instead.
digits

The number of digits to print.
tapprox

Whether or not to use an approximation to find the expected matrix. Highly recommended to be FALSE, as it takes much longer, and is an approximation.
summ

Whether the call is from the summary.arfima function. Should not be used except internally.
...

Optional arguments, currently not used.

Value

A list of lists (one for each mode) with components observed and/or expected.

Author(s)

JQ (Justin) Veenstra

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

See Also

summary.arfima, arfima

Examples

set.seed(1234)
sim <- arfima.sim(1000, model = list(dfrac = 0.4, phi = .8, theta = -0.5))
fit1 <- arfima(sim, order = c(1, 0, 1), back=TRUE)
fit2 <- arfima(sim, order = c(1, 0, 1), lmodel = "g", back=TRUE)
fit3 <- arfima(sim, order = c(1, 0, 1), lmodel = "h", back=TRUE)
fit1
fit2
fit3
vcov(fit1)
vcov(fit2)
vcov(fit2)

Weeds out fits from a call to arfima that are too close to each other.

Description

Weeds out fits from a call to arfima that are too close to each other.

Usage

weed(
  ans,
  type = c("A", "P", "B", "N"),
  walls = FALSE,
  eps2 = 0.025,
  eps3 = 0.01,
  adapt = TRUE,
  pn = 2
)

Arguments

ans

The result from a call to arfima.
type

The space to perform the weeding in. "A" is for operating parameters. "P" is in the PACF space. "B" performs weeding in both. "N" performs no weeding and is only used internally.
walls

If more than one mode is on a wall in the PACF space, all modes but the one with the highest log-likelihood on the same wall are deleted.
eps2

The maximum distance between modes that are close together for the mode with the lower log-likelihood to be weeded out. If adapt is TRUE (default) this value changes.
eps3

The minimum distance from a wall for a secondary mode to be weeded out, if walls are TRUE.
adapt

If TRUE, if dim is the dimensionality of the search, eps2 is changed to (1+eps2)dim−1(1 + eps2)^{dim} - 1.
pn

The p in the p-norm to be used in the weeding. p = 2 (default) is Euclidean distance.

Value

An object of class "arfima" with modes possibly weeded out.

Author(s)

JQ (Justin) Veenstra

See Also

arfima, distance

Examples

set.seed(1234)
sim <- arfima.sim(1000, model = list(theta = 0.9, dfrac = 0.4))
fit <- arfima(sim, order = c(0, 0, 1), autoweed = FALSE, back=TRUE)
fit
distance(fit)
fit1 <- weed(fit)
fit1
distance(fit1)