| Title: | Fractional ARIMA (and Other Long Memory) Time Series Modeling |
| Version: | 1.8-1 |
| Date: | 2022-08-18 |
| Author: | JQ (Justin) Veenstra [aut, cre], A.I. McLeod [aut] |
| Maintainer: | JQ Veenstra <jqveenstra@gmail.com> |
| Depends: | R (≥ 3.0.0), ltsa |
| Imports: | parallel |
| 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. |
| License: | MIT + file LICENSE |
| RoxygenNote: | 7.1.2 |
| NeedsCompilation: | yes |
| Packaged: | 2022-08-18 14:13:21 UTC; jq |
| Repository: | CRAN |
| Date/Publication: | 2022-08-19 10:40:02 UTC |
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 <jqveenstra@gmail.com>
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 |
k |
The penalty term to be used. See |
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)
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
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 |
phiseas |
The seasonal autoregressive parameters in vector form. |
thetaseas |
The seasonal moving average parameters in vector form. See
Details for differences from |
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 |
Hs |
The Hurst parameter for seasonal fractional Gaussian noise (FGN).
Should not be mixed with |
alpha |
The decay parameter for power-law autocovariance (PLA) noise.
Should not be mixed with |
alphas |
The decay parameter for seasonal power-law autocovariance
(PLA) noise. Should not be mixed with |
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
Examples
IdentInvertQ(phi = 0.3, theta = 0.3)
IdentInvertQ(phi = 1.2)
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
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)
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 |
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
|
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
|
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:
|
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 |
weedtype |
The type of weeding to be done. See |
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 |
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 |
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
|
sigma2 |
The desired variance for the innovations of the series. |
rand.gen |
The distribution of the innovations. Any distribution
recognized by |
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
|
... |
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
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()
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
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 |
tpacf |
If |
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 |
... |
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
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 |
phiseas |
The seasonal autoregressive parameters in vector form. |
thetaseas |
The seasonal moving average parameters in vector form. See
Details for differences from |
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
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))
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 |
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 |
dfs |
The seasonal fractional differencing parameter. |
H |
The Hurst parameter for fractional Gaussian noise (FGN). Should
not be mixed with |
Hs |
The Hurst parameter for seasonal fractional Gaussian noise (FGN).
Should not be mixed with |
alpha |
The decay parameter for power-law autocovariance (PLA) noise.
Should not be mixed with |
alphas |
The decay parameter for seasonal power-law autocovariance
(PLA) noise. Should not be mixed with |
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
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 |
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 |
dfs |
The seasonal fractional differencing parameter. |
H |
The Hurst parameter for fractional Gaussian noise (FGN). Should
not be mixed with |
Hs |
The Hurst parameter for seasonal fractional Gaussian noise (FGN).
Should not be mixed with |
alpha |
The decay parameter for power-law autocovariance (PLA) noise.
Should not be mixed with |
alphas |
The decay parameter for seasonal power-law autocovariance
(PLA) noise. Should not be mixed with |
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 |
... |
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
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 |
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 |
type |
See |
pch |
See |
xlab |
See |
ylab |
See |
main |
See |
xlim |
See |
ylim |
See |
tacf |
If |
maxlag |
The maximum lag for the plot |
lag0 |
Whether or not to plot lag 0 of the tacvfs/tacfs. Default
|
... |
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
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 |
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 |
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
|
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.use |
Directly set the number mentioned in |
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 |
digits |
The number of digits to print |
... |
Optional arguments. See |
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 |
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 |
... |
Optional arguments. See |
Value
The object is returned invisibly
Author(s)
JQ (Justin) Veenstra
See Also
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
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 |
reg |
Whether to extract the regression residuals instead. If
|
... |
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
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))
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 |
X |
The xreg matrix to add to the series, required if there is an xreg
argument in |
seed |
An optional seed that will be set before the simulation. If
|
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
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 |
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
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 |
... |
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
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 |
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 |
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 |
dfs |
The seasonal fractional differencing parameter. |
H |
The Hurst parameter for fractional Gaussian noise (FGN). Should
not be mixed with |
Hs |
The Hurst parameter for seasonal fractional Gaussian noise (FGN).
Should not be mixed with |
alpha |
The decay parameter for power-law autocovariance (PLA) noise.
Should not be mixed with |
alphas |
The decay parameter for seasonal power-law autocovariance
(PLA) noise. Should not be mixed with |
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 |
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 |
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 |
... |
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
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 |
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
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)