Package 'PPMiss'

Coefficients of an ARFIMA(p,d,q) model

Description

This function calculates the coefficients $c_k, k \geq 0$ corresponding to $\theta(z)\phi^{-1}(z)(1-z)^{-d} = \sum_{k = 0}^{\infty}c_k z^k$, up to a truncation point

Usage

arfima.coefs(ar = NULL, ma = NULL, d = 0, trunc = 1)

Arguments

ar	the coefficients of the autoregressive polinomial. Default is NULL
ma	the coefficients of the moving average polinomial. Default is null
d	the long memory parameter. Default is 0.
trunc	the truncation point. Default is 1.

Value

The coefficients values up to order ‘trunc’.

Examples

cks <- arfima.coefs(d = 0.3, trunc = 5)
cks

cks <- arfima.coefs(d = 0.1, trunc = 5, ar = 0.5, ma = 0.6)
cks

---

Long memory parameter estimation

Description

Let $\theta_h$ be the copula parameter associated to $(X_t,X_{t+h})$ and $\hat\theta_h$ be an estimate of $\theta_h$ based on pseudo observations. The long memory parameter $d$ is estimated by

$\hat d:=\mathrm{argmin}_{|d|<0.5}\bigg\{\sum_{h=s}^m \bigg|{\hat K_1}(\hat\theta_h-a)-\frac{\Gamma(1-d)}{\Gamma(d)}h^{2d-1}\bigg|^r\bigg\}, \quad r > 0$

.

Usage

d.fit(xt, copula = gauss, dCdtheta = dCtheta_gauss, theta.lower = -1,
  theta.upper = 1, optim.method = "Brent", method = "mpl", s = 1,
  m = 24, theta.start = 0.1, empirical = TRUE, r = 2, a = 0,
  d.interval = c(-0.5, 0.5))

Arguments

xt	a vector with the observed time series. Missing observations are allowed.
copula	an object of class ‘copula’. Readily available options are frank, amh, fgm and gauss. Other copulas can be used but the user must provide the corresponding dCdtheta. Default is gauss.
dCdtheta	a two parameter function that returns the limit of the copula derivative, with respect to $\theta$, as $\theta$ goes to $a$, where $a$ is such that $C_a(u,v)=uv$. Readily available options are dCtheta_frank, dCtheta_amh, dCtheta_fgm and dCtheta_gauss. Default is dCtheta_gauss.
theta.lower	the lower bound for $\theta$. Default is -1.
theta.upper	the upper bound for $\theta$. Default is 1.
optim.method	a character string specifying the optimization method. For all available options see optim. Default is ‘Brent’. See fitCopula for more details.
method	a character string specifying the copula parameter estimator used. This can be one of: ‘mpl’, ‘itau’, ‘irho’, ‘itau.mpl’ or ‘ml’. See fitCopula for details. Default is ‘mpl’.
s	integer. The smallest lag $h$ considered in the estimation. Default is 1.
m	integer. The largest lag $h$ considered in the estimation. Default is 24.
theta.start	starting value for the parameter optimization via optim.
empirical	logical. If TRUE, the sample estimators for the density and quantile functions are considered. Otherwhise, the gaussian density and quantile functions are used. Default is TRUE
r	the exponent used in the Minkowski distance used to calculate $\hat d$. Default is 2, the Euclidean distance.
a	the value of $\theta$ such that $\lim_{\theta \to a} C_\theta(u,v)=uv$, is the product (or independence) copula. Default is 0, which is the common value for the available copulas, namely, frank, amh, fgm and gauss.
d.interval	a vector of size 2 giving the lower and upper bound for the long memory parameter $d$. Default is c(-0.5,0.5).

Value

$\hat d$, the estimated value of $d$.

Examples

#-------------------------
# ARFIMA(0,d,0) process
#-------------------------
trunc <- 50000
n = 1000
cks <- arfima.coefs(d = 0.25, trunc = trunc)
eps <- rnorm(trunc+n)
x <- sapply(1:n, function(t) sum(cks*rev(eps[t:(t+trunc)])))

#----------------------
# Original time series
#-----------------------
# For Frank copula, -Inf < theta < Inf. However, "Brent" requires
# finite lower and upper bounds so we use c(-100, 100) here
d_frank <- d.fit(xt = x, copula = frank, dCdtheta = dCtheta_frank,
                 theta.lower = -100, theta.upper = 100)
d_amh <- d.fit(xt = x, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm <- d.fit(xt = x, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss <- d.fit(xt = x, copula = gauss, dCdtheta = dCtheta_gauss,
                 theta.lower = -1, theta.upper = 1)

c(FRANK = d_frank, AMH = d_amh, FGM = d_fgm, GAUSS = d_gauss)

#----------------------------
# Adding some missing values
#----------------------------
index <- sample(1:n, size = round(0.2*n))
xt <- x
xt[index] <- NA

d_frank_m <- d.fit(xt = xt, copula = frank,
                   dCdtheta = dCtheta_frank,
                   theta.lower = -100, theta.upper = 100)
d_amh_m <- d.fit(xt = xt, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm_m <- d.fit(xt = xt, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss_m <- d.fit(xt = xt, copula = gauss,
                   dCdtheta = dCtheta_gauss,
                   theta.lower = -1, theta.upper = 1)

data.frame(
  series = c("Complete", "Missing"),
  FRANK = c(d_frank, d_frank_m),
  AMH = c(d_amh, d_amh_m),
  FGM = c(d_fgm, d_fgm_m),
  GAUSS = c(d_gauss, d_gauss_m))

#-------------------------
# ARFIMA(1,d,1) process
#-------------------------
# For a faster algorithm to generate ARFIMA processes,
# see the package "arfima"
trunc <- 50000
cks <- arfima.coefs(d = 0.35, trunc = trunc, ar = -0.2, ma = 0.4)
n = 1000
eps <- rnorm(trunc+n)
x <- sapply(1:n, function(t) sum(cks*rev(eps[t:(t+trunc)])))

#----------------------
# Original time series
#-----------------------
# For Frank copula, -Inf < theta < Inf. However, "Brent" requires
# finite lower and upper bounds so we use c(-100, 100) here
d_frank <- d.fit(xt = x, copula = frank, dCdtheta = dCtheta_frank,
                 theta.lower = -100, theta.upper = 100)
d_amh <- d.fit(xt = x, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm <- d.fit(xt = x, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss <- d.fit(xt = x, copula = gauss, dCdtheta = dCtheta_gauss,
                 theta.lower = -1, theta.upper = 1)

c(FRANK = d_frank, AMH = d_amh, FGM = d_fgm, GAUSS = d_gauss)

#----------------------------
# Adding some missing values
#----------------------------
n = 1000
index <- sample(1:n, size = round(0.2*n))
xt <- x
xt[index] <- NA

d_frank_m <- d.fit(xt = xt, copula = frank,
                   dCdtheta = dCtheta_frank,
                   theta.lower = -100, theta.upper = 100)
d_amh_m <- d.fit(xt = xt, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm_m <- d.fit(xt = xt, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss_m <- d.fit(xt = xt, copula = gauss,
                   dCdtheta = dCtheta_gauss,
                   theta.lower = -1, theta.upper = 1)

data.frame(
  series = c("Complete", "Missing"),
  FRANK = c(d_frank, d_frank_m),
  AMH = c(d_amh, d_amh_m),
  FGM = c(d_fgm, d_fgm_m),
  GAUSS = c(d_gauss, d_gauss_m))

---

Constant K1

Description

Calculates an estimate for the constant $K_1$ given by

$K_1 = \int_0^1\int_0^1\frac{1}{l_0(u)l_n(v)}\lim_{\theta\to a}\frac{\partial C_{\theta}(u,v)}{\partial\theta}\,dudv,$

where $l_m(x):= F_m'\big(F_m^{(-1)}(x)\big)$, $a$ is such that $C_a(u,v)=uv$ (the product copula), and $\{F_n\}_{n \geq 0}$ is a sequence of absolutely continuous distribution functions.

Usage

k1fun(dCdtheta, fun, data, empirical, mean = 0, sd = 1)

Arguments

dCdtheta	a function providing the limit as $\theta \to a$ of the copula derivative with respect to $\theta$. For the readily available copulas, namely, frank, amh, fgm and gauss, $a=0$.
fun	optionally, a function providing an estimator for the probability density function.
data	the observed time series. Only used to obtain the quantile function when empirical = TRUE.
empirical	logical. If TRUE, the sample estimators for the density and quantile functions are considered. Otherwise, the gaussian density and quantile functions are used instead.
mean	the mean of the gaussian distribution. Only used if empirical = FALSE
sd	the standard deviation of the gaussian distribution. Only used if empirical = FALSE

Details

Here $F'$ and $F^{(-1)}$ are replaced by sample estimators for these functions or the gaussian density and quantile functions are used, depending on the context.

The function kdens is used as sample estimator of $F'$ and quantile is the sample estimator of $F^{(-1)}$.

Value

The value of $K_1$.

Examples

trunc = 50000
cks <- arfima.coefs(d = 0.25, trunc = trunc)
eps <- rnorm(trunc+1000)
x <- sapply(1:1000, function(t) sum(cks*rev(eps[t:(t+trunc)])))

# kernel density function
dfun <- kdens(x)

# calculating K1 using four copulas and empirical estimates for F' and F^{(-1)}
K1_frank_e <- k1fun(dCdtheta = dCtheta_frank, fun = dfun,
                 data = x, empirical = TRUE)
K1_amh_e <- k1fun(dCdtheta = dCtheta_amh, fun = dfun,
                 data = x, empirical = TRUE)
K1_fgm_e <- k1fun(dCdtheta = dCtheta_fgm, fun = dfun,
                 data = x, empirical = TRUE)
K1_gauss_e <- k1fun(dCdtheta = dCtheta_gauss, fun = dfun,
                 data = x, empirical = TRUE)

# calculating K1 using four copulas and gaussian marginals
K1_frank_g <- k1fun(dCdtheta = dCtheta_frank, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))
K1_amh_g <- k1fun(dCdtheta = dCtheta_amh, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))
K1_fgm_g <- k1fun(dCdtheta = dCtheta_fgm, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))
K1_gauss_g <- k1fun(dCdtheta = dCtheta_gauss, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))

# comparing results
 data.frame(MARGINAL = c("Empirical", "Gaussian"),
            FRANK = c(K1_frank_e, K1_frank_g),
            AMH = c(K1_amh_e,  K1_amh_g),
            FGM = c(K1_fgm_e, K1_fgm_g),
            GAUSS = c(K1_gauss_e, K1_gauss_g))

---

Kernel density estimator

Description

The probability density function $F'$ is estimated using a kernel density approach. More specifically, first $y_i = \hat{f}(x_i^\ast)$ is estimated using $T = 512$ (default for the function density) equally spaced points $x_i^\ast$, $1 \leq i \leq T$, in the interval $[x_{(1)} - 3b, x_{(n)} + 3b]$, where $b$ is the bandwidth for the Gaussian kernel density estimator, chosen by applying the Silverman's rule of thumb (the default procedure in density). A cubic spline interpolation (the default method for spline) is then applied to the pairs $\{(x_i^\ast, y_i)\}_{i=1}^T$ to obtain $\hat F_n'(x)$ for all $x \in [x_{(1)} - 3b, x_{(n)} + 3b]$.

Usage

kdens(x)

Arguments

x	the data from which the estimate is to be computed.

Value

a function that approximates the probability density function.

Examples

# creating a time series
trunc = 50000
cks <- arfima.coefs(d = 0.25, trunc = trunc)
eps <- rnorm(trunc+1000)
x <- sapply(1:1000, function(t) sum(cks*rev(eps[t:(t+trunc)])))

# kernel density function
dfun <- kdens(x)

# plot
curve(dfun, from = min(x), to = max(x))

---

Copula functions and the corresponding derivative limit.

Description

Implemented copulas and the corresponding derivative limit, as $\theta \to a$, where $a$ is such that $C_a(u,v)=uv$. An estimate for $\theta$ is obtained based on the copula function used. and the derivatives are used to obtain an estimate for $K_1$. The functions ‘frank’, ‘amh’, ‘fgm’ and ‘gauss’ are shortcuts for copula::frankCopula(), copula::amhCopula(), copula::fgmCopula() and copula::normalCopula() from package ‘copula’, respectively.

Usage

frank

dCtheta_frank(u, v)

amh

dCtheta_amh(u, v)

fgm

dCtheta_fgm(u, v)

gauss

dCtheta_gauss(u, v)

Arguments

u	a real number between 0 and 1.
v	a real number between 0 and 1.

Format

An object of class frankCopula of length 1.

An object of class amhCopula of length 1.

An object of class fgmCopula of length 1.

An object of class normalCopula of length 1.

Details

The constant $K_1$ is given by

$K_1 = \int_0^1\int_0^1\frac{1}{l_0(u)l_n(v)}\lim_{\theta\rightarrow a}\frac{\partial C_{\theta}(u,v)}{\partial\theta}\,dudv,$

where $I=[0,1]$, $l_m(x):= F_m'\big(F_m^{(-1)}(x)\big)$ and $\{F_n\}_{n \geq 0}$ is a sequence of absolutely continuous distribution functions

Value

Archimedean copula objects of class ‘frankCopula’, ‘amhCopula’ or a Farlie-Gumbel-Morgenstern copula object of class ‘fgmCopula’ or an elliptical copula object of class ‘normalCopula’. For details, see archmCopula, fgmCopula and ellipCopula.

The derivative functions return the limit, as $\theta \to 0$, of the derivative with respect to $\theta$, corresponding to the copula functions.

---