Package 'LSTS'

Title:	Locally Stationary Time Series
Description:	A set of functions that allow stationary analysis and locally stationary time series analysis.
Authors:	Ricardo Olea [aut, cph], Wilfredo Palma [aut, cph], Pilar Rubio [aut], Mauricio Vargas [aut, cre]
Maintainer:	Mauricio Vargas <[email protected]>
License:	Apache License (>= 2)
Version:	2.1
Built:	2024-11-08 04:12:38 UTC
Source:	https://github.com/pachadotdev/lsts

Help Index

Smooth Periodogram by Blocks
Ljung-Box Test Plot
Hessian Matrix
Kalman filter for locally stationary processes
Summary for Locally Stationary Time Series
Whittle estimator to Locally Stationary Time Series
Locally Stationary Whittle log-likelihood Function
Locally Stationary Whittle Log-likelihood sigma
Locally Stationary Whittle Log-likelihood theta
Average Araucaria Araucana Tree Ring Width
Periodogram function
Smoothing periodogram
Spectral Density
Diagnostic Plots for Time Series fits

Smooth Periodogram by Blocks

Description

Plots the contour plot of the smoothing periodogram of a time series, by blocks or windows.

Usage

block.smooth.periodogram(
  y,
  x = NULL,
  N = NULL,
  S = NULL,
  p = 0.25,
  spar.freq = 0,
  spar.time = 0
)
block.smooth.periodogram(
  y,
  x = NULL,
  N = NULL,
  S = NULL,
  p = 0.25,
  spar.freq = 0,
  spar.time = 0
)

Arguments

`y`	(type: numeric) data vector
`x`	(type: numeric) optional vector, if `x = NULL` then the function uses $(1,\ldots,n)$ where `n` is the length of `y`.
`N`	(type: numeric) value corresponding to the length of the window to compute periodogram. If `N=NULL` then the function will use $N = \textrm{trunc}(n^{0.8})$ , see Dahlhaus and Giraitis (1998) where $n$ is the length of the `y` vector.
`S`	(type: numeric) value corresponding to the lag with which will be taking the blocks or windows to calculate the periodogram.
`p`	(type: numeric) value used if it is desired that `S` is proportional to `N`. By default `p=0.25`, if `S` and `N` are not entered.
`spar.freq`	(type: numeric) smoothing parameter, typically (but not necessarily) in $(0,1]$ .
`spar.time`	(type: numeric) smoothing parameter, typically (but not necessarily) in $(0,1]$ .

Details

The number of windows of the function is $m = \textrm{trunc}((n-N)/S+1)$ , where trunc truncates de entered value and n is the length of the vector y. All windows are of the same length N, if this value isn't entered by user then is computed as $N=\textrm{trunc}(n^{0.8})$ (Dahlhaus). LSTS_spb computes the periodogram in each of the M windows and then smoothes it two times with smooth.spline function; the first time using spar.freq parameter and the second time with spar.time. These windows overlap between them.

Value

A ggplot object.

References

For more information on theoretical foundations and estimation methods see Dahlhaus R, others (1997). “Fitting time series models to nonstationary processes.” The annals of Statistics, 25(1), 1–37. Dahlhaus R, Giraitis L (1998). “On the optimal segment length for parameter estimates for locally stationary time series.” Journal of Time Series Analysis, 19(6), 629–655.

Examples

block.smooth.periodogram(malleco)
block.smooth.periodogram(malleco)

Ljung-Box Test Plot

Description

Plots the p-values Ljung-Box test.

Usage

Box.Ljung.Test(z, lag = NULL, main = NULL)
Box.Ljung.Test(z, lag = NULL, main = NULL)

Arguments

`z`	(type: numeric) data vector
`lag`	(type: numeric) the number of periods for the autocorrelation
`main`	(type: character) a title for the returned plot

Details

The Ljung-Box test is used to check if exists autocorrelation in a time series. The statistic is

$q = n(n+2)\cdot\sum_{j=1}^h \hat{\rho}(j)^2/(n-j)$

with n the number of observations and $\hat{\rho}(j)$ the autocorrelation coefficient in the sample when the lag is j. LSTS_lbtp computes $q$ and returns the p-values graph with lag j.

Value

A ggplot object.

References

For more information on theoretical foundations and estimation methods see Brockwell PJ, Davis RA, Calder MV (2002). Introduction to time series and forecasting, volume 2. Springer. Ljung GM, Box GE (1978). “On a measure of lack of fit in time series models.” Biometrika, 65(2), 297–303.

Examples

Box.Ljung.Test(malleco, lag = 5)
Box.Ljung.Test(malleco, lag = 5)

Hessian Matrix

Description

Numerical aproximation of the Hessian of a function.

Usage

hessian(f, x0, ...)
hessian(f, x0, ...)

Arguments

`f`	(type: numeric) name of function that defines log likelihood (or negative of it).
`x0`	(type: numeric) scalar or vector of parameters that give the point at which you want the hessian estimated (usually will be the mle).
`...`	Additional arguments to be passed to the function.

Details

Computes the numerical approximation of the Hessian of f, evaluated at x0. Usually needs to pass additional parameters (e.g. data). N.B. this uses no numerical sophistication.

Value

An $n \times n$ matrix of 2nd derivatives, where $n$ is the length of x0.

Examples

# Variance of the maximum likelihood estimator for mu parameter in
# gaussian data
loglik <- function(series, x, sd = 1) {
  -sum(log(dnorm(series, mean = x, sd = sd)))
}
sqrt(c(var(malleco) / length(malleco), diag(solve(hessian(
  f = loglik, x = mean(malleco), series = malleco,
  sd = sd(malleco)
)))))
# Variance of the maximum likelihood estimator for mu parameter in
# gaussian data
loglik <- function(series, x, sd = 1) {
  -sum(log(dnorm(series, mean = x, sd = sd)))
}
sqrt(c(var(malleco) / length(malleco), diag(solve(hessian(
  f = loglik, x = mean(malleco), series = malleco,
  sd = sd(malleco)
)))))

Kalman filter for locally stationary processes

Description

This function run the state-space equations for expansion infinite of moving average in processes LS-ARMA or LS-ARFIMA.

Usage

LS.kalman(
  series,
  start,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  m = NULL
)
LS.kalman(
  series,
  start,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  m = NULL
)

Arguments

`series`	(type: numeric) univariate time series.
`start`	(type: numeric) numeric vector, initial values for parameters to run the model.
`order`	(type: numeric) vector corresponding to `ARMA` model entered.
`ar.order`	(type: numeric) AR polimonial order.
`ma.order`	(type: numeric) MA polimonial order.
`sd.order`	(type: numeric) polinomial order noise scale factor.
`d.order`	(type: numeric) `d` polinomial order, where `d` is the `ARFIMA` parameter.
`include.d`	(type: numeric) logical argument for `ARFIMA` models. If `include.d=FALSE` then the model is an ARMA process.
`m`	(type: numeric) truncation order of the MA infinity process. By default $m = 0.25n^{0.8}$ where `n` the length of `series`.

Details

The model fit is done using the Whittle likelihood, while the generation of innovations is through Kalman Filter. Details about ar.order, ma.order, sd.order and d.order can be viewed in LS.whittle.

Value

A list with:

`residuals`	standard residuals.
`fitted_values`	model fitted values.
`delta`	variance prediction error.

References

For more information on theoretical foundations and estimation methods see Brockwell PJ, Davis RA, Calder MV (2002). Introduction to time series and forecasting, volume 2. Springer. Palma W (2007). Long-memory time series: theory and methods, volume 662. John Wiley \& Sons. Palma W, Olea R, Ferreira G (2013). “Estimation and forecasting of locally stationary processes.” Journal of Forecasting, 32(1), 86–96.

Examples

fit_kalman <- LS.kalman(malleco, start(malleco))
fit_kalman <- LS.kalman(malleco, start(malleco))

Summary for Locally Stationary Time Series

Description

Produces a summary of the results to Whittle estimator to Locally Stationary Time Series (LS.whittle function).

Usage

LS.summary(object)
LS.summary(object)

Arguments

object

(type: list) the output of LS.whittle function

Details

Calls the output from LS.whittle and computes the standard error and p-values to provide a detailed summary.

Value

A list with the following components:

`summary`	a resume table with estimate, std. error, z-value and p-value of the model.
`aic`	AIC of the model.
`npar`	number of parameters in the model.

Examples

fit_whittle <- LS.whittle(
  series = malleco, start = c(1, 1, 1, 1),
  order = c(p = 1, q = 0), ar.order = 1, sd.order = 1, N = 180, n.ahead = 10
)
LS.summary(fit_whittle)
fit_whittle <- LS.whittle(
  series = malleco, start = c(1, 1, 1, 1),
  order = c(p = 1, q = 0), ar.order = 1, sd.order = 1, N = 180, n.ahead = 10
)
LS.summary(fit_whittle)

Whittle estimator to Locally Stationary Time Series

Description

This function computes Whittle estimator to LS-ARMA and LS-ARFIMA models.

Usage

LS.whittle(
  series,
  start,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  control = list(),
  lower = -Inf,
  upper = Inf,
  m = NULL,
  n.ahead = 0
)
LS.whittle(
  series,
  start,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  control = list(),
  lower = -Inf,
  upper = Inf,
  m = NULL,
  n.ahead = 0
)

Arguments

`series`	(type: numeric) univariate time series.
`start`	(type: numeric) numeric vector, initial values for parameters to run the model.
`order`	(type: numeric) vector corresponding to `ARMA` model entered.
`ar.order`	(type: numeric) AR polimonial order.
`ma.order`	(type: numeric) MA polimonial order.
`sd.order`	(type: numeric) polinomial order noise scale factor.
`d.order`	(type: numeric) `d` polinomial order, where `d` is the `ARFIMA` parameter.
`include.d`	(type: numeric) logical argument for `ARFIMA` models. If `include.d=FALSE` then the model is an ARMA process.
`N`	(type: numeric) value corresponding to the length of the window to compute periodogram. If `N=NULL` then the function will use $N = \textrm{trunc}(n^{0.8})$ , see Dahlhaus (1998) where $n$ is the length of the `y` vector.
`S`	(type: numeric) value corresponding to the lag with which will go taking the blocks or windows.
`include.taper`	(type: logical) logical argument that by default is `TRUE`. See `periodogram`.
`control`	(type: list) A list of control parameters. More details in `nlminb` .
`lower`	(type: numeric) lower bound, replicated to be as long as `start`. If unspecified, all parameters are assumed to be lower unconstrained.
`upper`	(type: numeric) upper bound, replicated to be as long as `start`. If unspecified, all parameters are assumed to be upper unconstrained.
`m`	(type: numeric) truncation order of the MA infinity process, by default $m = 0.25n^{0.8}$ . Parameter used in `LSTS_kalman`.
`n.ahead`	(type: numeric) The number of steps ahead for which prediction is required. By default is zero.

Details

This function estimates the parameters in models: LS-ARMA

$\Phi(t/T, \, B)\, Y_{t, T} = \Theta(t/T,\, B)\,\sigma(t/T)\, \varepsilon_t$

and LS-ARFIMA

$\Phi(t/T, \, B)\, Y_{t, T} = \Theta(t/T,\, B)\, (1-B)^{-d(t/T)}\, \sigma(t/T)\, \varepsilon_t,$

with infinite moving average expansion

$Y_{t, T} = \sigma(t/T)\, \sum_{j=0}^{\infty} \psi(t/T)\,\varepsilon_t,$

for $t = 1,\ldots, T$ , where for $u = t/T \in [0,1]$ , $\Phi(u,B)=1+\phi_1(u)B +\cdots+\phi_p(u)B^p$ is an autoregressive polynomial, $\Theta(u, B) = 1 + \theta_1(u)B + \cdots + \theta_q(u)B^q$ is a moving average polynomial, $d(u)$ is a long-memory parameter, $\sigma(u)$ is a noise scale factor and $\{\varepsilon_t \}$ is a Gaussian white noise sequence with zero mean and unit variance. This class of models extends the well-known ARMA and ARFIMA process, which is obtained when the components $\Phi(u, B)$ , $\Theta(u, B)$ , $d(u)$ and $\sigma(u)$ do not depend on $u$ . The evolution of these models can be specified in terms of a general class of functions. For example, let $\{g_j(u)\}$ , $j = 1, 2, \ldots$ , be a basis for a space of smoothly varying functions and let $d_{\theta}(u)$ be the time-varying long-memory parameter in model LS-ARFIMA. Then we could write $d_{\theta}(u)$ in terms of the basis $\{g_j(u) = u^j\}$ as follows $d_{\theta}(u) = \sum_{j=0}^{k} \alpha_j\,g_j(u)$ for unknown values of $k$ and $\theta = (\alpha_0,\,\alpha_1,\,\ldots, \,\alpha_k)^{\prime}$ . In this situation, estimating $\theta$ involves determining $k$ and estimating the coefficients $\alpha_0,\,\alpha_1,\,\ldots, \,\alpha_k$ . LS.whittle optimizes LS.whittle.loglik as objective function using nlminb function, for both LS-ARMA (include.d=FALSE) and LS-ARFIMA (include.d=TRUE) models. Also computes Kalman filter with LS.kalman and this values are given in var.coef in the output.

Value

A list with the following components:

`coef`	The best set of parameters found.
`var.coef`	covariance matrix approximated for maximum likelihood estimator $\hat{\theta}$ of $\theta:=(\theta_1,\ldots,\theta_k)^{\prime}$ . This matrix is approximated by $H^{-1}/n$ , where $H$ is the Hessian matrix $[\partial^2 \ell(\theta)/\partial\theta_i \partial\theta_j]_{i,j=1}^{k}$ .
`loglik`	log-likelihood of `coef`, calculated with `LS.whittle`.
`aic`	Akaike'S ‘An Information Criterion’, for one fitted model LS-ARMA or LS-ARFIMA. The formula is $-2L + 2k/n$ , where L represents the log-likelihood, k represents the number of parameters in the fitted model and n is equal to the length of the `series`.
`series`	original time serie.
`residuals`	standard residuals.
`fitted.values`	model fitted values.
`pred`	predictions of the model.
`se`	the estimated standard errors.
`model`	A list representing the fitted model.

Examples

# Analysis by blocks of phi and sigma parameters
N <- 200
S <- 100
M <- trunc((length(malleco) - N) / S + 1)
table <- c()
for (j in 1:M) {
  x <- malleco[(1 + S * (j - 1)):(N + S * (j - 1))]
  table <- rbind(table, nlminb(
    start = c(0.65, 0.15), N = N,
    objective = LS.whittle.loglik,
    series = x, order = c(p = 1, q = 0)
  )$par)
}
u <- (N / 2 + S * (1:M - 1)) / length(malleco)
table <- as.data.frame(cbind(u, table))
colnames(table) <- c("u", "phi", "sigma")
# Start parameters
phi <- smooth.spline(table$phi, spar = 1, tol = 0.01)$y
fit.1 <- nls(phi ~ a0 + a1 * u, start = list(a0 = 0.65, a1 = 0.00))
sigma <- smooth.spline(table$sigma, spar = 1)$y
fit.2 <- nls(sigma ~ b0 + b1 * u, start = list(b0 = 0.65, b1 = 0.00))
fit_whittle <- LS.whittle(
  series = malleco, start = c(coef(fit.1), coef(fit.2)), order = c(p = 1, q = 0),
  ar.order = 1, sd.order = 1, N = 180, n.ahead = 10
)
# Analysis by blocks of phi and sigma parameters
N <- 200
S <- 100
M <- trunc((length(malleco) - N) / S + 1)
table <- c()
for (j in 1:M) {
  x <- malleco[(1 + S * (j - 1)):(N + S * (j - 1))]
  table <- rbind(table, nlminb(
    start = c(0.65, 0.15), N = N,
    objective = LS.whittle.loglik,
    series = x, order = c(p = 1, q = 0)
  )$par)
}
u <- (N / 2 + S * (1:M - 1)) / length(malleco)
table <- as.data.frame(cbind(u, table))
colnames(table) <- c("u", "phi", "sigma")
# Start parameters
phi <- smooth.spline(table$phi, spar = 1, tol = 0.01)$y
fit.1 <- nls(phi ~ a0 + a1 * u, start = list(a0 = 0.65, a1 = 0.00))
sigma <- smooth.spline(table$sigma, spar = 1)$y
fit.2 <- nls(sigma ~ b0 + b1 * u, start = list(b0 = 0.65, b1 = 0.00))
fit_whittle <- LS.whittle(
  series = malleco, start = c(coef(fit.1), coef(fit.2)), order = c(p = 1, q = 0),
  ar.order = 1, sd.order = 1, N = 180, n.ahead = 10
)

Locally Stationary Whittle log-likelihood Function

Description

This function computes Whittle estimator for LS-ARMA and LS-ARFIMA models, in data with mean zero. If mean is not zero, then it is subtracted to data.

Usage

LS.whittle.loglik(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE
)
LS.whittle.loglik(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE
)

Arguments

`x`	(type: numeric) parameter vector.
`series`	(type: numeric) univariate time series.
`order`	(type: numeric) vector corresponding to `ARMA` model entered.
`ar.order`	(type: numeric) AR polimonial order.
`ma.order`	(type: numeric) MA polimonial order.
`sd.order`	(type: numeric) polinomial order noise scale factor.
`d.order`	(type: numeric) `d` polinomial order, where `d` is the `ARFIMA` parameter.
`include.d`	(type: numeric) logical argument for `ARFIMA` models. If `include.d=FALSE` then the model is an ARMA process.
`N`	(type: numeric) value corresponding to the length of the window to compute periodogram. If `N=NULL` then the function will use $N = \textrm{trunc}(n^{0.8})$ , see Dahlhaus (1998) where $n$ is the length of the `y` vector.
`S`	(type: numeric) value corresponding to the lag with which will go taking the blocks or windows.
`include.taper`	(type: logical) logical argument that by default is `TRUE`. See `periodogram`.

Details

The estimation of the time-varying parameters can be carried out by means of the Whittle log-likelihood function proposed by Dahlhaus (1997),

$L_n(\theta) = \frac{1}{4\pi}\frac{1}{M} \int_{-\pi}^{\pi} \bigg\{log f_{\theta}(u_j,\lambda) + \frac{I_N(u_j, \lambda)}{f_{\theta}(u_j,\lambda)}\bigg\}\,d\lambda$

where $M$ is the number of blocks, $N$ the length of the series per block, $n =S(M-1)+N$ , $S$ is the shift from block to block, $u_j =t_j/n$ , $t_j =S(j-1)+N/2$ , $j =1,\ldots,M$ and $\lambda$ the Fourier frequencies in the block ( $2\,\pi\,k/N$ , $k = 1,\ldots, N$ ).

References

For more information on theoretical foundations and estimation methods see Brockwell PJ, Davis RA, Calder MV (2002). Introduction to time series and forecasting, volume 2. Springer. Palma W, Olea R, others (2010). “An efficient estimator for locally stationary Gaussian long-memory processes.” The Annals of Statistics, 38(5), 2958–2997.

Locally Stationary Whittle Log-likelihood sigma

Description

This function calculates log-likelihood with known $\theta$ , through LS.whittle.loglik function.

Usage

LS.whittle.loglik.sd(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  theta.par = numeric()
)
LS.whittle.loglik.sd(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  theta.par = numeric()
)

Arguments

`x`	(type: numeric) parameter vector.
`series`	(type: numeric) univariate time series.
`order`	(type: numeric) vector corresponding to `ARMA` model entered.
`ar.order`	(type: numeric) AR polimonial order.
`ma.order`	(type: numeric) MA polimonial order.
`sd.order`	(type: numeric) polinomial order noise scale factor.
`d.order`	(type: numeric) `d` polinomial order, where `d` is the `ARFIMA` parameter.
`include.d`	(type: numeric) logical argument for `ARFIMA` models. If `include.d=FALSE` then the model is an ARMA process.
`N`	(type: numeric) value corresponding to the length of the window to compute periodogram. If `N=NULL` then the function will use $N = \textrm{trunc}(n^{0.8})$ , see Dahlhaus (1998) where $n$ is the length of the `y` vector.
`S`	(type: numeric) value corresponding to the lag with which will go taking the blocks or windows.
`include.taper`	(type: logical) logical argument that by default is `TRUE`. See `periodogram`.
`theta.par`	(type: numeric) vector with the known parameters of the model.

Details

This function computes LS.whittle.loglik with x as x = c(theta.par, x).

Locally Stationary Whittle Log-likelihood theta

Description

Calculate the log-likelihood with $\sigma$ known, through LS.whittle.loglik function.

Usage

LS.whittle.loglik.theta(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  sd.par = 1
)
LS.whittle.loglik.theta(
  x,
  series,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  sd.par = 1
)

Arguments

`x`	(type: numeric) parameter vector.
`series`	(type: numeric) univariate time series.
`order`	(type: numeric) vector corresponding to `ARMA` model entered.
`ar.order`	(type: numeric) AR polimonial order.
`ma.order`	(type: numeric) MA polimonial order.
`sd.order`	(type: numeric) polinomial order noise scale factor.
`d.order`	(type: numeric) `d` polinomial order, where `d` is the `ARFIMA` parameter.
`include.d`	(type: numeric) logical argument for `ARFIMA` models. If `include.d=FALSE` then the model is an ARMA process.
`N`	(type: numeric) value corresponding to the length of the window to compute periodogram. If `N=NULL` then the function will use $N = \textrm{trunc}(n^{0.8})$ , see Dahlhaus (1998) where $n$ is the length of the `y` vector.
`S`	(type: numeric) value corresponding to the lag with which will go taking the blocks or windows.
`include.taper`	(type: logical) logical argument that by default is `TRUE`. See `periodogram`.
`sd.par`	(type: numeric) value corresponding to known variance.

Details

This function computes LS.whittle.loglik with x as x = c(x, sd.par).

Average Araucaria Araucana Tree Ring Width

Description

A ts object containing average annual ring width measured in milimiters for different Araucaria Araucana trees in the Malleco Region (Chile). The years of observation in this data cover the period 1242-1975.

Format

A time series object with 734 elements

Author(s)

National Oceanic and Atmospheric Administration (NOAA)

Periodogram function

Description

This function computes the periodogram from a stationary time serie. Returns the periodogram, its graph and the Fourier frequency.

Usage

periodogram(y, plot = TRUE, include.taper = FALSE)
periodogram(y, plot = TRUE, include.taper = FALSE)

Arguments

`y`	(type: numeric) data vector
`plot`	(type: logical) logical argument which allows to plot the periodogram. Defaults to TRUE.
`include.taper`	(type: logical) logical argument which by default is `FALSE`. If `include.taper=TRUE` then `y` is multiplied by $0.5(1 - \cos(2\pi(n-1)/n))$ (cosine bell).

Details

The tapered periodogram it is given by

$I(\lambda) = \frac{|D_n(\lambda)|^2}{2\pi H_{2,n}(0)}$

with $D(\lambda) = \sum_{s=0}^{n-1} h \left(\frac{s}{N}\right) y_{s+1}\, e^{-i\,\lambda\,s}$ , $H_{k,n} = \sum_{s=0}^{n-1}h \left(\frac{s}{N}\right)^k\, e^{-i\,\lambda\,s}$ and $\lambda$ are Fourier frequencies defined as $2\pi k/n$ , with $k = 1,\,\ldots,\, n$ . The data taper used is the cosine bell function, $h(x) = \frac{1}{2}[1-\cos(2\pi x)]$ . If the series has missing data, these are replaced by the average of the data and $n$ it is corrected by $n-N$, where $N$ is the amount of missing values of serie. The plot of the periodogram is periodogram values vs. $\lambda$ .

Value

A list with with the periodogram and the lambda values.

References

For more information on theoretical foundations and estimation methods see Brockwell PJ, Davis RA, Calder MV (2002). Introduction to time series and forecasting, volume 2. Springer. Dahlhaus R, others (1997). “Fitting time series models to nonstationary processes.” The annals of Statistics, 25(1), 1–37.

Examples

# AR(1) simulated
set.seed(1776)
ts.sim <- arima.sim(n = 1000, model = list(order = c(1, 0, 0), ar = 0.7))
per <- periodogram(ts.sim)
per$plot
# AR(1) simulated
set.seed(1776)
ts.sim <- arima.sim(n = 1000, model = list(order = c(1, 0, 0), ar = 0.7))
per <- periodogram(ts.sim)
per$plot

Smoothing periodogram

Description

This function returns the smoothing periodogram of a stationary time serie, its plot and its Fourier frequency.

Usage

smooth.periodogram(y, plot = TRUE, spar = 0)
smooth.periodogram(y, plot = TRUE, spar = 0)

Arguments

`y`	(type: numeric) data vector.
`plot`	(type: logical) logical argument which allows to plot the periodogram. Defaults to TRUE.
`spar`	(type: numeric) smoothing parameter, typically (but not necessarily) in $(0,1].$

Details

smooth.periodogram computes the periodogram from y vector and then smooth it with smoothing spline method, which basically approximates a curve using a cubic spline (see more details in smooth.spline). $\lambda$ is the Fourier frequency obtained through periodogram. It must have caution with the minimum length of y, because smooth.spline requires the entered vector has at least length 4 and the length of y does not equal to the length of the data of the periodogram that smooth.spline receives. If it presents problems with tol (tolerance), see smooth.spline.

Value

A list with with the smooth periodogram and the lambda values

Examples

# AR(1) simulated
require(ggplot2)
set.seed(1776)
ts.sim <- arima.sim(n = 1000, model = list(order = c(1, 0, 0), ar = 0.7))
per <- periodogram(ts.sim)
aux <- smooth.periodogram(ts.sim, plot = FALSE, spar = .7)
sm_p <- data.frame(x = aux$lambda, y = aux$smooth.periodogram)
sp_d <- data.frame(
  x = aux$lambda,
  y = spectral.density(ar = 0.7, lambda = aux$lambda)
)
g <- per$plot
g +
  geom_line(data = sm_p, aes(x, y), color = "#ff7f0e") +
  geom_line(data = sp_d, aes(x, y), color = "#d31244")
# AR(1) simulated
require(ggplot2)
set.seed(1776)
ts.sim <- arima.sim(n = 1000, model = list(order = c(1, 0, 0), ar = 0.7))
per <- periodogram(ts.sim)
aux <- smooth.periodogram(ts.sim, plot = FALSE, spar = .7)
sm_p <- data.frame(x = aux$lambda, y = aux$smooth.periodogram)
sp_d <- data.frame(
  x = aux$lambda,
  y = spectral.density(ar = 0.7, lambda = aux$lambda)
)
g <- per$plot
g +
  geom_line(data = sm_p, aes(x, y), color = "#ff7f0e") +
  geom_line(data = sp_d, aes(x, y), color = "#d31244")

Spectral Density

Description

Returns theoretical spectral density evaluated in ARMA and ARFIMA processes.

Usage

spectral.density(ar = numeric(), ma = numeric(), d = 0, sd = 1, lambda = NULL)
spectral.density(ar = numeric(), ma = numeric(), d = 0, sd = 1, lambda = NULL)

Arguments

`ar`	(type: numeric) AR vector. If the time serie doesn't have AR term then omit it. For more details see the examples.
`ma`	(type: numeric) MA vector. If the time serie doesn't have MA term then omit it. For more details see the examples.
`d`	(type: numeric) Long-memory parameter. If d is zero, then the process is ARMA(p,q).
`sd`	(type: numeric) Noise scale factor, by default is 1.
`lambda`	(type: numeric) $\lambda$ parameter on which the spectral density is calculated/computed. If `lambda=NULL` then it is considered a sequence between 0 and $\pi$ .

Details

The spectral density of an ARFIMA(p,d,q) processes is

$f(\lambda) = \frac{\sigma^2}{2\pi} \cdot \bigg(2\, \sin(\lambda/2)\bigg)^{-2d} \cdot \frac{\bigg|\theta\bigg(\exp\bigg(-i\lambda\bigg)\bigg)\bigg|^2} {\bigg|\phi\bigg(\exp\bigg(-i\lambda\bigg)\bigg)\bigg|^2}$

With $-\pi \le \lambda \le \pi$ and $-1 < d < 1/2$ . $|x|$ is the Mod of $x$ . LSTS_sd returns the values corresponding to $f(\lambda)$ . When d is zero, the spectral density corresponds to an ARMA(p,q).

Value

An unnamed vector of numeric class.

References

Examples

# Spectral Density AR(1)
require(ggplot2)
f <- spectral.density(ar = 0.5, lambda = malleco)
ggplot(data.frame(x = malleco, y = f)) +
  geom_line(aes(x = as.numeric(x), y = as.numeric(y))) +
  labs(x = "Frequency", y = "Spectral Density") +
  theme_minimal()
# Spectral Density AR(1)
require(ggplot2)
f <- spectral.density(ar = 0.5, lambda = malleco)
ggplot(data.frame(x = malleco, y = f)) +
  geom_line(aes(x = as.numeric(x), y = as.numeric(y))) +
  labs(x = "Frequency", y = "Spectral Density") +
  theme_minimal()

Diagnostic Plots for Time Series fits

Description

Plot time-series diagnostics.

Usage

ts.diag(x, lag = 10, band = qnorm(0.975)/sqrt(length(x)))
ts.diag(x, lag = 10, band = qnorm(0.975)/sqrt(length(x)))

Arguments

`x`	(type: numeric) residuals of the fitted time series model.
`lag`	(type: numeric) maximum lag at which to calculate the acf and Ljung-Box test. By default set to 10.
`band`	(type: numeric) absolute value for bandwidth in the the ACF plot. By default set to 'qnorm(0.975)/sqrt(n)' which approximates to 0.07 for malleco data (n = 734)

Details

This function plot the residuals, the autocorrelation function of the residuals (ACF) and the p-values of the Ljung-Box Test for all lags up to lag.

Value

A ggplot object.

Examples

ts.diag(malleco)
ts.diag(malleco)

Package 'LSTS'

Help Index

Smooth Periodogram by Blocks

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Ljung-Box Test Plot

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Hessian Matrix

Description

Usage

Arguments

Details

Value

See Also

Examples

Kalman filter for locally stationary processes

Description

Usage

Arguments

Details

Value

References

Examples

Summary for Locally Stationary Time Series

Description

Usage

Arguments

Details

Value

See Also

Examples

Whittle estimator to Locally Stationary Time Series

Description

Usage

Arguments

Details

Value

See Also

Examples

Locally Stationary Whittle log-likelihood Function

Description

Usage

Arguments

Details

References

See Also

Locally Stationary Whittle Log-likelihood sigma

Description

Usage

Arguments

Details

Locally Stationary Whittle Log-likelihood theta

Description

Usage

Arguments

Details

Average Araucaria Araucana Tree Ring Width

Description

Format

Author(s)

Periodogram function

Description

Usage

Arguments

Details

Value

References