corrected_est: Kernel Correction of the Standard Estimator.

In CovEsts: Nonparametric Estimators for Covariance Functions

Kernel Correction of the Standard Estimator.

Description

This function computes the standard autocovariance estimator and applies kernel correction to it,

\widehat{C}_{T}^{(a)}(h) = \widehat{C}(h) a_{T}(h),

where a_{T}(h) := a(h / N_{T}). It uses a kernel a(\cdot) which decays or vanishes to zero (depending on the type of kernel) where a(0) = 1. The rate or value at which the kernel vanishes is N_{T}, which is recommended to be of order 0.1 N, where N is the length of the observation window, however, one may need to play with this value.

Usage

corrected_est(
  X,
  kernel_name,
  kernel_params = c(),
  N_T = 0.1 * length(X),
  pd = TRUE,
  maxLag = length(X) - 1,
  type = "autocovariance",
  meanX = mean(X),
  custom_kernel = FALSE
)

Arguments

X	A vector representing observed values of the time series.
kernel_name	The name of the kernel function to be used. Possible values are: gaussian, exponential, wave, rational_quadratic, spherical, circular, bessel_j, matern, cauchy.
kernel_params	A vector of parameters of the kernel function. See kernel for parameters. In the case of gaussian, wave, rational_quadratic, spherical and circular, N_T takes the place of \theta. For kernels that require parameters other than \theta, such as the Matern kernel, those parameters are passed.
N_T	The range at which the kernel function vanishes at. Recommended to be 0.1 N when considering all lags. This parameter may be large for a lag small estimation lag.
pd	Whether a positive-definite estimate should be used. Defaults to TRUE.
maxLag	An optional parameter that determines the maximum lag to compute the estimated autocovariance function at. Defaults to length(X) - 1.
type	Compute either the 'autocovariance' or 'autocorrelation'. Defaults to 'autocovariance'.
meanX	The average value of X. Defaults to mean(X).
custom_kernel	If a custom kernel is to be used or not. Defaults to FALSE. See examples.

Details

The aim of this estimator is gradually bring the estimated values to zero through the use of a kernel multiplier. This can be useful when estimating an autocovariance function that is short-range dependent as estimators can have large fluctuations as the lag increases, or to deal with the wave artefacts for large lags, see Bilchouris and Olenko (2025). This estimator can be positive-definite depending on whether the choice of \widehat{C}(\cdot) and a are chosen to be positive-definite or not.

Value

A vector whose values are the kernel corrected autocovariance estimates.

References

Yaglom, AM (1987). Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results. Springer New York. https://doi.org/10.1007/978-1-4612-4628-2

Bilchouris, A. & Olenko, A (2025). On Nonparametric Estimation of Covariogram. Austrian Statistical Society (Vol. 54, Issue 1). https://doi.org/10.17713/ajs.v54i1.1975

Examples

X <- c(1, 2, 3)
corrected_est(X, "gaussian")

X <- rnorm(1000)
Y <- c(X[1], X[2])
for(i in 3:length(X)) { Y[i] <- X[i] - 0.3*X[i - 1] - 0.6*X[i - 2] }
plot(Y)
plot(corrected_est(Y, "bessel_j",
     kernel_params=c(0, 1), N_T=0.2*length(Y)))

# Custom kernel
my_kernel <- function(x, theta, params) {
  stopifnot(theta > 0, length(x) >= 1, all(x >= 0))
  return(sapply(x, function(t) ifelse(t == 0, 1,
         ifelse(t == Inf, 0,
         (sin((t^params[1]) / theta) / ((t^params[1]) / theta)) * cos((t^params[2]) / theta)))))
}
plot(corrected_est(Y,
     my_kernel, kernel_params=c(2, 0.25), custom_kernel = TRUE))

CovEsts documentation built on Sept. 10, 2025, 10:39 a.m.