adjusted_est: Compute the Kernel Regression Estimator.

In CovEsts: Nonparametric Estimators for Covariance Functions

Compute the Kernel Regression Estimator.

Description

This function computes the kernel regression estimator of the autocovariance function.

Usage

adjusted_est(
  X,
  x,
  t,
  b,
  kernel_name = c("gaussian", "wave", "rational_quadratic", "bessel_j"),
  kernel_params = c(),
  pd = TRUE,
  type = c("autocovariance", "autocorrelation"),
  meanX = mean(X),
  custom_kernel = FALSE,
  parallel = FALSE,
  ncores = parallel::detectCores() - 1,
  cl_export = NULL,
  cl = NULL
)

Arguments

X	A vector representing observed values of the time series.
x	A vector of lags.
t	The arguments at which the autocovariance function is calculated at.
b	Bandwidth parameter, greater than 0.
kernel_name	The name of the symmetric kernel (see kernel_symm_ec) function to be used. Possible values are: gaussian, wave, rational_quadratic, and bessel_j. Alternatively, a custom kernel function can be provided, see the examples.
kernel_params	A vector of parameters of the kernel function. See kernel_symm_ec for parameters.
pd	Whether a positive-definite estimate should be used. Defaults to TRUE.
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.
parallel	Whether or not the computations should be done in parallel or not. Defaults to FALSE.
ncores	The number of cores to be used in the parallel computations. Defaults to the number cores - 1 (threads if hyperthreading is available), calculated from parallel::detectCores() - 1.
cl_export	A vector of any additional functions or variables to export for parallel computations. This may be required if estimator is not within the package. Defaults to NULL.
cl	An optional cluster object created by parallel::makeCluster. Defaults to NULL, which creates a temporary PSOCK cluster.

Details

The kernel regression estimator of an autocovariance function is defined as

\hat{\rho}(t) = \left( \sum_{i=1}^{N} \sum_{j=1}^{N} \check{X}_{ij} K((t - (t_{i} - t_{j})) / b) \right) \left( \sum_{i=1}^{N} \sum_{j=1}^{N} K((t - (t_{i} - t_{j})) / b) \right)^{-1},

where \check{X}_{ij} = (X(t_{i}) - \bar{X}) (X(t_{j}) - \bar{X}).

If pd is TRUE, the estimator will be made positive-definite through the following procedure

1. Take the discrete Fourier cosine transform, \widehat{\mathcal{F}}(\theta), of the estimated autocovariance function

2. Compute a modified spectrum \widetilde{\mathcal{F}}(\theta) = \max(\widehat{\mathcal{F}}(\theta), 0) for all sample frequencies.

3. Perform the Fourier inversion to obtain a new estimator.

Value

A CovEsts S3 object (list) with the following values

acf

A numeric vector containing the autocovariance/autocorrelation estimates.

lags

A numeric vector containing the lag indices used to compute the estimates on.

est_type

The type of estimate, namely 'autocorrelation' or 'autocovariance', this depends on the type parameter.

est_used

The estimator function used, in this case, 'adjusted_est'.

References

Hall, P. & Patil, P. (1994). Properties of nonparametric estimators of autocovariance for stationary random fields. Probability Theory and Related Fields 99(3), 399-424. https://doi.org/10.1007/bf01199899

Hall, P., Fisher, N. I., & Hoffmann, B. (1994). On the nonparametric estimation of covariance functions. The Annals of Statistics 22(4), 2115-2134. https://doi.org/10.1214/aos/1176325774

Examples

X <- c(1, 2, 3, 4)
adjusted_est(X, 1:4, 1:3, 0.1, "gaussian")
my_kernel <- function(x, params) {
  stopifnot(params[1] > 0, length(x) >= 1)
  return(exp(-((abs(x) / params[1])^params[2])) * (2 * params[1] * gamma(1 + 1/params[2])))
}
adjusted_est(X, 1:4, 1:3, 0.1, my_kernel, c(0.25), custom_kernel = TRUE)
## Not run: 
library(parallel)
X <- c(1, 2, 3, 4)
my_cl <- makePSOCKcluster(2)
adjusted_est(X, 1:4, 1:3, 0.1, "gaussian", parallel = TRUE, cl = my_cl)
stopCluster(my_cl)

## End(Not run)

CovEsts documentation built on April 19, 2026, 5:06 p.m.