solve_spline: Objective Function for WLS.

In CovEsts: Nonparametric Estimators for Covariance Functions

Objective Function for WLS.

Description

This is the objective function to find the weights for each basis function in the minimising spline, see Choi, Li & Wang (2013, p. 617). The parameters must be nonnegative, so a penalty of 10^{12} is given if any parameters are negative. The weights are chosen as per Choi, Li & Wang (2013, p. 617).

Usage

solve_spline(par, splines_df, weights)

Arguments

par	A vector of initial parameters to used in the minimisation process.
splines_df	A data frame whose structure is defined in splines_df.
weights	A vector of weights, see the description.

Details

Let \mathbf{\beta} = (\beta_{0}, \dots, \beta_{m + p})^{\prime} be a vector of model coefficients, \{f_{1}^{(p - 1)} , \dots , f_{m + p}^{(p - 1)} \} be a set of completely monotone basis functions, and \widehat{C}(\cdot) be an estimated covariance function. As per Choi, Li & Wang (2013, p. 617), \mathbf{\beta} can be estimated via weighted-least squares,

\ \hat{\mathbf{\beta}}_{WLS} = {\arg\min}_{\beta_{j} \ge 0} \sum_{i=1}^{L} w_{i} \left(\widehat{C}(h_{i}) - \sum_{j = 1}^{m + p} \beta_{j} f_{j}^{(p - 1)}(h_{i}^{2}) \right)^{2} ,

where \{h_{1} , \dots , h_{L} \} is a set of lags and \{w_{1}, \dots , w_{L} \} is a set of weights. The set of weights is calculated in splines_est, and they are of the form w_{i} = (N - h_{i}) / ((1 - \widehat{C}(h_{i}))^{2}).

Value

The value of the objective function at those parameters.

References

Choi, I., Li, B. & Wang, X. (2013). Nonparametric Estimation of Spatial and Space-Time Covariance Function. JABES 18, 611-630. https://doi.org/10.1007/s13253-013-0152-z

Examples

taus <- get_taus(3, 2)
x <- seq(0, 2, by=0.25)
maxLag <- 4
splines_df <- splines_df(x[1:maxLag], 3, 2, taus)
splines_df$estCov <- exp(-splines_df$lags^2) + 0.001
# pars are the inital parameters used in the minimisation process.
pars <- c(0.5, 0.5, 0.5, 0.5, 0.5)
weights <- c()
X <- rnorm(50)
for(i in 0:(maxLag - 1)) {
  weights <- c(weights, (length(X) - i) / ( (1 - splines_df$estCov[i + 1])^2 ))
}
solve_spline(pars, splines_df, weights)

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