kare function

Kernel Adaptive Regression Estimator

Wrapper function which does some preparatory calculations and then calls the actual ``workhorse'' functions which do the main computations for kernel adaptive regression estimation of Eichner & Stute (2012). Finally, it structures and returns the obtained results. Summarizing information and technical details can be found in Eichner (2017).

kare(x.points, data, kernel = c("gaussian", "epanechnikov", "rectangular"),
  Sigma = seq(0.01, 10, length = 51), h = NULL, theta = NULL)

Arguments

• x.points: Vector of location(s) at which the regression estimate is to be computed.

• data: Data frame or list with one component named x which contains the vector of regressor values $x_1, \ldots, x_n$

  and one named y which holds the vector of pertaining response values $y_1, \ldots, y_n$ (in the corresponding order) of the data from which the estimate is to be computed at the values given in x.points. Pairs $(x_i, y_i)$

  with NA or an infinite value in a least one of their elements are removed (and a warning is issued).

• kernel: A character string naming the kernel to be used for the adaptive estimator. This must partially match one of "gaussian", "rectangular" or "epanechnikov", with default "gaussian", and may be abbreviated to a unique prefix. (Currently, this kernel is also used for the initial, non-adaptive Nadaraya-Watson regression estimator which enters into the estimators of bias and variance as described in the references.)

• Sigma: Vector of value(s) of the scale parameter $\sigma$. If of length 1 no adaptation is performed. Otherwise considered as the grid over which the optimization of the adaptive method will be performed. Defaults to seq(0.01, 10, length = 51).

• h: Numeric scalar for bandwidth $h$. Defaults to NULL and is then internally set to $n^{-1/5}$.

• theta: Numeric scalar for value of location parameter $\theta$. Defaults to NULL and is then internally set to the arithmetic mean of $x_1, \ldots, x_n$.

Returns

If length(x.points) = 1, a list of eight components with the following names and meanings:

x	Scalar $x$ -value in x.points at which the regression estimator was computed.
y	Estimated scalar value of $m(x)$ at point in x.points .
sigma.adap	The found scalar minimizer of the MSE-estimator, i.e., the adaptive smoothing parameter value.
msehat.min	The found scalar minimum of the MSE-estimator.
Sigma	Vector with the $\sigma$ -grid on which the minimization process was performed.
Bn	Vector with the estimator of bias on that $\sigma$ -grid.
Vn2	Ditto for the variance.
MSE	Ditto for the MSE.

If length(x.points) > 1, a list with the same component names as above, but then

x	Vector x.points with x-values at which the regression estimator was computed.
y	Vector of estimated values of $m(x)$ at the x-values in x.points .
sigma.adap	Vector of the found minimizers of the MSE-estimator, i.e., the adaptive smoothing parameter values.
msehat.min	Vector of the found minima of the MSE-estimator.
Sigma	Vector with the $\sigma$ -grid on which the minimization process was performed.
Bn	( length(Sigma) by length(x.points) )-matrix with the estimated values of the bias on the $\sigma$ -grid in their columns (which correspond to the x-values).
Vn2	Ditto for the variance.
MSE	Ditto for the MSE.

Examples

require(stats)

 # Regression function:
m <- function(x, x1 = 0, x2 = 8, a = 0.01, b = 0) {
 a * (x - x1) * (x - x2)^3 + b
}
 # Note: For a few details on m() see examples in ?nadwat.

x0 <- 5   # The point x_0 at which the MSE-optimal kernel adjusted
 # nonparametric estimation of m should take place. (Recall: for m's
 # default values a minimum is at 2, a point of inflection at 4, and
 # a saddle point an 8; an "arbitrary" point would, e.g., be at 5.)

n <- 100   # Sample size.
sdeps <- 1   # Std. dev. of the \epsilon_i: \sqrt(Var(Y|X=x))
             # (here: constant in x).
design.ctr <- x0 + 0.5   # "centre" and "scale" of the design, i.e.,
design.scl <- 1  # in the normal scenario below, expected value and
                 # std. dev. of the distribution of the x_i's.

set.seed(42)   # To guarantee reproducibility.
x <- rnorm(n, mean = design.ctr, sd = design.scl)   # x_1, ..., x_n
Y <- m(x) + rnorm(length(x), sd = sdeps)            # Y_1, ..., Y_n
data <- data.frame(x = x, y = Y)

 # Computing the kernel adaptive regression estimator values
 #**********************************************************
x.points <- seq(-3.3 * design.scl, 3.3 * design.scl, length = 101) +
   design.ctr  # x-grid on which to draw and estimate the regr. fct. m.

Sigma <- seq(0.01, 10, length = 51)   # \sigma-grid for minimization.
fit <- kare(x.points = x0, data = data, Sigma = Sigma)

## Not run:

 # Grafical display for the current data set
 #******************************************
 # Storing the curent settings of the graphics device
 # and changing its layout for the three plots to come:
op <- par(mfrow = c(3, 1), mar = c(3, 3, 2, 0.1)+0.1,
   mgp = c(1.5, 0.5, 0), tcl = -0.3, cex.main = 2)

 # The scatter plot of the "raw data":
plot(y ~ x, data = data, xlim = range(data$x, x.points),
   ylim = range(data$y, fit$y, na.rm = TRUE),
   main = bquote(n == .(n)), xlab = "x", ylab = "y")

 # The "true" regression function m:
lines(x.points, m(x.points), lty = 2)

 # The MSE-optimal kernel adjusted nonparametric regression estimator
 # at x_0, i.e., the point (x_0, \hat m_n(x_0)):
points(fit$x, fit$y, col = "red", pch = 4, cex = 2)

 # The legend for the "true" regression function m and for the point
 # (x_0, \hat m_n(x_0)):
legend("topleft", lty = c(2, NA), pch = c(NA, 4),
 col = c("black", "red"), bty = "n", cex = 1.2,
 legend = c(as.expression(bquote(paste("m  with  ",
                                       sigma(paste(Y, "|", X == x))
                                 == .(sdeps)))),
            as.expression(bquote(paste(hat(m)[n](x[0]), "  at  ",
                                       x[0] == .(x0))))))

 # Visualizing the estimators of (Bias_n(sigma))^2 and
 # Var_n(sigma) at x0 on the sigma-grid:
with(fit,
  matplot(Sigma, cbind(Bn*Bn, Vn2), type = "l", lty = 1:2,
   col = c("black", "red"), xlab = expression(sigma), ylab = ""))

 # The legend for (Bias_n(sigma))^2 and Var_n(sigma):
legend("topleft", lty = 1:2, col = c("black", "red"), bty = "n",
  legend = c(expression(paste(widehat(plain(Bias))[n]^2, (sigma))),
             expression(widehat(plain(Var))[n](sigma))),
  cex = 1.2)

 # Visualizing the estimator of MSE_n(sigma) at x0 on the sigma-grid
 # together with the point indicating the detected minimum, and a legend:
plot(fit$Sigma, fit$MSE, type = "l",
 xlab = expression(sigma), ylab = "")
points(fit$sigma.adap, fit$msehat.min, pch = 4, col = "red", cex = 2)
legend("topleft", lty = c(1, NA), pch = c(NA, 4),
 col = c("black", "red"), bty = "n", cex = 1.2,
 legend = c(expression(widehat(plain(MSE))[n](sigma)),
            substitute(group("(", list(plain(Minimizer),
                                       plain(Minimum)), ")")
                         == group("(", list(x, y), ")"),
                       list(x = signif(fit$sigma.adap, 4),
                            y = signif(fit$msehat.min, 4)))))

par(op) # Restoring the previous settings of the graphics device.
## End(Not run)

References

Eichner & Stute (2012) and Eichner (2017): see kader.