Graphic Comparison Of Discrete Kernels

library(kernopt)

Introduction

This vignette illustrates some functions provided by the kernopt package that implements discrete symmetric kernels, which were recently developed for modeling count data distributions.

The kernopt package computes that kernel at a target x for various values of observations z and a fixed bandwidth h by using the function discrete_kernel() included in this package.

Discrete Epanechnikov Kernel

For a positive integer \(h\in\mathbb{N}\setminus\{0\}\), the first example is a discrete symmetric Epanechnikov kernel with \(\mathcal{S}_{x}=\{x,...,x\pm h\}\) given by:

\[\begin{equation} K^{Epan}_{x,h}(y)= \left(a\bigg(\frac{x-y}{h}\bigg)^{2} + b\right){\bf 1}_{\{x,x\pm 1,...,x\pm h\}}(y), \end{equation}\]

with \(a=-b=3h/(1-4h^{2})\) (Chu, Henderson, and Parmeter 2017)

That kernel only involves the parameter \(h\), which is a positive integer that scales the distance between \(x\) and \(y\). The discrete Epanechnikov kernel is less often used in the literature, compared to its continuous counterpart.

Below is an illustration of the discrete Epanechnikov kernel for different values of h using the discrete_kernel()function.

x <- 5 # Target
z <- 0:10 # Observations (?)
h <- c(1, 2, 3, 4) # Set of Bandwidths
K_epan <- matrix(
  data = 0,
  nrow = length(z),
  ncol = length(h)
)
for (i in 1:length(h))
{
  K_epan[, i] <- discrete_kernel(kernel = "epanech", x, z, h[i])
}
plot(
  z,
  K_epan[, 1],
  xlab = "x",
  ylab = "Probability",
  ylim = c(0, 1),
  pch = 1
)
lines(z, K_epan[, 1], lty = 1)
for (i in 2:length(h))
{
  points(z, K_epan[, i], xlab = "z", pch = i)
  lines(z, K_epan[, i], lty = i)
}
legend(
  "topleft",
  c("h=1", "h=2", "h=3", "h=4"),
  lty = 1:4,
  pch = 1:4,
  cex = 1.6
)
Distribution of discrete symmetric Epanechnikov kernel at the target point x=5 for various bandwidth parameters h.
Distribution of discrete symmetric Epanechnikov kernel at the target point \(x=5\) for various bandwidth parameters \(h\).

Discrete Triangular Kernel

For \(h>0\) and non-negative integer \(a\in\mathbb{N}\), the second example is the discrete symmetric triangular kernel with \(\mathcal{S}_{x}=\{x,x\pm 1,...,x\pm a\}\), which has a pmf defined by \[\begin{equation*}\label{eq_TriangKern} K^{Triang}_{a;x,h}(y)= \frac{(a+1)^{h}}{C(a,h)}\left(1 - \frac{|y-x|^{h}}{(a+1)^{h}}\right){\bf 1}_{\{x,x\pm 1,...,x\pm a\}}(y), \end{equation*}\] with \(C(a,h)=(2a+1)(a+1)^{h} - 2\sum_{j=0}^{a}j^{h}\) (Kokonendji, Senga Kiessé, and Zocchi 2007). That kernel depends on two parameters \(h\) and \(a\). That kernel was implemented in the package through the function with its specific arguments as follows:

x <- 5
z <- 0:10
h <- c(0.1, 0.4, 1, 2)
a <- 1

K_trg <- matrix(
  data = 0,
  nrow = length(z),
  ncol = length(h)
)

for (i in 1:length(h))
{
  K_trg[, i] <- discrete_kernel(kernel = "triang", x, z, h[i], a)
}
plot(
  z,
  K_trg[, 1],
  xlab = "x",
  ylab = "Probability",
  ylim = c(0, 1),
  pch = 1
)
lines(z, K_trg[, 1], lty = 1)
for (i in 2:length(h))
{
  points(z, K_trg[, i], xlab = "z", pch = i)
  lines(z, K_trg[, i], lty = i)
}
legend(
  "topleft",
  c("h=0.1", "h=0.4", "h=1", "h=2"),
  lty = 1:4,
  pch = 1:4,
  cex = 1.6
)
Distribution of discrete symmetric triangular kernel (a=1) at the target point x=5 for various bandwidth parameters h.
Distribution of discrete symmetric triangular kernel (\(a=1\)) at the target point \(x=5\) for various bandwidth parameters \(h\).

The discrete symmetric triangular kernel is also available in the Ake package in R (Wansouwé, Somé, and Kokonendji 2016).

Discrete Optimal Kernel

The Discrete Optimal Kernel was proposed in (Senga Kiessé and Durrieu 2024).

For \(x\in \mathcal{S}\) and a fixed integer \(k\geq1\), the discrete symmetric “optimal” kernel on \(\mathcal{S}_{x}=\{x,x\pm 1,\ldots,x\pm k\}\) that minimises the mean integrate squared error of the estimator \(\widehat{f}_{K,h}\) in (\(\ref{eq:estim_fn}\)) is given by \[\begin{equation*}\label{eq:OptKern_k} K_{k;x,h}^{opt}(y) =\lambda\bigg(\frac{3k^{2}+3k-1}{5} -(x-y)^{2} \bigg)+\frac{h}{2k+1}, y\in\mathcal{S}_{x}, \end{equation*}\] where \(\lambda=15(1-h)/\big((2k+1)(4k^{2}+4k-3)\big)>0\) and \(3/5(1-1/k)<h<1\).

The bandwidth parameter \(h\) of the ``optimal’’ kernels \(K_{k;x,h}^{opt}\) is bounded with a lower bound \(h_{lower}= 3/5(1-1/k)\geq0\), which particularly depends on \(k\geq 1\). The distribution of that kernel can be plotted as follows:

x <- 5
z <- 0:10
h <- c(0.1, 0.4, 0.7, 0.9)
k <- 1

K_opt <- matrix(
  data = 0,
  nrow = length(z),
  ncol = length(h)
)

for (i in 1:length(h))
{
  K_opt[, i] <- discrete_kernel(kernel = "optimal", x, z, h[i], k)
}
plot(
  z,
  K_opt[, 1],
  xlab = "x",
  ylab = "Probability",
  ylim = c(0, 1),
  pch = 1
)
lines(z, K_opt[, 1], lty = 1)
for (i in 2:length(h))
{
  points(z, K_opt[, i], xlab = "z", pch = i)
  lines(z, K_opt[, i], lty = i)
}
legend(
  "topleft",
  c("h=0.1", "h=0.4", "h=0.7", "h=0.9"),
  lty = 1:4,
  pch = 1:4,
  cex = 1.6
)
Distribution of discrete symmetric “optimal” kernel (k=1) at the target point x=5 for various bandwidth parameters h.
Distribution of discrete symmetric “optimal” kernel (\(k=1\)) at the target point \(x=5\) for various bandwidth parameters \(h\).

Graphical comparison of the discrete kernels

oldpar <- par(mfrow = c(2, 2), mar = c(1, 1, 1, 1)) # 2 x 2 pictures on one plot
# 1,1 - Optimal (k=1, h) and Epanechnikov (h=1)
plot(
  x = z,
  # y = K_opt[, 1],
  xlab = "z",
  ylab = "Probability",
  ylim = c(0, 1),
  main = "Optimal (k=1, h) and Epanechnikov (h=1)",
  type = "o",
  pch = 1,
  # Symbol
  lty = 1,
  # Line type
  lwd = 2,
  # Line width
  cex = 1.6,
  # Magnification factor
  cex.axis = 1.2,
  # Magnification factor for axis
  cex.lab = 1.2,
  # Magnification factor for label
)

lines(
  z,
  K_opt[, 2],
  type = "o",
  pch = 2,
  # Symbol
  lty = 2,
  # Line type
  lwd = 2,
  # Line width
  col = "grey",
)

lines(
  z,
  K_opt[, 3],
  type = "o",
  pch = 3,
  lty = 3,
  lwd = 2
)


lines(
  z,
  K_epan[, 1],
  type = "o",
  pch = 4,
  lty = 4,
  lwd = 2,
  col = "grey"
)

legend(
  0,
  1,
  c("Opt. h=0.2", "Opt. h=0.7", "Opt. h=0.95", "Epan. h=1"),
  lty = c(1, 2, 3, 4),
  pch = c(1, 2, 3, 4),
  col = c("black", "grey", "black", "grey"),
  lwd = c(2, 2, 2, 2),
  cex = 1.2
)



# Triangular (a=1, h) and Epanechnikov (h=1)
z <- 0:10
x <- 5
a <- 1
h <- c(0.2, 0.7, 0.95)
K_trg <- matrix(0, length(z), length(h))
for (i in 1:length(h))
{
  K_trg[, i] <- discrete_triang(x, z, h[i], a)
}

plot(
  z,
  K_trg[, 1],
  xlab = "z",
  ylab = "Probability",
  main = "Triangular (a=1, h) and Epanechnikov (h=1)",
  ylim = c(0, 1),
  type = "o",
  pch = 1,
  lty = 1,
  lwd = 2,
  cex = 1.6,
  cex.axis = 1.2,
  cex.lab = 1.2
)

lines(
  z,
  K_trg[, 2],
  type = "o",
  pch = 2,
  lty = 2,
  lwd = 2,
  col = "grey"
)

lines(
  z,
  K_trg[, 3],
  type = "o",
  pch = 3,
  lty = 3,
  lwd = 2
)

lines(
  z,
  K_epan[, 1],
  type = "o",
  pch = 4,
  lty = 4,
  lwd = 2,
  col = "grey"
)

legend(
  0,
  1,
  c("Triang. h=0.2", "Triang. h=0.7", "Triang. h=0.95", "Epan. h=1"),
  lty = c(1, 2, 3, 4),
  pch = c(1, 2, 3, 4),
  col = c("black", "grey", "black", "grey"),
  lwd = c(2, 2, 2, 2),
  cex = 1.2
)

#  Optimal (k=2, h) and Epanechnikov (h=2)
# p=k=2
# opt
z <- 0:10
x <- 5
k <- 2
h <- c(0.5, 0.7, 0.95)
K_opt <- matrix(0, length(z), length(h))
for (i in 1:length(h))
{
  K_opt[, i] <- discrete_optimal(x, z, h[i], k)
}


plot(
  z,
  K_opt[, 1],
  xlab = "z",
  ylab = "Probability",
  ylim = c(0, 1),
  main = "Optimal (k=2, h) and Epanechnikov (h=2)",
  type = "o",
  pch = 1,
  lty = 1,
  lwd = 2,
  cex = 1.6,
  cex.axis = 1.2,
  cex.lab = 1.2
)

lines(
  z,
  K_opt[, 2],
  type = "o",
  pch = 2,
  lty = 2,
  lwd = 2,
  col = "grey"
)

lines(
  z,
  K_opt[, 3],
  type = "o",
  pch = 3,
  lty = 3,
  lwd = 2
)

lines(
  z,
  K_epan[, 2],
  type = "o",
  pch = 4,
  lty = 4,
  lwd = 2,
  col = "grey"
)

legend(
  0,
  1,
  c("Opt. h=0.5", "Opt. h=0.7", "Opt. h=0.95", "Epan. h=2"),
  lty = c(1, 2, 3, 4),
  pch = c(1, 2, 3, 4),
  col = c("black", "grey", "black", "grey"),
  lwd = c(2, 2, 2, 2),
  cex = 1.2
)

# Triangular (a=2, h) and Epanechnikov (h=2)
# triangular
z <- 0:10
x <- 5
a <- 2
h <- c(0.5, 0.7, 0.95)
K_trg <- matrix(0, length(z), length(h))
for (i in 1:length(h))
{
  K_trg[, i] <- discrete_triang(x, z, h[i], a)
}


plot(
  z,
  K_trg[, 1],
  xlab = "z",
  ylab = "Probability",
  main = "Triangular (a=2, h) and Epanechnikov (h=2)",
  ylim = c(0, 1),
  type = "o",
  pch = 1,
  lty = 1,
  lwd = 2,
  cex = 1.6,
  cex.axis = 1.2,
  cex.lab = 1.2
)

lines(
  z,
  K_trg[, 2],
  type = "o",
  pch = 2,
  lty = 2,
  lwd = 2,
  col = "grey"
)

lines(
  z,
  K_trg[, 3],
  type = "o",
  pch = 3,
  lty = 3,
  lwd = 2
)

lines(
  z,
  K_epan[, 2],
  type = "o",
  pch = 4,
  lty = 4,
  lwd = 2,
  col = "grey"
)

legend(
  0,
  1,
  c("Triang. h=0.5", "Triang. h=0.7", "Triang. h=0.95", "Epan. h=2"),
  lty = c(1, 2, 3, 4),
  pch = c(1, 2, 3, 4),
  col = c("black", "grey", "black", "grey"),
  lwd = c(2, 2, 2, 2),
  cex = 1.2
)


# Restore option
par(oldpar)
Chu, C-Y, D J Henderson, and C F Parmeter. 2017. “On Discrete Epanechnikov Kernel Functions.” Computational Statistics & Data Analysis 116: 79–105. https://doi.org/10.1016/j.csda.2017.07.003.
Kokonendji, CC, T Senga Kiessé, and S S Zocchi. 2007. “Discrete Triangular Distributions and Non-Parametric Estimation for Probability Mass Function.” Journal of Nonparametric Statistics 19 (6-8): 241–54. https://doi.org/10.1080/10485250701733747.
Senga Kiessé, Tristan, and Gilles Durrieu. 2024. “On a Discrete Symmetric Optimal Associated Kernel for Estimating Count Data Distributions.” Statistics & Probability Letters 208: 110078. https://doi.org/10.1016/j.spl.2024.110078.
Wansouwé, Wanbitching E., Sobom M. Somé, and Célestin C. Kokonendji. 2016. “Ake: An R Package for Discrete and Continuous Associated Kernel Estimations.” The R Journal 8 (2): 258–76. https://doi.org/10.32614/RJ-2016-045.