Applications on Real Data

SIMTAP project aimed to develop sustainable aquaculture production production system that, in particular, contributes to reduce fish feed inputs and resources consumption.
We used data from an experiment in which gilthead seabream (sparus aurata) were stocked in 1.6 m³ tanks at a density of 1.5 kg⋅m⁻³. Fish were reared for 46 days in a single recirculating aquaculture system composed of three rearing tanks. At the beginning of the experiment, a number n = 200 of the fish were individually weighed (dg), and their length at the caudal fork (mm) was measured.

The nonparametric kernel estimator was applied to provide smoothed count distributions of weight and length of fish measured. The performance of the nonparametric kernel estimator for modeling the empirical frequencies f₀ was evaluated by using Integrated Squared Error (ISE) given by ISE(h_cv) = ∑_{x ∈ ℕ}(f̂_{K, h_cv}(x) − f₀(x))².

# Data
data("fish_data", package = "kernopt")
y <- fish_data$weight
data1 <- as.data.frame(table(y))
x <- as.numeric(as.character(data1[, 1]))
freq_weight <- as.numeric(as.character(data1[, 2]))
f0 <- freq_weight / sum(freq_weight)

# Optimal kernel
H <- seq((max(y) - min(y)) / 200, (max(y) - min(y)) / 2, length.out = 50)
k <- 1
hcv_opt_k1 <- cv_bandwidth(kernel = "optimal", y, H, k = 1)
fn_opt_k1 <- estim_kernel(kernel = "optimal", x, hcv_opt_k1, y, k = 1)
ISE_opt_k1 <- sum((fn_opt_k1 - f0)^2)

# Epanechnikov
H <- seq(2, 10, 1)
hcv_epanech <- cv_bandwidth(kernel = "epanech", y, H)
fn_epanech <- estim_kernel(kernel = "epanech", x, hcv_epanech, y, k = NULL)
ISE_epanech <- sum((fn_epanech - f0)^2)

# Graph
oldpar <- par(mfrow = c(1, 2))
plot(
  x,
  f0,
  col = "black",
  axes = F,
  lwd = 3,
  ylab = "Probability",
  xlab = "Weight (dg)",
  ylim = c(0, 0.06),
  xlim = c(41, 132),
  type = "h",
  main = "(a) Optimal k=1 - ISE=0.002, (hcv=0.94)",
  cex.axis = 1.70,
  cex.lab = 1.70
)
axis(1,
  at = x,
  cex.axis = 1.70,
  cex.lab = 1.70
)
axis(2)
box()
points(
  x + 0.4,
  fn_opt_k1,
  lwd = 3,
  col = "grey",
  lty = 1,
  type = "h"
)
plot(
  x,
  f0,
  col = "black",
  axes = F,
  lwd = 3,
  ylab = "Probability",
  xlab = "Weight (dg)",
  ylim = c(0, 0.06),
  xlim = c(41, 132),
  type = "h",
  main = "(b) Epanechnikov - ISE=0.0033  (hcv=9)",
  cex.axis = 1.70,
  cex.lab = 1.70
)
axis(1,
  at = x,
  cex.axis = 1.70,
  cex.lab = 1.70
)
axis(2)
box()
points(
  x + 0.4,
  fn_epanech,
  lwd = 3,
  col = "grey",
  lty = 1,
  type = "h"
)

$Estimates (gray lines) of count distributions of weight (dg) (black lines) of fish by using optimal and Epanechnikov discrete kernels with bandwidth parameter ($h_{cv}$) from the cross-validation procedure. Integrated Squared Error (ISE) was also calculated.$

Estimates (gray lines) of count distributions of weight (dg) (black lines) of fish by using optimal and Epanechnikov discrete kernels with bandwidth parameter (h_cv) from the cross-validation procedure. Integrated Squared Error (ISE) was also calculated.

Applications on Real Data

Acknowledgments