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Kernel Density Estimation with Increasing Thickening

Generate sample data from a function

set.seed(42) x <- seq(-3, 3, length.out = 200) y <- sin(x) + rnorm(200, sd = 0.1) # Function with noise

Apply kernel smoothing with different thickening (bandwidths)

h_seq <- c(0.1, 0.3, 0.6) # Increasing kernel width = thicker approximation

Plot thickened approximations

plot(x, y, pch = 16, col = rgb(0, 0, 0, 0.2), main=”Function Approximation by three Kernels”) for (h in h_seq) { smoothed <- ksmooth(x, y, kernel=”normal”, bandwidth=h) lines(smoothed, col=rgb(1, 0, 0, 0.5)) } legend(“bottomright”, legend=paste(“h =”, h_seq), col=rgb(1, 0, 0, 0.5), lwd=2, bty = “n”)

Dynamic thickening

Generate sample data

set.seed(42) x <- seq(-3, 3, length.out = 200) # X values y <- sin(x) + rnorm(200, sd = 0.1) # Function with noise

Define a function that applies thickening (smoothing) with h

thickened_function <- function(x, y, h) { smoothed <- ksmooth(x, y, kernel = “normal”, bandwidth = h) return(smoothed) }

Define a range of h values for thickening

h_values <- seq(0.05, 0.6, length.out = 10) # Gradual thickening

Plot original noisy data

plot(x, y, pch = 16, col = rgb(0, 0, 0, 0.2), main = “Function Approximation with Thickening”, xlab = “x”, ylab = “Smoothed Value”)

Apply thickening dynamically by iterating over h

for (h in h_values) { smoothed <- thickened_function(x, y, h) lines(smoothed, col = rgb(1, 0, 0, h/0.6), lwd = 2) # Increasing opacity with h }

Add legend

legend(“bottomright”, legend = sprintf(“h = %.2f”, h_values[c(1, 5, 10)]), col = rgb(1, 0, 0, h_values[c(1, 5, 10)]/0.6), lwd = 2, bty = “n”, title=”bloody kernel”)

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