Kernel smoothing is a moving average smoother that uses a weight function, also referred to as kernel, to average the observations. The kernel smoothing function is estimated by

\[\hat f_t = \sum_{i=1}^b w_i(t)x_i\text{,}\]

where

\[w_i(t) = \frac{K \left(\frac{t-i}{b}\right)}{\sum_{i=1}^bK\left(\frac{t-i}{b}\right)}\]

are the weights and \(K(\cdot)\) is a kernel function, typically the normal kernel, \(K(z) = \frac{1}{\sqrt{2 \pi}}\exp(-z^2/2)\). The wider the bandwidth \(b\), the smoother the result. In R we apply the ksmooth() function for kernel smoothing.

library(xts)
load(url("https://userpage.fu-berlin.de/soga/data/r-data/Earth_Surface_Temperature.RData"))

dt <- index(temp_global)
y <- coredata(temp_global)
plot(dt, y,
  type = "l",
  col = "gray", xlab = "", ylab = "",
  main = "Kernel Smoothing"
)

lines(ksmooth(dt, y, "normal", bandwidth = 1),
  col = "red", type = "l"
)
lines(ksmooth(dt, y, "normal", bandwidth = 25),
  col = "green", type = "l"
)

legend("topleft",
  legend = c("b = 1", "b = 25"),
  col = c("red", "green"),
  lty = 1,
  cex = 0.6
)

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Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.