In a wide spectrum of applied fields, data is present as multivariate
circular data (toroidal data), rendering spurious the application of
many standard statistical tools. The package ridgetorus
implements Toroidal Ridge PCA (TR-PCA), an alternative to PCA suited to
toroidal data (García-Portugués and Prieto-Tirado
2023). Below are some simple examples of the use of TR-PCA
through ridge_pca(), the main function in
ridgetorus.
Texan wind
The following example illustrates the application of TR-PCA to a
small dataset comprised by wind directions measured at 6:00 and 7:00
from June 1, 2003 to June 30, 2003 at a weather station in Texas. The
direction is measured in radians in \([-\pi,
\pi)\) with \(-\pi/-\frac{\pi}{2}/0/\frac{\pi}{2}/\pi\)
representing the East/South/West/North/East directions.
library(ridgetorus)
#> Loading required package: Rcpp
data("wind")
plot(wind, xlim = c(-pi, pi), ylim = c(-pi, pi), axes = FALSE,
xlab = expression(theta[1]), ylab = expression(theta[2]))
sdetorus::torusAxis()
TR-PCA works by first modeling the data with the best-fitting model
among a Bivariate Wrapped Cauchy (Kato and Pewsey
2015) and a Bivariate Sine von Mises (Singh, Hnizdo, and Demchuk 2002). Then, the
density ridge (see Genovese et al. (2014))
is computed and its connected component through the mode(s) of the
distribution is extracted. In this case, this connected component
explains \(62\%\) of the variance of
the original data. The plot below shows how the scores are computed, and
how their signs are assigned. The blue asterisk stands for the Fréchet
mean of the sample through TR-PCA.
# Fit TR-PCA
fit <- ridge_pca(x = wind)
show_ridge_pca(fit)
# Variance explained
fit$var_exp
#> [1] 0.6191477 1.0000000
The produced scores allow the identification of outliers and
“rectify” the original sample.
plot(fit$scores, xlim = c(-pi, pi), ylim = c(-pi, pi), pch = 16, axes = FALSE,
xlab = "Scores 1", ylab = "Scores 2")
sdetorus::torusAxis()
Japanese earthquakes
The following dataset represents pre-earthquake direction of steepest
descent and the direction of lateral ground movement after an earthquake
in Noshiro, Japan. We use TR-PCA to unveil the behavior and explain the
variability of the earthquake phenomena.
data("earthquakes")
earthquakes <- sdetorus::toPiInt(earthquakes)
plot(earthquakes, xlim = c(-pi, pi), ylim = c(-pi, pi),
xlab = expression(theta[1]), ylab = expression(theta[2]), axes = FALSE)
sdetorus::torusAxis()
The dataset is modeled with the best-fitting model among a Bivariate
Wrapped Cauchy and a Bivariate Sine von Mises. Once the distribution is
determined, the connected component of the density ridge can be
computed. This flexible first principal component explains \(66\%\) of the variance and shows a clear
linear trend between both studied angles.
fit_earthquakes <- ridge_pca(x = earthquakes)
fit_earthquakes$var_exp
#> [1] 0.6554334 1.0000000
show_ridge_pca(fit_earthquakes)
plot(fit_earthquakes$scores, pch = 16, xlim = c(-pi, pi), axes = FALSE,
ylim = c(-pi, pi), xlab = "Scores 1", ylab = "Scores 2")
sdetorus::torusAxis()
