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https://github.com/egarpor/ridgetorus

PCA on the torus using density ridges. Software companion for "Toroidal PCA via density ridges"
https://github.com/egarpor/ridgetorus

circular-statistics dimensionality-reduction directional-statistics r

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PCA on the torus using density ridges. Software companion for "Toroidal PCA via density ridges"

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README 

        
---

output:

  md_document:

    variant: gfm

---



```{r, opts, echo = FALSE}

knitr::opts_chunk$set(

  collapse = TRUE, comment = "#>", fig.path = "README/README-",

  message = FALSE, warning = FALSE, fig.asp = 1, fig.align = 'center'

)

```



ridgetorus

==========



[![License:

GPLv3](https://img.shields.io/badge/license-GPLv3-blue.svg)](https://www.gnu.org/licenses/gpl-3.0)

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status](https://github.com/egarpor/ridgetorus/workflows/R-CMD-check/badge.svg)](https://github.com/egarpor/ridgetorus/actions)

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[![](https://www.r-pkg.org/badges/version/ridgetorus?color=green)](https://cran.r-project.org/package=ridgetorus)

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[![](http://cranlogs.r-pkg.org/badges/last-month/ridgetorus)](https://cran.r-project.org/package=ridgetorus)



## Overview



Implementation of principal component analysis on the two-dimensional torus $\mathbb{T}^2=[-\pi,\pi)^2$ via density ridges. Software companion for the paper "*Toroidal PCA via density ridges*" (García-Portugués and Prieto-Tirado, 2023).



## Installation



Get the latest version from GitHub:



```{r, install-devtools, eval = FALSE}

# Install the package

library(devtools)

install_github("egarpor/ridgetorus")



# Load package

library(ridgetorus)

```



```{r, load, echo = FALSE}

# Load package

library(ridgetorus)

```



## Usage



The main functionality of `ridgetorus` is the function `ridge_pca()`, which can be employed to do dimension reduction via the bivariate sine von Mises (Singh et al., 2002) and the bivariate wrapped Cauchy (Kato and Pewsey, 2015) models, as the following examples show.



### Bivariate sine von Mises



```{r, bvm}

# 1. Simulate data from r_bvm()

data <- r_bvm(n = 1000, mu = c(1, 2), kappa = c(5, 2, 1.5))



# 2. Do ridge_pca()

rpca <- ridge_pca(x = data, type = "bvm")



# 3. Plot simulated data with ridge fit using show_ridge_pca()

show_ridge_pca(rpca, col_data = "red")



# 4. Plot pairs plots of original data and scores with torus_pairs()

torus_pairs(data, col_data = "red", bwd = "EMI")

torus_pairs(rpca$scores, col_data = "red", bwd = "EMI", scales = rpca$scales)

```



### Bivariate wrapped Cauchy



```{r, bwc}

# 1. Simulate data from r_bwc()

data <- r_bwc(n = 1000, mu = c(-1, 2), xi = c(0.3, 0.6, 0.25))



# 2. Do ridge_pca()

rpca <- ridge_pca(x = data, type = "bwc")



# 3. Plot simulated data with ridge fit using show_ridge_pca()

show_ridge_pca(rpca, col_data = "red")



# 4. Plot pairs plots of original data and scores with torus_pairs()

torus_pairs(rpca$scores, col_data = "red", bwd = "EMI", scales = rpca$scales)

```



## Data application in oceanography



The data applications in García-Portugués and Prieto-Tirado (2023) can be reproduced through the script [data-application.R](https://github.com/egarpor/egarpor/blob/master/application/data-application.R). The code snippet below illustrates the toroidal PCA analysis onto currents of four zones at Santa Barbara strait. Zone A and B are on the northern coast of Santa Barbara Channel while zone C and D, are at the top and bottom ends of the interisland channel.



```{r, santabarbara}

# Load data

data("santabarbara")



# Example with zone A-B with automatic comparison between bvm and bwc

rpca_AB <- ridge_pca(x = santabarbara[c("A", "B")], type = "auto")

show_ridge_pca(fit = rpca_AB, col_data = "black", n_max = 1e3)

torus_pairs(santabarbara[c("A", "B")], col_data = "black")

torus_pairs(rpca_AB$scores, col_data = "black", scales = rpca_AB$scales)

rpca_AB$type

rpca_AB$var_exp



# Example with zone C-D with automatic comparison between bvm and bwc

rpca_CD <- ridge_pca(x = santabarbara[c("C", "D")], type = "auto")

show_ridge_pca(fit = rpca_CD, col_data = "black", n_max = 1e3)

torus_pairs(santabarbara[c("C", "D")], col_data = "black")

torus_pairs(rpca_CD$scores, col_data = "black", scales = rpca_CD$scales)

rpca_CD$type

rpca_CD$var_exp

```



It can be seen how the bivariate von Mises and the bivariate wrapped Cauchy are the most adequate fits for zones C--D and A--B, respectively. Toroidal PCA explains around 75% of the total variance in both cases, motivating its use for dimension reduction. The scores also transform the data distribution, reducing noise and allowing to check for groups or outliers, if any.



## References



García-Portugués, E. and Prieto-Tirado, A. (2023). Toroidal PCA via density ridges. *Statistics and Computing*, 33(5):107. [doi:10.1007/s11222-023-10273-9](https://doi.org/10.1007/s11222-023-10273-9).



Kato, S. and Pewsey, A. (2015). A Möbius transformation-induced distribution on the torus. *Biometrika*, 102(2):359--370. [doi:10.1093/biomet/asv003](https://doi.org/10.1093/biomet/asv003).



Singh, H., Hnizdo, V., and Demchuk, E. (2002). Probabilistic model for two dependent circular variables. *Biometrika*, 89(3):719--723. [doi:10.1093/biomet/89.3.719](https://doi.org/10.1093/biomet/89.3.719).