# testOTM

testOTM is an R package that computes multivariate ranks and quantiles defined through the theory of optimal transportation. It also provides several applications of these statistics, most notably a method for two-sample multivariate goodness-of-fit testing.

## Installation

You can install the released version of testOTM from CRAN with:

install.packages("testOTM")


You can install the development version from GitHub with:

# install.packages("devtools")
# devtools::install_github("Francis-Hsu/testOTM")


## Example

This is a basic example which shows you how to use testOTM to visualize the optimal transport map from $U[0, 1]^2$ to a (scaled) bivariate normal sample:

library(testOTM)

# generate bivariate normal data
p = 2
n = 100
Sigma = matrix(c(2, 1, 1, 2), 2, 2)
eS = eigen(Sigma, symmetric = TRUE)
X = t(eS$vectors %*% diag(sqrt(pmax(eS$values, 0)), p) %*% matrix(rnorm(p * n), p))

# compute the optimal transport map from U[0, 1]^2 to the data
# notice that the data will be scale into [0, 1] range
X.OTM = tos.fit(X)

# plot the restricted Voronoi diagram and the restricted Delaunay triangulation
par(mfrow = c(1, 2))
plot(X.OTM, which = "Both", draw.center = F, draw.map = T)


## Acknowledgment

The author is extremely grateful to Prof. Bodhisattva Sen and his student Promit Ghosal for their guidance in the development of this package. The author would also like to thank Dr. Bruno Lévy for his assistance with the Geogram library, and the TraME team, whose Rgeogram package provides inspirations to the early build of this package.

## Reference

• Aurenhammer, F. 1987. “Power Diagrams: Properties, Algorithms and Applications.” SIAM Journal on Computing 16 (1): 78–96. https://doi.org/10.1137/0216006.
• Brenier, Yann. 1991. “Polar Factorization and Monotone Rearrangement of Vector-Valued Functions.” Communications on Pure and Applied Mathematics 44 (4): 375–417. https://doi.org/10.1002/cpa.3160440402.
• Cheng, Siu-Wing, Tamal Krishna Dey, and Jonathan Richard Shewchuk. 2013. Delaunay Mesh Generation. Chapman; Hall/CRC.
• Chernozhukov, Victor, Alfred Galichon, Marc Hallin, and Marc Henry. 2017. “Monge–Kantorovich Depth, Quantiles, Ranks and Signs.” The Annals of Statistics 45 (1): 223–56. https://doi.org/10.1214/16-AOS1450.
• Ghosal, Promit, and Bodhisattva Sen. 2019. “Multivariate Ranks and Quantiles Using Optimal Transportation and Applications to Goodness-of-Fit Testing.” http://arxiv.org/abs/1905.05340.
• Inria, project ALICE-GRAPHYS. 2019. “Geogram: A Programming Library of Geometric Algorithms.” http://alice.loria.fr/software/geogram/doc/html/index.html.
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• Liu, Yang, Wenping Wang, Bruno Lévy, Feng Sun, Dong-Ming Yan, Lin Lu, and Chenglei Yang. 2009. “On Centroidal Voronoi Tessellation–Energy Smoothness and Fast Computation.” ACM Transactions on Graphics 28 (4): Article 101. https://doi.org/10.1145/1559755.1559758.
• McCann, Robert J. 1995. “Existence and Uniqueness of Monotone Measure-Preserving Maps.” Duke Math. J. 80 (2): 309–23. https://doi.org/10.1215/S0012-7094-95-08013-2.
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