# ChenMengjie/DepthDescent

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### Introduction

DepthDescent implements an algorithm to compute the deepest matrix estimator proposed in https://arxiv.org/abs/1506.00691. The core idea of the algorithm is to start with some robust scatter matrix estimator that is easy to compute, and then iteratively evaluate its depth on some direction subset of finite cardinality to approximate its matrix depth function and move towards the directions that can improve the depth.

### Installation

DepthDescent relies on the following R packages: Rcpp and RcppArmadillo. These package need to be pre-installed.

```install.packages ("Rcpp")

DepthDescent can be installed from github directly as follows:

```install.packages ("devtools")
library(devtools)
install_github("ChenMengjie/DepthDescent")```

### Tutorial

```p <- 10.
Sigma1 <- matrix(0, nrow = p, ncol = p)
for(i in 1:p){
for(j in 1:p){
Sigma1[i, j] <- 4*(0.5)^abs(i-j)
}
} # Simulated covariance matrix

require(MASS)
per <- 0.1 # Percentage of contamination
indicators <- sample(c(0, 1), n, replace = TRUE, prob = c(per, 1 - per))
outlier <- length(indicators[indicators == 0])
Z1 <- mvrnorm(n-outlier, rep(0, p), Sigma1)
if(per!=0){
Z2 <- matrix(10, ncol = p, nrow = outlier)
} else {
Z <- Z1
}

library(DepthDescent)
depth <- matrix_depth_by_descent(Z, n, p, K = 500, ntry = 500) #main function to obtain the scatter matrix
beta <- qnorm(3/4)^2  ### factor convert to covariance matrix for normal distribution, for t distribution beta = 0.72673^2
best <- depth[]
best <- best/beta  ### convert to covariance matrix
print(paste0("The operator norm estimation error is ", round(norm(best-Sigma1, "F"), 3)))```

K is the number of directions to use. ntry is the number of iterations to run.

### Author

Mengjie Chen (UChicago)

Bug report, comments or questions please send to mengjiechen@uchicago.edu.

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