densratio - An R Package for Density Ratio Estimation

1. Overview

Density ratio estimation is described as follows: for given two data samples $x$ and $y$ from unknown distributions $p_{nu}(x)$ and $p_{de}(y)$ respectively, estimate

\[ w(x) = \frac{p_{nu}(x)}{p_{de}(x)} \]

where $x$ and $y$ are $d$-dimensional real numbers.

The estimated density ratio function $w(x)$ can be used in many applications such as the inlier-based outlier detection[1], covariate shift adaptation[2] and etc[3].

The package densratio provides a function densratio() that returns a result has the function to estimate density ratio compute_density_ratio().

For example,

set.seed(3)
x <- rnorm(200, mean = 1, sd = 1/8)
y <- rnorm(200, mean = 1, sd = 1/2)

library(densratio)
result <- densratio(x, y)
result

## 
## Call:
## densratio(x = x, y = y, method = "uLSIF")
## 
## Kernel Information:
##   Kernel type:  Gaussian RBF 
##   Number of kernels:  100 
##   Bandwidth(sigma):  0.1 
##   Centers:  num [1:100, 1] 1.007 0.752 0.917 0.824 0.7 ...
## 
## Kernel Weights(alpha):
##   num [1:100] 0.4044 0.0479 0.1736 0.125 0.0597 ...
## 
## Regularization Parameter(lambda):  0.1 
## 
## The Function to Estimate Density Ratio:
##   compute_density_ratio()

In this case, the true density ratio $w(x)$ is known, so we can compare $w(x)$ with the estimated density ratio $\hat{w}(x)$.

true_density_ratio <- function(x) dnorm(x, 1, 1/8) / dnorm(x, 1, 1/2)
estimated_density_ratio <- result$compute_density_ratio

plot(true_density_ratio, xlim=c(-1, 3), lwd=2, col="red", xlab = "x", ylab = "Density Ratio")
plot(estimated_density_ratio, xlim=c(-1, 3), lwd=2, col="green", add=TRUE)
legend("topright", legend=c(expression(w(x)), expression(hat(w)(x))), col=2:3, lty=1, lwd=2, pch=NA)

2. How to Install

You can install the densratio package from CRAN.

install.packages("densratio")

You can also install the package from GitHub.

install.packages("devtools") # if you have not installed "devtools" package
devtools::install_github("hoxo-m/densratio")

The source code for densratio package is available on GitHub at

https://github.com/hoxo-m/densratio.

3. Details

3.1. Basics

The package provides densratio() that the result has the function to estimate density ratio.

For data samples x and y,

library(densratio)

result <- densratio(x, y)

In this case, result$compute_density_ratio() is the function to compute estimated density ratio.

3.2. Methods

densratio() has method parameter that you can pass "uLSIF" or "KLIEP".

uLSIF (unconstrained Least-Squares Importance Fitting) is the default method. This method estimates density ratio by minimizing the squared loss. You can find more information in Hido(2011)[1].
KLIEP (Kullback-Leibler Importance Estimation Procedure) is the anothor method. This method estimates density ratio by minimizing Kullback-Leibler divergence. You can find more information in Sugiyama(2007)[2].

The both methods assume that the denity ratio is represented by linear model:

\[ w(x) = \alpha_1 K(x, c_1) + \alpha_2 K(x, c_2) + ... + \alpha_b K(x, c_b) \]

where

\[ K(x, c) = \exp\left(-\frac{\|x - c\|^2}{2 \sigma ^ 2}\right) \]

is the Gaussian RBF.

densratio() performs the two main jobs:

First, deciding kernel parameter $\sigma$ by cross validation,
Second, optimizing kernel weights $\alpha$.

As the result, you can obtain compute_density_ratio().

3.3. Result and Paremeter Settings

densratio() outputs the result like as follows:

## 
## Call:
## densratio(x = x, y = y, method = "uLSIF")
## 
## Kernel Information:
##   Kernel type:  Gaussian RBF 
##   Number of kernels:  100 
##   Bandwidth(sigma):  0.1 
##   Centers:  num [1:100, 1] 1.007 0.752 0.917 0.824 0.7 ...
## 
## Kernel Weights(alpha):
##   num [1:100] 0.4044 0.0479 0.1736 0.125 0.0597 ...
## 
## Regularization Parameter(lambda):  0.1 
## 
## The Function to Estimate Density Ratio:
##   compute_density_ratio()

Kernel type is fixed by Gaussian RBF.
Number of kernels is the number of kernels in the linear model. You can change by setting kernel_num parameter. In default, kernel_num = 100.
Bandwidth(sigma) is the Gaussian kernel bandwidth. In default, sigma = "auto", the algorithms automatically select the optimal value by cross validation. If you set sigma a single number, it will be used. If you set a numeric vector, the algorithms select the optimal value in them by cross validation.
Centers are centers of Gaussian kernels in the linear model. These are selected at random from the data sample x underlying a numerator distribution p_nu(x). You can find the whole values in result$kernel_info$centers.
Kernel weights is the alpha parameters in the linear model. It was optimaized by the algorithms. You can find the whole values in result$alpha.
The funtion to estimate density ratio is named compute_density_ratio().

4. Multi Dimensional Data Samples

In the above, the input data samples x and y were one dimensional. densratio() allows to input multidimensional data samples as matrix.

For example,

library(densratio)
library(mvtnorm)

set.seed(71)
x <- rmvnorm(300, mean = c(1, 1), sigma = diag(1/8, 2))
y <- rmvnorm(300, mean = c(1, 1), sigma = diag(1/2, 2))

result <- densratio(x, y)
result

## 
## Call:
## densratio(x = x, y = y, method = "uLSIF")
## 
## Kernel Information:
##   Kernel type:  Gaussian RBF 
##   Number of kernels:  100 
##   Bandwidth(sigma):  0.316 
##   Centers:  num [1:100, 1:2] 1.178 0.863 1.453 0.961 0.831 ...
## 
## Kernel Weights(alpha):
##   num [1:100] 0.145 0.128 0.138 0.187 0.303 ...
## 
## Regularization Parameter(lambda):  0.1 
## 
## The Function to Estimate Density Ratio:
##   compute_density_ratio()

Also in this case, we can compare the true density ratio with the estimated density ratio.

true_density_ratio <- function(x) {
  dmvnorm(x, mean = c(1, 1), sigma = diag(1/8, 2)) /
    dmvnorm(x, mean = c(1, 1), sigma = diag(1/2, 2))
}
estimated_density_ratio <- result$compute_density_ratio

N <- 20
range <- seq(0, 2, length.out = N)
input <- expand.grid(range, range)
z_true <- matrix(true_density_ratio(input), nrow = N)
z_hat <- matrix(estimated_density_ratio(input), nrow = N)

old_par <- par(mfrow = c(1, 2))
contour(range, range, z_true, main = "True Density Ratio")
contour(range, range, z_hat, main = "Estimated Density Ratio")

par(old_par)

The dimensions of x and y must be same.