## 1. Overview

Density ratio estimation is described as follows: for given two data samples $x$ and $y$ from unknown distributions $p(x)$ and $q(y)$ respectively, estimate $$w(x) = \frac{p(x)}{q(x)}$$ 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] and covariate shift adaptation [2]. Other useful applications about density ratio estimation were summarized by Sugiyama et al. (2012) [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 ...
##
## 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)

w_hat <- result$compute_density_ratio(y) plot(y, w_hat) ### 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 algorithm estimates density ratio by minimizing the squared loss. You can find more information in Hido et al. (2011) [1]. • KLIEP (Kullback-Leibler Importance Estimation Procedure) is the anothor method. This algorithm estimates density ratio by minimizing Kullback-Leibler divergence. You can find more information in Sugiyama et al. (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): ## ## The Function to Estimate Density Ratio: ## compute_density_ratio() • Kernel type is fixed by Gaussian RBF. • The 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 number, that 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 are alpha parameters in the linear model. It is 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)

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

The dimensions of x and y must be same.

## 5. References

[1] Hido, S., Tsuboi, Y., Kashima, H., Sugiyama, M., & Kanamori, T. Statistical outlier detection using direct density ratio estimation. Knowledge and Information Systems 2011.

[2] Sugiyama, M., Nakajima, S., Kashima, H., von Bünau, P. & Kawanabe, M. Direct importance estimation with model selection and its application to covariate shift adaptation. NIPS 2007.

[3] Sugiyama, M., Suzuki, T. & Kanamori, T. Density Ratio Estimation in Machine Learning. Cambridge University Press 2012.