1. Overview
Density ratio estimation is described as follows: for given two data samples \(x1\) and \(x2\) from unknown distributions \(p(x)\) and \(q(x)\) respectively, estimate
\[
w(x) = \frac{p(x)}{q(x)}
\]
where \(x1\) and \(x2\) are \(d\)-dimensional real numbers.
The estimated density ratio function \(w(x)\) can be used in many applications such as anomaly detection [Hido et al. 2011], change-point detection [Liu et al. 2013], and covariate shift adaptation [Sugiyama et al. 2007]. Other useful applications about density ratio estimation were summarized by [Sugiyama et al. 2012].
The package densratio provides a function densratio() that returns an object with a method to estimate density ratio as compute_density_ratio().
For example,
The function densratio() estimates the density ratio of \(p(x)\) to \(q(x)\), \[
w(x) = \frac{p(x)}{q(x)} = \frac{\rm{Norm}(1, 1/8)}{\rm{Norm}(1, 1/2)}
\] and provides a function to compute estimated density ratio.
The densratio object has a function compute_density_ratio() that can compute density ratio \(\hat{w}(x) \simeq p(x)/q(x)\) for any \(d\)-dimensional input \(x\) (here \(d=1\)).
In this case, the true density ratio \(w(x) = p(x)/q(x) = \rm{Norm}(1, 1/8) / \rm{Norm}(1, 1/2)\) 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)
plot(true_density_ratio, xlim=c(0, 2), lwd=2, col="red", xlab = "x", ylab = "Density Ratio")
plot(densratio_obj$compute_density_ratio, xlim=c(0, 2), lwd=2, col="green", add=TRUE)
legend("topright", legend=c(expression(w(x)), expression(hat(w)(x))), col=2:3, lty=1, lwd=2, pch=NA)
3. Details
3.1. Basics
The package provides densratio(). The function returns an object that has a function to compute estimated density ratio.
For data samples x1 and x2,
In this case, densratio_obj$compute_density_ratio() can compute estimated density ratio.
3.2. Methods
densratio() has method argument that you can pass "uLSIF", "RuSLIF", 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 [Kanamori et al. 2009] and [Hido et al. 2011].
- RuLSIF (Relative unconstrained Least-Squares Importance Fitting). This algorithm estimates relative density ratio by minimizing the squared loss. You can find more information in [Yamada et al. 2011] and [Liu et al. 2013].
- KLIEP (Kullback-Leibler Importance Estimation Procedure). This algorithm estimates density ratio by minimizing Kullback-Leibler divergence. You can find more information in [Sugiyama et al. 2007].
The methods assume that density ratio are represented by linear model:
\[
w(x) = \theta_1 K(x, c_1) + \theta_2 K(x, c_2) + ... + \theta_b K(x, c_b)
\]
where
\[
K(x, c) = \exp\left(-\frac{\|x - c\|^2}{2 \sigma ^ 2}\right)
\]
is the Gaussian (RBF) kernel.
densratio() performs the following:
- Decides kernel parameter \(\sigma\) by cross-validation,
- Optimizes the kernel weights \(\theta\) (in other words, find the optimal coefficients of the linear model), and
- The parameters \(\sigma\) and \(\theta\) are saved into
densratio object, and are used when to compute density ratio in the call compute_density_ratio().
3.3. Result and Arguments
You can display information of densratio objects. Moreover, you can change some conditions to specify arguments of densratio().
##
## Call:
## densratio(x1 = x1, x2 = x2, method = "uLSIF")
##
## Kernel Information:
## Kernel type: Gaussian
## Number of kernels: 100
## Bandwidth (sigma): 0.1
## Centers: num [1:100, 1] 0.907 1.093 1.18 1.136 1.046 ...
##
## Kernel Weights:
## num [1:100] 0.067455 0.040045 0.000459 0.016849 0.067084 ...
##
## Regularization Parameter (lambda): 1
##
## Function to Estimate Density Ratio:
## compute_density_ratio()
- Kernel type is fixed as Gaussian.
- Number of kernels is the number of kernels in the linear model. You can change by setting
kernel_num argument. In default, kernel_num = 100.
- Bandwidth (sigma) is the Gaussian kernel bandwidth. In default,
sigma = "auto", the algorithm automatically select an optimal value by cross validation. If you set sigma a number, that will be used. If you set sigma a numeric vector, the algorithm select an 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
x1 underlying a numerator distribution \(p(x)\). You can find the whole values in result$kernel_info$centers.
- Kernel Weights are
theta parameters in the linear kernel model. You can find these values in result$kernel_weights.
- Function to Estimate Density Ratio is named
compute_density_ratio().
4. Multi Dimensional Data Samples
So far, the input data samples x1 and x2 were one dimensional. densratio() allows to input multidimensional data samples as matrix, as long as their dimensions are the same.
For example,
##
## Call:
## densratio(x1 = x1, x2 = x2, method = "uLSIF")
##
## Kernel Information:
## Kernel type: Gaussian
## Number of kernels: 100
## Bandwidth (sigma): 0.316
## Centers: num [1:100, 1:2] 1.257 0.758 1.122 1.3 1.386 ...
##
## Kernel Weights:
## num [1:100] 0.0756 0.0986 0.059 0.0797 0.0421 ...
##
## Regularization Parameter (lambda): 0.3162278
##
## Function to Estimate Density Ratio:
## compute_density_ratio()
In this case, as well, we can compare the true density ratio with the estimated density ratio.
true_density_ratio <- function(x) {
dmvnorm(x, mean = c(1, 1), sigma = diag(1/8, 2)) /
dmvnorm(x, mean = c(1, 1), sigma = diag(1/2, 2))
}
N <- 20
range <- seq(0, 2, length.out = N)
input <- expand.grid(range, range)
w_true <- matrix(true_density_ratio(input), nrow = N)
w_hat <- matrix(densratio_obj_d2$compute_density_ratio(input), nrow = N)
par(mfrow = c(1, 2))
contour(range, range, w_true, main = "True Density Ratio")
contour(range, range, w_hat, main = "Estimated Density Ratio")
