chenxingqiang / densratio_py

A Python Package for Density Ratio Estimation

A Python Package for Density Ratio Estimation

Koji MAKIYAMA (@hoxo_m)

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) = 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,

from numpy import random
from scipy.stats import norm
from densratio import densratio

random.seed(1)
x = norm.rvs(size = 200, loc = 1, scale = 1./8)
y = norm.rvs(size = 200, loc = 1, scale = 1./2)
result = densratio(x, y)
print(result)

#> Method: uLSIF
#> 
#> Kernel Information:
#>   Kernel type: Gaussian RBF
#>   Number of kernels: 100
#>   Bandwidth(sigma): 0.1
#>   Centers: array([ 1.04719547, 0.98294441, 1.06190142, 1.14145367, 1.18276349,..
#> 
#> Kernel Weights(alpha):
#>   array([ 0.25340775, 0.40951303, 0.27002649, 0.17998821, 0.14524305,..
#> 
#> Regularization Parameter(lambda): 0.1
#> 
#> The Function to Estimate Density Ratio:
#>   compute_density_ratio(x)

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

from matplotlib import pyplot as plt
from numpy import linspace

def true_density_ratio(x):
    return norm.pdf(x, 1, 1./8) / norm.pdf(x, 1, 1./2)

def estimated_density_ratio(x):
    return result.compute_density_ratio(x)

x = linspace(-1, 3, 400)
plt.plot(x, true_density_ratio(x), "r-", lw=5, alpha=0.6, label="w(x)")
plt.plot(x, estimated_density_ratio(x), "k-", lw=2, label="w-hat(x)")
plt.legend(loc="best", frameon=False)
plt.xlabel("x")
plt.ylabel("Density Ratio")
plt.show()

2. How to Install

You can install the package from PyPI.

$ pip install densratio

Also, You can install the package from GitHub.

$ pip install git+https://github.com/hoxo-m/densratio_py.git

The source code for densratio package is available on GitHub at

• https://github.com/hoxo-m/densratio_py.

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 x and y,

from scipy.stats import norm
from densratio import densratio

x = norm.rvs(size = 200, loc = 1, scale = 1./8)
y = norm.rvs(size = 200, loc = 1, scale = 1./2)
result = densratio(x, y)

In this case, result.compute_density_ratio() can compute estimated density ratio.

from matplotlib import pyplot as plt

density_ratio = result.compute_density_ratio(y)

plt.plot(y, density_ratio, "o")
plt.xlabel("x")
plt.ylabel("Density Ratio")
plt.show()

3.2. The Method

The package estimates density ratio by the uLSIF method.

• 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].

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

w(x) = alpha1 * K(x, c1) + alpha2 * K(x, c2) + ... + alphab * K(x, cb)

where K(x, c) = exp(- ||x - c||^2 / (2 * sigma ^ 2)) 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 Parameter Settings

densratio() outputs the result like as follows:

#> Method: uLSIF
#> 
#> Kernel Information:
#>   Kernel type: Gaussian RBF
#>   Number of kernels: 100
#>   Bandwidth(sigma): 0.1
#>   Centers: array([ 1.04719547, 0.98294441, 1.06190142, 1.14145367, 1.18276349,..
#> 
#> Kernel Weights(alpha):
#>   array([ 0.25340775, 0.40951303, 0.27002649, 0.17998821, 0.14524305,..
#> 
#> Regularization Parameter(lambda): 0.1
#> 
#> The Function to Estimate Density Ratio:
#>   compute_density_ratio(x)

• Method is fixed by uLSIF.
• 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 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 array, 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 x underlying a numerator distribution p(x). You can find the whole values in result.kernel_info.centers.
• Kernel weights(alpha) are alpha parameters in the linear model. It is optimaized by the algorithm. 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

So far, we have deal with one-dimensional data samples x and y. densratio() allows to input multidimensional data samples as numpy.ndarray or numpy.matrix.

For example,

from numpy import random
from scipy.stats import multivariate_normal
from densratio import densratio

random.seed(3)
x = multivariate_normal.rvs(size = 3000, mean = [1, 1], cov = [[1./8, 0], [0, 1./8]])
y = multivariate_normal.rvs(size = 3000, mean = [1, 1], cov = [[1./2, 0], [0, 1./2]])
result = densratio(x, y)
print(result)

#> Method: uLSIF
#> 
#> Kernel Information:
#>   Kernel type: Gaussian RBF
#>   Number of kernels: 100
#>   Bandwidth(sigma): 0.316227766017
#>   Centers: array([[ 1.06481062, 1.09768368],..
#> 
#> Kernel Weights(alpha):
#>   array([ 0.19009526, 0.18680918, 0.20764146, 0.19914001, 0.21313236,..
#> 
#> Regularization Parameter(lambda): 0.0316227766017
#> 
#> The Function to Estimate Density Ratio:
#>   compute_density_ratio(x)

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

from matplotlib import pyplot as plt
from numpy import linspace, dstack, meshgrid, concatenate

def true_density_ratio(x):
    return multivariate_normal.pdf(x, [1., 1.], [[1./8, 0], [0, 1./8]]) / \
           multivariate_normal.pdf(x, [1., 1.], [[1./2, 0], [0, 1./2]])

def estimated_density_ratio(x):
    return result.compute_density_ratio(x)

range_ = linspace(0, 2, 200)
grid = concatenate(dstack(meshgrid(range_, range_)))
levels = [0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4.5]

plt.subplot(1, 2, 1)
plt.contourf(range_, range_, true_density_ratio(grid).reshape(200, 200), levels)
plt.colorbar()
plt.title("True Density Ratio")
plt.subplot(1, 2, 2)
plt.contourf(range_, range_, estimated_density_ratio(grid).reshape(200, 200), levels)
plt.colorbar()
plt.title("Estimated Density Ratio")
plt.show()

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.