Koji Makiyama (@hoxo-m), Ameya Daigavane (@ameya98), and Krzysztof Mierzejewski (@mierzejk)

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) = 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 the inlier-based outlier detection [1] and covariate shift adaptation [2]. Other useful applications for density ratio estimation were summarized by Sugiyama et al. (2012) in [3].

The package densratio provides a function densratio() that returns an object with a method to estimate density ratio as compute_density_ratio().

Further, the alpha-relative density ratio p(x)/(alpha * p(x) + (1 - alpha) * q(x)) (where alpha is in the range [0, 1]) can also be estimated. When alpha is 0, this reduces to the ordinary density ratio w(x). The alpha-relative PE-divergence and KL-divergence between p(x) and q(x) are also computed.

For example,

import numpy as np
from scipy.stats import norm
from densratio import densratio

np.random.seed(1)
x = norm.rvs(size=500, loc=0, scale=1./8)
y = norm.rvs(size=500, loc=0, scale=1./2)
alpha = 0.1
densratio_obj = densratio(x, y, alpha=alpha)
print(densratio_obj)

gives the following output:

#> Method: RuLSIF
#> 
#> Alpha: 0.1
#> 
#> Kernel Information:
#>   Kernel type: Gaussian
#>   Number of kernels: 100
#>   Bandwidth(sigma): 0.1
#>   Centers: array([[-0.09591373],..
#> 
#> Kernel Weights (theta):
#>   array([0.04990797, 0.0550548 , 0.04784736, 0.04951904, 0.04840418,..
#> 
#> Regularization Parameter (lambda): 0.1
#> 
#> Alpha-Relative PE-Divergence: 0.618794133598705
#> 
#> Alpha-Relative KL-Divergence: 0.7037648129307483
#> 
#> 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). The code below gives the plot shown above.

from matplotlib import pyplot as plt
from numpy import linspace

def true_alpha_density_ratio(sample):
    return norm.pdf(sample, 0, 1./8) / (alpha * norm.pdf(sample, 0, 1./8) + (1 - alpha) * norm.pdf(sample, 0, 1./2))

def estimated_alpha_density_ratio(sample):
    return densratio_obj.compute_density_ratio(sample)

sample_points = np.linspace(-1, 3, 400)
plt.plot(sample_points, true_alpha_density_ratio(sample_points), 'b-', label='True Alpha-Relative Density Ratio')
plt.plot(sample_points, estimated_alpha_density_ratio(sample_points), 'r-', label='Estimated Alpha-Relative Density Ratio')
plt.title("Alpha-Relative Density Ratio - Normal Random Variables (alpha={:03.2f})".format(alpha))
plt.legend()
plt.show()

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.

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()

The package estimates density ratio by the RuLSIF method.

RuLSIF (Relative unconstrained Least-Squares Importance Fitting) estimates the alpha-relative density ratio by minimizing the squared loss between the true and estimated alpha-relative ratios. You can find more information in Hido et al. (2011) [1] and Liu et al (2013) [4].

The method assumes that the alpha-relative density ratio is represented by a linear kernel model:

w(x) = theta1 * K(x, c1) + theta2 * K(x, c2) + ... + thetab * K(x, cb) where K(x, c) = exp(- ||x - c||^2 / (2 * sigma ^ 2)) is the Gaussian RBF kernel.

densratio() performs the following: - Decides kernel parameter sigma by cross-validation. - Optimizes for kernel weights theta. - Computes the alpha-relative PE-divergence and KL-divergence from the learned alpha-relative ratio.

As the result, you can obtain compute_density_ratio(), which will compute the alpha-relative density ratio at the passed coordinates.

densratio() outputs the result like as follows:

#> Method: RuLSIF
#> 
#> Alpha: 0
#> 
#> Kernel Information:
#>   Kernel type: Gaussian
#>   Number of kernels: 100
#>   Bandwidth(sigma): 0.1
#>   Centers: array([[0.92113356],..
#> 
#> Kernel Weights (theta):
#>   array([0.08848922, 0.03377533, 0.0753727 , 0.06141277, 0.02543963,..
#> 
#> Regularization Parameter (lambda): 1.0
#> 
#> Alpha-Relative PE-Divergence: 0.9635169300831041
#> 
#> Alpha-Relative KL-Divergence: 0.838826626547327
#> 
#> Function to Estimate Density Ratio:
#>   compute_density_ratio(x)
#> 

• Method is fixed as RuLSIF.
• Kernel type is fixed as 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(theta) are theta parameters in the linear kernel model. You can find these values in result.theta.
• The function to estimate the alpha-relative density ratio is named compute_density_ratio().

When working out Gaussian kernels, linear algebra calculations can be done either with numpy or numba packages. The densratio.set_compute_kernel_target is a function that accepts a single str argument to globally target a specified engine:

• numpy - numpy broadcasting optimized. It must be noted the underlying BLAS library (e.g. Intel's MKL) can take advantage of multi threading model.
• cpu - numba generalized universal function single thread optimized.
• parallel - numba generalized universal function multi thread optimized. Please be advised all threading layer specifics apply.

densration defaults to cpu when numba is available (to take heed of multi threading technicalities), or numpy otherwise.

Although numba is not a requirement of densratio_py, its version ≥ 0.45.1 is necessary to set the calculation engine to cpu or parallel.

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, as long as their dimensions are the same.

For example,

from scipy.stats import multivariate_normal
from densratio import densratio

np.random.seed(1)
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]])
alpha = 0
densratio_obj = densratio(x, y, alpha=alpha, sigma_range=[0.1, 0.3, 0.5, 0.7, 1], lambda_range=[0.01, 0.02, 0.03, 0.04, 0.05])
print(densratio_obj)

gives the following output:

#> Method: RuLSIF
#> 
#> Alpha: 0
#> 
#> Kernel Information:
#>   Kernel type: Gaussian
#>   Number of kernels: 100
#>   Bandwidth(sigma): 0.3
#>   Centers: array([[1.01477443, 1.38864061],..
#> 
#> Kernel Weights (theta):
#>   array([0.06151164, 0.08012094, 0.10467369, 0.13868176, 0.14917063,..
#> 
#> Regularization Parameter (lambda): 0.04
#> 
#> Alpha-Relative PE-Divergence: 0.653615870855595
#> 
#> Alpha-Relative KL-Divergence: 0.6214285743087565
#> 
#> Function to Estimate Density Ratio:
#>   compute_density_ratio(x)
#> 

In this case, as well, 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_alpha_density_ratio(x):
    return multivariate_normal.pdf(x, [1., 1.], [[1. / 8, 0], [0, 1. / 8]]) / \
           (alpha * multivariate_normal.pdf(x, [1., 1.], [[1. / 8, 0], [0, 1. / 8]]) + (1 - alpha) * multivariate_normal.pdf(x, [1., 1.], [[1. / 2, 0], [0, 1. / 2]]))

def estimated_alpha_density_ratio(x):
    return densratio_obj.compute_density_ratio(x)

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

plt.figure(figsize=(10, 4))
plt.subplot(1, 2, 1)
plt.contourf(range_, range_, true_alpha_density_ratio(grid).reshape(200, 200), levels)
#> <matplotlib.contour.QuadContourSet object at 0x0000022E950202E0>
plt.colorbar()
#> <matplotlib.colorbar.Colorbar object at 0x0000022E9500DA80>
plt.title("True Alpha-Relative Density Ratio")
plt.subplot(1, 2, 2)
plt.contourf(range_, range_, estimated_alpha_density_ratio(grid).reshape(200, 200), levels)
#> <matplotlib.contour.QuadContourSet object at 0x0000022E942C8EE0>
plt.colorbar()
#> <matplotlib.colorbar.Colorbar object at 0x0000022E95095150>
plt.title("Estimated Alpha-Relative Density Ratio")
plt.show()

This is a supplementary function, that by applying the following linear transformation changes the lower 0 bound (infimum) of its values argument in the range of [0, 1] to [alpha, 1] :

The function returns an array whose values are elements of the [alpha, alpha^-1] bounded set; the upper alpha^-1 bound (supremum) is due to properties of the alpha-relative density ratio estimator. As long as the outcome is comprised of positive values only, when the densratio_obj.alpha_normalize function is applied to the densratio_obj.compute_density_ratio result, it renders the density ratio estimation eligible to be further processed on the logarithmic scale.

To preserve vital estimator properties, there are 2 invariants always met by the function results:

1. The number of unique values must not changed.
2. The order of output values remains the same as in the input argument (as determined by numpy.argsort).

Due to floating‑point arithmetic, if two consecutive input numbers that are not equal are to be transformed to the same outcome, in order to satisfy the constraints in the list above and yield different values as well, the numpy.nextafter is employed in the direction of 0.
By virtue of this implementation approach, in extreme cases there is a possibility of output values that:

• are less than alpha, or
• are nonpositive.

Should it happen, a relevant warning is issued.

[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.

[4] Liu, S., Yamada, M., Collier, N., & Sugiyama, M. Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation Neural Networks, 2013.

• densratio for R https://github.com/hoxo-m/densratio
• pykliep https://github.com/srome/pykliep