The densityratio package estimates the density ratio r(x) = p(x) / q(x) from two-samples x1 and x2 generated from two unknown distributions p(x), q(x), respectively, where x1 and x2 are d-dimensional real numbers.
The densityratio provides a densratio() class that returns an object with a function for estimating the density ratio and methods for getting the parameters of the density ratio estimation.
See the quick start for detailed usage.
The estimated density ratio function r(x) can be used in many applications such as the covariate shift adaptation, outlier detection, and change point detection. See related works.
You can install the package from GitHub
$ pip install git+https://github.com/JohnYKiyo/density_ratio_estimation.git
densityratio requires:
- Python (>= 3.6)
- Jax (>=0.1.57)
- Jaxlib (>=0.1.37)
The densityratio package includes the works (Jax, Jaxlib) that are distributed in the Apache License 2.0
Generate two samples that follow the normal distribution of N(0,1) and N(1,2), respectively.
The code below gives the above output:
import densityratio
import numpy as np
import scipy.stats
import matplotlib.pyplot as plt
m = [0.,1]
s = [1.0,2.0]
np.random.seed(10)
x1 = np.random.normal(loc= m[0], scale = s[0], size = 500)
x2 = np.random.normal(loc= m[1], scale = s[1], size = 500)
fig = plt.figure(figsize=[5,3])
ax = fig.add_subplot(111)
ax.set_title('distribution of samples')
edges = np.arange(-10,10,0.2)
a = ax.hist(x1, bins = edges, color='b', alpha = 0.5, label='x1')
b = ax.hist(x2, bins = edges, color='m', alpha = 0.5, label='x2')
ylim = 1.1*np.max([a[0],b[0]])
ax.set_ylim(0,ylim)
ax.legend(fontsize = 10, bbox_to_anchor = (1,1), loc='upper right', borderaxespad = 1)
plt.tight_layout()
fig.savefig('pic/QS1_samples.png')Pass two samples to the densratio and it will be calculated automatically.
dens = densityratio.densratio(x1,x2)In this case, the true density ratio r(x) is known, so we can compare r(x) with the estimated density ratio \hat{r}(x).
The code below gives the above output:
x = np.linspace(-10,10,100)
x1_pdf = lambda x: scipy.stats.norm.pdf(x,loc=m[0],scale=s[0])
x2_pdf = lambda x: scipy.stats.norm.pdf(x,loc=m[1],scale=s[1])
r = lambda x: x1_pdf(x)/x2_pdf(x)
plt.figure()
plt.plot(x,r(x),'r',label='true ratio')
plt.plot(x,dens(x),'gray',label='estimated ratio')
plt.xlim(-10,10)
plt.legend()
plt.savefig('pic/QS1_ratio.png')Here is this notebook
For example, Generate two samples that follow the 2-dimentional normal distribution of N(\mu_1,\Sigma_1) and N(\mu_2,\Sigma_2), respectively, and estimate the relative-densityratio [7]
r_\alpha (x) = p(x)/(\alpha p(x)+(1-\alpha)q(x)).
The code below gives the above output:
import densityratio
import numpy as np
import pandas as pd
import scipy.stats
import matplotlib.pyplot as plt
import seaborn as sns
m = [[0.,0.],[0,0.5]]
s = [np.array([[0.5,0],
[0,0.5]]),
np.array([[2.0,0.5],
[0.5,2.0]])]
np.random.seed(10)
x1 = np.random.multivariate_normal(m[0], s[0], size = 2000)
x2 = np.random.multivariate_normal(m[1], s[1], size = 2000)
df = pd.DataFrame({'x':np.r_[x1[:,0],x2[:,0]],'y':np.r_[x1[:,1],x2[:,1]],'label':['x1']*len(x1)+['x2']*len(x2)})
fig = plt.figure(figsize=(9,4))
ax = fig.add_subplot(111)
sns.scatterplot(data=df, x='x', y='y', alpha=0.5,hue='label',ax=ax)
fig.savefig('pic/QS2_samples.png')If a value is set for sigma, lambda, that value is used, and if a numerical array is set, the densityratio() selects the optimum value by CV.
The alpha is a parameter that can adjust the mixing ratio, and is set in the range of [0,1.].
The kernel_number is the number of kernels in the linear model.
dens = densityratio.densratio(x1,x2,sigma=[0.1, 0.3, 0.5, 0.7, 1., 2., 5.], lamb=[0.01, 0.02, 0.03, 0.04, 0.05],kernel_num=200,alpha=0.2)In this case, the true density ratio r(x) with the estimated density ratio \hat{r}(x).
The code below gives the above output:
X,Y = np.meshgrid(np.linspace(-5,5,100),np.linspace(-5,5,100))
data = np.c_[X.ravel(),Y.ravel()]
x1_pdf = lambda x: scipy.stats.multivariate_normal.pdf(x,m[0],s[0])
x2_pdf = lambda x: scipy.stats.multivariate_normal.pdf(x,m[1],s[1])
Z_true = x1_pdf(data)/(0.2*x1_pdf(data)+0.8*x2_pdf(data))
fig = plt.figure(figsize=[8,4])
ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)
levels=np.arange(0,4,0.5)
ax1.set_title(r'true $\alpha$ relative-densityratio: $\alpha=0.2$')
ax2.set_title(r'estimated $\alpha$ relative-densityratio: $\alpha=0.2$')
a = ax1.contourf(X,Y,Z_true.reshape(100,100),levels=levels)
b = ax2.contourf(X,Y,dens(np.c_[X.ravel(),Y.ravel()]).reshape(100,100),levels=levels)
fig.colorbar(a, ax=ax1)
fig.colorbar(b, ax=ax2)
fig.savefig('pic/QS2_ratio.png')Here is this notebook
See here
MIT License (see LICENSE).
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