thetakde 0.3.2

A package for the theta kernel density estimator

Theta Kernel Density Estimation

This repository contains a Python implementation of the fast Theta Kernel Density Estimator (KDE) from [2] with the optimal bandwidth selection method from [1].

Introduction

Kernel Density Estimation (KDE) is a non-parametric method for estimating an unknown probability density function from a dataset of independent and identically distributed observations. The theta KDE [2] allows estimation of densities on bounded intervals in a way that achieves consistent estimation at the boundaries. It is highly efficient due to its use of the discrete cosine transform (DCT). A judicious choice of the bandwidth parameter ensures that the theta KDE avoids both oversmoothing and undersmoothing. This is accomplished through direct minimization of the mean integrated square error (MISE), as described in [1].

Function

kde(data, m=2**16, MIN=None, MAX=None, sig=None, resamp=False)

This function performs the theta kernel density estimation on the input data.

Parameters:

• data (numpy array): Input data for density estimation.
• m (int, optional): Number of grid points. Default is 2**16.
• MIN (float, optional): Minimum value of the data range.
• MAX (float, optional): Maximum value of the data range.
• sig (float, optional): Bandwidth parameter. If not provided, it will be optimized.
• resamp (bool, optional): Set to True if the data has repeated values, e.g., due to resampling. Default is False.

Returns:

• density (numpy array): Estimated density values.
• mesh (numpy array): Grid points corresponding to the density values.
• sig (float): Optimized bandwidth parameter.

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Usage

Install the ThetaKDE package from the Python Package Index to access thetakde module:

pip install ThetaKDE

Example

Generate data from an exponential distribution and compare the KDE with the true PDF.

import numpy as np
from matplotlib import pyplot as plt
from thetakde import kde

# Generate exponential data
data = -np.log(np.random.uniform(size=5000))

# Estimate density
density, mesh, sig = kde(data)

# Plot
plt.plot(mesh, density,'r', label='KDE')
plt.plot(mesh, np.exp(-mesh), 'b--', label='True PDF')
plt.legend()
plt.show()

References

[1] Z.I. Botev, D.P. Kroese, and T. Taimre (2025). Data Science and Machine Learning: Mathematical and Statistical Methods, Second Edition. Chapman & Hall/CRC, Boca Raton.

[2] Z.I. Botev, J.F. Grotowski, and D.P. Kroese (2010). Kernel density estimation via diffusion. Annals of Statistics, 38(5), 2916–2957.

@Book{DSML2e,
  author = {Z. I. Botev and D.P. Kroese and T. Taimre},
  title = {Data Science and Machine Learning: Mathematical and Statistical Methods},
  publisher = {Chapman \&  Hall/CRC},
  year = {2025},
  address = {Boca Raton},
  edition = {2nd}
}

@ARTICLE{thetakde,
  author = {Z. I. Botev and J. F. Grotowski and D. P. Kroese},
  title = {{Kernel density estimation via diffusion}},
  journal = {Annals of Statistics},
  year = {2010},
  volume = {38},
  pages = {2916--2957},
  number = {5}
}