Density Estimation Using Kernel Density

Introduction

In this lab, we will explore density estimation, which is a technique used to estimate the probability density function of a random variable. Specifically, we will focus on kernel density estimation, which is a non-parametric method for estimating the density.

VM Tips

After the VM startup is done, click the top left corner to switch to the Notebook tab to access Jupyter Notebook for practice.

Sometimes, you may need to wait a few seconds for Jupyter Notebook to finish loading. The validation of operations cannot be automated because of limitations in Jupyter Notebook.

If you face issues during learning, feel free to ask Labby. Provide feedback after the session, and we will promptly resolve the problem for you.

Skills Graph

Import the necessary libraries

First, we need to import the libraries that we will be using for density estimation. We will use the KernelDensity estimator from the sklearn.neighbors module, and the numpy library for data manipulation.

from sklearn.neighbors import KernelDensity
import numpy as np

Generate some sample data

Next, we will generate some sample data to perform density estimation on. For the purpose of this lab, let's generate a 1-dimensional dataset with 100 points. We will use a normal distribution to generate the data.

np.random.seed(0)
X = np.random.normal(0, 1, 100).reshape(-1, 1)

Fit a kernel density estimator

Now, we will create an instance of the KernelDensity estimator and fit it to our data. We can choose the type of kernel and bandwidth for the estimator. For example, we can use a Gaussian kernel and set the bandwidth to 0.2.

kde = KernelDensity(kernel='gaussian', bandwidth=0.2).fit(X)

Score the samples

After fitting the estimator, we can use the score_samples method to compute the log-likelihood of the samples under the estimated density function. This will give us a measure of how likely each sample is according to the density estimate.

scores = kde.score_samples(X)

Visualize the density estimate

Finally, we can visualize the density estimate using a histogram and the estimated density function. We can plot the histogram of the original data as well as the estimated density function.

import matplotlib.pyplot as plt

bins = np.linspace(-5, 5, 50)
plt.hist(X, bins=bins, density=True, alpha=0.5, label='Histogram')
plt.plot(X, np.exp(scores), color='red', label='Kernel Density Estimate')
plt.legend()
plt.show()

Summary

In this lab, we learned how to perform kernel density estimation using the KernelDensity estimator from scikit-learn. Kernel density estimation is a powerful technique for estimating the probability density function of a random variable. By fitting a kernel density estimator to a dataset, we can estimate the underlying density and visualize it using a histogram and the estimated density function. This allows us to gain insights into the distribution of the data and make probabilistic predictions.