# Introduction to Kernel density estimation.

Before starting let’s get some background on ** Estimators, **they're classified into two classes

*Parametric**Non-*P*arametric*

**Parametric **make assumptions about the population from which a sample of data is drawn. Often this assumption is that the population is normally distributed, i.e. bell-shaped. This assumption allows the development of a theory that allows us to draw inferences about the population based on a sample taken from it.

The other family of estimators is **Non-Parametric **this set of distribution makes no distributional assumptions no fixed structure and depends upon all the data points to reach an estimate. **Kernel density estimators** belong to this class.

So why Kernel Density Estimation let us see how histograms are just not sufficient.

Histograms are *not smooth**, depend on the width of the bins and the endpoints of the bins, *This is where ** kernel density estimators** alleviate the problem.

**let’s see how histogram are affected by bins**

#Importing libraries

import pandas as pd

import numpy as np

import seaborn as sns

import matplotlib.pyplot as plt

import pylab

from scipy.stats.distributions import norm# Plotting a normal distribution with different bins

mu, sigma = 0, 0.1 # mean and standard deviation

s = np.random.normal(mu, sigma, 1000)#plotting the different bins

from matplotlib.pyplot import figure

figure(num=None, figsize=(10, 8), dpi=80, facecolor='w', edgecolor='k')

plt.hist(s,bins=10,label="10")

plt.hist(s,bins=50,label="50",color="green")

plt.hist(s,bins=300,label="300",color="orange")

plt.hist(s,bins=500,label="500",color="white")

plt.show()

So we see in the above visualization how bin changes the normalization look

So how do we overcome this?

To remove the dependence on the endpoints of the bins, kernel estimators center a kernel function at each data point. We place a kernel function on every data point to get the density estimates. Just like in high school getting the value of the function on at a given point of x

y = f(x)

## Kernal function

**Kernel Function typically has these following properties**

- Everywhere non-negative:
**K(x)≥0 ∀ x∈X** - Symmetric :
**K(x) = K(-x) ∀ x∈X** - Decreasing :
**K`(x) ≤ 0 ∀ x >0**

`sns.kdeplot(x,data2=None,bw=.4,color=”yellow”,label=”gaussian”,kernel=”gau”) `

sns.kdeplot(x,data2=None,bw=.4,color=”black”,label=”biw”,kernel=”biw”)

sns.kdeplot(x,data2=None,bw=.4,color=”red”,label=”cos”,kernel=”cos”)

sns.kdeplot(x,data2=None,bw=.4,color=”green”,label=”epa”,kernel=”epa”)

sns.kdeplot(x,data2=None,bw=.4,color=”blue”,label=”tri”,kernel=”tri”)

sns.kdeplot(x,data2=None,bw=.4,color=”green”,label=”triw”,kernel=”triw”)

The quality of a kernel estimate depends less on the shape of the *K* than on the value of *its bandwidth *** h**. It’s important to choose the most appropriate bandwidth as a value that is too small or too large is not useful.

`x = np.concatenate([norm(-1, 1.).rvs(400),norm(1, 0.3).rvs(100)])`

sns.kdeplot(x,data2=None ,bw=2,color="yellow",label="bw:2")

sns.kdeplot(x,data2=None ,bw=1,color="red",label="bw: 0.2")

sns.kdeplot(x,data2=None ,bw=.5,color ="blue",label="bw: 0.5")

sns.kdeplot(x,data2=None ,bw=.3,color="green",label="bw: 0.3")

sns.kdeplot(x,data2=None ,bw=.1,color="grey",label="bw: 0.1")

sns.kdeplot(x,data2=None ,bw=.05,color="grey",label="bw: 0.05")

plt.legend();

The smoothing bandwidth h plays a key role in the quality of KDE. Here is an example of applying different h to the dataset we see that when ** h** is too small (the gray curve), there are many wiggly structures on our density curve this is under smoothing

**.**On the other hand, when

**is too large (the yellow curve), we see that the two bumps are smoothed out. This situation is called over smoothing–some important structures are obscured by the huge amount of smoothing.**

*h*# Bandwidth selection methods, univariate case

Subjective choice

The natural way for choosing **ℎ** is to plot out several curves and choose the estimate that best matches one’s prior (subjective) ideas, However, this method is not practical in high-dimensional data.

Maximum likelihood cross-validation

Reference to a standard distribution

Conclusion

The idea of Kernel Density Estimators is to give you an idea about the distribution.

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