A kernel distribution is a non-parametric distribution of a random variable which can be used when parametric distribution cannot properly model the data.

1. 1
Non-Parametric Probability
Distribution for fitting data
by
Sarkar Nikhil Chandra, M.S.
PhD Student

2. 2
Problem Statement
Q. How to model univariate data with a non-parametric probability
distribution?

3. 3
Objective
• To fit a probability distribution that can model the univariate
data.

4. 4
Probability Distributions
Probability distributions
Discrete Continuous
Probability mass
function
Probability density
function
e.g. Binomial
Poisson
Parametric Non-parametric
e.g. Kernele.g. Normal
Lognormal
Gamma
Weibull
…….

5. 5
Univariate Data
Desired Speed
35.304
36.568
37.257
39.541
39.563
39.742
39.755
39.813
39.983
40.004
40.063
40.13
40.409
40.498
40.884
40.985
41.032
41.06
41.225
42
Sample
This is a sample of N = 100 univariate data for desired speed of different cars in a
specific road.

6. 6
Histogram, theoretical densities, empirical and
theoretical CDFs
Figure 1: Goodness-of-fit plot for various distributions fitted to desired speed (a) histogram and theoretical densities;
(b) empirical and theoretical CDFs.
Let’s first check some of the parametric distributions with data.
Remark: From observation we assume that not any of the parametric distributions
well fitted with the data.
(a) (b)

7. 7
Non-parametric Distribution

8. 8
Kernel Distribution
Kernel Density Estimator:
𝑓ℎ 𝑥 =
1
𝑛ℎ
𝑖=1
𝑛
𝐾(
𝑥 − 𝑥𝑖
ℎ
) ; −∞ < 𝑥 < ∞
where n is the sample size; K(.) is the kernel smoothing function, h is the bandwidth.
A kernel distribution is an example of non-parametric probability
distribution of a random variable. We can consider this distribution,
when parametric distribution cannot properly model the data.

9. 9
Kernel Probability Distribution
Figure 2: Probability density function for Kernel distribution of desired speed.

10. 10
Kernel Smoothing Function
• The kernel smoothing function 𝐾 . is a real valued function basically
refers to the shape of the probability distributions for each value in the
data.
It satisfies the following properties:
(i) 𝐾 𝑥 ≥ 0 ; ∀ 𝑥
(ii) −∞
∞
𝐾 𝑥 𝑑𝑥 = 1
• There are several options available for defining the kernel smoothing
function. For example,
Normal kernel : 𝐾 𝑥 =
1
2𝜋
𝑒−
1
2
𝑥2
; −∞ ≤ 𝑥 ≤ ∞
Epanechnikov kernel : 𝐾 𝑥 =
3
4
1 − 𝑥2 ; −1 ≤ 𝑥 ≤ 1

11. 11
Kernel Smoothing Function
Figure 3: Different smoothing functions for Kernel distribution.

12. 12
Kernel Probability Distribution
Figure 4: Kernel probability density function for speed with different kernel smoothing functions.

13. 13
Empirical and Kernel CDFs
Figure 5. Empirical and theoretical CDFs for Kernel distribution of desired speed.
From observation we conclude that kernel probabilistic distribution
well fitted with the data.

14. 14
Thank you !!!