This document discusses kernel density estimation, which is a non-parametric way to estimate the probability density function of a random variable. It describes how to calculate the kernel density estimate using a kernel function and bandwidth parameter. It also discusses how to select the optimal bandwidth by minimizing the mean integrated square error, and how this can be done using a standard family of distributions for smoothing. The document provides density estimates for monthly, biweekly, and daily Indian monsoon rainfall data using this kernel density estimation technique.

1. Density Estimation of Indian Monsoon Rainfall
May 11, 2015
Density Estimation of Indian Monsoon Rainfall May 11, 2015 1 / 11

2. Kernel Density Estimation
Let X1, X2, ..., Xn denote a sample size of n from a random variable
with density f. The symbol f will be used to denote the density
estimation of f.
The kernel density estimate of f at the point x is given by
fh(x) =
n
i=1
K(
x−Xi
h
)
nh
xεR
where the kernel function K : R −→ R is any function which satisfies
K(x)dx = 1 and the parameter h is known as the window width or
bandwidth.
Density Estimation of Indian Monsoon Rainfall May 11, 2015 2 / 11

3. In practice, the kernel K is generally chosen to be a unimodal
probability density symmetric about zero. In this case, K satisfies the
conditions
K(x)dx = 1
xk(x)dx = 0
x2
k(x)dx > 0
A popular choice for K is Gaussian Kernel,given by
K(x) =
exp( −x2
2
)
√
2π
Density Estimation of Indian Monsoon Rainfall May 11, 2015 3 / 11

4. Bias and Variance
Assuming that the density is sufficiently smooth and that the kernel
has finite fourth moment, it can be show using the Taylor series that
Bias(fh(x)) = E(f (x)) − f (x) = h2
2
f ”(x) x2
k(x)dx + o(h2
)
Var(fh(x)) = E(f (x) − E(f (x)))2
=
k(x)2dx
nh
f (x) + o( 1
nh
)
MISE(fh(x)) = E (fh(x) − f (x))2
dx (Mean Integrated Square Error)
Density Estimation of Indian Monsoon Rainfall May 11, 2015 4 / 11

5. Bandwidth Selection
The ideal value of h, by minimizing approximate MISE
1
4
h4
k2
2 f ”(x)2
dx +
k(x)2dx
nh
where k2 = x2
k(x)dx
can be shown by simple calculus to be hopt, where
hopt = n
−1
5 k
−2
5
2 ( k(x)2
dx)
1
5 ( f ”(x)2
dx)
−1
5
Density Estimation of Indian Monsoon Rainfall May 11, 2015 5 / 11

6. Smoothing
A very easy and natural approach is to use a standard family of
distributions to assign a value to the term f ”(x)2
dx in the
expression of hopt. In this discussion we are assuming the smoothing
to be normal distribution with mean 0 and variance σ2
.
f ”(x)2
dx =
φ”(x)2dx
σ5 = 0.212
σ5
where φ is standard normal density
Density Estimation of Indian Monsoon Rainfall May 11, 2015 6 / 11

7. If Gaussian Kernel is used then
hopt = (4π)
−1
10
3
8
(π)
−1
2 σ(n)
−1
5 = 1.06σn
−1
5
Estimate σ from the data by usual standard deviation.
Density Estimation of Indian Monsoon Rainfall May 11, 2015 7 / 11

8. Monthly Rainfall Pdf
Density Estimation of Indian Monsoon Rainfall May 11, 2015 8 / 11

9. First Two Weeks Rainfall Pdf
Density Estimation of Indian Monsoon Rainfall May 11, 2015 9 / 11

10. Second Two Weeks Rainfall Pdf
Density Estimation of Indian Monsoon Rainfall May 11, 2015 10 / 11

11. Daily Rainfall Pdf
Density Estimation of Indian Monsoon Rainfall May 11, 2015 11 / 11