Non-Parametric Estimation - Introduction to Pattern Recognition - Lecture Slides, Slides of Advanced Algorithms

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Recap

We are discussing non-parametric estimation ofdensity functions.

PR NPTEL course – p.1/

Recap

We are discussing non-parametric estimation ofdensity functions.

Here we do not assume any form for the densityfunction.

PR NPTEL course – p.2/

The choice of size of

V

is critical for getting good

estimates.

PR NPTEL course – p.4/

The choice of size of

V

is critical for getting good

estimates.

As discussed in last class there are two possibilities:

PR NPTEL course – p.5/

The choice of size of

V

is critical for getting good

estimates.

As discussed in last class there are two possibilities:

we can fix

V

and compute

k

(Parzen Window or

kernel-density estimate)

we can fix

k

and compute

V

(k-nearest neighbour

density estimate)

PR NPTEL course – p.7/

The choice of size of

V

is critical for getting good

estimates.

As discussed in last class there are two possibilities:

we can fix

V

and compute

k

(Parzen Window or

kernel-density estimate)

we can fix

k

and compute

V

(k-nearest neighbour

density estimate)

We discuss these in this class

PR NPTEL course – p.8/

Parzen Windows

We first consider the Parzen window method.

Define a function

φ

d

by

φ

u

if

u

i

, i

, d

otherwise

where

u

u

1

, u

d

T

.

PR NPTEL course – p.10/

Parzen Windows

We first consider the Parzen window method.

Define a function

φ

d

by

φ

u

if

u

i

, i

, d

otherwise

where

u

u

1

, u

d

T

.

This defines a unit hypercube in

d

centered at origin.

PR NPTEL course – p.11/

φ

u

−

u

0

h

is a hypercube of side

h

centered at

u

0

.

PR NPTEL course – p.13/

φ

u

−

u

0

h

is a hypercube of side

h

centered at

u

0

.

Let

D

x

1

, x

n

be the data samples.

PR NPTEL course – p.14/

φ

u

−

u

0

h

is a hypercube of side

h

centered at

u

0

.

Let

D

x

1

, x

n

be the data samples.

Then, for any

x

,

φ

x

−

x

i

h

would be

only if

x

i

falls in a

hypercube of side

h

centered at

x

.

Hence the number of data points falling in ahypercube of side

h

centered at

x

is

k

n

i

=

φ

x

x

i

h

PR NPTEL course – p.16/

Hypercube of side

h

in

d

has volume

h

d

.

PR NPTEL course – p.17/

Hypercube of side

h

in

d

has volume

h

d

.

Hence we can write our estimated density function as

f

x

(^1) n

n

i

=

h

d

φ

x

x

i

h

Known as Parzen window estimate.

PR NPTEL course – p.19/

Hypercube of side

h

in

d

has volume

h

d

.

Hence we can write our estimated density function as

f

x

(^1) n

n

i

=

h

d

φ

x

x

i

h

Known as Parzen window estimate.

If we store all

x

i

, we can compute

f

x

at any

x

.

PR NPTEL course – p.20/