Detection Theory

Chapter 3. Statistical Decision Theory I.

Isael Diaz

Oct 26th 2010

Outline

• Neyman-Pearson Theorem

• Detector Performance

• Irrelevant Data

• Minimum Probability of Error

• Bayes Risk

• Multiple Hypothesis Testing

Chapter Goal

• State basic statistical groundwork for detectors design

• Address simple hypothesis , where PDFs are known (Chapter 6 will cover the case of unknown PDFs)

• Introduction to the classical approach based on Neyman-Pearson and Bayes Risk– Bayesian methods employ prior knowledge of the hypotheses

– Neyman-Pearson has no prior knowledge

Bayes vs Newman-Pearson

• The particular method employed depends upon our willingness to incorporate prior knowledge about the probabilities of ocurrence of the various hypotheses

• Typical cases– Bayes : Communication systems, Pattern recognition

– NP : Sonar, Radar

Nomenclature

2 2( , ) : G a u s s ia n P D F w ith m e a n a n d v a r ia n c e

: H y p o th e s is i

( | ) : P ro b a b i li ty o f o b s e rv in g x u n d e r a s s u m p rio n H

( ; ) : P ro b a b i li ty o f d e te c t in g H , w h e n i t i s H

( | ) : P r

i

i i

i j i j

i j

H

p x H

P H H

P H H

o b a b i li ty o f d e te c t in g H , h e n H w a s o b s e rv e d

( ) : L ik e lih o o d o f x

i jw

L x

Nomenclature cont...

• Note that is the probability of deciding Hi, if Hj is true.

• While assumes that the outcome of a probabilistic experiment is observed to be Hj and that the probability of deciding Hi is conditioned to that outcome.

( | )i j

P H H

( ; )i j

P H H

Neyman-Pearson Theorem

Neyman-Pearson Theorem cont...

• It is no possible to reduce both error probabilities simultaneously

0

1

: [ 0 ] [0 ]

: [ 0 ] [ 0 ] [0 ]

H x w

H x s w

• In Neyman-Pearson the approach is to hold one error probability while minimizing the other.

Neyman-Pearson Theorem cont...

• We wish to maximize the probability of detection

• And constraint the probability of false alarm

• By choosing the threshold

1 1( ; )

DP P H H

1 0( ; )

F AP P H H

γ

Neyman-Pearson Theorem cont...

• To maximize the probability of detection for a given probability of false alarm decide for H1 if

• with

1

0

( ; )( )

( ; )

p x HL x

p x H

Likelihood ratio test

0

{ : ( ) }

( ; )

x L x

F AP p x H d x

Likelihood Ratio

• The LR test rejects the null hypothesis if the value of this statistic is too small.

• Intuition: large Pfa -> small gamma -> not much larger p(x,H1) than p(x;H0)

• Limiting case: Pfa->1

• Limiting case: Pfa->0

1

0

( ; )( )

( ; )

p x HL x

p x H

Neyman-Pearson Theorem cont...

• Example 1

• Maximize

• Calculate the threshold so that3

1 0F A

P

1 1( ; )

DP P H H

Neyman-Pearson Theorem Example 1

(1)

(2)

(3)

ex p ( )

γ '

γ ' 3(4)

(5)

(6)

(7)

(8)

Question

Answer is: Only if we allow for a larger Pfa.

NP gives the optimal detector!

By allowing a Pfa of 0.5, the Pd is increased to 0.85

How can Pd be improved?

System Model

• In the General case

• Where the signal s[n]=A for A>0 and w[n] is WGN with variance σ².

• It can be shown that the sum of x is a sufficient statistic

• The summation of x[n] is sufficient.

Detector Performance

• In a more general definition

Or

where

Energy-to-noise-ratio

Deflection Coefficient

Detector Performance cont...

• An alternative way to present the detection performance

Receiver Operating Characteristic (ROC)

γ ' in c reas in g

γ '

Irrelevant data

• Some data not belonging to the desired signal might actually be useful.

• The observed data-set is

• If

• Then the reference noise can be used to improve the detection. The perfect detector would become

[ 0 ] [ 0 ]2

R

AT x w

{ [0 ], [1], ..., [ 1], [0 ], [1], ..., [ 1]R R R

x x x N w w w N

0

[ ] [ ]:

[ ]R

x n w nH

w w n

1

[ ] [ ]:

[ ]R

x n A w nH

w w n

Irrelevant data cont...

• Generalizing, using the NP theorem

• If they are un-correlated

• Then, x2 is irrelevant and

1 2 1( , ) ( )L x x L x

Minimum Probability Error

• The probability of error is defined with Bayesian paradigm as

• Minimizing the probability of error, we decide H1 if

• If both prior probabilities are equal we have

Minimum Probability Error example

• An On-off key communication system , where we transmit either

• The probability of transmitting 0 or A is the same ½. Then, in order to minimize the probability of error γ=1

0 1[ ] 0 o r [ ]S n S n A

Minimum Probability Error example ...

1A n a lo g o u s to N P , w e d ec id e if

2AH x

Then we calculate the Pe

Deflection Coefficient

The probability error decreases monoto

nically with respect of the deflection

coefficient

Bayes Risk

• Bayes Risk assigns costs to the type of errors, denotes the cost if we decide but is true. If no error is made, no cost is assigned.

• The detector that minimizes Bayes risk is the following

i jC

iH

jH

Multiple Hyphoteses testing

• If we have a communication system, M-ary PAM (pulse amplitud modulation) with M=3.

Multiple Hyphoteses testing cont...

• The conditional PDF describing the communication system is given by

• In order to maximize p(x|Hi), we can minimize

Multiple Hyphoteses testing cont...

• Then

• Therefore

• There are six types of Pe, so Instead of calculating Pe, we calculate Pc=1-Pe, the probability of correct desicion

• And finally

Multiple Hyphoteses testing cont...

Multiple Hyphoteses testing cont...

Binary Case PAM Case

2

2

4

3 4e

N AP Q

2

24

e

N AP Q

In the digital communications world, there is a difference in the

energy per symbol, which is not considered in this example.

Chapter Problems

• Warm up problems: 3.1, 3.2

• Problems to be solved in class: 3.4, 3.8, 3.15, 3.18, 3.21