0.1 Detection performance criteria (Page 3/3)

Elements of detection theory Page 3 / 3

This optimization problem can be solved using Lagrange multipliers ; we seek to find the decision rule that maximizes $F P_{D} λ P_{F} α$ where $λ$ is a positive Lagrange multiplier. This optimization technique amounts to finding the decision rule that maximizes $F$ , then finding the value of the multiplier that allows the criterion toinge the detection probability in competition with false-alrm probabilitiesin excess of the criterion value. As is usual in the derivation of optimum decision rules, we maximize thesequantities with respect to the decision regions. Expressing $P_{D}$ and $P_{F}$ in terms of them, we have

F r Z_{1} p R ℳ_{1} r λ r Z_{1} p R ℳ_{0} r α λ α r Z_{1} (p R ℳ_{1} r λ p R ℳ_{0} r)

To maximize this quantity with respect to

Z_{1}

, we need only to integrate over those regions of

r

where the integrand is positive). The region

Z_{1}

thus corresponds to those values of

r

where

p R ℳ_{1} r λ p R ℳ_{0} r

and the resulting decision rule is

p R ℳ_{1} r p R ℳ_{0} r ≷_{ℳ_{0}}^{ℳ_{1}} λ

The ubiquitous likelihood ratio test again appears; it is indeed the fundamental quantity in hypothesis testing. Using either the logarithm of the likelihoodratio or the sufficient statistic, this result can be expressed as

Λ r ≷_{ℳ_{0}}^{ℳ_{1}} λ

ϒ r ≷_{ℳ_{0}}^{ℳ_{1}} γ

We have not as yet found a value for the threshold. The false-alarm probability can be expressed in terms of theNeyman-Pearson threshold in two (useful) ways.

P_{F} Λ λ

∞ p Λ ℳ 0 Λ ϒ γ ∞ p ϒ ℳ 0 ϒ

One of these implicit equations must be solved for the threshold by setting

P_{F}

equal to

α

. The selection of which to use is usually based on pragmatic considerations: the easiest to compute. From theprevious discussion of the relationship between the detection and false-alarm probabilities, we find that to maximize

P_{D}

we must allow

α

to be as large as possible while remaining less than

α

. Thus, we want to find the smallest value of

λ

consistent with the constraint. Computation of the threshold isproblem-dependent, but a solution always exists.

An important application of the likelihood ratio test occurs when $R$ is a Gaussian random vector for each model. Suppose the models correspond to Gaussian random vectorshaving different mean values but sharing the same covariance.

$ℳ_{0}$ : $R N 0 σ 2 I$
$ℳ_{1}$ : $R N m σ 2 I$

R

is of dimension

L

and has statistically independent, equi-variance components. The vector of means

m m_{0} … m_{L - 1}

distinguishes the two models. The likelihood functions associated this problem are

p R ℳ_{0} r l 0 L 1 1 2 σ 2 1 2 r_{l} σ 2

p R ℳ_{1} r l 0 L 1 1 2 σ 2 1 2 r_{l} m_{l} σ 2

The likelihood ratio

Λ r

becomes

Λ r l 0 L 1 1 2 r_{l} m_{l} σ 2 l 0 L 1 1 2 r_{l} σ 2

This expression for the likelihood ratio is complicated. In the Gaussian case (and many others), we usethe logarithm the reduce the complexity of the likelihood ratio and form a sufficient statistic.

Λ r l 0 L 1 -1 2 r_{l} m_{l} 2 σ 2 1 2 r_{l} 2 σ 2 1 σ 2 l 0 L 1 m_{l} r_{l} 1 2 σ 2 l 0 L 1 m_{l} 2

The likelihood ratio test then has the much simpler, but equivalent form

l 0 L 1 m_{l} r_{l} ≷_{ℳ_{0}}^{ℳ_{1}} σ 2 η 1 2 l 0 L 1 m_{l} 2

To focus on the model evaluation aspects of this problem, let's assume the means equal each other and are a positive constant:

m_{l} m 0

What would happen if the mean were negative?

We now have

l 0 L 1 r_{l} ≷_{ℳ_{0}}^{ℳ_{1}} σ 2 m η L m 2

Note that all that need be known about the observations

r_{l}

is their sum. This quantity is the sufficient statistic for the Gaussian problem:

ϒ r r_{l}

and

γ σ 2 η m L m 2

When trying to compute the probability of error or the threshold in the Neyman-Pearson criterion, we must find theconditional probability density of one of the decision statistics: the likelihood ratio, the log-likelihood, or thesufficient statistic. The log-likelihood and the sufficient statistic are quite similar in this problem, but clearly weshould use the latter. One practical property of the sufficient statistic is that it usually simplifiescomputations. For this Gaussian example, the sufficient statistic is a Gaussian random variable under each model.

$ℳ_{0}$ : $ϒ r N 0 L σ 2$
$ℳ_{1}$ : $ϒ r N L m L σ 2$

To find the probability of error from , we must evaluate the area under a Gaussian probability density function. These integrals aresuccinctly expressed in terms of

Q x

, which denotes the probability that a unit-variance, zero-mean Gaussian random variable exceeds

x

Q x α x

∞ 1 2 σ 2 α 2 2 As

1 Q x Q x

, the probability of error can be written as

P_{e} π_{1} Q L m γ L σ π_{0} Q γ L σ

An interesting special case occurs when

π_{0} 1 2 π_{1}

. In this case,

γ L m 2

and the probability of error becomes

P_{e} Q L m 2 σ

Q ·

is a monotonically decreasing function, the probability of error decreases with increasing values of theratio

L m 2 σ

. However, as shown in ,

Q ·

decreases in a nonlinear fashion. Thus, increasing

m

by a factor of two may decrease the probability of error by a larger or a smaller factor; the amount of change depends on the initial value of theratio.

To find the threshold for the Neyman-Pearson test from the expressions given on , we need the area under a Gaussian density.

P_{F} Q γ L σ 2 α

Q ·

is a monotonic and continuous function, we can set

α

equal to the criterion value

α

with the result

γ L σ Q α

where

Q ·

denotes the inverse function of

Q ·

. The solution of this equation cannot be performed analytically as no closed form expressionexists for

Q ·

(much less its inverse function). The criterionvalue must be found from tables or numerical routines. Because Gaussian problems arise frequently, the accompanying table provides numeric values for this quantity at the decade points.

$x$	$Q x$
$10 -1$	1.281
$10 -2$	2.396
$10 -3$	3.090
$10 -4$	3.719
$10 -5$	4.265
$10 -6$	4.754

The table displays interesting values for

Q ·

that can be used to determine thresholds in the Neyman-Pearson variant of the likelihood ratio test.Note how little the inverse function changes for decade changes in its argument;

Q ·

is indeed very nonlinear. The detection probability of the Neyman-Pearson decision rule is given by

P_{D} Q Q α L m σ

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Source: OpenStax, Elements of detection theory. OpenStax CNX. Jun 22, 2008 Download for free at http://cnx.org/content/col10531/1.5

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