Hypothesis Testing

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# Hypothesis Testing - PowerPoint PPT Presentation

Hypothesis Testing. Presenting: Lihu Berman. Agenda. Basic concepts Neyman-Pearson lemma UMP Invariance CFAR. X is a random vector with distribution. is a parameter, belonging to the parameter space. disjoint covering of the parameter space. denotes the hypothesis that.

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## Hypothesis Testing

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Presentation Transcript

### Hypothesis Testing

Presenting:

Lihu Berman

Agenda
• Basic concepts
• Neyman-Pearson lemma
• UMP
• Invariance
• CFAR

Xis a random vector with distribution

is a parameter, belonging to the parameter space

disjoint covering of the parameter space

denotes the hypothesis that

Binary test of hypotheses:

vs.

M-ary test:

vs.

vs.

Basic concepts

If

then

is said to be a simple hypothesis

vs.

simple vs. composite hypotheses

Example:

A two-sided test: the alternative lies on both sides of

vs.

A one-sided test (for scalar ):

Basic concepts (cont.)

Otherwise, it is said to be a composite hypothesis

RADAR – Is there a target or not ?

Physical model – is the coin we are tossing fair or not ?

If the measurement is in the Acceptance region – is accepted

If it is in the Rejection region – is rejected, and is accepted.

is a disjoint covering of the measurement space

Basic concepts (cont.)

simple:

composite:

i.e. the worst case

simple:

composite:

Basic concepts (cont.)

Probability of False-Alarm (a.k.a. Size):

Detection probability (a.k.a. Power):

The best test of size has the largest among all tests of that size

Basic concepts (cont.)

Receiving Operating Characteristics (ROC):

Chance line

Let:

and let denote the density function of X, then:

Is the most powerful test of size for testing which of the two

simple hypotheses is in force, for some

The Neyman-Pearson lemma

Note 1: If then the most powerful test is:

Note 2: Introduce the likelihood function:

Then the most powerful test can be written as:

The Neyman-Pearson (cont.)

Note 3: Choosing the threshold k.

Denote by the probability density of the likelihood function

under , then:

Note 4: If is not continuous (i.e. )

Then the previous equation might not work! In that case, use the test:

Toss a coin, and

The Neyman-Pearson (cont.)

Source

Mapper

‘1’ = Enemy spotted. ‘0’ = All is clear.

Prior probabilities unknown !

Binary comm. in AWGN
Binary comm. in AWGN

Natural logarithm is monotone, enabling the use of Log-Likelihood

Binary comm. (cont.)

Assume equal energies: and define

UMP Tests

The Neyman-Pearson lemma holds for simple hypotheses.

Uniformly Most Powerful tests generalize to composite hypotheses

Consider scalar R.Vs whose PDFs are parameterized by scalar

Karlin-Rubin Theorem (for UMP one-sided tests):

If the likelihood-ratio is monotone non-decreasing in x for any

pair

, then the test:

Is the UMP test of size for testing

UMP Tests (cont.)

Proof: begin with fixed values

By the Neyman-Pearson lemma, the most powerful test of size for

testing is:

As likelihood is monotone, we may replace it with the threshold test

UMP Tests (cont.)

making the most powerful test of size for testing

the composite alternative vs. the simple hypothesis

Consider now the power function

At . For any because is

more powerful than the test

A similar argument holds for any

UMP Tests (cont.)

Thus, we conclude that is non-decreasing

Consequently, is also a test whose size satisfies

Finally, no test with size can have power , as it would

UMP Tests (cont.)

The statistic T(x) is sufficient for if and only if

Fisher-Neyman factorization theorem:

The statistic T(x) is sufficient for if and only if

A note on sufficiency

No other statistic which can be calculated from the same sample

provides any additional information as to the value of the parameter

One can write the likelihood-ratio in terms of the sufficient statistic

Theorem: the one-parameter exponential family of distributions with

density:

has a monotone likelihood ratio in the sufficient statistic

provided that is non-decreasing

Proof:

UMP Tests (cont.)

UMP one-sided tests exist for a host of problems !

Therefore, the test

is the Uniformly Most Powerful test of size for testing

UMP Tests (cont.)

Source

Mapper

So what?! Let us continue with the log-likelihood as before…

Oops

Invariance

Intuitively: search for a statistic that is invariant to the nuisance parameter

Invariance (cont.)

Project the measurement on the subspace orthogonal to the disturbance!

Optimal

signals ?

The measurement is distributed as

Invariance (cont.)

Revisit the previous example (AWGN channel with unknown bias)

Another example

Consider the group of transformations:

The hypothesis test problem is invariant to G

Invariance (another example)

What statistic is invariant to the scale of S ?

The angle between the measurement and the signal-subspace

(or the subspace orthogonal to it: )

In fact, Z is a maximal invariant statistic to

a broader group G’, that includes also

rotation in the subspace.

G’ is specifically appropriate for channels

that introduce rotation in as well as gain

Invariance (another example)
Invariance (UMPI & summary)
• Invariance may be used to compress measurements into statistics of low dimensionality, that satisfy invariance conditions.
• It is often possible to find a UMP test within the class of invariant tests.
• Steps when applying invariance:

1. Find a meaningful group of transformations, for which the hypothesis testing problem is invariant.

2. Find a maximal invariant statistic M, and construct a likelihood ratio test.

3. If M has a monotone likelihood ratio, then the test is UMPI for testing one sided hypotheses of the form

• Note: Sufficiency principals may facilitate this process.
CFAR (introductory example)

Project the measurement on the signal space. A UMP test !

The False-Alarm Rate is Constant. Thus: CFAR

Redraw the problem as:

CFAR (cont.)

Utilize Invariance !!

As before:

Change slightly:

independent

CFAR (cont.)

is unknown !!!

Thus, we can set a threshold in the test:

in order to obtain

Furthermore, as the likelihood ratio for non-central t is monotone, this

test is UMPI for testing in the distribution

when is unknown !

CFAR (cont.)

CFAR !

CFAR (cont.)

The actual probability of detection depends on the actual value of the SNR

Summary
• Basic concepts
• Neyman-Pearson lemma
• UMP
• Invariance
• CFAR