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Detection Theory Composite tests
Detection Theory
Chapter 5: Correction
Detection Theory
Thu
I claimed that the above, which is the most general case, was captured by the below
Thu
Chapter 5: Correction
Thu
I claimed that the above, which is the most general case, was captured by the below
Thu Thu
Argument was
Chapter 5: Correction
Thu
I claimed that the above, which is the most general case, was captured by the below
Thu Thu
Slides have been corrected
This is not correct, since it is limited to the case that C2-C1 is positive semi-definite
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Thu
Consider the case when the value of A is unknown, but assume A>0
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Thu
Consider the case when the value of A is unknown, but assume A>0
Thu UMP: An optimal test no matter the value of A – similar concept to MVU
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Thu
Consider the case when the value of A is unknown, but assume A>0
Thu UMP: An optimal test no matter the value of A – similar concept to MVU
Strategy to get UMPs: 1. Design test as if A is known 2. Show that test does not need knowledge of the value A
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Thu
Step 1: Design test as if A is known
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 1: Design test as if A is known
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 1: Design test as if A is known
Cancel multiplicative constants Remove ”exp” by taking logarithm Cancel x2[n]
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 1: Design test as if A is known
Manipulate a bit….
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 1: Design test as if A is known
scale
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 1: Design test as if A is known
Test statistic is not dependent on A Threshold seems to be, but is not
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 2: Show that test does not need knowledge of the value A
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 2: Show that test does not need knowledge of the value A
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Step 2: Show that test does not need knowledge of the value A
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Threshold does not depend on PFA
Step 2: Show that test does not need knowledge of the value A
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Compute PD
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Compute PD
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Compute PD
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Compute PD
Performance depends on A
Chapter 6: UMP - Uniformly most powerful tests
Detection Theory
Recap
A test is UMP if it, for all possible values of the unknown parameter(s), maximzes PD for given PFA
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
Step 1: Design test as if A is known
Same steps as before
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
Step 1: Design test as if A is known
Same steps as before
Next thing was to divide with A This changes inequality with A<0
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
Step 1: Design test as if A is known
Same steps as before
Next thing was to divide with A This changes inequality with A<0 <
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
<
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
<
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
<
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
<
Chapter 6: One-sided vs. Two sided
Detection Theory
Consider now: A<0
Chapter 6: One-sided vs. Two sided
Detection Theory
This means problems, since test can not be implemented
For A>0, decide H1 if For A<0, decide H1 if
Chapter 6: One-sided vs. Two sided
Detection Theory
This means problems, since test can not be implemented
For A>0, decide H1 if For A<0, decide H1 if
UMP exists (one sided)
UMP does not exist (two sided)
Chapter 6: One-sided vs. Two sided
Detection Theory
This means problems, since test can not be implemented
For A>0, decide H1 if For A<0, decide H1 if
UMP exists (one sided)
UMP does not exist (two sided)
An educated guess would be to decide H1 if This will turn out to be well motivated by the GLRT that comes shortly
Chapter 6: Karlin-Rubin Thm - A condition for UMP
Detection Theory
If the likelihood ratio is monotonic in the test T(x)
and it is known that then Detect H1, if T(x) > γ is UMP
Detection Theory
If the likelihood ratio is monotonic in the test T(x)
and it is known that then Detect H1, if T(x) > γ is UMP
This follows directly from the Neyman-Pearson theorem
Chapter 6: Karlin-Rubin Thm - A condition for UMP
Detection Theory
Application: Exponential family
Chapter 6: Karlin-Rubin Thm - A condition for UMP
Detection Theory
Application: Exponential family
Likelihood ratio
Chapter 6: Karlin-Rubin Thm - A condition for UMP
0
Detection Theory
Application: Exponential family
Likelihood ratio: If p(θ) is increasing, then LLR is monotonic in
Chapter 6: Karlin-Rubin Thm - A condition for UMP
0
Detection Theory
Application: Exponential family
Likelihood ratio: If p(θ) is increasing, then LLR is monotonic in
In our case (DC level), we have p(θ) = θ/σ2
Chapter 6: Karlin-Rubin Thm - A condition for UMP
0
Chapter 6: Composite tesiting – Bayesian approach
Detection Theory
With likelihoods containing unknown parameters,
We can integrate away the unknown
Chapter 6: Composite tesiting – Bayesian approach
Detection Theory
With likelihoods containing unknown parameters,
We can integrate away the unknown
A case that is very common and fully doable is x=Hθ+w, with Gaussian matrix H
Chapter 6: Composite tesiting – Bayesian approach
Detection Theory
With likelihoods containing unknown parameters,
We can integrate away the unknown
If prior is unknown, use a non-informative one (See Estimation theory book)
Detection Theory
Chapter 6: GLRT – Finite data records
The Generalized Likelihood ratio test is heuristic for finite data records, but can be proven optimal asymptotically in the size of the data record
𝐴 = 𝜋𝑟2
Where 𝜽 1 is the MLE of θ under H1, 𝜽 0 is the MLE of θ under H0
Detection Theory
Chapter 6: GLRT – Finite data records
Example: Non-coherent detection
𝐴 = 𝜋𝑟2
Detection Theory
Chapter 6: GLRT – Finite data records
Example: Non-coherent detection
𝐴 = 𝜋𝑟2
GLRT replaces H with its ML estimate
Detection Theory
Chapter 6: GLRT – Finite data records
Example: Non-coherent detection
𝐴 = 𝜋𝑟2
Detection Theory
Chapter 6: GLRT – Finite data records
Example: Non-coherent detection
𝐴 = 𝜋𝑟2
𝐴 = 𝜋𝑟2
Detection Theory
Chapter 6: GLRT – Finite data records
Example:
𝐴 = 𝜋𝑟2 GRLT is
𝐴 = 𝜋𝑟2
Detection Theory
Chapter 6: GLRT – Finite data records
Example:
𝐴 = 𝜋𝑟2 GRLT is
But from estimation theory, we have that the MLE of A is
𝐴 = 𝜋𝑟2
Detection Theory
Chapter 6: GLRT – Finite data records
Example:
𝐴 = 𝜋𝑟2 GRLT is
But from estimation theory, we have that the MLE of A is
Thus
Detection Theory
Chapter 6: GLRT – Finite data records
Example:
Taking logs, and simplification gives
Detection Theory
Chapter 6: GLRT – Finite data records
Example:
Taking logs, and simplification gives
Detection Theory
Chapter 6: GLRT – Finite data records
Example:
Taking logs, and simplification gives
Thus,
Detection Theory
Chapter 6: GLRT – Large data records
Large in this case does not mean that we use Szegö and the Fourier transform.
In this case, we consider large N, but with independent measurements
Two assumptions: 1. Signal is weak
2. MLE attains asymptotic form
Detection Theory
Chapter 6: GLRT – Large data records
Large in this case does not mean that we use Szegö and the Fourier transform.
In this case, we consider large N, but with independent measurements
Two assumptions: 1. Signal is weak Means that A is not enormous. Reasonable, otherwise problem is simple
2. MLE attains asymptotic form
Detection Theory
Chapter 6: GLRT – Large data records
Large in this case does not mean that we use Szegö and the Fourier transform.
In this case, we consider large N, but with independent measurements
Two assumptions: 1. Signal is weak Means that A is not enormous. Reasonable, otherwise problem is simple
2. MLE attains asymptotic form From Estimation theory
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Setup
Parameter vector to be detected
Differ for H0 and H1
Equal for H0 and H1 (e.g. noise variance)
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Setup
Parameter vector to be detected
Differ for H0 and H1
Equal for H0 and H1 (e.g. noise variance)
Hypotheses to test for
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Setup
Parameter vector to be detected
Differ for H0 and H1
Equal for H0 and H1 (e.g. noise variance)
Hypotheses to test for
Definition of GLRT Note that MLEs of θs differ under H0 and H1
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Chi-2 variable, r DoF
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Non-central Chi-2 variable, r DoF
Chi-2 variable, r DoF
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Non-central Chi-2 variable, r DoF
Chi-2 variable, r DoF
True value of θr under H1
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Non-central Chi-2 variable, r DoF
Chi-2 variable, r DoF
True value of θr under H1
True value of θs under H1 / H0
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Non-central Chi-2 variable, r DoF
Chi-2 variable, r DoF
True value of θr under H1
True value of θs under H1 / H0
Fisher Inform, Doesn’t depend on H1 or H0
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Non-central Chi-2 variable, r DoF
Chi-2 variable, r DoF
True value of θr under H1
True value of θs under H1 / H0
Fisher Inform, Doesn’t depend on H1 or H0
Fisher information matrix: one does not need to think about H0 or H1. Think like this: Given x, what is the Fisher info for θr ,θs
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Cancels with no nusiance parameters
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Cancels with no nusiance parameters
Since Fisher is pos. def., λ is degraded by nuisance
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Cancels with no nusiance parameters
Since Fisher is pos. def., λ is degraded by nuisance
Larger λ separates the pdfs more, Thus better PD with larger λ
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
Cancels with no nusiance parameters
Since Fisher is pos. def., λ is degraded by nuisance
Larger λ separates the pdfs more, Thus better PD with larger λ
So, not surprisingly, nuisance degrades our detection capability
Detection Theory
Chapter 6: GLRT – Large data records
Theorem
Statement
𝐴 = 𝜋𝑟2
No nuisance
Note: The test is still difficult, since it is still given by and we need to find the MLEs
