1. Introduction to Signal Detection

2. Outline
94/10/14

3. Given some data and some number of probability dis-tributions from which the data might have been sampled, we want to determined which distribution was in effect at the time the data were taken.
The solution rests on the branch of mathematics called decision theory.
For engineering, it concerns that the detecting the presence of a target in some region of a radar surveillance area.
94/10/14

4. Decision theory
In communication systems, some source of message information produces a bit 0 or 1，which is distorted in the transmission and corrupted by noise in the channel and the receiver.
The task of design a receiver which will determine whether a 0 or 1 was the intended message.
The design rests on a model of the system which specifies the probability distributions of the received signal in two message cases.
94/10/14

5. Decision theory
The task of the receiver is to select between these two distributions, given the receiver data.
There are three related problems to be solved.
specified the model of the system.
design the receiver, under some mathematization of the desire for the best receiver.
evaluate its performance on the average.
94/10/14

6. Statement of Hypothesis Problem
the available some data which we model as a random process because some elements in the source of the data are not describable in certain due to absent of information or data are corrupted with noise of one kind of another, which is not deterministically predictable.
Interested in determining which of a number of situations gave rise to the data at hand. We will specify some number of hypotheses Hi i=0,m-1, among which we believe there is one which describes the state of affairs at the time the data were produced.
94/10/14

7. Statement of Hypothesis Problem
Let Hi specifically refer to m probabilistic models
By processing the data set y at hand, we want to determine which of the models Hi was in effect to produce the data in question.
The result of such processing will be a decision Dj that the data should be associated with Hj.
94/10/14

8. Statement of Hypothesis Problem
Give Hi,I=0,m-1, we want to determine how to arrive at the decision Dj which best fits the case and evaluate how well the strategy performs on average.
m=M, M-hypothesis testing m=2, Binary Hypothesis testing problem
Binary Hypothesis :
H0:null hypothesis
H1:alternative hypothesis
94/10/14

9. Formulation of Binary Hypothesis Testing
two possible hypotheses or ，corresponding to two possible probability distributions and ,respectively on the observation space (T ,G)
A decision rule  for verse is any partition of the observation set  into and , we choose
when y  j，
94/10/14

10. Formulation of Binary Hypothesis Testing
The decision rule  as a function on  given by
the values of  for a given y is the index of the hypothesis accepted by 
94/10/14

11. Formulation of Binary Hypothesis Testing
Y:observation. Scale or multi-dimensional
Acceptance region / critical region rejection region
Devide the data into M region, …… must include all point of y –space to make sure any event must make decision
94/10/14

12. Formulation of Binary Hypothesis Testing
Parametric decision theory: probability distributions corresponding to the hypotheses are certain function of known forms. Possible with parameters having unknowing value.
Simple: hypothesis have no parameters with unknown values.
Composite: one or some parameters whose values are unspecified
Non-parametric decision theory: too difficult to define probability distributions corresponding to the hypotheses.
Distribution free.
94/10/14

13. Formulation of Binary Hypothesis Testing
When decision is making for a binary hypothesis testing problem, one of four outcomes can happen.
true；choose
true ; choose
Type I Error
Type II Error
94/10/14

14. Formulation of Binary Hypothesis Testing
Type I error : error of the first kind, false alarm rate，the size of the test in statistical work.
Type II error, error of the second kind, Probility of miss detection/ Probability of missing
The probability of detection, the power of the test on statistic work
94/10/14

15. Decision criterion assumption:
The occurrences of hypothesis and are governed by probability assignments, denoted by ,
Priori probability
Represents the observer’s information about the occurrences of hypothesis before the testing
Is unconditioned on observation Y.
To attach some relative importance to the possible courses of action.
Cost assignment
The first subscript indicates the decision chosen
The second subscript indicates the hypothesis that was true.
94/10/14

16. Condition risk for hypothesis
Decision Criterion
Condition risk for hypothesis
the average or expected cost incurred by decision rule  when hypothesis is true.
The decision rule is designed such that on average the cost will be as small as possible.
94/10/14

17. Bayes Hypothesis Testing
Bayes risk
94/10/14

18. Bayes Hypothesis Testing
Bayes rule:the optimal decision rule is one that minimizes, over all decision rules, the Bayes risk
If the Pj has density pj for j=0,1, and then
The first term is const. and positive
In order to to get the minimum risk ,the second term should be negative such that r(), Baysian risk, is a minimum over all 1
94/10/14

19. Bayes Hypothesis Testing
Decision region for Min. Bayes risk
Assume , in general, that the cost of correctly is less than the cost of incorrectly rejecting , i. e thus the decision region can be rewritten as
94/10/14

20. Bayes Hypothesis Testing
Decision region for Min. Bayes risk
L(y) the likelihood ratio/ likelihood ratio statistic between and .
test threshold
94/10/14

21. Bayes Hypothesis Testing
The Bayes decision rule corresponding to decision region compute the likelihood ratio for the observation y and make its decision by comparing the ratio to the threshold , a Bayes rule is
94/10/14

22. Bays Hypothesis Testing
The minimum probability of error
In general, the cost assignment is defined as
The Bayes risk for a decision rule  with the critical region is given by
The bayes risk is the average probability error incurred by the decision rule 
Because “c” is a constant, to minimize the the Bayes risk is the same strategy as to minimize the probability error Pe itself.
94/10/14

23. Bays Hypothesis Testing
The minimum probability of error
The likelihood ratio test with the threshold minimizes for the cost structure defined as above, it is thus a minimum probability-of-error decision scheme.
Thus the decision rule
Unless priori probabilities can be defined, the minimum probability error decision rule can be implemented.
94/10/14

24. MAP-Maximum a posteriori probability criterion
The Bayes formula implies that the conditional probability that hypothesis is true given the random observation Y takes on value y is given
:the average density of Y
:the posterior or a posteriori probability of two hypothesis
The critical region of the Bayes rule
The average posterior cost incurred by choosing hypothesis Hi given Y equals y
94/10/14

25. MAP-Maximum a posteriori probability criterion
The Bayes rule makes its decision by choosing the hypothesis that yields the minimum posterior cost
For the uniform cost assignment C00=C11=0, C01=C10=1, the decision rule become
It is difficult to find out the posteriori probability
It is more nature to find out the priori probability
94/10/14

26. MAP-Maximum a posteriori probability criterion
Adopting Bayes rule
L(y): is defined a likelihood function and the test is likelihood ratio test
 is defined as the threshold value
The MAP decision has the maximum a posterior probability of having occurred given that Y=y, which is the same as the minimum probability-of-error decision
94/10/14

27. Example assume if then L(y)>1, choose H1
If then L(Y) <1, choose H0. for y=0, then the ratio test is 1 so choose H0
94/10/14

28. Example the likelihood ratio test for the Bayes decision rule
taking the logarithm
94/10/14

29. Example The Bayes test is For the minimum-probability-of-error ln()=0
The probability of Detection and Probability of False alarm
94/10/14

30. Remarks
Test statistic is the likelihood ratio, and threshold is the function of cost and pri-porbability.
Minimizing the Bayes risk is possible only when the cost function and pri-probability are known.
Change the Pri-probability
the threshold value changes
the conditional probabilities are also changed, PF、PD、PM，due to the region of integration of H0、H1 are changed,
the curve of cost function of Bayes test is changed.
Usually,
We don’t know the pri-probability of two Hypotheses.
We don’t want to assume the pri-probability, however, the costs have been defined.
94/10/14

31. Remarks
For a communication system, the designed of a decision rule may not have control over or access to the mechanism that generates the state of nature.
when a receiver is defined, the threshold is defined, then, the conditional probabilities are defined. Therefore,
The average or bayes risk is not an acceptable design criterion.
A single decision rule would not minimize the average risk for every possible prior distribution.
94/10/14

32. Minmax Hypothesis Testing
Find the decision rule that will minimizes, over all decision , the maximum of the conditional risk ,
a possible design criterion is
94/10/14

33. Minmax Hypothesis Testing
When a threshold is defined,
a decision  is defined,
the average risk is only functional of pri-probability---a straight line.
where corresponds the a certain unknown pri-probability which minimize the Bayes risk.
the maximum value occurs at either and the maximum value is
Minimize
94/10/14

34. Minmax Hypothesis Testing
Define is the minimum possible Bayes risk for the prior probability ,and
only the Bayes rules can possible be minimax rules
The prior value that maximizes ，and is constant over ，, a decision rule with equal conditional risk—equalizer rule，the prior value is called as least-favorable prior.
The minimax rule is a Bayes rule for the least-favorable prior
94/10/14

35. Minmax Hypothesis Testing
Recall the Bayes risk
The minmax decision rule is to choose such that the Conditional risk is the same under Ho and H1 are true hypotheses, respectively.
Special case
94/10/14

36. Remarks Bayes decision rule Minmax decision rule: In practical sense,
Knowing the prior probability of hypothesis, the cost function, minimize the expected the overall cost or the average risk
Minmax decision rule:
The priors are not assumed known and the optimum was defined in term of minimizing the maximum of the conditional expected costs under the two hypotheses.
In practical sense,
the imposition of a specific cost structure on the decision made in testing is not possible or desirable.
The design of a test for H0 and H1 involves a trade-off between the probabilities of the two types of errors. One can always be made arbitrarily small at the expense of the other.
94/10/14

37. Neyman-Pearson Hypothesis Testing
The Neyman-Pearson decision rule:
to place a bound on the false-alarm rate/probability and then to minimize the miss probability within this constraint.
the Neyman-Pearson design criterion is
where  is the level or significance level of the test.
The NP design goal is to find the most powerful--level test of H0. It recognize a basic asymmetry in importance of the two hypotheses.
94/10/14

38. Neyman-Pearson Hypothesis Testing
Optimality :if the decision rule satisfying and let be any decision rule of the form
where
94/10/14

39. Neyman-Pearson Hypothesis Testing
and design a test to maximize probability of detection under this constain.
Using Lagrange multipliers ,
Where and
The first term is positive, To minimize F, the second term should be negative.
94/10/14

40. Q & A
94/10/14