Statistical Detection Theory I

Statistical Detection Theory I - Detection and Estimation...

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Unformatted text preview: Detection and Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.1 Statistical Detection Theory (I) ~ Neyman-Pearson Theorem Observe a realization of a random variable whose pdf is either N (0,1) or N (1,1). Binary Hypotheses: H : µ = 0 null hypothesis H 1 : µ = 1 alternative hypothesis or H : x[0] = w[0] null hypothesis H 1 : x[0] = 1 + w[0] alternative hypothesis A reasonable approach is to decide H 1 if x[0] > 1/2. Threshold = 1/2 Type I error : We decide H 1 but H is true. ( False Alarm ) Type II error : We decide H but H 1 is true. ( Miss ) Type I error probability: P(H 1 ;H ) Type II error probability: P(H ;H 1 ) P(H i ;H j ): the probability of deciding H i when H j is true. It is not possible to reduce both error probabilities simultaneously. Detection & Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.2 As the threshold changes, one error increases while the other decreases. P(H 1 ;H ) Probability of false alarm ( P FA ) P(H 1 ;H 1 ) = 1 – P(H ;H 1 ) Probability of detection (P D ) Example, if P FA = 10-3 , we have γ = 3. With this choice, we have Detection & Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.3 Neyman-Pearson (NP) Approach: We wish to maximize P D subject to the constraint P FA = α . General operation of a detector: decide either H or H 1 based on an observed set of data {x[0], x[1], .., x[N-1]} R 1 = { x : decide H 1 or reject H } critical region R = { x : decide H or reject H 1 } α : significance level or size of the test. P D : the power of the test. There exist many critical regions that has the same significance level. ⇒ choose the one that maximizes P D . Remarks: 9 The critical region that attains the maximum power is the best critical region . 9 The NP theorem tells us how to choose R 1 if we are given p( x ;H ), p( x ;H 1 ), and α . Detection & Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.4 (Proof) We use Lagrangian multipliers to maximize P D for a given P...
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Statistical Detection Theory I - Detection and Estimation...

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