Fundamentals of Detection Theory#
Binary Hypothesis Detection from a Sample#
Binary hypothesis detection is a fundamental concept in statistical signal processing and decision theory. This chapter introduces the basic principles and mathematical frameworks for detecting one of two possible hypotheses based on observed data. The focus is on defining the problem and setting up the rules for making decisions under uncertainty.
Definitions of Hypothesis Testing#
Random Variable (\(\mathbf{y}\)): Consider a random variable \(\mathbf{y}\) that represents the observed data. The distribution of \(\mathbf{y}\) depends on one of two possible hypotheses:
\(H_0\) (null hypothesis)
\(H_1\) (alternative hypothesis) The goal is to determine which hypothesis is true based on the observation \(y\).
Conditional Probability: The likelihood of observing a particular value \(y\) under each hypothesis is given by the conditional probability \(P(y|H_j)\), where \(j \in \{0, 1\}\). This represents the probability of observing \(y\) given that hypothesis \(H_j\) is true.
Examples of \(H_0\) and \(H_1\)#
Cognitive Radio (White Space Detection):
\(H_0\): No primary user present, only noise (\(y = n\)) — white space available. This hypothesis typically represents the absence of the desired signal or the presence of noise only. In this case, \(P_{00}\), the probability of correctly deciding \(H_0\) when \(H_0\) is true, is often referred to as the probability of correct rejection.
\(H_1\): Primary user present, signal and noise (\(y = s + n\)) — band occupied. This hypothesis represents the presence of the signal we aim to detect.
Jamming Detection:
\(H_0\): No jamming, normal communication signal plus noise (\(y = s + n\)).
\(H_1\): Jamming present, communication signal plus noise and jamming signal (\(y = s + n + j\)).
GAN (Generative Adversarial Networks):
\(H_0\): Generated (fake) data, output from the generator.
\(H_1\): Real data, actual sample from the dataset (generated data is considered as real data).
Decision Rule (\(\delta(y)\))#
Objective: The central objective is to devise a decision rule \(\delta(y)\) that uses the observed data \(y\) to decide whether \(H_0\) or \(H_1\) is true.
Decision Rule Structure:
\[\begin{split} \delta(y) = \begin{cases} H_1, & \text{if } y \in R_1 \\ H_0, & \text{if } y \in R_0 \end{cases} \end{split}\]Here:
\(R_1\) is the observation region where \(H_1\) is declared.
\(R_0\) is the observation region where \(H_0\) is declared.
Decision Outcomes and Associated Probabilities#
The effectiveness of the hypothesis test is evaluated by the probabilities of the various outcomes, which are summarized in the following table:
True Hypothesis |
Decision |
Description |
Probability |
|---|---|---|---|
\( H_0 \) |
\( H_0 \) |
\( P_{00} = \int_{R_0} P(y \mid H_0) \, dy \) |
|
\( H_0 \) |
\( H_1 \) |
False alarm (Type I Error) |
\( P_f = P_{10} = \int_{R_1} P(y \mid H_0) \, dy \) |
\( H_1 \) |
\( H_1 \) |
Detection (Correct detection) |
\( P_d = P_{11} = \int_{R_1} P(y \mid H_1) \, dy \) |
\( H_1 \) |
\( H_0 \) |
Miss (Type II Error) |
\( P_m = P_{01} = \int_{R_0} P(y \mid H_1) \, dy \) |
\(P_{00}\): The probability of correctly deciding \(H_0\) when \(H_0\) is true.
\(P_f = P_{10}\): The probability of deciding \(H_1\) when \(H_0\) is true. This is known as a false alarm or Type I error.
\(P_d = P_{11}\): The probability of correctly deciding \(H_1\) when \(H_1\) is true, also known as the probability of detection.
\(P_m = P_{01}\): The probability of deciding \(H_0\) when \(H_1\) is true, known as a miss or Type II error.
Importance of Decision Rules
The effectiveness of a hypothesis test is determined by how well the decision rule \(\delta(y)\) can distinguish between \(H_0\) and \(H_1\). A well-designed rule maximizes \(P_d\) while minimizing \(P_f\) and \(P_m\).
References#
The contents of the sections in this chapter are based on the following materials.
T. Schonhoff and A. Giordano, Detection and Estimation Theory and its Applications. Prentice Hall, 2006 , Chapter 4