In the realm of electrical engineering and signal processing, we often encounter situations where we need to make decisions based on noisy or uncertain data. One fundamental tool for tackling these scenarios is binary hypothesis testing. This framework helps us choose between two competing hypotheses, denoted as H1 and H2, by analyzing the available observations.
The Problem:
Imagine you're trying to detect a faint signal amidst background noise. You have two possible hypotheses:
You receive some observations, denoted by y, which are influenced by the presence or absence of the signal. Your task is to determine which hypothesis is more likely given the observed data.
Key Elements:
To make an informed decision, we need the following information:
Decision Rules:
Based on the observed data y, we need to decide which hypothesis to accept. This is achieved through a decision rule, which typically involves comparing a "decision statistic" derived from the data to a threshold. The choice of threshold influences the trade-off between false positives (accepting H1 when H2 is true) and false negatives (accepting H2 when H1 is true).
Receiver Operating Characteristic (ROC) Curve:
The ROC curve is a powerful tool for visualizing the performance of different decision rules. It plots the true positive rate (sensitivity) against the false positive rate (1 - specificity) for various threshold values. The ideal ROC curve lies close to the top-left corner, indicating high sensitivity and high specificity.
M-ary Hypothesis Testing:
Binary hypothesis testing is a special case of M-ary hypothesis testing, where we have M possible hypotheses (M > 2). This framework is useful for situations involving multiple possibilities, such as classifying different types of signals or identifying multiple targets in radar systems.
Applications:
Binary hypothesis testing finds widespread application in various engineering fields, including:
Summary:
Binary hypothesis testing is a fundamental tool for making decisions based on uncertain data. It provides a framework for evaluating the relative likelihoods of two hypotheses and selecting the most probable one. The ROC curve is an essential visual aid for understanding the performance of different decision rules. This framework extends to the more general case of M-ary hypothesis testing, enabling us to make decisions among multiple possibilities.
Instructions: Choose the best answer for each question.
1. What is the primary goal of binary hypothesis testing? (a) To calculate the probability of each hypothesis being true. (b) To determine which of two hypotheses is more likely given the observed data. (c) To predict the future outcome based on the observed data. (d) To estimate the parameters of a statistical model.
(b) To determine which of two hypotheses is more likely given the observed data.
2. Which of the following is NOT a key element in binary hypothesis testing? (a) Prior probabilities of each hypothesis. (b) Likelihood functions for each hypothesis. (c) Decision rule based on observed data. (d) The probability distribution of the noise affecting the data.
(d) The probability distribution of the noise affecting the data.
3. What does the Receiver Operating Characteristic (ROC) curve visualize? (a) The relationship between the true positive rate and the false positive rate for different decision thresholds. (b) The distribution of the observed data under each hypothesis. (c) The accuracy of a specific decision rule. (d) The likelihood of each hypothesis being true.
(a) The relationship between the true positive rate and the false positive rate for different decision thresholds.
4. In M-ary hypothesis testing, how many hypotheses are considered? (a) 1 (b) 2 (c) More than 2 (d) It depends on the specific problem.
(c) More than 2
5. Which of the following is NOT a typical application of binary hypothesis testing? (a) Detecting a specific word in a speech signal. (b) Identifying a defective component in a machine. (c) Predicting the stock market price. (d) Distinguishing between different types of cancer cells.
(c) Predicting the stock market price.
Problem:
A medical device is designed to detect the presence of a specific disease in patients. The device measures a certain biological marker in the blood. Two hypotheses are considered:
The measured marker value, y, can be modeled as a Gaussian random variable:
where N(μ, σ²) denotes a normal distribution with mean μ and variance σ².
Task:
**1. Likelihood functions:** * **p(y|H1) = (1/√(2π)) * exp(-(y-10)²/2) ** * **p(y|H2) = (1/√(2π)) * exp(-(y-5)²/2) ** **2. Decision rule:** The decision rule is based on comparing the likelihood ratio to a threshold, *T*: * **If p(y|H1) / p(y|H2) > T, then decide H1 (disease present)** * **If p(y|H1) / p(y|H2) ≤ T, then decide H2 (disease absent)** To minimize the probability of error, we can choose *T* to be the point where the two likelihood functions intersect. This point is found by setting p(y|H1) / p(y|H2) = 1 and solving for *y*. This yields *y* = 7.5. Therefore, the decision rule is: * **If y > 7.5, then decide H1 (disease present)** * **If y ≤ 7.5, then decide H2 (disease absent)** **3. Probability of false positive and false negative for T = 7.5:** * **False Positive:** Probability of deciding H1 (disease present) when H2 (disease absent) is true. This is the area under the curve of p(y|H2) for y > 7.5. * P(False Positive) = 1 - Φ((7.5 - 5)/1) = 1 - Φ(2.5) ≈ 0.0062 * **False Negative:** Probability of deciding H2 (disease absent) when H1 (disease present) is true. This is the area under the curve of p(y|H1) for y ≤ 7.5. * P(False Negative) = Φ((7.5 - 10)/1) = Φ(-2.5) ≈ 0.0062 **Note:** Φ(z) denotes the cumulative distribution function of the standard normal distribution.
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