Line 36: | Line 36: | ||
=='''False Alarm Rate and Miss Rate'''== | =='''False Alarm Rate and Miss Rate'''== | ||
From statistical references [1, 2], these two hypotheses make one of two types of error, named Type I Error and Type II Error. If <math> \theta \in \Theta_0 </math> but the hypothesis test incorrectly decides to reject <math> H_0 </math>, then the test has made a '''Type I Error'''. If, on the other hand, <math> \theta \in \Theta_1 </math> but the test decides to accept <math> H_0 </math> a '''Type II Error''' has been made. | From statistical references [1, 2], these two hypotheses make one of two types of error, named Type I Error and Type II Error. If <math> \theta \in \Theta_0 </math> but the hypothesis test incorrectly decides to reject <math> H_0 </math>, then the test has made a '''Type I Error'''. If, on the other hand, <math> \theta \in \Theta_1 </math> but the test decides to accept <math> H_0 </math> a '''Type II Error''' has been made. | ||
+ | |||
+ | <center> | ||
+ | <table style="width:300px"> | ||
+ | <tr> | ||
+ | <td></td> | ||
+ | <td>Decision <br> Accpet <math> H_0 </math> </td> | ||
+ | <td>Decision <br> Reject <math> H_0 </math></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>Eve</td> | ||
+ | <td>Jackson</td> | ||
+ | <td>94</td> | ||
+ | </tr> | ||
+ | </table | ||
+ | |||
+ | |||
\begin{table}[htbp!] | \begin{table}[htbp!] |
Revision as of 21:24, 30 April 2014
Neyman-Pearson Lemma and Receiver Operating Characteristic Curve
A slecture by ECE student Soonam Lee
Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.
Contents
Introduction
The purpose of this slecture is understanding Neyman-Pearson Lemma and Receiver Operating Characteristic (ROC) curve from theory to application. The fundamental theories stem from statistics and these can be used for signal detection and classification. In order to firmly understand these concepts, we will review statistical concept first including False alarm and Miss. After that, we will consider Bayes' decision rule using cost function. Next, we visit Neyman-Pearson test and minimax test. Lastly, we will discuss ROC curves. Note that we only consider two classes case in this slecture, but the concept can be extended to multiple classes.
The General Two Classes Case Problem
Before starting our discussion, we have the following setup:
- $ X $ a measure random variable, random vector, or random process
- $ x \in \mathcal{X} $ is a realization of $ X $
- $ \theta \in \Theta $ are unknown parameters
- $ f(x; \theta) $ is pdf of $ X $ (a known function)
Two distinct hypotheses on $ \theta $ such that $ H_0: \theta \in \Theta_0 $ versus $ H_1: \theta \in \Theta_1 $ where $ \Theta_0 $, $ \Theta_1 $ is partition of $ \Theta $ into two disjoint regions
False Alarm Rate and Miss Rate
From statistical references [1, 2], these two hypotheses make one of two types of error, named Type I Error and Type II Error. If $ \theta \in \Theta_0 $ but the hypothesis test incorrectly decides to reject $ H_0 $, then the test has made a Type I Error. If, on the other hand, $ \theta \in \Theta_1 $ but the test decides to accept $ H_0 $ a Type II Error has been made.
Decision Accpet $ H_0 $ |
Decision Reject $ H_0 $ |
|
Eve | Jackson | 94 |