Introduction to Signal Detection and Classification. Phani Chavali
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1 Introduction to Signal Detection and Classification Phani Chavali
2 Outline Detection Problem Performance Measures Receiver Operating Characteristics (ROC) F-Test - Test Linear Discriminant Analysis (LDA)
3 Statistical Detection Problem The classical detection problem is to decide which of the many hypothesis is true using the noisy observations. Binary hypothesis testing
4 Examples
5 Examples
6
7 Likelihood ratio test Compute the likelihood ratio under the two hypothesis. Compare the likelihood ratio to a threshold. If the ratio is greater than the threshold, choose the alternate hypothesis; else choose the null hypothesis. Likelihood ratio is called as test-statistic.
8 Performance Analysis In binary hypothesis testing, the following events are possible: Accordingly, we assign four probabilities:
9 Performance Analysis Performance is determined by two kinds of probabilities. Probability of false alarm Probability of miss
10 We cannot minimize both probabilities simultaneously
11 Receiver operating characteristic Graph of probability of detection Vs the probability of false alarm. In this class, we plot the graphs as a function of the threshold.
12 F - test F-Test is a statistical test where the test statistic follows an F-distribution. Most commonly used scenario: The observations come from several distributions. Null hypothesis corresponds to the case where means of all the distributions are same. Alternate hypothesis corresponds to the case where the at least one mean is different.
13 How to do an F-test Compute the F-statistic Compare with a threshold. Between group variability refers to the variability between various groups. Within-group variability refers to variability in each group
14 - Test - Test is similar to the F-test and it is used in models whose purpose is to predict the future values using the observed values.
15 Classification Classification is the assignment of the data to one of pre-defined classes. Given noisy observations, we need to classify them into one of the several classes. Statistically, classification is equivalent to M- ary hypothesis test.
16 Methods There are many procedures to do this. Linear Discriminant Analysis (LDA) K-NN Logistic and probit regression Support Vector Machines and many more We will learn about LDA today.
17 Features Given the observation, find a feature (similar to the test-statistic or the likelihood ratio) that separates various classes. The goal is to find which is the best feature. Examples from the same class should have similar feature values Examples from different classes have different feature values
18 More Properties Some times it is better to select several features, instead of just one.
19 Linear discriminant analysis What class information may be useful? Between-class distance Distance between the centroids of different classes Within-class distance Accumulated distance of an instance to the centroid of its class
20 How it works? Linear discriminant analysis (LDA) finds most discriminant projection by maximizing between-class distance and minimizing withinclass distance
21 Some Theory (1) Assume we have a set of D-dimensional samples (x1, x2, xn), N1 of which belong to class ω1 and N2 to class ω2 We seek to obtain a scalar y by projecting the samples x onto a line Of all the possible lines we would like to select the one that maximizes the separability of the scalars
22 Some Theory(2) In order to find a good projection vector, we need to define a measure of separation The mean vector of each class in x-space and y-space is We could then choose the distance between the projected means as our objective function
23 Some Theory(3) However, the distance between projected means is not a good measure since it does not account for the standard deviation within classes
24 Some Theory(4) We use the difference between the means, normalized by a measure of the within-class variance For each class we define the scatter, an equivalent of the variance, as The linear discriminant is defined as the linear function that maximizes the criterion function
25 Some Theory(5) Therfore, we are finding most discriminant projection by maximizing between-class distance and minimizing within-class distance
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