Description
. Find and plot the decision boundary between class ω1 and ω2. Assume P(ω1) = P(ω2).
ω1 = [1,6; 3,4; 3,8; 5,6]
ω2 = [3,0; 1,-2;3,-4;5,-2]
Q2. Find and plot the decision boundary between class ω1 and ω2. Assume P(ω1) =0.3; P(ω2)=0.7
ω1 = [1,-1; 2,-5; 3,-6; 4,-10; 5,-12; 6,-15]
ω2 = [-1,1; -2,5; -3,6; -4,10, -5,12; -6, 15]
Q3. Find and plot the decision boundary between class ω1 and ω2. Assume P(ω1) = P(ω2).
ω1 = [2,6; 3,4; 3,8; 4,6]
ω2 = [3,0; 1,-2; 3,-4; 5,-2]
Q4. Implement Bayes Classifier for Iris Dataset.
Dataset Specifications:
Total number of samples = 150
Number of classes = 3 (Iris setosa, Iris virginica, and Iris versicolor) Number of samples in each class = 50
Use the following information to design classifier:
Number of training feature vectors ( first 40 in each class) = 40
Number of test feature vectors ( remaining 10 in each class) = 10
Number of dimensions = 4
Feature vector = <sepal length, sepal width, petal length, petal width>
If the samples follow a multivariate normal density, find the accuracy of classification for the test feature vectors.
Q5. Use only two features: Petal Length and Petal Width, for 3 class classification and draw the decision boundary between them (2 dimension, 3 regions also called as multi-class problem)
Q6. Consider the 128- dimensional feature vectors given in the “face feature vectors.csv” file. Use this information to design and implement a Bayes Classifier.
Dataset Specifications:
Total number of samples = 800
Number of classes = 2 ( labelled as “male” and “female”)
Samples from “1 to 400” belongs to class “male”
Samples from “401 to 800” belongs to class “female”
Number of samples per class = 400
Use the following information to design classifier:
Number of test feature vectors ( first 5 in each class) = 5
Number of training feature vectors ( remaining 395 in each class) = 395 Number of dimensions = 128
Design of Bayes Classifier
Given,
Iris dataset
𝑋 =< 𝑥1 , 𝑥2, 𝑥3, 𝑥4 >
Number of classes= 𝜔1, 𝜔2, 𝜔3 ; c=3
N=150; n(𝜔1)=n(𝜔1)=n(𝜔1)=50 Bayes Rule:
Find 𝑃(𝜔𝑖|𝑋) = 𝑃(𝑋|𝜔𝑃(𝑖𝑋)).𝑃(𝜔𝑖)
𝑃(𝑋) is a constant for all classes; so it can be ignored.
Steps to follow in Iris Classification:
- Find apriori probability 𝑃(𝜔𝑖) = 𝑛(𝑁𝜔𝑖) =
- Find 𝑃(𝑋|𝜔𝑖), it’s multivariate class, by following normal density
𝑃(𝑋|𝜔𝑖) = (2𝜋)𝑑/21 |𝛴𝑖|1/2 𝑒𝑥𝑝 [− {(𝑋 − µ𝑖)𝑡𝛴𝑖−1(𝑋 − µ𝑖)}]
2 a. Find the mean vector
2 b. Find the covariance matrix, 𝛴𝑖
2 c. Find the |𝛴𝑖| and |𝛴𝑖|−1
- Find 𝑃(𝜔1|𝑋), 𝑃(𝜔2|𝑋) 𝑎𝑛𝑑 𝑃(𝜔3|𝑋). Find the maximum and assign 𝑋 to that class. Also, plot the accuracy for : i) Separate classes ii) Overall performance
- Find the discriminant function and draw the decision surface between the classes.
