PR-Assignment 2 Design of Bayes Classifier Solved

. 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:

 

1.  Find apriori probability  𝑃(𝜔𝑖) = 𝑛 (𝑁𝜔𝑖) =   

2.  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

3.  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

4.  Find the discriminant function and draw the decision surface between the classes.