Experiment-5-Implementation-of-XOR-using-RBF

AIM:

To classify the Binary input patterns of XOR data by implementing Radial Basis Function Neural Networks.

EQUIPMENTS REQUIRED:

Hardware – PCs Anaconda – Python 3.7 Installation / Google Colab /Jupiter Notebook

RELATED THEORETICAL CONCEPT:

Exclusive or is a logical operation that outputs true when the inputs differ.For the XOR gate, the TRUTH table will be as follows XOR truth table

XOR is a classification problem, as it renders binary distinct outputs. If we plot the INPUTS vs OUTPUTS for the XOR gate, as shown in figure below

The graph plots the two inputs corresponding to their output. Visualizing this plot, we can see that it is impossible to separate the different outputs (1 and 0) using a linear equation.

A Radial Basis Function Network (RBFN) is a particular type of neural network. The RBFN approach is more intuitive than MLP. An RBFN performs classification by measuring the input’s similarity to examples from the training set. Each RBFN neuron stores a “prototype”, which is just one of the examples from the training set. When we want to classify a new input, each neuron computes the Euclidean distance between the input and its prototype. Thus, if the input more closely resembles the class A prototypes than the class B prototypes, it is classified as class A ,else class B.

A Neural network with input layer, one hidden layer with Radial Basis function and a single node output layer (as shown in figure below) will be able to classify the binary data according to XOR output.

The RBF of hidden neuron as gaussian function

ALGORIHM:

Step 1 :

Initialize the input patterns for XOR Gate

Step 2:

Initialize the desired output of the XOR Gate

Step 3:

Define the function for RBF and function for prediction.

Step 4:

Plot the graphs with inputs

Step 5:

Find the weights

Step 6:

Plot the graph with transformed inputs using RBF

Step 7 :

Test for the XOR patterns.

PROGRAM:

import numpy as np
import matplotlib.pyplot as plt

def gaussian_rbf(x, landmark, gamma=1):
    return np.exp(-gamma * np.linalg.norm(x - landmark)**2)

def predict_matrix(point, weights):
    gaussian_rbf_0 = gaussian_rbf(np.array(point), mu1)
    gaussian_rbf_1 = gaussian_rbf(np.array(point), mu2)
    A = np.array([gaussian_rbf_0, gaussian_rbf_1, 1])
    return np.round(A.dot(weights))

x1 = np.array([0, 0, 1, 1])
x2 = np.array([0, 1, 0, 1])
ys = np.array([0, 1, 1, 0])

plt.figure(figsize=(13, 5))
plt.subplot(1, 2, 1)
plt.scatter((x1[0], x1[3]), (x2[0], x2[3]), label="Class_0")
plt.scatter((x1[1], x1[2]), (x2[1], x2[2]), label="Class_1")
plt.xlabel("X1")
plt.ylabel("X2")
plt.title("Linearly Inseparable")
plt.legend()

# centers
mu1 = np.array([0, 1])
mu2 = np.array([1, 0])

from_1 = [gaussian_rbf(i, mu1) for i in zip(x1, x2)]
from_2 = [gaussian_rbf(i, mu2) for i in zip(x1, x2)]

A = []

for i, j in zip(from_1, from_2):
    temp = []
    temp.append(i)
    temp.append(j)
    temp.append(1)
    A.append(temp)
    
A = np.array(A)
W = np.linalg.inv(A.T.dot(A)).dot(A.T).dot(ys)
print(np.round(A.dot(W)))
print(ys)
print("Weights:",W)

plt.figure(figsize=(13, 5))
plt.subplot(1, 2, 2)
plt.scatter(from_1[0], from_2[0], label="Class_0")
plt.scatter(from_1[1], from_2[1], label="Class_1")
plt.scatter(from_1[2], from_2[2], label="Class_1")
plt.scatter(from_1[3], from_2[3], label="Class_0")
plt.plot([0, 0.95], [0.95, 0])
plt.annotate("Seperating hyperplane", xy=(0.5, 0.5), xytext=(0.5, 0.5))
plt.xlabel("µ1")
plt.ylabel("µ2")
plt.title("Transformed Inputs")
plt.legend()

print(f"Input:{np.array([0, 0])}, Predicted: {predict_matrix(np.array([0, 0]), W)}")
print(f"Input:{np.array([0, 1])}, Predicted: {predict_matrix(np.array([0, 1]), W)}")
print(f"Input:{np.array([1, 0])}, Predicted: {predict_matrix(np.array([1, 0]), W)}")
print(f"Input:{np.array([1, 1])}, Predicted: {predict_matrix(np.array([1, 1]), W)}")

subalakshmivenkat / experiment-5-implementation-of-xor-using-rbf Goto Github PK