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Relational.py
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Relational.py
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import numpy as np
import math
import matplotlib.pyplot as plt
class Relational_Data:
""" Here, assumptions are made that there exists a non-linear mapping which projects the data into a Euclidean Space
thus, there is a Dissimilarity matrix (interpreted Euclidean Space X_E), which contains the pairwise dissimilarities
between the data objects. The matrix D is symmetric, where D_ij = D_ji and D_ii = 0. Also, unlike Vectorial data
there is no direct prototype involved but, they are described indirectly by the coefficient Alpha_j so, by
updating Alpha virtual prototypes are generated. """
def __init__(self, file, number_of_prototype):
# A M x M Dissimilarity matrix.
self.D = np.loadtxt(file, delimiter=',')
self.Distance = np.asarray(self.D)
# Number of clusters k.
self.number_of_prototypes = number_of_prototype
# Length of the Dissimilarity matrix.
self.length = len(self.Distance)
# Degree of Membership values.
self.mem = 1.2
# Degree of Typicality values.
self.typ = 1.5
# Constant Scalars.
self.sig = 0.5
self.a = 0.8
self.b = 0.2
self.gamma = 1.0
# An M x k matrix contains the degree of D_i belonging to cluster k.
self.beta = np.zeros((self.length, self.number_of_prototypes))
# Initialisation of Membership and Typicality lists.
self.typicality_matrix = []
self.membership_matrix = []
self.prototype = np.zeros((self.number_of_prototypes, self.length))
# Initialisation of Centering MxM Matrix
self.CenteringData = np.zeros((self.length, self.length))
# Functions to be called initially.
self.initializeTypicalityMatrix()
self.initializeMembershipMatrix()
self.generate_coefficient()
self.DoubleCentering()
def generate_coefficient(self):
""" This function generates the coefficient Alpha which should range (0,1], with the condition that the sum of
every single row should be equal to 1.
:return: k x M matrix
"""
number_of_rows = self.Distance.shape[0]
random_indices = np.random.choice(number_of_rows, size=self.number_of_prototypes, replace=False)
self.Alpha = self.Distance[random_indices, :]
summ = np.sum(self.Alpha, axis=1)
self.Alpha = np.divide(self.Alpha.T, summ)
self.Alpha = self.Alpha.T
return self.Alpha
def generate_prototype(self):
""" The prototypes are updated indirectly using the updated Alpha values.
:return: The matrix of k x M of actual prototype.
"""
for j in range(self.number_of_prototypes):
self.prototype[j] = 0
for i in range(self.length):
self.prototype[j] += self.Alpha[j][i] * self.Distance[i]
def distanceDataNeuron(self, i, j):
""" The distances between the D_i and Alpha_j for Relational data can be calculated using this function.
:param i: index of the Dissimilarity matrix.
:param j: index of the coefficient Alpha.
:return: Distance between the D_i and Alpha_j.
"""
first_term = 0.0
normalise_term = 0.5 * np.dot(self.Alpha[j].T, np.dot(self.Distance, self.Alpha[j]))
for l in range(self.length):
first_term += self.Alpha[j][l] * self.Distance[i][l]
distance_ij = first_term - normalise_term
return distance_ij
def CalculateCentering(self, data_r, data_s):
""" Takes the indexes of the data and returns the value at that particular location.
:param data_r: index of Dissimilarity data.
:param data_s: index of the Dissimilarity data.
:return: Value after double centering the data at index data_r and data_s.
"""
second_term = 0.0
third_term = 0.0
fourth_term = 0.0
first_term = self.Distance[data_r][data_s]
for r in range(self.length):
second_term += self.Distance[r][data_s]
for s in range(self.length):
third_term += self.Distance[data_r][s]
for r in range(self.length):
for s in range(self.length):
fourth_term += self.Distance[r][s]
self.CenteringData[data_r][data_s] = -0.5 * (
first_term - (1 / self.length) * second_term - (1 / self.length) * third_term + (
1 / (self.length * self.length) * fourth_term))
def DoubleCentering(self):
""" Double centering the data means any row or any column should sum to 0. """
for vr in range(self.length):
for vs in range(self.length):
self.CalculateCentering(vr, vs)
def distance_prototype(self, j, k):
""" Calculates the distance between the Alphas.
:param j: jth index of the Alpha.
:param k: kth index of the Alpha.
:return: Distance between the Alpha at j and Alpha at k.
"""
distance_proJK = 0.0
for vr in range(self.length):
for vs in range(self.length):
centering_value = self.CenteringData[vr][vs]
first_term = self.Alpha[j][vr] * self.Alpha[j][vs]
second_term = 2 * self.Alpha[j][vr] * self.Alpha[k][vs]
third_term = self.Alpha[k][vr] * self.Alpha[k][vs]
pro_value = (first_term + third_term) - second_term
distance_proJK += centering_value * pro_value
return distance_proJK
def local_loss(self, i, j):
"""
@:var second_term: float value which contains value of typicality[i][l] to the power of typicality constant
and then multiplied with scalar b.
@:var first term: float value which contains value of membership[i][l] to the power of membership constant
and then multiplied with scalar a.
:return: a M x k matrix.
"""
local_error = 0.0
for l in range(self.number_of_prototypes):
local_error += self.neighborhood_function(j, l) * self.distanceDataNeuron(i, l)
return local_error
def neighborhood_function(self, j, l):
"""
:param j: contains the index of jth Alpha.
:param l: contains the index of lth Alpha.
@:var rank_lj: int value returned by the ranking function.
@:var sigma: a double value.
@:var denominator: a double value.
@:var numerator: int value containing (rank_lj)^2.
@:var rank_neighborhood: double value containing the neighborhood value.
:return: the neighborhood range according to the winning Alpha.
"""
rank_jl = self.rank_ij(j, l)
sigma = 1 / math.sqrt(2 * math.pi * self.sig * self.sig)
denominator = 2 * self.sig * self.sig
numerator = rank_jl * rank_jl
rank_neighborhood = sigma * (math.pow(math.e, -(numerator / denominator)))
return rank_neighborhood
def rank_ij(self, j, l):
"""
calculates the rank of the lth Alpha according to jth Alpha.
@:param j: contains the index of jth Alpha.
@:param l: contains the index of lth Alpha.
@:var prototype_l: contains the 1 x M vector with index j.
@:var prototype_j: contains the 1 x M vector with index l.
@:var rank: everytime adds 1, when the d(l, j) - d(l, k) > 0 else 0.
@:returns: the rank of lth Alpha according to jth Alpha.
"""
rank = 0
distance_lj = self.distance_prototype(l, j)
for k in range(self.number_of_prototypes):
if (distance_lj - self.distance_prototype(l, k)) > 0:
rank = rank + 1
return rank
def beta_ij(self):
"""
@:var second_term: float value which contains value of typicality[i][j] to the power of typicality constant
and then multiplied with scalar b.
@:var first term: float value which contains value of membership[i][j] to the power of membership constant
and then multiplied with scalar a.
:return: a M x k matrix.
"""
for i in range(self.length):
for j in range(self.number_of_prototypes):
typ = self.b * (math.pow(self.typicality_matrix[i][j], self.typ))
mem = self.a * (math.pow(self.membership_matrix[i][j], self.mem))
self.beta[i][j] = mem + typ
def initializeMembershipMatrix(self):
"""
@:var random_num_list: contains the list of size K with random numbers in range [0.0 , 1.0).
@:var summation: contains the sum of list.
@:var temp_list: contains the list which sums up to 1.
:return: a N x K membership matrix.
"""
for i in range(self.length):
random_num_list = [np.random.random() for j in range(self.number_of_prototypes)]
summation = sum(random_num_list)
temp_list = [num / summation for num in random_num_list]
self.membership_matrix.append(temp_list)
return self.membership_matrix
def initializeTypicalityMatrix(self):
"""
@:var random_number_list: contains the list of size K with random numbers in range [0.0 , 1.0).
:return: a N x K typicality matrix.
"""
for i in range(self.length):
random_num_list = [np.random.random() for j in range (self.number_of_prototypes)]
self.typicality_matrix.append(random_num_list)
return self.typicality_matrix
def updateMembershipMatrix(self, i, j):
"""
updates the membership of the data at index i according to the Alpha at index j.
:param i: index of the ith data.
:param j: index of the jth Alpha.
@var power_m: float value.
@var i: contains data at index i.
@var j: contains neuron at index j.
@var numerator: calculates the numerator part of updating the membership equation.
@var denominator: calculates the denominator part of updating the membership equation.
@:var div: contains the result after division of the numerator with denominator.
"""
div = 0.0
power_mem = float(1 / (self.mem - 1))
numerator = self.local_loss(i, j)
for l in range(self.number_of_prototypes):
denominator = self.local_loss(i, l)
div += pow(numerator / denominator, power_mem)
temp = 1 / div
self.membership_matrix[i][j] = temp
def updateTypicalityMatrix(self, i, j):
"""
:param i: index of the ith data.
:param j: index of the jth neuron.
@var num: calculates the numerator part of updating the typicality equation.
@var den: calculates the denominator part of updating the typicality equation
@var div: contains the result after division of the numerator with denominator.
"""
power_typ = 1 / (self.typ - 1)
num = self.b * self.local_loss(i, j)
den = self.gamma
div = 1 + math.pow((num / den), power_typ)
temp = 1 / div
self.typicality_matrix[i][j] = temp
def normaliseAlpha(self):
""" Everytime the coefficients are updates, there is a need for normalisation of these coefficient
so that the every row sum should be equal to one
@:var sum_alpha: takes the sum of row"""
for p in range(self.number_of_prototypes):
sum_alpha = 0.0
for k in range(self.length):
sum_alpha += self.Alpha[p][k]
for j in range(self.length):
self.Alpha[p][j] = self.Alpha[p][j] / sum_alpha
def visualise_update(self):
""" When the coefficients gets updates, this functions visualises where the prototypes are moving.
:return: An image as output which contains scatter plots.
"""
plt.scatter(self.Distance[:, 2], self.Distance[:, 1])
plt.scatter(self.prototype[:, 2], self.prototype[:, 1], marker='x', c='r')
plt.show()
plt.close()
def update_coefficient(self, p, q):
""" This function takes coefficient matrix and updates the values of the coefficient using the Stochastic
gradient descent learning.
:param q: index of the qth column of the coefficient matrix.
:param p: index of the pth row of the coefficient matrix.
"""
learning_rate = 0.002
self.beta_ij()
third_term = 0.0
final_term = 0.0
for i in range(self.length):
beta_ip = self.beta[i][p]
for l in range(self.number_of_prototypes):
cal_neighborhood = self.neighborhood_function(p, l)
third_term += cal_neighborhood * (self.Distance[i] - (self.Distance[q] * self.Alpha[p]))
final_term += beta_ip * third_term
third_term = 0.0
self.Alpha[p] -= learning_rate * final_term
class RelationalNeuralGas:
def __init__(self, data, epochs):
"""
:param data: contains the dataset which is unlabelled.
:param epochs: number of epochs to update the prototype.
@:var self.dec_sigma: factor to which the sigma value should be decreased.
"""
self.epoch = epochs
self.dec_sigma = 0.002
for e in range(self.epoch):
print('Prototype update at epoch ' + str(e + 1))
print(dataset.Alpha)
for p in range(dataset.number_of_prototypes):
for q in range(dataset.length):
dataset.update_coefficient(p, q)
dataset.normaliseAlpha()
dataset.visualise_update()
dataset.generate_prototype()
self.updateMembershipTypicality()
if (dataset.sig - self.dec_sigma) > 0:
dataset.sig = dataset.sig - self.dec_sigma
def updateMembershipTypicality(self):
"""
updateMemTyp(): updates the membership values and typicality values after the every
iteration of the prototype update.
"""
for i in range(dataset.length):
for j in range(dataset.number_of_prototypes):
dataset.updateMembershipMatrix(i, j)
dataset.updateTypicalityMatrix(i, j)
dataset = Relational_Data(r'C:\Users\amitk\Desktop\Programming_Project\D.csv', 3)
neuralgas = RelationalNeuralGas(dataset, 7)