1. Name: Shivakumar Koppad
Date: 12/03/2015
Submitted to: Dr. Richard Conway
Description

2. Two samplefiles G.dat and I.datwith genuine and imposter sample scores
respectively were given and the following should be obtained from those.
 Plot the scoredistributions for both
 DET curve
 Determining EER
 Operation point to minimize the cost
MATLAB 2013a has been used to show the requirements
Scores Distributioncurve
The following code has been used to generate the graph to plot the score for
distribution curve.
%Name: Shivakumar Koppad
clc;
clear all;
close all;
% Importing data
load('G.dat');
load('I.dat');
% Implementing G file
gen=[0 0 0 0 0 0 0 0 0 0];
for g=1:1:1200
if 0<=G(g)<0.1
gen(1)=gen(1)+1;
elseif 0.1<=G(g)<0.2
gen(2)=gen(2)+1;
elseif 0.2<=G(g)<0.3
gen(3)=gen(3)+1;
elseif 0.3<=G(g)<0.4
gen(4)=gen(4)+1;
elseif 0.4<=G(g)<0.5
gen(5)=gen(5)+1;
elseif 0.5<=G(g)<0.6
gen(6)=gen(6)+1;
elseif 0.6<=G(g)<0.7
gen(7)=gen(7)+1;
elseif 0.7<=G(g)<0.8
gen(8)=gen(8)+1;

3. elseif 0.8<=G(g)<0.9
gen(9)=gen(9)+1;
else
gen(10)=gen(10)+1;
end
end
% Implementing I file
imp=[0 0 0 0 0 0 0 0 0 0];
for i=1:1:1200
if 0<=I(i)<0.1
imp(1)=imp(1)+1;
elseif 0.1<=I(i)<0.2
imp(2)=imp(2)+1;
elseif 0.2<=I(i)<0.3
imp(3)=imp(3)+1;
elseif 0.3<=I(i)<0.4
imp(4)=imp(4)+1;
elseif 0.4<=I(i)<0.5
imp(5)=imp(5)+1;
elseif 0.5<=I(i)<0.6
imp(6)=imp(6)+1;
elseif 0.6<=I(i)<0.7
imp(7)=imp(7)+1;
elseif 0.7<=I(i)<0.8
imp(8)=imp(8)+1;
elseif 0.8<=I(i)<0.9
imp(9)=imp(9)+1;
else
imp(10)=imp(10)+1;
end
end
% Defining Score
Score=[0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9];
hold on;
plot(Score,imp,'color','r');
plot(Score,gen,'color','b');
hold off;
xlabel('Scores')
ylabel('No. of users')
title('Distribution Diagram')
legend('genuine','imposter')
% End
Result:
In the below output of the graph, X-axis represents the scores and Y-axis
represents the count for both genuine and imposter users.

4. DET curve
Detection Error Tradeoff curves areROC (receiver operating characteristic) type
curves showing therange of operating points of systems performing detection
tasks as a threshold is varied to alter the miss and false alarmrates and plotted
using a normal deviate scale for each axis. DET curves have the property that if
the underlying scoredistributions for the two types of trials are normal, the curve
becomes a straightline. They havebeen widely used to present the performance
characteristics of speaker recognition systems
The below code is used to generate the DET curve. A new function name called
DET curveis created to generate the graph and to find the EER equal error rate on
the curve.
%Name: Shivakumar Koppad
%ID: 14043513
%Title: Assignment 1
%
%Submitted to: Dr.Richard Conway

5. function [TPR,FPR] = DETCurve(G,I)
load('G.dat');
load('I.dat');
% Creates a matrix
TPR = zeros(1, 1000);
FPR = zeros(1, 1000);
P = 1200;
N = 1200;
index = 0;
% Assume the threshold as 0.005
for threshold = 0:0.005:1
TP = 0;
FP = 0;
%Provides the genuine users count
for i = 1:1200
if (G(i) >= threshold)
TP = TP + 1;
end
end
% Provides the Imposters count
for i1 = 1:1200
if(I(i1) >= threshold)
FP = FP + 1;
end
end
index = index + 1;
% Calculating true positive rate
TPR(index) = TP/P;
% Calculating false positive rate
FPR(index) = FP/N;
end
% Calculating false negative rate(FNR) using TPR+FNR=1
FNR = (1-TPR);
x = 0:0.1:1;
y = x;
[x(i),y(i)] = polyxpoly(x,y,FPR,FNR);
fprintf('EER(X): %d n', x(i));
fprintf('EER(Y): %d n', y(i));
plot(FPR,FNR,'LineWidth',2, 'color','g');
hold on;
plot(x,y,x,1-y, 'color','r');
plot (x(i),y(i),'X','MarkerSize',10, 'LineWidth', 2,'Color','b');
hold off;
title('DET CURVE');
xlabel('False Positive Rate (FPR) ');
ylabel('False Neagtive Rate (FNR) ');
end

6. Result
The below is the DET curve generated

7. Equal error rate
A biometric security systempredetermines the threshold values for its false
acceptance rate and its false rejection rate, and when the rates are equal, the
common value is referred to as the equal error rate. The value indicates that the
proportion of falseacceptances is equal to the proportion of false rejections. The
lower the equal error rate value, the higher the accuracy of the biometric system.
Fromabove figurewe can determine EER fromthe DET curve
EER(X, Y) = (FPR, FNR) = (0.08611, 0.08611)
Cost minimization
OP_min_costis created to find a suitable point on the DET curve which minimizes
the cost
FromBayesian statistics, the Cost (C) is calculated as:
C=CFAPIFAR(T) + CFRPGFRR(T)
Where,
CFA is the costof a single false accept
CFR is the cost of a single false reject
PI is probability of imposter users
PG is probability of genuine users
Given
Cost of False accept rate is 17
Cost of false reject rate is 27
Probability of Imposter users is 0.5
Probability of Genuine users is 0.5 (Weare assuming equal probabilities)
C = 17*0.5*FAR(T) +27*0.5*FRR(T)

8. C = 8.5*FAR(T) +13.5*FRR(T)
13.5*FRR(T) =C- 8.5*FAR(T)
FRR(T) = [C- 8.5*FAR(T)]/13.5
FRR(T) = -0.6296*FAR(T) +C/13.5
This is in the form of y= mx+c
The following code has been used to generate the graph which can be used to
find FRR(T) and FAR(T) the point operated to minimize the cost

9. function [FAR, FRR] = OPcost(G,I)
load('G.dat');
load('I.dat');
%Creates a matrix
FRR = zeros(1, 1000);
FAR = zeros(1, 1000);
i = 1200;
j = 1200;
index = 0;
%Assume the threshold as 0.001
for threshold = 0:0.001:1
FN = 0;
FP = 0;
for i = 1:1200
if (G(i) <= threshold)
FN = FN + 1;
end
end
for i1 = 1:1000
if(I(i1) >= threshold)
FP = FP + 1;
end
end
index = index + 1;
FRR(index) = FN/i;
FAR(index) = FP/j;
end
hold on;
plot(FAR,FRR,'LineWidth',1,'color','c');
%To start with assume the minimum cost as 100
min = 100;
min_new=0;
for i1=1:1000
if 17*0.5*FAR(i1) + 27*0.5*FRR(i1) < min
min = 17*0.5*FAR(i1) + 27*0.5*FRR(i1);
min_new = i1;
end
end
plot(FAR(min_new),FRR(min_new),'x','MarkerSize',14);
Xmin = 0:0.05:1;
Ymin = (-0.33)*Xmin + 0.06989*min;
plot (Xmin,Ymin,'color', 'm');
hold off;
title('Operating point to minimise cost');
xlabel('FAR ');
ylabel('FRR ');

10. Result
According to the graph
FAR(T) = 0.02333
FRR(T) = 0.1092
Now,
13.5*FRR(T) =C- 8.5*FAR(T)
13.5*0.1092=C-8.5*0.02333
1.4742=c-0.198305
Therefore, C = 1.6725
So the cost C is 1.6725