RECEIVER OPERATING CHARACTERISTICS (ROC) FOR BINARY HYPOTHESIS TESTING

PROBLEM SETUP:

  1)  Binary hypothesis testing problem

  2)  Null hypothesis: S = zeros(10,1)

  3)  Alternate hypothesis: S = ones(10,1)

  4)  Probabilistic transition mechanism: Multidimensional Gaussian with mean vector S and covariance matrix equal to scaled identity

  5)  SNR d² defined as S^TQ^-1S where Q is the covariance matrix of the multidimensional Gaussian distribution

MATLAB CODE:

clc;

clear all;

close all;

s=ones(10,1);

nos=zeros(10,1);

sigma=[sqrt(20) sqrt(10) sqrt(5)];

imax=10000;                %NUMBER OF EXPERIMENTS

threshold=-50:2:50;        %THRESHOLD RANGE OF LOG-LIKELIHOOD RATIO(THEORETICAL RANGE FROM -INFINITY TO +INFINITY)

for j=1:length(threshold)  %LOOP TO VARY THRESHOLD

   

    numdetect1(j)=0;       %NUMBER OF DETECTIONS

    numfa1(j)=0;           %NUMBER OF FALSE ALARMS

    numdetect2(j)=0;

    numfa2(j)=0;

    numdetect3(j)=0;

    numfa3(j)=0;

    notst(j)=0;             %NUMBER OF TIMES ALTERNATE HYPOTHESIS IS TRUE

    notsnt(j)=0;            %NUMBER OF TIMES NULL HYPOTHESIS IS TRUE

   

    for i=1:imax            %EXPERIMENT NUMBER i

       

        n=randn(10,1);

        n1=sigma(1)*n;      %GENERATE NOISE FOR i-th EXPT WITH DIFFERNT POWERS CORRESPONDING TO DIFFERENT SNRs

        n2=sigma(2)*n;

        n3=sigma(3)*n;

       

        sgnlpresent=randi(2)-1;    %RANDOM GENERATION OF HYPOTHESES

       

        if sgnlpresent==1

            notst(j)=notst(j)+1;

            r1=s+n1;               %PROBABILISTIC TRANSITION MECHANISM FOR ALTERNATE HYPOTHESIS FOR DIFFERENT SNRs

            r2=s+n2;

            r3=s+n3;

        else

            notsnt(j)=notsnt(j)+1;

            r1=nos+n1;

            r2=nos+n2;             %PROBABILISTIC TRANSITITON MECHANISM FOR NULL HYPOTHESIS FOR DIFFERENT SNRs

            r3=nos+n3;

        end                        

       

        suffstat1=transpose(r1)*s;       %COMPUTE SUFFICIENT STATISTIC FROM RECEIVED OBSERVATIONS

        suffstat2=transpose(r2)*s;

        suffstat3=transpose(r3)*s;       %STATISTIC DERIVED USING BAYE'S MINIMUM RISK ANALYSIS

       

        if suffstat1>threshold(j)

            detect1=1;

        else                                 %THRESHOLDING FOR DECESION DEVICE

            detect1=0;

        end          

        if detect1==1 && sgnlpresent==1

            numdetect1(j)=numdetect1(j)+1; 

        end                                 

        if detect1==1 && sgnlpresent==0

            numfa1(j)=numfa1(j)+1;

        end          

        pd1(j)=numdetect1(j)/notst(j);

        pf1(j)=numfa1(j)/notsnt(j);

       

        if suffstat2>threshold(j)

            detect2=1;

        else

            detect2=0;

        end          

        if detect2==1 && sgnlpresent==1

            numdetect2(j)=numdetect2(j)+1;

        end                                 %COMPUTING Pr(False Alarm) and Pr(Detection) FOR DIFFERENT SNRs

        if detect2==1 && sgnlpresent==0

            numfa2(j)=numfa2(j)+1;

        end          

        pd2(j)=numdetect2(j)/notst(j);

        pf2(j)=numfa2(j)/notsnt(j);

       

        if suffstat3>threshold(j)

            detect3=1;

        else

            detect3=0;

        end          

        if detect3==1 && sgnlpresent==1

            numdetect3(j)=numdetect3(j)+1;

        end

        if detect3==1 && sgnlpresent==0

            numfa3(j)=numfa3(j)+1;

        end          

        pd3(j)=numdetect3(j)/notst(j);

        pf3(j)=numfa3(j)/notsnt(j);

    end

end

figure(1);

plot(pf1,pd1,pf2,pd2,pf3,pd3,'linewidth',1.5);

title('RECEIVER OPERATING CHARACTERISTICS FOR DIFFERENT VALUES OF SNR');

xlabel('PROBABILITY OF FALSE ALARM');

ylabel('PROBABILITY OF DETECTION');

legend('d2=0.5','d2=1','d2=2');

grid;

ROC PLOT:

SOME PHILOSOPHY:

   1)  The statistic is derived in a generic framework using Bayes’ criterion. The result can be applied with minor modifications to problems whose null hypothesis, alternate hypothesis and probabilistic transition mechanism follow the model described in the beginning. I have tried this code to detect image signals in the presence of noise and it works properly.