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Muon-Lifetime/muon.m
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num=xlsread('muon.xlsx'); | |
x=(num(:,1)); | |
x=x./1000; | |
step=0.4; | |
X=0:step:max(x); | |
figure(1) | |
h=histogram(x,X); | |
xlabel('Decay Time (us)') | |
ylabel('Observed Counts') | |
hold on | |
y=h.Values; | |
XX=(step/2):step:max(x); | |
XX=XX(1:end-1); | |
errorbar(XX,y,sqrt(y),'o') | |
%linear and nonlinear fit before background subtraction | |
%plot(XX,log(y),'o') | |
fun1=@(v) v(1)*exp(-v(2)*XX)-y; | |
x0=[4000,1]; | |
[v,resnorm,residual,exitflag,output,lambda,jacobian]=lsqnonlin(fun1,x0); | |
Jacobian = full(jacobian); | |
varv=resnorm*inv(Jacobian'*Jacobian)/length(XX); | |
stdv=sqrt(diag(varv)); | |
Y1=v(1)*exp(-v(2)*XX); | |
plot(XX,Y1,'g') | |
legend('Observed Decays','Uncertainty','Best Fit Without Background Subtraction') | |
title('Nonlinear Regression with no Consideration of Bacground Effects') | |
ylabel('Bin Count') | |
xlabel('Decay Time (microseconds)') | |
%hold on | |
%plot(XX,y) | |
%% | |
%linear and nonlinear fit after background subtraction | |
clearvars v resnorm residual stdv jacobian | |
figure(2) | |
count=sum(y(XX>12)); | |
avg=count/(length(y(XX>12))); | |
Y=y-avg; | |
Y(Y<0)=0; | |
%plot(XX,log(Y),'o') | |
h=histogram(x,X); | |
xlabel('Decay Time (us)') | |
ylabel('Observed Counts') | |
hold on | |
y=h.Values; | |
errorbar(XX,y,sqrt(y),'o') | |
%fun2=@(v) v(1)*exp(-v(2)*XX)+v(3);%-y; | |
%v=[50,5,1]; | |
%fun2=@(v,x) (v(1)*exp(-v(2)*x)+v(3)); | |
%nlinfit(XX,y,fun2,X0) | |
fun2=@(v) v(1)*exp(-v(2)*XX)+v(3)-y; | |
X0=[4000,5,10]; | |
[v,resnorm,residual,exitflag,output,lambda,jacobian]=lsqnonlin(fun2,X0); | |
ci=nlparci(v,residual,'Jacobian',jacobian); | |
v=v'; | |
Jacobian = full(jacobian); | |
varv=resnorm*inv(Jacobian'*Jacobian)/length(XX); | |
stdv=sqrt(diag(varv)); | |
Y=v(1)*exp(-v(2)*XX)+v(3); | |
YY=log(v(1))-v(2)*XX+log(v(3)); | |
plot(XX,Y,'g') | |
legend('Observed Decays','Uncertainty','Best Fit With Background Subtraction') | |
title('Nonlinear Regression with Consideration of Bacground Effects') | |
ylabel('Bin Count') | |
xlabel('Decay Time (microseconds)') | |
%hold on | |
%plot(XX,y) | |
%% Maximum likelihood | |
%without background subtraction | |
lambda=0.275:0.0001:0.282; | |
%lambda=0.1:0.01:0.6; | |
P=zeros(length(lambda),length(x)); | |
mP=zeros(1,length(lambda)); | |
for i=1:length(lambda) | |
for j=1:length(x) | |
P(i,j)=lambda(i)*exp(-lambda(i)*x(j)); | |
end | |
end | |
P=log(P); | |
for i=1:length(lambda) | |
mP(i)=sum(P(i,:)); | |
end | |
figure(5) | |
plot(lambda,mP) | |
hold on | |
yy=(max(mP(:))-0.5)*ones(1,length(lambda)); | |
plot(lambda,yy) | |
vs=(mP((mP-yy>-0.04 & mP-yy<0.04))); | |
Vs=zeros(1,2); | |
for i=1:length(vs) | |
Vs(i)=lambda(mP==vs(i)); | |
end | |
plot(Vs,yy(1:2),'o') | |
lambda1=lambda(mP==max(mP)); | |
siga=abs(lambda1-Vs(1)); | |
sigb=abs(lambda1-Vs(2)); | |
yy1=get(gca,'ylim'); | |
plot([lambda1,lambda1],yy1); | |
title('Maximum Likelihood Approach with no Consideration of Background') | |
xlabel('Lambda (s^-1)') | |
ylabel('Sum of log of Probabilities') | |
legend('Sum of log of Probabilities','Maximum-0.5') | |
%% with background subtraction | |
lambda=0.48:0.0002:0.50; | |
b=0.0095:0.00002:0.0101; | |
P=zeros(length(lambda),length(b),length(x)); | |
mP=zeros(length(lambda),length(b)); | |
for i=1:length(lambda) | |
for j=1:length(b) | |
for k=1:length(x) | |
P(i,j,k)=lambda(i)*(1-b(j)*20)*exp(-lambda(i)*x(k))+b(j); | |
end | |
end | |
end | |
P=log(P); | |
for i=1:length(lambda) | |
for j=1:length(b) | |
mP(i,j)=sum(P(i,j,:)); | |
end | |
end | |
figure(6) | |
[XXX,YYY]=meshgrid(b,lambda); | |
surf(XXX,YYY,mP) | |
xlabel('Background') | |
ylabel('Lambda (s^-1)') | |
zlabel('Sum of log of Probabilities') | |
hold on | |
M=max(mP(:)); | |
[row,col]=find(mP==M); | |
lambda2=lambda(row); | |
background=b(col); | |
%% Uncertainty | |
L=M-0.5; | |
[row,col]=find(mP>L-0.03 & mP<L+0.03); | |
xp=b(col); | |
yp=lambda(row); | |
zp=L*ones(length(row),length(row)); | |
xp=xp'; yp=yp'; | |
k=boundary(xp,yp); | |
LL=length(k); | |
zp2=L*ones(LL,1); | |
plot3(xp(k),yp(k),zp2,'r','Linewidth',2) | |
title('Maximum Likelihood Approach with Consideration of Background') | |
hold off | |
figure(7) | |
contourf(YYY,XXX,mP) | |
hold on | |
scatter(yp,xp,'b') | |
xlabel('Lambda (s^-1)') | |
ylabel('Background') | |
zlabel('Sum of log of Probabilities') | |
yy1=get(gca,'ylim'); | |
h1=plot([lambda2,lambda2],yy1); | |
sig1=abs(lambda2-min(yp)); | |
sig2=abs(lambda2-max(yp)); | |
ul=xp(yp==max(yp)); | |
ll=xp(yp==min(yp)); | |
h2=plot([lambda2-sig1,lambda2],[ll(1),ll(1)]); | |
h3=plot([lambda2+sig2,lambda2],[ul(3),ul(3)]); | |
%legend('','''Value of Lambda','Sigma 1','Sigma 2') | |
legend([h1 h2 h3],{'Value of Lambda Determined','Sigma (-)','Sigma (+)'}) | |
title('Uncertainties in the Multidimensional Maximum Likelihood Approach') | |