Contents

tic
% by Jesse Smith for analysis of data taken in Physics307L Poisson distribution lab
% I've saved the output of the MCA program (except for the first few lines of text)
% as variables in a Matlab data file.  The variables are 'jk1ms','jk2ms', etc.

clear;

Load Data as Matrices

load jk.mat; % Load the Matlab data file
dwelltimes = [1 2 4 8 10 20 40 80 100 200 400 800 1000 2000 4000];
data1 = [jk1ms(:,2), jk2ms(:,2), jk4ms(:,2), jk8ms(:,2)];
data2 = [jk10ms(:,2), jk20ms(:,2), jk40ms(:,2)];
data3 = jk80ms(:,2);
data4 = [jk100ms(:,2), jk200ms(:,2), jk400ms(:,2), jk800ms(:,2)];
data5 = [jk1s(:,2), jk2s(:,2), jk4s(:,2)];

Calculate Std. Devs. and Means of Raw Data

stdevs = [std(data1) std(data2) std(data3) std(data4) std(data5)];
% means = [mean(data1) mean(data2) mean(data3) mean(data4) mean(data5)];

Plot Data

Make figures of raw MCA Output

scrsz = get(0,'ScreenSize');
figure('Position',[10, 50, scrsz(3)*0.6, scrsz(4)*0.6]);
subplot(2,2,1);
plot(jk1ms(:,1),jk1ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 1ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,2);
plot(jk2ms(:,1),jk2ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 2ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,3);
plot(jk4ms(:,1),jk4ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 4ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,4);
plot(jk8ms(:,1),jk8ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 8ms');
xlabel('Channel');
ylabel('Number of Events');

figure('Position',[10, 50, scrsz(3)*0.6, scrsz(4)*0.6]);
subplot(2,2,1);
plot(jk10ms(:,1),jk10ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 10ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,2);
plot(jk20ms(:,1),jk20ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 20ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,3);
plot(jk40ms(:,1),jk40ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 40ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,4);
plot(jk80ms(:,1),jk80ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 80ms');
xlabel('Channel');
ylabel('Number of Events');

figure('Position',[10, 50, scrsz(3)*0.6, scrsz(4)*0.6]);
subplot(2,2,1);
plot(jk100ms(:,1),jk100ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 100ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,2);
plot(jk200ms(:,1),jk200ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 200ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,3);
plot(jk400ms(:,1),jk400ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 400ms');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,4);
plot(jk800ms(:,1),jk800ms(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 800ms');
xlabel('Channel');
ylabel('Number of Events');

figure('Position',[10, 50, scrsz(3)*0.6, scrsz(4)*0.6]);
subplot(2,2,1);
plot(jk1s(:,1),jk1s(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 1s');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,2);
plot(jk2s(:,1),jk2s(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 2s');
xlabel('Channel');
ylabel('Number of Events');

subplot(2,2,3);
plot(jk4s(:,1),jk4s(:,2),'s','MarkerSize',4);
title('Raw Data, Dwell Time 4s');
xlabel('Channel');
ylabel('Number of Events');
clear scrsz;
clear jk1ms jk2ms jk4ms jk8ms jk10ms jk20ms jk40ms jk80ms;
clear jk100ms jk200ms jk400ms jk800ms jk1s jk2s jk4s;

Lambda Estimation

find estimated lambda for each set of data

lambdas = [poissfit(data1) poissfit(data2) poissfit(data3) poissfit(data4) poissfit(data5)];
%   determines the estimated lambda (max. likelihood of number of events)

%test for reasonableness: plot dwell times vs. lambda
figure;
loglog(dwelltimes,lambdas,'x-');
title('Lambda vs. Dwell Time');
xlabel('Dwell Time (milliseconds)');
ylabel('Lambda');

Probability Distrubtion Calculations

calculate stuff to plot and compare: poisson pdf 'prediction', gauss pdf 'prediction', and data histogram

nu1 = linspace(0,4,5);
nu2 = linspace(0,6,7);
nu3 = linspace(0,16,17);
nu4 = linspace(0,50,51);

nu1mat = (nu1'*ones(1,4))';
nu2mat = (nu2'*ones(1,4))';
nu3mat = (nu3'*ones(1,4))';
nu4mat = (nu4'*ones(1,3))';

lambdas1=lambdas(1:4)'*ones(1,5);
lambdas2=lambdas(5:8)'*ones(1,7);
lambdas3=lambdas(9:12)'*ones(1,17);
lambdas4=lambdas(13:15)'*ones(1,51);

poiss1 = poisspdf(nu1mat,lambdas1);
poiss2 = poisspdf(nu2mat,lambdas2);
poiss3 = poisspdf(nu3mat,lambdas3);
poiss4 = poisspdf(nu4mat,lambdas4);

norm1 = normpdf(nu1mat,lambdas1,lambdas1.^0.5);
norm2 = normpdf(nu2mat,lambdas2,lambdas2.^0.5);
norm3 = normpdf(nu3mat,lambdas3,lambdas3.^0.5);
norm4 = normpdf(nu4mat,lambdas4,lambdas4.^0.5);

datahist1 = (histc(data1,nu1)./length(data1))';
datahist2 = (histc(data2,nu2)./length(data2))';
datahist3 = (histc(data3,nu2)./length(data3))';
datahist4 = (histc(data4,nu3)./length(data4))';
datahist5 = (histc(data5,nu4)./length(data5))';

bignu1mat = (linspace(0,4,100)'*ones(1,4))';
bignu2mat = (linspace(0,6,100)'*ones(1,4))';
bignu3mat = (linspace(0,16,100)'*ones(1,4))';
bignu4mat = (linspace(0,50,100)'*ones(1,3))';

means1 = lambdas(1:4)'*ones(1,100);
means2 = lambdas(5:8)'*ones(1,100);
means3 = lambdas(9:12)'*ones(1,100);
means4 = lambdas(13:15)'*ones(1,100);

stdevs1 = stdevs(1:4)'*ones(1,100);
stdevs2 = stdevs(5:8)'*ones(1,100);
stdevs3 = stdevs(9:12)'*ones(1,100);
stdevs4 = stdevs(13:15)'*ones(1,100);

norm1plot = normpdf(bignu1mat,means1,stdevs1);
norm2plot = normpdf(bignu2mat,means2,stdevs2);
norm3plot = normpdf(bignu3mat,means3,stdevs3);
norm4plot = normpdf(bignu4mat,means4,stdevs4);

clear lambdas1 lambdas2 lambdas3 lambdas4
clear nu1mat nu2mat nu3mat nu4mat
clear stdevs1 stdevs2 stdevs3 stdevs4
clear bignu1mat bignu2mat bignu3mat bignu4mat

Figures

Make some figures: Poisson PDF, Gaussian PDF and Data vs. Number of Events

%call file 'fig.m' to make plots:
fig(nu1,poiss1(1,:),datahist1(1,:),linspace(0,4,100),norm1plot(1,:),'1ms',lambdas(1));
fig(nu1,poiss1(2,:),datahist1(2,:),linspace(0,4,100),norm1plot(2,:),'2ms',lambdas(2));
fig(nu1,poiss1(3,:),datahist1(3,:),linspace(0,4,100),norm1plot(3,:),'4ms',lambdas(3));
fig(nu1,poiss1(4,:),datahist1(4,:),linspace(0,4,100),norm1plot(4,:),'8ms',lambdas(4));
fig(nu2,poiss2(1,:),datahist2(1,:),linspace(0,6,100),norm2plot(1,:),'10ms',lambdas(5));
fig(nu2,poiss2(2,:),datahist2(2,:),linspace(0,6,100),norm2plot(2,:),'20ms',lambdas(6));
fig(nu2,poiss2(3,:),datahist2(3,:),linspace(0,6,100),norm2plot(3,:),'40ms',lambdas(7));
fig(nu2,poiss2(4,:),datahist3(1,:),linspace(0,6,100),norm2plot(4,:),'80ms',lambdas(8));
fig(nu3,poiss3(1,:),datahist4(1,:),linspace(0,16,100),norm3plot(1,:),'100ms',lambdas(9));
fig(nu3,poiss3(2,:),datahist4(2,:),linspace(0,16,100),norm3plot(2,:),'200ms',lambdas(10));
fig(nu3,poiss3(3,:),datahist4(3,:),linspace(0,16,100),norm3plot(3,:),'400ms',lambdas(11));
fig(nu3,poiss3(4,:),datahist4(4,:),linspace(0,16,100),norm3plot(4,:),'800ms',lambdas(12));
fig(nu4,poiss4(1,:),datahist5(1,:),linspace(0,50,100),norm4plot(1,:),'1s',lambdas(13));
fig(nu4,poiss4(2,:),datahist5(2,:),linspace(0,50,100),norm4plot(2,:),'2s',lambdas(14));
fig(nu4,poiss4(3,:),datahist5(3,:),linspace(0,50,100),norm4plot(3,:),'4s',lambdas(15));

clear norm1plot norm2plot norm3plot norm4plot

Goodness of Fit

%Examine "goodness of fit" (chi-square) for Poisson distribution vs. data
%and Gaussian distribution vs. data Calls the "fit" function, defined in
%"fit.m", which will return the sum of differencecs for the Gaussian PDF
%prediction from the data histogram, and the sum of differences for the
%Poisson PDF from the data histogram.  It also plots the differences vs.
%estimated lambda in a log-log plot.

gausschisquare = [fit(norm1,datahist1,stdevs(1:4)), fit(norm2(1:3,:),datahist2,stdevs(5:7)), fit(norm2(4,:),datahist3,stdevs(8)), fit(norm3,datahist4,stdevs(9:12)), fit(norm4,datahist5,stdevs(13:15))];
poisschisquare = [fit(poiss1,datahist1,stdevs(1:4)), fit(poiss2(1:3,:),datahist2,stdevs(5:7)), fit(poiss2(4,:),datahist3,stdevs(8)), fit(poiss3,datahist4,stdevs(9:12)), fit(poiss4,datahist5,stdevs(13:15))];

figure;
loglog(lambdas,gausschisquare,'x-',lambdas,poisschisquare,'s-');
legend('Gauss. Diff.','Poiss. Diff.');
title('\chi^2 vs. \lambda');
xlabel('\lambda');
ylabel('\chi^2');

Compare Std. Devs. of Raw Data with Std. Devs. of Poisson Distribution

In Poisson Distribution, std. deviation is sqrt(lambda). I'd like to check whether this fits the standard deviations of my raw numbers (0 events, 0 events, 1 event, 0 events, 0 events, etc.)

figure;
loglog(dwelltimes,lambdas.^0.5,'x',dwelltimes,stdevs,'s');
title('Standard Deviations vs. Dwell Time');
xlabel('Dwell Times (ms)');
ylabel('Standard Deviation');
legend('Poisson Distribution Std.Dev.','Raw Data Std.Dev.');
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Published with MATLAB® 7.4