Physics307L F09:People/Trujillo/LAB NOTEBOOK/071003

firstweek

Raw Data Files

 * [[Image:Dwell20ms.asc ]]
 * [[Image:Dwell40ms.asc]]
 * [[Image:Dwell80ms.asc]]
 * [[Image:Dwell100ms.asc]]
 * [[Image:Dwell200ms.asc]]
 * [[Image:Dwell400ms.asc]]
 * [[Image:Dwell800ms.asc]]
 * [[Image:Dwell1s.asc]]
 * [[Image:Dwell10s.asc]]

10ms Oct 03 2007    03:07:41 am     Elt: 000000 Seconds. Real Time: 000000

ID: No spectrum identifier defined

Memory Size: 16384 Chls Conversion Gain: 1024  Adc Offset: 0000 Chls

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Octave script for analysis & plotting
chop off the first ten lines of the data files before use

rename variable and filenames as necessary. Probably only needs minor tweeks so that it can run in matlab. #octave
 * 1) a sort of diary of commands to automate the data processing day to day
 * 2) some of octave's functions do not work in matlab,and vice
 * 3) versa. Some functions may work, but not in the same way
 * 4) Most things work, though.


 * 1) Tomas Mondragon

load LAB3_D10.ASC;    #The load function is intended load LAB3_D20.ASC;    #to read files saved by octave. load LAB3_D40.ASC;    #Load can also read .mat files, load LAB3_D80.ASC;    #but only versions saved by matlab load LAB3_D100.ASC;   #version 4. It can also read text files. load LAB3_D200.ASC;   #if no variable name info is in the file load LAB3_D400.ASC;   #it names the variable after the file name load LAB3_D800.ASC; load LAB3_D1S.ASC; load LAB3_D10S.ASC;

dwell10ms=LAB3_D10(:,2);    #load put what was on LAB3_D10.ASC into the dwell20ms=LAB3_D20(:,2);    #variable LAB3_D10. I only care about the second dwell40ms=LAB3_D40(:,2);    #column of the data, so here I extract it to an dwell80ms=LAB3_D80(:,2);     #appropriately named variable. dwell100ms=LAB3_D100(:,2); dwell200ms=LAB3_D200(:,2);  #The first column of LAB3_D10 is the channel number dwell400ms=LAB3_D400(:,2);  #but a more appropriate name would be the bin index. dwell800ms=LAB3_D800(:,2);  #The second column is a count of how many events fell dwell1s=LAB3_D1S(:,2);      #into the bin. The third column isn't important for this dwell10s=LAB3_D10S(:,2);    #experiment, but knowing what it is will allow one to take #advantage PCA3's region of interest feature. It just marks #where one has marked ROI's. for example, 4 indicates the bin or channel #lies within ROI 4


 * 1) idea:use max(dwell10s) as max bin for all so grapfh are same x scale


 * [y,x]=hist(data,bincenters) is a a function that sets up bins whose values are centered
 * 1) at the values given by the vector bincenter and counts the number of times the values in
 * 2) data fall into each of the bins. y contains the frequncy counts and x contains the
 * 3) corresponding bin index. bar(x,y) will plot the histogram of data. In general, x=bincenters,
 * 4) so one can shorten [y,x]=hist(data,bincenters) to y=hist(data,bincenters) and use
 * 5) bar(bincenters,y) to plot the same thing.

bincenters=0:max(dwell10s);

freq10ms=hist(dwell10ms,bincenters); freq20ms=hist(dwell20ms,bincenters); freq40ms=hist(dwell40ms,bincenters); freq80ms=hist(dwell80ms,bincenters); freq100ms=hist(dwell100ms,bincenters); freq200ms=hist(dwell200ms,bincenters); freq400ms=hist(dwell400ms,bincenters); freq800ms=hist(dwell800ms,bincenters); freq1s=hist(dwell1s,bincenters); freq10s=hist(dwell10s,bincenters);

bar(bincenters,freq10ms) title("frequency counts for dwell time=10ms") xlabel("number of events occuring during dwell time") ylabel("frequency count") pause replot bar(bincenters,freq20ms) title("frequency counts for dwell time=20ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq40ms) title("frequency counts for dwell time=40ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq80ms) title("frequency counts for dwell time=80ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq100ms) title("frequency counts for dwell time=100ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq200ms) title("frequency counts for dwell time=200ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq400ms) title("frequency counts for dwell time=400ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq800ms) title("frequency counts for dwell time=800ms") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq1s) title("frequency counts for dwell time=1s") %xlabel("number of events occuring during dwell time") %ylabel("frequency count") pause replot bar(bincenters,freq10s) title("frequency counts for dwell time=10s") %xlabel("number of events occuring during dwell time") %ylabel("frequency count")
 * 1) plots. press any key to move to next plot

Compilation of plots
The plot that the script above produced with the data I obtained are shown in sequence below. As the dwell time increases, the most often occurring event count increases and the frequency vs. event count plots drift from something resembling a poisson distibution of a rarely occuring event to a the poisson distribution of a more common event, which resembles a gaussian. I should point out that something interesting happened while we we counting events with a dwell of 200 ms. Our equipment recorded one instance where 27 events happened in that small dwell time. The mode of the data for dwell time=200ms was 0 events, and were were rarely getting counts more than or equal to 5.

In Dr. Gold's lab manual, the poisson distribution is a model for the results of experiments that count random events that occur at a definite average rate and the frequency that one should expect to get a certain count number. If the expected average count is $$\lambda$$, the probability that one will obtain the result of $$k$$ after performing the counting experiment is

$$f(k;\lambda)=\frac{\lambda^k e^{-\lambda}}{k!}$$

So, to fit my data, I will have to normalize my data and find a suitable $$\lambda$$ for each data set.

First off, normalize. If the frequency count of an event count $$k$$ is $$x_k$$, find some normalizing constant $$N$$ so that $$\sum_{k=\operatorname{min}\ k}^{\operatorname{max}\ k} \frac{x_k}{N}=1$$

Then, find $$\lambda$$ such that$$\sum_{k=\operatorname{min}\ k}^{\operatorname{max}\ k}\left(f(k,\lambda)-\frac{x_k}{N} \right)^2$$ is a minimum

Duurrrrr... $$N=256$$ because 256 tests were performed each time.

Through an iterative process, I determine $$\lambda$$ to be 0.0261205 for dwell=10ms. Ack, I won't go any further, I'll just stop at 6 significant digits for the others too.
 * 10ms,0.0261205
 * 20ms,0.0736898
 * 40ms,0.185892
 * 80ms,0.457154
 * 100ms,0.468385
 * 200ms,0.937453
 * 400ms,2.58578
 * 800ms,5.74562
 * 1s,7.27233
 * 10s,73.9022



how good are these fits, since I decided to stop at 6 sig figs (quite arbitrarily). I suppose I shall quantify this by taking the average of how far off the actual data is from the fit.

I think the formula is $$Error=\frac{\sqrt{\sum_{k=\operatorname{min}\ k}^{\operatorname{max}\ k}\left(f(k,\lambda)-\frac{x_k}{N} \right)^2}}{\sqrt{N}\sqrt{N-1}}$$
 * 10ms, 0.0261205, 5.45905*10^-6
 * 20ms, 0.0736898, 1.19310*10^-4
 * 40ms, 0.185892, 3.28259*10^-4
 * 80ms, 0.457154, 5.95464*10^-4
 * 100ms, 0.468385, 3.15888*10^-4
 * 200ms, 0.937453, 5.23153*10^-4
 * 400ms, 2.58578, 5.22450*10^-4
 * 800ms, 5.74562, 2.97039*10^-4
 * 1s, 7.27233, 2.59775*10^-4
 * 10s, 73.9022, 2.96150*10^-4

The poisson distribution fits count data of random events that happen at a definite average rate. Therefore $$\frac{\lambda}{dwell time}=some constant$$ A plot of $$\lambda$$ vs. dwell time should fit on a line. to do this, I use Dr. Gold's linefit matlab function, modified for use with Octave. (www-hep.phys.unm.edu is down at the moment, maybe provide link when it comes back up or upload modded program here?) slope=7.39233+/-0.00003 intercept= -0.048099+/-0.000005 cov(m,b)=0.000000 correlation=0.999983 point error estimate=0.231612 chisq/ndf=265341 So, with our equipment set up the way it was (sorry, no data on this, oops!) we were observing events that occured at an average rate of 7.39 events per second.