The following is an excerpt from the execution of MaxSas as it is analyzing the supplied test distribution BIMODAL.SAS, a bimodal distribution with two Gaussian peaks: one at 80 Angstroms with a sigma of 20 A and the other at 200 A with a sigma of 60 A. The Gaussian at 200 A is half the height of the one at 80 A. For a further discussion of this distribution, see [Culverwell, 1986]. The exact volume fraction of the original distribution was not specified.
In the section to follow, all user responses will be all in upper case where appropriate and will be followed by a {CR}, signifying that the user has pressed the return key. {CR} by itself signifies that the user has accepted the default answer to the question, as shown in <default>. Almost all questions have a default response, the only exception being the output file names which may take any value EXCEPT blank or the same as the input file name. Initially, all the defaults are preset to the proper answers for the supplied test distribution BIMODAL.SAS. If you type in a new value, that will value will become the default the next time that the question is asked.
The example to follow will be interrupted from time to time for explanations. These will be isolated between rows of “====” signs. For further explanations of each question which is asked, see the appropriate section appearing elsewhere in this document.
Now here’s the excerpt from the beginning of the run. Remember that all user responses are on a single line and are terminated with a {CR}.
$ RUN MaxSas{CR}
Size distributions from SAS data using the maximum entropy criterion
version: 3.1 (PRJ) , 7 February 1990
Input file? <Quit>
Bimodal.Sas{CR}
Output file?
Bimodal.Out{CR}
Minimum q-vector? [1/A] < 1.000000000000000E-008>
{CR}
Maximum q-vector? [1/A] < 100.000000000000 >
{CR}
Scattering contrast? [10^28 m^-4] < 1.00000000000000 >
{CR}
Factor to convert data to 1/cm? < 1.00000000000000 >
{CR}
Error scaling factor? < 1.00000000000000 >
{CR}
Background? < 0.000000000000000E+000>
{CR}
Spheroids: D x D x vD, Aspect ratio (v)? < 1.00000000000000 >
{CR}
Bin step scale? (1=Linear, 2=Log) < 1>
{CR}
Number of histogram bins? < 40>
{CR}
Maximum value of D? [A] < 400.000000000000 >
{CR}
Minimum value of D? [A] < 10.0000000000000 >
{CR}
Maximum number of iterations? < 20>
{CR}
Reading from file: Bimodal.Sas
38 points were read from the file
38 points were selected from the data
Preparation of the GRID function...
Setting BASE constant at 1.000000000000000E-012
MaxEnt routine beginning ...
To recap what has happened so far, the program was started and the input data file BIMODAL.SAS was specified for analysis. All the default answers appeared to be acceptable so the program read 38 points from the file, of which all 38 points were retained for the analysis. The program informs you that it has set an internal array called BASE to 1.0E-12. BASE is the initial guess for the distribution. If there is a value in the output distribution that is comparable with this number, then that particular histogram bin has no significant information. Consider then that BASE is the “featureless” distribution and rest assured that it is quite flat.
At this point, the program has gotten all the adjustable parameters that it needs and is now proceeding to attempt to solve the problem.
The fitting routine is an iterative one. If there are N histogram bins and P data points then the computation time for each iteration will be on the order of (very approximate) Order(2N+P). (This is why both N and P are bounded.)
All sorts of information will begin to scroll on the screen at an alarming rate as the Maximimum Entropy routine, or MaxEnt, (see [Skilling, 1984] for details on what’s happening) begins to extract statistically significant information from the SAS data. With each iteration, several different types of screen plots are drawn, each describing the data extracted so far. The different plots are titled:
The first two plots will not appear until the third iteration. They are intended to keep you informed about the progress of the MaxEnt routine. The residuals (difference between MaxEnt fit and input data normalized to the input errors) are plotted as a function of data point subscript number, not Q-vector. The distribution is weighted by the bin width, dD(i) = D(i+1) - D(i), and is also plotted as a function of diametral bin subscript number. All plots will be scaled to fit within the screen boundaries.
The last information to appear in each iteration will be a report, such as:
# 2 of 20, n = 38
test = 0.19101, Entropy = 3.4668138
SQRT((Chi^2)/n): target = 12.14204774 % off = 26.2698
f-vector: sum = 0.00452313 % change = 0.4663
The report is saying that for the second iteration out of 20, where there are 38 intensity data points, the difference between the entropy and ChiSquared gradients is 0.19101, the entropy of the distribution just plotted is 3.4668138 (whose units are exact), the target for the SQRT((ChiSquared)/n), where ChiSquared is derived from intensity calculated from the distribution just plotted, of 12.14204774 was missed by 26.2698%, and the total volume fraction of scatterers in the distribution just plotted is 0.452313 %, which did not change much from the previous iteration. The iterations will continue.
Here is the screen output from the last iteration:
LOG (ChiSq) vs. iteration number
1 point(s) per column
0.405168291335942 units per row
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|==========O|
-----------
Entropy vs. iteration number
1 point(s) per column
5.229508533621462E-002 units per row
-----------
|===========|
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-----------
Residuals
1 point(s) per column
0.315439585860872 standard deviations per row
--------------------------------------
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|====O==O======O====O=====O============|
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|OO OO O O |
|======O=========================OO====|
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--------------------------------------
Distribution
1 point(s) per column
6.975972282206744E-005 units per row
------------------------------------------
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|OOOOO=======================OOOOOOOOOOOOOO|
------------------------------------------
# 11 of 20, n = 38
test = 0.01587, Entropy = 3.1391598
SQRT((Chi^2)/n): target = 1.00000000 % off = 0.0504
f-vector: sum = 0.00809492 % change = 4.0445
The problem has been solved in 11 iterations of the MaxEnt routine. The two criteria for solution are that TEST <= 0.05 (5%) and that the SQRT((Chi^2)/n) target be met within 0.5%. Observe that the volume fraction has not changed very much from the previous iteration.
Here is the summary screen output of the analysis:
Input file: Bimodal.Sas
Volume weighted size dist.: V(r)N(r) versus r
2.63513513513514 units per column
1.395194456441349E-005 units per row
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|OO O O O O OO O O O O O O O O O|
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standardized residuals vs. point number
1 point(s) per column
0.315439585860872 standard deviations per row
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|====O==O======O====O=====O============|
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|OO OO O O |
|======O=========================OO====|
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--------------------------------------
Input data: Bimodal.Sas
Contrast = 1.0000000 x 10^28 m^-4.
spheroid: D x D x D* 1.00000000000000
Data conversion factor to 1/cm = 1.00000E+00
Error scaling factor = 1.00000E+00
Histogram bins are distributed in an increasing algebraic series.
Minimum particle dimension D = 10.00 A.
Maximum particle dimension D = 400.00 A.
Number of histogram bins = 40.
Maximum number of iterations allowed = 20.
Program left MaxEnt routine after 11 iterations.
Target chi-squared (# data points) = 38.
Best value of chi-squared achieved = 38.038279.
Entropy of the final distribution = 3.1389744.
Entropy of a flat distribution = 3.6888795.
Total particles = 1.57329E+16 per cubic cm.
Total volume fraction of all scatterers = 0.008094741.
Part of distribution smaller than 10.00 A = 0.00000000%.
Part of distribution larger than 400.00 A = 0.00215387%.
Volume-weighted mode D value = 70.00000 A.
Volume-weighted mean D value = 153.25044 A.
Volume-weighted std. deviation = 72.92106 A.
Number-weighted mode D value = 70.00000 A.
Number-weighted mean D value = 83.89447 A.
Number-weighted std. deviation = 33.47500 A.
Minimum Q-vector = 7.4935700E-03 1/A.
Maximum Q-vector = 9.9937470E-02 1/A.
User-specified background = 0.000000000 input data units
Suggested background = 0.000064250 input data units
StDev of shift in background = 0.000330552 input data units
New background should give ChiSq = 36.6534794792953
By now, the tabular data of the size distribution and the intensity fit have been written to the data file BIMODAL.OUT. The summary analysis of the distribution has also been written to the output file.
The program has found about 0.8% by volume of scatterers. The ChiSquared matches the number of intensity points (a requirement for solution) and the background suggested is not far from the background used (with respect to the sigma of the last intensity of 0.000123 1/cm). Observe that the standard deviation of the suggested shift in the background is much larger than the suggested shift.
It appears that we have a reasonable solution in hand. Don’t be too hasty to believe it yet. In order to test the stability of the answer, analyze the BIMODAL.SAS data again, using all the same parameters as before (just take the defaults) except use the background (0.000064251 1/cm) suggested by MaxSas.
Therefore, you should respond affirmatively to the Stability Check question. Note the default answer on the Stability Check question is “N”.
The change in ChiSquared should be < 5%.
Run the Stability Check? (Y/<N>)
Y{CR}
Setting BASE constant at 1.000000000000000E-12
MaxEnt routine beginning ...
For this documentation, the results of the stability check will not be shown. There is not much change in the volume fraction after the stability check (about 1% or so) for the supplied test distribution.
After the Stability Check, you should get this question again. This time, respond like the following to quit the program.
The change in ChiSquared should be < 5%.
Run the Stability Check? (Y/<N>)
{CR}
The program is finished.
The output file is: BIMODAL.MAX
Size distributions from SAS data using the maximum entropy criterion
version: 3.1 (PRJ) , 7 February 1990
Input file? <Quit>
{CR}
$