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From: Thomas Mattison <mattison@ph...>  20060301 22:18:19

Hi I have made a bunch of changes to gnuplot fitting in fit.c. Generally, they are flagged by comments containing "tsm feb06". In many cases, old code is still there for reference but commented out. This should probably be cleaned up before a release. Some of the changes have uservariables to control them. I have not yet figured out how to commit the changes, so they aren't out there yet. 1. Error message for undefined function evaluation The old message in call_gnuplot() just said "Undefined value during function evaluation." I added the data point number, x, y, z, and parameter values. 2. Check for error value equal to zero If the user supplies errors but any of them are zero, there will be a divide by zero, trashing the fit. I added a check in fit_command(), which prints the data point number, x, y, z, and error, then aborts back to command line. 3. Made new message when fit stops because lambda is too large Previously, the message in regress() said that the fit had converged in this case which wasn't always true. Other output unchanged. 4. Zerochange in chisquare is now "BETTER" rather than "WORSE" Previously, if the exact minimum of chisquare was found, so the chisquare change was zero, this was called "WORSE" in marquardt(). The loop in regress() didn't exit, and further iterations were done, which were also "WORSE" and increased lambda until the upper limit on lambda or iterations was reached. This was a waste of time, and potentially confusing, for no benefit. 5. Final parameter cleanup fixes The gnuplot internal variables for the fit parameters may contain values from an iteration that made the chisquare worse rather than better, if the loop in regress() terminates due to maximum iterations or maximum lambda rather than convergence. There was code at the end of regress() that restores the internal variable for only the last parameter, for a different reason (the last variable is altered in call_gnuplot() to calculate derivatives, but not restored there). I changed the code at the end of regress() to restore _all_ internal variables, from the array that contains the parameters that gave the best chisquare so far. I also put a copy of the lastparameterfix code into call_gnuplot(), where it logically belonged anyway. Without this, if the user interrupted the fit and tried to plot the current function on top of the data, the last internal parameter was not correct. 6. Changed convergence criterion, with new uservariable FIT_LIMIT_ABS. The new simpler criterion in regress() is absolute reduction in chisquare for an iteration of less than epsilon*chisquare plus epsilonAbs. Default for new internal variable epsilonAbs is zero, usersettable by FIT_LIMIT_ABS. Default for the existing internal variable epsilon is 1.0e4, usersettable by FIT_LIMIT (unchanged). In most cases, the old convergence criterion was _relative_ change in chisquare of less than epsilon. But if the chisquare was less than NEARLY_ZERO = 1.0e30, (which could happen if fitting data with magnitude less than 1.0e15 with no errors), the old criterion was _absolute_ change in chisquare less than epsilon. Any change in chisquare would probably be less than epsilon in these cases, so the fitter would announce convergence immediately rather than finding the minimum. Users would probably prefer the default relative convergence criterion in these cases, but there was no way for them to impose it. All they could do was to adjust FIT_LIMIT, which would only let them adjust the _absolute_ convergence criterion, which would require them to guess what their minimum really was. With the new criterion, the default convergence criterion is always relative no matter what the chisquare is, but users now have the flexibility of adding an absolute convergence criterion through FIT_LIMIT_ABS. The new scheme defaults to doing the same thing as the old scheme in almost all cases, is more useful in some (rare) cases, is easier to understand, and is more flexible. The loadfile newfit0.dem tries to fit a line to the file newfit0.dat. With old gnuplot, the fit gives up when the parameters are still pretty bad, with the new convergence criteria the fit continues until it is right. 7. New oneline progressreport, revert by FIT_CLASSIC_PROGRESS = 1 The old report was many lines per iteration, so you could not hold many iterations in the scroll buffer, and it was hard to see how chisquare, parameters, and lambda were changing. It also said nothing about iterations where the chisquare increased (except to print an asterisk). It printed "WSSR" with no explanation that this means weighted sum of squared residuals, rather than the more common term "chisquare". The new report is one line per iteration, with everything in neat columns so it's easy to track the progress. The first column is the chisquare, the second is the change in chisquare divided by the convergence limit, so convergence means a value between 1 and 0, the lambda value, and one column per parameter. It also shows iterations where the chisquare increases. Implementation is through new function show_fit1(), called by regress() [and now also by marquardt()]. Both console output and fit.log file are affected by the change. The user can revert to the old progress report show_fit() by setting FIT_CLASSIC_PROGRESS = 1. 8. New final fit parameter report format, revert by FIT_CLASSIC_RESULT = 1 The old default final report label "final sum of squares of residuals" was misleading because if the user supplied errors, the number printed was the chisquare (_weighted_ sum of squared residuals or WSSR). The new report labels are "chisquared, "degrees of freedom," and "chisq/ndf" when the user supplies errors. If the user did not supply errors, it prints "Sum of squared residuals" and "degrees of freedom", and not SSR/ndf because it is not particularly meaningful in that case. In both cases, it prints an errorrescaling factor (square root of internal chisquare per degree of freedom, calculated with error=1 if errors are not supplied). A new internal variable errColumn is set to 1 in fit_command() if the user supplied errors, otherwise it is zero. If the user supplied errors, both the raw and rescaled parameter errors are printed. If the user did not supply errors, only the rescaled parameter errors are printed. In both cases, the same thing goes to the fit.log file. The old printing code was moved into new function PrintResults(), and the new printing code is in PrintResults2(). A new internal variable classicResults controls which function is called by regress(). It defaults to 0 (new results) but setting FIT_CLASSIC_RESULT = 1 restores the old results report. The loadfile newfit1.dem fits lines to the files newfit1.dat and newfit1a.dat, switching back and forth between the old and new progress and results reports. 9. Errorrescaling control The old default was to always rescale parameter errors by the square root of the "chisquare" per degree of freedom (unless there were zero degrees of freedom, in which case errors were considered undefined). While this is the only sensible thing to do if the user does not supply errors to the fit, it is not the only sensible thing to do if errors are supplied. The parameter errors are calculated in regress() as before, but now they are only rescaled by the square root of chisquare per degree of freedom if new internal variable rescaleErrs is true (nonzero). The rescaleErrs variable is set in fit_command() along with errColumn, but at the moment it is always set to 1 whether the user supplied errors or not. This means that the internal variables for parameter errors (not default, but available through recompiling with a preprocessor option) and the (nondefault) oldstyle result report always give only rescaled errors, as before, unless someone changes the source and recompiles. The new default results report from FitResults2() checks rescaleErrs and errColumn so it can print the right values (both raw and rescaled if the user supplied errors, only rescaled if the user did not supply errors) no matter how rescaleErrs is set. 10. Derivativestep algorithm improvement, revert by FIT_CLASSIC_DRV_STEP = 1 The fit needs derivatives with respect to parameters of the prediction at each data point. These are calculated numerically by changing the parameters by a small amount. The old step size was DELTA=0.001 times the parameter value (or 1e33 if the parameter is less than 1e30). In some cases, this step is so large that nonlinearities in the function make the derivative inaccurate. This slows or prevents convergence, and may make the location of the minimum wrong. In other cases, the step is so small that the predicted function value does not change to machine precision, so the calculated derivative is zero. This usually causes a singular matrix error. There is an optimal step size that balances roundoff error and nonlinearities (see the discussion of numerical derivatives in Numerical Recipes). It requires an estimate of the roundoff error of the function, and an estimate of the second derivative. The function that we ultimately care about is the chisquare. We can estimate the roundoff error from the data values, and the matrices that the fit accumulates anyway can be used to estimate the second derivative. The calculation adapts to the data, function, and parameter values. In most cases, the step size is much less than 0.001 times the parameter value, although it is larger when it needs to be to avoid roundoff errors. The new algorithm is implemented by allocating (and freeing) in marquardt() a new array drvStepDA, initialized by calling new InitDrvSteps(), updated during convergence by calling new CalcDrvSteps()and changing calculate() to use the array. The new algorithm is the default, but the old algorithm (reimplemented in the new functions) can be restored by setting FIT_CLASSIC_DRV_STEP = 1. The load file newfit2.dem shows a case where the new derivative step algorithm works but the old algorithm step is too small, resulting in a singular matrix error. The load file newfit3.dem shows a case where the old derivative step is too large so the fit does not really converge, while the new algorithm is OK. 11. Improved treatment of parameters near zero Previously, an initial parameter value less than NEARLY_ZERO = 1.0e30 was silently changed to 1.0E30. But in some cases it is natural for parameters to be this small or smaller. Rather than silently overriding the user's input when near zero, the new version uses any initial value with no change. The exception is that an initial value of EXACTLY zero is disallowed. The reason is that the numerical derivative algorithm needs at least the order of magnitude of the parameter for initialization. There is a message telling the user to provide a nonzero value of the right magnitude, or the magnitude of the expected error if the expected value is zero. The old code was removed from fit_command(), the new code is CheckParms(), called at the start of regress(). The old behavior of creating a new variable with initial value 1.0 when a fit parameter doesn't yet exist is retained, but a print statement was added to createdvar() advising the user that it is better to provide a nonzero explicit value. 12. Parameter step size limit, controlled by FIT_MAX_PAR_STEP For nonlinear functions, particularly when the initial parameter values are far from optimum, the fit may try to change the parameters to values where the function is undefined, which terminates the fit with an error (at least in the new code, the user gets more information about the parameter values and data values that caused the undefined result!). The only thing the user could do was to change the starting point and pray. A new function LimitParSteps() called by marquardt() to scale down the steps in all parameters so the ratio of any parameter change to its value is no larger than internal variable maxParStep. The new user variable FIT_MAX_PAR_STEP controls maxParStep. The default value of maxParStep is 1.5, which allows a factor of 2.5 increase per iteration, and also for the sign to change. Note that maxParStep values of less than 1.0 would not allow the parameter to change sign, which could either be useful, or dangerous. 13. Scaleindependence through multiplicative lambda, revert by FIT_CLASSIC_LAMBDA The lambda parameter in the LevenbergMarquardt algorithm reduces the step size if the calculated step actually increases the chisquare. The amount of reduction for the different parameters depends on details of the implementation. One common implementation is "multiplicative", which is dimensionless and is insensitive to parameter scale differences. The implementation in gnuplot is "additive" which makes the performance sensitive to the relative scale of parameters and errors. Additive lambda requires a somewhat arbitrary initial value calculation, while multiplicative lambda can be simply initialized to 0.01. The interpretation of multiplicative lambda is easy (much larger than 1 means the step sizes are reduced, much less than 1 means the step sizes are "ideal"), while it is hard to interpret the value of additive lambda. My preference is for therefore for multiplicative lambda, although there are certainly combinations of data, function, and starting point where additive lambda will work and multiplicative lambda will fail. The implementation is through changes to marquardt() in both initialization and usage of lambda. The default is now multiplicative lambda initialized to 0.01, but additive lambda is still available by setting FIT_CLASSIC_LAMBDA = 1. Users can set FIT_START_LAMBDA to change the initial lambda value in either case. But for additive lambda, supplying a FIT_START_LAMBDA value overrides the initial value calculation, and there is no way to turn it back on without restarting gnuplot (this has always been true). The load file newfit4.dem shows a case where a simple line fit gives the wrong answer with additive lambda (because it thinks it is converged when it has not), and multiplicative lambda gives the right answer. 14. "Skeptical" chisquare for nonlinear fits, controlled by FIT_SKEPTICAL In a nonlinear fit, the influence of a given data point on the fit parameters (which depends on the parameter derivatives at the data point) can be highly sensitive to the values of the parameters. This can result in slow convergence, unless the initial parameters are chosen very close to the (unknown) desired minimum. An example is a function like a*exp(b*x), where the derivatives at large x can be orders of magnitude larger than at small x. This interferes with convergence, because LevenbergMarquardt refuses to take a step if the chisquare goes up, so if it gets close to the data at high x, even tiny parameter changes make the chisquare worse (unless they are balanced in a nonlinear way). These problems can be reduced in some cases by adjusting the weights of data points to cancel out the dependence of the derivatives on the parameters. I call this "skeptical chisquare" because it doesn't take the initial parameter values as seriously. In some cases, the convergence time goes from thousands of iterations to only a few. However, the location of the minimum of the skeptical chisquare will be close to the minimum of the normal chisquare if the functional form can reproduce the data well, but it will not necessarily be very close if the function does not reproduce the data, at least over the region being fit. Also, the errors calculated with the skeptical chisquare are not very meaningful. For these reasons, a skeptical chisquare fit should only be used to determine better initial values for a normal chisquare fit. There are ifstatements in marquardt() that test new variable skepticalChisq, and rescale the internal errors. The default is 0 for normal chisquare. The user can turn it on by FIT_SKEPTICAL = 1, and off by FIT_SKEPTICAL = 0. The load file newfit5.dem gives an example with a*exp(b*x) where "convergence" takes thousands of iterations with normal chisquare, even starting with both parameters close to right, but only a few with skeptical chisquare, and the final refit with normal chisquare takes only a few iterations. 15. Monte Carlo search for initial fit parameters There are many problems where fits only converge when the initial parameters are rather close to the solution, and it can be tedious finding such a starting point. To make this easier, I have implemented a Monte Carlo search for the minimum chisquare over a userdefined range, which is used as the initial parameters for the normal LevenbergMarquardt fit. The range information is input using the "via file" mechanism. The "via file" parser in fit_command() was modified to allow (but not require) two values per line. If any lines have two values, a Monte Carlo search for starting parameter values is done. Parameters with only a single value are held constant for the Monte Carlo search, but still vary in the final fit. The existing "# FIXED" syntax for nonparameter constants was not changed. The default Monte Carlo search is 1000 iterations. The user can change this by setting FIT_MONTE_CARLO_TRIALS. The default behavior is to continue with a regular fit from the best chisquare point. But if FIT_MONTE_CARLO_TRIALS is negative, after (minus) that many trials, control goes back to the user without a fit, but with the internal parameter values set to the best trial. This lets the user plot the function with these values as a diagnostic. The random number generator is not reseeded, so repeating the search with the same number of iterations on the same function and data always gives the same result. The load file newfit6.dem shows sinewave data where fits fail unless the frequency parameter is quite close to the right value. The Monte Carlo search does quite a good job in 1000 iterations, which takes only seconds, and the fit converges well from this starting point. I compared the new fitting code results with the old version on fit.dem. The first fit starts with both parameters initialized to zero, which is now illegal, so I changed fit.dem to initialize them to 0.001. When the switches are set to revert to the old algorithm, the results are nearly identical. The first fit takes a few extra iterations because of maxParStep. With the default changes to the derivative and lambda algorithms, the results are noticably different on the hemisphere fit. I traced this to the fact that the function has a square root, and for some of the data points, the argument is negative. The fit only uses the real part of the function, so it is zero for those data points. So the chisquare actually has multiple minima, depending on how many data points give negative square roots. The modified fit.dem now makes a plot of the argument of the square root, and also defines a new fit function with a Taylor expansion of the square root that is defined for all arguments. With the modified fit function, the new and old versions give the same result (and both converge in far fewer iterations, implying that most of the iterations were being confused by the multiple minima). The new version generally tends to converge in somewhat fewer iterations, although for some of the cases in fit.dem, the new version takes more iterations. Cheers Prof. Thomas Mattison, Dept. of Physics & Astronomy, Univ. of British Columbia Present Address: Stanford Linear Accelerator Center 2575 Sand Hill Road, Menlo Park, CA, 94025 Building 48 (Research Office Building), Mail Station MS35 Office: ROB231 Phone: 6509265342 Fax: 6509268522 
From: Thomas Mattison <mattison@ph...>  20060301 22:16:14

Hi I have made a bunch of improvements to gnuplot fitting. They are in src/fit.c, plus some changes to demo/fit.dem. They are described in a separate message. How do I commit the changes? I started by CVS downloading gnuplot 4.1, made my changes, and have done a CVS update since making my changes, so I should be in synch with everyone else. But when I tried to CVS commit, I got a broken pipe abort. I'm working on a Linux system. I have also made some new short demo files to demonstrate the reason for and effect of the improvements. These would be new files, and they are really for developers rather than users. How should they be handled? Finally, the documentation should be changed to reflect the changes in fitting. What is the proper way to edit the source for documentation, and to test the changes? Cheers Prof. Thomas Mattison, Dept. of Physics & Astronomy, Univ. of British Columbia Present Address: Stanford Linear Accelerator Center 2575 Sand Hill Road, Menlo Park, CA, 94025 Building 48 (Research Office Building), Mail Station MS35 Office: ROB231 Phone: 6509265342 Fax: 6509268522 
From: Ethan A Merritt <merritt@u.washington.edu>  20060301 16:56:59

On Wednesday 01 March 2006 03:32 am, Petr Mikulik wrote: > > Thus yet another missing compatibility are dashed styls  set terminal > {no}dashed or set termoption {no}dashed .. and then dashed styles ... Right. It's the same issue. Only a few terminal types can draw dashed lines. Adding the command itself is easy; making it do something useful is another question.  Ethan A Merritt Biomolecular Structure Center University of Washington, Seattle 981957742 
From: Petr Mikulik <mikulik@ph...>  20060301 11:32:40

>> plot sin(x) with points 4 >> plot sin(x) with points 8 > > This proposal may seem charming, but neglects the wellconsidered > reasons why gnuplot uses numbered point and line types instead of named > ones. The question that this fails to answer is: what is a terminal > driver supposed to do if I request a named point type that it simply > doesn't have? Throw a tantrum, commit seppuku, read my mind, or what? > > The single most important design principle of gnuplot has always been > absolute script compatibility  scripts were never allowed to just > fail just because you run them on a different platform's gnuplot, or on > a different terminal driver. And where possible, they should generate > usable output. In recent years, the tendency goes to compatibility in terminal output. We really want this. Thus yet another missing compatibility are dashed styls  set terminal {no}dashed or set termoption {no}dashed .. and then dashed styles ...  PM 
From: Daniel J Sebald <daniel.sebald@ie...>  20060301 09:27:32

[Forgot to CC: the list on this...] Daniel J Sebald wrote: > HansBernhard Br=F6ker wrote: >=20 >> Daniel J Sebald wrote: >> >>> It would be nice to give names to the symbols so that either the name= =20 >>> or the number can be used. For example: >>> >>> plot sin(x) with points square >>> plot sin(x) with points triangle >>> >>> instead of >>> >>> plot sin(x) with points 4 >>> plot sin(x) with points 8 >> >> >> >> This proposal may seem charming, >=20 >=20 > You aren't riffing off my Lucky Charms humor, are you? :) >=20 >> but neglects the wellconsidered reasons why gnuplot uses numbered=20 >> point and line types instead of named ones. The question that this=20 >> fails to answer is: what is a terminal driver supposed to do if I=20 >> request a named point type that it simply doesn't have? Throw a=20 >> tantrum, commit seppuku, read my mind, or what? >=20 >=20 > Read the user's mind... I'll get working on that patch right now. >=20 > Well, of course there is that matter to deal with. There may be more=20 > than one way of addressing this, but ultimately it would mean gracefull= y=20 > substituting symbols; and in a unique way. (Don't want to substitute=20 > with a symbol that is used on the plot already.) >=20 > One method would be for gnuplot to have a fixed name to number mapping=20 > as I showed in my previous email. Then inside the terminal driver is=20 > another array of translations. For example to gnuplot the square might= =20 > be #4, but the internal table for the terminal driver looks up array=20 > offset 4 and finds that it is supposed to use #6 for its library call. = =20 > Something like that. Then there might be an array for substitutes if=20 > the terminal driver doesn't have the square symbol. It might issue a=20 > warning (just the first time) "Do not have 'square' symbol, substitutin= g=20 > with 'Maltese cross'". I suppose a table of strings would need to be=20 > saved for that... which brings me to the second method. >=20 > Dan >=20 =20 Dan Sebald phone: 608 256 7718 email: daniel DOT sebald AT ieee DOT org URL: http://webpages DOT charter DOT net/dsebald/ 
From:
<broeker@ph...>  20060301 09:09:27

[Forgot to CC: the list on this...] Daniel J Sebald wrote: > It would be nice to give names to the symbols so that either the name or > the number can be used. For example: > > plot sin(x) with points square > plot sin(x) with points triangle > > instead of > > plot sin(x) with points 4 > plot sin(x) with points 8 This proposal may seem charming, but neglects the wellconsidered reasons why gnuplot uses numbered point and line types instead of named ones. The question that this fails to answer is: what is a terminal driver supposed to do if I request a named point type that it simply doesn't have? Throw a tantrum, commit seppuku, read my mind, or what? The single most important design principle of gnuplot has always been absolute script compatibility  scripts were never allowed to just fail just because you run them on a different platform's gnuplot, or on a different terminal driver. And where possible, they should generate usable output. It's this universality that the number point and line types achieve. Replace them by names, and compatibility goes down the drain. 