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11.8.5 Curve Fitting Details

Calc's internal least-squares fitter can only handle multilinear models. More precisely, it can handle any model of the form ‘a f(x,y,z) + b g(x,y,z) + c h(x,y,z)’, where ‘a,b,c’ are the parameters and ‘x,y,z’ are the independent variables (of course there can be any number of each, not just three).

In a simple multilinear or polynomial fit, it is easy to see how to convert the model into this form. For example, if the model is ‘a + b x + c x^2’, then ‘f(x) = 1’, ‘g(x) = x’, and ‘h(x) = x^2’ are suitable functions.

For most other models, Calc uses a variety of algebraic manipulations to try to put the problem into the form

     Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)

where ‘Y,A,B,C,F,G,H’ are arbitrary functions. It computes ‘Y’, ‘F’, ‘G’, and ‘H’ for all the data points, does a standard linear fit to find the values of ‘A’, ‘B’, and ‘C’, then uses the equation solver to solve for ‘a,b,c’ in terms of ‘A,B,C’.

A remarkable number of models can be cast into this general form. We'll look at two examples here to see how it works. The power-law model ‘y = a x^b’ with two independent variables and two parameters can be rewritten as follows:

     y = a x^b
     y = a exp(b ln(x))
     y = exp(ln(a) + b ln(x))
     ln(y) = ln(a) + b ln(x)

which matches the desired form with Y = ln(y)’, A = ln(a)’, ‘F = 1’, ‘B = b’, and G = ln(x)’. Calc thus computes the logarithms of your ‘y’ and ‘x’ values, does a linear fit for ‘A’ and ‘B’, then solves to get a = exp(A)’ and ‘b = B’.

Another interesting example is the “quadratic” model, which can be handled by expanding according to the distributive law.

     y = a + b*(x - c)^2
     y = a + b c^2 - 2 b c x + b x^2

which matches with ‘Y = y’, ‘A = a + b c^2’, ‘F = 1’, ‘B = -2 b c’, ‘G = x’ (the -2 factor could just as easily have been put into ‘G’ instead of ‘B’), ‘C = b’, and ‘H = x^2’.

The Gaussian model looks quite complicated, but a closer examination shows that it's actually similar to the quadratic model but with an exponential that can be brought to the top and moved into ‘Y’.

The logistic models cannot be put into general linear form. For these models, and the Hubbert linearization, Calc computes a rough approximation for the parameters, then uses the Levenberg-Marquardt iterative method to refine the approximations.

Another model that cannot be put into general linear form is a Gaussian with a constant background added on, i.e., ‘d’ + the regular Gaussian formula. If you have a model like this, your best bet is to replace enough of your parameters with constants to make the model linearizable, then adjust the constants manually by doing a series of fits. You can compare the fits by graphing them, by examining the goodness-of-fit measures returned by I a F, or by some other method suitable to your application. Note that some models can be linearized in several ways. The Gaussian-plus-d model can be linearized by setting ‘d’ (the background) to a constant, or by setting ‘b’ (the standard deviation) and ‘c’ (the mean) to constants.

To fit a model with constants substituted for some parameters, just store suitable values in those parameter variables, then omit them from the list of parameters when you answer the variables prompt.

A last desperate step would be to use the general-purpose minimize function rather than fit. After all, both functions solve the problem of minimizing an expression (the chi^2’ sum) by adjusting certain parameters in the expression. The a F command is able to use a vastly more efficient algorithm due to its special knowledge about linear chi-square sums, but the a N command can do the same thing by brute force.

A compromise would be to pick out a few parameters without which the fit is linearizable, and use minimize on a call to fit which efficiently takes care of the rest of the parameters. The thing to be minimized would be the value of chi^2’ returned as the fifth result of the xfit function:

     minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)

where gaus represents the Gaussian model with background, data represents the data matrix, and guess represents the initial guess for ‘d’ that minimize requires. This operation will only be, shall we say, extraordinarily slow rather than astronomically slow (as would be the case if minimize were used by itself to solve the problem).

The I a F [xfit] command is somewhat trickier when nonlinear models are used. The second item in the result is the vector of “raw” parameters ‘A’, ‘B’, ‘C’. The covariance matrix is written in terms of those raw parameters. The fifth item is a vector of filter expressions. This is the empty vector ‘[]’ if the raw parameters were the same as the requested parameters, i.e., if ‘A = a’, ‘B = b’, and so on (which is always true if the model is already linear in the parameters as written, e.g., for polynomial fits). If the parameters had to be rearranged, the fifth item is instead a vector of one formula per parameter in the original model. The raw parameters are expressed in these “filter” formulas as ‘fitdummy(1)’ for ‘A’, ‘fitdummy(2)’ for ‘B’, and so on.

When Calc needs to modify the model to return the result, it replaces ‘fitdummy(1)’ in all the filters with the first item in the raw parameters list, and so on for the other raw parameters, then evaluates the resulting filter formulas to get the actual parameter values to be substituted into the original model. In the case of H a F and I a F where the parameters must be error forms, Calc uses the square roots of the diagonal entries of the covariance matrix as error values for the raw parameters, then lets Calc's standard error-form arithmetic take it from there.

If you use I a F with a nonlinear model, be sure to remember that the covariance matrix is in terms of the raw parameters, not the actual requested parameters. It's up to you to figure out how to interpret the covariances in the presence of nontrivial filter functions.

Things are also complicated when the input contains error forms. Suppose there are three independent and dependent variables, ‘x’, ‘y’, and ‘z’, one or more of which are error forms in the data. Calc combines all the error values by taking the square root of the sum of the squares of the errors. It then changes ‘x’ and ‘y’ to be plain numbers, and makes ‘z’ into an error form with this combined error. The ‘Y(x,y,z)’ part of the linearized model is evaluated, and the result should be an error form. The error part of that result is used for sigma_i’ for the data point. If for some reason ‘Y(x,y,z)’ does not return an error form, the combined error from ‘z’ is used directly for sigma_i’. Finally, ‘z’ is also stripped of its error for use in computing ‘F(x,y,z)’, ‘G(x,y,z)’ and so on; the righthand side of the linearized model is computed in regular arithmetic with no error forms.

(While these rules may seem complicated, they are designed to do the most reasonable thing in the typical case that ‘Y(x,y,z)’ depends only on the dependent variable ‘z’, and in fact is often simply equal to ‘z’. For common cases like polynomials and multilinear models, the combined error is simply used as the sigma’ for the data point with no further ado.)

It may be the case that the model you wish to use is linearizable, but Calc's built-in rules are unable to figure it out. Calc uses its algebraic rewrite mechanism to linearize a model. The rewrite rules are kept in the variable FitRules. You can edit this variable using the s e FitRules command; in fact, there is a special s F command just for editing FitRules. See Operations on Variables.

See Rewrite Rules, for a discussion of rewrite rules.

Calc uses FitRules as follows. First, it converts the model to an equation if necessary and encloses the model equation in a call to the function fitmodel (which is not actually a defined function in Calc; it is only used as a placeholder by the rewrite rules). Parameter variables are renamed to function calls ‘fitparam(1)’, ‘fitparam(2)’, and so on, and independent variables are renamed to ‘fitvar(1)’, ‘fitvar(2)’, etc. The dependent variable is the highest-numbered fitvar. For example, the power law model ‘a x^b’ is converted to ‘y = a x^b’, then to

     fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))

Calc then applies the rewrites as if by ‘C-u 0 a r FitRules’. (The zero prefix means that rewriting should continue until no further changes are possible.)

When rewriting is complete, the fitmodel call should have been replaced by a fitsystem call that looks like this:

     fitsystem(Y, FGH, abc)

where Y is a formula that describes the function ‘Y(x,y,z)’, FGH is the vector of formulas ‘[F(x,y,z), G(x,y,z), H(x,y,z)]’, and abc is the vector of parameter filters which refer to the raw parameters as ‘fitdummy(1)’ for ‘A’, ‘fitdummy(2)’ for ‘B’, etc. While the number of raw parameters (the length of the FGH vector) is usually the same as the number of original parameters (the length of the abc vector), this is not required.

The power law model eventually boils down to

     fitsystem(ln(fitvar(2)),
               [1, ln(fitvar(1))],
               [exp(fitdummy(1)), fitdummy(2)])

The actual implementation of FitRules is complicated; it proceeds in four phases. First, common rearrangements are done to try to bring linear terms together and to isolate functions like exp and ln either all the way “out” (so that they can be put into Y) or all the way “in” (so that they can be put into abc or FGH). In particular, all non-constant powers are converted to logs-and-exponentials form, and the distributive law is used to expand products of sums. Quotients are rewritten to use the ‘fitinv’ function, where ‘fitinv(x)’ represents ‘1/x’ while the FitRules are operating. (The use of fitinv makes recognition of linear-looking forms easier.) If you modify FitRules, you will probably only need to modify the rules for this phase.

Phase two, whose rules can actually also apply during phases one and three, first rewrites fitmodel to a two-argument form ‘fitmodel(Y, model)’, where Y is initially zero and model has been changed from ‘a=b’ to ‘a-b’ form. It then tries to peel off invertible functions from the outside of model and put them into Y instead, calling the equation solver to invert the functions. Finally, when this is no longer possible, the fitmodel is changed to a four-argument fitsystem, where the fourth argument is model and the FGH and abc vectors are initially empty. (The last vector is really ABC, corresponding to raw parameters, for now.)

Phase three converts a sum of items in the model to a sum of ‘fitpart(a, b, c)’ terms which represent terms ‘a*b*c’ of the sum, where a is all factors that do not involve any variables, b is all factors that involve only parameters, and c is the factors that involve only independent variables. (If this decomposition is not possible, the rule set will not complete and Calc will complain that the model is too complex.) Then fitparts with equal b or c components are merged back together using the distributive law in order to minimize the number of raw parameters needed.

Phase four moves the fitpart terms into the FGH and ABC vectors. Also, some of the algebraic expansions that were done in phase 1 are undone now to make the formulas more computationally efficient. Finally, it calls the solver one more time to convert the ABC vector to an abc vector, and removes the fourth model argument (which by now will be zero) to obtain the three-argument fitsystem that the linear least-squares solver wants to see.

Two functions which are useful in connection with FitRules are ‘hasfitparams(x)’ and ‘hasfitvars(x)’, which check whether ‘x’ refers to any parameters or independent variables, respectively. Specifically, these functions return “true” if the argument contains any fitparam (or fitvar) function calls, and “false” otherwise. (Recall that “true” means a nonzero number, and “false” means zero. The actual nonzero number returned is the largest n from all the ‘fitparam(n)’s or ‘fitvar(n)’s, respectively, that appear in the formula.)

The fit function in algebraic notation normally takes four arguments, ‘fit(model, vars, params, data)’, where model is the model formula as it would be typed after a F ', vars is the independent variable or a vector of independent variables, params likewise gives the parameter(s), and data is the data matrix. Note that the length of vars must be equal to the number of rows in data if model is an equation, or one less than the number of rows if model is a plain formula. (Actually, a name for the dependent variable is allowed but will be ignored in the plain-formula case.)

If params is omitted, the parameters are all variables in model except those that appear in vars. If vars is also omitted, Calc sorts all the variables that appear in model alphabetically and uses the higher ones for vars and the lower ones for params.

Alternatively, ‘fit(modelvec, data)’ is allowed where modelvec is a 2- or 3-vector describing the model and variables, as discussed previously.

If Calc is unable to do the fit, the fit function is left in symbolic form, ordinarily with an explanatory message. The message will be “Model expression is too complex” if the linearizer was unable to put the model into the required form.

The efit (corresponding to H a F) and xfit (for I a F) functions are completely analogous.