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fitter - Tool

	Package	utility
	Module	fitting

Postscript file available

fitting tool

Constructors

fitter

Construct fitting tool

Functions

chi2	get the chi squared
constraint	get missing rank constraint equations
covariance	get the covariance matrix
deficiency	get missing rank
done	Remove the tool's resources
error	get errors in unknowns
fitavg	Get an average
fitpoly	Solve polynomial equations
fitspoly	Solve polynomial equationswith large x-range
fitted	Test Levenberg-Marquardt loop
fitter	Create a sub-tool
functional	Solve a general fitting problem
getstate	Obtain the state of the fitter
id	identification of tool
init	Initialize a fitting tool for use or re-use
linear	Solve a linear fitting problem
rank	get solution rank
reset	Reset the tool to its initialized state
sd	get standard deviation per unit weight
set	(Re-)initialize selected fitting tool properties
solution	get solution
stddev	get mean error (standard deviation per observation)
type	Obtain the type of the tool

A fitting tool has properties that can be set at creation; by the init method, by the set method and implicity by the use of the linear and functional methods. Properties are:

• n: the number of unknowns. In some specialized cases this property will be set automatically. All other properties have always to be set by the user.
• type: the type of condition equation:
  • real: n real factors for n real unknowns
  • complex: n complex factors for n complex unknowns
  • conjugate: 2n complex factors c, for n complex unknowns. The condition equations will be assumed to be:
    c₀.z₀ + c₁.z₀^* +...
  • asreal: n complex factors. The condition equations:
    real (c₀).real (z₀) + i(imag(c₀).imag(z₀)) +...
  • separable: 2n complex factors. Condition equations:
    c₀.real (z₀) + c₁.imag(z₀) +...
  The types can be given, in general, as minimum-match strings.
• colfac: collinearity factor (used in singular value decomposition to test dependencies)
• lmfac: Levenberg-Marquardt factor for non-linear solutions.

Example
A few basic examples: averaging a set of real numbers, and a set of complex numbers:

- include 'fitting.g'                                           # 1
T
- y := 0.1*1:5                                                  # 2
- f := dfs.compiled('p')                                        # 3
- dfit.linear(f,[],y);                                          # 4
T
- dfit.solution();                                              # 5
0.3 
- dfit.getstate()                                               # 6
[n=1, typ=0, colfac=1e-08, lmfac=0.001, type=real] 
- dfit.rank()                                                   # 7
1 
- dfit.sd()
0.158114 
- dfit.stddev()
0.158114 
- dfit.chi2()
0.1 
- dfit.covariance()                                             # 8
0.2 
- dfit.error()
0.0707107
- dfit.set(type='complex');                                     # 9
T 
- dfit.getstate()                
[n=1, typ=1, colfac=1e-08, lmfac=0.001, type=complex] 
- dfit.linear(dfs.compiled('p*x'), [1:6]*0 + 1, [1:6] + 1.5i*(1:6)) # 10
T 
- dfit.solution()                                               # 11
3.5+5.25i
# a one step option:
- dfit.linear(dfs.poly(), 1:5, 1:5); dfit.solution()            # 12
T 
3+0i
- dfit.set(type='real');                                        # 13
T 
- dfit.linear(dfs.poly(), 1:5, 1:5); dfit.solution()            # 14
T 
3
- dfit.set(type='complex')                                      #15
T

1.

Gives access to the fitting and creates the default fitting tool. It also includes the functionals tool.

2.

initialize the default fitter for 1 unknown (real by default)

3.

Create normal equations from 5 (simple) condition equations:
x = i

4.

Solve the normal equations

5.

Show the solution

6.

what is the fitter's state?

7.

the rank shows the number of dependencies. In this case there is no missing rank in the set of condition equations (i.e. they can be solved). The following functions show the information available after a solution. chi2 is the $\chi^{2}_{}$; sd the standard deviation per observed point; mu the standard deviation per unit weight.

8.

the covariance and errors in the solved unknowns can be obtained. The errors are the covariance scaled with the standard deviation

9.

Make a complex fitter (re-use) with two simultaneous sets of equations

10.

Make normal equations (Note the multiple condition equations in call) Effectively the first argument (the coefficients of the parameter to be solved) is a vector of length 5, all elements value 1; the second is a matrix, with for each condition equation two values. The values are:

   - [1:5,2*(1:5)]+1i*(1:10)
   [1+1i 2+2i 3+3i 4+4i 5+5i 2+6i 4+7i 6+8i 8+9i 10+10i]

where alternate values belong to the same set of observables

11.

solve

12.

show solution (Note the indices)

13.

do a fit for functional polynomial of zeroth order

This example shows the general usage of the fitting tool for linear equations:

• create and/or initialize a fitter
• use one or more make() methods to create the normal equations
• use fit() to solve the equations
• use information methods to get the solution information

Some specialized fit methods are available to combine the above in one call.

The above, simple, example, it is assumed that the data will fit a zero degree polynomial (i.e. just an average). It is just as easy to fit through the same data a higher degree polynomial. Let us first make the observed values:

- y := [1:10]

The simple way to fit an average here is (note we are still in complex type mode, but easy to correct):

- dfit.fitavg(y)
T 
- dfit.solution()
5.5+0i 
- dfit.set(type='real')
T 
- dfit.fitavg(y)
T 
- dfit.solution()
5.5

Another way is to make use of a zero degree polynomial, specified as a = y:

- x:=array(1,10)
- dfit.fitpoly(0,x,y) 
T 
- dfit.solution()
5.5

What would a 2-degree polynomial through the data look like ( a + bx + cx² = y):

- dfit.fitpoly(2,1:10,y)
T 
- dfit.solution()     
[0 1 0]

Non-linear fitting can be done in the same way; with slightly different calls for the full-blown case. For the generic case just replace the linear with functional, and a non-linear method will be used, using the Levenberg-Marquardt method.

The major difference is that the functional provided as model should have initial parameter estimates set.

Note that, of course, you can write your own non-linear solution by providing an initial parameter estimate in the functional to be fitted, use the linear method, and upfate the parameters until some criterium is met.

Tools

fitter	fitting tool
functionfitter	Tool to do simple fitting of numeric data

Constructors

fitter

Construct fitting tool

Functions

chi2	get the chi squared
constraint	get missing rank constraint equations
covariance	get the covariance matrix
deficiency	get missing rank
done	Remove the tool's resources
error	get errors in unknowns
fitavg	Get an average
fitpoly	Solve polynomial equations
fitspoly	Solve polynomial equationswith large x-range
fitted	Test Levenberg-Marquardt loop
fitter	Create a sub-tool
functional	Solve a general fitting problem
getstate	Obtain the state of the fitter
id	identification of tool
init	Initialize a fitting tool for use or re-use
linear	Solve a linear fitting problem
rank	get solution rank
reset	Reset the tool to its initialized state
sd	get standard deviation per unit weight
set	(Re-)initialize selected fitting tool properties
solution	get solution
stddev	get mean error (standard deviation per observation)
type	Obtain the type of the tool