Fityk 0.8.8 - User's Manual

The GUI window of fityk consists of (from the top): menu bar, toolbar, main plot, auxiliary plot, output window, input field, status bar and of sidebar at right-hand side. The input field allows you to type and execute commands in a similar way as is done in the CLI version. The output window (which is configurable through a pop-up menu) shows the results. Incidentally, all GUI commands are converted into text and are visible in the output window, providing a simple way to learn the syntax.

The main plot can display data points, model that is to be fitted to the data and component functions of the model. Use the pop-up menu (click right button on the plot) to configure it. Some properties of the plot (e.g. colors of data points) can be changed using the sidebar.

One of the most useful things which can be displayed by the auxiliary plot is the difference between the data and the model (also controlled by a pop-up menu). Hopefully, a quick look at this menu and a minute or two's worth of experiments will show the potential of this auxiliary plot.

Configuration of the GUI (visible windows, colors, etc.) can be saved using GUI → Save current config. Two different configurations can be saved, which allows easy changing of colors for printing. On Unix platforms, these configurations are stored in a file in the user's home directory. On Windows - they are stored in the registry (perhaps in the future they will also be stored in a file).

The usage of the mouse on menu, dialog windows, input field and output window is (hopefully) intuitive, so the only remaining topic to be discussed here is how to effectively use the mouse on plots.

Let us start with the auxiliary plot. The right button displays a pop-up menu with a range of options, while the left allows you to select the range to be displayed on the x-axis. Clicking with the middle button (or with left button and Shift pressed simultaneously) will zoom out to display all data.

On the main plot, the meaning of the left and right mouse button depends on current mode (selected using either the toolbar or menu). There are hints on the status bar. In normal mode, the left button is used for zooming and the right invokes the pop-up menu. The same behaviour can be obtained in any mode by pressing Ctrl (or Alt.). The middle button can be used to select a rectangle that you want to zoom in to. If an operation has two steps, such as rectangle zooming (i.e. first you press a button to select the first corner, then move the mouse and release the button to select the second corner of the rectangle), this can be cancelled by pressing another button when the first one is pressed.

The basic file format is ascii text file with every line corresponding to one data point. If there are more than two columns of numbers, it can be specified which columns corresponds to x and y, and, optionally, also sigma. Numbers in line can be separated by whitespace, commas or semicolons. Lines that can't be read as numbers are ignored.

The modified version of xylib library is used to read data from file. New formats can be easily added.

Points are loaded from files using the command

where dataslot should be replaced with @0, unless many datasets are to be used simultaneously (for details see: the section called “Working with multiple datasets”), filetype and options usually can be omitted (in most of the cases the filetype can be detected automatically, all supported filetypes are listed at the end of this section), xcol, ycol, scol (supported only in text file) are columns corresponding to x, y and std. dev. of y. A column number of 0 generates a number increasing (from zero) with each point. block is supported by formats with multiple blocks of data.

If the filename contains blank characters, a semicolon or comma, it should be put inside single quotation marks (together with colon-separated indices, if any).

Multiple y columns and/or blocks can be specified, see the examples below.

      @0 < foo.vms
      @0 < foo.fii text first-line-header
      @0 < foo.dat:1:4:: # x,y - 1st and 4th columns
      @0 < foo.dat:1:3,4:: # load two dataset (with y in columns 3,4)
      @0 < foo.dat:1:3..5:: # load three dataset (with y in columns 3,4,5)
      @0 < foo.dat:1:4..6,2:: # load four dataset (y: 4,5,6,2)
      @0 < foo.dat:1:2..:: # load 2nd and all the next columns as y
      @0 < foo.dat:1:2:3: # read std. dev. of y from 3rd column
      @0 < foo.dat:0:1:: # x - 0,1,2,..., y - first column
      @0 < foo.raw::::0,1 # load two first blocks of data (as one dataset)
     

Information about loaded data can be obtained with: info data in dataslot

We often have the situation that only a part of the data from a file is of interest. We should be able to exclude selected points from fitting and all computations. Every point can be either active or inactive. This can be done with the command A=... (see the section called “Data transformations” for details) or with a mouse-click in the GUI. The idea of active and inactive points is simple: only the active ones are subject to fitting and peak-finding, inactive ones are neglected in these cases.

When fitting data, we assume that only the y coordinate is subject to statistical errors in measurement. This is a common assumption. To see how the y standard deviation influences fitting (optimization), look at the weighted sum of squared residuals formula in the section called “Nonlinear optimization ”. We can also think about weights of points - every point has a weight assigned, that is equal

Standard deviation of points can be read from file together with the x and y coordinates. Otherwise, it is set either to max(sqrt(y), 1.0) or to 1, depending on the value of data-default-sigma option. Setting std. dev. as a square root of the value is common and has theoretical ground when y is the number of independent events. You can always change standard deviation, e.g. make it equal for every point with command: S=1. See the section called “Data transformations” for details.

Every data point has four properties: x coordinate, y coordinate, standard deviation of y and active/inactive flag. Lower case letters x, y, s, a stand for these properties before transformation, and upper case X, Y, S, A for the same properties after transformation. M stands for the number of points. Data can be transformed using assignments. Command Y=-y will change the sign of the y coordinate of every point. You can also apply transformation to selected points: Y[3]=1.2 will change point with index 3 (which is 4th point, because first has index 0), and Y[3..6]=1.2 will do the same for points with indices 3, 4, 5, but not 6. Y[2...]=1.2 will apply the transformation to points with index 2 and above. You can guess what Y[..6]=1.2 does. Most of operations are executed sequentially for points from the first to the last one. n stands for the index of currently transformed point. The sequance of commands: M=500; x=n/100; y=sin(x) will generate the sinusoid dataset with 500 points.

If you have more than one dataset, you have to specify explicitly which dataset transformation applies to. See the section called “Working with multiple datasets” for details.

Expressions can contain real numbers in normal or scientific format (e.g. 1.23e5), constant pi, binary operators: +, -, *, /, ^, one argument functions: sqrt, exp, log10, ln, sin, cos, tan, sinh, cosh, tanh, atan, asin, acos, erf, erfc, gamma, lgamma (=ln(|gamma|)), abs, round (rounds to the nearest integer), two argument functions: min2, max2 (e.g. max2(3,5) will give 5), randuniform(a, b) (random number from interval (a, b)), randnormal(mu, sigma) (random number from normal distribution), voigt(a, b) (see below) and ternary ?: operator: condition ? expression1 : expression2, which performs expression1 if condition is true and expression2 otherwise. Conditions can be built using boolean operators and comparisions: AND, OR, NOT, >, >=, <, <=, ==, != (or <>), TRUE, FALSE.

The voigt function above has formula:

The value of a data expression can be shown using the command info, see examples at the end of this section.

t[x=expression], where t=x,y,s,a,X,Y,S,A gives a linear interpolation of t between two points (or the value of first/last point if the given x is outside the current data range).

Two numbers that differ less than epsilon i.e. abs(a-b)<epsilon, are considered equal. Indices are also computed in real number domain, and then rounded to the nearest integer.

Transformations can be joined with comma (,), e.g. X=y, Y=x swaps axes.

Before and after executing transformations, points are always sorted according to their x coordinate. You can change the order of points using order=t, where t is one of x, y, s, a, -x, -y, -s, -a. Clearly, this only makes sense for a sequence of transformations (joined with comma) as after finishing each transformation, points will be reordered again.

Points can be deleted using the following syntax: delete[index-or-range] or delete(condition) and created simply by increasing value of M.

There are two parametrized functions: spline and interpolate. The general syntax is: parametrizedfunc [param1, param2](expression) e.g. spline[22.1, 37.9, 48.1, 17.2, 93.0, 20.7](x) will give the value of a cubic spline interpolation through points (22.1, 37.9), (48.1, 17.2), ... in x. Function interpolation is similar, but gives a polyline interpolation. Spline function is used for manual background subtraction via the GUI.

There are also aggragate functions: min, max, sum, avg, stddev, darea. They have two forms. In the simpler one: aggragatefunc (expression), the value of expression in brackets is calculated for all points. min gives the smallest value, max the largest, sum, avg and stddev give the sum of all values, arithmetic mean and standard deviation, respectively. True value in data expression is represented numerically by 1., and false by 0, so sum can be also used to count points that fulfil given criteria.

darea gives the sum of expressions calculated using formulae: t*(x[n+1]-x[n-1])/2, where t is the value of the expression in brackets. darea(y) gives the area under interpolated data points, and can be used to normalize the area.

The second form: aggragatefunc (expression if condition) takes into account only points for which the condition is true.

A few examples:

     Y[1...] = Y[n-1] + y[n] # integrate

     x[...-1] = (x[n]+x[n+1])/2;  # reduces
     y[...-1] = y[n]+y[n+1];      # two times
     delete(n%2==1)               # number of points

     delete(not a) # delete inactive points

     X = 4*pi * sin(x/2*pi/180) / 1.54051 # changes x scale (2theta -> Q)

     # make equal step, keep the number of points the same
     X = x[0] + n * (x[M-1]-x[0]) / (M-1),  Y = y[x=X], S = s[x=X], A = a[x=X]

     # take the first 2000 points, average them and subtract as background
     Y = y - avg(y if n<2000)

     # fityk can also be used as a simple calculator
     i 2+2 #4
     i sin(pi/4)+cos(pi/4) #1.41421
     i gamma(10) #362880

     # examples of aggregate functions
     i max(y) # the largest y value
     i sum(y>avg(y)) # the number of points which have y value greater than arithmetic mean
     Y = y / darea(y) # normalize data area
     i darea(y-F(x) if 20<x<25)
     

There is also another kind of transformations, dataset tranformations, which operate on a whole dataset, not single points. The syntax (for one dataset) is: @0 = dstransformation @0, where dstransformation can be one of:

information in this section are not often used in practice. Read it after reading the section called “Model ”.

Variables ($foo) and functions (%bar) can be used in data transformations, and a current value of data expression can be assigned to the variable. Values of the function parameters (e.g. %fun.a0) and pseudo-parameters Center, Height, FWHM and Area (e.g. %fun.Area) can also be used. Pseudo-parameters are supported only by functions, which know how to calculate these properties.

Some properties of functions can be calculated using functions numarea, findx and extremum.

numarea(%f, x1, x2, n) gives area integrated numerically from x1 to x2 using trapezoidal rule with n equal steps.

findx(%f, x1, x2, y) finds x in interval (x1, x2) such that %f(x)=y using bisection method combined with Newton-Raphson method. It is a requirement that %f(x1) < y < %f(x2).

extremum(%f, x1, x2) finds x in interval (x1, x2) such that %f'(x)=0 using bisection method. It is a requirement that %f'(x1) and %f'(x2) have different signs.

A few examples:

      $foo = {y[0]} # data expression can be used in variable assignment
      $foo2 = {y[0] in @0}  # dataset can be given if necessary
      Y = y / $foo  # and variables can be used in data transformation

      Y = y - %f(x) # subtracts function %f from data

      Y = y - @0.F(x) # subtracts all functions in F

      Z += Constant(~0)  # fit constant x-correction (this can be caused...
      fit                # ...by a shift in scale of the instrument collecting data),
      X = x + @0.Z(x)  # ...remove it from the dataset,
      Z = 0            # ...and clear the x-correction in the model

      info numarea(%fun, 0, 100, 10000) # shows area of function %fun
      info %fun.Area  # it is not always supported

      info %_1(extremum(%_1, 40, 50)) # shows extremum value

      # calculate FWHM numerically, value 50 can be tuned
      $c = {%f.Center}
      i findx(%f, $c, $c+50, %f.Height/2) - findx(%f, $c, $c-50, %f.Height/2)
      i %f.FWHM # should give almost the same.
     

Let us call a set of data that usually comes from one file - a dataset. All operations described above assume only one dataset. If there are more datasets created, it must be explicitly stated which dataset the command is being applied to, e.g. M=500 in @0. Datasets have numbers and are referenced by '@' with the number, e.g. @3. @* means all datasets, (e.g. Y=y/10 in @*).

To load dataset from file, use one of commands:

The first one uses existing data slot and the second one creates a new slot. Using @+ increases the number of datasets, and command delete @n decreases it.

The syntax

can be used to duplicate a dataset (@+ = @n), to create new dataset as a sum of two or more existing sets (@+ = @n + @m + ...), to perform dataset transformations (@n = dataset_transformation @n), etc. A sum of datasets contains all points from all component datasets. If you want to merge points with the same x value, use one of dataset transformations: @+ = sum_same_x @n + @m + ....

Each dataset has a separate model, that can be fitted to the data. This is explained in the next chapter.

Each dataset also has a title (it does not have to be unique, however). When loading file, a title is automatically created, either using the filename or by reading it from the file (depending on the format of the file). Titles can be changed using the command set @n.title=new-title . To see the current title of the dataset, use info title in @n.

It is possible to show values of a data expression calculated for each dataset. Example: i+ avg(y) in @*.

Command

can export data to an ASCII TSV (tab separated values) file. To export data in a 3-column (x, y and standard deviation) format, use info @n (x, y, s) > file. If a is not listed in the list of columns, such as in this example, only the active points are exported.

All expressions that can be used on the right-hand side of data transformations can also be used in the column list. Additionally, F and Z can be used with dataset prefix, e.g. info @0 (n+1, x, y, F(x), y-F(x), Z(x), %foo(x), a, sin(pi*x)+y^2) > bar.tsv.

The model S (the function that is fitted to the data) is computed as a sum of component functions, like Gaussians or polynomials. To avoid confusion we will always use name model when referring to the total function fitted to data. The name function will be used only when referring to a component function. , where f_i is a function of x, and depends on a vector of parameters a. This vector contains all fitted parameters. Because we often have the situation, that the error in the x coordinate of data points can be modeled with function z(x; a), we introduce this term to the model:

where . Note that the same x-correction z(x) is used in all functions f_i.

Now we will have a closer look at f_i functions. Every function f_i has a type chosen from the function types available in the program. The same is true about functions z_i. One of these types is the Gaussian. It has the following formula:

There are three parameters of Gaussian. These parameters do not depend on x. There must be one variable bound to each parameter.

Variables in Fityk have names prefixed with the dollar symbol ($). A variable is created by assigning a value to it, e.g. $foo=~5.3 or $c=3.1 or $bar=5*sin($foo). $foo is here a so-called simple variable. It is created by assigning to it real number prefixed with ~. The `~' means that the value assigned to the variable can be changed when fitting the model to the data. For people familiar with optimization techniques: the number of defined simple variables is the number of dimensions of space we are looking for the optimum in. In the above example, the variable $c is actually a constant. $bar depends on the value of $foo. When $foo changes, the value of $bar also changes. Compound variables can be build using operators +, -, *, /, ^ and the functions sqrt, exp, log10, ln, sin, cos, tan, sinh, cosh, tanh, atan, asin, acos, erf, erfc, lgamma, abs, voigt. This is a subset of the functions used in data transformations.

Every simple parameter has a value and, optionally, domain. The domain is used only by the fitting algorithms which need to randomly initialize or change variables. Genetic Algorithms are a good example.

Variables can be used in data tranformations, e.g. Y=y/$a.

The value of the data expression can be used in the variable definition, but it must be inside braces, e.g. $bleh={M} or, to create a simple variable: $bleh=~{M}.

Sometimes it is useful to freeze a variable, i.e. to prevent it from changing while fitting. There is no special syntax for it, but it can be done using data expressions in this way:

      $a = ~12.3 # $a is fittable
      $a = {$a}  # $a is not fittable
      $a = ~{$a}  # $a is fittable again
     

It is also possible to define a variable as e.g. $bleh=~9.1*exp(~2). In this case two simple variables (with values 9.1 and 2) will be created automatically. Automatically created variables are named $_1, $_2, $_3, and so on.

Variables can be deleted using the command delete $variable.

Some fitting algorithms need to randomize the parameters of the fitted function (i.e. simple variables). For this purpose, the simple variable can have a specified domain. Note that the domain does not imply any constraints on the value the variable can have -- it is only a hint for fitting methods such as the Nelder-Mead simplex or Genetic Algorithms. Further information on how the domain is used in these methods is contained in the appropriate fitting description. The syntax is as follows:

      $a = ~12.3 [11 +- 5] # center and width of the domain is given

      $b = ~12.3 [ +- 5] # if the center of the domain is not specified,
                         # current value of the variable is used
     

If the domain is not specified, the value of variable-domain-percent option is used (domain is +/- value-of-variable * value-of-the-option / 100)

Let us go back to functions. Function types have names that start with upper case letter, e.g. Linear or Voigt. Functions (i.e. function instances) have names prefixed with a percent symbol, e.g. %func. Every function has a type and variables bound to its parameters.

To see a list of available function types, use the command info types. You can also use the command info typename, e.g. info Pearson7 to see the names of the parameters, default values and formulae.

Functions can be created by giving the type and the correct number of comma-separated variables in brackets, e.g. %f = Gaussian(~66254., ~24.7, ~0.264) or %f = Gaussian(~6e4, $ctr, $b+$c). Every expression which is valid on the right-hand side of a variable assignment, can also be used as a variable. If it is not simply a name of a variable, an automatic variable is created. In the last example two variables are created (value 60000 and the sum).

The second way is to give named parameters of a function, in any order, e.g. %f = Gaussian(height=~66254., hwhm=~0.264, center=~24.7) Function types can can have specified default values for some parameters, so this assignment is also valid: %f = Pearson7(height=~66254., center=~24.7, fwhm=~0.264) , although the shape parameter of Pearson7 is not given.

A deep copy of function (i.e. all variables that it depends on are also copied) can be made using the command %function =copy(%anotherfunction)

Functions can be also created with the command guess, as described in the section called “Guessing peak location ”.

You can change a variable bound to any of the function parameters in this manner:

      =-> %f = Pearson7(height=~66254., center=~24.7, fwhm=~0.264)
      New function %f was created.
      =-> %f.center=~24.8
      =-> $h = ~66254
      =-> %f.height=$h
      =-> info %f
      %f = Pearson7($h, $_5, $_3, $_4)
      =-> $h = ~60000 # variables are kept by name, so this also changes %f
      =-> %p1.center = %p2.center + 3 # keep fixed distance between %p1 and %p2
     

Functions can be deleted using the command delete %function.

User-defined function types can be created using command define, and then used in the same way as built-in functions. The name of new type must start with an upper-case letter, contain only letters and digits, have at least two characters and must not be the same as the name of built-in function. Defined functions can be undefined using command undefine.

The name of a UDF should be followed by parameters in brackets (see examples). Names of parameters should contain only lower-case alphanumeric characters and the underscore (_), and start with lowercase letter. The name "x" is reserved, do not put it into parameter list, just use it on the right-hand side of the definition.

Each parameter can have a specified default value. To allow adding a peak with the command guess, the default value is given as an expression which can then be calculated for a known "height", "center", "fwhm" and "area". If the name itself is one of the following: "height", "center", "fwhm, "area" or "hwhm", default value is deduced (in case of "hwhm" it is "fwhm/2").

UDFs can be defined either by giving a full formula, or as a sum of already defined functions, with possible re-parametrization (see GaussianArea and GLSum below for the example of the latter). When giving a full formula, right-hand side of the equality sign is similar to the definiton of variable, but the formula can also depend on x. Hopefully the examples below will make the syntax clear.

How it works (you can skip this paragraph): the formula is parsed, derivatives of the formula are calculated symbolically, all expressions are simplified (but there is a lot of space for optimization here), bytecode is created for a kind of virtual machine, and when fitting, the VM calculates the value of the function and derivatives for every point. Common Subexpression Elimination is not implemented yet, I suppose it will noticeably speed up UDFs.

Hint: use the init file for often-used definitions. See the section called “Invoking fityk ” for details.

Examples:

  # first how some built-in functions could be defined
  define MyGaussian(height, center, hwhm) = height*exp(-ln(2)*((x-center)/hwhm)^2)
  define MyLorentzian(height, center, hwhm) = height/(1+((x-center)/hwhm)^2)
  define MyCubic(a0=height,a1=0, a2=0, a3=0) = a0 + a1*x + a2*x^2 + a3*x^3

  # supersonic beam arrival time distribution
  define SuBeArTiDi(c, s, v0, dv) = c*(s/x)^3*exp(-(((s/x)-v0)/dv)^2)/x


  # area-based Gaussian can be defined as modification of built-in Gaussian
  # (it is the same as built-in GaussianA function)
  define GaussianArea(area, center, hwhm) = Gaussian(area/fwhm/sqrt(pi*ln(2)), center, hwhm)

  # sum of Gaussian and Lorentzian, a.k.a PseudoVoigt (should be in one line)
  define GLSum(height, center, hwhm, shape) = Gaussian(height*(1-shape), center, hwhm)
  + Lorentzian(height*shape, center, hwhm)

  # to change definition of UDF, first undefine previous definition
  undefine GaussianArea
     

With default settings, the value of every function is calculated at every point. Functions such as Gaussian often have non-neglectible values only in a small fraction of all points. To speed up the calculation, set the option cut-function-level to a non-zero value. For each function the range with values greater than cut-function-level will be estimated, and all values outside of this range are considered to be equal zero. Note that not all functions support this optimization.

If you have a number of loaded dataset, and the functions in different datasets do not share parameters, it is faster to fit the datasets sequentially (fit @0; fit @1; ...) then parallelly (fit @*).

Each defined simple-variable slows down the fitting, although this is often negligible.

As already discussed, each dataset has a separate model that can be fitted to the data. As can be seen from the formula above, the model is defined as a set functions f_i and the set of functions z_i. These sets are named F and Z respectively. The model is constructed by specifying names of functions in these two sets.

In many cases x-correction Z can safely be ignored. The fitted curve is thus the sum of all functions in F.

Command F += %function adds %function to F, command Z += %function adds %function to Z. To remove %function from F (or Z) either do F -= %function or delete %function (del %function). If there is more than one dataset, F and Z must be prefixed with the dataset number (e.g. @1.F += %function ). The following syntax is also valid:

  # create and add funtion to F
  %g = Gaussian(height=~66254., hwhm=~0.264, center=~24.7)
  @0.F += %g
  # create automatically named funtion and add it to F
  @0.F += Gaussian(height=~66254., hwhm=~0.264, center=~24.7)
  # clear F
  @0.F = 0
  # clear F and put three functions in it
  @0.F = %a + %b + %c
  # show info about the first and the last function in @0.F
  info @0.F[0], @0.F[-1]
  # the same as %bcp = copy(%b)
  %bcp = copy(@0.F[1])
  # make @1.F the exact (shallow) copy of @0.F
  @1.F = @0.F
  # make @1.F a deep copy of @0.F (all functions and variables
  # are duplicated).
  @1.F = copy(@0.F) 

The model can be exported as data points, using the syntax described in the section called “Exporting data”, or as mathematical formulae, using the command info formula in @n > filename. Some primitive simplifications are applied to the formula. To prevent it, put plus sign (+) after "info". The style of the formula output, governed by the formula-export-style option, can be either "normal" (exp(-x^2)) or "gnuplot" (exp(-x**2)).

Peak parameters can be exported using the command info peaks in @n > filename. Put the plus sign (+) after "info" to also export symmetric errors of the parameters. "@*" will export formulae or parameters used in all datasets to the same file.

It is often required to keep the width or shape of peaks constant for all peaks in the dataset. To change the variables bound to parameters with a given name for all functions in F, use the command: F.param=variable . Examples:

  F.hwhm=$foo # hwhm's of all functions in F that have parameter hwhm will be
              # equal to $foo. (hwhm here means half-width-at-half-maximum)
  F.shape=%_1.shape  # variable bound to shape of peak %_1 is bound
                     # also to shapes of all functions in F
  F.hwhm=~0.2  # For every function in F a new variable is created and bound
               # to parameter hwhm. All parameters are independent. 

It is possible to guess peak location and add it to F with the command: %name = guess PeakType [x1:x2] in @n , e.g. guess Gaussian [22.1:30.5] in @0. If the range is omitted, the whole dataset will be searched. Name of the function is optional. Some of parameters can be specified with syntax parameter=variable, e.g. guess PseudoVoigt [22.1:30.5] center=$ctr, shape=~0.3 in @0.

As an exception, if the range is omitted and the parameter center is given, the peak is searched around the center, +/- value of the option guess-at-center-pm.

Fityk offers only a primitive algorithm for peak-detection. It looks for the highest point in a given range, and than tries to find the width of the peak.

If the highest point is found near the boundary of the given range, it is very probable that it is not the peak top, and, if the option can-cancel-guess is set to true, the guess is cancelled.

There are two real-number options related to guess: height-correction and width-correction. The default value of them is 1. The guessed height and width are multiplied by the values of these options respectively.

If you are using the GUI, most of the available information can be displayed with mouse clicks. Alternatively, you can use the info command. Using info+ instead of info sometimes displays more verbose information.

Below is the list of arguments of info+ related to this chapter. The full list is in the section called “info: show information”

This is the core. We have a set of observations (data points), to which we want to fit a model that depends on adjustable parameters. Let me quote Numerical Recipes, chapter 15.0, page 656 (if you do not know the book, visit http://www.nr.com):

Our function of merit is WSSR - the weighted sum of squared residuals, also called chi-square:

Weights are based on standard deviations, . You can learn why squares of residuals are minimized e.g. from chapter 15.1 of Numerical Recipes. So we are looking for a global minimum of chi². This field of numerical research (looking for a minimum or maximum) is usually called optimization; it is non-linear and global optimization. Fityk implements three very different optimization methods. All are well-known and described in many standard textbooks.

The standard deviations of the best-fit parameters are given by the square root of the corresponding diagonal elements of the covariance matrix. The covariance matrix is based on standard deviations of data points. Formulae can be found e.g. in GSL Manual , chapter Linear regression. Overview (weighted data version).

To fit model to data, use command

The plus sign (+) prevents initialization of the fitting method. It is used to continue the previous fitting where it left off. All non-linear fitting methods are iterative. number-of-iterations is the maximum number of iterations. There are also other stopping criteria, so that the number of executed iterations can be smaller.

fit [...] in @* fits all datasets simultaneously.

Fitting methods can be set using the set command: set fitting-method = method, where method is one of: Levenberg-Marquardt, Nelder-Mead-simplex, Genetic-Algorithms.

All non-linear fitting methods are iterative, and there are two common stopping criteria. The first is the number of iterations and can be specified after the fit command. The second is the number of evaluations of the objective function (WSSR), specified by the value of option max-wssr-evaluations (0=unlimited). It is approximately proportional to time of computations, because most of time in fitting process is taken by evaluating WSSR. There are also other criteria, different for each method.

If you give too small n to fit command, and fit is stopped because of it, not because of convergence, it makes sense to use fit+ command to process further iterations. [TODO: how to stop fit interactively]

Setting set autoplot = on-fit-iteration will draw a plot after every iteration, to visualize progress. (see autoplot)

Information about goodness-of-fit can be displayed using info fit. To see symmetric errors use info errors, and info+ errors additionally shows the variance-covariance matrix.

Available methods can be mixed together, e.g. it is sensible to obtain initial parameter estimates using the Simplex method, and then fit it using Levenberg-Marquardt.

Values of all parameters are stored before and after fitting (if they changed). This enables simple undo/redo functionality. If in the meantime some functions or variables where added or removed, the program can still load the old parameters, but the result can be unexpected. The following history-related commands are provided:

This is a standard nonlinear least-squares routine, and involves computing the first derivatives of functions. For a description of the L-M method see Numerical Recipes, chapter 15.5 or Siegmund Brandt Data Analysis, chapter 10.15. Essentially, it combines an inverse-Hessian method with a steepest descent method by introducing a lambda factor. When lambda is equal to 0, the method is equivalent to the inverse-Hessian method. When lambda increases, the shift vector is rotated toward the direction of steepest descent and the length of the shift vector decreases. (The shift vector is a vector that is added to the parameter vector.) If a better fit is found on iteration, lambda is decreased - it is divided by the value of lm-lambda-down-factor option (default: 10). Otherwise, lambda is multiplied by the value of lm-lambda-up-factor (default: 10). The initial lambda value is equal to lm-lambda-start (default: 0.0001).

The Marquardt method has two stopping criteria other than the common criteria. If it happens twice in sequence, that the relative change of the value of the objective function (WSSR) is smaller then the value of the lm-stop-rel-change option, the fit is considered to have converged and is stopped. Additionally, if lambda is greater than the value of the lm-max-lambda option (default: 10^15), - usually when due to limited numerical precision WSSR is no longer changing, the fitting is also stopped.

To quote chapter 4.8.3, p. 86 of Peter Gans Data Fitting in the Chemical Sciences by the Method of Least Squares

This method is also described in previously mentioned Numerical Recipes (chapter 10.4) and Data Analysis (chapter 10.8).

There are a few options for tuning this method. One of these is a stopping criterium nm-convergence. If the value of the expression 2(M-m)/(M+m), where M and m are the values of the worst and best vertices respectively (values of objective functions of vertices, to be precise!), is smaller then the value of nm-convergence option, fitting is stopped. In other words, fitting is stopped if all vertices are almost at the same level.

The remaining options are related to initialization of the simplex. Before starting iterations, we have to choose a set of points in space of the parameters, called vertices. Unless the option nm-move-all is set, one of these points will be the current point - values that parameters have at this moment. All but this one are drawn as follows: each parameter of each vertex is drawn separately. It is drawn from a distribution that has its center in the center of the domain of the parameter, and a width proportional to both width of the domain and value of the nm-move-factor parameter. Distribution shape can be set using the option nm-distribution as one of: uniform, gaussian, lorentzian and bound. The last one causes the value of the parameter to be either the greatest or smallest value in the domain of the parameter - one of two bounds of the domain (assuming that nm-move-factor is equal 1).

[TODO]

In the GUI version there is hardly ever a need to use this command directly.

The command plot controls visualization of data and the model. It is used to plot a given area - in GUI it is plotted in the program's main window, in CLI the popular program gnuplot is used, if available.

xrange and yrange have one of two following syntaxes:

The second is just a dot (.), and it implies that the appropriate range is not to be changed.

Examples:

   plot [20.4:50] [10:20] # show x from 20.4 to 50 and y from 10 to 20

   plot [20.4:] # x from 20.4 to the end,
                # y range will be adjusted to encompass all data

   plot . [:10] # x range will not be changed, y from the lowest point to 10
   plot [:] [:] # all data will be shown
   plot         # all data will be shown
   plot . .     # nothing changes
     

The value of the option autoplot changes the automatic plotting behaviour. By default, the plot is refreshed automatically after changing the data or the model. It is also possible to visualize each iteration of the fitting method by replotting the peaks after every iteration.

First, there is an option verbosity (not related to command info) which sets the amount of messages displayed when executing commands.

If you are using the GUI, most information can be displayed with mouse clicks. Alternatively, you can use the info command. Using the info+ instead of info sometimes displays more detailed information.

The output of info can be redirected to file using info args > filename syntax to truncate the file or info args >> filename to append to the file.

The following arguments are recognized:

info der shows derivatives of given function.

      =-> info der sin(a) + 3*exp(b/a)
      f(a, b) = sin(a)+3*exp(b/a)
      df / d a = cos(a)-3*exp(b/a)*b/a^2
      df / d b = 3*exp(b/a)/a
     

All commands given during program execution are stored in memory. They can be listed using the command: info commands [n:m] or written to file: info commands [n:m] > filename . To put all commands executed so far during the session into the file foo.fit, type info commands[:] > foo.fit. With the plus sign (+) (i.e. info+ commands [n:m] ) information about the exit status of each command will be added.

To log commands to a file when they are executed, use: commands > filename or, to log also the output: commands+ > filename . To stop logging, use: commands > /dev/null .

Scripts can be executed using the command: commands < filename . It is possible to execute only selected lines from the script: commands < filename[n:m]

There is also a command dump > filename, which writes the current state of the program together with all datasets to a single .fit file.

Command sleep sec makes the program wait sec seconds, before continuing.

The command quit works as expected. If this command is found in a script it quits the program, not only the script.

Commands that start with ! are passed (without '!') to system() call.

To add a built-in function, you have to change the source of the program and then recompile it. Users who want to do this should be able to compile the program from source and know the basics of C, C++ or another programming language.

The description that follows is not complete. If something is not clear, you can always send me e-mail, etc.

"fp" you can see in fityk source means a real (floating point) number (typedef double fp).

The name of your function should start with uppercase letter and contain only letters and digits. Let us add function Foo with the formula: Foo(height, center, hwhm) = height/(1+((x-center)/hwhm)^2). C++ class representing Foo will be named FuncFoo.

In src/func.cpp you will find a list of functions:

       ...
       FACTORY_FUNC(Polynomial6)
       FACTORY_FUNC(Gaussian)
       ...
      

Now, add:

       FACTORY_FUNC(Foo)
      

Then find another list:

       ...
       FuncPolynomial6::formula,
       FuncGaussian::formula,
       ...
      

and add the line

      FuncFoo::formula,
     

Note that in the second list all items but the last are followed by comma.

In the file src/bfunc.h you can now begin writing the definition of your class:

      class FuncFoo : public Function
      {
          DECLARE_FUNC_OBLIGATORY_METHODS(Foo)
     

If you want to make some calculations every time parameters of the function are changed, you can do it in method do_precomputations. This possibility is provided for calculating expressions, which do not depend on x. Write the declaration here:

     void do_precomputations(std::vector<Variable*> const &variables);
     

and provide a proper definition of this method in src/bfunc.cpp.

If you want to optimize the calculation of your function by neglecting its value outside of a given range (see option cut-function-level in the program), you will need to use the method:

      bool get_nonzero_range (fp level, fp &left, fp &right) const;
     

This method takes the level below which the value of the function can be approximated by zero, and should set the left and right variables to proper values of x, such that if x<left or x>right than |f(x)|<level. If the function sets left and right, it should return true.

If your function does not have a "center" parameter, and there is a center-like point where you want the peak top to be drawn, write:

      bool has_center() const { return true; }
      fp center() const { return vv[1]; }
     

In the second line, between return and the semicolon, there is an expression for the x coordinate of peak top; vv[0] is the first parameter of function, vv[1] is the second, etc.

Finally, close the definition of the class with:

      };
     

Now go to file src/bfunc.cpp.

Write the function formula in this way:

      const char *FuncFoo::formula
      = "Foo(height, center, hwhm) = height/(1+((x-center)/hwhm)^2)";
     

The syntax of the formula is the similar as that of the UDF, but for built-in functions only the left hand side of the formula is parsed. The right hand side is for documentation only.

Write how to calculate the value of the function:

      FUNC_CALCULATE_VALUE_BEGIN(Foo)
          fp xa1a2 = (x - vv[1]) / vv[2];
          fp inv_denomin = 1. / (1 + xa1a2 * xa1a2);
      FUNC_CALCULATE_VALUE_END(vv[0] * inv_denomin)
     

The expression at the end (i.e. vv[0]*inv_denomin) is the calculated value. xa1xa2 and inv_denomin are variables introduced to simplify the expression. Note the "fp" (you can also use "double") at the beginning and semicolon at the end of both lines. The meaning of vv has already been explained. Usually it is more difficult to calculate derivatives:

      FUNC_CALCULATE_VALUE_DERIV_BEGIN(Foo)
          fp xa1a2 = (x - vv[1]) / vv[2];
          fp inv_denomin = 1. / (1 + xa1a2 * xa1a2);
          dy_dv[0] = inv_denomin;
          fp dcenter = 2 * vv[0] * xa1a2 / vv[2] * inv_denomin * inv_denomin;
          dy_dv[1] = dcenter;
          dy_dv[2] = dcenter * xa1a2;
          dy_dx = -dcenter;
      FUNC_CALCULATE_VALUE_DERIV_END(vv[0] * inv_denomin)
     

You must set derivatives dy_dv[n] for n=0,1,...,(number of parameters of your function - 1) and dy_dx. In the last brackets there is a value of the function again.

If you declared do_precomputations or get_nonzero_range methods, do not forget to write definitions for them.

After compilation of the program check if the derivatives are calculated correctly using command "info dF(x)", e.g. i dF(30.1). You can also use numarea, findx and extremum (see the section called “Functions and variables in data transformation” for details) to verify center, area, height and FWHM properties.

Hope this helps. Do not hesistate to change this description or ask questions if you have any. Consider sharing your function with other users (using FitykWiki or mailing list).