# 1. Beginners Guide / Help

Analysis of data consists of several steps that can be done with Jscatter :

- Reading of measured data
- Define a model
- Fit your model (test if it describes data)
- Save results !!!

These are described in the following.

Some places to start with Python/Numpy/SciPy :

- Python-Numpy-Tutorial A crash course with contents as :

> - Basic data types, Containers (Lists, Dictionaries, Sets, Tuples), Functions, Classes
> - Numpy, Arrays, indexing, Datatypes, Array math, Broadcasting

- The Python Tutorial
- Numpy Quickstart tutorial
- SciPy : high-level scientific computing

How to use, get help and command expansion

Jscatter works in a terminal/shell (open it) as basis without a graphical user interface (GUI).

For convenience we use Ipython (colors highlighting, history, …), start it typing “ipython” in your shell.

Optionally Jscatter can also be run in an environment like Jupyter (see Jupyter Notebook) or try a Jscatter live demo in a Jupyter Notebook at .

In a shell some simple things help the user.

- The Python docstring of a function can be retrieved by help(command)
- The ‘TAB’ can be used to expand a command or get the list of possible methods/attributes
- Just typing the name of an object returns a short representation e.g. the content of a dataarray.
- js.showDoc() opens the local documentation in a web browser.

    import jscatter as js
    i5 = js.dL(js.examples.datapath + '/iqt_1hho.dat')

    # try with some data from below
    help(i5.fit)

    # command completion by *TAB*
    i5.a    # write this and press 2x *TAB* to get *append, aslist, attr*

    # show string representation
    i5

    js.showDoc()

## 1.1. What are dataArray/dataList

dataArray is a container for matrix like data like a spreadsheet. Additional to the entries in the spreadsheet we have attributes describing metadata like the temperature of the measurement related to the spreadsheet data accessible as `data.temperature`.

dataList is a container for a list of dataArrays with variable size.

dataArray/dataList have additional methods to treat the contained data. This might be simple mathematical operations like add a value to a column or summing, averaging, adding another array of data as inherited from numpy arrays or methods for fitting and interpolation.

## 1.2. First basic examples

This shows the basic reading and fitting of some data as a starting point.

Details of all commands are explained later or in respective chapters.

### 1.2.1. A parabolic fit

A simple example without additional stuff as a skeleton. Additional possibilities are shown later.

    import jscatter as js
    import numpy as np

    # load data into a dataArray
    data=js.dA(js.examples.datapath+'/exampledata0.dat')

    # Look at some attributes
    data.attr            # shows a list of attributes as below
    data.temperature     # contains the temperature found in the file
    data.comment         # contains a comment

    # define model, q will get a list of values (numpy array)
    def parabola(q,a,b,c):
       return (q-a)**2+b*q+c

    # fit it defining free parameters, fixed parameters and map model names to data names
    # ('X' is always x axis)
    data.fit( model=parabola ,freepar={'a':2,'b':4}, fixpar={'c':-20}, mapNames={'q':'X'})
    data.showlastErrPlot()

    # One may use the 'temperature' in the data for parameter 'c'
    data.fit( model=parabola ,freepar={'a':2,'b':4}, fixpar={}, mapNames={'q':'X','c':'temperature'})

    # make nice plot with data and fit
    p=js.grace(1,1)
    p.plot(data)
    p.plot(data.lastfit,line=1,symbol=0)       # add result in lastfit
    p.title('Noisy parabola')

### 1.2.2. A bit more complex data

Here we introduce more possibilities and concepts how to treat the data and introduce models.

- Use a model from the libraries and change it to your needs.
- Fit with a condition
- Simulate fit result with changed parameters.
- Use a vectorized quadrature to include the distribution of a parameter.
- Access the fit result

The used data relate to small angle scattering of spherical particles in solvent with a polydispersity in radius

    import jscatter as js
    import numpy as np

    # load data
    data=js.dA(js.examples.datapath+'/exampledata.dat')

    # plot data
    p=js.grace(1,1)
    p.plot(data,le='my data')
    p.yaxis(scale='l',label='I(Q) / cm\S-1',size=1.5)
    p.xaxis(min=0.1,max=4,scale='l',label='Q / nm\S-1',size=1.5)
    p.title('Measurement of spheres')

    # use Guinier model from module formfactor (shortcut ff) as first size estimate in limited range q<1/Rg
    # fit it (with start values) and show result
    data.fit(js.ff.guinier,freepar={'Rg':5,'A':1},
                           fixpar={},
                           mapNames={'q':'X'},
                           condition=lambda a:a.X<1/7.)

    # show fit result (reused in next fits)
    data.showlastErrPlot(yscale='log',fitlinecolor=5)

    # compare to full dataset in above plot
    # modelValues recalcs fit result with optional changed parameters (here q)
    p.plot( data.modelValues(q=np.r_[0.1:1:0.02]), line=[1,3,2], symbol=0, legend='fit Guinier')
    p.legend(x=0.8,y=1e6)          # update legend with position

    # define monodisperse sphere model adding amplitude and background
    def sphere(q,R,A,bgr):
       result=A*js.ff.sphere(q=q,radius=R).Y+bgr
       return result

    # use sphere model (keywords as *freepar* can be dropped if order of keywords is preserved)
    data.fit(sphere,{'R':5,'A':1,'bgr':50},{},{'q':'X'})

    p.plot(   data.modelValues(q=data.X),   li=[1,3,3],sy=0,le='fit sphere')
    p.legend()

    # define better sphere model with polydisperse radius
    def psphere(q,R,A,bgr,dR):
       # dR is radius width of a Gaussian centered at R (see doc of pDA)
       result=A*js.formel.pDA(js.ff.sphere,dR,'radius',q=q,radius=R).Y+bgr
       return result

    # use the polydisperse sphere model
    data.fit(psphere,{'R':5,'A':1,'bgr':50,'dR':1},{},{'q':'X'})

    p.plot(  data.modelValues(q=np.r_[0.1:4:0.04]),  li=[1,3,4],sy=0,le='fit polydisperse sphere')
    p.legend(x=0.8,y=1e6)
    p.xaxis(min=0.1,max=4)
    # p.save('myevaluation.agr')
    # p.save('myevaluation.png')

    # look at result with parameters with errors
    data.lastfit          # model data
    data.lastfit.R        # fit parameter
    data.lastfit.R_err    # fit parameter 1-sigma error
    data.lastfit.dR
    data.lastfit.dR_err

## 1.3. Reading ASCII files

- dataArray (js.dA) reads one dataset from a file.

      # create single dataArray from file
      dat=js.dA('singledatafilename.dat')            # take dataset in file
      dat=js.dA('multidatafilename.dat',index=3)     # take second dataset in file

- dataList (js.dL) reads all datasets from one/multiple files (may differ in shape)

      # create dataList
      data=js.dL('../data/latest_0001.dat')          # read all dataset in file
      data=js.dL('../data/latest_0001.dat',index=2)  # select only the second dataset in the file
      data.append(dat)                               # append single dataArray read from above

  Read all, filter later Create dataList from multiple files (uses glob patterns)

      data=js.dL('../data/latest*.dat')              # reads all '.dat' files starting with 'latest' in data folder.
      # filter according to attributes or name
      data2=data.filter(lambda a:a.name.startswith('latest_conc2') & (a.Temp>100))
      data3=data[1:-1:3]                             # drop first and last and use only each 3rd

      # same with glob (using name patterns as ?,*) treating
      import glob
      data=js.dL()  # empty dataList
      for filename in glob.glob('../data/latest*.dat'):
         data.append(filename)                       # you may add options

      # add attributes e.g. from filename or comments
      for dat in data:
         dat.Temp=float(dat.name.split('_')[2])     # extract attribute from name (which is always the filename read from)
         dat.pressure=dat.comment[1].split()[1]     # if second comment line contains 'pressure 1000 mbar'

A common problem is how to read ASCII files with data as the format is often not very intuitive designed. Often there is additional metadata before or after a matrix like block or in the filename.

Jscatter uses a simple concept to classify lines :

- 2 numbers at the beginning of a line are data (matrix like data block).
- a name followed by a number (and more) is an attribute with name and content.
- everything else is comment (but can later be converted to an attribute).
- The filename is always stored in attribute `.name`

> A new dataArray is created if, while reading a file, a data block with a parameter block (preceded or appended) is found or a keyword indicates it (see `dataList`).

Often it is just necessary to replace some characters to fit into this concept. This can be done during reading using some simple options in dataArray/dataList creation:

- replace={‘old’:’new’,’,’:’.’} ==> replace char and strings
- skiplines=lambda words: any(w in words for w in [‘’,’ ‘,’NAN’,’‘*]) ==> skip complete bad lines
- takeline=’ATOM’ ==> select specific lines
- ignore =’#’ ==> skip lines starting with this character
- usecols =[1,2,5] ==> select specific columns
- lines2parameter=[2,3,4] ==> use these data lines as comment and not as data.

See `dataArray()` for all options and how to use them.

If there is more information in comments or filename this can be extracted by using the comment lines.
  - data.getfromcomment(‘nameatfirstcolumn’) ==> extract a list of words in this line
  - data.name ==> filename, see below examples.

How to extract information from comment lines

    # lines as 'pressure 1013 14' are used automatically to set an attribute as .pressure
    # if data.name is a string as 'adh_Temp273_conc02.dat' extract Temp and conc like
    temp=data.name.split('_')
    data.Temp=float(temp[1][4:])
    data.conc=float(temp[2][4:6])
    #
    # if same line is in comment use
    temp=data.comment[0].split('_')
    # or use data.getfromComment(...)

Some examples and how to read them

filename: data1_273K_10mM.dat (e.g. Instrument JNSE@MLZ, Garching)

    this is just a comment or description of the data
    temp     293
    pressure 1013 14
    detectorsetting up
    name     temp1bsa
    0.854979E-01  0.178301E+03  0.383044E+02
    0.882382E-01  0.156139E+03  0.135279E+02
    0.909785E-01  0.150313E+03  0.110681E+02
    0.937188E-01  0.147430E+03  0.954762E+01
    0.964591E-01  0.141615E+03  0.846613E+01
    0.991995E-01  0.141024E+03  0.750891E+01
    0.101940E+00  0.135792E+03  0.685011E+01
    0.104680E+00  0.140996E+03  0.607993E+01

Read by

    data=js.dA('data1_273K_10mM.dat')
    data.getfromComment('detectorsetting')           # creates attribute detectorsetting with string value 'up' found in comments
    data.Temp=float(data.name.split('_')[1][:-1])    # extracts the temperature from filename
    data.conc=float(data.name.split('_')[2][:-2])    # same for concentration
    data.pressure[0]                                 # use pressure value 1013 # this was created automatically
    data.Temp                                        # use temperature value   # this was created explicit

NSE measurement from IN15 at ILL Grenoble

    ftime          E_SUM           EERR_SUM        EQ_0.0596       EERRQ_0.0596    EQ_0.0662       EERRQ_0.0662    EQ_0.0728       EERRQ_0.0728    EQ_0.0793       EERRQ_0.0793    EQ_0.0859       EERRQ_0.0859
    Amplitude       -1.0000e+00      0.0000e+00      3.3149e+00      1.9984e-03      3.4203e+00      2.0375e-03      3.2560e+00      1.9803e-03      2.7188e+00      1.8161e-03      1.8634e+00      1.5032e-03
    Polarisation    -1.0000e+00      0.0000e+00      2.3719e+00      4.4403e-03      2.3723e+00      4.6673e-03      2.1675e+00      4.6726e-03      1.7156e+00      4.4392e-03      1.1127e+00      3.7890e-03
     0.0000e+00      1.0000e+00      1.0318e-03      1.0000e+00      1.9261e-03      1.0000e+00      2.0252e-03      1.0000e+00      2.2186e-03      1.0000e+00      2.6615e-03      1.0000e+00      3.4992e-03
     2.2428e-01      9.7447e-01      3.4201e-03      9.7363e-01      6.3708e-03      9.7026e-01      6.6990e-03      9.8392e-01      7.3605e-03      9.8819e-01      8.8623e-03      9.5632e-01      1.1831e-02
     2.9474e-01      9.8425e-01      3.3694e-03      9.9020e-01      6.1962e-03      9.7785e-01      6.5809e-03      9.9125e-01      7.2723e-03      9.8005e-01      8.8698e-03      9.9022e-01      1.1909e-02
     3.6520e-01      9.7910e-01      3.3071e-03      9.8269e-01      6.0875e-03      9.8190e-01      6.4363e-03      9.7275e-01      7.1155e-03      9.8566e-01      8.7117e-03      9.7766e-01      1.1829e-02
     5.0612e-01      9.7927e-01      3.2226e-03      9.7898e-01      5.9112e-03      9.7517e-01      6.2379e-03      9.8108e-01      6.9563e-03      9.8669e-01      8.5569e-03      9.8611e-01      1.1557e-02
     ...

Read by

    # column 1,2 are averages over following columns. First line contains q values
    data=js.dL()   # empty dataList
    temp=js.dA('017112345.txt')         # read all then sort later
    for i in [3,5,7,9]:
       data.append(temp[[0,i,i+1]])
       data[-1].Amplitude=temp.Amplitude[i-1:i+1]
       data[-1].Polarisation=temp.Polarisation[i-1:i+1]
       data[-1].q=float(temp.comment[0].split()[i].split('_')[1])

aspirin.pdb: Atomic coordinates for aspirin (AIN from Protein Data Bank, PDB ):

    Header
    Remarks blabla
    Remarks in pdb files are sometimes more than 100 lines
    ATOM      1  O1  AIN A   1       1.731   0.062  -2.912  1.00 10.00           O
    ATOM      2  C7  AIN A   1       1.411   0.021  -1.604  1.00 10.00           C
    ATOM      3  O2  AIN A   1       2.289   0.006  -0.764  1.00 10.00           O
    ATOM      4  C3  AIN A   1      -0.003  -0.006  -1.191  1.00 10.00           C
    ATOM      5  C4  AIN A   1      -1.016   0.010  -2.153  1.00 10.00           C
    ATOM      6  C5  AIN A   1      -2.337  -0.015  -1.761  1.00 10.00           C
    ATOM      7  C6  AIN A   1      -2.666  -0.063  -0.416  1.00 10.00           C
    ATOM      8  C1  AIN A   1      -1.675  -0.085   0.544  1.00 10.00           C
    ATOM      9  C2  AIN A   1      -0.340  -0.060   0.168  1.00 10.00           C
    ATOM     10  O3  AIN A   1       0.634  -0.083   1.111  1.00 10.00           O
    ATOM     11  C8  AIN A   1       0.314   0.035   2.410  1.00 10.00           C
    ATOM     12  O4  AIN A   1      -0.824   0.277   2.732  1.00 10.00           O
    ATOM     13  C9  AIN A   1       1.376  -0.134   3.466  1.00 10.00           C
    ATOM     14  HO1 AIN A   1       2.659   0.080  -3.183  1.00 10.00           H
    ATOM     15  H4  AIN A   1      -0.765   0.047  -3.203  1.00 10.00           H
    ATOM     16  H5  AIN A   1      -3.119   0.001  -2.505  1.00 10.00           H
    ATOM     17  H6  AIN A   1      -3.704  -0.082  -0.117  1.00 10.00           H
    ATOM     18  H1  AIN A   1      -1.939  -0.123   1.591  1.00 10.00           H
    ATOM     19  H91 AIN A   1       0.931  -0.004   4.453  1.00 10.00           H
    ATOM     20  H92 AIN A   1       1.807  -1.133   3.391  1.00 10.00           H
    ATOM     21  H93 AIN A   1       2.158   0.610   3.318  1.00 10.00           H
    CONECT    1    2   14 may appear at the end
    HETATOM lines may appear at the end
    END

Read by (several methods):

    # 1.
    # take 'ATOM' lines, but only column 6-8 as x,y,z coordinates.
    js.dA('AIN_ideal.pdb',takeline='ATOM',replace={'ATOM':'0'},usecols=[6,7,8])
    # 2.
    # replace 'ATOM' string by number and set XYZ for convenience
    js.dA('AIN_ideal.pdb',replace={'ATOM':'0'},usecols=[6,7,8],XYeYeX=[0,1,None,None,2])
    # 3.
    # only the Oxygen atoms
    js.dA('AIN_ideal.pdb',takeline=lambda w:(w[0]=='ATOM') & (w[2][0]=='O'),replace={'ATOM':'0'},usecols=[6,7,8])
    # 4.
    # using regular expressions we can decode the atom specifier into a scattering length
    import re
    rHO=re.compile('HO\d') # 14 is HO1
    rH=re.compile('H\d+')  # represents something like 'H11' or 'H1' see regular expressions
    rC=re.compile('C\d+')
    rO=re.compile('O\d+')
    # replace atom specifier by number and use it as last column
    ain=js.dA('AIN_ideal.pdb',replace={'ATOM':'0',rC:1,rH:5,rO:2,rHO:5},usecols=[6,7,8,2],XYeYeX=[0,1,None,None,2])
    # 5.
    # read only atoms and use it to retrieve atom data from js.formel.Elements
    atoms=js.dA('AIN_ideal.pdb',replace={'ATOM':'0'},usecols=[2],XYeYeX=[0,1,None,None,2])[0].array
    al=[js.formel.Elements[a[0].lower()] for a in atoms]

data2.txt:

    # this is just a comment or description of the data
    # temp     ;    293
    # pressure ; 1013 14  bar
    # name     ; temp1bsa
    &doit
    0,854979E-01  0,178301E+03  0,383044E+02
    0,882382E-01  0,156139E+03  0,135279E+02
    0,909785E-01  *             0,110681E+02
    0,937188E-01  0,147430E+03  0,954762E+01
    0,964591E-01  0,141615E+03  0,846613E+01
    nan           nan           0

Read by

    # ignore is by default '#', so switch it of
    # skip lines with non numbers in data
    # replace some char by others or remove by replacing with empty string ''.
    js.dA('data2.txt',replace={'#':'',';':'',',':'.'},skiplines=[‘*’,'nan'],ignore='' )

pdh format used in some SAXS instruments (first real data point is line 4):

    SAXS BOX
          2057         0         0         0         0         0         0         0
      0.000000E+00   3.053389E+02   0.000000E+00   1.000000E+00   1.541800E-01
      0.000000E+00   1.332462E+00   0.000000E+00   0.000000E+00   0.000000E+00
    -1.069281E-01   2.277691E+03   1.168599E+00
    -1.037351E-01   2.239132E+03   1.275602E+00
    -1.005422E-01   2.239534E+03   1.068182E+00
    -9.734922E-02   2.219594E+03   1.102175E+00
    ......

Read by:

    # this saves the prepended lines in attribute line_2,...
    empty=js.dA('exampleData/buffer_averaged_corrected_despiked.pdh',usecols=[0,1],lines2parameter=[2,3,4])
    # next just ignores the first lines (and last 50) and uses every second line,
    empty=js.dA('exampleData/buffer_averaged_corrected_despiked.pdh',usecols=[0,1],block=[5,-50,2])

Read csv data by (comma separated list)

    js.dA('data2.txt',replace={',':' '})
    # If tabs separate the columns
    js.dA('data2.txt',replace={',':' ','\t':' '})

Get a list of files in a folder with specific names

    import glob
    files=glob.glob('latest*.dat')     # files starting with 'latest' and ending '.dat'
    files=glob.glob('latest???.dat')     # files starting with 'latest' and ending '.dat' and 3 char in between

## 1.4. Creating from numpy arrays

This demonstrates how to create dataArrays form calculated data with error:

    #
    x=np.r_[0:10:0.5]                 # a list of values
    D,A,q=0.45,0.99,1.2               # parameters
    data=js.dA(np.vstack([x,  np.exp(-q**2*D*x)+np.random.rand(len(x))*0.05,  x*0+0.05]))
    data.diffusioncoefficient=D
    data.amplitude=A
    data.wavevector=q

    # alternative (diffusion with noise and error )
    data=js.dA(np.c_[x,np.exp(-q**2*D*x)*0.05,x*0+0.05].T)
    f=lambda xx,DD,qq,e:np.exp(-qq**2*DD*xx)+np.random.rand(len(x))*e
    data=js.dA(np.c_[x,f(x,D,q,0.05),np.zeros_like(x)+0.05].T)

## 1.5. Manipulating dataArray/dataList

Changing values uses the same syntax as in numpy arrays with all available methods and additional .X,.Y…

The columns used for .X,.Y,.Z,.W and respective errors can be changed by .setColumnIndex(ix=2,iy=4,iey=None) (default ix,iy,iey = 0,1,2 as one expects this). Later .Y is used during fits as dependent variable.

dataList elements should be changed individually as dataArray (this can be done in loops)

    i7=js.dL(js.examples.datapath+'/polymer.dat')
    for ii in i7:
       ii.X/=10          # change scale
       ii.Y/=ii.conc     # normalising by concentration
       ii.Y=-np.log(ii.Y)*2
    i1=js.dA(js.examples.datapath+'/a0_336.dat')
    # all the same to multiply .X by 2
    i1.X*=2
    i1[0]*=2
    i1[0]=i1[0]*2        # most clear writing
    # multiply each second Y value by 2 (using advanced numpy indexing)
    i1[1,::2]=i1[1,::2]*2
    # now more strange: each second gets the value from following *2+1
    # unlimited possibilities to manipulate data :-)
    i1[1,::2]=i1[1,1::2]*2+1 + i1[2,1::2]

    # making a Kratky plot
    p=js.grace()
    i1k=i1.copy()
    i1k.Y=i1k.Y*i1k.X**2
    p.plot(i1k)
    # or
    p.plot(i1.X*10,i1.Y*i1.X**2)

## 1.6. Indexing dataArray/dataList and reducing

Basic Slicing and Indexing/Advanced Indexing/Slicing works as described at numpy indexing .

This means accessing parts of the dataArray/dataList by indexing with integers, boolean masks or arrays to extract/manipulate a subset of the data.

[A,B,C] in the following describes A dataList, B dataArray columns and C values in columns.

The start:end:step notation is used. With missing value start equals 0, end the last and step 1.

    i5=js.dL(js.examples.datapath+'/iqt_1hho.dat')
    # remove first 2 and last 2 datapoints in all dataArrays
    i6=i5[:,:,:2:-2]
    # remove first column and use 1,2,3 columns in all dataArrays
    i6=i5[:,1:4,:]
    # use each second element in dataList and remove last 2 datapoints in all dataArrays
    i6=i5[::2,:,:-2]
    # You can loop over the dataArrays
    for dat in i5:
       dat.X=dat.X*10
    # select a subset by explicit list
    i7=i5[[2,3,5,6,]]

Reducing data to a lower number of values is done by data.prune (see `dataList` )

prune reduces e.g by 2000 points by averaging in intervals to get 100 points.

    i7=js.dL(js.examples.datapath+'/a0_336.dat')
    # mean values in interval [0.1,4] with 100 points distributed on logscale
    i7_2=i7.prune(lower=0.1,upper=4,number=100,kind='log') #type='mean' is default

dataList can be filtered to use a subset e.g. with restricted attribute values as .q>1 or temp>300 .

    i5=js.dL(js.examples.datapath+'/iqt_1hho.dat')
    i6=i5.filter(lambda a:a.q<2)
    i6=i5.filter(lambda a:a.q in [1,2,3,4,5])
    i6=i5.filter(temp=300)         # automatically sorted for these attributes

This demonstrates how to filter data values according to some rule.

    x=np.r_[0:10:0.5]
    D,A,q=0.45,0.99,1.2               # parameters
    rand=np.random.randn(len(x))      # the noise on the signal
    data=js.dA(np.vstack([x,np.exp(-q**2*D*x)+rand*0.05,x*0+0.05,rand])) # generate data with noise
    # select like this
    newdata=data[:,data[3]>0]         # take only positive noise in column 3
    newdata=data[:,data.X>2]          # X>2
    newdata=data[:,data.Y<0.9]        # Y<0.9

## 1.7. Fitting experimental data

We need:

- Data need to be read and prepared.

> - Data may be a single dataset (usually in a dataArray) or several of these (in a dataList) like multiple measurements with same or changing parameters (e.g. wavevectors). Coordinates are in .X and values in .Y
> - 2D data (e.g. a detector image with 2 dimensions) need to be transformed to coordinates .X,*.Z* with values in .Y. This also gives pixels coordinates in an image a physical interpretation as e.g. wavevectors. See examples 2D fitting and Fitting the 2D scattering of a lattice
> - Attributes need to be extracted from read data (from Comments or filename or from a list..) In the below example the Temperature is stored in the data as attribute.

- A model, which can be defined in different ways. See below or in How to build simple models for different ways.

  Please avoid using lists as parameters as list are used to discriminate between common parameters (a single float) and individual fit parameters (a list of float for each) in dataList.

- Fit algorithm We use methods from the scipy.optimize module that are incorporated in the .fit method of dataArray/dataList. .fit supports different fit algorithms (see dataList.fit Examples how to choose and about speed differences):

> - method=’leastsq’ (default) is a wrapper around MINPACK’s lmdif and lmder, which is a modification of the Levenberg-Marquardt algorithm. This is what you usually expect by “fitting” including error bars (and a covariance matrix for experts….). Errors are 1-sigma errors as they are calculated from the covariance matrix and not directly depend on the errors of the .eY. Still the relative weight of values according to .eY is relevant.
> - method=’BFGS’ Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. Not as fast as ‘leastsq’ but gives also error bars.
> - method=’differential_evolution’ is a global optimization method using iterative improving candidate solutions. In general it needs a large number of function calls but may find a global minimum.
> - method=‘Nelder-Mead’,’CG’, ….. other optimization methods. These are slower converging than ‘leastsq’ and give no error bars. Some require a gradient function ore more. They are more for advanced users if someone really knows why using it in special cases.

Warning

Save results !!!

The resulting parameters with errors are in .lastfit. Save the fit result !!! I regularly have to ask PhD students “What are the errors ?” and they repeat all their work again.

Fit results without Errors are meaningless.

If the fit finishes it tells if there was success or if it failed. For error messages the final parameters are no valid fit results!!! Please always check this.

A typical example of several datasets from a SAXS measurement (polymer in solution)

    import jscatter as js
    import numpy as np

    # read data
    data=js.dL(js.examples.datapath+'/polymer.dat')
    # merge equal Temperatures each measured with two detector distances
    data.mergeAttribut('Temp',limit=0.01,isort='X')

    # define model
    # q will get the X values from your data as numpy ndarray.
    def gCpower(q,I0,Rg,A,beta,bgr):
        """Model Gaussian chain  + power law and background"""
        gc=js.ff.gaussianChain(q=q,Rg=Rg)
        # add power law and background
        gc.Y=I0*gc.Y+A*q**beta+bgr
        # add attributes for later documentation, these are additional content of lastfit (see below)
        gc.A=A
        gc.I0=I0
        gc.bgr=bgr
        gc.beta=beta
        gc.comment=['gaussianChain with power law and bgr','a second comment']
        return gc

    data.makeErrPlot(yscale='l',xscale='l')    # additional errorplot with intermediate output
    data.setlimit(bgr=[0,1])                   # upper and lower soft limit

    # here we use individual parameter ([]) for all except a common beta ( no [] )
    # please try removing the [] and play with it :-)
    # mapNames tells that q is *.X* (maps model names to data names )
    # condition limits the range to fit (may also contain something like (a.Y>0))
    data.fit(model=gCpower,
             freepar={'I0':[0.1],'Rg':[3],'A':[1],'bgr':[0.01],'beta':-3},
             fixpar={},
             mapNames={'q':'X'},
             condition =lambda a:(a.X>0.05) & (a.X<4))

    # to fix a parameter move it to fixpar dict (bgr is automatically extended)
    data.fit(model=gCpower,
             freepar={'I0':[0.1],'Rg':[3],'A':[1]},
             fixpar={'bgr':[0.001, 0.0008, 0.0009],'beta':-4},
             mapNames={'q':'X'},
             condition =lambda a:(a.X>0.05) & (a.X<4))

    # result parameter and error (example)
    data.lastfit.Rg
    data.lastfit.Rg_err
    # as result dataArray
    result=js.dA(np.c_[data.Tempmean,data.lastfit.Rg,data.lastfit.Rg_err].T)
    # plot it
    p=js.grace()
    p.plot(result)

    # save the fit result including parameters, errors and covariance matrix
    # and your model description
    data.lastfit.save('polymer_fitDebye.dat')

You may want to fit dataArray in a dataList individually. Do it in a loop.

    # from the above
    for dat in data:
       dat.fit(model=gCpower,
            freepar={'I0':0.1,'Rg':3,'A':1,},
            fixpar={'bgr':0.001,'beta':-3},
            mapNames={'q':'X'},
            condition =lambda a:(a.X>0.05) & (a.X<4))

    # each dataArray has its own .lastfit, .errPlot and so on
    data[0].showlastErrPlot()
    # for saving with individual names
    for dat in data:
       dat.lastfit.save('fitresult_Temp%2g.dat' %(dat.Tempmean))

## 1.8. Why Fits may fail

If your fit fails it is most not an error of the fit algorithm. Read the message at the end of the fit it gives a hint what happened.

- If your fit results in a not converging solution or maximum steps reached then its not a valid fit result. Decrease tolerance, increase maxstep or reduce number of parameter to get a valid result. Try more reasonable start parameters.
- Your model may have dependent parameters. Then the gradient cannot be evaluated. Think of it as a valley with a flat ground. Then you have a line as minimum but you ask for a point.
- Your starting parameters are way of and within the first try the algorithm finds no improvement. This may happen if you have a dominating function of high power and bad starting parameters. Choose better ones.
- You may run into a local minimum which also depends on the noise in your data. Try different start parameter or a global optimization method.
- Play with the starting parameters and get an idea how parameters influence your function. This helps to get an idea what goes wrong.

And finally :

- You have chosen the wrong model ( not correlated to your measurement), units are wrong by orders of magnitude, missing contributions, ….. So read the docs of the models and maybe choose a better one.

## 1.9. Plot experimental data and fit result

    # plot data
    p=js.grace()
    p.plot(data,legend='measured data')
    p.xaxis(min=0.07,max=4,scale='l',label='Q / nm\S-1')
    p.yaxis(scale='l',label='I(Q) / a.u.')
    # plot the result of the fit
    p.plot(data.lastfit,symbol=0,line=[1,1,4],legend='fit Rg=$radiusOfGyration I0=$I0')
    p.legend()

    p1=js.grace()
    # Tempmean because of previous mergeAttribut; otherwise data.Temp
    p1.plot(data.Tempmean,data.lastfit.Rg,data.lastfit.Rg_err)
    p1.xaxis(label='Temperature / C')
    p1.yaxis(label='Rg / nm')

## 1.10. Save data and fit results

Jscatter saves files in a ASCII format including attributes that can be reread including the attributes (See first example above and dataArray help). In this way no information is lost.

    data.save('filename.dat')
    # later read them again
    data=js.dA('filename.dat')  # retrieves all attributes

If needed, the raw numpy array can be saved (see numpy.savetxt). All attribute information is lost.

    np.savetxt('test.dat',data.array.T)

Save fit results by saving the .lastfit attribute (it is in general NOT automatically saved with the above)

    data.lastfit.save('fitresult.filename.dat')

## 1.11. Additional stuff

Creating grids as list of points in 3D [Nx3] e.g. for 2D scattering images or for integration

    # inspect the later grids by
    js.mpl.scatter3d(qxyz)

    N=10  # number of points
    d=0.1 # distance between grid points

    # traditional numpy way
    # create 2D lattice and append 3rd dimension (2N+1 points in each direction)
    qxy=np.mgrid[-d:d:N*1j, -d:d:N*1j].reshape(2,-1).T
    qxyz=np.c_[qxy,np.zeros(qxy.shape[0])]

    # 2D using 3D lattice
    qxyz=js.sf.sqLattice(d,[N,N]).XYZ     # 2D lattice from square lattice
    qxyz=js.sf.scLattice(d,[N,N,0]).XYZ   # 3D lattice Z collapsed
    qxyz=js.sf.hexLattice(d,[N,N,0]).XYZ  # 3D lattice hexagonal Z collapsed

    # quasi 2D, collapsed dimension but multiple atoms in unit cell
    # cube grid with second point in unit cell (2.5D if second point has Z dimension)
    qxyz=js.sf.rhombicLattice([[1,0,0],[0,1,0],[0,0,1]],[3,3,0],[[-0.1,-0.1,0],[0.1,0.1,0]],[1,2]).XYZ

    # 3D grid
    qxyz=js.sf.scLattice(d,[N,N,N]).XYZ

    # grid of pseudorandom points
    qxyz=js.formel.randomPointsInCube(100, dim=3)  # 3D
    qxyz=np.c_[js.formel.randomPointsInCube(N**2,dim=2),np.zeros(N**2)]  # 2D plane in 3D
    points=js.parallel.randomPointsOnSphere(1500)
    qxyz=js.formel.rphitheta2xyz(points[points[:,1]>0])  # select half sphere

    # rotate if needed
    rotaxis=[0,0,1]       # rotation around y axis
    R=js.formel.rotationMatrix(rotaxis,np.deg2rad(-120))    # -120° rotation matrix
    Rqxyz=np.einsum('ij,kj->ki',R,qxyz)