# Fitting Absorption Spectra¶

This tool can be used to fit absorption spectra, particularly those
generated using the
`AbsorptionSpectrum`

.
For more details on its uses and implementation please see (Egan et al. (2013)). If you find this tool useful we
encourage you to cite accordingly.

## Loading an Absorption Spectrum¶

To load an absorption spectrum created by
`AbsorptionSpectrum`

,
specify the output file name. It is advisable to use either an .h5
or .fits file, rather than an ascii file to save the spectrum as rounding
errors produced in saving to a ascii file will negatively impact fit quality.

```
f = h5py.File('spectrum.h5')
wavelength = f["wavelength"][:]
flux = f['flux'][:]
f.close()
```

## Specifying Species Properties¶

Before fitting a spectrum, you must specify the properties of all the species included when generating the spectrum.

The physical properties needed for each species are the rest wavelength, f-value, gamma value, and atomic mass. These will be the same values as used to generate the initial absorption spectrum. These values are given in list form as some species generate multiple lines (as in the OVI doublet). The number of lines is also specified on its own.

To fine tune the fitting procedure and give results in a minimal number of optimizing steps, we specify expected maximum and minimum values for the column density, doppler parameter, and redshift. These values can be well outside the range of expected values for a typical line and are mostly to prevent the algorithm from fitting to negative values or becoming numerically unstable.

Common initial guesses for doppler parameter and column density should also be given. These values will not affect the specific values generated by the fitting algorithm, provided they are in a reasonably appropriate range (ie: within the range given by the max and min values for the parameter).

For a spectrum containing both the H Lyman-alpha line and the OVI doublet, we set up a fit as shown below.

```
HI_parameters = {'name':'HI',
'f': [.4164],
'Gamma':[6.265E8],
'wavelength':[1215.67],
'numLines':1,
'maxN': 1E22, 'minN':1E11,
'maxb': 300, 'minb':1,
'maxz': 6, 'minz':0,
'init_b':30,
'init_N':1E14}
OVI_parameters = {'name':'OVI',
'f':[.1325,.06580],
'Gamma':[4.148E8,4.076E8],
'wavelength':[1031.9261,1037.6167],
'numLines':2,
'maxN':1E17,'minN':1E11,
'maxb':300, 'minb':1,
'maxz':6, 'minz':0,
'init_b':20,
'init_N':1E12}
speciesDicts = {'HI':HI_parameters,'OVI':OVI_parameters}
```

## Generating Fit of Spectrum¶

After loading a spectrum and specifying the properties of the species used to generate the spectrum, an appropriate fit can be generated.

```
from trident.absorption_spectrum.absorption_spectrum_fit import generate_total_fit
orderFits = ['OVI','HI']
fitted_lines, fitted_flux = generate_total_fit(wavelength,
flux, orderFits, speciesDicts)
```

The orderFits variable is used to determine in what order the species should be fitted. This may affect the results of the resulting fit, as lines may be fit as an incorrect species. For best results, it is recommended to fit species the generate multiple lines first, as a fit will only be accepted if all of the lines are fit appropriately using a single set of parameters. At the moment no cross correlation between lines of different species is performed.

The parameters of the lines that are needed to fit the spectrum are contained
in the `fitted_lines`

variable. Each species given in `orderFits`

will
be a key in the `fitted_lines`

dictionary. The entry for each species
key will be another dictionary containing entries for ‘N’,’b’,’z’, and
‘group#’ which are the column density, doppler parameter, redshift,
and associate line complex respectively. The i ^{th} line
of a given species is then given by the parameters `N[i]`

, `b[i]`

,
and `z[i]`

and is part of the same complex (and was fitted at the same time)
as all lines with the same group number as `group#[i]`

.

The `fitted_flux`

is an ndarray of the same size as `flux`

and
`wavelength`

that contains the cumulative absorption spectrum generated
by the lines contained in `fitted_lines`

.

## Saving a Spectrum Fit¶

Saving the results of a fitted spectrum for further analysis is accomplished automatically using the h5 file format. A group is made for each species that is fit, and each species group has a group for the corresponding N, b, z, and group# values.

## Procedure for Generating Fits¶

To generate a fit for a spectrum
`generate_total_fit()`

is called.
This function controls the identification of line complexes, the fit
of a series of absorption lines for each appropriate species, checks of
those fits, and returns the results of the fits.

## Finding Line Complexes¶

Line complexes are found using the `_find_complexes`

function. The process by which line complexes are found involves walking
through the array of flux in order from minimum to maximum wavelength, and
finding series of spatially contiguous cells whose flux is less than some
limit. These regions are then checked in terms of an additional flux limit
and size. The bounds of all the passing regions are then listed and returned.
Those bounds that cover an exceptionally large region of wavelength space will
be broken up if a suitable cut point is found. This method is only appropriate
for noiseless spectra.

The optional parameter `complexLim`

(default = 0.999), controls the limit
that triggers the identification of a spatially contiguous region of flux
that could be a line complex. This number should be very close to 1 but not
exactly equal. It should also be at least an order of magnitude closer to 1
than the later discussed `fitLim`

parameter, because a line complex where
the flux of the trough is very close to the flux of the edge can be incredibly
unstable when optimizing.

The `fitLim`

parameter controls what is the maximum flux that the trough
of the region can have and still be considered a line complex. This
effectively controls the sensitivity to very low column absorbers. Default
value is `fitLim`

= 0.99. If a region is identified where the flux of the
trough is greater than this value, the region is simply ignored.

The `minLength`

parameter controls the minimum number of array elements
that an identified region must have. This value must be greater than or
equal to 3 as there are a minimum of 3 free parameters that must be fit.
Default is `minLength`

= 3.

The `maxLength`

parameter controls the maximum number of array elements
that an identified region can have before it is split into separate regions.
Default is `maxLength`

= 1000. This should be adjusted based on the
resolution of the spectrum to remain appropriate. The value correspond
to a wavelength of roughly 50 angstroms.

The `splitLim`

parameter controls how exceptionally large regions are split.
When such a region is identified by having more array elements than
`maxLength`

, the point of maximum flux (or minimum absorption) in the
middle two quartiles is identified. If that point has a flux greater than
or equal to `splitLim`

, then two separate complexes are created: one from
the lower wavelength edge to the minimum absorption point and the other from
the minimum absorption point to the higher wavelength edge. The default
value is `splitLim`

=.99, but it should not drastically affect results, so
long as the value is reasonably close to 1.

## Fitting a Line Complex¶

After a complex is identified, it is fitted by iteratively adding and optimizing a set of Voigt Profiles for a particular species until the region is considered successfully fit. The optimizing is accomplished using scipy’s least squares optimizer. This requires an initial estimate of the parameters to be fit (column density, b-value, redshift) for each line.

Each time a line is added, the guess of the parameters is based on the difference between the line complex and the fit so far. For the first line this just means the initial guess is based solely on the flux of the line complex. The column density is given by the initial column density given in the species parameters dictionary. If the line is saturated (some portion of the flux with a value less than .1) than the larger initial column density guess is chosen. If the flux is relatively high (all values >.9) than the smaller initial guess is given. These values are chosen to make optimization faster and more stable by being closer to the actual value, but the final results of fitting should not depend on them as they merely provide a starting point.

After the parameters for a line are optimized for the first time, the optimized parameters are then used for the initial guess on subsequent iterations with more lines.

The complex is considered successfully fit when the sum of the squares of
the difference between the flux generated from the fit and the desired flux
profile is less than `errBound`

. `errBound`

is related to the optional
parameter to
`generate_total_fit()`

`maxAvgError`

by the number of array elements in the region such that
`errBound`

= number of elements * `maxAvgError`

.

There are several other conditions under which the cycle of adding and optimizing lines will halt. If the error of the optimized fit from adding a line is an order of magnitude worse than the error of the fit without that line, then it is assumed that the fitting has become unstable and the latest line is removed. Lines are also prevented from being added if the total number of lines is greater than the number of elements in the flux array being fit divided by 3. This is because there must not be more free parameters in a fit than the number of points to constrain them.

## Checking Fit Results¶

After an acceptable fit for a region is determined, there are several steps the algorithm must go through to validate the fits.

First, the parameters must be in a reasonable range. This is a check to make sure that the optimization did not become unstable and generate a fit that diverges wildly outside the region where the fit was performed. This way, even if particular complex cannot be fit, the rest of the spectrum fitting still behaves as expected. The range of acceptability for each parameter is given in the species parameter dictionary. These are merely broad limits that will prevent numerical instability rather than physical limits.

In cases where a single species generates multiple lines (as in the OVI doublet), the fits are then checked for higher wavelength lines. Originally the fits are generated only considering the lowest wavelength fit to a region. This is because we perform the fitting of complexes in order from the lowest wavelength to the highest, so any contribution to a complex being fit must come from the lower wavelength as the higher wavelength contributions would already have been subtracted out after fitting the lower wavelength.

## Saturated Lyman Alpha Fitting Tools¶

In cases where a large or saturated line (there exists a point in the complex
where the flux is less than .1) fails to be fit properly at first pass, a
more robust set of fitting tools is used to try and remedy the situation.
The basic approach is to simply try a much wider range of initial parameter
guesses in order to find the true optimization minimum, rather than getting
stuck in a local minimum. A set of hard coded initial parameter guesses
for Lyman alpha lines is given by the `_get_test_lines`

function
Also included in these parameter guesses is an an initial guess of a high
column cool line overlapping a lower column warm line, indictive of a
broad Lyman alpha (BLA) absorber.