Central concepts

Note that emgfit is tailored to the needs of analyzing high-precision time-of-flight mass spectra. However, both the available selection of hyper-exponentially-modified Gaussian (Hyper-EMG) line shapes as well as the implemented statistical techniques could be used as powerful tools for analyzing spectroscopic data sets from a variety of fields.

The peak and spectrum classes

The data analysis approach of emgfit is highly object oriented. The central objects that users interact with are instances of the spectrum and peak classes. A spectrum object is instantiated by importing a data set. All relevant information about this data set is stored in attributes of the spectrum object. Initially, the user adds a number of peak objects to the spectrum which are then stored as a list in the spectrum’s peaks attribute. Each peak object holds specific information about a given peak (e.g. peak position, peak area, …). A table with all relevant information about the peaks in the spectrum can be viewed with the show_peak_properties() spectrum method.

The different peaks are fitted and more and more information is obtained by progressively calling different methods on the spectrum object. An outline of a typical analysis with emgfit is given in the tutorial. Once a peak has been fitted the obtained ModelResult (or “fit result”) is stored at the corresponding position in the spectrum’s fit_results list. Comprehensive lists of the available methods and attributes of the spectrum and peak classes are compiled along with detailed usage information in the spectrum and peak sections of the API docs. The spectrum and peak classes are tailored to the analysis needs of multi-reflection time-of-flight mass spectrometry. However, the statistical techniques incorporated in the spectrum class could be applied to analyses of spectroscopic data from various other fields. If specific needs emerge other specialized classes could be derived from the above.

Adding peaks to a spectrum

The easiest way to add peaks to a spectrum is to use emgfit’s automatic peak detection method detect_peaks(). This method applies some smoothing to the spectrum and then detects peaks via minima in the second derivative of the smoothed data. This approach yields a high sensitivity in the identification of overlapping or low-intensity peaks. Increased sensitivity can be achieved by adapting the method’s tuning parameters such as the minimal threshold for peak detection to the specifics of the given data set - see the method docs for the available options.

Alternatively or additionally, peaks can be added manually with the add_peak() method. By default, markers of the associated peaks are added to plots of spectrum data. Peaks can be removed from a spectrum object using the remove_peaks() method. To avoid ambiguities peaks should only be added or removed in the initial analysis stage, i.e. before the shape calibration or any other fits have been performed.

Assigning species to peaks and fetching literature mass values

The following attributes can be used to select a peak:

  • Peak index (i.e. index in the peaks list)

  • Peak marker position x_pos

  • Ionic species label (if assigned)

The peak index and x_pos are always defined as soon as a peak is added to a spectrum. The optional species attribute can either be set in add_peak() or with assign_species(). The species labels must follow the :-notation of chemical substances. As soon as a species is assigned to a peak the corresponding literature mass and its uncertainty are automatically fetched from the atomic mass evaluation (AME) mass database. When an AME mass value is not purely based on experimental data the peak’s extrapolated attribute is set to True. By default, literature values are grabbed from AME2020. Optionally, you can switch to values from AME2016. To revert to AME2016 for all peaks, set default_lit_src=’AME2016’ when instantiating the spectrum object. To only use AME2016 values for certain peaks, use the lit_src option when defining the peak species with the add_peak() or assign_species(). Whenever literature values are fetched from AME2016 this is automatically indicated by adding ‘lit_src: AME2016’ to the respective peak comment. User-defined literature values can be specified with the add_peak() and set_lit_values() methods.

Hyper-EMG distributions

A core feature of emgfit is its numerically robust implementation of hyper-exponentially-modified Gaussian (hyper-EMG) probability density functions. Exponentially-modified Gaussian distributions have been demonstrated to be a powerful tool for fitting spectroscopic data from various fields including mass spectrometry 1, alpha-particle spectrometry 2 and chromatography 3. Hyper-EMG distributions \(h_\mathrm{emg}(x)\) as introduced in 1 are mixture models that allow the convolution of a Gaussian with an arbitrary number of left-hand and right-hand exponential tails, respectively:

\[h_\mathrm{emg}(x; \mu, \sigma, \Theta, \eta_-, \tau_-, \eta_+, \tau_+) = \Theta h_\mathrm{-emg}(x; \mu, \sigma, \eta_-, \tau_-) + (1-\Theta) h_\mathrm{+emg}(x; \mu, \sigma, \eta_+, \tau_+).\]

where \(0 \leq \Theta \leq 1\) is the mixing weight that determines the relative contribution of the negative and positive skewed EMG distributions, \(h_\mathrm{-emg}\) and \(h_\mathrm{+emg}\), respectively. The latter are defined as:

\[ \begin{align}\begin{aligned}h_\mathrm{-emg}(x; \mu, \sigma, \eta_-, \tau_-) = \sum_{i=1}^{N_-}{\frac{\eta_{-i}}{2\tau_{-i}} \exp{\left( \left(\frac{\sigma}{\sqrt{2}\tau_{-i}}\right)^2 + \frac{x-\mu}{\sqrt{2}\tau_{-i}}\right)} \mathrm{erfc}\left(\frac{\sigma}{\sqrt{2}\tau_{-i}} + \frac{x-\mu}{\sqrt{2}\sigma}\right)},\\h_\mathrm{+emg}(x; \mu, \sigma, \eta_+, \tau_+) = \sum_{i=1}^{N_+}{\frac{\eta_{+i}}{2\tau_{+i}} \exp{\left(\left(\frac{\sigma}{\sqrt{2}\tau_{+i}}\right)^2 - \frac{x-\mu}{\sqrt{2}\tau_{+i}}\right)} \mathrm{erfc}\left(\frac{\sigma}{\sqrt{2}\tau_{+i}} - \frac{x-\mu}{\sqrt{2}\sigma}\right)}.\end{aligned}\end{align} \]

\(N_{-}\) and \(N_{+}\) are referred to as the negative and positive tail order. \(\mu=\mu_G\) denotes the mean and \(\sigma=\sigma_G\) the standard deviation of the underlying Gaussian distribution. The decay constants of the left- and right-handed exponential tails are given by \(\tau_-=(\tau_{-1},\tau_{-2},...,\tau_{-N_-})\) & \(\tau_+=(\tau_{+1},\tau_{+2},...,\tau_{+N_+})\), respectively. The negative and positive tail weights are denoted by \(\eta_-=(\eta_{-1},\eta_{-2},...,\eta_{-N_-})\) & \(\eta_+=(\eta_{+1},\eta_{+2},...,\eta_{+N_+})\), respectively, and obey the following normalizations:

\[ \begin{align}\begin{aligned}\sum_{i=1}^{N_-}{\eta_\mathrm{-i}} = 1,\\\sum_{i=1}^{N_+}{\eta_\mathrm{+i}} = 1.\end{aligned}\end{align} \]

For information on the numerical implementation of hyper-EMG distributions see emgfit.emg_funcs.

For fits of actual count data, the normalized \(h_\mathrm{emg}(x)\) distribution is multiplied by an amplitude parameter (\(A\) or amp) and optionally a uniform baseline parameter (\(c_\mathrm{bkg} or \)) is added:

\[H_\mathrm{emg} = A \cdot h_\mathrm{emg}(x) + c_\mathrm{bkg}\]

Available fit models

All supported (single peak) fit models or peak shapes are defined in the emgfit.fit_models module. Currently, the following models are available:

where the numbers in brackets indicate the negative and positive tail orders, i.e. the number of exponential tails added to the left and right side of the peak, respectively. All fit models in emgfit are expressed using lmfit’s Model class. This interface is used to define appropriate parameter bounds and ensure the normalization of the negative and positive tail weights (eta_p and eta_m parameters) of Hyper-EMG models. For more details on the above fit models see the API docs of the emgfit.fit_models module.

Multi-peak fits

If multiple peaks are to be fitted at once a suitable multi-peak model is automatically created within the spectrum class by adding a suitable number of single-peak models. In multi-peak fits, the values of the shape (or scale) parameters of all peaks are enforced to be identical, only the amplitude and position parameters are allowed to differ. In multi-reflection time-of-flight mass spectrometry the width of peaks acquired with a given instrumental resolution scales linearly with the peak’s centroid mass. Simultaneously fitting peaks with significantly different mass centroids therefore requires a mass-dependent rescaling of the shape parameters to the respective peak’s mass. So far analysis practice has shown that the required scaling corrections for isobaric peaks are significantly smaller than the typical relative errors of the corresponding shape parameters. Since emgfit (currently) only supports fits of isobaric species such a mass-rescaling has not been implemented in the package. Support for fitting non-isobaric mass peaks in the same spectrum might be added in the future.

Peak fitting approach

Peak fits with emgfit are executed by the internal peakfit() method which builds on lmfit’s Model interface. However, usually the user only interacts with higher level methods (e.g. determine_peak_shape() or fit_peaks()) that internally call peakfit(). Initial parameter values are determined as follows:

  • The initial value of the peak amplitude (amp parameter) is estimated from the product of the number of counts in the bin at the peak’s marker position x_pos and the initial value of the standard deviation of the Gaussian peak component (sigma). This product is multiplied by an empirical proportionality factor. The factor is somewhat peak-shape dependent but has been found to work well for a variety of hyper-EMG peaks. If user intervention still becomes necessary, the par_hint_args option of the peakfit() method can be used to overwrite the initial value of the peak amplitude.

  • In the case of a Gaussian, the peak centroid is initialized at the peak marker position x_pos. For a hyper-EMG fit, the initial centroid of the underlying Gaussian (denoted mu or \(\mu\)) is calculated by rearranging the equation for the mode (i.e. the x-position with maximum probability):

    \[\begin{split}\mu = x_{m} - \theta\sum_{i=1}^{N_-}\eta_{-i}\left(\sqrt{2}\sigma \cdot\mathrm{erfcxinv}\left( \frac{\tau_{-i}}{\sigma} \sqrt{\frac{2}{\pi}}\right) - \frac{\sigma^2}{\tau_{-i}} \right) \\ + (1-\theta)\sum_{i=1}^{N_-}\eta_{+i}\left(\sqrt{2}\sigma \cdot\mathrm{erfcxinv}\left( \frac{\tau_{+i}}{\sigma} \sqrt{\frac{2}{\pi}}\right) - \frac{\sigma^2}{\tau_{-i}} \right),\end{split}\]

    where the mode \(x_{m}\) is estimated by the peak marker position x_pos and \({N_-}\) and \({N_+}\) denote the order of negative and positive exponential tails, respectively.

  • If the shape parameters have not already been determined in a preceding peak-shape calibration there is two possibilities for their initialization. By default, a set of suitable initial values is then derived by re-scaling the shape parameters for a representative peak at mass unit 100 to the mass of the given spectrum. The default parameters at mass 100 u are defined in the emgfit.fit_models.create_default_init_pars() function. Alternatively, the shape parameter values can be user-defined with the init_pars argument.

  • If fitting of a uniform background has been activated with the vary_baseline argument, the baseline amplitude parameter bkg_c is initialized at the minimal number of counts found in any bin + 0.1 counts. The slight offset of 0.1 counts circumvents possible convergence problems due to the parameter starting right on its lower bound of zero counts.

Fits are performed by minimizing either of the following cost functions:

  • chi-square: This variance weighted cost function is commonly known as Pearson’s chi squared statistic and defined as:

    \[\chi^2_P = \sum_i \frac{(f(x_i) - y_i)^2}{f(x_i)+\epsilon},\]

    where \(x_i\) and \(y_i\) denote the center and contained counts of the i-th bin, respectively. On each iteration the variances of the residuals are estimated using the latest model predictions: \(\sigma_i^2 \approx f(x_i)\). The inclusion of the small constant \(\epsilon = 1e-10\) ensures numerical robustness as \(f(x_i)\) approaches zero and only causes a negligibly small bias in the parameter estimates. The iteratively re-calculated weights result in improved behavior in low-count situations.

  • MLE: With this cost function a binned maximum likelihood estimation is performed by minimizing the (doubled) negative log-likelihood ratio, also known as Cash-statistic 4:

    \[L = 2\sum_i \left[ f(x_i) - y_i + y_i \ln{\left(\frac{y_i}{f(x_i)}\right)} \right].\]

    The assumption that the counts in each bin follow a Poisson (instead of a normal) distribution makes this method applicable to count data with very low statistics. When a non-scalar minimization algorithm is used (e.g. least_squares) the above optimization problem is rephrased into a least-squares problem by minimizing the square roots of the (positive semidefinite) summands in the above equation. See the notes section of the docs of peakfit() for more details.

A number of different optimization algorithms are available to perform the minimization. In principle, any of the algorithms listed under lmfit’s fitting methods can be used by passing the respective method name to the method option if emgfit’s fitting routines. By default, the least_squares minimizer is used.

Peak-shape calibration

The peak-shape calibration is performed with the determine_peak_shape() method and offers a way to reduce the number of parameters varied in the peak-of-interest fit(s). This not only increases the robustness and computational speed of multi-peak fits but can also enhance the sensitivity for detecting unidentified overlapping peaks.

In the peak-shape calibration an ideally well separated, high-statistics peak is fitted to obtain a suitable peak shape to describe the data. We refer to all parameters that determine the shape of a single peak in the absence of background as shape parameters. In the case of a Gaussian peak model the only shape parameter is given by the standard deviation \(\sigma\). The shape parameters of a hyper-EMG model function are given by:

  • the standard deviation \(\sigma\) of the underlying Gaussian,

  • the left-right mixture weight \(\Theta\),

  • the weights for the positive and negative exponential tails, \(\eta_{-i}\) & \(\eta_{+i}\), respectively,

  • and their corresponding decay constants \(\tau_{-i}\) & \(\tau_{+i}\), respectively,

where i = 1, 2, 3, … indicates the tail order. emgfit assumes that all peaks in a spectrum have been acquired with a fixed instrumental resolution and exhibit the same theoretical peak shape. In multi-reflection time-of-flight mass spectrometry this assumption is not strictly satisfied since at a given resolving power the peak widths exhibit a linear scaling with mass. However, since emgfit is currently only intended for isobaric peaks the required scale corrections of shape parameters are usually only on the sub-percent level and hence negligible compared to the typical uncertainties in determining these parameters in the shape calibration fit. Therefore, an identical peak shape is enforced for all simultaneously fitted peaks. A mass-dependent re-scaling of the scale parameters might be added in the future.

Before the peak-shape calibration the user must decide which of the Available fit models best describes the data. To aid in this process the determine_peak_shape() method comes with an automatic model selection feature. Therein, chi-square fits with increasingly complicated model functions are performed on the shape calibration peak, starting from a regular Gaussian up to Hyper-EMG functions of successively increasing tail order. To avoid overfitting, models with any best-fit shape parameters agreeing with zero within 1:math:sigma confidence are excluded from selection. Amongst the remaining models, the one yielding the lowest chi-square per degree of freedom is selected. Alternatively, this feature can be skipped by setting the vary_tail_order option to False and a peak shape can be defined manually with the fit_model option of determine_peak_shape().

Once a peak-shape calibration has been established, all subsequent fits will, by default, be performed with this fixed peak-shape, only varying the peak amplitudes, peak positions and (if applicable) the amplitude of the uniform background. If fits with a varying peak shape are desired the vary_shape option of the peakfit() method must be set to True. The imperfect knowledge of the exact peak shape can be associated with an additional uncertainty in the determination of the peak’s mass centroid and peak area. To include these contributions in the uncertainty budget, emgfit provides specialized methods to quantify the Peak-shape uncertainties.

Mass recalibration and calculation of final mass values

Before being imported into emgfit mass spectra must have undergone a preliminary mass calibration. This initial mass scale will persist throughout the entire analysis process and will be used as the x-axis for all plots of spectrum data. In multi-reflection time-of-flight mass spectrometry the initial mass scale is usually established using the following calibration equation 5:

\[\frac{m}{z} = c \frac{(t-t_0)^2}{(1+Nb)^2},\]

where \(\frac{m}{z}\) denotes the mass-to-charge ratio of the ion, t is the measured time of flight of the ion \(t_0\) marks a small time offset due to electronic delays and N is the number of revolutions the ion has undergone. Since N is easy to infer, the factors c and b and the time offset \(t_0\) remain as the calibration constants to be determined.

There is a number of ways to determine the above calibration constants. To ensure high precision in the final mass values a second mass calibration - the so-called mass re-calibration - must be performed in emgfit. This removes any systematics that could arise when different procedures are used to determine the calibrant peak position in the initial calibration and the positions of peaks of interest in the final fitting 5. Further, it renders the specific choice of the peak position parameter irrelevant as long as the same convention is followed for all peaks. In fact, instead of using the mean or mode of the full hyper-EMG distribution (\(\mu_\mathrm{emg}\)) emgfit uses the mean of the underlying Gaussian (\(\mu\)) to establish peak positions.

In the mass recalibration a calibrant peak with a well-known (ionic) literature mass \(m_{cal, lit}\) is fitted and the obtained peak position \((m/z)_{cal, fit}\) is used to calculate the spectrum’s mass recalibration factor defined as:

\[\gamma_\mathrm{recal} = \frac{(m/z)_\mathrm{cal,lit}}{(m/z)_\mathrm{cal,fit}} = \frac{m_\mathrm{cal,lit}}{m_\mathrm{cal,fit}},\]

The calibrant peak can either be fitted individually upfront via the fit_calibrant() method or the calibrant fit can be performed simultaneous with the ion-of-interest fits using the index_mass_calib or species_mass_calib options of the fit_peaks() method. The relative uncertainty of the recalibration factor (“recalibration uncertainty”) is given by the literature mass uncertainty \(\Delta m_\mathrm{cal, lit}\) and the statistical uncertainty of the calibrant fit result \(\Delta m_\mathrm{cal, fit}\):

\[\frac{\Delta \gamma_\mathrm{recal}}{\gamma_\mathrm{recal}} = \sqrt{ \left(\frac{\Delta m_\mathrm{cal, lit}}{m_\mathrm{cal, lit}} \right)^2 + \left(\frac{\Delta m_\mathrm{cal, fit}}{m_\mathrm{cal, fit}} \right)^2}.\]

The final ionic masses m_ion are calculated as:

\[m_\mathrm{ion} = \frac{(m/z)_\mathrm{cal, lit}}{(m/z)_\mathrm{cal, fit}} \cdot (m/z)_\mathrm{fit} \cdot z = \gamma_\mathrm{recal} \cdot (m/z)_\mathrm{fit} \cdot z.\]

The relative uncertainty of the final mass values is given by adding the statistical mass uncertainty, the recalibration uncertainty and the peak-shape mass uncertainty in quadrature:

\[\frac{\Delta m_\mathrm{ion}}{m_\mathrm{ion}} = \sqrt{ \left(\left(\frac{\Delta m}{m}\right)_\mathrm{stat} \right)^2 + \left(\frac{\Delta \gamma_\mathrm{recal}}{\gamma_\mathrm{recal}} \right)^2 + \left( \left(\frac{\Delta m}{m}\right)_\mathrm{PS} \right)^2 }.\]

Note that in the above, \(m\) refers to ionic rather than atomic masses. The atomic mass excess (atomic_ME_keV peak attribute) and its uncertainty are calculated from \(m_\mathrm{ion}\) from the following relations:

\[ \begin{align}\begin{aligned}\mathrm{ME}= m_\mathrm{ion} + z\cdot m_e - A \cdot u\\\Delta\mathrm{ME} = \mathrm{ME} \cdot \frac{\Delta m_\mathrm{ion}}{m_\mathrm{ion}},\end{aligned}\end{align} \]

where A denotes the atomic mass number. Note that the above neglects the atomic binding energy of the stripped electrons, as well as the uncertainties of the electron mass and the atomic mass unit \(u\). Assuming singly or doubly charged ions, these contributions lie well below 1 keV.

Fitting peaks of interest

Peaks of interest are fitted with the fit_peaks() method of the spectrum class. By default, this method fits all defined peaks in the spectrum. Alternatively, a specific mass range or specific neighboring peaks to fit can be selected. It is the user’s choice whether all peaks are treated at once or whether fit_peaks() is run multiple times on single peaks or subgroups of peaks.

Estimation of statistical uncertainties

With emgfit the statistical uncertainties of peak centroids can be estimated in two different ways:

  1. The default approach exploits the scaling of the statistical uncertainty of the mean of a Gaussian or hyper-EMG distribution with the number of counts in the peak \(N_\mathrm{counts}\):

    \[\Delta \left(m/z\right)_\mathrm{stat} = A_\mathrm{stat} \frac{\mathrm{FWHM}}{\sqrt{N_\mathrm{counts}}}.\]

    In the case of a Gaussian \(A_\mathrm{stat}\) is simply given by \(A_\mathrm{stat,G} = 1/(2\sqrt{2\ln{2}}) = 0.425\). For hyper-EMG distributions the respective constant of proportionality \(A_\mathrm{stat,emg}\) is typically larger and depends on the specific peak shape 5. emgfit’s determine_A_stat_emg() method can be used to estimate \(A_\mathrm{stat,emg}\) for a specific peak shape via non-parametric bootstrapping of a reference peak with decent statistics (see method docs for details). The updated \(A_\mathrm{stat,emg}\) factor will be used in subsequent fits to calculate the stat. mass errors with the above equation. If determine_A_stat_emg() is not run a default value of \(A_\mathrm{stat,emg} = 0.52\) 5 is used.

  2. Alternatively, the statistical uncertainty can be estimated after the peak fitting with the get_errors_from_resampling() method. This routine follows the approach outlined in 5 and does not use a reference peak but determines the statistical mass uncertainty for each peak of interest individually. This is done by re-performing the fit on many synthetic spectra obtained by resampling from the best-fit model curve (parametric bootstrap). Assuming that the data is well-described by the chosen fit model this method yields refined estimates of the statistical uncertainties that account for departures from the simple scaling law above (as possible e.g. for strongly overlapping peaks).

Since the computational overhead of the second approach is usually rather small this method is oftentimes preferable. Note that the second method also yields estimates of the statistical uncertainty of the peak areas, whereas the first approach only yields stat. mass errors and requires the area errors to be independently estimated from the covariance matrix provided by lmfit (which can be problematic for MLE fits).

Peak-shape uncertainties

Peak-shape uncertainties quantify the effect of shape parameter uncertainties obtained in a preceding peak-shape calibration on the final mass values and peak areas obtained in ion-of-interest fits.

When ion-of-interest fits are performed with a fixed peak-shape, the uncertainties of shape parameters obtained in the peak-shape calibration can cause additional uncertainties in the final mass and peak area values. Consequently, these so-called peak-shape uncertainties must be carefully estimated and propagated into the final mass and area uncertainties. emgfit provides two ways to estimate the peak-shape uncertainties of the peak areas and the mass values m_ion:

1. A quick peak-shape (PS) estimation is automatically performed in fits with fit_peaks() and fit_calibrant() by calling the internal _eval_peakshape_errors() method. This routine adapts the approach of 5 and re-performs a given fit a number of times, each time changing a different shape parameter by plus and minus its 1:math:sigma confidence interval, respectively, while keeping all other shape parameters fixed. For each shape parameter, the larger of the two shifts in a peak’s mass and area is recorded and the peak-shape uncertainty is estimated for each peak by summing those values in quadrature. Mind that the considered mass shifts are corrected for the respective shifts of the calibrant peak position.

2. The above approach implicitly assumes that the shape parameters follow normal posterior distributions and neglects any correlations between shape parameters. Since these assumptions are oftentimes violated, refined PS error estimates can be obtained with emgfit’s get_MC_peakshape_errors() method. This re-performs a given fit many times with a variety of different peak-shapes. For the used peak shapes to be representative of all line shapes supported by the data the full shape parameter posterior distributions are upfront estimated by Markov-Chain Monte Carlo (MCMC) sampling. Assuming a sufficiently large subset of these MCMC parameter sets is used to refit the data, the resulting PS errors account for non-normal parameter distributions (typically found when a parameter is near its bounds) and parameter correlations. Since this approach is computationally expensive it makes heavy use of parallel processing. If appropriate MCMC sampling has already been performed in the peak-shape calibration (with the map_par_covar option) those samples will be re-used in the Monte Carlo PS uncertainty estimation. If get_MC_peakshape_errors() is run the peak properties table is updated with the refined uncertainties and the new values are marked in color to clearly indicate the way they were estimated.

Saving fit traces

Obtained fit curves and the underlying count data can be exported to a TXT file using the save_fit_trace() method of the spectrum class.

Saving fit results

All critical results obtained in the analysis of a spectrum can be saved with the save_results() spectrum method. This routine saves the analysis results to an XLSX-file with three worksheets containing:

  1. General properties of the spectrum, such as the input filename, the fit model used in the peak-shape calibration and the obtained mass recalibration factor. For details on what the respective parameters refer to see the attribute list of the spectrum class.

  2. The peak properties table with the attributes of all peaks as well as images of all best-fit curves. Check the attribute list of the peak class for short descriptions of what the different columns contain.

  3. The eff_mass_shifts dictionary holding for each peak the larger of the two effective mass shifts obtained when varying each shape parameter by +-1:math:sigma in the default peak-shape error estimation. These shifts are irrelevant for peaks whose peak-shape uncertainties have been estimated with the get_MC_peakshape_errors() routine.

Additionally, the spectrum’s peak-shape calibration parameters and their uncertainties are saved to a separate TXT-file and plots with the obtained fit curves are saved to PNG-files (optional).

:-notation of chemical substances

emgfit follows the :-notation of chemical compounds. The chemical composition of an ion is denoted as a single string in which the constituting isotopes are separated by a colon (:). Each isotope is denoted as 'n(El)A' where El is the corresponding element symbol, n denotes the number of atoms of the given isotope and A is the respective atomic mass number. In the case n = 1, the number indication n can be omitted. The charge state of the ion is indicated by subtracting the desired number of electrons from the atomic species (i.e. ':-1e' for singly charged cations, ':-2e' for doubly charged cations etc.). Once the ionic species of a peak is assigned emgfit automatically fetches the respective literature value from the AME2020 6 (or the AME2016 7) mass database. The subtraction of the electron is important since otherwise the atomic instead of the ionic mass is used for subsequent calculations. The calculated literature mass values do not account for electron binding energies which can in most applications safely be neglected for singly and doubly charged ions. emgfit does currently not interface with an isomer database. However, isomers can be marked by appending an 'm' or 'm0' up to 'm9' to the end of an isotope substring (see last example below). The literature mass (and mass error) of an isomer are automatically calculated from the respective ground-state AME mass when the excitation energy is passed to the Ex (and Ex_error) option of the relevant spectrum methods (e.g. add_peak() & assign_species()). Further, literature values can be fully user-defined with the set_lit_values() & method.

Examples:

  • The most abundant isotope of the hydronium cation \(H_{3}O^{+}\) can be denoted as '3H1:1O16:-1e' or '3H1:O16:-e' or '1O16:3H1:-1e' or …

  • The most abundant isotope of the ammonium cation \(N H_{4}^{+}\) can be denoted as '4H1:1N14:-1e' or '4H1:N14:-e' or 'N14:4H1:-1e' or …

  • The proton is denoted as '1H1:-1e' or 'H1:-1e' or 'H1:-e'.

  • A Indium-127 ion in the second isomeric state is denoted as '1In127m1:-1e'

Creating simulated spectra

The functions in the emgfit.sample module allow the fast creation of synthetic spectrum data by extending random variate sampling with Scipy’s exponnorm class to hyper-EMG distributions. The functions in this module can serve as a valuable tool for Monte Carlo studies with count data.

Blind analysis

Premature comparison of fit results to literature values can lead to biased results. To avoid that user bias (consciously or unconsciously) enters the final mass values emgfit incorporates the option to blind the obtained mass values and peak positions during the analysis process. Blindfolding is activated with the set_blinded_peaks() method. The option to only blind specific peaks of interest leaves the possibility to use less interesting peaks with well-known literature masses as accuracy checks. The blinding is automatically lifted once the processing of the spectrum is finalized and the results are exported.

References

1(1,2)

Purushothaman, S., et al. “Hyper-EMG: A new probability distribution function composed of Exponentially Modified Gaussian distributions to analyze asymmetric peak shapes in high-resolution time-of-flight mass spectrometry.” International Journal of Mass Spectrometry 421 (2017): 245-254.

2

Pommé, S., and B. Caro Marroyo. “Improved peak shape fitting in alpha spectra.” Applied Radiation and Isotopes 96 (2015): 148-153.

3

Naish, Pamela J., and S. Hartwell. “Exponentially modified Gaussian functions — a good model for chromatographic peaks in isocratic HPLC?.” Chromatographia 26.1 (1988): 285-296.

4

Cash, Webster. “Parameter estimation in astronomy through application of the likelihood ratio.” The Astrophysical Journal 228 (1979): 939-947.

5(1,2,3,4,5,6)

San Andrés, Samuel Ayet, et al. “High-resolution, accurate multiple-reflection time-of-flight mass spectrometry for short-lived, exotic nuclei of a few events in their ground and low-lying isomeric states.” Physical Review C 99.6 (2019): 064313.

6

Wang, M., et al. “The AME2020 atomic mass evaluation (II). Tables, graphs and references.” Chinese Physics C 45 (2021): 030003.

7

Wang, M., et al. “The AME2016 atomic mass evaluation (II). Tables, graphs and references.” Chinese Physics C 41.3 (2017): 030003.