Fisher

Base Class

Definition of the FisherForecast interface and a number of general utility functions.

class fisher.fisher_forecast.FisherForecast

Class that collects and contains information for making Fisher-matrix type forecasts for observations of various types. Forms a base class for fully analytical versions that may be application specific.

size

Number of parameters expected by this model.

Type

int

stokes

If this is a Stokes I model (‘I’) or a polarized model (‘full’).

Type

str

prior_sigma_list

List of standard deviations associated with the parameter priors.

Type

list

covar

Internal space for the computation of the covariance matrix.

Type

numpy.ndarray

argument_hash

MD5 hash object indicating last state of covariance computation. Used to determine if the covariance needs to be recomputed.

Type

str

add_gaussian_prior(pindex, sigma, verbosity=0)

Add a Gaussian prior on the parameter with the specified index.

Parameters

• pindex (int) – Index of the parameter on which to place the prior.

• sigma (float) – Standard deviation of the Gaussian prior to apply.

• verbosity (int) – Verbosity level. Default: 0.

add_gaussian_prior_list(sigma_list, verbosity=0)

Add a Gaussian priors on all of the model parameters.

Parameters

• sigma_list (list) – List of standard deviations of the Gaussian priors to apply.

• verbosity (int) – Verbosity level. Default: 0.

check_gradients(obs, p, h=None, verbosity=0)

Numerically evaluates the gradients using the visibilities function and compares them to the gradients returned by the visibility_gradients function. Numerical differentiation is 2nd-order, centered finite differences. Results are output to standard out. For best results, the user should pass some option for the step size h to use for numerical differentiation.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.) at which to check gradients..

• p (list) – List of parameter values.

• h (float,list) – List of small steps which define the finite differences. If None, uses 1e-4*p. If a float, uses h*p. If a list, sets each step size separately.

• verbosity (int) – Verbosity level. Default: 0.

display_image(p, limits=None, shape=None, verbosity=0, **kwargs)

Generate and plot an image for the parent visibility model evaluated at a given set of parameter values. The user is responsible for setting the plot limits and shape intelligently. Note that this uses FFTs and assumes that the resolution and field of view are sufficient to adequately resolve model image features.

Parameters

• p (list) – List of parameter values.

• limits (float,list) – Limits on the field of view in which to construct the image in uas. Either a single float, in which case the limits are symmetric and set to [-limits,limits,-limits,limits], or a list of floats. Default: 100.

• shape (int,list) – Dimensions of the image. If an int, the dimensions are equal. If a two-element list of ints, sets the dimensions to the two values. Default: 256.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Handles to the figure, axes, and colorbar.

Return type

(matplotlib.pyplot.figure,matplotlib.pyplot.axes,matplotlib.pyplot.colorbar)

fisher_covar(obs, p, verbosity=0, **kwargs)

Returns the Fisher matrix as defined in the accompanying documentation. Intelligently avoids recomputation if the observation and parameters are unchanged.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of model parameters at which to compute the Fisher matrix.

• verbosity (int) – Verbosity level. Default: 0.

Returns

The Fisher matrix.

Return type

(numpy.ndarray)

generate_image(p, limits=None, shape=None, verbosity=0)

Generate and return an image for the parent visibility model evaluated at a given set of parameter values. The user is responsible for setting the plot limits and shape intelligently. Note that this uses FFTs and assumes that the resolution and field of view are sufficient to adequately resolve model image features.

Parameters

• p (list) – List of parameter values.

• limits (float,list) – Limits on the field of view in which to construct the image in uas. Either a single float, in which case the limits are symmetric and set to [-limits,limits,-limits,limits], or a list of floats. Default: 100.

• shape (int,list) – Dimensions of the image. If an int, the dimensions are equal. If a two-element list of ints, sets the dimensions to the two values. Default: 256.

• verbosity (int) – Verbosity level. Default: 0.

Returns

x positions of the pixel centers in uas, y positions of the pixel centers in uas, intensities at pixel centers in Jy/uas^2.

Return type

(numpy.ndarray,numpy.ndarray,numpy.ndarray)

inverse_fisher_covar(obs, p, verbosity=0, **kwargs)

Returns the inverse of the Fisher matrix as defined in the accompanying documentation. Intelligently avoids recomputation if the observation and parameters are unchanged.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of model parameters at which to compute the Fisher matrix.

• verbosity (int) – Verbosity level. Default: 0.

Returns

The inverse of the Fisher matrix.

Return type

(numpy.ndarray)

joint_biparameter_chisq(obs, p, i1, i2, kind='marginalized', verbosity=0, **kwargs)

Computes the ensemble-averaged 2nd-order contribution to the chi-square for two parameters after fixing or marginalizing over all others.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• i1 (int) – Index of first parameter.

• i2 (int) – Index of second parameter.

• kind (str) – Choice of what to do with other parameters. Choices are ‘marginalized’, ‘fixed’. Default: ‘marginalized’.

• verbosity (int) – Verbosity level. Default: 0.

Returns

\(p_{i1}\), \(p_{i2}\), and \(\chi^2\) on a grid of points that covers the inner \(4.5\Sigma\) region.

Return type

(numpy.ndarray,numpy.ndarray,numpy.ndarray)

marginalized_covariance(obs, p, ilist=None, verbosity=0, **kwargs)

Computes the covariance for a subset of model parameters, marginalized over all parameters outside of the subset.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• ilist (int,list) – Index or list of indicies for which to compute the marginalized uncertainties. If None will return a list of all marginalized uncertainties. Default: None.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Marginalized covariance for desired parameter subset.

Return type

(numpy.ndarray)

marginalized_uncertainties(obs, p, ilist=None, verbosity=0, **kwargs)

Computes the uncertainties on a subset of model parameters, marginalized over all others. Note that this is not the marginalized covariance; each uncertainty is for a single parameter, marginalizing out all others.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• ilist (int,list) – Index or list of indicies for which to compute the marginalized uncertainties. If None will return a list of all marginalized uncertainties. Default: None.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Marginalized parameter uncertainty or list of marginalized uncertainties of desired parameters.

Return type

(float/list)

parameter_labels(verbosity=0)

User-defined function in child classes that returns a list of of parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

plot_1d_forecast(obs, p, i1, kind='marginalized', labels=None, fig=None, axes=None, colors=None, alphas=0.25, verbosity=0, **kwargs)

Plot the marginalized posterior for a single parameter for a given observation.

Parameters

• obs (list,Obsdata) – An Obsdata object or list of Obsdata objects containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• i1 (int) – Index of parameter to be plotted.

• kind (str) – Choice of what to do with other parameters. Choices are ‘marginalized’, ‘fixed’. Default: ‘marginalized’.

• labels (list,str) – A list of labels for each Obsdata object. When fewer labels than observations are provided, the first set of observations are labeled. Default: None.

• fig (matplotlib.figure.Figure) – Figure on which to place plot. If None, a new figure will be created. Default: None.

• axes (matplotlib.axes.Axes) – Axes on which to place plot. If None, a new axes will be created. Default: None.

• colors (list,str) – A color or list of colors for the plots. When fewer colors than observations are provided, they will be cycled through. If None, a default list will be used. Default: None.

• alphas (list,float) – An alpha value or list of alpha values for the filled portion of the plots. Default: 0.25.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Handles to the figure and axes objects in the plot.

Return type

(matplotlib.figure.Figure, matplotlib.axes.Axes)

plot_2d_forecast(obs, p, i1, i2, kind='marginalized', labels=None, fig=None, axes=None, colors=None, cmaps=None, alphas=0.75, verbosity=0, **kwargs)

Plot the joint marginalized posterior for a pair of parameters for a given observation.

Parameters

• obs (list,Obsdata) – An Obsdata object or list of Obsdata objects containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• i1 (int) – Index of first parameter to be plotted.

• i2 (int) – Index of second parameter to be plotted.

• kind (str) – Choice of what to do with other parameters. Choices are ‘marginalized’, ‘fixed’. Default: ‘marginalized’.

• labels (list,str) – A list of labels for each Obsdata object. When fewer labels than observations are provided, the first set of observations are labeled. Default: None.

• fig (matplotlib.figure.Figure) – Figure on which to place plot. If None, a new figure will be created. Default: None.

• axes (matplotlib.axes.Axes) – Axes on which to place plot. If None, a new axes will be created. Default: None.

• colors (list,str) – A color or list of colors for the plots. When fewer colors than observations are provided, they will be cycled through. If None, a default list will be used. Default: None.

• cmaps (list,matplotlib.colors.Colormap) – A colormap or list of colormaps for the plots. When fewer colormaps than observations are provided, they will be cycled through. If None, a default list will be used. Default: None.

• alphas (list,float) – An alpha value or list of alpha values for the filled portion of the plots. Default: 0.75.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Handles to the figure and axes objects in the plot.

Return type

(matplotlib.figure.Figure, matplotlib.axes.Axes)

plot_triangle_forecast(obs, p, ilist=None, kind='marginalized', labels=None, colors=None, cmaps=None, alphas=0.75, fig=None, axes=None, figsize=None, axis_location=None, verbosity=0, **kwargs)

Generate a triangle plot of joint posteriors for a selected list of parameter indexes.

Parameters

• obs (list,Obsdata) – An Obsdata object or list of Obsdata objects containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• ilist (list) – List of indicies of parameters to include in triangle plot. If None, includes all parameters. Default: None.

• kind (str) – Choice of what to do with other parameters. Choices are ‘marginalized’, ‘fixed’. Default: ‘marginalized’.

• labels (list,str) – A list of labels for each Obsdata object. When fewer labels than observations are provided, the first set of observations are labeled. Default: None.

• colors (list,str) – A color or list of colors for the plots. When fewer colors than observations are provided, they will be cycled through. If None, a default list will be used. Default: None.

• cmaps (list,matplotlib.colors.Colormap) – A colormap or list of colormaps for the plots. When fewer colormaps than observations are provided, they will be cycled through. If None, a default list will be used. Default: None.

• alphas (list,float) – An alpha value or list of alpha values for the filled portion of the plots. Default: 0.75.

• fig (matplotlib.figure.Figure) – Figure on which to place plot. If None, a new figure will be created. Default: None.

• axes (matplotlib.axes.Axes) – Axes on which to place plot. If None, a new axes will be created. Default: None.

• figsize (list) – Figure size in inches. If None, attempts a guess. Default: None.

• axis_location (list) – Location parameters for triangle plot region. If None, attempts a guess. Default: None.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Handles to the figure and dictionary of axes objects in the plot.

Return type

(matplotlib.figure.Figure, dict)

sample_posterior(obs, p, N, ilist=None, verbosity=0, **kwargs)

Generates a chain of samples appropriate for post-processing as done in MCMC analyses. Such chains may be useful for computing the posterior on complicated non-linear quantities.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• N (int) – Number of samples to generate.

• ilist (list) – List of parameters to sample, marginalizing over all others. If None, all parameters are sampled. Default: None.

Returns

Two-dimensional array containing chain of parameter samples.

Return type

(numpy.ndarray)

uncertainties(obs, p, ilist=None, verbosity=0, **kwargs)

Computes the uncertainties on a subset of model parameters, both fixing and marginalizing over all others.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• ilist (int,list) – Index or list of indicies for which to compute the marginalized uncertainties. If None will return a list of all marginalized uncertainties. Default: None.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Parameter uncertainty or list of marginalized uncertainties of desired parameters, marginalized parameter uncertainty or list of marginalized uncertainties of desired parameters.

Return type

(float/list,float/list)

uniparameter_uncertainties(obs, p, ilist=None, verbosity=0, **kwargs)

Computes the uncertainties on a subset of model parameters, fixing all others. Note that this is not a covariance; each uncertainty is for a single parameter, fixing all others. This is probably not what is wanted for forecasting, see marginalized_uncertainties.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of parameter values at which to compute uncertainties.

• ilist (int,list) – Index or list of indicies for which to compute the uncertainties. If None will return a list of all uncertainties. Default: None.

• verbosity (int) – Verbosity level. Default: 0.

Returns

Parameter uncertainty or list of marginalized uncertainties of desired parameters.

Return type

(float/list)

visibilities(obs, p, verbosity=0)

User-defined function in child classes that generates visibilities associated with a given model image object.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.)

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0, **kwargs)

User-defined function in child classes that generates visibility gradients associated with a given model image object.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.)

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities gradients computed at observations.

Return type

(numpy.ndarray)

Station Gain Mitigation

A set of FisherForecast childe classes that facilitate including gain uncertainties.

class fisher.ff_complex_gains.FF_complex_gains(ff, marg_method='vMv')

FisherForecast with complex gain reconstruction with multiple epochs. This is usually the FisherForecast object that should be used to investigate the impact of uncertain station gains.

Parameters

• ff (FisherForecast) – A FisherForecast object to which we wish to add gains.

• marg_method (str) – Method used for intermediate marginalization. Options are ‘covar’ and ‘vMv’. Default: ‘covar’.

ff

The FisherForecast object before gain reconstruction.

Type

FisherForecast

gain_epochs

2D array containing the start and end times for each gain solution epoch.

Type

numpy.ndarray

gain_amplitude_priors

list of standard deviations of the log-normal priors on the gain amplitudes. Default: 10.

Type

list

gain_phase_priors

list of standard deviations on the normal priors on the gain phases. Default: 30.

Type

list

gain_ratio_amplitude_priors

For polarized gains, list of the log-normal priors on the gain amplitude ratios. Default: 1e-10.

Type

list

gain_ratio_phase_priors

For polarized gains, list of the normal priors on the gain phase differences. Default: 1e-10.

Type

list

marg_method

Method used for intermediate marginalization. Options are ‘covar’ and ‘vMv’. Based on preliminary tests, ‘vMv’ is more accurate for the gain marginalization, though this may not be the case for models with very many parameters.

Type

str

fisher_covar(obs, p, verbosity=0, **kwargs)

Returns the Fisher matrix as defined in the accompanying documentation, marginalized over the complex station gains. Intelligently avoids recomputation if the observation and parameters are unchanged.

Parameters

• obs (ehtim.obsdata.Obsdata) – An Obsdata object containing the observation particulars.

• p (list) – List of model parameters at which to compute the Fisher matrix.

• verbosity (int) – Verbosity level. Default: 0.

Returns

The Fisher matrix.

Return type

(numpy.ndarray)

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

set_gain_amplitude_prior(sigma, station=None, verbosity=0)

Sets the log-normal priors on the gain amplitudes, either for a specified station or for all stations. Identical priors are assummed across the entire observation.

Parameters

• sigma (float) – Standard deviation of the log-amplitude.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

• verbosity (int) – Verbosity level. Default: 0.

set_gain_epochs(scans=False, gain_epochs=None)

Sets the gain solution intervals (gain epochs) to be used. If neither scans nor gain_epochs selected, will solve for gains on each unique timestamp in the data.

Parameters

• scans (bool) – If True, solves for independent gains by scan. Overrides explicit specification of gain solution intervals. Default: False.

• gain_epochs (nd.array) – 2D array containing the start and end times for each gain solution epoch. Default: None.

set_gain_phase_prior(sigma, station=None, verbosity=0)

Sets the normal priors on the gain phases, either for a specified station or for all stations. Usually, this should be the default (uniformative). Identical priors are assummed across the entire observation.

Parameters

• sigma (float) – Standard deviation of the phase.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

set_gain_ratio_amplitude_prior(sigma, station=None, verbosity=0)

Sets the log-normal priors on the R/L gain amplitude ratios, either for a specified station or for all stations. Only relevant for polarized models. Default: 1e-10 for all stations. Identical priors are assummed across the entire observation.

Parameters

• sigma (float) – Standard deviation of the log-amplitude ratio.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

set_gain_ratio_phase_prior(sigma, station=None, verbosity=0)

Sets the normal priors on the R/L gain phase differences, either for a specified station or for all stations. Only relevant for polarized models. Default: 1e-10 for all stations. Identical priors are assummed across the entire observation.

Parameters

• sigma (float) – Standard deviation of the phase.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

visibilities(obs, p, verbosity=0, **kwargs)

Complex visibilities associated with the underlying FisherForecast object evaluated at the data points in the given Obsdata object for the model with the given parameter values.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.)

• p (numpy.ndarray) – list of parameters for the model at which visiblities are desired.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0, **kwargs)

Gradients of the complex visibilities associated with the underlying FisherForecast object with respect to the model pararmeters evaluated at the data points in the given Obsdata object for the model with the given parameter values.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.)

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibility gradients computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_complex_gains.FF_complex_gains_single_epoch(ff)

FisherForecast with complex gain reconstruction assuming a single epoch. This is a helper class, and probably not the FisherForecast object that should be used to study the impact of gains. See FF_complex_gains.

Parameters

ff (FisherForecast) – A FisherForecast object to which we wish to add gains.

ff

The FisherForecast object before gain reconstruction.

Type

FisherForecast

plbls

List of parameter labels (str) including gains parameter names.

Type

list

gain_amplitude_priors

list of standard deviations of the log-normal priors on the gain amplitudes. Default: 10.

Type

list

gain_phase_priors

list of standard deviations on the normal priors on the gain phases. Default: 30.

Type

list

gain_ratio_amplitude_priors

For polarized gains, list of the log-normal priors on the gain amplitude ratios. Default: 1e-10.

Type

list

gain_ratio_phase_priors

For polarized gains, list of the normal priors on the gain phase differences. Default: 1e-10.

Type

list

parameter_labels(verbosity=0)

Parameter labels to be used in plotting, including those associated with the gains.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

set_gain_amplitude_prior(sigma, station=None)

Sets the log-normal priors on the gain amplitudes, either for a specified station or for all stations.

Parameters

• sigma (float) – Standard deviation of the log-amplitude.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

set_gain_phase_prior(sigma, station=None)

Sets the normal priors on the gain phases, either for a specified station or for all stations. Usually, this should be the default (uniformative).

Parameters

• sigma (float) – Standard deviation of the phase.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

set_gain_ratio_amplitude_prior(sigma, station=None)

Sets the log-normal priors on the R/L gain amplitude ratios, either for a specified station or for all stations. Only relevant for polarized models. Default: 1e-10 for all stations.

Parameters

• sigma (float) – Standard deviation of the log-amplitude ratio.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

set_gain_ratio_phase_prior(sigma, station=None)

Sets the normal priors on the R/L gain phase differences, either for a specified station or for all stations. Only relevant for polarized models. Default: 1e-10 for all stations.

Parameters

• sigma (float) – Standard deviation of the phase.

• station (str,list) – Station code of the station(s) for which the prior is to be set. If None, will set the prior on all stations. Default: None.

visibilities(obs, p, verbosity=0, **kwargs)

Complex visibilities associated with the underlying FisherForecast object evaluated at the data points in the given Obsdata object for the model with the given parameter values.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.)

• p (numpy.ndarray) – list of parameters for the model at which visiblities are desired.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0, **kwargs)

Gradients of the complex visibilities associated with the underlying FisherForecast object with respect to the model pararmeters evaluated at the data points in the given Obsdata object for the model with the given parameter values, including gradients with respect to the gain amplitudes, phases, and if this is a polarized model, R/L gain amplitude ratios and phase differences.

Parameters

• obs (ehtim.obsdata.Obsdata) – An ehtim Obsdata object with a particular set of observation details (u,v positions, times, etc.)

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibility gradients computed at observations.

Return type

(numpy.ndarray)

Models

Specific models implemented in ngEHTforecast as FisherForecast child classes. Each inherits all of the functionality of FisherForecast, and may be used anywhere a FisherForecast object is required.

class fisher.ff_models.FF_asymmetric_gaussian(stokes='I')

FisherForecast object for a noncircular Gaussian. Parameter vector is:

• p[0] … Total flux in Jy.

• p[1] … Symmetrized mean of the FHWM in the major and minor axes in uas: \({\rm FWHM}^{-2} = {\rm FMHM}_{\rm min}^{-2}+{\rm FWHM}_{\rm max}^{-2}\)

• p[2] … Asymmetry parameter, \(A\), expected to be in the range [0,1).

• p[3] … Position angle in radians of the major axis E of N.

In terms of these, \({\rm FWHM}_{\rm maj}={\rm FWHM}/\sqrt{1-A}\) and \({\rm FWHM}_{\rm min}={\rm FWHM}/\sqrt{1-A}\).

Parameters

stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with a noncircular Gaussian.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with a noncircular Gaussian.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_double_smoothed_delta_ring(stokes='I')

FisherForecast object for a delta-ring convolved with a circular Gaussian. Parameter vector is:

• p[0] … Flux of n=1 ring in Jy.

• p[1] … Diameter of n=1 ring in uas.

• p[2] … Twice the standard deviation of the Gaussian smoothing kernel in uas.

• p[3] … Ratio of flux in n=2 and n=1 rings.

• p[4] … Ratio of difference in the diameters between the n=1 and n=2 rings to the diameter of the n=1 ring.

• p[5] … Ratio of the standard deviations of the n=2 and n=1 rings.

Parameters

stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with Gaussian-convolved delta-ring.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with Gaussian-convolved delta-ring.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_model_image(img, stokes='I')

FisherForecast from ThemisyPy model_image objects. Uses FFTs and centered finite-difference to compute the visibilities and gradients. May not always produce sensible behaviors. Has not been extensively tested for some time.

Parameters

• img (model_image) – A ThemisPy model_image object.

• stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

img

The ThemisPy model_image object for which to make forecasts.

Type

model_image

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, limits=None, shape=None, padding=4, verbosity=0)

Generates visibilities associated with a given model image object. Note that this uses FFTs to create the visibilities and then interpolates.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• limits (list) – May be single number, list with four elements specifying in \(\mu as\) the symmetric limits or explicit limits in [xmin,xmax,ymin,ymax] order. Default: 100.

• shape (list) – May be single number or list with two elements indicating the number of pixels in the two directions. Default: 256.

• padding (int) – Factor by which to pad image with zeros prior to FFT. Default: 4.

• verbosity (int) – Verbosity level. 0 prints nothing. 1 prints various elements of the plotting process. Passed to model_image.generate_intensity_map(). Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, h=0.01, limits=None, shape=None, padding=4, verbosity=0)

Generates visibilities associated with a given model image object. Note that this uses FFTs to create the visibilities and then interpolates.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• h (float) – fractional step for finite-difference gradient computation. Default: 0.01.

• limits (list) – May be single number, list with four elements specifying in \(\mu as\) the symmetric limits or explicit limits in [xmin,xmax,ymin,ymax] order. Default: 100.

• shape (list) – May be single number or list with two elements indicating the number of pixels in the two directions. Default: 256.

• padding (int) – Factor by which to pad image with zeros prior to FFT. Default: 4.

• verbosity (int) – Verbosity level. 0 prints nothing. 1 prints various elements of the plotting process. Passed to model_image.generate_intensity_map(). Default: 0.

Returns

List of complex visibilities gradients computed at observation.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_smoothed_delta_ring(stokes='I')

FisherForecast object for a delta-ring convolved with a circular Gaussian. Parameter vector is:

• p[0] … Total flux in Jy.

• p[1] … Diameter in uas.

• p[2] … Twice the standard deviation of the Gaussian smoothing kernel in uas.

Parameters

stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with Gaussian-convolved delta-ring.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with Gaussian-convolved delta-ring.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_smoothed_delta_ring_themage(N, fov, stokes='I')

FisherForecast object for a splined raster (i.e., themage) plus a Gaussian-convolved delta-ring.

Parameter vector is the log of the intensity at the control points:

• p[0] ……. \(\ln(I[0,0])\)

• p[1] ……. \(\ln(I[1,0])\)

• …

• p[N-1] ….. \(\ln(I[N-1,0])\)

• p[N] ……. \(\ln(I[0,1])\)

• …

• p[N*N-1] … \(\ln(I[N-1,N-1])\)

• p[N*N+0] … Total flux in Jy.

• p[N*N+1] … Diameter in uas.

• p[N*N+2] … Twice the standard deviation of the Gaussian smoothing kernel in uas.

where each \(I[i,j]\) is measured in Jy/sr.

Parameters

• N (int) – Raster dimension (only supports square raster).

• fov (float) – Raster field of view in uas (only supports fixed rasters).

• stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

xcp

x-positions of raster control points.

Type

numpy.ndarray

ycp

y-positions of raster control points.

Type

numpy.ndarray

apx

Raster pixel size.

Type

float

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with splined raster + Gaussian-convolved delta-ring.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with splined raster + Gaussian-convolved delta-ring.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_splined_raster(N, fov, stokes='I')

FisherForecast object for a splined raster (i.e., themage). Parameter vector is the log of the intensity at the control points:

• p[0] ……. \(\ln(I[0,0])\)

• p[1] ……. \(\ln(I[1,0])\)

• …

• p[N-1] ….. \(\ln(I[N-1,0])\)

• p[N] ……. \(\ln(I[0,1])\)

• …

• p[N*N-1] … \(\ln(I[N-1,N-1])\)

where each \(I[i,j]\) is measured in Jy/sr.

Parameters

• N (int) – Raster dimension (only supports square raster).

• fov (float) – Raster field of view in uas (only supports fixed rasters).

• stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

xcp

x-positions of raster control points.

Type

numpy.ndarray

ycp

y-positions of raster control points.

Type

numpy.ndarray

dxcp

pixel spacing in x-direction

Type

float

dycp

pixel spacing in y-direction

Type

float

apx

Raster pixel size.

Type

float

generate_parameter_list(glob, verbosity=0, **kwargs)

Utility function to quickly and easily generate a set of raster values associated with various potential initialization schemes. These include: an existing FisherForecast object, a FisherForecast class name, an ehtim.image.Image object, or the name of a FITS file. If initializing from a FITS file, the image will be blurred to the raster resolution prior to setting the parameter list.

Parameters

• glob (str) – Can be an existing FisherForecast child object, the name of a FisherForecast child class, ehtim.image.Image object, or a string with the name of a FITS file.

• verbosity (int) – Verbosity level. Default: 0.

• **kwargs (dict) – Additional key-word arguments necesary to define the object. May include parameter values for the relevant FisherForecast object or arguments to the fisher.fisher_forecast.FisherForecast.generate_image() function.

Returns

Approximate parameter list for associated splined raster object.

Return type

p (list)

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with a splined raster model.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with a splined raster model.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_symmetric_gaussian(stokes='I')

FisherForecast object for a circular Gaussian. Parameter vector is:

• p[0] … Total flux in Jy.

• p[1] … FWHM in uas.

Parameters

stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with a circular Gaussian.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with a circular Gaussian.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

class fisher.ff_models.FF_thick_mring(m, mp=None, mc=None, stokes='I')

FisherForecast object for an m-ring model (based on ehtim). Parameter vector is:

• p[0] … DOM, PLEASE FILL THIS IN.

Parameters

• m (int) – Stokes I azimuthal Fourier series order.

• mp (int) – Linear polarization azimuthal Fourier series order. Only used if stokes==’full’. Default: None.

• mc (int) – Circular polarization azimuthal Fourier series order. Only used if stokes==’full’. Default: None.

• stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

m

Stokes I azimuthal Fourier series order.

Type

int

mp

Linear polarization azimuthal Fourier series order. Only used if stokes==’full’.

Type

int

mc

Circular polarization azimuthal Fourier series order. Only used if stokes==’full’.

Type

int

param_wrapper(p)

Converts parameters from a flattened list to an ehtim-style dictionary.

Parameters

p (float) – Parameter list as used by FisherForecast.

Returns

Dictionary containing parameter values as used by ehtim.

Return type

(dict)

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with m-ring model.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with m-ring model.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

fisher.ff_models.W_cubic_spline_1d(k, a=- 0.5)

Fourier domain convolution function for the approximate 1D cubic spline. Useful for the splined raster image classes.

Parameters

• k (numpy.ndarray) – Wavenumber.

• a (float) – Cubic spline control parameter. Default: -0.5.

Returns

1D array of values of the cubic spline weight function evaluated at each value of k.

Return type

(numpy.ndarray)

Model Combinations

Classes that can be used to construct new FisherForecast objects from others. The prototypical example is fisher.ff_metamodels.FF_Sum, which constructs a new model by directly summing the intensity maps of many previous models.

class fisher.ff_metamodels.FF_sum(ff_list=None, stokes='I')

FisherForecast object constructed from the sum of multiple FisherForecast objects. For example, a binary might be generated from the sum of two Gaussians.

The parameter vector is constructed from the concatenation of the parameter vectors from each individual object, with all objects after the first gaining a pair of offset parameters. That is:

• p[0] ………… Obj1 p[0]

• p[1] ………… Obj1 p[1]

• …

• p[n1] ……….. Obj2 p[0]

• p[n1+1] ……… Obj2 p[0]

• …

• p[n1+n2] …….. Obj2 dx

• p[n1+n2+1] …… Obj2 dy

• p[n1+n2+2] …… Obj3 p[0]

• …

• p[n1+n2+n3+2] … Obj3 dx

• p[n1+n2+n3+3] … Obj3 dy

• …

Prior lists are constructed similarly.

Parameters

• ff_list (list) – List of FisherForecast objects to be summed. Additional objects can be added later. Default: None.

• stokes (str) – Indicates if this is limited to Stokes I (‘I’) or include full polarization (‘full’). Default: ‘I’.

add(ff)

Adds a FisherForecast object to the sum.

Parameters

ff (FisherForecast) – An existing FisherForecast object to add.

parameter_labels(verbosity=0)

Parameter labels to be used in plotting.

Parameters

verbosity (int) – Verbosity level. Default: 0.

Returns

List of strings with parameter labels.

Return type

(list)

visibilities(obs, p, verbosity=0)

Generates visibilities associated with the summed model.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)

visibility_gradients(obs, p, verbosity=0)

Generates visibility gradients associated with the summed model.

Parameters

• obs (ehtim.Obsdata) – ehtim data object

• p (numpy.ndarray) – list of parameters for the model image used to create object.

• verbosity (int) – Verbosity level. Default: 0.

Returns

List of complex visibilities computed at observations.

Return type

(numpy.ndarray)