pyerrors
What is pyerrors?
pyerrors
is a python package for error computation and propagation of Markov chain Monte Carlo data.
It is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:
- automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
- treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
- coherent error propagation for data from different Markov chains.
- non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
- real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
More detailed examples can found in the GitHub repository .
If you use pyerrors
for research that leads to a publication please consider citing:
- Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. [arXiv:2209.14371 [hep-lat]].
- Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
- Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
and
- Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
where applicable.
There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
Basic example
import numpy as np
import pyerrors as pe
my_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object
my_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object
my_new_obs.gamma_method() # Estimate the statistical error
print(my_new_obs) # Print the result to stdout
> 0.31498(72)
The Obs
class
pyerrors
introduces a new datatype, Obs
, which simplifies error propagation and estimation for auto- and cross-correlated data.
An Obs
object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.
The samples can either be provided as python list or as numpy array.
The second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
import pyerrors as pe
my_obs = pe.Obs([samples], ['ensemble_name'])
Error propagation
When performing mathematical operations on Obs
objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion
$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha\,,$$
as introduced in arXiv:hep-lat/0306017.
The required derivatives $\bar{f}_\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
The Obs
class is designed such that mathematical numpy functions can be used on Obs
just as for regular floats.
import numpy as np
import pyerrors as pe
my_obs1 = pe.Obs([samples1], ['ensemble_name'])
my_obs2 = pe.Obs([samples2], ['ensemble_name'])
my_sum = my_obs1 + my_obs2
my_m_eff = np.log(my_obs1 / my_obs2)
iamzero = my_m_eff - my_m_eff
# Check that value and fluctuations are zero within machine precision
print(iamzero == 0.0)
> True
Error estimation
The error estimation within pyerrors
is based on the gamma method introduced in arXiv:hep-lat/0306017.
After having arrived at the derived quantity of interest the gamma_method
can be called as detailed in the following example.
my_sum.gamma_method()
print(my_sum)
> 1.70(57)
my_sum.details()
> Result 1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)
> t_int 2.71422900e+00 +/- 6.40320983e-01 S = 2.00
> 1000 samples in 1 ensemble:
> · Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)
We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988
$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$
The window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.
The standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method
as parameter.
my_sum.gamma_method(S=3.0)
my_sum.details()
> Result 1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)
> t_int 3.29909703e+00 +/- 9.77310102e-01 S = 3.00
> 1000 samples in 1 ensemble:
> · Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)
The integrated autocorrelation time $\tau_\mathrm{int}$ and the autocorrelation function $\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint
and pyerrors.obs.Obs.plot_rho
.
If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero. In this case the error estimate is identical to the sample standard error.
Exponential tails
Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\tau_\mathrm{exp}$, can be passed to the gamma_method
as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
my_sum.gamma_method(tau_exp=7.2)
my_sum.details()
> Result 1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)
> t_int 3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20, N_sigma = 1
> 1000 samples in 1 ensemble:
> · Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)
For the full API see pyerrors.obs.Obs.gamma_method
.
Multiple ensembles/replica
Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name
.
obs1 = pe.Obs([samples1], ['ensemble1'])
obs2 = pe.Obs([samples2], ['ensemble2'])
my_sum = obs1 + obs2
my_sum.details()
> Result 2.00697958e+00
> 1500 samples in 2 ensembles:
> · Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)
> · Ensemble 'ensemble2' : 500 configurations (from 1 to 500)
Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.
pyerrors
identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar |
in the name of the data set.
obs1 = pe.Obs([samples1], ['ensemble1|r01'])
obs2 = pe.Obs([samples2], ['ensemble1|r02'])
> my_sum = obs1 + obs2
> my_sum.details()
> Result 2.00697958e+00
> 1500 samples in 1 ensemble:
> · Ensemble 'ensemble1'
> · Replicum 'r01' : 1000 configurations (from 1 to 1000)
> · Replicum 'r02' : 500 configurations (from 1 to 500)
Error estimation for multiple ensembles
In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
pe.Obs.S_dict['ensemble1'] = 2.5
pe.Obs.tau_exp_dict['ensemble2'] = 8.0
pe.Obs.tau_exp_dict['ensemble3'] = 2.0
In case the gamma_method
is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.
Passing arguments to the gamma_method
still dominates over the dictionaries.
Irregular Monte Carlo chains
Obs
objects defined on irregular Monte Carlo chains can be initialized with the parameter idl
.
# Observable defined on configurations 20 to 519
obs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])
obs1.details()
> Result 9.98319881e-01
> 500 samples in 1 ensemble:
> · Ensemble 'ensemble1' : 500 configurations (from 20 to 519)
# Observable defined on every second configuration between 5 and 1003
obs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])
obs2.details()
> Result 9.99100712e-01
> 500 samples in 1 ensemble:
> · Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)
# Observable defined on configurations 2, 9, 28, 29 and 501
obs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])
obs3.details()
> Result 1.01718064e+00
> 5 samples in 1 ensemble:
> · Ensemble 'ensemble1' : 5 configurations (irregular range)
Obs
objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.
Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.
Make sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho
or pyerrors.obs.Obs.plot_tauint
.
For the full API see pyerrors.obs.Obs
.
Correlators
When one is not interested in single observables but correlation functions, pyerrors
offers the Corr
class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr
objects one needs to arrange the data as a list of Obs
my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])
print(my_corr)
> x0/a Corr(x0/a)
> ------------------
> 0 0.7957(80)
> 1 0.5156(51)
> 2 0.3227(33)
> 3 0.2041(21)
In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])
print(my_corr)
> x0/a Corr(x0/a)
> ------------------
> 0
> 1 0.7957(80)
> 2 0.5156(51)
> 3 0.3227(33)
> 4 0.2041(21)
> 5
The individual entries of a correlator can be accessed via slicing
print(my_corr[3])
> 0.3227(33)
Error propagation with the Corr
class works very similar to Obs
objects. Mathematical operations are overloaded and Corr
objects can be computed together with other Corr
objects, Obs
objects or real numbers and integers.
my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr
pyerrors
provides the user with a set of regularly used methods for the manipulation of correlator objects:
Corr.gamma_method
applies the gamma method to all entries of the correlator.Corr.m_eff
to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.Corr.deriv
returns the first derivative of the correlator asCorr
. Different discretizations of the numerical derivative are available.Corr.second_deriv
returns the second derivative of the correlator asCorr
. Different discretizations of the numerical derivative are available.Corr.symmetric
symmetrizes parity even correlations functions, assuming periodic boundary conditions.Corr.anti_symmetric
anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.Corr.T_symmetry
averages a correlator with its time symmetry partner, assuming fixed boundary conditions.Corr.plateau
extracts a plateau value from the correlator in a given range.Corr.roll
periodically shifts the correlator.Corr.reverse
reverses the time ordering of the correlator.Corr.correlate
constructs a disconnected correlation function from the correlator and anotherCorr
orObs
object.Corr.reweight
reweights the correlator.
pyerrors
can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP
).
For the full API see pyerrors.correlators.Corr
.
Complex valued observables
pyerrors
can handle complex valued observables via the class pyerrors.obs.CObs
.
CObs
are initialized with a real and an imaginary part which both can be Obs
valued.
my_real_part = pe.Obs([samples1], ['ensemble1'])
my_imag_part = pe.Obs([samples2], ['ensemble1'])
my_cobs = pe.CObs(my_real_part, my_imag_part)
my_cobs.gamma_method()
print(my_cobs)
> (0.9959(91)+0.659(28)j)
Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs
class.
my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)
my_derived_cobs.gamma_method()
print(my_derived_cobs)
> (1.668(23)+0.0j)
The Covobs
class
In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs
class allows to define such quantities in pyerrors
. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
This concept is built into the definition of Covobs
. In pyerrors
, external input is defined by $M$ mean values, a $M\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs
, since the second argument of this function is the covariance matrix of the Covobs
.
import pyerrors.obs as pe
mpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')
mpi.gamma_method()
mpi.details()
> Result 1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)
> pi^0 mass 5.00000000e-04
> 0 samples in 1 ensemble:
> · Covobs 'pi^0 mass'
The resulting object mpi
is an Obs
that contains a Covobs
. In the following, it may be handled as any other Obs
. The contribution of the covariance matrix to the error of an Obs
is determined from the $M \times M$ covariance matrix $\Sigma$ and the gradient of the Obs
with respect to the external quantities, which is the $1\times M$ Jacobian matrix $J$, via
$$s = \sqrt{J^T \Sigma J}\,,$$
where the Jacobian is computed for each derived quantity via automatic differentiation.
Correlated auxiliary data is defined similarly to above, e.g., via
RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')
print(RAP)
> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]
where RAP
now is a list of two Obs
that contains the two correlated parameters.
Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs
class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs
o
with respect to a covariance matrix with the identifying string k
may be accessed via
o.covobs[k].grad
Error propagation in iterative algorithms
pyerrors
supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.
Least squares fits
Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares
. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
Fit functions have to be of the following form
import autograd.numpy as anp
def func(a, x):
return a[1] * anp.exp(-a[0] * x)
It is important that numerical functions refer to autograd.numpy
instead of numpy
for the automatic differentiation in iterative algorithms to work properly.
Fits can then be performed via
fit_result = pe.fits.least_squares(x, y, func)
print("\n", fit_result)
> Fit with 2 parameters
> Method: Levenberg-Marquardt
> `ftol` termination condition is satisfied.
> chisquare/d.o.f.: 0.9593035785160936
> Goodness of fit:
> χ²/d.o.f. = 0.959304
> p-value = 0.5673
> Fit parameters:
> 0 0.0548(28)
> 1 1.933(64)
where x is a list
or numpy.array
of floats
and y is a list
or numpy.array
of Obs
.
Data stored in Corr
objects can be fitted directly using the Corr.fit
method.
my_corr = pe.Corr(y)
fit_result = my_corr.fit(func, fitrange=[12, 25])
this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
For fit functions with multiple independent variables the fit function can be of the form
def func(a, x):
(x1, x2) = x
return a[0] * x1 ** 2 + a[1] * x2
pyerrors
also supports correlated fits which can be triggered via the parameter correlated_fit=True
.
Details about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance
.
Direct visualizations of the performed fits can be triggered via resplot=True
or qqplot=True
. For all available options see pyerrors.fits.least_squares
.
Total least squares fits
pyerrors
can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares
. The syntax is identical to the standard least squares case, the only difference being that x
also has to be a list
or numpy.array
of Obs
.
For the full API see pyerrors.fits
for fits and pyerrors.roots
for finding roots of functions.
Matrix operations
pyerrors
provides wrappers for Obs
- and CObs
-valued matrix operations based on numpy.linalg
. The supported functions include:
inv
for the matrix inverse.cholseky
for the Cholesky decomposition.det
for the matrix determinant.eigh
for eigenvalues and eigenvectors of hermitean matrices.eig
for eigenvalues of general matrices.pinv
for the Moore-Penrose pseudoinverse.svd
for the singular-value-decomposition.
For the full API see pyerrors.linalg
.
Export data
The preferred exported file format within pyerrors
is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
- What observables are stored? Possibly: How exactly are they defined.
- How does each single ensemble or external quantity contribute to the error of the observable?
- Who did write the file when and on which machine?
This can be achieved by storing all information in one single file. The export routines of pyerrors
are written such that as much information as possible is written automatically as described in the following example
my_obs = pe.Obs([samples], ["test_ensemble"])
my_obs.tag = "My observable"
pe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")
# For a single observable one can equivalently use the class method dump
my_obs.dump("test_output_file", description="This file contains a test observable")
check = pe.input.json.load_json("test_output_file")
print(my_obs == check)
> True
The format also allows to directly write out the content of Corr
objects or lists and arrays of Obs
objects by passing the desired data to pyerrors.input.json.dump_to_json
.
json.gz format specification
The first entries of the file provide optional auxiliary information:
program
is a string that indicates which program was used to write the file.version
is a string that specifies the version of the format.who
is a string that specifies the user name of the creator of the file.date
is a string and contains the creation date of the file.host
is a string and contains the hostname of the machine where the file has been written.description
contains information on the content of the file. This field is not filled automatically inpyerrors
. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
The only necessary entry of the file is the field
-obsdata
, an array that contains the actual data.
Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs
, list
, numpy.ndarray
, Corr
. All Obs
inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata
, are treated independently. Each entry of the array obsdata
has the following required entries:
type
is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.value
is an array that contains the mean values of the Obs inside the structure. The following entries are optional:layout
is a string that specifies the layout of multi-dimensional structures. Examples are "2, 2" for a 2x2 dimensional matrix or "64, 4, 4" for a Corr with $T=64$ and 4x4 matrices on each time slices. "1" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).tag
is any JSON type. It contains additional information concerning the structure. Thetag
of anObs
inpyerrors
is written here.reweighted
is a Bool that may be used to specify, whether theObs
in the structure have been reweighted.data
is an array that contains the data from MC chains. We will define it below.cdata
is an array that contains the data from external quantities with an error (Covobs
inpyerrors
). We will define it below.
The array data
contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
id
, a string that contains the name of the ensemblereplica
, an array that contains an entry per replica of the ensemble.
Each entry of replica
contains
name
, a string that contains the name of the replica
deltas
, an array that contains the actual data.
Each entry in deltas
corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs
inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
The array cdata
contains information about the contribution of auxiliary observables, represented by Covobs
in pyerrors
, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
id
, a string that identifies the covariance matrixlayout
, a string that defines the dimensions of the $M\times M$ covariance matrix (has to be "M, M" or "1").cov
, an array that contains the $M\times M$ many entries of the covariance matrix, stored in row-major format.grad
, an array that contains N entries, one for eachObs
inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
1r''' 2# What is pyerrors? 3`pyerrors` is a python package for error computation and propagation of Markov chain Monte Carlo data. 4It is based on the gamma method [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). Some of its features are: 5- automatic differentiation for exact linear error propagation as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289) (partly based on the [autograd](https://github.com/HIPS/autograd) package). 6- treatment of slow modes in the simulation as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228). 7- coherent error propagation for data from different Markov chains. 8- non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289). 9- real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...). 10 11More detailed examples can found in the [GitHub repository](https://github.com/fjosw/pyerrors/tree/develop/examples) [](https://mybinder.org/v2/gh/fjosw/pyerrors/HEAD?labpath=examples). 12 13If you use `pyerrors` for research that leads to a publication please consider citing: 14- Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, *pyerrors: a python framework for error analysis of Monte Carlo data*. [arXiv:2209.14371 [hep-lat]]. 15- Ulli Wolff, *Monte Carlo errors with less errors*. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum). 16- Alberto Ramos, *Automatic differentiation for error analysis of Monte Carlo data*. Comput.Phys.Commun. 238 (2019) 19-35. 17 18and 19 20- Stefan Schaefer, Rainer Sommer, Francesco Virotta, *Critical slowing down and error analysis in lattice QCD simulations*. Nucl.Phys.B 845 (2011) 93-119. 21 22where applicable. 23 24There exist similar publicly available implementations of gamma method error analysis suites in [Fortran](https://gitlab.ift.uam-csic.es/alberto/aderrors), [Julia](https://gitlab.ift.uam-csic.es/alberto/aderrors.jl) and [Python](https://github.com/mbruno46/pyobs). 25 26## Basic example 27 28```python 29import numpy as np 30import pyerrors as pe 31 32my_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object 33my_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object 34my_new_obs.gamma_method() # Estimate the statistical error 35print(my_new_obs) # Print the result to stdout 36> 0.31498(72) 37``` 38 39# The `Obs` class 40 41`pyerrors` introduces a new datatype, `Obs`, which simplifies error propagation and estimation for auto- and cross-correlated data. 42An `Obs` object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain. 43The samples can either be provided as python list or as numpy array. 44The second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. **It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See [Multiple ensembles/replica](#Multiple-ensemblesreplica) for details.** 45 46```python 47import pyerrors as pe 48 49my_obs = pe.Obs([samples], ['ensemble_name']) 50``` 51 52## Error propagation 53 54When performing mathematical operations on `Obs` objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion 55$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha\,,$$ 56as introduced in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). 57The required derivatives $\bar{f}_\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289). 58 59The `Obs` class is designed such that mathematical numpy functions can be used on `Obs` just as for regular floats. 60 61```python 62import numpy as np 63import pyerrors as pe 64 65my_obs1 = pe.Obs([samples1], ['ensemble_name']) 66my_obs2 = pe.Obs([samples2], ['ensemble_name']) 67 68my_sum = my_obs1 + my_obs2 69 70my_m_eff = np.log(my_obs1 / my_obs2) 71 72iamzero = my_m_eff - my_m_eff 73# Check that value and fluctuations are zero within machine precision 74print(iamzero == 0.0) 75> True 76``` 77 78## Error estimation 79 80The error estimation within `pyerrors` is based on the gamma method introduced in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). 81After having arrived at the derived quantity of interest the `gamma_method` can be called as detailed in the following example. 82 83```python 84my_sum.gamma_method() 85print(my_sum) 86> 1.70(57) 87my_sum.details() 88> Result 1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%) 89> t_int 2.71422900e+00 +/- 6.40320983e-01 S = 2.00 90> 1000 samples in 1 ensemble: 91> · Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000) 92 93``` 94 95We use the following definition of the integrated autocorrelation time established in [Madras & Sokal 1988](https://link.springer.com/article/10.1007/BF01022990) 96$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$ 97The window $W$ is determined via the automatic windowing procedure described in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). 98The standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the `gamma_method` as parameter. 99 100```python 101my_sum.gamma_method(S=3.0) 102my_sum.details() 103> Result 1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%) 104> t_int 3.29909703e+00 +/- 9.77310102e-01 S = 3.00 105> 1000 samples in 1 ensemble: 106> · Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000) 107 108``` 109 110The integrated autocorrelation time $\tau_\mathrm{int}$ and the autocorrelation function $\rho(W)$ can be monitored via the methods `pyerrors.obs.Obs.plot_tauint` and `pyerrors.obs.Obs.plot_rho`. 111 112If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero. 113In this case the error estimate is identical to the sample standard error. 114 115### Exponential tails 116 117Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\rho$ as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228). The longest autocorrelation time in the history, $\tau_\mathrm{exp}$, can be passed to the `gamma_method` as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate. 118 119```python 120my_sum.gamma_method(tau_exp=7.2) 121my_sum.details() 122> Result 1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%) 123> t_int 3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20, N_sigma = 1 124> 1000 samples in 1 ensemble: 125> · Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000) 126``` 127 128For the full API see `pyerrors.obs.Obs.gamma_method`. 129 130## Multiple ensembles/replica 131 132Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their `name`. 133 134```python 135obs1 = pe.Obs([samples1], ['ensemble1']) 136obs2 = pe.Obs([samples2], ['ensemble2']) 137 138my_sum = obs1 + obs2 139my_sum.details() 140> Result 2.00697958e+00 141> 1500 samples in 2 ensembles: 142> · Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000) 143> · Ensemble 'ensemble2' : 500 configurations (from 1 to 500) 144``` 145Observables from the **same Monte Carlo chain** have to be initialized with the **same name** for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error. 146 147 148`pyerrors` identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar `|` in the name of the data set. 149 150```python 151obs1 = pe.Obs([samples1], ['ensemble1|r01']) 152obs2 = pe.Obs([samples2], ['ensemble1|r02']) 153 154> my_sum = obs1 + obs2 155> my_sum.details() 156> Result 2.00697958e+00 157> 1500 samples in 1 ensemble: 158> · Ensemble 'ensemble1' 159> · Replicum 'r01' : 1000 configurations (from 1 to 1000) 160> · Replicum 'r02' : 500 configurations (from 1 to 500) 161``` 162 163### Error estimation for multiple ensembles 164 165In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example. 166 167```python 168pe.Obs.S_dict['ensemble1'] = 2.5 169pe.Obs.tau_exp_dict['ensemble2'] = 8.0 170pe.Obs.tau_exp_dict['ensemble3'] = 2.0 171``` 172 173In case the `gamma_method` is called without any parameters it will use the values specified in the dictionaries for the respective ensembles. 174Passing arguments to the `gamma_method` still dominates over the dictionaries. 175 176 177## Irregular Monte Carlo chains 178 179`Obs` objects defined on irregular Monte Carlo chains can be initialized with the parameter `idl`. 180 181```python 182# Observable defined on configurations 20 to 519 183obs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)]) 184obs1.details() 185> Result 9.98319881e-01 186> 500 samples in 1 ensemble: 187> · Ensemble 'ensemble1' : 500 configurations (from 20 to 519) 188 189# Observable defined on every second configuration between 5 and 1003 190obs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)]) 191obs2.details() 192> Result 9.99100712e-01 193> 500 samples in 1 ensemble: 194> · Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2) 195 196# Observable defined on configurations 2, 9, 28, 29 and 501 197obs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]]) 198obs3.details() 199> Result 1.01718064e+00 200> 5 samples in 1 ensemble: 201> · Ensemble 'ensemble1' : 5 configurations (irregular range) 202 203``` 204 205`Obs` objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of. 206 207**Warning:** Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions. 208Make sure to check the autocorrelation time with e.g. `pyerrors.obs.Obs.plot_rho` or `pyerrors.obs.Obs.plot_tauint`. 209 210For the full API see `pyerrors.obs.Obs`. 211 212# Correlators 213When one is not interested in single observables but correlation functions, `pyerrors` offers the `Corr` class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a `Corr` objects one needs to arrange the data as a list of `Obs` 214```python 215my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3]) 216print(my_corr) 217> x0/a Corr(x0/a) 218> ------------------ 219> 0 0.7957(80) 220> 1 0.5156(51) 221> 2 0.3227(33) 222> 3 0.2041(21) 223``` 224In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced. 225```python 226my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1]) 227print(my_corr) 228> x0/a Corr(x0/a) 229> ------------------ 230> 0 231> 1 0.7957(80) 232> 2 0.5156(51) 233> 3 0.3227(33) 234> 4 0.2041(21) 235> 5 236``` 237The individual entries of a correlator can be accessed via slicing 238```python 239print(my_corr[3]) 240> 0.3227(33) 241``` 242Error propagation with the `Corr` class works very similar to `Obs` objects. Mathematical operations are overloaded and `Corr` objects can be computed together with other `Corr` objects, `Obs` objects or real numbers and integers. 243```python 244my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr 245``` 246 247`pyerrors` provides the user with a set of regularly used methods for the manipulation of correlator objects: 248- `Corr.gamma_method` applies the gamma method to all entries of the correlator. 249- `Corr.m_eff` to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available. 250- `Corr.deriv` returns the first derivative of the correlator as `Corr`. Different discretizations of the numerical derivative are available. 251- `Corr.second_deriv` returns the second derivative of the correlator as `Corr`. Different discretizations of the numerical derivative are available. 252- `Corr.symmetric` symmetrizes parity even correlations functions, assuming periodic boundary conditions. 253- `Corr.anti_symmetric` anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions. 254- `Corr.T_symmetry` averages a correlator with its time symmetry partner, assuming fixed boundary conditions. 255- `Corr.plateau` extracts a plateau value from the correlator in a given range. 256- `Corr.roll` periodically shifts the correlator. 257- `Corr.reverse` reverses the time ordering of the correlator. 258- `Corr.correlate` constructs a disconnected correlation function from the correlator and another `Corr` or `Obs` object. 259- `Corr.reweight` reweights the correlator. 260 261`pyerrors` can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see `pyerrors.correlators.Corr.GEVP`). 262 263For the full API see `pyerrors.correlators.Corr`. 264 265# Complex valued observables 266 267`pyerrors` can handle complex valued observables via the class `pyerrors.obs.CObs`. 268`CObs` are initialized with a real and an imaginary part which both can be `Obs` valued. 269 270```python 271my_real_part = pe.Obs([samples1], ['ensemble1']) 272my_imag_part = pe.Obs([samples2], ['ensemble1']) 273 274my_cobs = pe.CObs(my_real_part, my_imag_part) 275my_cobs.gamma_method() 276print(my_cobs) 277> (0.9959(91)+0.659(28)j) 278``` 279 280Elementary mathematical operations are overloaded and samples are properly propagated as for the `Obs` class. 281```python 282my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs) 283my_derived_cobs.gamma_method() 284print(my_derived_cobs) 285> (1.668(23)+0.0j) 286``` 287 288# The `Covobs` class 289In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The `Covobs` class allows to define such quantities in `pyerrors`. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix. 290 291This concept is built into the definition of `Covobs`. In `pyerrors`, external input is defined by $M$ mean values, a $M\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters `cov_Obs`, since the second argument of this function is the covariance matrix of the `Covobs`. 292 293```python 294import pyerrors.obs as pe 295 296mpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass') 297mpi.gamma_method() 298mpi.details() 299> Result 1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%) 300> pi^0 mass 5.00000000e-04 301> 0 samples in 1 ensemble: 302> · Covobs 'pi^0 mass' 303``` 304The resulting object `mpi` is an `Obs` that contains a `Covobs`. In the following, it may be handled as any other `Obs`. The contribution of the covariance matrix to the error of an `Obs` is determined from the $M \times M$ covariance matrix $\Sigma$ and the gradient of the `Obs` with respect to the external quantities, which is the $1\times M$ Jacobian matrix $J$, via 305$$s = \sqrt{J^T \Sigma J}\,,$$ 306where the Jacobian is computed for each derived quantity via automatic differentiation. 307 308Correlated auxiliary data is defined similarly to above, e.g., via 309```python 310RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)') 311print(RAP) 312> [Obs[16.7(1.9)], Obs[-19.0(3.3)]] 313``` 314where `RAP` now is a list of two `Obs` that contains the two correlated parameters. 315 316Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the `Covobs` class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an `Obs` `o` with respect to a covariance matrix with the identifying string `k` may be accessed via 317```python 318o.covobs[k].grad 319``` 320 321# Error propagation in iterative algorithms 322 323`pyerrors` supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289). 324 325## Least squares fits 326 327Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with `pyerrors.fits.least_squares`. As default solver the Levenberg-Marquardt algorithm implemented in [scipy](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html) is used. 328 329Fit functions have to be of the following form 330```python 331import autograd.numpy as anp 332 333def func(a, x): 334 return a[1] * anp.exp(-a[0] * x) 335``` 336**It is important that numerical functions refer to `autograd.numpy` instead of `numpy` for the automatic differentiation in iterative algorithms to work properly.** 337 338Fits can then be performed via 339```python 340fit_result = pe.fits.least_squares(x, y, func) 341print("\n", fit_result) 342> Fit with 2 parameters 343> Method: Levenberg-Marquardt 344> `ftol` termination condition is satisfied. 345> chisquare/d.o.f.: 0.9593035785160936 346 347> Goodness of fit: 348> χ²/d.o.f. = 0.959304 349> p-value = 0.5673 350> Fit parameters: 351> 0 0.0548(28) 352> 1 1.933(64) 353``` 354where x is a `list` or `numpy.array` of `floats` and y is a `list` or `numpy.array` of `Obs`. 355 356Data stored in `Corr` objects can be fitted directly using the `Corr.fit` method. 357```python 358my_corr = pe.Corr(y) 359fit_result = my_corr.fit(func, fitrange=[12, 25]) 360``` 361this can simplify working with absolute fit ranges and takes care of gaps in the data automatically. 362 363For fit functions with multiple independent variables the fit function can be of the form 364 365```python 366def func(a, x): 367 (x1, x2) = x 368 return a[0] * x1 ** 2 + a[1] * x2 369``` 370 371`pyerrors` also supports correlated fits which can be triggered via the parameter `correlated_fit=True`. 372Details about how the required covariance matrix is estimated can be found in `pyerrors.obs.covariance`. 373 374Direct visualizations of the performed fits can be triggered via `resplot=True` or `qqplot=True`. For all available options see `pyerrors.fits.least_squares`. 375 376## Total least squares fits 377`pyerrors` can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to orthogonal distance regression as implemented in [scipy](https://docs.scipy.org/doc/scipy/reference/odr.html), see `pyerrors.fits.least_squares`. The syntax is identical to the standard least squares case, the only difference being that `x` also has to be a `list` or `numpy.array` of `Obs`. 378 379For the full API see `pyerrors.fits` for fits and `pyerrors.roots` for finding roots of functions. 380 381# Matrix operations 382`pyerrors` provides wrappers for `Obs`- and `CObs`-valued matrix operations based on `numpy.linalg`. The supported functions include: 383- `inv` for the matrix inverse. 384- `cholseky` for the Cholesky decomposition. 385- `det` for the matrix determinant. 386- `eigh` for eigenvalues and eigenvectors of hermitean matrices. 387- `eig` for eigenvalues of general matrices. 388- `pinv` for the Moore-Penrose pseudoinverse. 389- `svd` for the singular-value-decomposition. 390 391For the full API see `pyerrors.linalg`. 392 393# Export data 394 395The preferred exported file format within `pyerrors` is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information: 396- What observables are stored? Possibly: How exactly are they defined. 397- How does each single ensemble or external quantity contribute to the error of the observable? 398- Who did write the file when and on which machine? 399 400This can be achieved by storing all information in one single file. The export routines of `pyerrors` are written such that as much information as possible is written automatically as described in the following example 401```python 402my_obs = pe.Obs([samples], ["test_ensemble"]) 403my_obs.tag = "My observable" 404 405pe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable") 406# For a single observable one can equivalently use the class method dump 407my_obs.dump("test_output_file", description="This file contains a test observable") 408 409check = pe.input.json.load_json("test_output_file") 410 411print(my_obs == check) 412> True 413``` 414The format also allows to directly write out the content of `Corr` objects or lists and arrays of `Obs` objects by passing the desired data to `pyerrors.input.json.dump_to_json`. 415 416## json.gz format specification 417The first entries of the file provide optional auxiliary information: 418- `program` is a string that indicates which program was used to write the file. 419- `version` is a string that specifies the version of the format. 420- `who` is a string that specifies the user name of the creator of the file. 421- `date` is a string and contains the creation date of the file. 422- `host` is a string and contains the hostname of the machine where the file has been written. 423- `description` contains information on the content of the file. This field is not filled automatically in `pyerrors`. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information. 424 425The only necessary entry of the file is the field 426-`obsdata`, an array that contains the actual data. 427 428Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of `Obs`, `list`, `numpy.ndarray`, `Corr`. All `Obs` inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array `obsdata`, are treated independently. Each entry of the array `obsdata` has the following required entries: 429- `type` is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs. 430- `value` is an array that contains the mean values of the Obs inside the structure. 431The following entries are optional: 432- `layout` is a string that specifies the layout of multi-dimensional structures. Examples are "2, 2" for a 2x2 dimensional matrix or "64, 4, 4" for a Corr with $T=64$ and 4x4 matrices on each time slices. "1" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below). 433- `tag` is any JSON type. It contains additional information concerning the structure. The `tag` of an `Obs` in `pyerrors` is written here. 434- `reweighted` is a Bool that may be used to specify, whether the `Obs` in the structure have been reweighted. 435- `data` is an array that contains the data from MC chains. We will define it below. 436- `cdata` is an array that contains the data from external quantities with an error (`Covobs` in `pyerrors`). We will define it below. 437 438The array `data` contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains: 439- `id`, a string that contains the name of the ensemble 440- `replica`, an array that contains an entry per replica of the ensemble. 441 442Each entry of `replica` contains 443`name`, a string that contains the name of the replica 444`deltas`, an array that contains the actual data. 445 446Each entry in `deltas` corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each `Obs` inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration. 447 448The array `cdata` contains information about the contribution of auxiliary observables, represented by `Covobs` in `pyerrors`, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains: 449- `id`, a string that identifies the covariance matrix 450- `layout`, a string that defines the dimensions of the $M\times M$ covariance matrix (has to be "M, M" or "1"). 451- `cov`, an array that contains the $M\times M$ many entries of the covariance matrix, stored in row-major format. 452- `grad`, an array that contains N entries, one for each `Obs` inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix. 453 454A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file. 455 456Julia I/O routines for the json.gz format, compatible with [ADerrors.jl](https://gitlab.ift.uam-csic.es/alberto/aderrors.jl), can be found [here](https://github.com/fjosw/ADjson.jl). 457''' 458from .obs import * 459from .correlators import * 460from .fits import * 461from .misc import * 462from . import dirac 463from . import input 464from . import linalg 465from . import mpm 466from . import roots 467 468from .version import __version__