# 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 liner 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...).

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.

    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 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_tauint`.

If the parameter $S$ is set to zero it is assumed that dataset does not exhibit any autocorrelation and the windowsize 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)

`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 computed 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 as `Corr`. Different discretizations of the numerical derivative are available.
- `Corr.second_deriv` returns the second derivative of the correlator as `Corr`. 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 another `Corr` or `Obs` 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 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)

# Optimization / fits / roots

`pyerrors.fits` `pyerrors.roots`

# Matrix operations

`pyerrors` provides wrappers for `Obs`-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

## Jackknife samples

For comparison with other analysis workflows `pyerrors` can generate jackknife samples from an `Obs` object or import jackknife samples into an `Obs` object. See `pyerrors.obs.Obs.export_jackknife` and `pyerrors.obs.import_jackknife` for details.

# Input

`pyerrors` includes an `input` submodule in which input routines and parsers for the output of various numerical programs are contained. For details see `pyerrors.input`.

View Source

    r'''
    # 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](https://arxiv.org/abs/hep-lat/0306017). Some of its features are:
    - automatic differentiation for exact liner 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).
    - treatment of slow modes in the simulation as suggested in [arXiv:1009.5228](https://arxiv.org/abs/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](https://arxiv.org/abs/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...).

    There 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).

    ## Basic example

    ```python
    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.

    ```python
    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 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](https://arxiv.org/abs/hep-lat/0306017).
    The 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).

    The `Obs` class is designed such that mathematical numpy functions can be used on `Obs` just as for regular floats.

    ```python
    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](https://arxiv.org/abs/hep-lat/0306017).
    After having arrived at the derived quantity of interest the `gamma_method` can be called as detailed in the following example.

    ```python
    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](https://link.springer.com/article/10.1007/BF01022990)
    $$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$
    The window $W$ is determined via the automatic windowing procedure described in [arXiv:hep-lat/0306017](https://arxiv.org/abs/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.

    ```python
    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_tauint`.

    If the parameter $S$ is set to zero it is assumed that dataset does not exhibit any autocorrelation and the windowsize 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](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.

    ```python
    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`.

    ```python
    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)
    ```

    `pyerrors` identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar `|` in the name of the data set.

    ```python
    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.

    ```python
    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`.

    ```python
    # 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 computed 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´
    ```python
    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.
    ```python
    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
    ```python
    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.
    ```python
    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 as `Corr`. Different discretizations of the numerical derivative are available.
    - `Corr.second_deriv` returns the second derivative of the correlator as `Corr`. 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 another `Corr` or `Obs` 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 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.

    ```python
    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.
    ```python
    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)
    ```

    # Optimization / fits / roots
    `pyerrors.fits`
    `pyerrors.roots`

    # Matrix operations
    `pyerrors` provides wrappers for `Obs`-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

    ## Jackknife samples
    For comparison with other analysis workflows `pyerrors` can generate jackknife samples from an `Obs` object or import jackknife samples into an `Obs` object.
    See `pyerrors.obs.Obs.export_jackknife` and `pyerrors.obs.import_jackknife` for details.

    # Input
    `pyerrors` includes an `input` submodule in which input routines and parsers for the output of various numerical programs are contained. For details see `pyerrors.input`.
    '''
    from .obs import *
    from .correlators import *
    from .fits import *
    from .misc import *
    from . import dirac
    from . import input
    from . import linalg
    from . import mpm
    from . import roots

    from .version import __version__