import warnings
import numpy as np
from numpy.linalg import LinAlgError
from scipy.optimize._numdiff import approx_derivative
import scipy.stats as stats

from refnx.util import ErrorProp as EP
from refnx._lib import flatten, approx_hess2
from refnx._lib import unique as f_unique
from refnx.dataset import Data1D
from refnx.analysis import (
    is_parameter,
    Parameter,
    possibly_create_parameter,
    is_parameters,
    Parameters,
    Interval,
    PDF,
)


class BaseObjective:
    """Don't necessarily have to use Parameters, could use np.array"""

    def __init__(
        self,
        p,
        logl,
        logp=None,
        fcn_args=(),
        fcn_kwds=None,
        name=None,
        weighted=True,
    ):
        self.name = name
        self.parameters = p
        self.nvary = len(p)
        self._logl = logl
        self._logp = logp
        self.fcn_args = fcn_args
        self.fcn_kwds = {}
        # give the BaseObjective a default value, so that it can be used in a
        # GlobalObjective
        self.weighted = weighted
        if fcn_kwds is not None:
            self.fcn_kwds = fcn_kwds

    def setp(self, pvals):
        """
        Set the parameters from pvals

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        """
        self.parameters[:] = pvals

    def nll(self, pvals=None):
        """
        Negative log-likelihood function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        nll : float
            negative log-likelihood

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        return -self.logl(vals)

    def logp(self, pvals=None):
        """
        Log-prior probability function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        logp : float
            log-prior probability

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        if callable(self._logp):
            return self._logp(vals, *self.fcn_args, **self.fcn_kwds)
        return 0

    def logl(self, pvals=None):
        """
        Log-likelihood probability function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        logl : float
            log-likelihood probability.

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        return self._logl(vals, *self.fcn_args, **self.fcn_kwds)

    def logpost(self, pvals=None):
        """
        Log-posterior probability function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        logpost : float
            log-probability.

        Notes
        -----
        The log probability is the sum is the sum of the log-prior and
        log-likelihood probabilities. Does not set the parameter attribute.

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        logpost = self.logp(vals)
        if not np.isfinite(logpost):
            return -np.inf
        logpost += self.logl(vals)
        return logpost

    def nlpost(self, pvals=None):
        """
        Negative log-posterior function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        nlpost : float
            negative log-posterior

        """
        return -self.logpost(pvals)

    def varying_parameters(self):
        """
        Returns
        -------
        varying_parameters : np.ndarray
            The parameters varying in this objective function.

        """
        return self.parameters

    def covar(self, target="nll"):
        """
        Estimates a covariance matrix based on numerical differentiation
        of either the negative log-likelihood or negative log-posterior
        probabilities.

        Parameters
        ----------
        target : str, {"nll", "nlpost"}

        Returns
        -------
        covar : np.ndarray
            The covariance matrix for the fitting system

        Notes
        -----
        Estimation of a covariance matrix can be susceptible to numeric
        instabilities. Critically evaluate the matrix before further use.
        """
        _pvals = np.array(self.varying_parameters())

        if target == "nll":
            fn = self.nll
        elif target == "nlpost":
            fn = self.nlpost

        try:
            # from statsmodels
            # the output from this for the test in test_objective.covar
            # is very similar to numdifftools.Hessian, or a chained version
            # of approx_derivative
            hess = approx_hess2(_pvals, fn)
            covar = np.linalg.inv(hess)
        except LinAlgError:
            sz = np.size(_pvals, 0)
            covar = np.full((sz, sz), np.inf)
        finally:
            self.setp(_pvals)

        return covar


class Objective(BaseObjective):
    """
    Objective function for using with curvefitters such as
    `refnx.analysis.curvefitter.CurveFitter`.

    Parameters
    ----------
    model : refnx.analysis.Model
        the generative model function. One can also provide an object that
        inherits `refnx.analysis.Model`.
    data : refnx.dataset.Data1D
        data to be analysed.
    lnsigma : float or refnx.analysis.Parameter, optional
        Used if the  experimental uncertainty (`data.y_err`) underestimated by
        a constant fractional amount. The experimental uncertainty is modified
        as:

        `s_n**2 = y_err**2 + exp(lnsigma * 2) * model**2`

        See `Objective.logl` for more details.
    use_weights : bool
        use experimental uncertainty in calculation of residuals and
        logl, if available. If this is set to False, then you should also
        set `self.lnsigma.vary = False`, it will have no effect on the fit.
    transform : callable, optional
        the model, data and data uncertainty are transformed by this
        function before calculating the likelihood/residuals. Has the
        signature `transform(data.x, y, y_err=None)`, returning the tuple
        (`transformed_y, transformed_y_err`).
    logp_extra : callable, optional
        user specifiable log-probability term. This contribution is in
        addition to the log-prior term of the `model` parameters, and
        `model.logp`, as well as the log-likelihood of the `data`. Has
        signature:
        `logp_extra(model, data)`. The `model` will already possess
        updated parameters. Beware of including the same log-probability
        terms more than once.
    auxiliary_params : {sequence, Parameters}, optional
        Extra Parameter objects that are involved with curvefitting, but
        aren't directly included as part of the `model`. See notes for more
        details.
    name : str
        Name for the objective.

    Notes
    -----
    For parallelisation `logp_extra` needs to be picklable.

    `auxiliary_params` are included in calculating the `Objective.logp`
    term, are present in `Objective.varying_parameters()`, and are modified by
    Curvefitter during an analysis. Their main purpose is to aid in making
    constraints in models.
    """

    def __init__(
        self,
        model,
        data,
        lnsigma=None,
        use_weights=True,
        transform=None,
        logp_extra=None,
        auxiliary_params=(),
        name=None,
    ):
        self.model = model
        # should be a Data1D instance
        if isinstance(data, Data1D):
            self.data = data
        else:
            self.data = Data1D(data=data)

        self.lnsigma = lnsigma
        if lnsigma is not None:
            self.lnsigma = possibly_create_parameter(lnsigma, "lnsigma")

        if isinstance(auxiliary_params, Parameters):
            self.auxiliary_params = auxiliary_params
        else:
            self.auxiliary_params = Parameters(auxiliary_params)

        self._use_weights = use_weights
        self.transform = transform
        self.logp_extra = logp_extra
        self.name = name
        if name is None:
            self.name = id(self)

    def __str__(self):
        s = ["{:_>80}".format("")]
        s.append("Objective - {0}".format(self.name))

        # dataset name
        if self.data.name is None:
            s.append("Dataset = {0}".format(self.data))
        else:
            s.append("Dataset = {0}".format(self.data.name))

        s.append("datapoints = {0}".format(self.npoints))
        s.append("chi2 = {0}".format(self.chisqr()))
        s.append("Weighted = {0}".format(self.weighted))
        s.append("Transform = {0}".format(self.transform))
        s.append(str(self.parameters))

        return "\n".join(s)

    def __repr__(self):
        return (
            "Objective({model!r}, {data!r},"
            " lnsigma={lnsigma!r},"
            " use_weights={_use_weights},"
            " transform={transform!r},"
            " logp_extra={logp_extra!r},"
            " name={name!r})".format(**self.__dict__)
        )

    @property
    def weighted(self):
        """
        **bool** Does the data have weights (`data.y_err`), and is the
        objective using them?

        """
        return self.data.weighted and self._use_weights

    @weighted.setter
    def weighted(self, use_weights):
        self._use_weights = bool(use_weights)

    @property
    def npoints(self):
        """
        **int** the number of points in the dataset.

        """
        return self.data.y.size

    def varying_parameters(self):
        """
        Returns
        -------
        varying_parameters : refnx.analysis.Parameters
            The varying Parameter objects allowed to vary during the fit.

        """
        # create and return a Parameters object because it has the
        # __array__ method, which allows one to quickly get numerical values.
        p = Parameters()
        p.data = list(f_unique(p for p in flatten(self.parameters) if p.vary))
        return p

    def _data_transform(self, model=None):
        """
        Returns
        -------
        y, y_err, model: tuple of np.ndarray
            The y data, its uncertainties, and the model, all put through the
             transform.
        """
        x = self.data.x
        y = self.data.y

        y_err = 1.0
        if self.weighted:
            y_err = self.data.y_err

        if self.transform is None:
            return y, y_err, model
        else:
            if model is not None:
                model, _ = self.transform(x, model)

            y, y_err = self.transform(x, y, y_err)
            if self.weighted:
                return y, y_err, model
            else:
                return y, 1, model

    def generative(self, pvals=None):
        """
        Calculate the generative (dependent variable) function associated with
        the model.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        model : np.ndarray

        """
        self.setp(pvals)
        return self.model(self.data.x, x_err=self.data.x_err)

    def residuals(self, pvals=None):
        """
        Calculates the residuals for a given fitting system.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        residuals : np.ndarray
            Residuals, `(data.y - model) / y_err`.

        """
        self.setp(pvals)

        model = self.model(self.data.x, x_err=self.data.x_err)
        # TODO add in varying parameter residuals? (z-scores...)

        y, y_err, model = self._data_transform(model)

        if self.lnsigma is not None:
            s_n = np.sqrt(
                y_err * y_err + np.exp(2 * float(self.lnsigma)) * model * model
            )
        else:
            s_n = y_err

        return (y - model) / s_n

    def chisqr(self, pvals=None):
        """
        Calculates the chi-squared value for a given fitting system.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        chisqr : np.ndarray
            Chi-squared value, `np.sum(residuals**2)`.

        """
        # TODO reduced chisqr? include z-scores for parameters? DOF?
        self.setp(pvals)
        res = self.residuals(None)
        return np.sum(res * res)

    @property
    def parameters(self):
        """
        :class:`refnx.analysis.Parameters`, all the Parameters contained in the
        fitting system.

        """
        if is_parameter(self.lnsigma):
            return self.lnsigma | self.auxiliary_params | self.model.parameters
        elif len(self.auxiliary_params):
            return self.auxiliary_params | self.model.parameters
        else:
            return self.model.parameters

    def setp(self, pvals):
        """
        Set the parameters from pvals.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        """
        if pvals is None:
            return

        # set here rather than delegating to a Parameters
        # object, because it may not necessarily be a
        # Parameters object
        _varying_parameters = self.varying_parameters()
        if len(pvals) == len(_varying_parameters):
            for idx, param in enumerate(_varying_parameters):
                param.value = pvals[idx]
            return

        # values supplied are enough to specify all parameter values
        # even those that are repeated
        flattened_parameters = list(flatten(self.parameters))
        if len(pvals) == len(flattened_parameters):
            for idx, param in enumerate(flattened_parameters):
                param.value = pvals[idx]
            return

        raise ValueError(
            f"Incorrect number of values supplied ({len(pvals)})"
            f", supply either the full number of parameters"
            f" ({len(flattened_parameters)}, or only the varying"
            f" parameters ({len(_varying_parameters)})."
        )

    def prior_transform(self, u):
        """
        Calculate the prior transform of the system.

        Transforms uniform random variates in the unit hypercube,
        `u ~ uniform[0.0, 1.0)`, to the parameter space of interest, according
        to the priors on the varying parameters.

        Parameters
        ----------
        u : array-like
            Size of the varying parameters

        Returns
        -------
        pvals : array-like
            Scaled parameter values

        Notes
        -----
        If a parameter has bounds, `x ~ Unif[-10, 10)` then the scaling from
        `u` to `x` is done as follows:

        .. code-block:: python

            x = 2. * u - 1.  # scale and shift to [-1., 1.)
            x *= 10.  # scale to [-10., 10.)

        """
        var_pars = self.varying_parameters()
        pvals = np.empty(len(var_pars), dtype=np.float64)

        for i, var_par in enumerate(var_pars):
            pvals[i] = var_par.bounds.invcdf(u[i])

        return pvals

    def logp(self, pvals=None):
        """
        Calculate the log-prior of the system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logp : float
            log-prior probability

        Notes
        -----
        The log-prior is calculated as:

        .. code-block:: python

            logp = np.sum(param.logp() for param in
                             self.varying_parameters())

        """
        self.setp(pvals)

        logp = np.sum(
            [
                param.logp()
                for param in f_unique(
                    p for p in flatten(self.parameters) if p.vary
                )
            ]
        )

        if not np.isfinite(logp):
            return -np.inf

        return logp

    def logl(self, pvals=None):
        """
        Calculate the log-likelhood of the system

        The major component of the log-likelhood probability is from the data.
        Extra potential terms are added on from the Model, `self.model.logp`,
        and the user specifiable `logp_extra` function.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logl : float
            log-likelihood probability

        Notes
        -----
        The log-likelihood is calculated as:

        .. code-block:: python

            logl = -0.5 * np.sum(((y - model) / s_n)**2
                                 + np.log(2 * pi * s_n**2))
            logp += self.model.logp()
            logp += self.logp_extra(self.model, self.data)

        where

        .. code-block:: python

            s_n**2 = y_err**2 + exp(2 * lnsigma) * model**2

        """
        self.setp(pvals)

        model = self.model(self.data.x, x_err=self.data.x_err)

        logl = 0.0

        y, y_err, model = self._data_transform(model)

        if self.lnsigma is not None:
            var_y = (
                y_err * y_err + np.exp(2 * float(self.lnsigma)) * model * model
            )
        else:
            var_y = y_err ** 2

        # TODO do something sensible if data isn't weighted
        if self.weighted:
            logl += np.log(2 * np.pi * var_y)

        logl += (y - model) ** 2 / var_y

        # nans play havoc
        if np.isnan(logl).any():
            raise RuntimeError("Objective.logl encountered a NaN.")

        # add on extra 'potential' terms from the model.
        extra_potential = self.model.logp()

        if self.logp_extra is not None:
            extra_potential += self.logp_extra(self.model, self.data)

        return -0.5 * np.sum(logl) + extra_potential

    def nll(self, pvals=None):
        """
        Negative log-likelihood function

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        nll : float
            negative log-likelihood

        """
        self.setp(pvals)
        return -self.logl()

    def logpost(self, pvals=None):
        """
        Calculate the log-probability of the curvefitting system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logpost : float
            log-probability

        Notes
        -----
        The overall log-probability is the sum of the log-prior and
        log-likelihood. The log-likelihood is not calculated if the log-prior
        is impossible (`logp == -np.inf`).

        """
        self.setp(pvals)
        logpost = self.logp()

        # only calculate the probability if the parameters have finite
        # log-prior
        if not np.isfinite(logpost):
            return -np.inf

        logpost += self.logl()
        return logpost

    def covar(self, target="residuals"):
        """
        Estimates the covariance matrix of the Objective.

        Parameters
        ----------
        target : {"residuals", "nll", "nlpost"}
            Specifies what approach should be used to estimate covariance.

        Returns
        -------
        covar : np.ndarray
            Covariance matrix

        Notes
        -----
        For most purposes the Jacobian of the `'residuals'` should be used to
        calculate the covariance matrix, estimated as J.T x J.
        If an Objective cannot calculate residuals then the covariance matrix
        can be estimated by inverting a Hessian matrix created from either the
        `'nll'` or `'nlpost'` methods.
        The default `'residuals'` approach falls back to `'nll'` if a problem
        is experienced.
        The default `'residuals'` setting is preferred as the other settings
        can sometimes experience instabilities during Hessian estimation with
        numerical differentiation.
        """
        if target == "residuals":
            try:
                covar = self._covar_from_residuals()
            except Exception:
                # fallback to "nll"
                target = "nll"

        if target in ["nll", "nlpost"]:
            covar = super().covar(target)

        pvar = np.diagonal(covar).copy()
        psingular = np.where(pvar == 0)[0]

        if len(psingular) > 0:
            var_params = self.varying_parameters()
            singular_params = [var_params[ps] for ps in psingular]

            raise LinAlgError(
                "The following Parameters have no effect on"
                " Objective.residuals, please consider fixing"
                " them.\n" + repr(singular_params)
            )

        return covar

    def _covar_from_residuals(self):
        _pvals = np.array(self.varying_parameters())

        used_residuals_scaler = False

        def fn_scaler(vals):
            return np.squeeze(self.residuals(_pvals * vals))

        try:
            # we should be able to calculate a Jacobian for a parameter whose
            # value is zero. However, the scaling approach won't work.
            # This will force Jacobian calculation by unscaled parameters
            if np.any(_pvals == 0):
                raise FloatingPointError()

            with np.errstate(invalid="raise"):
                jac = approx_derivative(fn_scaler, np.ones_like(_pvals))
            used_residuals_scaler = True
        except FloatingPointError:
            jac = approx_derivative(self.residuals, _pvals)
        finally:
            # using approx_derivative changes the state of the objective
            # parameters have to make sure they're set at the end
            self.setp(_pvals)

        # need to create this because GlobalObjective may not have
        # access to all the datapoints being fitted.
        n_datapoints = np.size(jac, 0)

        # covar = J.T x J

        # from scipy.optimize.minpack.py
        # eliminates singular parameters
        _, s, VT = np.linalg.svd(jac, full_matrices=False)
        threshold = np.finfo(float).eps * max(jac.shape) * s[0]
        s = s[s > threshold]
        VT = VT[: s.size]
        covar = np.dot(VT.T / s ** 2, VT)

        if used_residuals_scaler:
            # unwind the scaling.
            covar = covar * np.atleast_2d(_pvals) * np.atleast_2d(_pvals).T

        scale = 1.0
        # scale by reduced chi2 if experimental uncertainties weren't used.
        if not (self.weighted):
            scale = self.chisqr() / (
                n_datapoints - len(self.varying_parameters())
            )

        return covar * scale

    def pgen(self, ngen=1000, nburn=0, nthin=1):
        """
        Yield random parameter vectors from the MCMC samples. The objective
        state is not altered.

        Parameters
        ----------
        ngen : int, optional
            the number of samples to yield. The actual number of samples
            yielded is `min(ngen, chain.size)`
        nburn : int, optional
            discard this many steps from the start of the chain
        nthin : int, optional
            only accept every `nthin` samples from the chain

        Yields
        ------
        pvec : np.ndarray
            A randomly chosen parameter vector

        """
        yield from self.parameters.pgen(ngen=ngen, nburn=nburn, nthin=nthin)

    def plot(self, pvals=None, samples=0, parameter=None, fig=None):
        """
        Plot the data/model.

        Requires matplotlib be installed.

        Parameters
        ----------
        pvals : np.ndarray, optional
            Numeric values for the Parameter's that are varying
        samples: number
            If the objective has been sampled, how many samples you wish to
            plot on the graph.
        parameter: refnx.analysis.Parameter
            Creates an interactive plot for the Parameter in Jupyter. Requires
            ipywidgets be installed. Use with %matplotlib notebook/qt.
        fig: Figure instance, optional
            If `fig` is not supplied then a new figure is created. Otherwise
            the graph is created on the current axes on the supplied figure.

        Returns
        -------
        fig, ax : :class:`matplotlib.Figure`, :class:`matplotlib.Axes`
            `matplotlib` figure and axes objects.

        """
        self.setp(pvals)

        if fig is None:
            import matplotlib.pyplot as plt

            fig = plt.figure()
            ax = fig.add_subplot(111)
        else:
            ax = fig.gca()

        y, y_err, model = self._data_transform(model=self.generative())

        # add the data (in a transformed fashion)
        if self.weighted:
            ax.errorbar(
                self.data.x,
                y,
                y_err,
                color="blue",
                label=self.data.name,
                marker="o",
                ms=3,
                lw=0,
                elinewidth=2,
            )
        else:
            ax.scatter(self.data.x, y, color="blue", s=3, label=self.data.name)

        if samples > 0:
            saved_params = np.array(self.parameters)
            # Get a number of chains, chosen randomly, set the objective,
            # and plot the model.
            for pvec in self.pgen(ngen=samples):
                y, y_err, model = self._data_transform(
                    model=self.generative(pvec)
                )

                ax.plot(self.data.x, model, color="k", alpha=0.01)

            # put back saved_params
            self.setp(saved_params)

        # add the fit
        generative_plot = ax.plot(self.data.x, model, color="red", zorder=20)

        if parameter is None:
            return fig, ax

        # create an interactive plot in a Jupyter notebook.
        def f(val):
            if parameter is not None:
                parameter.value = float(val)
            y, y_err, model = self._data_transform(model=self.generative())
            generative_plot[0].set_data(self.data.x, model)
            fig.canvas.draw()

        import ipywidgets

        return fig, ax, ipywidgets.interact(f, val=float(parameter))

    def corner(self, **kwds):
        """
        Corner plot of the chains belonging to the Parameters.
        Requires the `corner` and `matplotlib` packages.

        Parameters
        ----------
        kwds: dict
            passed directly to the `corner.corner` function

        Returns
        -------
        fig : :class:`matplotlib.Figure` object.
        """
        import corner

        var_pars = self.varying_parameters()
        chain = np.array([par.chain for par in var_pars])
        labels = [par.name for par in var_pars]
        chain = chain.reshape(len(chain), -1).T
        kwds["labels"] = labels
        kwds["quantiles"] = [0.16, 0.5, 0.84]
        return corner.corner(chain, **kwds)


class GlobalObjective(Objective):
    """
    Global Objective function for simultaneous fitting with
    `refnx.analysis.CurveFitter`

    Parameters
    ----------
    objectives : list
        list of :class:`refnx.analysis.Objective` objects

    """

    def __init__(self, objectives):
        self.objectives = objectives
        weighted = [objective.weighted for objective in objectives]

        self._weighted = np.array(weighted, dtype=bool)

        if len(np.unique(self._weighted)) > 1:
            raise ValueError(
                "All the objectives must be either weighted or"
                " unweighted, you cannot have a mixture."
            )

    def __str__(self):
        s = ["{:_>80}".format("\n")]
        s.append("--Global Objective--")
        for obj in self.objectives:
            s.append(str(obj))
            s.append("\n")
        return "\n".join(s)

    def __repr__(self):
        return "GlobalObjective({0})".format(repr(self.objectives))

    @property
    def weighted(self):
        """
        **bool** do all the datasets have y_err, and are all the objectives
        wanting to use weights?

        """
        return self._weighted.all()

    @property
    def npoints(self):
        """
        **int** number of data points in all the objectives.

        """
        npoints = 0
        for objective in self.objectives:
            npoints += objective.npoints
        return npoints

    def residuals(self, pvals=None):
        """
        Concatenated residuals for each of the
        :meth:`refnx.analysis.Objective.residuals`.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        residuals : np.ndarray
            Concatenated :meth:`refnx.analysis.Objective.residuals`

        """
        self.setp(pvals)

        residuals = []
        for objective in self.objectives:
            residual = objective.residuals()
            residuals.append(residual)

        return np.concatenate(residuals)

    @property
    def parameters(self):
        """
        :class:`refnx.analysis.Parameters` associated with all the objectives.

        """
        # TODO this is probably going to be slow.
        # cache and update strategy?
        p = Parameters(name="global fitting parameters")

        for objective in self.objectives:
            p.append(objective.parameters)

        return p

    def logp(self, pvals=None):
        """
        Calculate the log-prior of the system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters, optional
            values for the varying or entire set of parameters

        Returns
        -------
        logp : float
            log-prior probability

        """
        self.setp(pvals)

        logp = 0.0
        for objective in self.objectives:
            logp += objective.logp()
            # shortcut if one of the priors is impossible
            if not np.isfinite(logp):
                return -np.inf

        return logp

    def logl(self, pvals=None):
        """
        Calculate the log-likelhood of the system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logl : float
            log-likelihood probability

        """
        self.setp(pvals)
        logl = 0.0

        for objective in self.objectives:
            logl += objective.logl()

        return logl

    def plot(self, pvals=None, samples=0, parameter=None, fig=None):
        """
        Plot the data/model for all the objectives in the GlobalObjective.

        Matplotlib must be installed to use this method.

        Parameters
        ----------
        pvals : np.ndarray, optional
            Numeric values for the Parameter's that are varying
        samples: number, optional
            If the objective has been sampled, how many samples you wish to
            plot on the graph.
        parameter: refnx.analysis.Parameter, optional
            Creates an interactive plot for the Parameter in Jupyter. Requires
            ipywidgets be installed. Use with %matplotlib notebook/qt.
        fig: Figure instance, optional
            If `fig` is not supplied then a new figure is created. Otherwise
            the graph is created on the current axes on the supplied figure.

        Returns
        -------
        fig, ax : :class:`matplotlib.Figure`, :class:`matplotlib.Axes`
            `matplotlib` figure and axes objects.

        """

        self.setp(pvals)

        if fig is None:
            import matplotlib.pyplot as plt

            fig = plt.figure()
            ax = fig.add_subplot(111)
        else:
            ax = fig.gca()

        generative_plots = []

        if samples > 0:
            saved_params = np.array(self.parameters)

            # Get a number of chains, chosen randomly, set the objectives,
            # and plot the model.
            for pvec in self.pgen(ngen=samples):
                self.setp(pvec)

                for objective in self.objectives:
                    y, y_err, model = objective._data_transform(
                        model=objective.generative()
                    )

                    ax.plot(objective.data.x, model, color="k", alpha=0.01)

            # put back saved_params
            self.setp(saved_params)

        for objective in self.objectives:
            # add the data (in a transformed fashion)
            y, y_err, model = objective._data_transform(
                model=objective.generative()
            )

            if objective.weighted:
                ax.errorbar(
                    objective.data.x,
                    y,
                    y_err,
                    label=objective.data.name,
                    ms=3,
                    lw=0,
                    elinewidth=2,
                    marker="o",
                )
            else:
                ax.scatter(objective.data.x, y, label=objective.data.name)

            # add the fit
            generative_plots.append(
                ax.plot(objective.data.x, model, color="r", lw=1.5, zorder=20)[
                    0
                ]
            )

        if parameter is None:
            return fig, ax

        # create an interactive plot in a Jupyter notebook.
        def f(val):
            if parameter is not None:
                parameter.value = float(val)
            for i, objective in enumerate(self.objectives):
                y, y_err, model = objective._data_transform(
                    model=objective.generative()
                )

                generative_plots[i].set_data(objective.data.x, model)
            fig.canvas.draw()

        import ipywidgets

        return fig, ax, ipywidgets.interact(f, val=float(parameter))

        return fig, ax


class Transform:
    r"""
    Mathematical transforms of numeric data.

    Parameters
    ----------
    form : None or str
        One of:

            - 'lin'
                No transform is made
            - 'logY'
                log10 transform
            - 'YX4'
                YX**4 transform
            - 'YX2'
                YX**2 transform
            - None
                No transform is made

    Notes
    -----
    You ask for a transform to be carried out by calling the Transform object
    directly.

    >>> x = np.linspace(0.01, 0.1, 11)
    >>> y = np.linspace(100, 1000, 11)
    >>> y_err = np.sqrt(y)
    >>> t = Transform('logY')
    >>> ty, te = t(x, y, y_err)
    >>> ty
    array([2.        , 2.2787536 , 2.44715803, 2.56820172, 2.66275783,
           2.74036269, 2.80617997, 2.86332286, 2.91381385, 2.95904139,
           3.        ])

    """

    def __init__(self, form):
        types = [None, "lin", "logY", "YX4", "YX2"]
        self.form = None

        if form in types:
            self.form = form
        else:
            raise ValueError(
                "The form parameter must be one of [None, 'lin',"
                " 'logY', 'YX4', 'YX2']"
            )

    def __repr__(self):
        return "Transform({0})".format(repr(self.form))

    def __call__(self, x, y, y_err=None):
        """
        Calculate the transformed data

        Parameters
        ----------
        x : array-like
            x-values
        y : array-like
            y-values
        y_err : array-like
            Uncertainties in `y` (standard deviation)

        Returns
        -------
        yt, et : tuple
            The transformed data

        Examples
        --------
        >>> x = np.linspace(0.01, 0.1, 11)
        >>> y = np.linspace(100, 1000, 11)
        >>> y_err = np.sqrt(y)
        >>> t = Transform('logY')
        >>> ty, te = t(x, y, y_err)
        >>> ty
        array([2.        , 2.2787536 , 2.44715803, 2.56820172, 2.66275783,
               2.74036269, 2.80617997, 2.86332286, 2.91381385, 2.95904139,
               3.        ])

        """
        return self.__transform(x, y, y_err=y_err)

    def __transform(self, x, y, y_err=None):
        r"""
        Transform the data passed in

        Parameters
        ----------
        x : array-like

        y : array-like

        y_err : array-like

        Returns
        -------
        yt, et : tuple
            The transformed data
        """

        if y_err is None:
            etemp = np.ones_like(y)
        else:
            etemp = y_err

        if self.form in ["lin", None]:
            yt = np.copy(y)
            et = np.copy(etemp)
        elif self.form == "logY":
            yt, et = EP.EPlog10(y, etemp)
            if not np.isfinite(yt).all():
                warnings.warn(
                    "Some of the transformed data was non-finite."
                    " Please check your datasets for points with zero or"
                    " negative values.",
                    RuntimeWarning,
                )
        elif self.form == "YX4":
            yt = y * np.power(x, 4)
            et = etemp * np.power(x, 4)
        elif self.form == "YX2":
            yt = y * np.power(x, 2)
            et = etemp * np.power(x, 2)
        if y_err is None:
            return yt, None
        else:
            return yt, et


def pymc3_model(objective):
    """
    Creates a pymc3 model from an Objective.

    Requires theano and pymc3 be installed. This is an experimental feature.

    Parameters
    ----------
    objective: refnx.analysis.Objective

    Returns
    -------
    model: pymc3.Model

    Notes
    -----
    The varying parameters are renamed 'p0', 'p1', etc, as it's vital in pymc3
    that all parameters have their own unique name.

    """
    import pymc3 as pm
    import theano.tensor as tt
    from refnx._lib._pymc3 import _LogLikeWithGrad

    basic_model = pm.Model()

    pars = objective.varying_parameters()
    wrapped_pars = []
    with basic_model:
        # Priors for unknown model parameters
        for i, par in enumerate(pars):
            name = "p%d" % i
            p = _to_pymc3_distribution(name, par)
            wrapped_pars.append(p)

        # Expected value of outcome
        try:
            # Likelihood (sampling distribution) of observations
            pm.Normal(
                "y_obs",
                mu=objective.generative,
                sigma=objective.data.y_err,
                observed=objective.data.y,
            )
        except Exception:
            # Falling back, theano autodiff won't work on function object
            theta = tt.as_tensor_variable(wrapped_pars)
            logl = _LogLikeWithGrad(objective.logl)
            pm.Potential("log-likelihood", logl(theta))

    return basic_model


def _to_pymc3_distribution(name, par):
    """
    Create a pymc3 continuous distribution from a Bounds object.

    Parameters
    ----------
    name : str
        Name of parameter
    par : refnx.analysis.Parameter
        The parameter to wrap

    Returns
    -------
    d : pymc3.Distribution
        The pymc3 distribution

    """
    import pymc3 as pm
    import theano.tensor as T
    from theano.compile.ops import as_op

    dist = par.bounds
    # interval and both lb, ub are finite
    if isinstance(dist, Interval) and np.isfinite([dist.lb, dist.ub]).all():
        return pm.Uniform(name, dist.lb, dist.ub)
    # no bounds
    elif (
        isinstance(dist, Interval)
        and np.isneginf(dist.lb)
        and np.isinf(dist.lb)
    ):
        return pm.Flat(name)
    # half open uniform
    elif isinstance(dist, Interval) and not np.isfinite(dist.lb):
        return dist.ub - pm.HalfFlat(name)
    # half open uniform
    elif isinstance(dist, Interval) and not np.isfinite(dist.ub):
        return dist.lb + pm.HalfFlat(name)

    # it's a PDF
    if isinstance(dist, PDF):
        dist_gen = getattr(dist.rv, "dist", None)

        if isinstance(dist.rv, stats.rv_continuous):
            dist_gen = dist.rv

        if isinstance(dist_gen, type(stats.uniform)):
            if hasattr(dist.rv, "args"):
                p = pm.Uniform(
                    name, dist.rv.args[0], dist.rv.args[1] + dist.rv.args[0]
                )
            else:
                p = pm.Uniform(name, 0, 1)
            return p

        # norm from scipy.stats
        if isinstance(dist_gen, type(stats.norm)):
            if hasattr(dist.rv, "args"):
                p = pm.Normal(name, mu=dist.rv.args[0], sigma=dist.rv.args[1])
            else:
                p = pm.Normal(name, mu=0, sigma=1)
            return p

    # not open, uniform, or normal, so fall back to DensityDist.
    d = as_op(itypes=[T.dscalar], otypes=[T.dscalar])(dist.logp)
    r = as_op(itypes=[T.dscalar], otypes=[T.dscalar])(dist.rvs)
    p = pm.DensityDist(name, d, random=r)

    return p

import warnings
import numpy as np
from numpy.linalg import LinAlgError
from scipy.optimize._numdiff import approx_derivative
import scipy.stats as stats

from refnx.util import ErrorProp as EP
from refnx._lib import flatten, approx_hess2
from refnx._lib import unique as f_unique
from refnx.dataset import Data1D
from refnx.analysis import (
    is_parameter,
    Parameter,
    possibly_create_parameter,
    is_parameters,
    Parameters,
    Interval,
    PDF,
)


class BaseObjective:
    """Don't necessarily have to use Parameters, could use np.array"""

    def __init__(
        self,
        p,
        logl,
        logp=None,
        fcn_args=(),
        fcn_kwds=None,
        name=None,
        weighted=True,
    ):
        self.name = name
        self.parameters = p
        self.nvary = len(p)
        self._logl = logl
        self._logp = logp
        self.fcn_args = fcn_args
        self.fcn_kwds = {}
        # give the BaseObjective a default value, so that it can be used in a
        # GlobalObjective
        self.weighted = weighted
        if fcn_kwds is not None:
            self.fcn_kwds = fcn_kwds

    def setp(self, pvals):
        """
        Set the parameters from pvals

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        """
        self.parameters[:] = pvals

    def nll(self, pvals=None):
        """
        Negative log-likelihood function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        nll : float
            negative log-likelihood

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        return -self.logl(vals)

    def logp(self, pvals=None):
        """
        Log-prior probability function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        logp : float
            log-prior probability

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        if callable(self._logp):
            return self._logp(vals, *self.fcn_args, **self.fcn_kwds)
        return 0

    def logl(self, pvals=None):
        """
        Log-likelihood probability function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        logl : float
            log-likelihood probability.

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        return self._logl(vals, *self.fcn_args, **self.fcn_kwds)

    def logpost(self, pvals=None):
        """
        Log-posterior probability function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        logpost : float
            log-probability.

        Notes
        -----
        The log probability is the sum is the sum of the log-prior and
        log-likelihood probabilities. Does not set the parameter attribute.

        """
        vals = self.parameters
        if pvals is not None:
            vals = pvals

        logpost = self.logp(vals)
        if not np.isfinite(logpost):
            return -np.inf
        logpost += self.logl(vals)
        return logpost

    def nlpost(self, pvals=None):
        """
        Negative log-posterior function

        Parameters
        ----------
        pvals : np.ndarray
            Array containing the values to be tested.

        Returns
        -------
        nlpost : float
            negative log-posterior

        """
        return -self.logpost(pvals)

    def varying_parameters(self):
        """
        Returns
        -------
        varying_parameters : np.ndarray
            The parameters varying in this objective function.

        """
        return self.parameters

    def covar(self, target="nll"):
        """
        Estimates a covariance matrix based on numerical differentiation
        of either the negative log-likelihood or negative log-posterior
        probabilities.

        Parameters
        ----------
        target : str, {"nll", "nlpost"}

        Returns
        -------
        covar : np.ndarray
            The covariance matrix for the fitting system

        Notes
        -----
        Estimation of a covariance matrix can be susceptible to numeric
        instabilities. Critically evaluate the matrix before further use.
        """
        _pvals = np.array(self.varying_parameters())

        if target == "nll":
            fn = self.nll
        elif target == "nlpost":
            fn = self.nlpost

        try:
            # from statsmodels
            # the output from this for the test in test_objective.covar
            # is very similar to numdifftools.Hessian, or a chained version
            # of approx_derivative
            hess = approx_hess2(_pvals, fn)
            covar = np.linalg.inv(hess)
        except LinAlgError:
            sz = np.size(_pvals, 0)
            covar = np.full((sz, sz), np.inf)
        finally:
            self.setp(_pvals)

        return covar


class Objective(BaseObjective):
    """
    Objective function for using with curvefitters such as
    `refnx.analysis.curvefitter.CurveFitter`.

    Parameters
    ----------
    model : refnx.analysis.Model
        the generative model function. One can also provide an object that
        inherits `refnx.analysis.Model`.
    data : refnx.dataset.Data1D
        data to be analysed.
    lnsigma : float or refnx.analysis.Parameter, optional
        Used if the  experimental uncertainty (`data.y_err`) underestimated by
        a constant fractional amount. The experimental uncertainty is modified
        as:

        `s_n**2 = y_err**2 + exp(lnsigma * 2) * model**2`

        See `Objective.logl` for more details.
    use_weights : bool
        use experimental uncertainty in calculation of residuals and
        logl, if available. If this is set to False, then you should also
        set `self.lnsigma.vary = False`, it will have no effect on the fit.
    transform : callable, optional
        the model, data and data uncertainty are transformed by this
        function before calculating the likelihood/residuals. Has the
        signature `transform(data.x, y, y_err=None)`, returning the tuple
        (`transformed_y, transformed_y_err`).
    logp_extra : callable, optional
        user specifiable log-probability term. This contribution is in
        addition to the log-prior term of the `model` parameters, and
        `model.logp`, as well as the log-likelihood of the `data`. Has
        signature:
        `logp_extra(model, data)`. The `model` will already possess
        updated parameters. Beware of including the same log-probability
        terms more than once.
    auxiliary_params : {sequence, Parameters}, optional
        Extra Parameter objects that are involved with curvefitting, but
        aren't directly included as part of the `model`. See notes for more
        details.
    name : str
        Name for the objective.

    Notes
    -----
    For parallelisation `logp_extra` needs to be picklable.

    `auxiliary_params` are included in calculating the `Objective.logp`
    term, are present in `Objective.varying_parameters()`, and are modified by
    Curvefitter during an analysis. Their main purpose is to aid in making
    constraints in models.
    """

    def __init__(
        self,
        model,
        data,
        lnsigma=None,
        use_weights=True,
        transform=None,
        logp_extra=None,
        auxiliary_params=(),
        name=None,
    ):
        self.model = model
        # should be a Data1D instance
        if isinstance(data, Data1D):
            self.data = data
        else:
            self.data = Data1D(data=data)

        self.lnsigma = lnsigma
        if lnsigma is not None:
            self.lnsigma = possibly_create_parameter(lnsigma, "lnsigma")

        if isinstance(auxiliary_params, Parameters):
            self.auxiliary_params = auxiliary_params
        else:
            self.auxiliary_params = Parameters(auxiliary_params)

        self._use_weights = use_weights
        self.transform = transform
        self.logp_extra = logp_extra
        self.name = name
        if name is None:
            self.name = id(self)

    def __str__(self):
        s = ["{:_>80}".format("")]
        s.append("Objective - {0}".format(self.name))

        # dataset name
        if self.data.name is None:
            s.append("Dataset = {0}".format(self.data))
        else:
            s.append("Dataset = {0}".format(self.data.name))

        s.append("datapoints = {0}".format(self.npoints))
        s.append("chi2 = {0}".format(self.chisqr()))
        s.append("Weighted = {0}".format(self.weighted))
        s.append("Transform = {0}".format(self.transform))
        s.append(str(self.parameters))

        return "\n".join(s)

    def __repr__(self):
        return (
            "Objective({model!r}, {data!r},"
            " lnsigma={lnsigma!r},"
            " use_weights={_use_weights},"
            " transform={transform!r},"
            " logp_extra={logp_extra!r},"
            " name={name!r})".format(**self.__dict__)
        )

    @property
    def weighted(self):
        """
        **bool** Does the data have weights (`data.y_err`), and is the
        objective using them?

        """
        return self.data.weighted and self._use_weights

    @weighted.setter
    def weighted(self, use_weights):
        self._use_weights = bool(use_weights)

    @property
    def npoints(self):
        """
        **int** the number of points in the dataset.

        """
        return self.data.y.size

    def varying_parameters(self):
        """
        Returns
        -------
        varying_parameters : refnx.analysis.Parameters
            The varying Parameter objects allowed to vary during the fit.

        """
        # create and return a Parameters object because it has the
        # __array__ method, which allows one to quickly get numerical values.
        p = Parameters()
        p.data = list(f_unique(p for p in flatten(self.parameters) if p.vary))
        return p

    def _data_transform(self, model=None):
        """
        Returns
        -------
        y, y_err, model: tuple of np.ndarray
            The y data, its uncertainties, and the model, all put through the
             transform.
        """
        x = self.data.x
        y = self.data.y

        y_err = 1.0
        if self.weighted:
            y_err = self.data.y_err

        if self.transform is None:
            return y, y_err, model
        else:
            if model is not None:
                model, _ = self.transform(x, model)

            y, y_err = self.transform(x, y, y_err)
            if self.weighted:
                return y, y_err, model
            else:
                return y, 1, model

    def generative(self, pvals=None):
        """
        Calculate the generative (dependent variable) function associated with
        the model.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        model : np.ndarray

        """
        self.setp(pvals)
        return self.model(self.data.x, x_err=self.data.x_err)

    def residuals(self, pvals=None):
        """
        Calculates the residuals for a given fitting system.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        residuals : np.ndarray
            Residuals, `(data.y - model) / y_err`.

        """
        self.setp(pvals)

        model = self.model(self.data.x, x_err=self.data.x_err)
        # TODO add in varying parameter residuals? (z-scores...)

        y, y_err, model = self._data_transform(model)

        if self.lnsigma is not None:
            s_n = np.sqrt(
                y_err * y_err + np.exp(2 * float(self.lnsigma)) * model * model
            )
        else:
            s_n = y_err

        return (y - model) / s_n

    def chisqr(self, pvals=None):
        """
        Calculates the chi-squared value for a given fitting system.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        chisqr : np.ndarray
            Chi-squared value, `np.sum(residuals**2)`.

        """
        # TODO reduced chisqr? include z-scores for parameters? DOF?
        self.setp(pvals)
        res = self.residuals(None)
        return np.sum(res * res)

    @property
    def parameters(self):
        """
        :class:`refnx.analysis.Parameters`, all the Parameters contained in the
        fitting system.

        """
        if is_parameter(self.lnsigma):
            return self.lnsigma | self.auxiliary_params | self.model.parameters
        elif len(self.auxiliary_params):
            return self.auxiliary_params | self.model.parameters
        else:
            return self.model.parameters

    def setp(self, pvals):
        """
        Set the parameters from pvals.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        """
        if pvals is None:
            return

        # set here rather than delegating to a Parameters
        # object, because it may not necessarily be a
        # Parameters object
        _varying_parameters = self.varying_parameters()
        if len(pvals) == len(_varying_parameters):
            for idx, param in enumerate(_varying_parameters):
                param.value = pvals[idx]
            return

        # values supplied are enough to specify all parameter values
        # even those that are repeated
        flattened_parameters = list(flatten(self.parameters))
        if len(pvals) == len(flattened_parameters):
            for idx, param in enumerate(flattened_parameters):
                param.value = pvals[idx]
            return

        raise ValueError(
            f"Incorrect number of values supplied ({len(pvals)})"
            f", supply either the full number of parameters"
            f" ({len(flattened_parameters)}, or only the varying"
            f" parameters ({len(_varying_parameters)})."
        )

    def prior_transform(self, u):
        """
        Calculate the prior transform of the system.

        Transforms uniform random variates in the unit hypercube,
        `u ~ uniform[0.0, 1.0)`, to the parameter space of interest, according
        to the priors on the varying parameters.

        Parameters
        ----------
        u : array-like
            Size of the varying parameters

        Returns
        -------
        pvals : array-like
            Scaled parameter values

        Notes
        -----
        If a parameter has bounds, `x ~ Unif[-10, 10)` then the scaling from
        `u` to `x` is done as follows:

        .. code-block:: python

            x = 2. * u - 1.  # scale and shift to [-1., 1.)
            x *= 10.  # scale to [-10., 10.)

        """
        var_pars = self.varying_parameters()
        pvals = np.empty(len(var_pars), dtype=np.float64)

        for i, var_par in enumerate(var_pars):
            pvals[i] = var_par.bounds.invcdf(u[i])

        return pvals

    def logp(self, pvals=None):
        """
        Calculate the log-prior of the system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logp : float
            log-prior probability

        Notes
        -----
        The log-prior is calculated as:

        .. code-block:: python

            logp = np.sum(param.logp() for param in
                             self.varying_parameters())

        """
        self.setp(pvals)

        logp = np.sum(
            [
                param.logp()
                for param in f_unique(
                    p for p in flatten(self.parameters) if p.vary
                )
            ]
        )

        if not np.isfinite(logp):
            return -np.inf

        return logp

    def logl(self, pvals=None):
        """
        Calculate the log-likelhood of the system

        The major component of the log-likelhood probability is from the data.
        Extra potential terms are added on from the Model, `self.model.logp`,
        and the user specifiable `logp_extra` function.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logl : float
            log-likelihood probability

        Notes
        -----
        The log-likelihood is calculated as:

        .. code-block:: python

            logl = -0.5 * np.sum(((y - model) / s_n)**2
                                 + np.log(2 * pi * s_n**2))
            logp += self.model.logp()
            logp += self.logp_extra(self.model, self.data)

        where

        .. code-block:: python

            s_n**2 = y_err**2 + exp(2 * lnsigma) * model**2

        """
        self.setp(pvals)

        model = self.model(self.data.x, x_err=self.data.x_err)

        logl = 0.0

        y, y_err, model = self._data_transform(model)

        if self.lnsigma is not None:
            var_y = (
                y_err * y_err + np.exp(2 * float(self.lnsigma)) * model * model
            )
        else:
            var_y = y_err ** 2

        # TODO do something sensible if data isn't weighted
        if self.weighted:
            logl += np.log(2 * np.pi * var_y)

        logl += (y - model) ** 2 / var_y

        # nans play havoc
        if np.isnan(logl).any():
            raise RuntimeError("Objective.logl encountered a NaN.")

        # add on extra 'potential' terms from the model.
        extra_potential = self.model.logp()

        if self.logp_extra is not None:
            extra_potential += self.logp_extra(self.model, self.data)

        return -0.5 * np.sum(logl) + extra_potential

    def nll(self, pvals=None):
        """
        Negative log-likelihood function

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        nll : float
            negative log-likelihood

        """
        self.setp(pvals)
        return -self.logl()

    def logpost(self, pvals=None):
        """
        Calculate the log-probability of the curvefitting system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logpost : float
            log-probability

        Notes
        -----
        The overall log-probability is the sum of the log-prior and
        log-likelihood. The log-likelihood is not calculated if the log-prior
        is impossible (`logp == -np.inf`).

        """
        self.setp(pvals)
        logpost = self.logp()

        # only calculate the probability if the parameters have finite
        # log-prior
        if not np.isfinite(logpost):
            return -np.inf

        logpost += self.logl()
        return logpost

    def covar(self, target="residuals"):
        """
        Estimates the covariance matrix of the Objective.

        Parameters
        ----------
        target : {"residuals", "nll", "nlpost"}
            Specifies what approach should be used to estimate covariance.

        Returns
        -------
        covar : np.ndarray
            Covariance matrix

        Notes
        -----
        For most purposes the Jacobian of the `'residuals'` should be used to
        calculate the covariance matrix, estimated as J.T x J.
        If an Objective cannot calculate residuals then the covariance matrix
        can be estimated by inverting a Hessian matrix created from either the
        `'nll'` or `'nlpost'` methods.
        The default `'residuals'` approach falls back to `'nll'` if a problem
        is experienced.
        The default `'residuals'` setting is preferred as the other settings
        can sometimes experience instabilities during Hessian estimation with
        numerical differentiation.
        """
        if target == "residuals":
            try:
                covar = self._covar_from_residuals()
            except Exception:
                # fallback to "nll"
                target = "nll"

        if target in ["nll", "nlpost"]:
            covar = super().covar(target)

        pvar = np.diagonal(covar).copy()
        psingular = np.where(pvar == 0)[0]

        if len(psingular) > 0:
            var_params = self.varying_parameters()
            singular_params = [var_params[ps] for ps in psingular]

            raise LinAlgError(
                "The following Parameters have no effect on"
                " Objective.residuals, please consider fixing"
                " them.\n" + repr(singular_params)
            )

        return covar

    def _covar_from_residuals(self):
        _pvals = np.array(self.varying_parameters())

        used_residuals_scaler = False

        def fn_scaler(vals):
            return np.squeeze(self.residuals(_pvals * vals))

        try:
            # we should be able to calculate a Jacobian for a parameter whose
            # value is zero. However, the scaling approach won't work.
            # This will force Jacobian calculation by unscaled parameters
            if np.any(_pvals == 0):
                raise FloatingPointError()

            with np.errstate(invalid="raise"):
                jac = approx_derivative(fn_scaler, np.ones_like(_pvals))
            used_residuals_scaler = True
        except FloatingPointError:
            jac = approx_derivative(self.residuals, _pvals)
        finally:
            # using approx_derivative changes the state of the objective
            # parameters have to make sure they're set at the end
            self.setp(_pvals)

        # need to create this because GlobalObjective may not have
        # access to all the datapoints being fitted.
        n_datapoints = np.size(jac, 0)

        # covar = J.T x J

        # from scipy.optimize.minpack.py
        # eliminates singular parameters
        _, s, VT = np.linalg.svd(jac, full_matrices=False)
        threshold = np.finfo(float).eps * max(jac.shape) * s[0]
        s = s[s > threshold]
        VT = VT[: s.size]
        covar = np.dot(VT.T / s ** 2, VT)

        if used_residuals_scaler:
            # unwind the scaling.
            covar = covar * np.atleast_2d(_pvals) * np.atleast_2d(_pvals).T

        scale = 1.0
        # scale by reduced chi2 if experimental uncertainties weren't used.
        if not (self.weighted):
            scale = self.chisqr() / (
                n_datapoints - len(self.varying_parameters())
            )

        return covar * scale

    def pgen(self, ngen=1000, nburn=0, nthin=1):
        """
        Yield random parameter vectors from the MCMC samples. The objective
        state is not altered.

        Parameters
        ----------
        ngen : int, optional
            the number of samples to yield. The actual number of samples
            yielded is `min(ngen, chain.size)`
        nburn : int, optional
            discard this many steps from the start of the chain
        nthin : int, optional
            only accept every `nthin` samples from the chain

        Yields
        ------
        pvec : np.ndarray
            A randomly chosen parameter vector

        """
        yield from self.parameters.pgen(ngen=ngen, nburn=nburn, nthin=nthin)

    def plot(self, pvals=None, samples=0, parameter=None, fig=None):
        """
        Plot the data/model.

        Requires matplotlib be installed.

        Parameters
        ----------
        pvals : np.ndarray, optional
            Numeric values for the Parameter's that are varying
        samples: number
            If the objective has been sampled, how many samples you wish to
            plot on the graph.
        parameter: refnx.analysis.Parameter
            Creates an interactive plot for the Parameter in Jupyter. Requires
            ipywidgets be installed. Use with %matplotlib notebook/qt.
        fig: Figure instance, optional
            If `fig` is not supplied then a new figure is created. Otherwise
            the graph is created on the current axes on the supplied figure.

        Returns
        -------
        fig, ax : :class:`matplotlib.Figure`, :class:`matplotlib.Axes`
            `matplotlib` figure and axes objects.

        """
        self.setp(pvals)

        if fig is None:
            import matplotlib.pyplot as plt

            fig = plt.figure()
            ax = fig.add_subplot(111)
        else:
            ax = fig.gca()

        y, y_err, model = self._data_transform(model=self.generative())

        # add the data (in a transformed fashion)
        if self.weighted:
            ax.errorbar(
                self.data.x,
                y,
                y_err,
                color="blue",
                label=self.data.name,
                marker="o",
                ms=3,
                lw=0,
                elinewidth=2,
            )
        else:
            ax.scatter(self.data.x, y, color="blue", s=3, label=self.data.name)

        if samples > 0:
            saved_params = np.array(self.parameters)
            # Get a number of chains, chosen randomly, set the objective,
            # and plot the model.
            for pvec in self.pgen(ngen=samples):
                y, y_err, model = self._data_transform(
                    model=self.generative(pvec)
                )

                ax.plot(self.data.x, model, color="k", alpha=0.01)

            # put back saved_params
            self.setp(saved_params)

        # add the fit
        generative_plot = ax.plot(self.data.x, model, color="red", zorder=20)

        if parameter is None:
            return fig, ax

        # create an interactive plot in a Jupyter notebook.
        def f(val):
            if parameter is not None:
                parameter.value = float(val)
            y, y_err, model = self._data_transform(model=self.generative())
            generative_plot[0].set_data(self.data.x, model)
            fig.canvas.draw()

        import ipywidgets

        return fig, ax, ipywidgets.interact(f, val=float(parameter))

    def corner(self, **kwds):
        """
        Corner plot of the chains belonging to the Parameters.
        Requires the `corner` and `matplotlib` packages.

        Parameters
        ----------
        kwds: dict
            passed directly to the `corner.corner` function

        Returns
        -------
        fig : :class:`matplotlib.Figure` object.
        """
        import corner

        var_pars = self.varying_parameters()
        chain = np.array([par.chain for par in var_pars])
        labels = [par.name for par in var_pars]
        chain = chain.reshape(len(chain), -1).T
        kwds["labels"] = labels
        kwds["quantiles"] = [0.16, 0.5, 0.84]
        return corner.corner(chain, **kwds)


class GlobalObjective(Objective):
    """
    Global Objective function for simultaneous fitting with
    `refnx.analysis.CurveFitter`

    Parameters
    ----------
    objectives : list
        list of :class:`refnx.analysis.Objective` objects

    """

    def __init__(self, objectives):
        self.objectives = objectives
        weighted = [objective.weighted for objective in objectives]

        self._weighted = np.array(weighted, dtype=bool)

        if len(np.unique(self._weighted)) > 1:
            raise ValueError(
                "All the objectives must be either weighted or"
                " unweighted, you cannot have a mixture."
            )

    def __str__(self):
        s = ["{:_>80}".format("\n")]
        s.append("--Global Objective--")
        for obj in self.objectives:
            s.append(str(obj))
            s.append("\n")
        return "\n".join(s)

    def __repr__(self):
        return "GlobalObjective({0})".format(repr(self.objectives))

    @property
    def weighted(self):
        """
        **bool** do all the datasets have y_err, and are all the objectives
        wanting to use weights?

        """
        return self._weighted.all()

    @property
    def npoints(self):
        """
        **int** number of data points in all the objectives.

        """
        npoints = 0
        for objective in self.objectives:
            npoints += objective.npoints
        return npoints

    def residuals(self, pvals=None):
        """
        Concatenated residuals for each of the
        :meth:`refnx.analysis.Objective.residuals`.

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        residuals : np.ndarray
            Concatenated :meth:`refnx.analysis.Objective.residuals`

        """
        self.setp(pvals)

        residuals = []
        for objective in self.objectives:
            residual = objective.residuals()
            residuals.append(residual)

        return np.concatenate(residuals)

    @property
    def parameters(self):
        """
        :class:`refnx.analysis.Parameters` associated with all the objectives.

        """
        # TODO this is probably going to be slow.
        # cache and update strategy?
        p = Parameters(name="global fitting parameters")

        for objective in self.objectives:
            p.append(objective.parameters)

        return p

    def logp(self, pvals=None):
        """
        Calculate the log-prior of the system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters, optional
            values for the varying or entire set of parameters

        Returns
        -------
        logp : float
            log-prior probability

        """
        self.setp(pvals)

        logp = 0.0
        for objective in self.objectives:
            logp += objective.logp()
            # shortcut if one of the priors is impossible
            if not np.isfinite(logp):
                return -np.inf

        return logp

    def logl(self, pvals=None):
        """
        Calculate the log-likelhood of the system

        Parameters
        ----------
        pvals : array-like or refnx.analysis.Parameters
            values for the varying or entire set of parameters

        Returns
        -------
        logl : float
            log-likelihood probability

        """
        self.setp(pvals)
        logl = 0.0

        for objective in self.objectives:
            logl += objective.logl()

        return logl

    def plot(self, pvals=None, samples=0, parameter=None, fig=None):
        """
        Plot the data/model for all the objectives in the GlobalObjective.

        Matplotlib must be installed to use this method.

        Parameters
        ----------
        pvals : np.ndarray, optional
            Numeric values for the Parameter's that are varying
        samples: number, optional
            If the objective has been sampled, how many samples you wish to
            plot on the graph.
        parameter: refnx.analysis.Parameter, optional
            Creates an interactive plot for the Parameter in Jupyter. Requires
            ipywidgets be installed. Use with %matplotlib notebook/qt.
        fig: Figure instance, optional
            If `fig` is not supplied then a new figure is created. Otherwise
            the graph is created on the current axes on the supplied figure.

        Returns
        -------
        fig, ax : :class:`matplotlib.Figure`, :class:`matplotlib.Axes`
            `matplotlib` figure and axes objects.

        """

        self.setp(pvals)

        if fig is None:
            import matplotlib.pyplot as plt

            fig = plt.figure()
            ax = fig.add_subplot(111)
        else:
            ax = fig.gca()

        generative_plots = []

        if samples > 0:
            saved_params = np.array(self.parameters)

            # Get a number of chains, chosen randomly, set the objectives,
            # and plot the model.
            for pvec in self.pgen(ngen=samples):
                self.setp(pvec)

                for objective in self.objectives:
                    y, y_err, model = objective._data_transform(
                        model=objective.generative()
                    )

                    ax.plot(objective.data.x, model, color="k", alpha=0.01)

            # put back saved_params
            self.setp(saved_params)

        for objective in self.objectives:
            # add the data (in a transformed fashion)
            y, y_err, model = objective._data_transform(
                model=objective.generative()
            )

            if objective.weighted:
                ax.errorbar(
                    objective.data.x,
                    y,
                    y_err,
                    label=objective.data.name,
                    ms=3,
                    lw=0,
                    elinewidth=2,
                    marker="o",
                )
            else:
                ax.scatter(objective.data.x, y, label=objective.data.name)

            # add the fit
            generative_plots.append(
                ax.plot(objective.data.x, model, color="r", lw=1.5, zorder=20)[
                    0
                ]
            )

        if parameter is None:
            return fig, ax

        # create an interactive plot in a Jupyter notebook.
        def f(val):
            if parameter is not None:
                parameter.value = float(val)
            for i, objective in enumerate(self.objectives):
                y, y_err, model = objective._data_transform(
                    model=objective.generative()
                )

                generative_plots[i].set_data(objective.data.x, model)
            fig.canvas.draw()

        import ipywidgets

        return fig, ax, ipywidgets.interact(f, val=float(parameter))

        return fig, ax


class Transform:
    r"""
    Mathematical transforms of numeric data.

    Parameters
    ----------
    form : None or str
        One of:

            - 'lin'
                No transform is made
            - 'logY'
                log10 transform
            - 'YX4'
                YX**4 transform
            - 'YX2'
                YX**2 transform
            - None
                No transform is made

    Notes
    -----
    You ask for a transform to be carried out by calling the Transform object
    directly.

    >>> x = np.linspace(0.01, 0.1, 11)
    >>> y = np.linspace(100, 1000, 11)
    >>> y_err = np.sqrt(y)
    >>> t = Transform('logY')
    >>> ty, te = t(x, y, y_err)
    >>> ty
    array([2.        , 2.2787536 , 2.44715803, 2.56820172, 2.66275783,
           2.74036269, 2.80617997, 2.86332286, 2.91381385, 2.95904139,
           3.        ])

    """

    def __init__(self, form):
        types = [None, "lin", "logY", "YX4", "YX2"]
        self.form = None

        if form in types:
            self.form = form
        else:
            raise ValueError(
                "The form parameter must be one of [None, 'lin',"
                " 'logY', 'YX4', 'YX2']"
            )

    def __repr__(self):
        return "Transform({0})".format(repr(self.form))

    def __call__(self, x, y, y_err=None):
        """
        Calculate the transformed data

        Parameters
        ----------
        x : array-like
            x-values
        y : array-like
            y-values
        y_err : array-like
            Uncertainties in `y` (standard deviation)

        Returns
        -------
        yt, et : tuple
            The transformed data

        Examples
        --------
        >>> x = np.linspace(0.01, 0.1, 11)
        >>> y = np.linspace(100, 1000, 11)
        >>> y_err = np.sqrt(y)
        >>> t = Transform('logY')
        >>> ty, te = t(x, y, y_err)
        >>> ty
        array([2.        , 2.2787536 , 2.44715803, 2.56820172, 2.66275783,
               2.74036269, 2.80617997, 2.86332286, 2.91381385, 2.95904139,
               3.        ])

        """
        return self.__transform(x, y, y_err=y_err)

    def __transform(self, x, y, y_err=None):
        r"""
        Transform the data passed in

        Parameters
        ----------
        x : array-like

        y : array-like

        y_err : array-like

        Returns
        -------
        yt, et : tuple
            The transformed data
        """

        if y_err is None:
            etemp = np.ones_like(y)
        else:
            etemp = y_err

        if self.form in ["lin", None]:
            yt = np.copy(y)
            et = np.copy(etemp)
        elif self.form == "logY":
            yt, et = EP.EPlog10(y, etemp)
            if not np.isfinite(yt).all():
                warnings.warn(
                    "Some of the transformed data was non-finite."
                    " Please check your datasets for points with zero or"
                    " negative values.",
                    RuntimeWarning,
                )
        elif self.form == "YX4":
            yt = y * np.power(x, 4)
            et = etemp * np.power(x, 4)
        elif self.form == "YX2":
            yt = y * np.power(x, 2)
            et = etemp * np.power(x, 2)
        if y_err is None:
            return yt, None
        else:
            return yt, et


def pymc3_model(objective):
    """
    Creates a pymc3 model from an Objective.

    Requires theano and pymc3 be installed. This is an experimental feature.

    Parameters
    ----------
    objective: refnx.analysis.Objective

    Returns
    -------
    model: pymc3.Model

    Notes
    -----
    The varying parameters are renamed 'p0', 'p1', etc, as it's vital in pymc3
    that all parameters have their own unique name.

    """
    import pymc3 as pm
    import theano.tensor as tt
    from refnx._lib._pymc3 import _LogLikeWithGrad

    basic_model = pm.Model()

    pars = objective.varying_parameters()
    wrapped_pars = []
    with basic_model:
        # Priors for unknown model parameters
        for i, par in enumerate(pars):
            name = "p%d" % i
            p = _to_pymc3_distribution(name, par)
            wrapped_pars.append(p)

        # Expected value of outcome
        try:
            # Likelihood (sampling distribution) of observations
            pm.Normal(
                "y_obs",
                mu=objective.generative,
                sigma=objective.data.y_err,
                observed=objective.data.y,
            )
        except Exception:
            # Falling back, theano autodiff won't work on function object
            theta = tt.as_tensor_variable(wrapped_pars)
            logl = _LogLikeWithGrad(objective.logl)
            pm.Potential("log-likelihood", logl(theta))

    return basic_model


def _to_pymc3_distribution(name, par):
    """
    Create a pymc3 continuous distribution from a Bounds object.

    Parameters
    ----------
    name : str
        Name of parameter
    par : refnx.analysis.Parameter
        The parameter to wrap

    Returns
    -------
    d : pymc3.Distribution
        The pymc3 distribution

    """
    import pymc3 as pm
    import theano.tensor as T
    from theano.compile.ops import as_op

    dist = par.bounds
    # interval and both lb, ub are finite
    if isinstance(dist, Interval) and np.isfinite([dist.lb, dist.ub]).all():
        return pm.Uniform(name, dist.lb, dist.ub)
    # no bounds
    elif (
        isinstance(dist, Interval)
        and np.isneginf(dist.lb)
        and np.isinf(dist.lb)
    ):
        return pm.Flat(name)
    # half open uniform
    elif isinstance(dist, Interval) and not np.isfinite(dist.lb):
        return dist.ub - pm.HalfFlat(name)
    # half open uniform
    elif isinstance(dist, Interval) and not np.isfinite(dist.ub):
        return dist.lb + pm.HalfFlat(name)

    # it's a PDF
    if isinstance(dist, PDF):
        dist_gen = getattr(dist.rv, "dist", None)

        if isinstance(dist.rv, stats.rv_continuous):
            dist_gen = dist.rv

        if isinstance(dist_gen, type(stats.uniform)):
            if hasattr(dist.rv, "args"):
                p = pm.Uniform(
                    name, dist.rv.args[0], dist.rv.args[1] + dist.rv.args[0]
                )
            else:
                p = pm.Uniform(name, 0, 1)
            return p

        # norm from scipy.stats
        if isinstance(dist_gen, type(stats.norm)):
            if hasattr(dist.rv, "args"):
                p = pm.Normal(name, mu=dist.rv.args[0], sigma=dist.rv.args[1])
            else:
                p = pm.Normal(name, mu=0, sigma=1)
            return p

    # not open, uniform, or normal, so fall back to DensityDist.
    d = as_op(itypes=[T.dscalar], otypes=[T.dscalar])(dist.logp)
    r = as_op(itypes=[T.dscalar], otypes=[T.dscalar])(dist.rvs)
    p = pm.DensityDist(name, d, random=r)

    return p

Don't necessarily have to use Parameters, could use np.array # give the BaseObjective a default value, so that it can be used in a # GlobalObjective Set the parameters from pvals          Parameters         ----------         pvals : np.ndarray             Array containing the values to be tested. Negative log-likelihood function          Parameters         ----------         pvals : np.ndarray             Array containing the values to be tested.          Returns         -------         nll : float             negative log-likelihood Log-prior probability function          Parameters         ----------         pvals : np.ndarray             Array containing the values to be tested.          Returns         -------         logp : float             log-prior probability Log-likelihood probability function          Parameters         ----------         pvals : np.ndarray             Array containing the values to be tested.          Returns         -------         logl : float             log-likelihood probability. Log-posterior probability function          Parameters         ----------         pvals : np.ndarray             Array containing the values to be tested.          Returns         -------         logpost : float             log-probability.          Notes         -----         The log probability is the sum is the sum of the log-prior and         log-likelihood probabilities. Does not set the parameter attribute. Negative log-posterior function          Parameters         ----------         pvals : np.ndarray             Array containing the values to be tested.          Returns         -------         nlpost : float             negative log-posterior Returns         -------         varying_parameters : np.ndarray             The parameters varying in this objective function. Estimates a covariance matrix based on numerical differentiation         of either the negative log-likelihood or negative log-posterior         probabilities.          Parameters         ----------         target : str, {"nll", "nlpost"}          Returns         -------         covar : np.ndarray             The covariance matrix for the fitting system          Notes         -----         Estimation of a covariance matrix can be susceptible to numeric         instabilities. Critically evaluate the matrix before further use. # from statsmodels # the output from this for the test in test_objective.covar # is very similar to numdifftools.Hessian, or a chained version # of approx_derivative Objective function for using with curvefitters such as     `refnx.analysis.curvefitter.CurveFitter`.      Parameters     ----------     model : refnx.analysis.Model         the generative model function. One can also provide an object that         inherits `refnx.analysis.Model`.     data : refnx.dataset.Data1D         data to be analysed.     lnsigma : float or refnx.analysis.Parameter, optional         Used if the  experimental uncertainty (`data.y_err`) underestimated by         a constant fractional amount. The experimental uncertainty is modified         as:          `s_n**2 = y_err**2 + exp(lnsigma * 2) * model**2`          See `Objective.logl` for more details.     use_weights : bool         use experimental uncertainty in calculation of residuals and         logl, if available. If this is set to False, then you should also         set `self.lnsigma.vary = False`, it will have no effect on the fit.     transform : callable, optional         the model, data and data uncertainty are transformed by this         function before calculating the likelihood/residuals. Has the         signature `transform(data.x, y, y_err=None)`, returning the tuple         (`transformed_y, transformed_y_err`).     logp_extra : callable, optional         user specifiable log-probability term. This contribution is in         addition to the log-prior term of the `model` parameters, and         `model.logp`, as well as the log-likelihood of the `data`. Has         signature:         `logp_extra(model, data)`. The `model` will already possess         updated parameters. Beware of including the same log-probability         terms more than once.     auxiliary_params : {sequence, Parameters}, optional         Extra Parameter objects that are involved with curvefitting, but         aren't directly included as part of the `model`. See notes for more         details.     name : str         Name for the objective.      Notes     -----     For parallelisation `logp_extra` needs to be picklable.      `auxiliary_params` are included in calculating the `Objective.logp`     term, are present in `Objective.varying_parameters()`, and are modified by     Curvefitter during an analysis. Their main purpose is to aid in making     constraints in models. # should be a Data1D instance # dataset name **bool** Does the data have weights (`data.y_err`), and is the         objective using them? **int** the number of points in the dataset. Returns         -------         varying_parameters : refnx.analysis.Parameters             The varying Parameter objects allowed to vary during the fit. # create and return a Parameters object because it has the # __array__ method, which allows one to quickly get numerical values. Returns         -------         y, y_err, model: tuple of np.ndarray             The y data, its uncertainties, and the model, all put through the              transform. Calculate the generative (dependent variable) function associated with         the model.          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         model : np.ndarray Calculates the residuals for a given fitting system.          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         residuals : np.ndarray             Residuals, `(data.y - model) / y_err`. # TODO add in varying parameter residuals? (z-scores...) Calculates the chi-squared value for a given fitting system.          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         chisqr : np.ndarray             Chi-squared value, `np.sum(residuals**2)`. # TODO reduced chisqr? include z-scores for parameters? DOF? :class:`refnx.analysis.Parameters`, all the Parameters contained in the         fitting system. Set the parameters from pvals.          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters # set here rather than delegating to a Parameters # object, because it may not necessarily be a # Parameters object # values supplied are enough to specify all parameter values # even those that are repeated Calculate the prior transform of the system.          Transforms uniform random variates in the unit hypercube,         `u ~ uniform[0.0, 1.0)`, to the parameter space of interest, according         to the priors on the varying parameters.          Parameters         ----------         u : array-like             Size of the varying parameters          Returns         -------         pvals : array-like             Scaled parameter values          Notes         -----         If a parameter has bounds, `x ~ Unif[-10, 10)` then the scaling from         `u` to `x` is done as follows:          .. code-block:: python              x = 2. * u - 1.  # scale and shift to [-1., 1.)             x *= 10.  # scale to [-10., 10.) Calculate the log-prior of the system          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         logp : float             log-prior probability          Notes         -----         The log-prior is calculated as:          .. code-block:: python              logp = np.sum(param.logp() for param in                              self.varying_parameters()) Calculate the log-likelhood of the system          The major component of the log-likelhood probability is from the data.         Extra potential terms are added on from the Model, `self.model.logp`,         and the user specifiable `logp_extra` function.          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         logl : float             log-likelihood probability          Notes         -----         The log-likelihood is calculated as:          .. code-block:: python              logl = -0.5 * np.sum(((y - model) / s_n)**2                                  + np.log(2 * pi * s_n**2))             logp += self.model.logp()             logp += self.logp_extra(self.model, self.data)          where          .. code-block:: python              s_n**2 = y_err**2 + exp(2 * lnsigma) * model**2 # TODO do something sensible if data isn't weighted # nans play havoc # add on extra 'potential' terms from the model. Negative log-likelihood function          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         nll : float             negative log-likelihood Calculate the log-probability of the curvefitting system          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         logpost : float             log-probability          Notes         -----         The overall log-probability is the sum of the log-prior and         log-likelihood. The log-likelihood is not calculated if the log-prior         is impossible (`logp == -np.inf`). # only calculate the probability if the parameters have finite # log-prior Estimates the covariance matrix of the Objective.          Parameters         ----------         target : {"residuals", "nll", "nlpost"}             Specifies what approach should be used to estimate covariance.          Returns         -------         covar : np.ndarray             Covariance matrix          Notes         -----         For most purposes the Jacobian of the `'residuals'` should be used to         calculate the covariance matrix, estimated as J.T x J.         If an Objective cannot calculate residuals then the covariance matrix         can be estimated by inverting a Hessian matrix created from either the         `'nll'` or `'nlpost'` methods.         The default `'residuals'` approach falls back to `'nll'` if a problem         is experienced.         The default `'residuals'` setting is preferred as the other settings         can sometimes experience instabilities during Hessian estimation with         numerical differentiation. # fallback to "nll" # we should be able to calculate a Jacobian for a parameter whose # value is zero. However, the scaling approach won't work. # This will force Jacobian calculation by unscaled parameters # using approx_derivative changes the state of the objective # parameters have to make sure they're set at the end # need to create this because GlobalObjective may not have # access to all the datapoints being fitted. # covar = J.T x J # from scipy.optimize.minpack.py # eliminates singular parameters # unwind the scaling. # scale by reduced chi2 if experimental uncertainties weren't used. Yield random parameter vectors from the MCMC samples. The objective         state is not altered.          Parameters         ----------         ngen : int, optional             the number of samples to yield. The actual number of samples             yielded is `min(ngen, chain.size)`         nburn : int, optional             discard this many steps from the start of the chain         nthin : int, optional             only accept every `nthin` samples from the chain          Yields         ------         pvec : np.ndarray             A randomly chosen parameter vector Plot the data/model.          Requires matplotlib be installed.          Parameters         ----------         pvals : np.ndarray, optional             Numeric values for the Parameter's that are varying         samples: number             If the objective has been sampled, how many samples you wish to             plot on the graph.         parameter: refnx.analysis.Parameter             Creates an interactive plot for the Parameter in Jupyter. Requires             ipywidgets be installed. Use with %matplotlib notebook/qt.         fig: Figure instance, optional             If `fig` is not supplied then a new figure is created. Otherwise             the graph is created on the current axes on the supplied figure.          Returns         -------         fig, ax : :class:`matplotlib.Figure`, :class:`matplotlib.Axes`             `matplotlib` figure and axes objects. # add the data (in a transformed fashion) # Get a number of chains, chosen randomly, set the objective, # and plot the model. # put back saved_params # add the fit # create an interactive plot in a Jupyter notebook. Corner plot of the chains belonging to the Parameters.         Requires the `corner` and `matplotlib` packages.          Parameters         ----------         kwds: dict             passed directly to the `corner.corner` function          Returns         -------         fig : :class:`matplotlib.Figure` object. Global Objective function for simultaneous fitting with     `refnx.analysis.CurveFitter`      Parameters     ----------     objectives : list         list of :class:`refnx.analysis.Objective` objects **bool** do all the datasets have y_err, and are all the objectives         wanting to use weights? **int** number of data points in all the objectives. Concatenated residuals for each of the         :meth:`refnx.analysis.Objective.residuals`.          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         residuals : np.ndarray             Concatenated :meth:`refnx.analysis.Objective.residuals` :class:`refnx.analysis.Parameters` associated with all the objectives. # TODO this is probably going to be slow. # cache and update strategy? Calculate the log-prior of the system          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters, optional             values for the varying or entire set of parameters          Returns         -------         logp : float             log-prior probability # shortcut if one of the priors is impossible Calculate the log-likelhood of the system          Parameters         ----------         pvals : array-like or refnx.analysis.Parameters             values for the varying or entire set of parameters          Returns         -------         logl : float             log-likelihood probability Plot the data/model for all the objectives in the GlobalObjective.          Matplotlib must be installed to use this method.          Parameters         ----------         pvals : np.ndarray, optional             Numeric values for the Parameter's that are varying         samples: number, optional             If the objective has been sampled, how many samples you wish to             plot on the graph.         parameter: refnx.analysis.Parameter, optional             Creates an interactive plot for the Parameter in Jupyter. Requires             ipywidgets be installed. Use with %matplotlib notebook/qt.         fig: Figure instance, optional             If `fig` is not supplied then a new figure is created. Otherwise             the graph is created on the current axes on the supplied figure.          Returns         -------         fig, ax : :class:`matplotlib.Figure`, :class:`matplotlib.Axes`             `matplotlib` figure and axes objects. # Get a number of chains, chosen randomly, set the objectives, # and plot the model. # put back saved_params # add the data (in a transformed fashion) # add the fit # create an interactive plot in a Jupyter notebook. Mathematical transforms of numeric data.      Parameters     ----------     form : None or str         One of:              - 'lin'                 No transform is made             - 'logY'                 log10 transform             - 'YX4'                 YX**4 transform             - 'YX2'                 YX**2 transform             - None                 No transform is made      Notes     -----     You ask for a transform to be carried out by calling the Transform object     directly.      >>> x = np.linspace(0.01, 0.1, 11)     >>> y = np.linspace(100, 1000, 11)     >>> y_err = np.sqrt(y)     >>> t = Transform('logY')     >>> ty, te = t(x, y, y_err)     >>> ty     array([2.        , 2.2787536 , 2.44715803, 2.56820172, 2.66275783,            2.74036269, 2.80617997, 2.86332286, 2.91381385, 2.95904139,            3.        ]) Calculate the transformed data          Parameters         ----------         x : array-like             x-values         y : array-like             y-values         y_err : array-like             Uncertainties in `y` (standard deviation)          Returns         -------         yt, et : tuple             The transformed data          Examples         --------         >>> x = np.linspace(0.01, 0.1, 11)         >>> y = np.linspace(100, 1000, 11)         >>> y_err = np.sqrt(y)         >>> t = Transform('logY')         >>> ty, te = t(x, y, y_err)         >>> ty         array([2.        , 2.2787536 , 2.44715803, 2.56820172, 2.66275783,                2.74036269, 2.80617997, 2.86332286, 2.91381385, 2.95904139,                3.        ]) Transform the data passed in          Parameters         ----------         x : array-like          y : array-like          y_err : array-like          Returns         -------         yt, et : tuple             The transformed data Creates a pymc3 model from an Objective.      Requires theano and pymc3 be installed. This is an experimental feature.      Parameters     ----------     objective: refnx.analysis.Objective      Returns     -------     model: pymc3.Model      Notes     -----     The varying parameters are renamed 'p0', 'p1', etc, as it's vital in pymc3     that all parameters have their own unique name. # Priors for unknown model parameters # Expected value of outcome # Likelihood (sampling distribution) of observations # Falling back, theano autodiff won't work on function object Create a pymc3 continuous distribution from a Bounds object.      Parameters     ----------     name : str         Name of parameter     par : refnx.analysis.Parameter         The parameter to wrap      Returns     -------     d : pymc3.Distribution         The pymc3 distribution # interval and both lb, ub are finite # no bounds # half open uniform # half open uniform # it's a PDF # norm from scipy.stats # not open, uniform, or normal, so fall back to DensityDist.