""" Standard cost functions to minimize for statistical fits. We provide these for convenience, so that you do not have to write your own for standard fits. The cost functions optionally use Numba to accelerate some calculations, if Numba is installed. What to use when ---------------- - Fit a normalised probability density to data - Data are not binned: :class:`UnbinnedNLL` - Data are binned: :class:`BinnedNLL`, also supports histogram of weighted samples - Fit a density to data, density is not normalised - Data are not binned: :class:`ExtendedUnbinnedNLL` - Data are binned: :class:`ExtendedBinnedNLL`, also supports histogram of weighted samples - Fit a template to binned data with bin-wise uncertainties on the template: :class:`Template`, which also supports weighted data and weighted templates - Fit of a function f(x) to (x, y, yerror) pairs with normal-distributed fluctuations. x is one- or multi-dimensional, y is one-dimensional. - y values contain no outliers: :class:`LeastSquares` - y values contain outliers: :class:`LeastSquares` with loss function set to "soft_l1" - Include constraints from external fits or apply regularisation: :class:`NormalConstraint` Combining cost functions ------------------------ All cost functions can be added, which generates a new combined cost function. Parameters with the same name are shared between component cost functions. Use this to constrain one or several parameters with different data sets and using different statistical models for each data set. Gaussian penalty terms can also be added to the cost function to introduce external knowledge about a parameter. Notes ----- The cost functions defined here have been optimized with knowledge about implementation details of Minuit to give the highest accucary and the most robust results, so they should perform well. If you have trouble with your own implementations, try these. The binned versions of the log-likelihood fits support weighted samples. For each bin of the histogram, the sum of weights and the sum of squared weights is needed then, see class documentation for details. """ from .util import ( describe, make_func_code, merge_signatures, PerformanceWarning, _smart_sampling, ) import numpy as np from collections.abc import Sequence import abc import typing as _tp import warnings CHISQUARE = 1.0 NEGATIVE_LOG_LIKELIHOOD = 0.5 # correct ArrayLike from numpy.typing generates horrible looking signatures # in python's help(), so we use this as a workaround _ArrayLike = _tp.Sequence def _safe_log(x): # guard against x = 0 return np.log(x + 1e-323) def _unbinned_nll(x): return -np.sum(_safe_log(x)) def _z_squared(y, ye, ym): z = (y - ym) / ye return z * z def _soft_l1_loss(z_sqr): return np.sum(2 * (np.sqrt(1 + z_sqr) - 1)) def _soft_l1_cost(y, ye, ym): return _soft_l1_loss(_z_squared(y, ye, ym)) def _replace_none(x, replacement): if x is None: return replacement return x class BohmZechTransform: """ Apply Bohm-Zech transform. See Bohm and Zech, NIMA 748 (2014) 1-6. :meta private: """ def __init__(self, val: _ArrayLike, var: _ArrayLike): """ Initialize transformer with data value and variance. Parameters ---------- val : array-like Observed values. var : array-like Estimated variance of observed values. """ val, var = np.atleast_1d(val, var) self._scale = np.ones_like(val) np.divide(val, var, out=self._scale, where=var > 0) self._obs = val * self._scale def __call__( self, val: _ArrayLike, var: _ArrayLike = None ) -> _tp.Union[ _tp.Tuple[np.ndarray, np.ndarray], _tp.Tuple[np.ndarray, np.ndarray, np.ndarray] ]: """ Return precomputed scaled data and scaled prediction. Parameters ---------- val : array-like Predicted values. var : array-like, optional Predicted variance. Returns ------- (obs, pred) or (obs, pred, pred_var) """ s = self._scale if var is None: return self._obs, val * s return self._obs, val * s, var * s**2 def chi2(y: _ArrayLike[float], ye: _ArrayLike[float], ym: _ArrayLike[float]) -> float: """ Compute (potentially) chi2-distributed cost. The value returned by this function is chi2-distributed, if the observed values are normally distributed around the expected values with the provided standard deviations. Parameters ---------- y : array-like Observed values. ye : array-like Uncertainties of values. ym : array-like Expected values. Returns ------- float Const function value. """ y, ye, ym = np.atleast_1d(y, ye, ym) return np.sum(_z_squared(y, ye, ym)) def multinominal_chi2(n: _ArrayLike[float], mu: _ArrayLike[float]) -> float: """ Compute asymptotically chi2-distributed cost for binomially-distributed data. See Baker & Cousins, NIM 221 (1984) 437-442. Parameters ---------- n : array-like Observed counts. mu : array-like Expected counts per bin. Must satisfy sum(mu) == sum(n). Returns ------- float Cost function value. Notes ----- The implementation makes the result asymptotically chi2-distributed and keeps the sum small near the minimum, which helps to maximise the numerical accuracy for Minuit. """ n, mu = np.atleast_1d(n, mu) return 2 * np.sum(n * (_safe_log(n) - _safe_log(mu))) def poisson_chi2(n: _ArrayLike[float], mu: _ArrayLike[float]) -> float: """ Compute asymptotically chi2-distributed cost for Poisson-distributed data. See Baker & Cousins, NIM 221 (1984) 437-442. Parameters ---------- n : array-like Observed counts. mu : array-like Expected counts. Returns ------- float Cost function value. Notes ----- The implementation makes the result asymptotically chi2-distributed, which helps to maximise the numerical accuracy for Minuit. """ n, mu = np.atleast_1d(n, mu) return 2 * np.sum(mu - n + n * (_safe_log(n) - _safe_log(mu))) def template_chi2_jsc( n: _ArrayLike[float], mu: _ArrayLike[float], mu_var: _ArrayLike[float] ) -> float: """ Compute asymptotically chi2-distributed cost for a template fit. J.S. Conway, PHYSTAT 2011, https://doi.org/10.48550/arXiv.1103.0354 Parameters ---------- n : array-like Observed counts. mu : array-like Expected counts. This is the sum of the normalised templates scaled with the component yields. Must be positive everywhere. mu_var : array-like Expected variance of mu. Must be positive everywhere. Returns ------- float Asymptotically chi-square-distributed test statistic. Notes ----- The implementation deviates slightly from the paper by making the result asymptotically chi2-distributed, which helps to maximise the numerical accuracy for Minuit. """ n, mu, mu_var = np.atleast_1d(n, mu, mu_var) beta_var = mu_var / mu**2 # Eq. 15 from https://doi.org/10.48550/arXiv.2206.12346 p = 0.5 - 0.5 * mu * beta_var beta = p + np.sqrt(p**2 + n * beta_var) return poisson_chi2(n, mu * beta) + np.sum( # type:ignore (beta - 1) ** 2 / beta_var ) def template_chi2_da( n: _ArrayLike[float], mu: _ArrayLike[float], mu_var: _ArrayLike[float] ) -> float: """ Compute asymptotically chi2-distributed cost for a template fit. H.P. Dembinski, A. Abdelmotteleb, https://doi.org/10.48550/arXiv.2206.12346 Parameters ---------- n : array-like Observed counts. mu : array-like Expected counts. This is the sum of the normalised templates scaled with the component yields. mu_var : array-like Expected variance of mu. Must be positive everywhere. Returns ------- float Asymptotically chi-square-distributed test statistic. """ n, mu, mu_var = np.atleast_1d(n, mu, mu_var) k = mu**2 / mu_var beta = (n + k) / (mu + k) return poisson_chi2(n, mu * beta) + poisson_chi2(k, k * beta) # type:ignore def template_nll_asy( n: _ArrayLike[float], mu: _ArrayLike[float], mu_var: _ArrayLike[float] ) -> float: """ Compute marginalized negative log-likelikihood for a template fit. This is the negative logarithm of equation 3.15 of the paper by C.A. Argüelles, A. Schneider, T. Yuan, https://doi.org/10.1007/JHEP06(2019)030. The authors use a Bayesian approach and integrate over the nuisance parameters. Like the other Barlow-Beeston-lite methods, this is an approximation. The resulting likelihood cannot be turned into an asymptotically chi-square distributed test statistic as detailed in Baker & Cousins, NIM 221 (1984) 437-442. Parameters ---------- n : array-like Observed counts. mu : array-like Expected counts. This is the sum of the normalised templates scaled with the component yields. mu_var : array-like Expected variance of mu. Must be positive everywhere. Returns ------- float Negative log-likelihood function value. """ from scipy.special import loggamma as lg n, mu, mu_var = np.atleast_1d(n, mu, mu_var) alpha = mu**2 / mu_var + 1 beta = mu / mu_var return -np.sum( alpha * np.log(beta) + lg(n + alpha) - (lg(n + 1) + (n + alpha) * np.log(1 + beta) + lg(alpha)) ) # If numba is available, use it to accelerate computations in float32 and float64 # precision. Fall back to plain numpy for float128 which is not currently supported # by numba. try: from numba import njit as _njit from numba.extending import overload as _overload @_overload(_safe_log, inline="always") def _ol_safe_log(x): return _safe_log # pragma: no cover @_overload(_z_squared, inline="always") def _ol_z_squared(y, ye, ym): return _z_squared # pragma: no cover _unbinned_nll_np = _unbinned_nll _unbinned_nll_nb = _njit( nogil=True, cache=True, error_model="numpy", )(_unbinned_nll_np) def _unbinned_nll(x): if x.dtype in (np.float32, np.float64): return _unbinned_nll_nb(x) # fallback to numpy for float128 return _unbinned_nll_np(x) _multinominal_chi2_np = multinominal_chi2 _multinominal_chi2_nb = _njit( nogil=True, cache=True, error_model="numpy", )(_multinominal_chi2_np) def multinominal_chi2(n: _ArrayLike[float], mu: _ArrayLike[float]) -> float: # noqa n, mu = np.atleast_1d(n, mu) # type:ignore if mu.dtype in (np.float32, np.float64): # type:ignore return _multinominal_chi2_nb(n, mu) # fallback to numpy for float128 return _multinominal_chi2_np(n, mu) multinominal_chi2.__doc__ = _multinominal_chi2_np.__doc__ _poisson_chi2_np = poisson_chi2 _poisson_chi2_nb = _njit( nogil=True, cache=True, error_model="numpy", )(_poisson_chi2_np) def poisson_chi2(n: _ArrayLike[float], mu: _ArrayLike[float]) -> float: # noqa n, mu = np.atleast_1d(n, mu) # type:ignore if mu.dtype in (np.float32, np.float64): # type:ignore return _poisson_chi2_nb(n, mu) # fallback to numpy for float128 return _poisson_chi2_np(n, mu) poisson_chi2.__doc__ = _poisson_chi2_np.__doc__ _chi2_np = chi2 _chi2_nb = _njit( nogil=True, cache=True, error_model="numpy", )(_chi2_np) def chi2( y: _ArrayLike[float], ye: _ArrayLike[float], ym: _ArrayLike[float] ) -> float: # noqa y, ye, ym = np.atleast_1d(y, ye, ym) # type:ignore if ym.dtype in (np.float32, np.float64): # type:ignore return _chi2_nb(y, ye, ym) # fallback to numpy for float128 return _chi2_np(y, ye, ym) chi2.__doc__ = _chi2_np.__doc__ _soft_l1_loss_np = _soft_l1_loss _soft_l1_loss_nb = _njit( nogil=True, cache=True, error_model="numpy", )(_soft_l1_loss_np) def _soft_l1_loss(z_sqr): if z_sqr.dtype in (np.float32, np.float64): return _soft_l1_loss_nb(z_sqr) # fallback to numpy for float128 return _soft_l1_loss_np(z_sqr) @_overload(_soft_l1_loss, inline="always") def _ol_soft_l1_loss(z_sqr): return _soft_l1_loss_np # pragma: no cover _soft_l1_cost_np = _soft_l1_cost _soft_l1_cost_nb = _njit( nogil=True, cache=True, error_model="numpy", )(_soft_l1_cost_np) def _soft_l1_cost(y, ye, ym): if ym.dtype in (np.float32, np.float64): return _soft_l1_cost_nb(y, ye, ym) # fallback to numpy for float128 return _soft_l1_cost_np(y, ye, ym) except ModuleNotFoundError: pass class Cost(abc.ABC): """ Base class for all cost functions. :meta private: """ __slots__ = ("_func_code", "_verbose") @property def errordef(self): """ For internal use. :meta private: """ return self._errordef() def _errordef(self): return CHISQUARE @property def func_code(self): """ For internal use. :meta private: """ return self._func_code @property def ndata(self): """ Return number of points in least-squares fits or bins in a binned fit. Infinity is returned if the cost function is unbinned. This is used by Minuit to compute the reduced chi2, a goodness-of-fit estimate. """ return self._ndata() @abc.abstractmethod def _ndata(self): NotImplemented # pragma: no cover @property def verbose(self): """ Access verbosity level. Set this to 1 to print all function calls with input and output. """ return self._verbose @verbose.setter def verbose(self, value: int): self._verbose = value def __init__(self, args: _tp.Tuple[str, ...], verbose: int): """For internal use.""" self._func_code = make_func_code(args) self._verbose = verbose def __add__(self, rhs): """ Add two cost functions to form a combined cost function. Returns ------- CostSum """ return CostSum(self, rhs) def __radd__(self, lhs): """ Add two cost functions to form a combined cost function. Returns ------- CostSum """ return CostSum(lhs, self) def __call__(self, *args): """ Evaluate the cost function. If verbose >= 1, print arguments and result. Parameters ---------- *args : float Parameter values. Returns ------- float """ r = self._call(args) if self.verbose >= 1: print(args, "->", r) return r class Constant(Cost): """ Cost function that represents a constant. If your cost function produces results that are far away from O(1), adding a constant that brings the value closer to zero may improve the numerical stability. """ __slots__ = "value" def __init__(self, value: float): """Initialize constant with a value.""" self.value = value super().__init__((), False) def _ndata(self): return 0 def _call(self, args): return self.value class CostSum(Cost, Sequence): """Sum of cost functions. Users do not need to create objects of this class themselves. They should just add cost functions, for example:: nll = UnbinnedNLL(...) lsq = LeastSquares(...) ncs = NormalConstraint(...) csum = nll + lsq + ncs CostSum is used to combine data from different experiments or to combine normal cost functions with penalty terms (see NormalConstraint). The parameters of CostSum are the union of all parameters of its constituents. Supports the sequence protocol to access the constituents. Warnings -------- CostSum does not support cost functions that accept a parameter array, because the function signature does not allow one to determine how many parameters are accepted by the function and which parameters overlap between different cost functions. """ __slots__ = "_items", "_maps" def __init__(self, *items): """Initialize with cost functions. Parameters ---------- *items : Cost Cost functions. May also be other CostSum functions. """ self._items = [] for item in items: if isinstance(item, CostSum): self._items += item._items elif isinstance(item, (int, float)): if item != 0: self._items.append(Constant(item)) else: self._items.append(item) args, self._maps = merge_signatures(self._items) super().__init__(args, max(c.verbose for c in self._items)) def _split(self, args): for component, cmap in zip(self._items, self._maps): component_args = tuple(args[i] for i in cmap) yield component, component_args def _call(self, args): r = 0.0 for comp, cargs in self._split(args): r += comp._call(cargs) / comp.errordef return r def _ndata(self): return sum(c.ndata for c in self._items) def __len__(self): """Return number of constituent cost functions.""" return self._items.__len__() def __getitem__(self, key): """Get constituent cost function by index.""" return self._items.__getitem__(key) def visualize( self, args: _ArrayLike, component_kwargs: _tp.Dict[int, _tp.Dict] = None ): """ Visualize data and model agreement (requires matplotlib). The visualization is drawn with matplotlib.pyplot into the current figure. Subplots are created to visualize each part of the cost function, the figure height is increased accordingly. Parts without a visualize method are silently ignored. Parameters ---------- args : array-like Parameter values. component_kwargs : dict of dicts, optional Dict that maps an index to dict of keyword arguments. This can be used to pass keyword arguments to a visualize method of a component with that index. **kwargs : Other keyword arguments are forwarded to all components. """ from matplotlib import pyplot as plt args = np.atleast_1d(args) n = sum(hasattr(comp, "visualize") for comp in self) w, h = plt.rcParams["figure.figsize"] fig, ax = plt.subplots(1, n, figsize=(w, h * n)) if component_kwargs is None: component_kwargs = {} i = 0 for k, (comp, cargs) in enumerate(self._split(args)): if not hasattr(comp, "visualize"): continue kwargs = component_kwargs.get(k, {}) plt.sca(ax[i]) comp.visualize(cargs, **kwargs) i += 1 class MaskedCost(Cost): """ Base class for cost functions that support data masking. :meta private: """ __slots__ = "_data", "_mask", "_masked" def __init__(self, args, data, verbose): """For internal use.""" self._data = data self._mask = None self._update_cache() Cost.__init__(self, args, verbose) @property def mask(self): """Boolean array, array of indices, or None. If not None, only values selected by the mask are considered. The mask acts on the first dimension of a value array, i.e. values[mask]. Default is None. """ return self._mask @mask.setter def mask(self, mask): self._mask = None if mask is None else np.asarray(mask) self._update_cache() @property def data(self): """Return data samples.""" return self._data @data.setter def data(self, value): self._data[...] = value self._update_cache() def _update_cache(self): self._masked = self._data[_replace_none(self._mask, ...)] class UnbinnedCost(MaskedCost): """ Base class for unbinned cost functions. :meta private: """ __slots__ = "_model", "_log" def __init__(self, data, model: _tp.Callable, verbose: int, log: bool): """For internal use.""" self._model = model self._log = log super().__init__(describe(model)[1:], _norm(data), verbose) @abc.abstractproperty def pdf(self): """Get probability density model.""" ... # pragma: no cover @abc.abstractproperty def scaled_pdf(self): """Get number density model.""" ... # pragma: no cover def _ndata(self): # unbinned likelihoods have infinite degrees of freedom return np.inf def visualize(self, args: _ArrayLike, model_points: int = 0): """ Visualize data and model agreement (requires matplotlib). The visualization is drawn with matplotlib.pyplot into the current axes. Parameters ---------- args : array-like Parameter values. model_points : int, optional How many points to use to draw the model. Default is 0, in this case an smart sampling algorithm selects the number of points. """ from matplotlib import pyplot as plt if self.data.ndim > 1: raise ValueError("visualize is not implemented for multi-dimensional data") n, xe = np.histogram(self.data, bins=50) cx = 0.5 * (xe[1:] + xe[:-1]) plt.errorbar(cx, n, n**0.5, fmt="ok") if model_points > 0: if xe[0] > 0 and xe[-1] / xe[0] > 1e2: xm = np.geomspace(xe[0], xe[-1], model_points) else: xm = np.linspace(xe[0], xe[-1], model_points) ym = self.scaled_pdf(xm, *args) else: xm, ym = _smart_sampling(lambda x: self.scaled_pdf(x, *args), xe[0], xe[-1]) dx = xe[1] - xe[0] plt.fill_between(xm, 0, ym * dx, fc="C0") class UnbinnedNLL(UnbinnedCost): """Unbinned negative log-likelihood. Use this if only the shape of the fitted PDF is of interest and the original unbinned data is available. The data can be one- or multi-dimensional. """ __slots__ = () @property def pdf(self): """Get probability density model.""" if self._log: return lambda *args: np.exp(self._model(*args)) return self._model @property def scaled_pdf(self): """Get number density model.""" scale = np.prod(self.data.shape) if self._log: return lambda *args: scale * np.exp(self._model(*args)) return lambda *args: scale * self._model(*args) def __init__( self, data: _ArrayLike, pdf: _tp.Callable, verbose: int = 0, log: bool = False, ): """ Initialize UnbinnedNLL with data and model. Parameters ---------- data : array-like Sample of observations. If the observations are multidimensional, data must have the shape (D, N), where D is the number of dimensions and N the number of data points. pdf : callable Probability density function of the form f(data, par0, [par1, ...]), where data is the data sample and par0, ... are model parameters. If the data are multivariate, data passed to f has shape (D, N), where D is the number of dimensions and N the number of data points. verbose : int, optional Verbosity level. 0: is no output (default). 1: print current args and negative log-likelihood value. log : bool, optional Distributions of the exponential family (normal, exponential, poisson, ...) allow one to compute the logarithm of the pdf directly, which is more accurate and efficient than effectively doing ``log(exp(logpdf))``. Set this to True, if the model returns the logarithm of the pdf instead of the pdf. Default is False. """ super().__init__(data, pdf, verbose, log) def _call(self, args): data = self._masked x = self._model(data, *args) x = _normalize_model_output(x) if self._log: return -2.0 * np.sum(x) return 2.0 * _unbinned_nll(x) class ExtendedUnbinnedNLL(UnbinnedCost): """Unbinned extended negative log-likelihood. Use this if shape and normalization of the fitted PDF are of interest and the original unbinned data is available. The data can be one- or multi-dimensional. """ __slots__ = () @property def pdf(self): """Get probability density model.""" if self._log: def fn(*args): n, x = self._model(*args) return np.exp(x) / n else: def fn(*args): n, x = self._model(*args) return x / n return fn @property def scaled_pdf(self): """Get density model.""" if self._log: return lambda *args: np.exp(self._model(*args)[1]) return lambda *args: self._model(*args)[1] def __init__( self, data: _ArrayLike, scaled_pdf: _tp.Callable, verbose: int = 0, log: bool = False, ): """ Initialize cost function with data and model. Parameters ---------- data : array-like Sample of observations. If the observations are multidimensional, data must have the shape (D, N), where D is the number of dimensions and N the number of data points. scaled_pdf : callable Density function of the form f(data, par0, [par1, ...]), where data is the sample and par0, ... are model parameters. Must return a tuple (, ). The first value is the density integrated over the data window, the interval that we consider for the fit. For example, if the data are exponentially distributed, but we fit only the interval (0, 5), then the first value is the density integrated from 0 to 5. If the data are multivariate, data passed to f has shape (D, N), where D is the number of dimensions and N the number of data points. verbose : int, optional Verbosity level. 0: is no output (default). 1: print current args and negative log-likelihood value. log : bool, optional Distributions of the exponential family (normal, exponential, poisson, ...) allow one to compute the logarithm of the pdf directly, which is more accurate and efficient than effectively doing ``log(exp(logpdf))``. Set this to True, if the model returns the logarithm of the density as the second argument instead of the density. Default is False. """ super().__init__(data, scaled_pdf, verbose, log) def _call(self, args): data = self._masked ns, x = self._model(data, *args) x = _normalize_model_output( x, "Model should return numpy array in second position" ) if self._log: return 2 * (ns - np.sum(x)) return 2 * (ns + _unbinned_nll(x)) class BinnedCost(MaskedCost): """ Base class for binned cost functions. :meta private: """ __slots__ = "_xe", "_ndim", "_bztrafo" n = MaskedCost.data @property def xe(self): """Access bin edges.""" return self._xe def __init__(self, args, n, xe, verbose, *updater): """For internal use.""" if not isinstance(xe, _tp.Iterable): raise ValueError("xe must be iterable") shape = _shape_from_xe(xe) self._ndim = len(shape) self._xe = _norm(xe) if self._ndim == 1 else tuple(_norm(xei) for xei in xe) n = _norm(n) is_weighted = n.ndim > self._ndim if n.ndim != (self._ndim + int(is_weighted)): raise ValueError("n must either have same dimension as xe or one extra") for i, xei in enumerate([self._xe] if self._ndim == 1 else self._xe): if len(xei) != n.shape[i] + 1: raise ValueError( f"n and xe have incompatible shapes along dimension {i}, " "xe must be longer by one element along each dimension" ) if is_weighted: if n.shape[-1] != 2: raise ValueError("n must have shape (..., 2)") self._bztrafo = BohmZechTransform(n[..., 0], n[..., 1]) else: self._bztrafo = None super().__init__(args, n, verbose) def _ndata(self): return np.prod(self._masked.shape[: self._ndim]) def _update_cache(self): super()._update_cache() if self._bztrafo: ma = _replace_none(self._mask, ...) self._bztrafo = BohmZechTransform(self._data[ma, 0], self._data[ma, 1]) def visualize(self, args: _tp.Sequence[float]): """ Visualize data and model agreement (requires matplotlib). The visualization is drawn with matplotlib.pyplot into the current axes. Parameters ---------- args : sequence of float Parameter values. """ from matplotlib import pyplot as plt if self._ndim > 1: raise ValueError("visualize is not implemented for multi-dimensional data") n = self._masked[..., 0] if self._bztrafo else self._masked ne = (self._masked[..., 1] if self._bztrafo else self._masked) ** 0.5 xe = self.xe cx = 0.5 * (xe[1:] + xe[:-1]) if self.mask is not None: cx = cx[self.mask] plt.errorbar(cx, n, ne, fmt="ok") mu = self._pred(args) plt.stairs(mu, xe, fill=True, color="C0") @abc.abstractmethod def _pred(self, args: _tp.Sequence[float]) -> np.ndarray: ... # pragma: no cover class BinnedCostWithModel(BinnedCost): """ Base class for binned cost functions with parametric model. :meta private: """ __slots__ = "_xe_shape", "_model", "_model_arg" def __init__(self, n, xe, model, verbose): """For internal use.""" self._model = model super().__init__(describe(model)[1:], n, xe, verbose) if self._ndim == 1: self._xe_shape = None self._model_arg = _norm(self.xe) else: self._xe_shape = tuple(len(xei) for xei in self.xe) self._model_arg = np.row_stack( [x.flatten() for x in np.meshgrid(*self.xe, indexing="ij")] ) def _pred(self, args: _tp.Sequence[float]): d = self._model(self._model_arg, *args) d = _normalize_model_output(d) if self._xe_shape is not None: d = d.reshape(self._xe_shape) for i in range(self._ndim): d = np.diff(d, axis=i) # differences can come out negative due to round-off error in subtraction, # we set negative values to zero d[d < 0] = 0 return d class Template(BinnedCost): """ Binned cost function for a template fit with uncertainties on the template. This cost function is for a mixture model. Samples originate from two or more components and we are interested in estimating the yield that originates from each component. In high-energy physics, one component is often a peaking signal over a smooth background component. Templates are shape estimates for these components which are obtained from Monte-Carlo simulation. Even if the Monte-Carlo simulation is exact, the templates introduce some uncertainty since the Monte-Carlo simulation produces only a finite sample of events that contribute to each template. This cost function takes that additional uncertainty into account. There are several ways to approach this problem. Barlow and Beeston [1]_ found an exact likelihood for this problem, with one nuisance parameter per component per bin. Solving this likelihood is somewhat challenging though. The Barlow-Beeston likelihood also does not handle the additional uncertainty in weighted templates unless the weights per bin are all equal. Other works [2]_ [3]_ [4]_ describe likelihoods that use only one nuisance parameter per bin, which is an approximation. Some marginalize over the nuisance parameters with some prior, while others profile over the nuisance parameter. This class implements several of these methods. The default method is the one which performs best under most conditions, according to current knowledge. The default may change if this assessment changes. The cost function returns an asymptotically chi-square distributed test statistic, except for the method "asy", where it is the negative logarithm of the marginalised likelihood instead. The standard transform [5]_ which we use convert likelihoods into test statistics only works for (profiled) likelihoods, not for likelihoods marginalized over a prior. All methods implemented here have been generalized to work with both weighted data and weighted templates, under the assumption that the weights are independent of the data. This is not the case for sWeights, and the uncertaintes for results obtained with sWeights will only be approximately correct [6]_. .. [1] Barlow and Beeston, Comput.Phys.Commun. 77 (1993) 219-228 .. [2] Conway, PHYSTAT 2011 proceeding, https://doi.org/10.48550/arXiv.1103.0354 .. [3] Argüelles, Schneider, Yuan, JHEP 06 (2019) 030 .. [4] Dembinski and Abdelmotteleb, https://doi.org/10.48550/arXiv.2206.12346 .. [5] Baker and Cousins, NIM 221 (1984) 437-442 .. [6] Langenbruch, Eur.Phys.J.C 82 (2022) 5, 393 """ __slots__ = "_bbl_data", "_impl" def __init__( self, n: _ArrayLike, xe: _ArrayLike, templates: _tp.Sequence[_tp.Sequence], name: _tp.Collection[str] = None, verbose: int = 0, method: str = "da", ): """ Initialize cost function with data and model. Parameters ---------- n : array-like Histogram counts. If this is an array with dimension D+1, where D is the number of histogram axes, then the last dimension must have two elements and is interpreted as pairs of sum of weights and sum of weights squared. xe : array-like or collection of array-like Bin edge locations, must be len(n) + 1, where n is the number of bins. If the histogram has more than one axis, xe must be a collection of the bin edge locations along each axis. templates : collection of array-like Collection of arrays, which contain the histogram counts of each template. The template histograms must use the same axes as the data histogram. If the counts are represented by an array with dimension D+1, where D is the number of histogram axes, then the last dimension must have two elements and is interpreted as pairs of sum of weights and sum of weights squared. name : collection of str, optional Optional name for the yield of each template. Must have length K. verbose : int, optional Verbosity level. 0: is no output (default). 1: print current args and negative log-likelihood value. method : {"jsc", "asy", "da"}, optional Which method to use. "jsc": Conway's method [2]_. "asy": ASY method [3]_. "da": DA method [4]_. Default is "da", which to current knowledge offers the best overall performance. The default may change in the future, so please set this parameter explicitly in code that has to be stable. For all methods except the "asy" method, the minimum value is chi-square distributed. """ M = len(templates) if M < 1: raise ValueError("at least one template is required") if name is None: name = [f"x{i}" for i in range(M)] else: if len(name) != M: raise ValueError("number of names must match number of templates") shape = _shape_from_xe(xe) ndim = len(shape) temp = [] temp_var = [] for ti in templates: t = _norm(ti) if t.ndim > ndim: # template is weighted if t.ndim != ndim + 1 or t.shape[:-1] != shape: raise ValueError("shapes of n and templates do not match") temp.append(t[..., 0]) temp_var.append(t[..., 1]) else: if t.ndim != ndim or t.shape != shape: raise ValueError("shapes of n and templates do not match") temp.append(t) temp_var.append(t) nt = [] nt_var = [] for t, tv in zip(temp, temp_var): f = 1 / np.sum(t) nt.append(t * f) nt_var.append(tv * f**2) self._bbl_data = (nt, nt_var) known_methods = { "jsc": template_chi2_jsc, "asy": template_nll_asy, "hpd": template_chi2_da, "da": template_chi2_da, } try: self._impl = known_methods[method] except KeyError: raise ValueError( f"method {method} is not understood, allowed values: {known_methods}" ) if method == "hpd": warnings.warn( "key 'hpd' is deprecated, please use 'da' instead", category=np.VisibleDeprecationWarning, stacklevel=2, ) super().__init__(name, n, xe, verbose) def _pred(self, args: _tp.Sequence[float]): ntemp, ntemp_var = self._bbl_data mu = 0 mu_var = 0 for a, nt, vnt in zip(args, ntemp, ntemp_var): mu += a * nt mu_var += a**2 * vnt return mu, mu_var def _call(self, args): mu, mu_var = self._pred(args) ma = self.mask if ma is not None: mu = mu[ma] mu_var = mu_var[ma] if self._bztrafo: n, mu, mu_var = self._bztrafo(mu, mu_var) else: n = self._masked ma = mu > 0 return self._impl(n[ma], mu[ma], mu_var[ma]) def _errordef(self): return NEGATIVE_LOG_LIKELIHOOD if self._impl is template_nll_asy else CHISQUARE def visualize(self, args: _ArrayLike): """ Visualize data and model agreement (requires matplotlib). The visualization is drawn with matplotlib.pyplot into the current axes. Parameters ---------- args : array-like Parameter values. """ from matplotlib import pyplot as plt if self._ndim > 1: raise ValueError("visualize is not implemented for multi-dimensional data") args = np.atleast_1d(args) n = self._masked[..., 0] if self._bztrafo else self._masked ne = (self._masked[..., 1] if self._bztrafo else self._masked) ** 0.5 xe = self.xe cx = 0.5 * (xe[1:] + xe[:-1]) if self.mask is not None: cx = cx[self.mask] plt.errorbar(cx, n, ne, fmt="ok") mu, mu_var = self._pred(args) mu_err = mu_var**0.5 plt.stairs(mu + mu_err, xe, baseline=mu - mu_err, fill=True, color="C0") class BinnedNLL(BinnedCostWithModel): """ Binned negative log-likelihood. Use this if only the shape of the fitted PDF is of interest and the data is binned. This cost function works with normal and weighted histograms. The histogram can be one- or multi-dimensional. The cost function has a minimum value that is asymptotically chi2-distributed. It is constructed from the log-likelihood assuming a multivariate-normal distribution and using the saturated model as a reference. """ __slots__ = () @property def cdf(self): """Get cumulative density function.""" return self._model def __init__( self, n: _ArrayLike, xe: _ArrayLike, cdf: _tp.Callable, verbose: int = 0 ): """ Initialize cost function with data and model. Parameters ---------- n : array-like Histogram counts. If this is an array with dimension D+1, where D is the number of histogram axes, then the last dimension must have two elements and is interpreted as pairs of sum of weights and sum of weights squared. xe : array-like or collection of array-like Bin edge locations, must be len(n) + 1, where n is the number of bins. If the histogram has more than one axis, xe must be a collection of the bin edge locations along each axis. cdf : callable Cumulative density function of the form f(xe, par0, par1, ..., parN), where xe is a bin edge and par0, ... are model parameters. The corresponding density must be normalized to unity over the space covered by the histogram. If the model is multivariate, xe must be an array-like with shape (D, N), where D is the dimension and N is the number of points where the model is evaluated. verbose : int, optional Verbosity level. 0: is no output (default). 1: print current args and negative log-likelihood value. """ super().__init__(n, xe, cdf, verbose) def _pred(self, args): p = super()._pred(args) ma = self.mask if ma is not None: p /= np.sum(p[ma]) # normalise probability of remaining bins scale = np.sum(self._masked[..., 0] if self._bztrafo else self._masked) return p * scale def _call(self, args): mu = self._pred(args) ma = self.mask if ma is not None: mu = mu[ma] if self._bztrafo: n, mu = self._bztrafo(mu) else: n = self._masked return multinominal_chi2(n, mu) class ExtendedBinnedNLL(BinnedCostWithModel): """ Binned extended negative log-likelihood. Use this if shape and normalization of the fitted PDF are of interest and the data is binned. This cost function works with normal and weighted histograms. The histogram can be one- or multi-dimensional. The cost function works for both weighted data. The cost function assumes that the weights are independent of the data. This is not the case for sWeights, and the uncertaintes for results obtained with sWeights will only be approximately correct, see C. Langenbruch, Eur.Phys.J.C 82 (2022) 5, 393. The cost function has a minimum value that is asymptotically chi2-distributed. It is constructed from the log-likelihood assuming a poisson distribution and using the saturated model as a reference. """ __slots__ = () @property def scaled_cdf(self): """Get integrated density model.""" return self._model def __init__( self, n: _ArrayLike, xe: _ArrayLike, scaled_cdf: _tp.Callable, verbose: int = 0, ): """ Initialize cost function with data and model. Parameters ---------- n : array-like Histogram counts. If this is an array with dimension D+1, where D is the number of histogram axes, then the last dimension must have two elements and is interpreted as pairs of sum of weights and sum of weights squared. xe : array-like or collection of array-like Bin edge locations, must be len(n) + 1, where n is the number of bins. If the histogram has more than one axis, xe must be a collection of the bin edge locations along each axis. scaled_cdf : callable Scaled Cumulative density function of the form f(xe, par0, [par1, ...]), where xe is a bin edge and par0, ... are model parameters. If the model is multivariate, xe must be an array-like with shape (D, N), where D is the dimension and N is the number of points where the model is evaluated. verbose : int, optional Verbosity level. 0: is no output (default). 1: print current args and negative log-likelihood value. """ super().__init__(n, xe, scaled_cdf, verbose) def _call(self, args): mu = self._pred(args) ma = self.mask if ma is not None: mu = mu[ma] if self._bztrafo: n, mu = self._bztrafo(mu) else: n = self._masked return poisson_chi2(n, mu) class LeastSquares(MaskedCost): """Least-squares cost function (aka chisquare function). Use this if you have data of the form (x, y +/- yerror), where x can be one-dimensional or multi-dimensional, but y is always one-dimensional. See :meth:`__init__` for details on how to use a multivariate model. """ __slots__ = "_loss", "_cost", "_model", "_ndim" @property def x(self): """Get explanatory variables.""" if self._ndim == 1: return self.data[:, 0] return self.data.T[: self._ndim] @x.setter def x(self, value): if self._ndim == 1: self.data[:, 0] = _norm(value) else: self.data[:, : self._ndim] = _norm(value).T self._update_cache() @property def y(self): """Get samples.""" return self.data[:, self._ndim] @y.setter def y(self, value): self.data[:, self._ndim] = _norm(value) self._update_cache() @property def yerror(self): """Get sample uncertainties.""" return self.data[:, self._ndim + 1] @yerror.setter def yerror(self, value): self.data[:, self._ndim + 1] = _norm(value) self._update_cache() @property def model(self): """Get model of the form y = f(x, par0, [par1, ...]).""" return self._model @property def loss(self): """Get loss function.""" return self._loss @loss.setter def loss(self, loss: _tp.Union[str, _tp.Callable[[np.ndarray], np.ndarray]]): self._loss = loss if isinstance(loss, str): if loss == "linear": self._cost = chi2 elif loss == "soft_l1": self._cost = _soft_l1_cost else: raise ValueError("unknown loss type: " + loss) else: assert hasattr(loss, "__call__") self._cost = lambda y, ye, ym: np.sum( loss(_z_squared(y, ye, ym)) # type:ignore ) def __init__( self, x: _ArrayLike, y: _ArrayLike, yerror: _ArrayLike, model: _tp.Callable, loss: _tp.Union[str, _tp.Callable] = "linear", verbose: int = 0, ): """ Initialize cost function with data and model. Parameters ---------- x : array-like Locations where the model is evaluated. If the model is multivariate, x must have shape (D, N), where D is the number of dimensions and N the number of data points. y : array-like Observed values. Must have the same length as x. yerror : array-like or float Estimated uncertainty of observed values. Must have same shape as y or be a scalar, which is then broadcasted to same shape as y. model : callable Function of the form f(x, par0, [par1, ...]) whose output is compared to observed values, where x is the location and par0, ... are model parameters. If the model is multivariate, x has shape (D, N), where D is the number of dimensions and N the number of data points. loss : str or callable, optional The loss function can be modified to make the fit robust against outliers, see scipy.optimize.least_squares for details. Only "linear" (default) and "soft_l1" are currently implemented, but users can pass any loss function as this argument. It should be a monotonic, twice differentiable function, which accepts the squared residual and returns a modified squared residual. verbose : int, optional Verbosity level. 0: is no output (default). 1: print current args and negative log-likelihood value. Notes ----- Alternative loss functions make the fit more robust against outliers by weakening the pull of outliers. The mechanical analog of a least-squares fit is a system with attractive forces. The points pull the model towards them with a force whose potential is given by :math:`rho(z)` for a squared-offset :math:`z`. The plot shows the standard potential in comparison with the weaker soft-l1 potential, in which outliers act with a constant force independent of their distance. .. plot:: plots/loss.py """ x = _norm(x) y = _norm(y) assert x.ndim >= 1 # guaranteed by _norm self._ndim = x.ndim self._model = model self.loss = loss x = np.atleast_2d(x) data = np.column_stack(np.broadcast_arrays(*x, y, yerror)) super().__init__(describe(self._model)[1:], data, verbose) def _call(self, args): x = self._masked.T[0] if self._ndim == 1 else self._masked.T[: self._ndim] y, yerror = self._masked.T[self._ndim :] ym = self._model(x, *args) ym = _normalize_model_output(ym) return self._cost(y, yerror, ym) def _ndata(self): return len(self._masked) def visualize(self, args: _ArrayLike, model_points: int = 0): """ Visualize data and model agreement (requires matplotlib). The visualization is drawn with matplotlib.pyplot into the current axes. Parameters ---------- args : array-like Parameter values. model_points : int, optional How many points to use to draw the model. Default is 0, in this case an smart sampling algorithm selects the number of points. """ from matplotlib import pyplot as plt if self._ndim > 1: raise ValueError("visualize is not implemented for multi-dimensional data") # TODO # - make linear or log-spacing configurable x, y, ye = self._masked.T plt.errorbar(x, y, ye, fmt="ok") if model_points > 0: if x[0] > 0 and x[-1] / x[0] > 1e2: xm = np.geomspace(x[0], x[-1], model_points) else: xm = np.linspace(x[0], x[-1], model_points) ym = self.model(xm, *args) else: xm, ym = _smart_sampling(lambda x: self.model(x, *args), x[0], x[-1]) plt.plot(xm, ym) class NormalConstraint(Cost): """ Gaussian penalty for one or several parameters. The Gaussian penalty acts like a pseudo-measurement of the parameter itself, based on a (multi-variate) normal distribution. Penalties can be set for one or several parameters at once (which is more efficient). When several parameter are constrained, one can specify the full covariance matrix of the parameters. Notes ----- It is sometimes necessary to add a weak penalty on a parameter to avoid instabilities in the fit. A typical example in high-energy physics is the fit of a signal peak above some background. If the amplitude of the peak vanishes, the shape parameters of the peak become unconstrained and the fit becomes unstable. This can be avoided by adding weak (large uncertainty) penalty on the shape parameters whose pull is negligible if the peak amplitude is non-zero. This class can also be used to approximately include external measurements of some parameters, if the original cost function is not available or too costly to compute. If the external measurement was performed in the asymptotic limit with a large sample, a Gaussian penalty is an accurate statistical representation of the external result. """ __slots__ = "_value", "_cov", "_covinv" def __init__( self, args: _tp.Union[str, _tp.Iterable[str]], value: _ArrayLike, error: _ArrayLike, ): """ Initialize the normal constraint with expected value(s) and error(s). Parameters ---------- args : str or sequence of str Parameter name(s). value : float or array-like Expected value(s). Must have same length as `args`. error : float or array-like Expected error(s). If 1D, must have same length as `args`. If 2D, must be the covariance matrix of the parameters. """ self._value = _norm(value) self._cov = _norm(error) if self._cov.ndim < 2: self._cov **= 2 self._covinv = _covinv(self._cov) tp_args = (args,) if isinstance(args, str) else tuple(args) super().__init__(tp_args, False) @property def covariance(self): """ Get expected covariance of parameters. Can be 1D (diagonal of covariance matrix) or 2D (full covariance matrix). """ return self._cov @covariance.setter def covariance(self, value): self._cov[:] = value self._covinv = _covinv(self._cov) @property def value(self): """Get expected parameter values.""" return self._value @value.setter def value(self, value): self._value[:] = value def _call(self, args): delta = self._value - args if self._covinv.ndim < 2: return np.sum(delta**2 * self._covinv) return np.einsum("i,ij,j", delta, self._covinv, delta) def _ndata(self): return len(self._value) def visualize(self, args: _ArrayLike): """ Visualize data and model agreement (requires matplotlib). The visualization is drawn with matplotlib.pyplot into the current axes. Parameters ---------- args : array-like Parameter values. """ from matplotlib import pyplot as plt args = np.atleast_1d(args) par = self.func_code.co_varnames val = self.value cov = self.covariance if cov.ndim == 2: cov = np.diag(cov) err = np.sqrt(cov) n = len(par) i = 0 max_pull = 0 for v, e, a in zip(val, err, args): pull = (a - v) / e max_pull = max(abs(pull), max_pull) plt.errorbar(pull, -i, 0, 1, fmt="o", color="C0") i += 1 plt.axvline(0, color="k") plt.xlim(-max_pull - 1.1, max_pull + 1.1) yaxis = plt.gca().yaxis yaxis.set_ticks(-np.arange(n)) yaxis.set_ticklabels(par) plt.ylim(-n + 0.5, 0.5) def _norm(value: _ArrayLike[float]) -> np.ndarray: value = np.atleast_1d(value) dtype = value.dtype if dtype.kind != "f": value = value.astype(np.float64) return value def _covinv(array): return np.linalg.inv(array) if array.ndim == 2 else 1.0 / array def _normalize_model_output(x, msg="Model should return numpy array"): if not isinstance(x, np.ndarray): warnings.warn( f"{msg}, but returns {type(x)}", PerformanceWarning, ) x = np.array(x) if x.dtype.kind != "f": return x.astype(float) return x def _shape_from_xe(xe): if isinstance(xe[0], _tp.Iterable): return tuple(len(xei) - 1 for xei in xe) return (len(xe) - 1,) _deprecated_content = { "BarlowBeestonLite": ("Template", Template), "barlow_beeston_lite_chi2_jsc": ("template_chi2_jsc", template_chi2_jsc), "barlow_beeston_lite_chi2_hpd": ("template_chi2_da", template_chi2_da), } def __getattr__(name: str) -> _tp.Any: if name in _deprecated_content: new_name, obj = _deprecated_content[name] warnings.warn( f"{name} was renamed to {new_name}, please import {new_name} instead", np.VisibleDeprecationWarning, stacklevel=2, ) return obj raise AttributeError