import math import numpy as np from scipy.special import betainc from scipy._lib._array_api import ( xp_capabilities, xp_ravel, array_namespace, xp_promote, xp_device, _length_nonmasked, is_torch, ) import scipy._lib.array_api_extra as xpx from scipy.stats._axis_nan_policy import _broadcast_arrays, _contains_nan def _quantile_iv(x, p, method, axis, nan_policy, keepdims, weights): xp = array_namespace(x, p, weights) if not xp.isdtype(xp.asarray(x).dtype, ('integral', 'real floating')): raise ValueError("`x` must have real dtype.") if not xp.isdtype(xp.asarray(p).dtype, 'real floating'): raise ValueError("`p` must have real floating dtype.") if not (weights is None or xp.isdtype(xp.asarray(weights).dtype, ('integral', 'real floating'))): raise ValueError("`weights` must have real dtype.") x, p, weights = xp_promote(x, p, weights, force_floating=True, xp=xp) p = xp.asarray(p, device=xp_device(x)) dtype = x.dtype axis_none = axis is None ndim = max(x.ndim, p.ndim) if axis_none: x = xp_ravel(x) p = xp_ravel(p) axis = 0 elif np.iterable(axis) or int(axis) != axis: message = "`axis` must be an integer or None." raise ValueError(message) elif (axis >= ndim) or (axis < -ndim): message = "`axis` is not compatible with the shapes of the inputs." raise ValueError(message) axis = int(axis) methods = {'inverted_cdf', 'averaged_inverted_cdf', 'closest_observation', 'hazen', 'interpolated_inverted_cdf', 'linear', 'median_unbiased', 'normal_unbiased', 'weibull', 'harrell-davis', '_lower', '_midpoint', '_higher', '_nearest', 'round_outward', 'round_inward', 'round_nearest'} if method not in methods: message = f"`method` must be one of {methods}" raise ValueError(message) no_weights = {'_lower', '_midpoint', '_higher', '_nearest', 'harrell-davis', 'round_nearest', 'round_inward', 'round_outward'} if weights is not None and method in no_weights: message = f"`method='{method}'` does not support `weights`." raise ValueError(message) contains_nans = _contains_nan(x, nan_policy, xp_omit_okay=True, xp=xp) if keepdims not in {None, True, False}: message = "If specified, `keepdims` must be True or False." raise ValueError(message) # If data has length zero along `axis`, the result will be an array of NaNs just # as if the data had length 1 along axis and were filled with NaNs. This is treated # naturally below whether `nan_policy` is `'propagate'` or `'omit'`. if x.shape[axis] == 0: shape = list(x.shape) shape[axis] = 1 x = xp.full(shape, xp.nan, dtype=dtype, device=xp_device(x)) if weights is None: y = xp.sort(x, axis=axis, stable=False) y, p = _broadcast_arrays((y, p), axis=axis) n_zero_weight = None else: x, weights = xp.broadcast_arrays(x, weights) i_zero_weight = (weights == 0) n_zero_weight = xp.count_nonzero(i_zero_weight, axis=axis, keepdims=True) x = xpx.at(x)[i_zero_weight].set(xp.inf, copy=True) i_y = xp.argsort(x, axis=axis, stable=False) y = xp.take_along_axis(x, i_y, axis=axis) weights = xp.take_along_axis(weights, i_y, axis=axis) y, p, weights, i_y, n_zero_weight = _broadcast_arrays( (y, p, weights, i_y, n_zero_weight), axis=axis) if (keepdims is False) and (p.shape[axis] != 1): message = "`keepdims` may be False only if the length of `p` along `axis` is 1." raise ValueError(message) keepdims = (p.shape[axis] != 1) if keepdims is None else keepdims y = xp.moveaxis(y, axis, -1) p = xp.moveaxis(p, axis, -1) weights = weights if weights is None else xp.moveaxis(weights, axis, -1) n_zero_weight = (n_zero_weight if n_zero_weight is None else xp.moveaxis(n_zero_weight, axis, -1)) n = _length_nonmasked(y, -1, xp=xp, keepdims=True) n = n if n_zero_weight is None else n - n_zero_weight if contains_nans: nans = xp.isnan(y) # Note that if length along `axis` were 0 to begin with, # it is now length 1 and filled with NaNs. if nan_policy == 'propagate': nan_out = xp.any(nans, axis=-1) else: # 'omit' n_int = n - xp.count_nonzero(nans, axis=-1, keepdims=True) n = xp.astype(n_int, dtype) # NaNs are produced only if slice is empty after removing NaNs nan_out = xp.any(n == 0, axis=-1) n = xpx.at(n, nan_out).set(y.shape[-1]) # avoids pytorch/pytorch#146211 if xp.any(nan_out): y = xp.asarray(y, copy=True) # ensure writable y = xpx.at(y, nan_out).set(xp.nan) elif xp.any(nans) and method == 'harrell-davis': y = xp.asarray(y, copy=True) # ensure writable y = xpx.at(y, nans).set(0) # any non-nan will prevent NaN from propagating n = xp.asarray(n, dtype=dtype, device=xp_device(y)) p_mask = (p > 1) | (p < 0) | xp.isnan(p) if xp.any(p_mask): p = xp.asarray(p, copy=True) p = xpx.at(p, p_mask).set(0.5) # these get NaN-ed out at the end return (y, p, method, axis, nan_policy, keepdims, n, axis_none, ndim, p_mask, weights, xp) @xp_capabilities(skip_backends=[("dask.array", "No take_along_axis yet.")], jax_jit=False) def quantile(x, p, *, method='linear', axis=0, nan_policy='propagate', keepdims=None, weights=None): """ Compute the p-th quantile of the data along the specified axis. Parameters ---------- x : array_like of real numbers Data array. p : array_like of float Probability or sequence of probabilities of the quantiles to compute. Values must be between 0 and 1 (inclusive). While `numpy.quantile` can only compute quantiles according to the Cartesian product of the first two arguments, this function enables calculation of quantiles at different probabilities for each axis slice by following broadcasting rules like those of `scipy.stats` reducing functions. See `axis`, `keepdims`, and the examples. method : str, default: 'linear' The method to use for estimating the quantile. The available options, numbered as they appear in [1]_, are: 1. 'inverted_cdf' 2. 'averaged_inverted_cdf' 3. 'closest_observation' 4. 'interpolated_inverted_cdf' 5. 'hazen' 6. 'weibull' 7. 'linear' (default) 8. 'median_unbiased' 9. 'normal_unbiased' 'harrell-davis' is also available to compute the quantile estimate according to [2]_. 'round_outward', 'round_inward', and 'round_nearest' are available for use in trimming and winsorizing data. See Notes for details. axis : int or None, default: 0 Axis along which the quantiles are computed. ``None`` ravels both `x` and `p` before performing the calculation, without checking whether the original shapes were compatible. As in other `scipy.stats` functions, a positive integer `axis` is resolved after prepending 1s to the shape of `x` or `p` as needed until the two arrays have the same dimensionality. When providing `x` and `p` with different dimensionality, consider using negative `axis` integers for clarity. nan_policy : str, default: 'propagate' Defines how to handle NaNs in the input data `x`. - ``propagate``: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding slice of the output will contain NaN(s). - ``omit``: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding slice of the output will contain NaN(s). - ``raise``: if a NaN is present, a ``ValueError`` will be raised. If NaNs are present in `p`, a ``ValueError`` will be raised. keepdims : bool, optional Consider the case in which `x` is 1-D and `p` is a scalar: the quantile is a reducing statistic, and the default behavior is to return a scalar. If `keepdims` is set to True, the axis will not be reduced away, and the result will be a 1-D array with one element. The general case is more subtle, since multiple quantiles may be requested for each axis-slice of `x`. For instance, if both `x` and `p` are 1-D and ``p.size > 1``, no axis can be reduced away; there must be an axis to contain the number of quantiles given by ``p.size``. Therefore: - By default, the axis will be reduced away if possible (i.e. if there is exactly one element of `p` per axis-slice of `x`). - If `keepdims` is set to True, the axis will not be reduced away. - If `keepdims` is set to False, the axis will be reduced away if possible, and an error will be raised otherwise. weights : array_like of finite, non-negative real numbers Frequency weights; e.g., for counting number weights, ``quantile(x, p, weights=weights)`` is equivalent to ``quantile(np.repeat(x, weights), p)``. Values other than finite counting numbers are accepted, but may not have valid statistical interpretations. Not compatible with ``method='harrell-davis'`` or those that begin with ``'round_'``. Returns ------- quantile : scalar or ndarray The resulting quantile(s). The dtype is the result dtype of `x` and `p`. See Also -------- numpy.quantile :ref:`outliers` Notes ----- Given a sample `x` from an underlying distribution, `quantile` provides a nonparametric estimate of the inverse cumulative distribution function. By default, this is done by interpolating between adjacent elements in ``y``, a sorted copy of `x`:: (1-g)*y[j] + g*y[j+1] where the index ``j`` and coefficient ``g`` are the integral and fractional components of ``p * (n-1)``, and ``n`` is the number of elements in the sample. This is a special case of Equation 1 of H&F [1]_. More generally, - ``j = (p*n + m - 1) // 1``, and - ``g = (p*n + m - 1) % 1``, where ``m`` may be defined according to several different conventions. The preferred convention may be selected using the ``method`` parameter: =============================== =============== =============== ``method`` number in H&F ``m`` =============================== =============== =============== ``interpolated_inverted_cdf`` 4 ``0`` ``hazen`` 5 ``1/2`` ``weibull`` 6 ``p`` ``linear`` (default) 7 ``1 - p`` ``median_unbiased`` 8 ``p/3 + 1/3`` ``normal_unbiased`` 9 ``p/4 + 3/8`` =============================== =============== =============== Note that indices ``j`` and ``j + 1`` are clipped to the range ``0`` to ``n - 1`` when the results of the formula would be outside the allowed range of non-negative indices. When ``j`` is clipped to zero, ``g`` is set to zero as well. The ``-1`` in the formulas for ``j`` and ``g`` accounts for Python's 0-based indexing. The table above includes only the estimators from [1]_ that are continuous functions of probability `p` (estimators 4-9). SciPy also provides the three discontinuous estimators from [1]_ (estimators 1-3), where ``j`` is defined as above, ``m`` is defined as follows, and ``g`` is ``0`` when ``index = p*n + m - 1`` is less than ``0`` and otherwise is defined below. 1. ``inverted_cdf``: ``m = 0`` and ``g = int(index - j > 0)`` 2. ``averaged_inverted_cdf``: ``m = 0`` and ``g = (1 + int(index - j > 0)) / 2`` 3. ``closest_observation``: ``m = -1/2`` and ``g = 1 - int((index == j) & (j%2 == 1))`` Note that for methods ``inverted_cdf`` and ``averaged_inverted_cdf``, only the relative proportions of tied observations (and relative weights) affect the results; for all other methods, the total number of observations (and absolute weights) matter. A different strategy for computing quantiles from [2]_, ``method='harrell-davis'``, uses a weighted combination of all elements. The weights are computed as: .. math:: w_{n, i} = I_{i/n}(a, b) - I_{(i - 1)/n}(a, b) where :math:`n` is the number of elements in the sample, :math:`i` are the indices :math:`1, 2, ..., n-1, n` of the sorted elements, :math:`a = p (n + 1)`, :math:`b = (1 - p)(n + 1)`, :math:`p` is the probability of the quantile, and :math:`I` is the regularized, lower incomplete beta function (`scipy.special.betainc`). ``method='round_nearest'`` is equivalent to indexing ``y[j]``, where:: j = int(np.round(p*n) if p < 0.5 else np.round(n*p - 1)) This is useful when winsorizing data: replacing ``p*n`` of the most extreme observations with the next most extreme observation. ``method='round_outward'`` adjusts the direction of rounding to winsorize fewer elements:: j = int(np.floor(p*n) if p < 0.5 else np.ceil(n*p - 1)) and ``method='round_inward'`` rounds to winsorize more elements:: j = int(np.ceil(p*n) if p < 0.5 else np.floor(n*p - 1)) These methods are also useful for trimming data: removing ``p*n`` of the most extreme observations. See :ref:`outliers` for example applications. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.asarray([[10, 8, 7, 5, 4], ... [0, 1, 2, 3, 5]]) Take the median of each row. >>> stats.quantile(x, 0.5, axis=-1) array([7., 2.]) Take a different quantile for each row. >>> stats.quantile(x, [[0.25], [0.75]], axis=-1, keepdims=True) array([[5.], [3.]]) Take multiple quantiles for each row. >>> stats.quantile(x, [0.25, 0.75], axis=-1) array([[5., 8.], [1., 3.]]) Take different quantiles for each row. >>> p = np.asarray([[0.25, 0.75], ... [0.5, 1.0]]) >>> stats.quantile(x, p, axis=-1) array([[5., 8.], [2., 5.]]) Take different quantiles for each column. >>> stats.quantile(x.T, p.T, axis=0) array([[5., 2.], [8., 5.]]) References ---------- .. [1] R. J. Hyndman and Y. Fan, "Sample quantiles in statistical packages," The American Statistician, 50(4), pp. 361-365, 1996 .. [2] Harrell, Frank E., and C. E. Davis. "A new distribution-free quantile estimator." Biometrika 69.3 (1982): 635-640. """ # Input validation / standardization temp = _quantile_iv(x, p, method, axis, nan_policy, keepdims, weights) (y, p, method, axis, nan_policy, keepdims, n, axis_none, ndim, p_mask, weights, xp) = temp if method in {'inverted_cdf', 'averaged_inverted_cdf', 'closest_observation', 'hazen', 'interpolated_inverted_cdf', 'linear', 'median_unbiased', 'normal_unbiased', 'weibull'}: res = _quantile_hf(y, p, n, method, weights, xp) elif method in {'harrell-davis'}: res = _quantile_hd(y, p, n, xp) elif method in {'_lower', '_midpoint', '_higher', '_nearest'}: res = _quantile_bc(y, p, n, method, xp) else: # method.startswith('round'): res = _quantile_winsor(y, p, n, method, xp) res = xpx.at(res, p_mask).set(xp.nan) # Reshape per axis/keepdims if axis_none and keepdims: shape = (1,)*(ndim - 1) + res.shape res = xp.reshape(res, shape) axis = -1 res = xp.moveaxis(res, -1, axis) if not keepdims: res = xp.squeeze(res, axis=axis) return res[()] if res.ndim == 0 else res def _quantile_hf(y, p, n, method, weights, xp): ms = dict(inverted_cdf=0, averaged_inverted_cdf=0, closest_observation=-0.5, interpolated_inverted_cdf=0, hazen=0.5, weibull=p, linear=1 - p, median_unbiased=p/3 + 1/3, normal_unbiased=p/4 + 3/8) m = ms[method] if weights is None: jg = p * n + m jp1 = jg // 1 j = jp1 - 1 else: cumulative_weights = xp.cumulative_sum(weights, axis=-1) n_int = xp.asarray(n, dtype=xp.int64) n_int = xp.broadcast_to(n_int, cumulative_weights.shape[:-1] + (1,)) total_weight = xp.take_along_axis(cumulative_weights, n_int-1, axis=-1) jg = p * total_weight + m jp1 = _xp_searchsorted(cumulative_weights, jg, side='right') j = _xp_searchsorted(cumulative_weights, jg-1, side='right') j, jp1 = xp.astype(j, y.dtype), xp.astype(jp1, y.dtype) g = jg % 1 if method == 'inverted_cdf': g = xp.astype((g > 0), jg.dtype) elif method == 'averaged_inverted_cdf': g = (1 + xp.astype((g > 0), jg.dtype)) / 2 elif method == 'closest_observation': g = (1 - xp.astype((g == 0) & (j % 2 == 1), jg.dtype)) if method in {'inverted_cdf', 'averaged_inverted_cdf', 'closest_observation'}: g = xp.asarray(g) g = xpx.at(g, jg < 0).set(0) g = xpx.at(g)[j < 0].set(0) j = xp.clip(j, 0., n - 1) jp1 = xp.clip(jp1, 0., n - 1) return ((1 - g) * xp.take_along_axis(y, xp.astype(j, xp.int64), axis=-1) + g * xp.take_along_axis(y, xp.astype(jp1, xp.int64), axis=-1)) def _quantile_hd(y, p, n, xp): # RE axis handling: We need to perform a reducing operation over rows of `y` for # each element in the corresponding row of `p` (a la Cartesian product). Strategy: # move rows of `p` to an axis at the front that is orthogonal to all the rest, # perform the reducing operating over the last axis, then move the front axis back # to the end. p = xp.moveaxis(p, -1, 0)[..., xp.newaxis] a = p * (n + 1) b = (1 - p) * (n + 1) i = xp.arange(y.shape[-1] + 1, dtype=y.dtype, device=xp_device(y)) w = betainc(a, b, i / n) w = w[..., 1:] - w[..., :-1] w = xpx.at(w, xp.isnan(w)).set(0) res = xp.vecdot(w, y, axis=-1) return xp.moveaxis(res, 0, -1) def _quantile_winsor(y, p, n, method, xp): ops = dict(round_outward=(xp.floor, xp.ceil), round_inward=(xp.ceil, xp.floor), round_nearest=(xp.round, xp.round)) op_left, op_right = ops[method] j = xp.where(p < 0.5, op_left(p*n), op_right(n*p - 1)) return xp.take_along_axis(y, xp.astype(j, xp.int64), axis=-1) def _quantile_bc(y, p, n, method, xp): # Methods retained for backward compatibility. NumPy documentation is not # quite right about what these methods do: if `p * (n - 1)` is integral, # that is used as the index. See numpy/numpy#28910. ij = p * (n - 1) if method == '_midpoint': return (xp.take_along_axis(y, xp.astype(xp.floor(ij), xp.int64), axis=-1) + xp.take_along_axis(y, xp.astype(xp.ceil(ij), xp.int64), axis=-1)) / 2 elif method == '_lower': k = xp.floor(ij) elif method == '_higher': k = xp.ceil(ij) elif method == '_nearest': k = xp.round(ij) return xp.take_along_axis(y, xp.astype(k, xp.int64), axis=-1) @xp_capabilities(skip_backends=[("dask.array", "No take_along_axis yet.")]) def _xp_searchsorted(x, y, *, side='left', xp=None): # Vectorized xp.searchsorted. Search is performed along last axis. The shape of the # output is that of `y`, broadcasting the batch dimensions with those of `x` if # necessary. xp = array_namespace(x, y) if xp is None else xp xp_default_int = xp.asarray(1).dtype y_0d = xp.asarray(y).ndim == 0 x, y = _broadcast_arrays((x, y), axis=-1, xp=xp) x_1d = x.ndim <= 1 if x_1d or is_torch(xp): y = xp.reshape(y, ()) if (y_0d and x_1d) else y out = xp.searchsorted(x, y, side=side) out = xp.astype(out, xp_default_int, copy=False) return out a = xp.full(y.shape, 0, device=xp_device(x)) if x.shape[-1] == 0: return a n = xp.count_nonzero(~xp.isnan(x), axis=-1, keepdims=True) b = xp.broadcast_to(n, y.shape) compare = xp.less_equal if side == 'left' else xp.less # while xp.any(b - a > 1): # refactored to for loop with ~log2(n) iterations for JAX JIT for i in range(int(math.log2(x.shape[-1])) + 1): c = (a + b) // 2 x0 = xp.take_along_axis(x, c, axis=-1) j = compare(y, x0) b = xp.where(j, c, b) a = xp.where(j, a, c) out = xp.where(compare(y, xp.min(x, axis=-1, keepdims=True)), 0, b) out = xp.where(xp.isnan(y), x.shape[-1], out) if side == 'right' else out out = xp.astype(out, xp_default_int, copy=False) return out