Source code for mars.tensor.random.gumbel

#!/usr/bin/env python
# -*- coding: utf-8 -*-
# Copyright 1999-2020 Alibaba Group Holding Ltd.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# See the License for the specific language governing permissions and
# limitations under the License.

import numpy as np

from ... import opcodes as OperandDef
from ...serialize import AnyField
from .core import TensorRandomOperandMixin, handle_array, TensorDistribution

class TensorGumbel(TensorDistribution, TensorRandomOperandMixin):
    __slots__ = '_loc', '_scale', '_size'
    _input_fields_ = ['_loc', '_scale']
    _op_type_ = OperandDef.RAND_GUMBEL

    _loc = AnyField('loc')
    _scale = AnyField('scale')
    _func_name = 'gumbel'

    def __init__(self, state=None, size=None, dtype=None, gpu=None, **kw):
        dtype = np.dtype(dtype) if dtype is not None else dtype
        super().__init__(_state=state, _size=size, _dtype=dtype, _gpu=gpu, **kw)

    def loc(self):
        return self._loc

    def scale(self):
        return self._scale

    def __call__(self, loc, scale, chunk_size=None):
        return self.new_tensor([loc, scale], None, raw_chunk_size=chunk_size)

[docs]def gumbel(random_state, loc=0.0, scale=1.0, size=None, chunk_size=None, gpu=None, dtype=None): r""" Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. chunk_size : int or tuple of int or tuple of ints, optional Desired chunk size on each dimension gpu : bool, optional Allocate the tensor on GPU if True, False as default dtype : data-type, optional Data-type of the returned tensor. Returns ------- out : Tensor or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> import mars.tensor as mt >>> mu, beta = 0, 0.1 # location and scale >>> s = mt.random.gumbel(mu, beta, 1000).execute() Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import numpy as np >>> count, bins, ignored = plt.hist(s, 30, normed=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = mt.random.normal(mu, beta, 1000) ... means.append(a.mean().execute()) ... maxima.append(a.max().execute()) >>> count, bins, ignored = plt.hist(maxima, 30, normed=True) >>> beta = mt.std(maxima) * mt.sqrt(6) / mt.pi >>> mu = mt.mean(maxima) - 0.57721*beta >>> plt.plot(bins, ((1/beta)*mt.exp(-(bins - mu)/beta) ... * mt.exp(-mt.exp(-(bins - mu)/beta))).execute(), ... linewidth=2, color='r') >>> plt.plot(bins, (1/(beta * mt.sqrt(2 * mt.pi)) ... * mt.exp(-(bins - mu)**2 / (2 * beta**2))).execute(), ... linewidth=2, color='g') >>> """ if dtype is None: dtype = np.random.RandomState().gumbel( handle_array(loc), handle_array(scale), size=(0,)).dtype size = random_state._handle_size(size) op = TensorGumbel(state=random_state.to_numpy(), size=size, gpu=gpu, dtype=dtype) return op(loc, scale, chunk_size=chunk_size)