#!python #cython: wraparound=False, nonecheck=False, boundscheck=False, cdivision=True, language_level=3 import operator import warnings from collections.abc import Sequence from cpython.pycapsule cimport PyCapsule_IsValid, PyCapsule_GetPointer from cpython cimport (Py_INCREF, PyFloat_AsDouble) from cpython.mem cimport PyMem_Malloc, PyMem_Free cimport cython import numpy as np cimport numpy as np from numpy.core.multiarray import normalize_axis_index from .c_distributions cimport * from libc cimport string from libc.math cimport sqrt from libc.stdint cimport (uint8_t, uint16_t, uint32_t, uint64_t, int32_t, int64_t, INT64_MAX, SIZE_MAX) from ._bounded_integers cimport (_rand_bool, _rand_int32, _rand_int64, _rand_int16, _rand_int8, _rand_uint64, _rand_uint32, _rand_uint16, _rand_uint8, _gen_mask) from ._pcg64 import PCG64 from numpy.random cimport bitgen_t from ._common cimport (POISSON_LAM_MAX, CONS_POSITIVE, CONS_NONE, CONS_NON_NEGATIVE, CONS_BOUNDED_0_1, CONS_BOUNDED_GT_0_1, CONS_BOUNDED_LT_0_1, CONS_GT_1, CONS_POSITIVE_NOT_NAN, CONS_POISSON, double_fill, cont, kahan_sum, cont_broadcast_3, float_fill, cont_f, check_array_constraint, check_constraint, disc, discrete_broadcast_iii, validate_output_shape ) cdef extern from "numpy/arrayobject.h": int PyArray_ResolveWritebackIfCopy(np.ndarray) int PyArray_FailUnlessWriteable(np.PyArrayObject *obj, const char *name) except -1 object PyArray_FromArray(np.PyArrayObject *, np.PyArray_Descr *, int) enum: NPY_ARRAY_WRITEBACKIFCOPY np.import_array() cdef int64_t _safe_sum_nonneg_int64(size_t num_colors, int64_t *colors): """ Sum the values in the array `colors`. Return -1 if an overflow occurs. The values in *colors are assumed to be nonnegative. """ cdef size_t i cdef int64_t sum sum = 0 for i in range(num_colors): if colors[i] > INT64_MAX - sum: return -1 sum += colors[i] return sum cdef inline void _shuffle_raw_wrap(bitgen_t *bitgen, np.npy_intp n, np.npy_intp first, np.npy_intp itemsize, np.npy_intp stride, char* data, char* buf) nogil: # We trick gcc into providing a specialized implementation for # the most common case, yielding a ~33% performance improvement. # Note that apparently, only one branch can ever be specialized. if itemsize == sizeof(np.npy_intp): _shuffle_raw(bitgen, n, first, sizeof(np.npy_intp), stride, data, buf) else: _shuffle_raw(bitgen, n, first, itemsize, stride, data, buf) cdef inline void _shuffle_raw(bitgen_t *bitgen, np.npy_intp n, np.npy_intp first, np.npy_intp itemsize, np.npy_intp stride, char* data, char* buf) nogil: """ Parameters ---------- bitgen Pointer to a bitgen_t instance. n Number of elements in data first First observation to shuffle. Shuffles n-1, n-2, ..., first, so that when first=1 the entire array is shuffled itemsize Size in bytes of item stride Array stride data Location of data buf Location of buffer (itemsize) """ cdef np.npy_intp i, j for i in reversed(range(first, n)): j = random_interval(bitgen, i) string.memcpy(buf, data + j * stride, itemsize) string.memcpy(data + j * stride, data + i * stride, itemsize) string.memcpy(data + i * stride, buf, itemsize) cdef inline void _shuffle_int(bitgen_t *bitgen, np.npy_intp n, np.npy_intp first, int64_t* data) nogil: """ Parameters ---------- bitgen Pointer to a bitgen_t instance. n Number of elements in data first First observation to shuffle. Shuffles n-1, n-2, ..., first, so that when first=1 the entire array is shuffled data Location of data """ cdef np.npy_intp i, j cdef int64_t temp for i in reversed(range(first, n)): j = random_bounded_uint64(bitgen, 0, i, 0, 0) temp = data[j] data[j] = data[i] data[i] = temp cdef bint _check_bit_generator(object bitgen): """Check if an object satisfies the BitGenerator interface. """ if not hasattr(bitgen, "capsule"): return False cdef const char *name = "BitGenerator" return PyCapsule_IsValid(bitgen.capsule, name) cdef class Generator: """ Generator(bit_generator) Container for the BitGenerators. ``Generator`` exposes a number of methods for generating random numbers drawn from a variety of probability distributions. In addition to the distribution-specific arguments, each method takes a keyword argument `size` that defaults to ``None``. If `size` is ``None``, then a single value is generated and returned. If `size` is an integer, then a 1-D array filled with generated values is returned. If `size` is a tuple, then an array with that shape is filled and returned. The function :func:`numpy.random.default_rng` will instantiate a `Generator` with numpy's default `BitGenerator`. **No Compatibility Guarantee** ``Generator`` does not provide a version compatibility guarantee. In particular, as better algorithms evolve the bit stream may change. Parameters ---------- bit_generator : BitGenerator BitGenerator to use as the core generator. Notes ----- The Python stdlib module `random` contains pseudo-random number generator with a number of methods that are similar to the ones available in ``Generator``. It uses Mersenne Twister, and this bit generator can be accessed using ``MT19937``. ``Generator``, besides being NumPy-aware, has the advantage that it provides a much larger number of probability distributions to choose from. Examples -------- >>> from numpy.random import Generator, PCG64 >>> rng = Generator(PCG64()) >>> rng.standard_normal() -0.203 # random See Also -------- default_rng : Recommended constructor for `Generator`. """ cdef public object _bit_generator cdef bitgen_t _bitgen cdef binomial_t _binomial cdef object lock _poisson_lam_max = POISSON_LAM_MAX def __init__(self, bit_generator): self._bit_generator = bit_generator capsule = bit_generator.capsule cdef const char *name = "BitGenerator" if not PyCapsule_IsValid(capsule, name): raise ValueError("Invalid bit generator. The bit generator must " "be instantiated.") self._bitgen = ( PyCapsule_GetPointer(capsule, name))[0] self.lock = bit_generator.lock def __repr__(self): return self.__str__() + ' at 0x{:X}'.format(id(self)) def __str__(self): _str = self.__class__.__name__ _str += '(' + self.bit_generator.__class__.__name__ + ')' return _str # Pickling support: def __getstate__(self): return self.bit_generator.state def __setstate__(self, state): self.bit_generator.state = state def __reduce__(self): ctor, name_tpl, state = self._bit_generator.__reduce__() from ._pickle import __generator_ctor # Requirements of __generator_ctor are (name, ctor) return __generator_ctor, (name_tpl[0], ctor), state @property def bit_generator(self): """ Gets the bit generator instance used by the generator Returns ------- bit_generator : BitGenerator The bit generator instance used by the generator """ return self._bit_generator def spawn(self, int n_children): """ spawn(n_children) Create new independent child generators. See :ref:`seedsequence-spawn` for additional notes on spawning children. .. versionadded:: 1.25.0 Parameters ---------- n_children : int Returns ------- child_generators : list of Generators Raises ------ TypeError When the underlying SeedSequence does not implement spawning. See Also -------- random.BitGenerator.spawn, random.SeedSequence.spawn : Equivalent method on the bit generator and seed sequence. bit_generator : The bit generator instance used by the generator. Examples -------- Starting from a seeded default generator: >>> # High quality entropy created with: f"0x{secrets.randbits(128):x}" >>> entropy = 0x3034c61a9ae04ff8cb62ab8ec2c4b501 >>> rng = np.random.default_rng(entropy) Create two new generators for example for parallel executation: >>> child_rng1, child_rng2 = rng.spawn(2) Drawn numbers from each are independent but derived from the initial seeding entropy: >>> rng.uniform(), child_rng1.uniform(), child_rng2.uniform() (0.19029263503854454, 0.9475673279178444, 0.4702687338396767) It is safe to spawn additional children from the original ``rng`` or the children: >>> more_child_rngs = rng.spawn(20) >>> nested_spawn = child_rng1.spawn(20) """ return [type(self)(g) for g in self._bit_generator.spawn(n_children)] def random(self, size=None, dtype=np.float64, out=None): """ random(size=None, dtype=np.float64, out=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` use `uniform` or multiply the output of `random` by ``(b - a)`` and add ``a``:: (b - a) * random() + a Parameters ---------- 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. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). See Also -------- uniform : Draw samples from the parameterized uniform distribution. Examples -------- >>> rng = np.random.default_rng() >>> rng.random() 0.47108547995356098 # random >>> type(rng.random()) >>> rng.random((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random Three-by-two array of random numbers from [-5, 0): >>> 5 * rng.random((3, 2)) - 5 array([[-3.99149989, -0.52338984], # random [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) """ cdef double temp _dtype = np.dtype(dtype) if _dtype == np.float64: return double_fill(&random_standard_uniform_fill, &self._bitgen, size, self.lock, out) elif _dtype == np.float32: return float_fill(&random_standard_uniform_fill_f, &self._bitgen, size, self.lock, out) else: raise TypeError('Unsupported dtype %r for random' % _dtype) def beta(self, a, b, size=None): """ beta(a, b, size=None) Draw samples from a Beta distribution. The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function .. math:: f(x; a,b) = \\frac{1}{B(\\alpha, \\beta)} x^{\\alpha - 1} (1 - x)^{\\beta - 1}, where the normalization, B, is the beta function, .. math:: B(\\alpha, \\beta) = \\int_0^1 t^{\\alpha - 1} (1 - t)^{\\beta - 1} dt. It is often seen in Bayesian inference and order statistics. Parameters ---------- a : float or array_like of floats Alpha, positive (>0). b : float or array_like of floats Beta, positive (>0). 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 ``a`` and ``b`` are both scalars. Otherwise, ``np.broadcast(a, b).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized beta distribution. """ return cont(&random_beta, &self._bitgen, size, self.lock, 2, a, 'a', CONS_POSITIVE, b, 'b', CONS_POSITIVE, 0.0, '', CONS_NONE, None) def exponential(self, scale=1.0, size=None): """ exponential(scale=1.0, size=None) Draw samples from an exponential distribution. Its probability density function is .. math:: f(x; \\frac{1}{\\beta}) = \\frac{1}{\\beta} \\exp(-\\frac{x}{\\beta}), for ``x > 0`` and 0 elsewhere. :math:`\\beta` is the scale parameter, which is the inverse of the rate parameter :math:`\\lambda = 1/\\beta`. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_. The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_. Parameters ---------- scale : float or array_like of floats The scale parameter, :math:`\\beta = 1/\\lambda`. Must be non-negative. 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 ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized exponential distribution. Examples -------- A real world example: Assume a company has 10000 customer support agents and the average time between customer calls is 4 minutes. >>> n = 10000 >>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n) What is the probability that a customer will call in the next 4 to 5 minutes? >>> x = ((time_between_calls < 5).sum())/n >>> y = ((time_between_calls < 4).sum())/n >>> x-y 0.08 # may vary References ---------- .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57. .. [2] Wikipedia, "Poisson process", https://en.wikipedia.org/wiki/Poisson_process .. [3] Wikipedia, "Exponential distribution", https://en.wikipedia.org/wiki/Exponential_distribution """ return cont(&random_exponential, &self._bitgen, size, self.lock, 1, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def standard_exponential(self, size=None, dtype=np.float64, method='zig', out=None): """ standard_exponential(size=None, dtype=np.float64, method='zig', out=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. Parameters ---------- 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. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. method : str, optional Either 'inv' or 'zig'. 'inv' uses the default inverse CDF method. 'zig' uses the much faster Ziggurat method of Marsaglia and Tsang. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray Drawn samples. Examples -------- Output a 3x8000 array: >>> n = np.random.default_rng().standard_exponential((3, 8000)) """ _dtype = np.dtype(dtype) if _dtype == np.float64: if method == 'zig': return double_fill(&random_standard_exponential_fill, &self._bitgen, size, self.lock, out) else: return double_fill(&random_standard_exponential_inv_fill, &self._bitgen, size, self.lock, out) elif _dtype == np.float32: if method == 'zig': return float_fill(&random_standard_exponential_fill_f, &self._bitgen, size, self.lock, out) else: return float_fill(&random_standard_exponential_inv_fill_f, &self._bitgen, size, self.lock, out) else: raise TypeError('Unsupported dtype %r for standard_exponential' % _dtype) def integers(self, low, high=None, size=None, dtype=np.int64, endpoint=False): """ integers(low, high=None, size=None, dtype=np.int64, endpoint=False) Return random integers from `low` (inclusive) to `high` (exclusive), or if endpoint=True, `low` (inclusive) to `high` (inclusive). Replaces `RandomState.randint` (with endpoint=False) and `RandomState.random_integers` (with endpoint=True) Return random integers from the "discrete uniform" distribution of the specified dtype. If `high` is None (the default), then results are from 0 to `low`. Parameters ---------- low : int or array-like of ints Lowest (signed) integers to be drawn from the distribution (unless ``high=None``, in which case this parameter is 0 and this value is used for `high`). high : int or array-like of ints, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). If array-like, must contain integer values 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. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. Byteorder must be native. The default value is np.int64. endpoint : bool, optional If true, sample from the interval [low, high] instead of the default [low, high) Defaults to False Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. Notes ----- When using broadcasting with uint64 dtypes, the maximum value (2**64) cannot be represented as a standard integer type. The high array (or low if high is None) must have object dtype, e.g., array([2**64]). Examples -------- >>> rng = np.random.default_rng() >>> rng.integers(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random >>> rng.integers(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> rng.integers(5, size=(2, 4)) array([[4, 0, 2, 1], [3, 2, 2, 0]]) # random Generate a 1 x 3 array with 3 different upper bounds >>> rng.integers(1, [3, 5, 10]) array([2, 2, 9]) # random Generate a 1 by 3 array with 3 different lower bounds >>> rng.integers([1, 5, 7], 10) array([9, 8, 7]) # random Generate a 2 by 4 array using broadcasting with dtype of uint8 >>> rng.integers([1, 3, 5, 7], [[10], [20]], dtype=np.uint8) array([[ 8, 6, 9, 7], [ 1, 16, 9, 12]], dtype=uint8) # random References ---------- .. [1] Daniel Lemire., "Fast Random Integer Generation in an Interval", ACM Transactions on Modeling and Computer Simulation 29 (1), 2019, http://arxiv.org/abs/1805.10941. """ if high is None: high = low low = 0 _dtype = np.dtype(dtype) # Implementation detail: the old API used a masked method to generate # bounded uniform integers. Lemire's method is preferable since it is # faster. randomgen allows a choice, we will always use the faster one. cdef bint _masked = False if _dtype == np.int32: ret = _rand_int32(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.int64: ret = _rand_int64(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.int16: ret = _rand_int16(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.int8: ret = _rand_int8(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.uint64: ret = _rand_uint64(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.uint32: ret = _rand_uint32(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.uint16: ret = _rand_uint16(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.uint8: ret = _rand_uint8(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif _dtype == np.bool_: ret = _rand_bool(low, high, size, _masked, endpoint, &self._bitgen, self.lock) elif not _dtype.isnative: raise ValueError('Providing a dtype with a non-native byteorder ' 'is not supported. If you require ' 'platform-independent byteorder, call byteswap ' 'when required.') else: raise TypeError('Unsupported dtype %r for integers' % _dtype) if size is None and dtype in (bool, int): if np.array(ret).shape == (): return dtype(ret) return ret def bytes(self, np.npy_intp length): """ bytes(length) Return random bytes. Parameters ---------- length : int Number of random bytes. Returns ------- out : bytes String of length `length`. Examples -------- >>> np.random.default_rng().bytes(10) b'\\xfeC\\x9b\\x86\\x17\\xf2\\xa1\\xafcp' # random """ cdef Py_ssize_t n_uint32 = ((length - 1) // 4 + 1) # Interpret the uint32s as little-endian to convert them to bytes # consistently. return self.integers(0, 4294967296, size=n_uint32, dtype=np.uint32).astype('>> rng = np.random.default_rng() >>> rng.choice(5, 3) array([0, 3, 4]) # random >>> #This is equivalent to rng.integers(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> rng.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) # random Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> rng.choice(5, 3, replace=False) array([3,1,0]) # random >>> #This is equivalent to rng.permutation(np.arange(5))[:3] Generate a uniform random sample from a 2-D array along the first axis (the default), without replacement: >>> rng.choice([[0, 1, 2], [3, 4, 5], [6, 7, 8]], 2, replace=False) array([[3, 4, 5], # random [0, 1, 2]]) Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> rng.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) # random Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> rng.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random dtype='np.PyArray_FROM_OTF( p, np.NPY_DOUBLE, np.NPY_ALIGNED | np.NPY_ARRAY_C_CONTIGUOUS) pix = np.PyArray_DATA(p) if p.ndim != 1: raise ValueError("p must be 1-dimensional") if p.size != pop_size: raise ValueError("a and p must have same size") p_sum = kahan_sum(pix, d) if np.isnan(p_sum): raise ValueError("probabilities contain NaN") if np.logical_or.reduce(p < 0): raise ValueError("probabilities are not non-negative") if abs(p_sum - 1.) > atol: raise ValueError("probabilities do not sum to 1") # `shape == None` means `shape == ()`, but with scalar unpacking at the # end is_scalar = size is None if not is_scalar: shape = size size = np.prod(shape, dtype=np.intp) else: shape = () size = 1 # Actual sampling if replace: if p is not None: cdf = p.cumsum() cdf /= cdf[-1] uniform_samples = self.random(shape) idx = cdf.searchsorted(uniform_samples, side='right') # searchsorted returns a scalar idx = np.array(idx, copy=False, dtype=np.int64) else: idx = self.integers(0, pop_size, size=shape, dtype=np.int64) else: if size > pop_size: raise ValueError("Cannot take a larger sample than " "population when replace is False") elif size < 0: raise ValueError("negative dimensions are not allowed") if p is not None: if np.count_nonzero(p > 0) < size: raise ValueError("Fewer non-zero entries in p than size") n_uniq = 0 p = p.copy() found = np.zeros(shape, dtype=np.int64) flat_found = found.ravel() while n_uniq < size: x = self.random((size - n_uniq,)) if n_uniq > 0: p[flat_found[0:n_uniq]] = 0 cdf = np.cumsum(p) cdf /= cdf[-1] new = cdf.searchsorted(x, side='right') _, unique_indices = np.unique(new, return_index=True) unique_indices.sort() new = new.take(unique_indices) flat_found[n_uniq:n_uniq + new.size] = new n_uniq += new.size idx = found else: size_i = size pop_size_i = pop_size # This is a heuristic tuning. should be improvable if shuffle: cutoff = 50 else: cutoff = 20 if pop_size_i > 10000 and (size_i > (pop_size_i // cutoff)): # Tail shuffle size elements idx = np.PyArray_Arange(0, pop_size_i, 1, np.NPY_INT64) idx_data = (idx).data with self.lock, nogil: _shuffle_int(&self._bitgen, pop_size_i, max(pop_size_i - size_i, 1), idx_data) # Copy to allow potentially large array backing idx to be gc idx = idx[(pop_size - size):].copy() else: # Floyd's algorithm idx = np.empty(size, dtype=np.int64) idx_data = np.PyArray_DATA(idx) # smallest power of 2 larger than 1.2 * size set_size = (1.2 * size_i) mask = _gen_mask(set_size) set_size = 1 + mask hash_set = np.full(set_size, -1, np.uint64) with self.lock, cython.wraparound(False), nogil: for j in range(pop_size_i - size_i, pop_size_i): val = random_bounded_uint64(&self._bitgen, 0, j, 0, 0) loc = val & mask while hash_set[loc] != -1 and hash_set[loc] != val: loc = (loc + 1) & mask if hash_set[loc] == -1: # then val not in hash_set hash_set[loc] = val idx_data[j - pop_size_i + size_i] = val else: # we need to insert j instead loc = j & mask while hash_set[loc] != -1: loc = (loc + 1) & mask hash_set[loc] = j idx_data[j - pop_size_i + size_i] = j if shuffle: _shuffle_int(&self._bitgen, size_i, 1, idx_data) idx.shape = shape if is_scalar and isinstance(idx, np.ndarray): # In most cases a scalar will have been made an array idx = idx.item(0) # Use samples as indices for a if a is array-like if a.ndim == 0: return idx if not is_scalar and idx.ndim == 0: # If size == () then the user requested a 0-d array as opposed to # a scalar object when size is None. However a[idx] is always a # scalar and not an array. So this makes sure the result is an # array, taking into account that np.array(item) may not work # for object arrays. res = np.empty((), dtype=a.dtype) res[()] = a[idx] return res # asarray downcasts on 32-bit platforms, always safe # no-op on 64-bit platforms return a.take(np.asarray(idx, dtype=np.intp), axis=axis) def uniform(self, low=0.0, high=1.0, size=None): """ uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. high - low must be non-negative. The default value is 1.0. 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 ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- integers : Discrete uniform distribution, yielding integers. random : Floats uniformly distributed over ``[0, 1)``. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \\frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. Examples -------- Draw samples from the distribution: >>> s = np.random.default_rng().uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 15, density=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() """ cdef bint is_scalar = True cdef np.ndarray alow, ahigh, arange cdef double _low, _high, rng cdef object temp alow = np.PyArray_FROM_OTF(low, np.NPY_DOUBLE, np.NPY_ALIGNED) ahigh = np.PyArray_FROM_OTF(high, np.NPY_DOUBLE, np.NPY_ALIGNED) if np.PyArray_NDIM(alow) == np.PyArray_NDIM(ahigh) == 0: _low = PyFloat_AsDouble(low) _high = PyFloat_AsDouble(high) rng = _high - _low if not np.isfinite(rng): raise OverflowError('high - low range exceeds valid bounds') return cont(&random_uniform, &self._bitgen, size, self.lock, 2, _low, '', CONS_NONE, rng, 'high - low', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) temp = np.subtract(ahigh, alow) # needed to get around Pyrex's automatic reference-counting # rules because EnsureArray steals a reference Py_INCREF(temp) arange = np.PyArray_EnsureArray(temp) if not np.all(np.isfinite(arange)): raise OverflowError('Range exceeds valid bounds') return cont(&random_uniform, &self._bitgen, size, self.lock, 2, alow, '', CONS_NONE, arange, 'high - low', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) # Complicated, continuous distributions: def standard_normal(self, size=None, dtype=np.float64, out=None): """ standard_normal(size=None, dtype=np.float64, out=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). Parameters ---------- 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. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray A floating-point array of shape ``size`` of drawn samples, or a single sample if ``size`` was not specified. See Also -------- normal : Equivalent function with additional ``loc`` and ``scale`` arguments for setting the mean and standard deviation. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use one of:: mu + sigma * rng.standard_normal(size=...) rng.normal(mu, sigma, size=...) Examples -------- >>> rng = np.random.default_rng() >>> rng.standard_normal() 2.1923875335537315 # random >>> s = rng.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random -0.38672696, -0.4685006 ]) # random >>> s.shape (8000,) >>> s = rng.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * rng.standard_normal(size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random """ _dtype = np.dtype(dtype) if _dtype == np.float64: return double_fill(&random_standard_normal_fill, &self._bitgen, size, self.lock, out) elif _dtype == np.float32: return float_fill(&random_standard_normal_fill_f, &self._bitgen, size, self.lock, out) else: raise TypeError('Unsupported dtype %r for standard_normal' % _dtype) def normal(self, loc=0.0, scale=1.0, size=None): """ normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative. 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. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \\frac{1}{\\sqrt{ 2 \\pi \\sigma^2 }} e^{ - \\frac{ (x - \\mu)^2 } {2 \\sigma^2} }, where :math:`\\mu` is the mean and :math:`\\sigma` the standard deviation. The square of the standard deviation, :math:`\\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \\sigma` and :math:`x - \\sigma` [2]_). This implies that :meth:`normal` is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> s = np.random.default_rng().normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) 0.0 # may vary >>> abs(sigma - np.std(s, ddof=1)) 0.0 # may vary Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> np.random.default_rng().normal(3, 2.5, size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random """ return cont(&random_normal, &self._bitgen, size, self.lock, 2, loc, '', CONS_NONE, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def standard_gamma(self, shape, size=None, dtype=np.float64, out=None): """ standard_gamma(shape, size=None, dtype=np.float64, out=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. Parameters ---------- shape : float or array_like of floats Parameter, must be non-negative. 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 ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\\frac{e^{-x/\\theta}}{\\theta^k\\Gamma(k)}, where :math:`k` is the shape and :math:`\\theta` the scale, and :math:`\\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> s = np.random.default_rng().standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() """ cdef void *func _dtype = np.dtype(dtype) if _dtype == np.float64: return cont(&random_standard_gamma, &self._bitgen, size, self.lock, 1, shape, 'shape', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, out) if _dtype == np.float32: return cont_f(&random_standard_gamma_f, &self._bitgen, size, self.lock, shape, 'shape', CONS_NON_NEGATIVE, out) else: raise TypeError('Unsupported dtype %r for standard_gamma' % _dtype) def gamma(self, shape, scale=1.0, size=None): """ gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Must be non-negative. scale : float or array_like of floats, optional The scale of the gamma distribution. Must be non-negative. Default is equal to 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 ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\\frac{e^{-x/\\theta}}{\\theta^k\\Gamma(k)}, where :math:`k` is the shape and :math:`\\theta` the scale, and :math:`\\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> s = np.random.default_rng().gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() """ return cont(&random_gamma, &self._bitgen, size, self.lock, 2, shape, 'shape', CONS_NON_NEGATIVE, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def f(self, dfnum, dfden, size=None): """ f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters must be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. Parameters ---------- dfnum : float or array_like of floats Degrees of freedom in numerator, must be > 0. dfden : float or array_like of float Degrees of freedom in denominator, must be > 0. 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 ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", https://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> s = np.random.default_rng().f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> np.sort(s)[-10] 7.61988120985 # random So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. """ return cont(&random_f, &self._bitgen, size, self.lock, 2, dfnum, 'dfnum', CONS_POSITIVE, dfden, 'dfden', CONS_POSITIVE, 0.0, '', CONS_NONE, None) def noncentral_f(self, dfnum, dfden, nonc, size=None): """ noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. Parameters ---------- dfnum : float or array_like of floats Numerator degrees of freedom, must be > 0. .. versionchanged:: 1.14.0 Earlier NumPy versions required dfnum > 1. dfden : float or array_like of floats Denominator degrees of freedom, must be > 0. nonc : float or array_like of floats Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0. 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 ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", https://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> rng = np.random.default_rng() >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = rng.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, density=True) >>> c_vals = rng.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, density=True) >>> import matplotlib.pyplot as plt >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() """ return cont(&random_noncentral_f, &self._bitgen, size, self.lock, 3, dfnum, 'dfnum', CONS_POSITIVE, dfden, 'dfden', CONS_POSITIVE, nonc, 'nonc', CONS_NON_NEGATIVE, None) def chisquare(self, df, size=None): """ chisquare(df, size=None) Draw samples from a chi-square distribution. When `df` independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing. Parameters ---------- df : float or array_like of floats Number of degrees of freedom, must be > 0. 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 ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \\sum_{i=0}^{\\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \\sim \\chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \\frac{(1/2)^{k/2}}{\\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\\Gamma` is the gamma function, .. math:: \\Gamma(x) = \\int_0^{-\\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> np.random.default_rng().chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random """ return cont(&random_chisquare, &self._bitgen, size, self.lock, 1, df, 'df', CONS_POSITIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def noncentral_chisquare(self, df, nonc, size=None): """ noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\\chi^2` distribution is a generalization of the :math:`\\chi^2` distribution. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. .. versionchanged:: 1.10.0 Earlier NumPy versions required dfnum > 1. nonc : float or array_like of floats Non-centrality, must be non-negative. 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 ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \\sum^{\\infty}_{i=0} \\frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. References ---------- .. [1] Wikipedia, "Noncentral chi-squared distribution" https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution Examples -------- Draw values from the distribution and plot the histogram >>> rng = np.random.default_rng() >>> import matplotlib.pyplot as plt >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(rng.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> values2 = plt.hist(rng.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() """ return cont(&random_noncentral_chisquare, &self._bitgen, size, self.lock, 2, df, 'df', CONS_POSITIVE, nonc, 'nonc', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def standard_cauchy(self, size=None): """ standard_cauchy(size=None) Draw samples from a standard Cauchy distribution with mode = 0. Also known as the Lorentz distribution. Parameters ---------- 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. Default is None, in which case a single value is returned. Returns ------- samples : ndarray or scalar The drawn samples. Notes ----- The probability density function for the full Cauchy distribution is .. math:: P(x; x_0, \\gamma) = \\frac{1}{\\pi \\gamma \\bigl[ 1+ (\\frac{x-x_0}{\\gamma})^2 \\bigr] } and the Standard Cauchy distribution just sets :math:`x_0=0` and :math:`\\gamma=1` The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails. References ---------- .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy Distribution", https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html .. [3] Wikipedia, "Cauchy distribution" https://en.wikipedia.org/wiki/Cauchy_distribution Examples -------- Draw samples and plot the distribution: >>> import matplotlib.pyplot as plt >>> s = np.random.default_rng().standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() """ return cont(&random_standard_cauchy, &self._bitgen, size, self.lock, 0, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def standard_t(self, df, size=None): """ standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. 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 ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \\frac{\\Gamma(\\frac{df+1}{2})}{\\sqrt{\\pi df} \\Gamma(\\frac{df}{2})}\\Bigl( 1+\\frac{x^2}{df} \\Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" https://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \\ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed. Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root). >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> t -2.8207540608310198 We draw 1000000 samples from Student's t distribution with the adequate degrees of freedom. >>> import matplotlib.pyplot as plt >>> s = np.random.default_rng().standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) Does our t statistic land in one of the two critical regions found at both tails of the distribution? >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) 0.018318 #random < 0.05, statistic is in critical region The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold. Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation. """ return cont(&random_standard_t, &self._bitgen, size, self.lock, 1, df, 'df', CONS_POSITIVE, 0, '', CONS_NONE, 0, '', CONS_NONE, None) def vonmises(self, mu, kappa, size=None): """ vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. 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 ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \\frac{e^{\\kappa cos(x-\\mu)}}{2\\pi I_0(\\kappa)}, where :math:`\\mu` is the mode and :math:`\\kappa` the dispersion, and :math:`I_0(\\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> s = np.random.default_rng().vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 # doctest: +SKIP >>> plt.hist(s, 50, density=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP >>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() """ return cont(&random_vonmises, &self._bitgen, size, self.lock, 2, mu, 'mu', CONS_NONE, kappa, 'kappa', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def pareto(self, a, size=None): """ pareto(a, size=None) Draw samples from a Pareto II or Lomax distribution with specified shape. The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter ``m`` (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is ``mu``, where the standard Pareto distribution has location ``mu = 1``. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero. The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the "80-20 rule". In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range. Parameters ---------- a : float or array_like of floats Shape of the distribution. Must be positive. 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 ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Pareto distribution. See Also -------- scipy.stats.lomax : probability density function, distribution or cumulative density function, etc. scipy.stats.genpareto : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Pareto distribution is .. math:: p(x) = \\frac{am^a}{x^{a+1}} where :math:`a` is the shape and :math:`m` the scale. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called "fat-tailed" distributions. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", https://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a, m = 3., 2. # shape and mode >>> s = (np.random.default_rng().pareto(a, 1000) + 1) * m Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, density=True) >>> fit = a*m**a / bins**(a+1) >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r') >>> plt.show() """ return cont(&random_pareto, &self._bitgen, size, self.lock, 1, a, 'a', CONS_POSITIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def weibull(self, a, size=None): """ weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\\lambda` is just :math:`X = \\lambda(-ln(U))^{1/a}`. Parameters ---------- a : float or array_like of floats Shape parameter of the distribution. Must be nonnegative. 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 ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \\frac{a} {\\lambda}(\\frac{x}{\\lambda})^{a-1}e^{-(x/\\lambda)^a}, where :math:`a` is the shape and :math:`\\lambda` the scale. The function has its peak (the mode) at :math:`\\lambda(\\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", https://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> a = 5. # shape >>> s = rng.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.arange(1,100.)/50. >>> def weib(x,n,a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, ignored = plt.hist(rng.weibull(5.,1000)) >>> x = np.arange(1,100.)/50. >>> scale = count.max()/weib(x, 1., 5.).max() >>> plt.plot(x, weib(x, 1., 5.)*scale) >>> plt.show() """ return cont(&random_weibull, &self._bitgen, size, self.lock, 1, a, 'a', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def power(self, a, size=None): """ power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Must be non-negative. 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 ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a <= 0. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \\le x \\le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> a = 5. # shape >>> samples = 1000 >>> s = rng.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats # doctest: +SKIP >>> rvs = rng.power(5, 1000000) >>> rvsp = rng.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP >>> plt.figure() >>> plt.hist(rvs, bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of 1 + Generator.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of stats.pareto(5)') """ return cont(&random_power, &self._bitgen, size, self.lock, 1, a, 'a', CONS_POSITIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def laplace(self, loc=0.0, scale=1.0, size=None): """ laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\\lambda`, the exponential decay. Default is 1. Must be non- negative. 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. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. Notes ----- It has the probability density function .. math:: f(x; \\mu, \\lambda) = \\frac{1}{2\\lambda} \\exp\\left(-\\frac{|x - \\mu|}{\\lambda}\\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", https://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> s = np.random.default_rng().laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) """ return cont(&random_laplace, &self._bitgen, size, self.lock, 2, loc, 'loc', CONS_NONE, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def gumbel(self, loc=0.0, scale=1.0, size=None): """ gumbel(loc=0.0, scale=1.0, size=None) 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. Must be non- negative. 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. Returns ------- out : ndarray 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: >>> rng = np.random.default_rng() >>> mu, beta = 0, 0.1 # location and scale >>> s = rng.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() 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 = rng.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, ignored = plt.hist(maxima, 30, density=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() """ return cont(&random_gumbel, &self._bitgen, size, self.lock, 2, loc, 'loc', CONS_NONE, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def logistic(self, loc=0.0, scale=1.0, size=None): """ logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Must be non-negative. 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. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = P(x) = \\frac{e^{-(x-\\mu)/s}}{s(1+e^{-(x-\\mu)/s})^2}, where :math:`\\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", https://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> s = np.random.default_rng().logistic(loc, scale, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=50) # plot against distribution >>> def logist(x, loc, scale): ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2) >>> lgst_val = logist(bins, loc, scale) >>> plt.plot(bins, lgst_val * count.max() / lgst_val.max()) >>> plt.show() """ return cont(&random_logistic, &self._bitgen, size, self.lock, 2, loc, 'loc', CONS_NONE, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def lognormal(self, mean=0.0, sigma=1.0, size=None): """ lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Must be non-negative. 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 ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \\frac{1}{\\sigma x \\sqrt{2\\pi}} e^{(-\\frac{(ln(x)-\\mu)^2}{2\\sigma^2})} where :math:`\\mu` is the mean and :math:`\\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = rng.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> rng = rng >>> b = [] >>> for i in range(1000): ... a = 10. + rng.standard_normal(100) ... b.append(np.prod(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() """ return cont(&random_lognormal, &self._bitgen, size, self.lock, 2, mean, 'mean', CONS_NONE, sigma, 'sigma', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, None) def rayleigh(self, scale=1.0, size=None): """ rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\\chi` and Weibull distributions are generalizations of the Rayleigh. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Must be non-negative. 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 ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \\frac{x}{scale^2}e^{\\frac{-x^2}{2 \\cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" https://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> from matplotlib.pyplot import hist >>> rng = np.random.default_rng() >>> values = hist(rng.rayleigh(3, 100000), bins=200, density=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = rng.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 # random """ return cont(&random_rayleigh, &self._bitgen, size, self.lock, 1, scale, 'scale', CONS_NON_NEGATIVE, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE, None) def wald(self, mean, scale, size=None): """ wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. Parameters ---------- mean : float or array_like of floats Distribution mean, must be > 0. scale : float or array_like of floats Scale parameter, must be > 0. 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 ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \\sqrt{\\frac{scale}{2\\pi x^3}}e^ \\frac{-scale(x-mean)^2}{2\\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Inverse Gaussian distribution" https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.default_rng().wald(3, 2, 100000), bins=200, density=True) >>> plt.show() """ return cont(&random_wald, &self._bitgen, size, self.lock, 2, mean, 'mean', CONS_POSITIVE, scale, 'scale', CONS_POSITIVE, 0.0, '', CONS_NONE, None) def triangular(self, left, mode, right, size=None): """ triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value must fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, must be larger than `left`. 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 ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \\begin{cases} \\frac{2(x-l)}{(r-l)(m-l)}& \\text{for $l \\leq x \\leq m$},\\\\ \\frac{2(r-x)}{(r-l)(r-m)}& \\text{for $m \\leq x \\leq r$},\\\\ 0& \\text{otherwise}. \\end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" https://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.default_rng().triangular(-3, 0, 8, 100000), bins=200, ... density=True) >>> plt.show() """ cdef bint is_scalar = True cdef double fleft, fmode, fright cdef np.ndarray oleft, omode, oright oleft = np.PyArray_FROM_OTF(left, np.NPY_DOUBLE, np.NPY_ALIGNED) omode = np.PyArray_FROM_OTF(mode, np.NPY_DOUBLE, np.NPY_ALIGNED) oright = np.PyArray_FROM_OTF(right, np.NPY_DOUBLE, np.NPY_ALIGNED) if np.PyArray_NDIM(oleft) == np.PyArray_NDIM(omode) == np.PyArray_NDIM(oright) == 0: fleft = PyFloat_AsDouble(left) fright = PyFloat_AsDouble(right) fmode = PyFloat_AsDouble(mode) if fleft > fmode: raise ValueError("left > mode") if fmode > fright: raise ValueError("mode > right") if fleft == fright: raise ValueError("left == right") return cont(&random_triangular, &self._bitgen, size, self.lock, 3, fleft, '', CONS_NONE, fmode, '', CONS_NONE, fright, '', CONS_NONE, None) if np.any(np.greater(oleft, omode)): raise ValueError("left > mode") if np.any(np.greater(omode, oright)): raise ValueError("mode > right") if np.any(np.equal(oleft, oright)): raise ValueError("left == right") return cont_broadcast_3(&random_triangular, &self._bitgen, size, self.lock, oleft, '', CONS_NONE, omode, '', CONS_NONE, oright, '', CONS_NONE) # Complicated, discrete distributions: def binomial(self, n, p, size=None): """ binomial(n, p, size=None) Draw samples from a binomial distribution. Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use) Parameters ---------- n : int or array_like of ints Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers. p : float or array_like of floats Parameter of the distribution, >= 0 and <=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 ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials. See Also -------- scipy.stats.binom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the binomial distribution is .. math:: P(N) = \\binom{n}{N}p^N(1-p)^{n-N}, where :math:`n` is the number of trials, :math:`p` is the probability of success, and :math:`N` is the number of successes. When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics with R", Springer-Verlag, 2002. .. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html .. [5] Wikipedia, "Binomial distribution", https://en.wikipedia.org/wiki/Binomial_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> n, p = 10, .5 # number of trials, probability of each trial >>> s = rng.binomial(n, p, 1000) # result of flipping a coin 10 times, tested 1000 times. A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening? Let's do 20,000 trials of the model, and count the number that generate zero positive results. >>> sum(rng.binomial(9, 0.1, 20000) == 0)/20000. # answer = 0.38885, or 39%. """ # Uses a custom implementation since self._binomial is required cdef double _dp = 0 cdef int64_t _in = 0 cdef bint is_scalar = True cdef np.npy_intp i, cnt cdef np.ndarray randoms cdef np.int64_t *randoms_data cdef np.broadcast it p_arr = np.PyArray_FROM_OTF(p, np.NPY_DOUBLE, np.NPY_ALIGNED) is_scalar = is_scalar and np.PyArray_NDIM(p_arr) == 0 n_arr = np.PyArray_FROM_OTF(n, np.NPY_INT64, np.NPY_ALIGNED) is_scalar = is_scalar and np.PyArray_NDIM(n_arr) == 0 if not is_scalar: check_array_constraint(p_arr, 'p', CONS_BOUNDED_0_1) check_array_constraint(n_arr, 'n', CONS_NON_NEGATIVE) if size is not None: randoms = np.empty(size, np.int64) else: it = np.PyArray_MultiIterNew2(p_arr, n_arr) randoms = np.empty(it.shape, np.int64) cnt = np.PyArray_SIZE(randoms) it = np.PyArray_MultiIterNew3(randoms, p_arr, n_arr) validate_output_shape(it.shape, randoms) with self.lock, nogil: for i in range(cnt): _dp = (np.PyArray_MultiIter_DATA(it, 1))[0] _in = (np.PyArray_MultiIter_DATA(it, 2))[0] (np.PyArray_MultiIter_DATA(it, 0))[0] = random_binomial(&self._bitgen, _dp, _in, &self._binomial) np.PyArray_MultiIter_NEXT(it) return randoms _dp = PyFloat_AsDouble(p) _in = n check_constraint(_dp, 'p', CONS_BOUNDED_0_1) check_constraint(_in, 'n', CONS_NON_NEGATIVE) if size is None: with self.lock: return random_binomial(&self._bitgen, _dp, _in, &self._binomial) randoms = np.empty(size, np.int64) cnt = np.PyArray_SIZE(randoms) randoms_data = np.PyArray_DATA(randoms) with self.lock, nogil: for i in range(cnt): randoms_data[i] = random_binomial(&self._bitgen, _dp, _in, &self._binomial) return randoms def negative_binomial(self, n, p, size=None): """ negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` successes and `p` probability of success where `n` is > 0 and `p` is in the interval (0, 1]. Parameters ---------- n : float or array_like of floats Parameter of the distribution, > 0. p : float or array_like of floats Parameter of the distribution. Must satisfy 0 < p <= 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 ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached. Notes ----- The probability mass function of the negative binomial distribution is .. math:: P(N;n,p) = \\frac{\\Gamma(N+n)}{N!\\Gamma(n)}p^{n}(1-p)^{N}, where :math:`n` is the number of successes, :math:`p` is the probability of success, :math:`N+n` is the number of trials, and :math:`\\Gamma` is the gamma function. When :math:`n` is an integer, :math:`\\frac{\\Gamma(N+n)}{N!\\Gamma(n)} = \\binom{N+n-1}{N}`, which is the more common form of this term in the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. Because this method internally calls ``Generator.poisson`` with an intermediate random value, a ValueError is raised when the choice of :math:`n` and :math:`p` would result in the mean + 10 sigma of the sampled intermediate distribution exceeding the max acceptable value of the ``Generator.poisson`` method. This happens when :math:`p` is too low (a lot of failures happen for every success) and :math:`n` is too big ( a lot of successes are allowed). Therefore, the :math:`n` and :math:`p` values must satisfy the constraint: .. math:: n\\frac{1-p}{p}+10n\\sqrt{n}\\frac{1-p}{p}<2^{63}-1-10\\sqrt{2^{63}-1}, Where the left side of the equation is the derived mean + 10 sigma of a sample from the gamma distribution internally used as the :math:`lam` parameter of a poisson sample, and the right side of the equation is the constraint for maximum value of :math:`lam` in ``Generator.poisson``. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", https://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> s = np.random.default_rng().negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): # doctest: +SKIP ... probability = sum(snp.PyArray_FROM_OTF(p, np.NPY_DOUBLE, np.NPY_ALIGNED) is_scalar = is_scalar and np.PyArray_NDIM(p_arr) == 0 n_arr = np.PyArray_FROM_OTF(n, np.NPY_DOUBLE, np.NPY_ALIGNED) is_scalar = is_scalar and np.PyArray_NDIM(n_arr) == 0 if not is_scalar: check_array_constraint(n_arr, 'n', CONS_POSITIVE_NOT_NAN) check_array_constraint(p_arr, 'p', CONS_BOUNDED_GT_0_1) # Check that the choice of negative_binomial parameters won't result in a # call to the poisson distribution function with a value of lam too large. max_lam_arr = (1 - p_arr) / p_arr * (n_arr + 10 * np.sqrt(n_arr)) if np.any(np.greater(max_lam_arr, POISSON_LAM_MAX)): raise ValueError("n too large or p too small, see Generator.negative_binomial Notes") else: _dn = np.PyArray_DATA(n_arr) _dp = np.PyArray_DATA(p_arr) check_constraint(_dn[0], 'n', CONS_POSITIVE_NOT_NAN) check_constraint(_dp[0], 'p', CONS_BOUNDED_GT_0_1) # Check that the choice of negative_binomial parameters won't result in a # call to the poisson distribution function with a value of lam too large. _dmax_lam = (1 - _dp[0]) / _dp[0] * (_dn[0] + 10 * sqrt(_dn[0])) if _dmax_lam > POISSON_LAM_MAX: raise ValueError("n too large or p too small, see Generator.negative_binomial Notes") return disc(&random_negative_binomial, &self._bitgen, size, self.lock, 2, 0, n_arr, 'n', CONS_NONE, p_arr, 'p', CONS_NONE, 0.0, '', CONS_NONE) def poisson(self, lam=1.0, size=None): """ poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. Parameters ---------- lam : float or array_like of floats Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size. 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 ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. Notes ----- The Poisson distribution .. math:: f(k; \\lambda)=\\frac{\\lambda^k e^{-\\lambda}}{k!} For events with an expected separation :math:`\\lambda` the Poisson distribution :math:`f(k; \\lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\\lambda`. Because the output is limited to the range of the C int64 type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", https://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> import numpy as np >>> rng = np.random.default_rng() >>> s = rng.poisson(5, 10000) Display histogram of the sample: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 14, density=True) >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = rng.poisson(lam=(100., 500.), size=(100, 2)) """ return disc(&random_poisson, &self._bitgen, size, self.lock, 1, 0, lam, 'lam', CONS_POISSON, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE) def zipf(self, a, size=None): """ zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. Parameters ---------- a : float or array_like of floats Distribution parameter. Must be greater than 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 ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the Zipf distribution is .. math:: p(k) = \\frac{k^{-a}}{\\zeta(a)}, for integers :math:`k \geq 1`, where :math:`\\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 4.0 >>> n = 20000 >>> s = np.random.default_rng().zipf(a, size=n) Display the histogram of the samples, along with the expected histogram based on the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import zeta # doctest: +SKIP `bincount` provides a fast histogram for small integers. >>> count = np.bincount(s) >>> k = np.arange(1, s.max() + 1) >>> plt.bar(k, count[1:], alpha=0.5, label='sample count') >>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5, ... label='expected count') # doctest: +SKIP >>> plt.semilogy() >>> plt.grid(alpha=0.4) >>> plt.legend() >>> plt.title(f'Zipf sample, a={a}, size={n}') >>> plt.show() """ return disc(&random_zipf, &self._bitgen, size, self.lock, 1, 0, a, 'a', CONS_GT_1, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE) def geometric(self, p, size=None): """ geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. 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 ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. Examples -------- Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35: >>> z = np.random.default_rng().geometric(p=0.35, size=10000) How many trials succeeded after a single run? >>> (z == 1).sum() / 10000. 0.34889999999999999 # random """ return disc(&random_geometric, &self._bitgen, size, self.lock, 1, 0, p, 'p', CONS_BOUNDED_GT_0_1, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE) def hypergeometric(self, ngood, nbad, nsample, size=None): """ hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, `ngood` (ways to make a good selection), `nbad` (ways to make a bad selection), and `nsample` (number of items sampled, which is less than or equal to the sum ``ngood + nbad``). Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative and less than 10**9. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative and less than 10**9. nsample : int or array_like of ints Number of items sampled. Must be nonnegative and less than ``ngood + nbad``. 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 `ngood`, `nbad`, and `nsample` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size `nsample` taken from a set of `ngood` good items and `nbad` bad items. See Also -------- multivariate_hypergeometric : Draw samples from the multivariate hypergeometric distribution. scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \\frac{\\binom{g}{x}\\binom{b}{n-x}}{\\binom{g+b}{n}}, where :math:`0 \\le x \\le n` and :math:`n-b \\le x \\le g` for P(x) the probability of ``x`` good results in the drawn sample, g = `ngood`, b = `nbad`, and n = `nsample`. Consider an urn with black and white marbles in it, `ngood` of them are black and `nbad` are white. If you draw `nsample` balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. The arguments `ngood` and `nbad` each must be less than `10**9`. For extremely large arguments, the algorithm that is used to compute the samples [4]_ breaks down because of loss of precision in floating point calculations. For such large values, if `nsample` is not also large, the distribution can be approximated with the binomial distribution, `binomial(n=nsample, p=ngood/(ngood + nbad))`. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", https://en.wikipedia.org/wiki/Hypergeometric_distribution .. [4] Stadlober, Ernst, "The ratio of uniforms approach for generating discrete random variates", Journal of Computational and Applied Mathematics, 31, pp. 181-189 (1990). Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = rng.hypergeometric(ngood, nbad, nsamp, 1000) >>> from matplotlib.pyplot import hist >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = rng.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! """ DEF HYPERGEOM_MAX = 10**9 cdef bint is_scalar = True cdef np.ndarray ongood, onbad, onsample cdef int64_t lngood, lnbad, lnsample ongood = np.PyArray_FROM_OTF(ngood, np.NPY_INT64, np.NPY_ALIGNED) onbad = np.PyArray_FROM_OTF(nbad, np.NPY_INT64, np.NPY_ALIGNED) onsample = np.PyArray_FROM_OTF(nsample, np.NPY_INT64, np.NPY_ALIGNED) if np.PyArray_NDIM(ongood) == np.PyArray_NDIM(onbad) == np.PyArray_NDIM(onsample) == 0: lngood = ngood lnbad = nbad lnsample = nsample if lngood >= HYPERGEOM_MAX or lnbad >= HYPERGEOM_MAX: raise ValueError("both ngood and nbad must be less than %d" % HYPERGEOM_MAX) if lngood + lnbad < lnsample: raise ValueError("ngood + nbad < nsample") return disc(&random_hypergeometric, &self._bitgen, size, self.lock, 0, 3, lngood, 'ngood', CONS_NON_NEGATIVE, lnbad, 'nbad', CONS_NON_NEGATIVE, lnsample, 'nsample', CONS_NON_NEGATIVE) if np.any(ongood >= HYPERGEOM_MAX) or np.any(onbad >= HYPERGEOM_MAX): raise ValueError("both ngood and nbad must be less than %d" % HYPERGEOM_MAX) if np.any(np.less(np.add(ongood, onbad), onsample)): raise ValueError("ngood + nbad < nsample") return discrete_broadcast_iii(&random_hypergeometric, &self._bitgen, size, self.lock, ongood, 'ngood', CONS_NON_NEGATIVE, onbad, 'nbad', CONS_NON_NEGATIVE, onsample, 'nsample', CONS_NON_NEGATIVE) def logseries(self, p, size=None): """ logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 <= ``p`` < 1. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range [0, 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 ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. Notes ----- The probability mass function for the Log Series distribution is .. math:: P(k) = \\frac{-p^k}{k \\ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", https://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> s = np.random.default_rng().logseries(a, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s) # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*np.log(1-p)) >>> plt.plot(bins, logseries(bins, a) * count.max()/ ... logseries(bins, a).max(), 'r') >>> plt.show() """ return disc(&random_logseries, &self._bitgen, size, self.lock, 1, 0, p, 'p', CONS_BOUNDED_LT_0_1, 0.0, '', CONS_NONE, 0.0, '', CONS_NONE) # Multivariate distributions: def multivariate_normal(self, mean, cov, size=None, check_valid='warn', tol=1e-8, *, method='svd'): """ multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8, *, method='svd') Draw random samples from a multivariate normal distribution. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or "center") and variance (the squared standard deviation, or "width") of the one-dimensional normal distribution. Parameters ---------- mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``. If no shape is specified, a single (`N`-D) sample is returned. check_valid : { 'warn', 'raise', 'ignore' }, optional Behavior when the covariance matrix is not positive semidefinite. tol : float, optional Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check. method : { 'svd', 'eigh', 'cholesky'}, optional The cov input is used to compute a factor matrix A such that ``A @ A.T = cov``. This argument is used to select the method used to compute the factor matrix A. The default method 'svd' is the slowest, while 'cholesky' is the fastest but less robust than the slowest method. The method `eigh` uses eigen decomposition to compute A and is faster than svd but slower than cholesky. .. versionadded:: 1.18.0 Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. Notes ----- The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution. Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its "spread"). Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements, and only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> x, y = np.random.default_rng().multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. This function internally uses linear algebra routines, and thus results may not be identical (even up to precision) across architectures, OSes, or even builds. For example, this is likely if ``cov`` has multiple equal singular values and ``method`` is ``'svd'`` (default). In this case, ``method='cholesky'`` may be more robust. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> rng = np.random.default_rng() >>> x = rng.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) We can use a different method other than the default to factorize cov: >>> y = rng.multivariate_normal(mean, cov, (3, 3), method='cholesky') >>> y.shape (3, 3, 2) Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465. >>> cov = np.array([[6, -3], [-3, 3.5]]) >>> pts = rng.multivariate_normal([0, 0], cov, size=800) Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values: >>> pts.mean(axis=0) array([ 0.0326911 , -0.01280782]) # may vary >>> np.cov(pts.T) array([[ 5.96202397, -2.85602287], [-2.85602287, 3.47613949]]) # may vary >>> np.corrcoef(pts.T)[0, 1] -0.6273591314603949 # may vary We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample. >>> import matplotlib.pyplot as plt >>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5) >>> plt.axis('equal') >>> plt.grid() >>> plt.show() """ if method not in {'eigh', 'svd', 'cholesky'}: raise ValueError( "method must be one of {'eigh', 'svd', 'cholesky'}") # Check preconditions on arguments mean = np.array(mean) cov = np.array(cov) if (np.issubdtype(mean.dtype, np.complexfloating) or np.issubdtype(cov.dtype, np.complexfloating)): raise TypeError("mean and cov must not be complex") if size is None: shape = [] elif isinstance(size, (int, long, np.integer)): shape = [size] else: shape = size if len(mean.shape) != 1: raise ValueError("mean must be 1 dimensional") if (len(cov.shape) != 2) or (cov.shape[0] != cov.shape[1]): raise ValueError("cov must be 2 dimensional and square") if mean.shape[0] != cov.shape[0]: raise ValueError("mean and cov must have same length") # Compute shape of output and create a matrix of independent # standard normally distributed random numbers. The matrix has rows # with the same length as mean and as many rows are necessary to # form a matrix of shape final_shape. final_shape = list(shape[:]) final_shape.append(mean.shape[0]) x = self.standard_normal(final_shape).reshape(-1, mean.shape[0]) # Transform matrix of standard normals into matrix where each row # contains multivariate normals with the desired covariance. # Compute A such that dot(transpose(A),A) == cov. # Then the matrix products of the rows of x and A has the desired # covariance. Note that sqrt(s)*v where (u,s,v) is the singular value # decomposition of cov is such an A. # # Also check that cov is positive-semidefinite. If so, the u.T and v # matrices should be equal up to roundoff error if cov is # symmetric and the singular value of the corresponding row is # not zero. We continue to use the SVD rather than Cholesky in # order to preserve current outputs. Note that symmetry has not # been checked. # GH10839, ensure double to make tol meaningful cov = cov.astype(np.double) if method == 'svd': from numpy.linalg import svd (u, s, vh) = svd(cov) elif method == 'eigh': from numpy.linalg import eigh # could call linalg.svd(hermitian=True), but that calculates a vh we don't need (s, u) = eigh(cov) else: from numpy.linalg import cholesky l = cholesky(cov) # make sure check_valid is ignored when method == 'cholesky' # since the decomposition will have failed if cov is not valid. if check_valid != 'ignore' and method != 'cholesky': if check_valid != 'warn' and check_valid != 'raise': raise ValueError( "check_valid must equal 'warn', 'raise', or 'ignore'") if method == 'svd': psd = np.allclose(np.dot(vh.T * s, vh), cov, rtol=tol, atol=tol) else: psd = not np.any(s < -tol) if not psd: if check_valid == 'warn': warnings.warn("covariance is not symmetric positive-semidefinite.", RuntimeWarning) else: raise ValueError("covariance is not symmetric positive-semidefinite.") if method == 'cholesky': _factor = l elif method == 'eigh': # if check_valid == 'ignore' we need to ensure that np.sqrt does not # return a NaN if s is a very small negative number that is # approximately zero or when the covariance is not positive-semidefinite _factor = u * np.sqrt(abs(s)) else: _factor = u * np.sqrt(s) x = mean + x @ _factor.T x.shape = tuple(final_shape) return x def multinomial(self, object n, object pvals, size=None): """ multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. Parameters ---------- n : int or array-like of ints Number of experiments. pvals : array-like of floats Probabilities of each of the ``p`` different outcomes with shape ``(k0, k1, ..., kn, p)``. Each element ``pvals[i,j,...,:]`` must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[..., :-1], axis=-1) <= 1.0``. Must have at least 1 dimension where pvals.shape[-1] > 0. 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 each with ``p`` elements. Default is None where the output size is determined by the broadcast shape of ``n`` and all by the final dimension of ``pvals``, which is denoted as ``b=(b0, b1, ..., bq)``. If size is not None, then it must be compatible with the broadcast shape ``b``. Specifically, size must have ``q`` or more elements and size[-(q-j):] must equal ``bj``. Returns ------- out : ndarray The drawn samples, of shape size, if provided. When size is provided, the output shape is size + (p,) If not specified, the shape is determined by the broadcast shape of ``n`` and ``pvals``, ``(b0, b1, ..., bq)`` augmented with the dimension of the multinomial, ``p``, so that that output shape is ``(b0, b1, ..., bq, p)``. Each entry ``out[i,j,...,:]`` is a ``p``-dimensional value drawn from the distribution. .. versionchanged:: 1.22.0 Added support for broadcasting `pvals` against `n` Examples -------- Throw a dice 20 times: >>> rng = np.random.default_rng() >>> rng.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) # random It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> rng.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], [2, 4, 3, 4, 0, 7]]) # random For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. Now, do one experiment throwing the dice 10 time, and 10 times again, and another throwing the dice 20 times, and 20 times again: >>> rng.multinomial([[10], [20]], [1/6.]*6, size=(2, 2)) array([[[2, 4, 0, 1, 2, 1], [1, 3, 0, 3, 1, 2]], [[1, 4, 4, 4, 4, 3], [3, 3, 2, 5, 5, 2]]]) # random The first array shows the outcomes of throwing the dice 10 times, and the second shows the outcomes from throwing the dice 20 times. A loaded die is more likely to land on number 6: >>> rng.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) # random Simulate 10 throws of a 4-sided die and 20 throws of a 6-sided die >>> rng.multinomial([10, 20],[[1/4]*4 + [0]*2, [1/6]*6]) array([[2, 1, 4, 3, 0, 0], [3, 3, 3, 6, 1, 4]], dtype=int64) # random Generate categorical random variates from two categories where the first has 3 outcomes and the second has 2. >>> rng.multinomial(1, [[.1, .5, .4 ], [.3, .7, .0]]) array([[0, 0, 1], [0, 1, 0]], dtype=int64) # random ``argmax(axis=-1)`` is then used to return the categories. >>> pvals = [[.1, .5, .4 ], [.3, .7, .0]] >>> rvs = rng.multinomial(1, pvals, size=(4,2)) >>> rvs.argmax(axis=-1) array([[0, 1], [2, 0], [2, 1], [2, 0]], dtype=int64) # random The same output dimension can be produced using broadcasting. >>> rvs = rng.multinomial([[1]] * 4, pvals) >>> rvs.argmax(axis=-1) array([[0, 1], [2, 0], [2, 1], [2, 0]], dtype=int64) # random The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> rng.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) # random not like: >>> rng.multinomial(100, [1.0, 2.0]) # WRONG Traceback (most recent call last): ValueError: pvals < 0, pvals > 1 or pvals contains NaNs """ cdef np.npy_intp d, i, sz, offset, pi cdef np.ndarray parr, mnarr, on, temp_arr cdef double *pix cdef int ndim cdef int64_t *mnix cdef int64_t ni cdef np.broadcast it on = np.PyArray_FROM_OTF(n, np.NPY_INT64, np.NPY_ARRAY_ALIGNED | np.NPY_ARRAY_C_CONTIGUOUS) parr = np.PyArray_FROM_OTF(pvals, np.NPY_DOUBLE, np.NPY_ARRAY_ALIGNED | np.NPY_ARRAY_C_CONTIGUOUS) ndim = parr.ndim d = parr.shape[ndim - 1] if ndim >= 1 else 0 if d == 0: raise ValueError( "pvals must have at least 1 dimension and the last dimension " "of pvals must be greater than 0." ) check_array_constraint(parr, 'pvals', CONS_BOUNDED_0_1) pix = np.PyArray_DATA(parr) sz = np.PyArray_SIZE(parr) # Cython 0.29.20 would not correctly translate the range-based for # loop to a C for loop # for offset in range(0, sz, d): offset = 0 while offset < sz: if kahan_sum(pix + offset, d-1) > (1.0 + 1e-12): # When floating, but not float dtype, and close, improve the error # 1.0001 works for float16 and float32 slice_repr = "[:-1]" if ndim == 1 else "[...,:-1]" if (isinstance(pvals, np.ndarray) and np.issubdtype(pvals.dtype, np.floating) and pvals.dtype != float and pvals.sum() < 1.0001): msg = (f"sum(pvals{slice_repr}.astype(np.float64)) > 1.0." " The pvals array is cast to 64-bit floating" " point prior to checking the sum. Precision " "changes when casting may cause problems even " "if the sum of the original pvals is valid.") else: msg = f"sum(pvals{slice_repr}) > 1.0" raise ValueError(msg) offset += d if np.PyArray_NDIM(on) != 0 or ndim > 1: # vector check_array_constraint(on, 'n', CONS_NON_NEGATIVE) # This provides the offsets to use in the C-contig parr when # broadcasting offsets = np.arange( 0, np.PyArray_SIZE(parr), d, dtype=np.intp ).reshape((parr).shape[:ndim - 1]) if size is None: it = np.PyArray_MultiIterNew2(on, offsets) else: temp = np.empty(size, dtype=np.int8) temp_arr = temp it = np.PyArray_MultiIterNew3(on, offsets, temp_arr) # Validate size and the broadcast shape try: size = (operator.index(size),) except: size = tuple(size) # This test verifies that an axis with dim 1 in size has not # been increased by broadcasting with the input if it.shape != size: raise ValueError( f"Output size {size} is not compatible with " f"broadcast dimensions of inputs {it.shape}." ) shape = it.shape + (d,) multin = np.zeros(shape, dtype=np.int64) mnarr = multin mnix = np.PyArray_DATA(mnarr) offset = 0 sz = it.size with self.lock, nogil: for i in range(sz): ni = (np.PyArray_MultiIter_DATA(it, 0))[0] pi = (np.PyArray_MultiIter_DATA(it, 1))[0] random_multinomial(&self._bitgen, ni, &mnix[offset], &pix[pi], d, &self._binomial) offset += d np.PyArray_MultiIter_NEXT(it) return multin if size is None: shape = (d,) else: try: shape = (operator.index(size), d) except: shape = tuple(size) + (d,) multin = np.zeros(shape, dtype=np.int64) mnarr = multin mnix = np.PyArray_DATA(mnarr) sz = np.PyArray_SIZE(mnarr) ni = n check_constraint(ni, 'n', CONS_NON_NEGATIVE) offset = 0 with self.lock, nogil: for i in range(sz // d): random_multinomial(&self._bitgen, ni, &mnix[offset], pix, d, &self._binomial) offset += d return multin def multivariate_hypergeometric(self, object colors, object nsample, size=None, method='marginals'): """ multivariate_hypergeometric(colors, nsample, size=None, method='marginals') Generate variates from a multivariate hypergeometric distribution. The multivariate hypergeometric distribution is a generalization of the hypergeometric distribution. Choose ``nsample`` items at random without replacement from a collection with ``N`` distinct types. ``N`` is the length of ``colors``, and the values in ``colors`` are the number of occurrences of that type in the collection. The total number of items in the collection is ``sum(colors)``. Each random variate generated by this function is a vector of length ``N`` holding the counts of the different types that occurred in the ``nsample`` items. The name ``colors`` comes from a common description of the distribution: it is the probability distribution of the number of marbles of each color selected without replacement from an urn containing marbles of different colors; ``colors[i]`` is the number of marbles in the urn with color ``i``. Parameters ---------- colors : sequence of integers The number of each type of item in the collection from which a sample is drawn. The values in ``colors`` must be nonnegative. To avoid loss of precision in the algorithm, ``sum(colors)`` must be less than ``10**9`` when `method` is "marginals". nsample : int The number of items selected. ``nsample`` must not be greater than ``sum(colors)``. size : int or tuple of ints, optional The number of variates to generate, either an integer or a tuple holding the shape of the array of variates. If the given size is, e.g., ``(k, m)``, then ``k * m`` variates are drawn, where one variate is a vector of length ``len(colors)``, and the return value has shape ``(k, m, len(colors))``. If `size` is an integer, the output has shape ``(size, len(colors))``. Default is None, in which case a single variate is returned as an array with shape ``(len(colors),)``. method : string, optional Specify the algorithm that is used to generate the variates. Must be 'count' or 'marginals' (the default). See the Notes for a description of the methods. Returns ------- variates : ndarray Array of variates drawn from the multivariate hypergeometric distribution. See Also -------- hypergeometric : Draw samples from the (univariate) hypergeometric distribution. Notes ----- The two methods do not return the same sequence of variates. The "count" algorithm is roughly equivalent to the following numpy code:: choices = np.repeat(np.arange(len(colors)), colors) selection = np.random.choice(choices, nsample, replace=False) variate = np.bincount(selection, minlength=len(colors)) The "count" algorithm uses a temporary array of integers with length ``sum(colors)``. The "marginals" algorithm generates a variate by using repeated calls to the univariate hypergeometric sampler. It is roughly equivalent to:: variate = np.zeros(len(colors), dtype=np.int64) # `remaining` is the cumulative sum of `colors` from the last # element to the first; e.g. if `colors` is [3, 1, 5], then # `remaining` is [9, 6, 5]. remaining = np.cumsum(colors[::-1])[::-1] for i in range(len(colors)-1): if nsample < 1: break variate[i] = hypergeometric(colors[i], remaining[i+1], nsample) nsample -= variate[i] variate[-1] = nsample The default method is "marginals". For some cases (e.g. when `colors` contains relatively small integers), the "count" method can be significantly faster than the "marginals" method. If performance of the algorithm is important, test the two methods with typical inputs to decide which works best. .. versionadded:: 1.18.0 Examples -------- >>> colors = [16, 8, 4] >>> seed = 4861946401452 >>> gen = np.random.Generator(np.random.PCG64(seed)) >>> gen.multivariate_hypergeometric(colors, 6) array([5, 0, 1]) >>> gen.multivariate_hypergeometric(colors, 6, size=3) array([[5, 0, 1], [2, 2, 2], [3, 3, 0]]) >>> gen.multivariate_hypergeometric(colors, 6, size=(2, 2)) array([[[3, 2, 1], [3, 2, 1]], [[4, 1, 1], [3, 2, 1]]]) """ cdef int64_t nsamp cdef size_t num_colors cdef int64_t total cdef int64_t *colors_ptr cdef int64_t max_index cdef size_t num_variates cdef int64_t *variates_ptr cdef int result if method not in ['count', 'marginals']: raise ValueError('method must be "count" or "marginals".') try: operator.index(nsample) except TypeError: raise ValueError('nsample must be an integer') if nsample < 0: raise ValueError("nsample must be nonnegative.") if nsample > INT64_MAX: raise ValueError("nsample must not exceed %d" % INT64_MAX) nsamp = nsample # Validation of colors, a 1-d sequence of nonnegative integers. invalid_colors = False try: colors = np.asarray(colors) if colors.ndim != 1: invalid_colors = True elif colors.size > 0 and not np.issubdtype(colors.dtype, np.integer): invalid_colors = True elif np.any((colors < 0) | (colors > INT64_MAX)): invalid_colors = True except ValueError: invalid_colors = True if invalid_colors: raise ValueError('colors must be a one-dimensional sequence ' 'of nonnegative integers not exceeding %d.' % INT64_MAX) colors = np.ascontiguousarray(colors, dtype=np.int64) num_colors = colors.size colors_ptr = np.PyArray_DATA(colors) total = _safe_sum_nonneg_int64(num_colors, colors_ptr) if total == -1: raise ValueError("sum(colors) must not exceed the maximum value " "of a 64 bit signed integer (%d)" % INT64_MAX) if method == 'marginals' and total >= 1000000000: raise ValueError('When method is "marginals", sum(colors) must ' 'be less than 1000000000.') # The C code that implements the 'count' method will malloc an # array of size total*sizeof(size_t). Here we ensure that that # product does not overflow. if SIZE_MAX > INT64_MAX: max_index = INT64_MAX // sizeof(size_t) else: max_index = SIZE_MAX // sizeof(size_t) if method == 'count' and total > max_index: raise ValueError("When method is 'count', sum(colors) must not " "exceed %d" % max_index) if nsamp > total: raise ValueError("nsample > sum(colors)") # Figure out the shape of the return array. if size is None: shape = (num_colors,) elif np.isscalar(size): shape = (size, num_colors) else: shape = tuple(size) + (num_colors,) variates = np.zeros(shape, dtype=np.int64) if num_colors == 0: return variates # One variate is a vector of length num_colors. num_variates = variates.size // num_colors variates_ptr = np.PyArray_DATA(variates) if method == 'count': with self.lock, nogil: result = random_multivariate_hypergeometric_count(&self._bitgen, total, num_colors, colors_ptr, nsamp, num_variates, variates_ptr) if result == -1: raise MemoryError("Insufficient memory for multivariate_" "hypergeometric with method='count' and " "sum(colors)=%d" % total) else: with self.lock, nogil: random_multivariate_hypergeometric_marginals(&self._bitgen, total, num_colors, colors_ptr, nsamp, num_variates, variates_ptr) return variates def dirichlet(self, object alpha, size=None): """ dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference. Parameters ---------- alpha : sequence of floats, length k Parameter of the distribution (length ``k`` for sample of length ``k``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n)``, then ``m * n * k`` samples are drawn. Default is None, in which case a vector of length ``k`` is returned. Returns ------- samples : ndarray, The drawn samples, of shape ``(size, k)``. Raises ------ ValueError If any value in ``alpha`` is less than zero Notes ----- The Dirichlet distribution is a distribution over vectors :math:`x` that fulfil the conditions :math:`x_i>0` and :math:`\\sum_{i=1}^k x_i = 1`. The probability density function :math:`p` of a Dirichlet-distributed random vector :math:`X` is proportional to .. math:: p(x) \\propto \\prod_{i=1}^{k}{x^{\\alpha_i-1}_i}, where :math:`\\alpha` is a vector containing the positive concentration parameters. The method uses the following property for computation: let :math:`Y` be a random vector which has components that follow a standard gamma distribution, then :math:`X = \\frac{1}{\\sum_{i=1}^k{Y_i}} Y` is Dirichlet-distributed References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, http://www.inference.org.uk/mackay/itila/ .. [2] Wikipedia, "Dirichlet distribution", https://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> s = np.random.default_rng().dirichlet((10, 5, 3), 20).transpose() >>> import matplotlib.pyplot as plt >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") """ # ================= # Pure python algo # ================= # alpha = N.atleast_1d(alpha) # k = alpha.size # if n == 1: # val = N.zeros(k) # for i in range(k): # val[i] = sgamma(alpha[i], n) # val /= N.sum(val) # else: # val = N.zeros((k, n)) # for i in range(k): # val[i] = sgamma(alpha[i], n) # val /= N.sum(val, axis = 0) # val = val.T # return val cdef np.npy_intp k, totsize, i, j cdef np.ndarray alpha_arr, val_arr, alpha_csum_arr cdef double csum cdef double *alpha_data cdef double *alpha_csum_data cdef double *val_data cdef double acc, invacc, v k = len(alpha) alpha_arr = np.PyArray_FROMANY( alpha, np.NPY_DOUBLE, 1, 1, np.NPY_ARRAY_ALIGNED | np.NPY_ARRAY_C_CONTIGUOUS) if np.any(np.less(alpha_arr, 0)): raise ValueError('alpha < 0') alpha_data = np.PyArray_DATA(alpha_arr) if size is None: shape = (k,) else: try: shape = (operator.index(size), k) except: shape = tuple(size) + (k,) diric = np.zeros(shape, np.float64) val_arr = diric val_data= np.PyArray_DATA(val_arr) i = 0 totsize = np.PyArray_SIZE(val_arr) # Select one of the following two algorithms for the generation # of Dirichlet random variates (RVs) # # A) Small alpha case: Use the stick-breaking approach with beta # random variates (RVs). # B) Standard case: Perform unit normalisation of a vector # of gamma random variates # # A) prevents NaNs resulting from 0/0 that may occur in B) # when all values in the vector ':math:\\alpha' are smaller # than 1, then there is a nonzero probability that all # generated gamma RVs will be 0. When that happens, the # normalization process ends up computing 0/0, giving nan. A) # does not use divisions, so that a situation in which 0/0 has # to be computed cannot occur. A) is slower than B) as # generation of beta RVs is slower than generation of gamma # RVs. A) is selected whenever `alpha.max() < t`, where `t < # 1` is a threshold that controls the probability of # generating a NaN value when B) is used. For a given # threshold `t` this probability can be bounded by # `gammainc(t, d)` where `gammainc` is the regularized # incomplete gamma function and `d` is the smallest positive # floating point number that can be represented with a given # precision. For the chosen threshold `t=0.1` this probability # is smaller than `1.8e-31` for double precision floating # point numbers. if (k > 0) and (alpha_arr.max() < 0.1): # Small alpha case: Use stick-breaking approach with beta # random variates (RVs). # alpha_csum_data will hold the cumulative sum, right to # left, of alpha_arr. # Use a numpy array for memory management only. We could just as # well have malloc'd alpha_csum_data. alpha_arr is a C-contiguous # double array, therefore so is alpha_csum_arr. alpha_csum_arr = np.empty_like(alpha_arr) alpha_csum_data = np.PyArray_DATA(alpha_csum_arr) csum = 0.0 for j in range(k - 1, -1, -1): csum += alpha_data[j] alpha_csum_data[j] = csum with self.lock, nogil: while i < totsize: acc = 1. for j in range(k - 1): v = random_beta(&self._bitgen, alpha_data[j], alpha_csum_data[j + 1]) val_data[i + j] = acc * v acc *= (1. - v) val_data[i + k - 1] = acc i = i + k else: # Standard case: Unit normalisation of a vector of gamma random # variates with self.lock, nogil: while i < totsize: acc = 0. for j in range(k): val_data[i + j] = random_standard_gamma(&self._bitgen, alpha_data[j]) acc = acc + val_data[i + j] invacc = 1. / acc for j in range(k): val_data[i + j] = val_data[i + j] * invacc i = i + k return diric def permuted(self, object x, *, axis=None, out=None): """ permuted(x, axis=None, out=None) Randomly permute `x` along axis `axis`. Unlike `shuffle`, each slice along the given axis is shuffled independently of the others. Parameters ---------- x : array_like, at least one-dimensional Array to be shuffled. axis : int, optional Slices of `x` in this axis are shuffled. Each slice is shuffled independently of the others. If `axis` is None, the flattened array is shuffled. out : ndarray, optional If given, this is the destination of the shuffled array. If `out` is None, a shuffled copy of the array is returned. Returns ------- ndarray If `out` is None, a shuffled copy of `x` is returned. Otherwise, the shuffled array is stored in `out`, and `out` is returned See Also -------- shuffle permutation Notes ----- An important distinction between methods ``shuffle`` and ``permuted`` is how they both treat the ``axis`` parameter which can be found at :ref:`generator-handling-axis-parameter`. Examples -------- Create a `numpy.random.Generator` instance: >>> rng = np.random.default_rng() Create a test array: >>> x = np.arange(24).reshape(3, 8) >>> x array([[ 0, 1, 2, 3, 4, 5, 6, 7], [ 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23]]) Shuffle the rows of `x`: >>> y = rng.permuted(x, axis=1) >>> y array([[ 4, 3, 6, 7, 1, 2, 5, 0], # random [15, 10, 14, 9, 12, 11, 8, 13], [17, 16, 20, 21, 18, 22, 23, 19]]) `x` has not been modified: >>> x array([[ 0, 1, 2, 3, 4, 5, 6, 7], [ 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23]]) To shuffle the rows of `x` in-place, pass `x` as the `out` parameter: >>> y = rng.permuted(x, axis=1, out=x) >>> x array([[ 3, 0, 4, 7, 1, 6, 2, 5], # random [ 8, 14, 13, 9, 12, 11, 15, 10], [17, 18, 16, 22, 19, 23, 20, 21]]) Note that when the ``out`` parameter is given, the return value is ``out``: >>> y is x True """ cdef int ax cdef np.npy_intp axlen, axstride, itemsize cdef void *buf cdef np.flatiter it cdef np.ndarray to_shuffle cdef int status cdef int flags x = np.asarray(x) if out is None: out = x.copy(order='K') else: if type(out) is not np.ndarray: raise TypeError('out must be a numpy array') PyArray_FailUnlessWriteable(out, "out") if out.shape != x.shape: raise ValueError('out must have the same shape as x') np.copyto(out, x, casting='safe') if axis is None: if x.ndim > 1: if not (np.PyArray_FLAGS(out) & (np.NPY_ARRAY_C_CONTIGUOUS | np.NPY_ARRAY_F_CONTIGUOUS)): flags = (np.NPY_ARRAY_C_CONTIGUOUS | NPY_ARRAY_WRITEBACKIFCOPY) to_shuffle = PyArray_FromArray(out, NULL, flags) self.shuffle(to_shuffle.ravel(order='K')) # Because we only execute this block if out is not # contiguous, we know this call will always result in a # copy of to_shuffle back to out. I.e. status will be 1. status = PyArray_ResolveWritebackIfCopy(to_shuffle) assert status == 1 else: # out is n-d with n > 1, but is either C- or F-contiguous, # so we know out.ravel(order='A') is a view. self.shuffle(out.ravel(order='A')) else: # out is 1-d self.shuffle(out) return out ax = normalize_axis_index(axis, np.ndim(out)) itemsize = out.itemsize axlen = out.shape[ax] axstride = out.strides[ax] it = np.PyArray_IterAllButAxis(out, &ax) buf = PyMem_Malloc(itemsize) if buf == NULL: raise MemoryError('memory allocation failed in permuted') if out.dtype.hasobject: # Keep the GIL when shuffling an object array. with self.lock: while np.PyArray_ITER_NOTDONE(it): _shuffle_raw_wrap(&self._bitgen, axlen, 0, itemsize, axstride, np.PyArray_ITER_DATA(it), buf) np.PyArray_ITER_NEXT(it) else: # out is not an object array, so we can release the GIL. with self.lock, nogil: while np.PyArray_ITER_NOTDONE(it): _shuffle_raw_wrap(&self._bitgen, axlen, 0, itemsize, axstride, np.PyArray_ITER_DATA(it), buf) np.PyArray_ITER_NEXT(it) PyMem_Free(buf) return out def shuffle(self, object x, axis=0): """ shuffle(x, axis=0) Modify an array or sequence in-place by shuffling its contents. The order of sub-arrays is changed but their contents remains the same. Parameters ---------- x : ndarray or MutableSequence The array, list or mutable sequence to be shuffled. axis : int, optional The axis which `x` is shuffled along. Default is 0. It is only supported on `ndarray` objects. Returns ------- None See Also -------- permuted permutation Notes ----- An important distinction between methods ``shuffle`` and ``permuted`` is how they both treat the ``axis`` parameter which can be found at :ref:`generator-handling-axis-parameter`. Examples -------- >>> rng = np.random.default_rng() >>> arr = np.arange(10) >>> arr array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> rng.shuffle(arr) >>> arr array([2, 0, 7, 5, 1, 4, 8, 9, 3, 6]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> arr array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> rng.shuffle(arr) >>> arr array([[3, 4, 5], # random [6, 7, 8], [0, 1, 2]]) >>> arr = np.arange(9).reshape((3, 3)) >>> arr array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> rng.shuffle(arr, axis=1) >>> arr array([[2, 0, 1], # random [5, 3, 4], [8, 6, 7]]) """ cdef: np.npy_intp i, j, n = len(x), stride, itemsize char* x_ptr char* buf_ptr if isinstance(x, np.ndarray): if not x.flags.writeable: raise ValueError('array is read-only') # Only call ndim on ndarrays, see GH 18142 axis = normalize_axis_index(axis, np.ndim(x)) if type(x) is np.ndarray and x.ndim == 1 and x.size: # Fast, statically typed path: shuffle the underlying buffer. # Only for non-empty, 1d objects of class ndarray (subclasses such # as MaskedArrays may not support this approach). x_ptr = np.PyArray_BYTES(x) stride = x.strides[0] itemsize = x.dtype.itemsize # As the array x could contain python objects we use a buffer # of bytes for the swaps to avoid leaving one of the objects # within the buffer and erroneously decrementing it's refcount # when the function exits. buf = np.empty(itemsize, dtype=np.int8) # GC'd at function exit buf_ptr = np.PyArray_BYTES(buf) if x.dtype.hasobject: with self.lock: _shuffle_raw_wrap(&self._bitgen, n, 1, itemsize, stride, x_ptr, buf_ptr) else: # Same as above, but the GIL is released. with self.lock, nogil: _shuffle_raw_wrap(&self._bitgen, n, 1, itemsize, stride, x_ptr, buf_ptr) elif isinstance(x, np.ndarray): if x.size == 0: # shuffling is a no-op return x = np.swapaxes(x, 0, axis) buf = np.empty_like(x[0, ...]) with self.lock: for i in reversed(range(1, len(x))): j = random_interval(&self._bitgen, i) if i == j: # i == j is not needed and memcpy is undefined. continue buf[...] = x[j, ...] x[j, ...] = x[i, ...] x[i, ...] = buf else: # Untyped path. if not isinstance(x, Sequence): # See gh-18206. We may decide to deprecate here in the future. warnings.warn( f"you are shuffling a '{type(x).__name__}' object " "which is not a subclass of 'Sequence'; " "`shuffle` is not guaranteed to behave correctly. " "E.g., non-numpy array/tensor objects with view semantics " "may contain duplicates after shuffling.", UserWarning, stacklevel=1) # Cython does not add a level if axis != 0: raise NotImplementedError("Axis argument is only supported " "on ndarray objects") with self.lock: for i in reversed(range(1, n)): j = random_interval(&self._bitgen, i) x[i], x[j] = x[j], x[i] def permutation(self, object x, axis=0): """ permutation(x, axis=0) Randomly permute a sequence, or return a permuted range. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. axis : int, optional The axis which `x` is shuffled along. Default is 0. Returns ------- out : ndarray Permuted sequence or array range. Examples -------- >>> rng = np.random.default_rng() >>> rng.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random >>> rng.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> rng.permutation(arr) array([[6, 7, 8], # random [0, 1, 2], [3, 4, 5]]) >>> rng.permutation("abc") Traceback (most recent call last): ... numpy.exceptions.AxisError: axis 0 is out of bounds for array of dimension 0 >>> arr = np.arange(9).reshape((3, 3)) >>> rng.permutation(arr, axis=1) array([[0, 2, 1], # random [3, 5, 4], [6, 8, 7]]) """ if isinstance(x, (int, np.integer)): arr = np.arange(x) self.shuffle(arr) return arr arr = np.asarray(x) axis = normalize_axis_index(axis, arr.ndim) # shuffle has fast-path for 1-d if arr.ndim == 1: # Return a copy if same memory if np.may_share_memory(arr, x): arr = np.array(arr) self.shuffle(arr) return arr # Shuffle index array, dtype to ensure fast path idx = np.arange(arr.shape[axis], dtype=np.intp) self.shuffle(idx) slices = [slice(None)]*arr.ndim slices[axis] = idx return arr[tuple(slices)] @cython.embedsignature(True) def default_rng(seed=None): """Construct a new Generator with the default BitGenerator (PCG64). Parameters ---------- seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the `BitGenerator`. If None, then fresh, unpredictable entropy will be pulled from the OS. If an ``int`` or ``array_like[ints]`` is passed, then it will be passed to `SeedSequence` to derive the initial `BitGenerator` state. One may also pass in a `SeedSequence` instance. Additionally, when passed a `BitGenerator`, it will be wrapped by `Generator`. If passed a `Generator`, it will be returned unaltered. Returns ------- Generator The initialized generator object. Notes ----- If ``seed`` is not a `BitGenerator` or a `Generator`, a new `BitGenerator` is instantiated. This function does not manage a default global instance. See :ref:`seeding_and_entropy` for more information about seeding. Examples -------- ``default_rng`` is the recommended constructor for the random number class ``Generator``. Here are several ways we can construct a random number generator using ``default_rng`` and the ``Generator`` class. Here we use ``default_rng`` to generate a random float: >>> import numpy as np >>> rng = np.random.default_rng(12345) >>> print(rng) Generator(PCG64) >>> rfloat = rng.random() >>> rfloat 0.22733602246716966 >>> type(rfloat) Here we use ``default_rng`` to generate 3 random integers between 0 (inclusive) and 10 (exclusive): >>> import numpy as np >>> rng = np.random.default_rng(12345) >>> rints = rng.integers(low=0, high=10, size=3) >>> rints array([6, 2, 7]) >>> type(rints[0]) Here we specify a seed so that we have reproducible results: >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> print(rng) Generator(PCG64) >>> arr1 = rng.random((3, 3)) >>> arr1 array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) If we exit and restart our Python interpreter, we'll see that we generate the same random numbers again: >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> arr2 = rng.random((3, 3)) >>> arr2 array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) """ if _check_bit_generator(seed): # We were passed a BitGenerator, so just wrap it up. return Generator(seed) elif isinstance(seed, Generator): # Pass through a Generator. return seed # Otherwise we need to instantiate a new BitGenerator and Generator as # normal. return Generator(PCG64(seed))