westpa.oldtools.aframe package

westpa.oldtools.aframe

WEST Analyis framework – an unholy mess of classes exploiting each other

class westpa.oldtools.aframe.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

exception westpa.oldtools.aframe.ArgumentError(*args, **kwargs)

Bases: RuntimeError

class westpa.oldtools.aframe.WESTAnalysisTool

Bases: object

add_args(parser, upcall=True)

Add arguments to a parser common to all analyses of this type.

process_args(args, upcall=True)

open_analysis_backing()

close_analysis_backing()

require_analysis_group(groupname, replace=False)

class westpa.oldtools.aframe.IterRangeMixin

Bases: AnalysisMixin

A mixin for limiting the range of data considered for a given analysis. This should go after DataManagerMixin

add_args(parser, upcall=True)

process_args(args, upcall=True)

check_iter_range()

iter_block_iter()

Return an iterable of (block_first,block_last+1) over the blocks of iterations selected by –first/–last/–step. NOTE WELL that the second of the pair follows Python iterator conventions and returns one past the last element of the block.

n_iter_blocks()

Return the number of blocks of iterations (as returned by iter_block_iter) selected by –first/–last/–step.

record_data_iter_range(h5object, first_iter=None, last_iter=None)

Store attributes first_iter and last_iter on the given HDF5 object (group/dataset)

record_data_iter_step(h5object, iter_step=None)

Store attribute iter_step on the given HDF5 object (group/dataset).

check_data_iter_range_least(h5object, first_iter=None, last_iter=None)

Check that the given HDF5 object contains (as denoted by its first_iter/last_iter attributes) at least the data range specified.

check_data_iter_range_equal(h5object, first_iter=None, last_iter=None)

Check that the given HDF5 object contains per-iteration data for exactly the specified iterations (as denoted by the object’s first_iter and last_iter attributes

check_data_iter_step_conformant(h5object, iter_step=None)

Check that the given HDF5 object contains per-iteration data at an iteration stride suitable for extracting data with the given stride. (In other words, is the given iter_step a multiple of the stride with which data was recorded.)

check_data_iter_step_equal(h5object, iter_step=None)

Check that the given HDF5 object contains per-iteration data at an iteration stride the same as that specified.

slice_per_iter_data(dataset, first_iter=None, last_iter=None, iter_step=None, axis=0)

Return the subset of the given dataset corresponding to the given iteration range and stride. Unless otherwise specified, the first dimension of the dataset is the one sliced.

iter_range(first_iter=None, last_iter=None, iter_step=None)

class westpa.oldtools.aframe.WESTDataReaderMixin

Bases: AnalysisMixin

A mixin for analysis requiring access to the HDF5 files generated during a WEST run.

add_args(parser, upcall=True)

process_args(args, upcall=True)

clear_run_cache()

property cache_pcoords

Whether or not to cache progress coordinate data. While caching this data can significantly speed up some analysis operations, this requires copious RAM.

Setting this to False when it was formerly True will release any cached data.

get_summary_table()

get_iter_group(n_iter)

Return the HDF5 group corresponding to n_iter

get_segments(n_iter, include_pcoords=True)

Return all segments present in iteration n_iter

get_segments_by_id(n_iter, seg_ids, include_pcoords=True)

Get segments from the data manager, employing caching where possible

get_children(segment, include_pcoords=True)

get_seg_index(n_iter)

get_wtg_parent_array(n_iter)

get_parent_array(n_iter)

get_pcoord_array(n_iter)

get_pcoord_dataset(n_iter)

get_pcoords(n_iter, seg_ids)

get_seg_ids(n_iter, bool_array=None)

get_created_seg_ids(n_iter)

Return a list of seg_ids corresponding to segments which were created for the given iteration (are not continuations).

max_iter_segs_in_range(first_iter, last_iter)

Return the maximum number of segments present in any iteration in the range selected

total_segs_in_range(first_iter, last_iter)

Return the total number of segments present in all iterations in the range selected

get_pcoord_len(n_iter)

Get the length of the progress coordinate array for the given iteration.

get_total_time(first_iter=None, last_iter=None, dt=None)

Return the total amount of simulation time spanned between first_iter and last_iter (inclusive).

class westpa.oldtools.aframe.ExtDataReaderMixin

Bases: AnalysisMixin

An external data reader, primarily designed for reading brute force data, but also suitable for any auxiliary datasets required for analysis.

default_chunksize = 8192

add_args(parser, upcall=True)

process_args(args, upcall=True)

is_npy(filename)

load_npy_or_text(filename)

Load an array from an existing .npy file, or read a text file and convert to a NumPy array. In either case, return a NumPy array. If a pickled NumPy dataset is found, memory-map it read-only. If the specified file does not contain a pickled NumPy array, attempt to read the file using numpy.loadtxt(filename).

text_to_h5dataset(fileobj, group, dsname, dtype=<class 'numpy.float64'>, skiprows=0, usecols=None, chunksize=None)

Read text-format data from the given filename or file-like object fileobj and write to a newly-created dataset called dsname in the HDF5 group group. The data is stored as type dtype. By default, the shape is taken as (number of lines, number of columns); columns can be omitted by specifying a list for usecols, and lines can be skipped by using skiprows. Data is read in chunks of chunksize rows.

npy_to_h5dataset(array, group, dsname, usecols=None, chunksize=None)

Store the given array into a newly-created dataset named dsname in the HDF5 group group, optionally only storing a subset of columns. Data is written chunksize rows at a time, allowing very large memory-mapped arrays to be copied.

class westpa.oldtools.aframe.BFDataManager

Bases: AnalysisMixin

A class to manage brute force trajectory data. The primary purpose is to read in and manage brute force progress coordinate data for one or more trajectories. The trajectories need not be the same length, but they do need to have the same time spacing for progress coordinate values.

traj_index_dtype = dtype([('pcoord_len', '<u8'), ('source_data', 'O')])

add_args(parser, upcall=True)

process_args(args, upcall=True)

update_traj_index(traj_id, pcoord_len, source_data)

get_traj_group(traj_id)

create_traj_group()

get_n_trajs()

get_traj_len(traj_id)

get_max_traj_len()

get_pcoord_array(traj_id)

get_pcoord_dataset(traj_id)

require_bf_h5file()

close_bf_h5file()

class westpa.oldtools.aframe.BinningMixin

Bases: AnalysisMixin

A mixin for performing binning on WEST data.

add_args(parser, upcall=True)

process_args(args, upcall=True)

mapper_from_expr(expr)

write_bin_labels(dest, header='# bin labels:\n', format='# bin {bin_index:{max_iwidth}d} -- {label!s}\n')

Print labels for all bins in self.mapper to dest. If provided, header is printed before any labels. The format string specifies how bin labels are to be printed. Valid entries are:

• bin_index – the zero-based index of the bin

• label – the label, as obtained by bin.label

• max_iwidth – the maximum width (in characters) of the bin index, for pretty alignment

require_binning_group()

delete_binning_group()

record_data_binhash(h5object)

Record the identity hash for self.mapper as an attribute on the given HDF5 object (group or dataset)

check_data_binhash(h5object)

Check whether the recorded bin identity hash on the given HDF5 object matches the identity hash for self.mapper

assign_to_bins()

Assign WEST segment data to bins. Requires the DataReader mixin to be in the inheritance tree

require_bin_assignments()

get_bin_assignments(first_iter=None, last_iter=None)

get_bin_populations(first_iter=None, last_iter=None)

class westpa.oldtools.aframe.MCBSMixin

Bases: AnalysisMixin

add_args(parser, upcall=True)

process_args(args, upcall=True)

calc_mcbs_nsets(alpha=None)

calc_ci_bound_indices(n_sets=None, alpha=None)

class westpa.oldtools.aframe.TrajWalker(data_reader, history_chunksize=100)

Bases: object

A class to perform analysis by walking the trajectory tree. A stack is used rather than recursion, or else the highest number of iterations capable of being considered would be the same as the Python recursion limit.

trace_to_root(n_iter, seg_id)

Trace the given segment back to its starting point, returning a list of Segment objects describing the entire trajectory.

get_trajectory_roots(first_iter, last_iter, include_pcoords=True)

Get segments which start new trajectories. If min_iter or max_iter is specified, restrict the set of iterations within which the search is conducted.

get_initial_nodes(first_iter, last_iter, include_pcoords=True)

Get segments with which to begin a tree walk – those alive or created within [first_iter,last_iter].

trace_trajectories(first_iter, last_iter, callable, include_pcoords=True, cargs=None, ckwargs=None, get_state=None, set_state=None)

Walk the trajectory tree depth-first, calling

callable(segment, children, history, *cargs, **ckwargs) for each segment

visited. segment is the segment being visited, children is that segment’s children, history is the chain of segments leading to segment (not including segment). get_state and set_state are used to record and reset, respectively, any state specific to callable when a new branch is traversed.

class westpa.oldtools.aframe.TransitionAnalysisMixin

Bases: AnalysisMixin

require_transitions_group()

delete_transitions_group()

get_transitions_ds()

add_args(parser, upcall=True)

process_args(args, upcall=True)

require_transitions()

find_transitions()

class westpa.oldtools.aframe.TransitionEventAccumulator(n_bins, output_group, calc_fpts=True)

Bases: object

index_dtype

alias of uint64

count_dtype

alias of uint64

weight_dtype

alias of float64

output_tdat_chunksize = 4096

tdat_buffersize = 524288

max_acc = 32768

clear()

clear_state()

get_state()

set_state(state_dict)

record_transition_data(tdat)

Update running statistics and write transition data to HDF5 (with buffering)

flush_transition_data()

Flush any unwritten output that may be present

start_accumulation(assignments, weights, bin_pops, traj=0, n_iter=0)

continue_accumulation(assignments, weights, bin_pops, traj=0, n_iter=0)

class westpa.oldtools.aframe.BFTransitionAnalysisMixin

Bases: TransitionAnalysisMixin

require_transitions()

find_transitions(chunksize=65536)

class westpa.oldtools.aframe.KineticsAnalysisMixin

Bases: AnalysisMixin

add_args(parser, upcall=True)

process_args(args, upcall=True)

parse_bin_range(range_string)

check_bin_selection(n_bins=None)

Check to see that the bin ranges selected by the user conform to the available bins (i.e., bin indices are within the permissible range). Also assigns the complete bin range if the user has not explicitly limited the bins to be considered.

property selected_bin_pair_iter

class westpa.oldtools.aframe.CommonOutputMixin

Bases: AnalysisMixin

add_common_output_args(parser_or_group)

process_common_output_args(args)

class westpa.oldtools.aframe.PlottingMixin

Bases: AnalysisMixin

require_matplotlib()

westpa.oldtools.aframe.atool module

class westpa.oldtools.aframe.atool.WESTAnalysisTool

Bases: object

add_args(parser, upcall=True)

Add arguments to a parser common to all analyses of this type.

process_args(args, upcall=True)

open_analysis_backing()

close_analysis_backing()

require_analysis_group(groupname, replace=False)

westpa.oldtools.aframe.base_mixin module

exception westpa.oldtools.aframe.base_mixin.ArgumentError(*args, **kwargs)

Bases: RuntimeError

class westpa.oldtools.aframe.base_mixin.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

westpa.oldtools.aframe.binning module

class westpa.oldtools.aframe.binning.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

class westpa.oldtools.aframe.binning.BinningMixin

Bases: AnalysisMixin

A mixin for performing binning on WEST data.

add_args(parser, upcall=True)

process_args(args, upcall=True)

mapper_from_expr(expr)

write_bin_labels(dest, header='# bin labels:\n', format='# bin {bin_index:{max_iwidth}d} -- {label!s}\n')

Print labels for all bins in self.mapper to dest. If provided, header is printed before any labels. The format string specifies how bin labels are to be printed. Valid entries are:

• bin_index – the zero-based index of the bin

• label – the label, as obtained by bin.label

• max_iwidth – the maximum width (in characters) of the bin index, for pretty alignment

require_binning_group()

delete_binning_group()

record_data_binhash(h5object)

Record the identity hash for self.mapper as an attribute on the given HDF5 object (group or dataset)

check_data_binhash(h5object)

Check whether the recorded bin identity hash on the given HDF5 object matches the identity hash for self.mapper

assign_to_bins()

Assign WEST segment data to bins. Requires the DataReader mixin to be in the inheritance tree

require_bin_assignments()

get_bin_assignments(first_iter=None, last_iter=None)

get_bin_populations(first_iter=None, last_iter=None)

westpa.oldtools.aframe.data_reader module

class westpa.oldtools.aframe.data_reader.Segment(n_iter=None, seg_id=None, weight=None, endpoint_type=None, parent_id=None, wtg_parent_ids=None, pcoord=None, status=None, walltime=None, cputime=None, data=None)

Bases: object

A class wrapping segment data that must be passed through the work manager or data manager. Most fields are self-explanatory. One item worth noting is that a negative parent ID means that the segment starts from the initial state with ID -(segment.parent_id+1)

SEG_STATUS_UNSET = 0

SEG_STATUS_PREPARED = 1

SEG_STATUS_COMPLETE = 2

SEG_STATUS_FAILED = 3

SEG_INITPOINT_UNSET = 0

SEG_INITPOINT_CONTINUES = 1

SEG_INITPOINT_NEWTRAJ = 2

SEG_ENDPOINT_UNSET = 0

SEG_ENDPOINT_CONTINUES = 1

SEG_ENDPOINT_MERGED = 2

SEG_ENDPOINT_RECYCLED = 3

statuses = {'SEG_STATUS_COMPLETE': 2, 'SEG_STATUS_FAILED': 3, 'SEG_STATUS_PREPARED': 1, 'SEG_STATUS_UNSET': 0}

initpoint_types = {'SEG_INITPOINT_CONTINUES': 1, 'SEG_INITPOINT_NEWTRAJ': 2, 'SEG_INITPOINT_UNSET': 0}

endpoint_types = {'SEG_ENDPOINT_CONTINUES': 1, 'SEG_ENDPOINT_MERGED': 2, 'SEG_ENDPOINT_RECYCLED': 3, 'SEG_ENDPOINT_UNSET': 0}

status_names = {0: 'SEG_STATUS_UNSET', 1: 'SEG_STATUS_PREPARED', 2: 'SEG_STATUS_COMPLETE', 3: 'SEG_STATUS_FAILED'}

initpoint_type_names = {0: 'SEG_INITPOINT_UNSET', 1: 'SEG_INITPOINT_CONTINUES', 2: 'SEG_INITPOINT_NEWTRAJ'}

endpoint_type_names = {0: 'SEG_ENDPOINT_UNSET', 1: 'SEG_ENDPOINT_CONTINUES', 2: 'SEG_ENDPOINT_MERGED', 3: 'SEG_ENDPOINT_RECYCLED'}

static initial_pcoord(segment)

Return the initial progress coordinate point of this segment.

static final_pcoord(segment)

Return the final progress coordinate point of this segment.

property initpoint_type

property initial_state_id

property status_text

property endpoint_type_text

class westpa.oldtools.aframe.data_reader.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

westpa.oldtools.aframe.data_reader.parse_int_list(list_string)

Parse a simple list consisting of integers or ranges of integers separated by commas. Ranges are specified as min:max, and include the maximum value (unlike Python’s range). Duplicate values are ignored. Returns the result as a sorted list. Raises ValueError if the list cannot be parsed.

class westpa.oldtools.aframe.data_reader.WESTDataReaderMixin

Bases: AnalysisMixin

A mixin for analysis requiring access to the HDF5 files generated during a WEST run.

add_args(parser, upcall=True)

process_args(args, upcall=True)

clear_run_cache()

property cache_pcoords

Whether or not to cache progress coordinate data. While caching this data can significantly speed up some analysis operations, this requires copious RAM.

Setting this to False when it was formerly True will release any cached data.

get_summary_table()

get_iter_group(n_iter)

Return the HDF5 group corresponding to n_iter

get_segments(n_iter, include_pcoords=True)

Return all segments present in iteration n_iter

get_segments_by_id(n_iter, seg_ids, include_pcoords=True)

Get segments from the data manager, employing caching where possible

get_children(segment, include_pcoords=True)

get_seg_index(n_iter)

get_wtg_parent_array(n_iter)

get_parent_array(n_iter)

get_pcoord_array(n_iter)

get_pcoord_dataset(n_iter)

get_pcoords(n_iter, seg_ids)

get_seg_ids(n_iter, bool_array=None)

get_created_seg_ids(n_iter)

Return a list of seg_ids corresponding to segments which were created for the given iteration (are not continuations).

max_iter_segs_in_range(first_iter, last_iter)

Return the maximum number of segments present in any iteration in the range selected

total_segs_in_range(first_iter, last_iter)

Return the total number of segments present in all iterations in the range selected

get_pcoord_len(n_iter)

Get the length of the progress coordinate array for the given iteration.

get_total_time(first_iter=None, last_iter=None, dt=None)

Return the total amount of simulation time spanned between first_iter and last_iter (inclusive).

class westpa.oldtools.aframe.data_reader.ExtDataReaderMixin

Bases: AnalysisMixin

An external data reader, primarily designed for reading brute force data, but also suitable for any auxiliary datasets required for analysis.

default_chunksize = 8192

add_args(parser, upcall=True)

process_args(args, upcall=True)

is_npy(filename)

load_npy_or_text(filename)

Load an array from an existing .npy file, or read a text file and convert to a NumPy array. In either case, return a NumPy array. If a pickled NumPy dataset is found, memory-map it read-only. If the specified file does not contain a pickled NumPy array, attempt to read the file using numpy.loadtxt(filename).

text_to_h5dataset(fileobj, group, dsname, dtype=<class 'numpy.float64'>, skiprows=0, usecols=None, chunksize=None)

Read text-format data from the given filename or file-like object fileobj and write to a newly-created dataset called dsname in the HDF5 group group. The data is stored as type dtype. By default, the shape is taken as (number of lines, number of columns); columns can be omitted by specifying a list for usecols, and lines can be skipped by using skiprows. Data is read in chunks of chunksize rows.

npy_to_h5dataset(array, group, dsname, usecols=None, chunksize=None)

Store the given array into a newly-created dataset named dsname in the HDF5 group group, optionally only storing a subset of columns. Data is written chunksize rows at a time, allowing very large memory-mapped arrays to be copied.

class westpa.oldtools.aframe.data_reader.BFDataManager

Bases: AnalysisMixin

A class to manage brute force trajectory data. The primary purpose is to read in and manage brute force progress coordinate data for one or more trajectories. The trajectories need not be the same length, but they do need to have the same time spacing for progress coordinate values.

traj_index_dtype = dtype([('pcoord_len', '<u8'), ('source_data', 'O')])

add_args(parser, upcall=True)

process_args(args, upcall=True)

update_traj_index(traj_id, pcoord_len, source_data)

get_traj_group(traj_id)

create_traj_group()

get_n_trajs()

get_traj_len(traj_id)

get_max_traj_len()

get_pcoord_array(traj_id)

get_pcoord_dataset(traj_id)

require_bf_h5file()

close_bf_h5file()

westpa.oldtools.aframe.iter_range module

class westpa.oldtools.aframe.iter_range.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

exception westpa.oldtools.aframe.iter_range.ArgumentError(*args, **kwargs)

Bases: RuntimeError

class westpa.oldtools.aframe.iter_range.IterRangeMixin

Bases: AnalysisMixin

A mixin for limiting the range of data considered for a given analysis. This should go after DataManagerMixin

add_args(parser, upcall=True)

process_args(args, upcall=True)

check_iter_range()

iter_block_iter()

Return an iterable of (block_first,block_last+1) over the blocks of iterations selected by –first/–last/–step. NOTE WELL that the second of the pair follows Python iterator conventions and returns one past the last element of the block.

n_iter_blocks()

Return the number of blocks of iterations (as returned by iter_block_iter) selected by –first/–last/–step.

record_data_iter_range(h5object, first_iter=None, last_iter=None)

Store attributes first_iter and last_iter on the given HDF5 object (group/dataset)

record_data_iter_step(h5object, iter_step=None)

Store attribute iter_step on the given HDF5 object (group/dataset).

check_data_iter_range_least(h5object, first_iter=None, last_iter=None)

Check that the given HDF5 object contains (as denoted by its first_iter/last_iter attributes) at least the data range specified.

check_data_iter_range_equal(h5object, first_iter=None, last_iter=None)

Check that the given HDF5 object contains per-iteration data for exactly the specified iterations (as denoted by the object’s first_iter and last_iter attributes

check_data_iter_step_conformant(h5object, iter_step=None)

Check that the given HDF5 object contains per-iteration data at an iteration stride suitable for extracting data with the given stride. (In other words, is the given iter_step a multiple of the stride with which data was recorded.)

check_data_iter_step_equal(h5object, iter_step=None)

Check that the given HDF5 object contains per-iteration data at an iteration stride the same as that specified.

slice_per_iter_data(dataset, first_iter=None, last_iter=None, iter_step=None, axis=0)

Return the subset of the given dataset corresponding to the given iteration range and stride. Unless otherwise specified, the first dimension of the dataset is the one sliced.

iter_range(first_iter=None, last_iter=None, iter_step=None)

westpa.oldtools.aframe.kinetics module

class westpa.oldtools.aframe.kinetics.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

class westpa.oldtools.aframe.kinetics.KineticsAnalysisMixin

Bases: AnalysisMixin

add_args(parser, upcall=True)

process_args(args, upcall=True)

parse_bin_range(range_string)

check_bin_selection(n_bins=None)

Check to see that the bin ranges selected by the user conform to the available bins (i.e., bin indices are within the permissible range). Also assigns the complete bin range if the user has not explicitly limited the bins to be considered.

property selected_bin_pair_iter

westpa.oldtools.aframe.mcbs module

Tools for Monte Carlo bootstrap error analysis

class westpa.oldtools.aframe.mcbs.Generator(bit_generator)

Bases: object

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 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 :external+python:mod:`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.

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

\[f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1},\]

where the normalization, B, is the beta function,

\[B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.\]

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 – Drawn samples from the parameterized beta distribution.

Return type:

ndarray or scalar

Examples

The beta distribution has mean a/(a+b). If a == b and both are > 1, the distribution is symmetric with mean 0.5.

>>> rng = np.random.default_rng()
>>> a, b, size = 2.0, 2.0, 10000
>>> sample = rng.beta(a=a, b=b, size=size)
>>> np.mean(sample)
0.5047328775385895  # may vary

Otherwise the distribution is skewed left or right according to whether a or b is greater. The distribution is mirror symmetric. See for example:

>>> a, b, size = 2, 7, 10000
>>> sample_left = rng.beta(a=a, b=b, size=size)
>>> sample_right = rng.beta(a=b, b=a, size=size)
>>> m_left, m_right = np.mean(sample_left), np.mean(sample_right)
>>> print(m_left, m_right)
0.2238596793678923 0.7774613834041182  # may vary
>>> print(m_left - a/(a+b))
0.001637457145670096  # may vary
>>> print(m_right - b/(a+b))
-0.0003163943736596009  # may vary

Display the histogram of the two samples:

>>> import matplotlib.pyplot as plt
>>> plt.hist([sample_left, sample_right],
...          50, density=True, histtype='bar')
>>> plt.show()

References

[1]

Wikipedia, “Beta distribution”, https://en.wikipedia.org/wiki/Beta_distribution

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 – Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.

Return type:

ndarray or scalar

See also

scipy.stats.binom

probability density function, distribution or cumulative density function, etc.

Notes

The probability mass function (PMF) for the binomial distribution is

\[P(N) = \binom{n}{N}p^N(1-p)^{n-N},\]

where \(n\) is the number of trials, \(p\) is the probability of success, and \(N\) 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. https://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, size = 10, .5, 10000
>>> s = rng.binomial(n, p, 10000)

Assume a company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of p=0.1. All nine wells fail. What is the probability of that happening?

Over size = 20,000 trials the probability of this happening is on average:

>>> n, p, size = 9, 0.1, 20000
>>> np.sum(rng.binomial(n=n, p=p, size=size) == 0)/size
0.39015  # may vary

The following can be used to visualize a sample with n=100, p=0.4 and the corresponding probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.stats import binom
>>> n, p, size = 100, 0.4, 10000
>>> sample = rng.binomial(n, p, size=size)
>>> count, bins, _ = plt.hist(sample, 30, density=True)
>>> x = np.arange(n)
>>> y = binom.pmf(x, n, p)
>>> plt.plot(x, y, linewidth=2, color='r')

bit_generator

Gets the bit generator instance used by the generator

Returns:

bit_generator – The bit generator instance used by the generator

Return type:

BitGenerator

bytes(length)

Return random bytes.

Parameters:

length (int) – Number of random bytes.

Returns:

out – String of length length.

Return type:

bytes

Notes

This function generates random bytes from a discrete uniform distribution. The generated bytes are independent from the CPU’s native endianness.

Examples

>>> rng = np.random.default_rng()
>>> rng.bytes(10)
b'\xfeC\x9b\x86\x17\xf2\xa1\xafcp'  # random

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 – Drawn samples from the parameterized chi-square distribution.

Return type:

ndarray or scalar

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:

\[Q = \sum_{i=1}^{\mathtt{df}} X^2_i\]

is chi-square distributed, denoted

\[Q \sim \chi^2_k.\]

The probability density function of the chi-squared distribution is

\[p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2},\]

where \(\Gamma\) is the gamma function,

\[\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.\]

References

[1]

NIST “Engineering Statistics Handbook” https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

Examples

>>> rng = np.random.default_rng()
>>> rng.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random

The distribution of a chi-square random variable with 20 degrees of freedom looks as follows:

>>> import matplotlib.pyplot as plt
>>> import scipy.stats as stats
>>> s = rng.chisquare(20, 10000)
>>> count, bins, _ = plt.hist(s, 30, density=True)
>>> x = np.linspace(0, 60, 1000)
>>> plt.plot(x, stats.chi2.pdf(x, df=20))
>>> plt.xlim([0, 60])
>>> plt.show()

choice(a, size=None, replace=True, p=None, axis=0, shuffle=True)

Generates a random sample from a given array

Parameters:

• a ({array_like, int}) – If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated from np.arange(a).

• size ({int, tuple[int]}, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn from the 1-d a. If a has more than one dimension, the size shape will be inserted into the axis dimension, so the output ndim will be a.ndim - 1 + len(size). Default is None, in which case a single value is returned.

• replace (bool, optional) – Whether the sample is with or without replacement. Default is True, meaning that a value of a can be selected multiple times.

• p (1-D array_like, optional) – The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in a.

• axis (int, optional) – The axis along which the selection is performed. The default, 0, selects by row.

• shuffle (bool, optional) – Whether the sample is shuffled when sampling without replacement. Default is True, False provides a speedup.

Returns:

samples – The generated random samples

Return type:

single item or ndarray

Raises:

ValueError – If a is an int and less than zero, if p is not 1-dimensional, if a is array-like with a size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size.

See also

integers, shuffle, permutation

Notes

Setting user-specified probabilities through p uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of p is 1 / len(a).

p must sum to 1 when cast to float64. To ensure this, you may wish to normalize using p = p / np.sum(p, dtype=float).

When passing a as an integer type and size is not specified, the return type is a native Python int.

Examples

Generate a uniform random sample from np.arange(5) of size 3:

>>> 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='<U11')

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 – The drawn samples, of shape (size, k).

Return type:

ndarray,

Raises:

ValueError – If any value in alpha is less than zero

Notes

The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).

The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where \(\alpha\) is a vector containing the positive concentration parameters.

The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed

References

[1]

David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, https://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.

>>> rng = np.random.default_rng()
>>> s = 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")

exponential(scale=1.0, size=None)

Draw samples from an exponential distribution.

Its probability density function is

\[f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),\]

for x > 0 and 0 elsewhere. \(\beta\) is the scale parameter, which is the inverse of the rate parameter \(\lambda = 1/\beta\). 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, \(\beta = 1/\lambda\). 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 – Drawn samples from the parameterized exponential distribution.

Return type:

ndarray or scalar

Examples

Assume a company has 10000 customer support agents and the time between customer calls is exponentially distributed and that the average time between customer calls is 4 minutes.

>>> scale, size = 4, 10000
>>> rng = np.random.default_rng()
>>> time_between_calls = rng.exponential(scale=scale, size=size)

What is the probability that a customer will call in the next 4 to 5 minutes?

>>> x = ((time_between_calls < 5).sum())/size
>>> y = ((time_between_calls < 4).sum())/size
>>> x - y
0.08  # may vary

The corresponding distribution can be visualized as follows:

>>> import matplotlib.pyplot as plt
>>> scale, size = 4, 10000
>>> rng = np.random.default_rng()
>>> sample = rng.exponential(scale=scale, size=size)
>>> count, bins, _ = plt.hist(sample, 30, density=True)
>>> plt.plot(bins, scale**(-1)*np.exp(-scale**-1*bins), linewidth=2, color='r')
>>> plt.show()

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

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 – Drawn samples from the parameterized Fisher distribution.

Return type:

ndarray or scalar

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
>>> rng = np.random.default_rng()
>>> s = 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.

The corresponding probability density function for n = 20 and m = 20 is:

>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> dfnum, dfden, size = 20, 20, 10000
>>> s = rng.f(dfnum=dfnum, dfden=dfden, size=size)
>>> bins, density, _ = plt.hist(s, 30, density=True)
>>> x = np.linspace(0, 5, 1000)
>>> plt.plot(x, stats.f.pdf(x, dfnum, dfden))
>>> plt.xlim([0, 5])
>>> plt.show()

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 – Drawn samples from the parameterized gamma distribution.

Return type:

ndarray or scalar

See also

scipy.stats.gamma

probability density function, distribution or cumulative density function, etc.

Notes

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) 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. https://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)
>>> rng = np.random.default_rng()
>>> s = 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
>>> count, bins, _ = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()

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

\[f(k) = (1 - p)^{k - 1} p\]

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 – Drawn samples from the parameterized geometric distribution.

Return type:

ndarray or scalar

References

[1]

Wikipedia, “Geometric distribution”, https://en.wikipedia.org/wiki/Geometric_distribution

Examples

Draw 10,000 values from the geometric distribution, with the probability of an individual success equal to p = 0.35:

>>> p, size = 0.35, 10000
>>> rng = np.random.default_rng()
>>> sample = rng.geometric(p=p, size=size)

What proportion of trials succeeded after a single run?

>>> (sample == 1).sum()/size
0.34889999999999999  # may vary

The geometric distribution with p=0.35 looks as follows:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(sample, bins=30, density=True)
>>> plt.plot(bins, (1-p)**(bins-1)*p)
>>> plt.xlim([0, 25])
>>> plt.show()

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 – Drawn samples from the parameterized Gumbel distribution.

Return type:

ndarray or scalar

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

\[p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}},\]

where \(\mu\) is the mode, a location parameter, and \(\beta\) 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 \(\mu + 0.57721\beta\) and a variance of \(\frac{\pi^2}{6}\beta^2\).

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, _ = 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, _ = 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()

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 – 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.

Return type:

ndarray or scalar

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 mass function (PMF) for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where \(0 \le x \le n\) and \(n-b \le x \le g\)

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. https://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!

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 – size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.

Return type:

int or ndarray of ints

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, https://arxiv.org/abs/1805.10941.

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, \(\mu\), of the distribution peak. Default is 0.

• scale (float or array_like of floats, optional) – \(\lambda\), 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 – Drawn samples from the parameterized Laplace distribution.

Return type:

ndarray or scalar

Notes

It has the probability density function

\[f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right).\]

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. https://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.
>>> rng = np.random.default_rng()
>>> s = rng.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = 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)

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 – Drawn samples from the parameterized logistic distribution.

Return type:

ndarray or scalar

See also

scipy.stats.logistic

probability density function, distribution or cumulative density function, etc.

Notes

The probability density for the Logistic distribution is

\[P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},\]

where \(\mu\) = location and \(s\) = 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. https://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
>>> rng = np.random.default_rng()
>>> s = rng.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, bins=50, label='Sampled data')

# plot sampled data against the exact distribution

>>> def logistic(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> logistic_values  = logistic(bins, loc, scale)
>>> bin_spacing = np.mean(np.diff(bins))
>>> plt.plot(bins, logistic_values  * bin_spacing * s.size, label='Logistic PDF')
>>> plt.legend()
>>> plt.show()

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 – Drawn samples from the parameterized log-normal distribution.

Return type:

ndarray or scalar

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:

\[p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}\]

where \(\mu\) is the mean and \(\sigma\) 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, _ = 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, _ = 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()

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 – Drawn samples from the parameterized logarithmic series distribution.

Return type:

ndarray or scalar

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

\[P(k) = \frac{-p^k}{k \ln(1-p)},\]

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
>>> rng = np.random.default_rng()
>>> s = rng.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> bins = np.arange(-.5, max(s) + .5 )
>>> count, bins, _ = plt.hist(s, bins=bins, label='Sample count')

Plot against the distribution:

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> centres = np.arange(1, max(s) + 1)
>>> plt.plot(centres, logseries(centres, a) * s.size, 'r', label='logseries PMF')
>>> plt.legend()
>>> plt.show()

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 – 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.

Return type:

ndarray

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

multivariate_hypergeometric(colors, 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 – Array of variates drawn from the multivariate hypergeometric distribution.

Return type:

ndarray

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.

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]]])

multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-08, *, 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.

Returns:

out – 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.

Return type:

ndarray

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, \(X = [x_1, x_2, ..., x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (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 the variables are independent, and the probability density contours have their axes aligned with the coordinate axes:

>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
>>> x, y = 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()

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 – 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.

Return type:

ndarray or scalar

Notes

The probability mass function of the negative binomial distribution is

\[P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},\]

where \(n\) is the number of successes, \(p\) is the probability of success, \(N+n\) is the number of trials, and \(\Gamma\) is the gamma function. When \(n\) is an integer, \(\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}\), 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 \(n\) and \(p\) 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 \(p\) is too low (a lot of failures happen for every success) and \(n\) is too big ( a lot of successes are allowed). Therefore, the \(n\) and \(p\) values must satisfy the constraint:

\[n\frac{1-p}{p}+10n\sqrt{n}\frac{1-p}{p}<2^{63}-1-10\sqrt{2^{63}-1},\]

Where the left side of the equation is the derived mean + 10 sigma of a sample from the gamma distribution internally used as the \(lam\) parameter of a poisson sample, and the right side of the equation is the constraint for maximum value of \(lam\) in Generator.poisson.

References

[1]

Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. https://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.?

>>> rng = np.random.default_rng()
>>> s = rng.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11):
...    probability = sum(s<i) / 100000.
...    print(i, "wells drilled, probability of one success =", probability)

noncentral_chisquare(df, nonc, size=None)

Draw samples from a noncentral chi-square distribution.

The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.

Parameters:

• df (float or array_like of floats) – Degrees of freedom, must be > 0.

• 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 – Drawn samples from the parameterized noncentral chi-square distribution.

Return type:

ndarray or scalar

Notes

The probability density function for the noncentral Chi-square distribution is

\[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

where \(Y_{q}\) 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()

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.

• 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 – Drawn samples from the parameterized noncentral Fisher distribution.

Return type:

ndarray or scalar

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. https://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()

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 – Drawn samples from the parameterized normal distribution.

Return type:

ndarray or scalar

See also

scipy.stats.norm

probability density function, distribution or cumulative density function, etc.

Notes

The probability density for the Gaussian distribution is

\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]

where \(\mu\) is the mean and \(\sigma\) the standard deviation. The square of the standard deviation, \(\sigma^2\), 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 \(x + \sigma\) and \(x - \sigma\) [2]_). This implies that 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
>>> rng = np.random.default_rng()
>>> s = rng.normal(mu, sigma, 1000)

Verify the mean and the standard deviation:

>>> 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, _ = 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:

>>> rng = np.random.default_rng()
>>> 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

pareto(a, size=None)

Draw samples from a Pareto II (AKA Lomax) distribution with specified shape.

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 – Drawn samples from the Pareto II distribution.

Return type:

ndarray or scalar

See also

scipy.stats.pareto

Pareto I distribution

scipy.stats.lomax

Lomax (Pareto II) distribution

scipy.stats.genpareto

Generalized Pareto distribution

Notes

The probability density for the Pareto II distribution is

\[p(x) = \frac{a}{(x+1)^{a+1}} , x \ge 0\]

where \(a > 0\) is the shape.

The Pareto II distribution is a shifted and scaled version of the Pareto I distribution, which can be found in scipy.stats.pareto.

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 = 3.
>>> rng = np.random.default_rng()
>>> s = rng.pareto(a, 10000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 3, 50)
>>> pdf = a / (x+1)**(a+1)
>>> plt.hist(s, bins=x, density=True, label='histogram')
>>> plt.plot(x, pdf, linewidth=2, color='r', label='pdf')
>>> plt.xlim(x.min(), x.max())
>>> plt.legend()
>>> plt.show()

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 – Permuted sequence or array range.

Return type:

ndarray

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]])

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:

If out is None, a shuffled copy of x is returned. Otherwise, the shuffled array is stored in out, and out is returned

Return type:

ndarray

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 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

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 – Drawn samples from the parameterized Poisson distribution.

Return type:

ndarray or scalar

Notes

The probability mass function (PMF) of Poisson distribution is

\[f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}\]

For events with an expected separation \(\lambda\) the Poisson distribution \(f(k; \lambda)\) describes the probability of \(k\) events occurring within the observed interval \(\lambda\).

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. https://mathworld.wolfram.com/PoissonDistribution.html

[2]

Wikipedia, “Poisson distribution”, https://en.wikipedia.org/wiki/Poisson_distribution

Examples

Draw samples from the distribution:

>>> rng = np.random.default_rng()
>>> lam, size = 5, 10000
>>> s = rng.poisson(lam=lam, size=size)

Verify the mean and variance, which should be approximately lam:

>>> s.mean(), s.var()
(4.9917 5.1088311)  # may vary

Display the histogram and probability mass function:

>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> x = np.arange(0, 21)
>>> pmf = stats.poisson.pmf(x, mu=lam)
>>> plt.hist(s, bins=x, density=True, width=0.5)
>>> plt.stem(x, pmf, 'C1-')
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = rng.poisson(lam=(100., 500.), size=(100, 2))

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 – Drawn samples from the parameterized power distribution.

Return type:

ndarray or scalar

Raises:

ValueError – If a <= 0.

Notes

The probability density function is

\[P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.\]

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, _ = 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
>>> rvs = rng.power(5, 1000000)
>>> rvsp = rng.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)

>>> plt.figure()
>>> plt.hist(rvs, bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('power(5)')

>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of 1 + Generator.pareto(5)')

>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of stats.pareto(5)')

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 \(Unif[a, b), b > a\) 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 – Array of random floats of shape size (unless size=None, in which case a single float is returned).

Return type:

float or ndarray of floats

See also

uniform

Draw samples from the parameterized uniform distribution.

Examples

>>> rng = np.random.default_rng()
>>> rng.random()
0.47108547995356098 # random
>>> type(rng.random())
<class 'float'>
>>> 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]])

rayleigh(scale=1.0, size=None)

Draw samples from a Rayleigh distribution.

The \(\chi\) 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 – Drawn samples from the parameterized Rayleigh distribution.

Return type:

ndarray or scalar

Notes

The probability density function for the Rayleigh distribution is

\[P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}\]

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

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.

Return type:

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 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]])

spawn(n_children)

Create new independent child generators.

See seedsequence-spawn for additional notes on spawning children.

Added in version 1.25.0.

Parameters:

n_children (int)

Returns:

child_generators

Return type:

list of Generators

Raises:

TypeError – When the underlying SeedSequence does not implement spawning.

See also

random.BitGenerator.spawn, random.SeedSequence.spawn

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 execution:

>>> 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)

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 – The drawn samples.

Return type:

ndarray or scalar

Notes

The probability density function for the full Cauchy distribution is

\[P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] }\]

and the Standard Cauchy distribution just sets \(x_0=0\) and \(\gamma=1\)

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. https://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
>>> rng = np.random.default_rng()
>>> s = rng.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()

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 – Drawn samples.

Return type:

float or ndarray

Examples

Output a 3x8000 array:

>>> rng = np.random.default_rng()
>>> n = rng.standard_exponential((3, 8000))

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 – Drawn samples from the parameterized standard gamma distribution.

Return type:

ndarray or scalar

See also

scipy.stats.gamma

probability density function, distribution or cumulative density function, etc.

Notes

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) 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. https://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
>>> rng = np.random.default_rng()
>>> s = 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
>>> count, bins, _ = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/
...                       (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()

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 – A floating-point array of shape size of drawn samples, or a single sample if size was not specified.

Return type:

float or ndarray

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

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 – Drawn samples from the parameterized standard Student’s t distribution.

Return type:

ndarray or scalar

Notes

The probability density function for the t distribution is

\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]

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
>>> rng = np.random.default_rng()
>>> s = 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.

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 – Drawn samples from the parameterized triangular distribution.

Return type:

ndarray or scalar

Notes

The probability density function for the triangular distribution is

\[\begin{split}P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases}\end{split}\]

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
>>> rng = np.random.default_rng()
>>> h = plt.hist(rng.triangular(-3, 0, 8, 100000), bins=200,
...              density=True)
>>> plt.show()

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 – Drawn samples from the parameterized uniform distribution.

Return type:

ndarray or scalar

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

\[p(x) = \frac{1}{b - a}\]

anywhere within the interval [a, b), and zero elsewhere.

When high == low, values of low will be returned.

Examples

Draw samples from the distribution:

>>> rng = np.random.default_rng()
>>> s = 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, _ = plt.hist(s, 15, density=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()

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 concentration (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) – Concentration 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 – Drawn samples from the parameterized von Mises distribution.

Return type:

ndarray or scalar

See also

scipy.stats.vonmises

probability density function, distribution, or cumulative density function, etc.

Notes

The probability density for the von Mises distribution is

\[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

where \(\mu\) is the mode and \(\kappa\) the concentration, and \(I_0(\kappa)\) 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 concentration
>>> rng = np.random.default_rng()
>>> s = 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
>>> 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))
>>> plt.plot(x, y, linewidth=2, color='r')
>>> plt.show()

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 – Drawn samples from the parameterized Wald distribution.

Return type:

ndarray or scalar

Notes

The probability density function for the Wald distribution is

\[P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x}\]

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
>>> rng = np.random.default_rng()
>>> h = plt.hist(rng.wald(3, 2, 100000), bins=200, density=True)
>>> plt.show()

weibull(a, size=None)

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

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 – Drawn samples from the parameterized Weibull distribution.

Return type:

ndarray or scalar

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

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

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
>>> def weibull(x, n, a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, _ = plt.hist(rng.weibull(5., 1000))
>>> x = np.linspace(0, 2, 1000)
>>> bin_spacing = np.mean(np.diff(bins))
>>> plt.plot(x, weibull(x, 1., 5.) * bin_spacing * s.size, label='Weibull PDF')
>>> plt.legend()
>>> plt.show()

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 – Drawn samples from the parameterized Zipf distribution.

Return type:

ndarray or scalar

See also

scipy.stats.zipf

probability density function, distribution, or cumulative density function, etc.

Notes

The probability mass function (PMF) for the Zipf distribution is

\[p(k) = \frac{k^{-a}}{\zeta(a)},\]

for integers \(k \geq 1\), where \(\zeta\) 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
>>> rng = np.random.default_rng()
>>> s = 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

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')
>>> plt.semilogy()
>>> plt.grid(alpha=0.4)
>>> plt.legend()
>>> plt.title(f'Zipf sample, a={a}, size={n}')
>>> plt.show()

class westpa.oldtools.aframe.mcbs.MT19937(seed=None)

Bases: BitGenerator

Container for the Mersenne Twister pseudo-random number generator.

Parameters:

seed ({None, int, array_like[ints], SeedSequence}, 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.

lock

Lock instance that is shared so that the same bit git generator can be used in multiple Generators without corrupting the state. Code that generates values from a bit generator should hold the bit generator’s lock.

Type:

threading.Lock

Notes

MT19937 provides a capsule containing function pointers that produce doubles, and unsigned 32 and 64- bit integers [1]_. These are not directly consumable in Python and must be consumed by a Generator or similar object that supports low-level access.

The Python stdlib module “random” also contains a Mersenne Twister pseudo-random number generator.

State and Seeding

The MT19937 state vector consists of a 624-element array of 32-bit unsigned integers plus a single integer value between 0 and 624 that indexes the current position within the main array.

The input seed is processed by SeedSequence to fill the whole state. The first element is reset such that only its most significant bit is set.

Parallel Features

The preferred way to use a BitGenerator in parallel applications is to use the SeedSequence.spawn method to obtain entropy values, and to use these to generate new BitGenerators:

>>> from numpy.random import Generator, MT19937, SeedSequence
>>> sg = SeedSequence(1234)
>>> rg = [Generator(MT19937(s)) for s in sg.spawn(10)]

Another method is to use MT19937.jumped which advances the state as-if \(2^{128}\) random numbers have been generated ([1]_, [2]_). This allows the original sequence to be split so that distinct segments can be used in each worker process. All generators should be chained to ensure that the segments come from the same sequence.

>>> from numpy.random import Generator, MT19937, SeedSequence
>>> sg = SeedSequence(1234)
>>> bit_generator = MT19937(sg)
>>> rg = []
>>> for _ in range(10):
...    rg.append(Generator(bit_generator))
...    # Chain the BitGenerators
...    bit_generator = bit_generator.jumped()

Compatibility Guarantee

MT19937 makes a guarantee that a fixed seed will always produce the same random integer stream.

References

[1]

Hiroshi Haramoto, Makoto Matsumoto, and Pierre L’Ecuyer, “A Fast Jump Ahead Algorithm for Linear Recurrences in a Polynomial Space”, Sequences and Their Applications - SETA, 290–298, 2008.

[2]

Hiroshi Haramoto, Makoto Matsumoto, Takuji Nishimura, François Panneton, Pierre L’Ecuyer, “Efficient Jump Ahead for F2-Linear Random Number Generators”, INFORMS JOURNAL ON COMPUTING, Vol. 20, No. 3, Summer 2008, pp. 385-390.

jumped(jumps=1)

Returns a new bit generator with the state jumped

The state of the returned bit generator is jumped as-if 2**(128 * jumps) random numbers have been generated.

Parameters:

jumps (integer, positive) – Number of times to jump the state of the bit generator returned

Returns:

bit_generator – New instance of generator jumped iter times

Return type:

MT19937

Notes

The jump step is computed using a modified version of Matsumoto’s implementation of Horner’s method. The step polynomial is precomputed to perform 2**128 steps. The jumped state has been verified to match the state produced using Matsumoto’s original code.

References

[1]

Matsumoto, M, Generating multiple disjoint streams of pseudorandom number sequences. Accessed on: May 6, 2020. http://www.math.sci.hiroshima-u.ac.jp/m-mat/MT/JUMP/

[2]

Hiroshi Haramoto, Makoto Matsumoto, Takuji Nishimura, François Panneton, Pierre L’Ecuyer, “Efficient Jump Ahead for F2-Linear Random Number Generators”, INFORMS JOURNAL ON COMPUTING, Vol. 20, No. 3, Summer 2008, pp. 385-390.

state

Get or set the PRNG state

Returns:

state – Dictionary containing the information required to describe the state of the PRNG

Return type:

dict

class westpa.oldtools.aframe.mcbs.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

westpa.oldtools.aframe.mcbs.msort(input_array)

class westpa.oldtools.aframe.mcbs.MCBSMixin

Bases: AnalysisMixin

add_args(parser, upcall=True)

process_args(args, upcall=True)

calc_mcbs_nsets(alpha=None)

calc_ci_bound_indices(n_sets=None, alpha=None)

westpa.oldtools.aframe.mcbs.calc_mcbs_nsets(alpha)

Return a bootstrap data set size appropriate for the given confidence level.

westpa.oldtools.aframe.mcbs.calc_ci_bound_indices(n_sets, alpha)

westpa.oldtools.aframe.mcbs.bootstrap_ci_ll(estimator, data, alpha, n_sets, storage, sort, eargs=(), ekwargs={}, fhat=None)

Low-level routine for calculating bootstrap error estimates. Arguments and return values are as those for bootstrap_ci, except that no argument is optional except additional arguments for the estimator (eargs, ekwargs). data must be an array (or subclass), and an additional array storage must be provided, which must be appropriately shaped and typed to hold n_sets results from estimator. Further, if the value fhat of the estimator must be pre-calculated to allocate storage, then its value may be passed; otherwise, estimator(data,*eargs,**kwargs) will be called to calculate it.

westpa.oldtools.aframe.mcbs.bootstrap_ci(estimator, data, alpha, n_sets=None, sort=<function msort>, eargs=(), ekwargs={})

Perform a Monte Carlo bootstrap of a (1-alpha) confidence interval for the given estimator. Returns (fhat, ci_lower, ci_upper), where fhat is the result of estimator(data, *eargs, **ekwargs), and ci_lower and ci_upper are the lower and upper bounds of the surrounding confidence interval, calculated by calling estimator(syndata, *eargs, **ekwargs) on each synthetic data set syndata. If n_sets is provided, that is the number of synthetic data sets generated, otherwise an appropriate size is selected automatically (see calc_mcbs_nsets()).

sort, if given, is applied to sort the results of calling estimator on each synthetic data set prior to obtaining the confidence interval. This function must sort on the last index.

Individual entries in synthetic data sets are selected by the first index of data, allowing this function to be used on arrays of multidimensional data.

Returns (fhat, lb, ub, ub-lb, abs((ub-lb)/fhat), and max(ub-fhat,fhat-lb)) (that is, the estimated value, the lower and upper bounds of the confidence interval, the width of the confidence interval, the relative width of the confidence interval, and the symmetrized error bar of the confidence interval).

westpa.oldtools.aframe.output module

class westpa.oldtools.aframe.output.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

class westpa.oldtools.aframe.output.CommonOutputMixin

Bases: AnalysisMixin

add_common_output_args(parser_or_group)

process_common_output_args(args)

westpa.oldtools.aframe.plotting module

class westpa.oldtools.aframe.plotting.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

class westpa.oldtools.aframe.plotting.PlottingMixin

Bases: AnalysisMixin

require_matplotlib()

westpa.oldtools.aframe.trajwalker module

class westpa.oldtools.aframe.trajwalker.TrajWalker(data_reader, history_chunksize=100)

Bases: object

A class to perform analysis by walking the trajectory tree. A stack is used rather than recursion, or else the highest number of iterations capable of being considered would be the same as the Python recursion limit.

trace_to_root(n_iter, seg_id)

Trace the given segment back to its starting point, returning a list of Segment objects describing the entire trajectory.

get_trajectory_roots(first_iter, last_iter, include_pcoords=True)

Get segments which start new trajectories. If min_iter or max_iter is specified, restrict the set of iterations within which the search is conducted.

get_initial_nodes(first_iter, last_iter, include_pcoords=True)

Get segments with which to begin a tree walk – those alive or created within [first_iter,last_iter].

trace_trajectories(first_iter, last_iter, callable, include_pcoords=True, cargs=None, ckwargs=None, get_state=None, set_state=None)

Walk the trajectory tree depth-first, calling

callable(segment, children, history, *cargs, **ckwargs) for each segment

visited. segment is the segment being visited, children is that segment’s children, history is the chain of segments leading to segment (not including segment). get_state and set_state are used to record and reset, respectively, any state specific to callable when a new branch is traversed.

westpa.oldtools.aframe.transitions module

class westpa.oldtools.aframe.transitions.AnalysisMixin

Bases: object

add_args(parser, upcall=True)

process_args(args, upcall=True)

class westpa.oldtools.aframe.transitions.TrajWalker(data_reader, history_chunksize=100)

Bases: object

A class to perform analysis by walking the trajectory tree. A stack is used rather than recursion, or else the highest number of iterations capable of being considered would be the same as the Python recursion limit.

trace_to_root(n_iter, seg_id)

Trace the given segment back to its starting point, returning a list of Segment objects describing the entire trajectory.

get_trajectory_roots(first_iter, last_iter, include_pcoords=True)

Get segments which start new trajectories. If min_iter or max_iter is specified, restrict the set of iterations within which the search is conducted.

get_initial_nodes(first_iter, last_iter, include_pcoords=True)

Get segments with which to begin a tree walk – those alive or created within [first_iter,last_iter].

trace_trajectories(first_iter, last_iter, callable, include_pcoords=True, cargs=None, ckwargs=None, get_state=None, set_state=None)

Walk the trajectory tree depth-first, calling

callable(segment, children, history, *cargs, **ckwargs) for each segment

visited. segment is the segment being visited, children is that segment’s children, history is the chain of segments leading to segment (not including segment). get_state and set_state are used to record and reset, respectively, any state specific to callable when a new branch is traversed.

class westpa.oldtools.aframe.transitions.TransitionEventAccumulator(n_bins, output_group, calc_fpts=True)

Bases: object

index_dtype

alias of uint64

count_dtype

alias of uint64

weight_dtype

alias of float64

output_tdat_chunksize = 4096

tdat_buffersize = 524288

max_acc = 32768

clear()

clear_state()

get_state()

set_state(state_dict)

record_transition_data(tdat)

Update running statistics and write transition data to HDF5 (with buffering)

flush_transition_data()

Flush any unwritten output that may be present

start_accumulation(assignments, weights, bin_pops, traj=0, n_iter=0)

continue_accumulation(assignments, weights, bin_pops, traj=0, n_iter=0)

class westpa.oldtools.aframe.transitions.TransitionAnalysisMixin

Bases: AnalysisMixin

require_transitions_group()

delete_transitions_group()

get_transitions_ds()

add_args(parser, upcall=True)

process_args(args, upcall=True)

require_transitions()

find_transitions()

class westpa.oldtools.aframe.transitions.BFTransitionAnalysisMixin

Bases: TransitionAnalysisMixin

require_transitions()

find_transitions(chunksize=65536)