| |
- add_docstring(...)
- docstring(obj, docstring)
Add a docstring to a built-in obj if possible.
If the obj already has a docstring raise a RuntimeError
If this routine does not know how to add a docstring to the object
raise a TypeError
- alterdot(...)
- Change `dot`, `vdot`, and `innerproduct` to use accelerated BLAS functions.
Typically, as a user of Numpy, you do not explicitly call this function. If
Numpy is built with an accelerated BLAS, this function is automatically
called when Numpy is imported.
When Numpy is built with an accelerated BLAS like ATLAS, these functions
are replaced to make use of the faster implementations. The faster
implementations only affect float32, float64, complex64, and complex128
arrays. Furthermore, the BLAS API only includes matrix-matrix,
matrix-vector, and vector-vector products. Products of arrays with larger
dimensionalities use the built in functions and are not accelerated.
See Also
--------
restoredot : `restoredot` undoes the effects of `alterdot`.
- arange(...)
- arange([start,] stop[, step,], dtype=None)
Return evenly spaced values within a given interval.
Values are generated within the half-open interval ``[start, stop)``
(in other words, the interval including `start` but excluding `stop`).
For integer arguments the function is equivalent to the Python built-in
`range <http://docs.python.org/lib/built-in-funcs.html>`_ function,
but returns a ndarray rather than a list.
Parameters
----------
start : number, optional
Start of interval. The interval includes this value. The default
start value is 0.
stop : number
End of interval. The interval does not include this value.
step : number, optional
Spacing between values. For any output `out`, this is the distance
between two adjacent values, ``out[i+1] - out[i]``. The default
step size is 1. If `step` is specified, `start` must also be given.
dtype : dtype
The type of the output array. If `dtype` is not given, infer the data
type from the other input arguments.
Returns
-------
out : ndarray
Array of evenly spaced values.
For floating point arguments, the length of the result is
``ceil((stop - start)/step)``. Because of floating point overflow,
this rule may result in the last element of `out` being greater
than `stop`.
See Also
--------
linspace : Evenly spaced numbers with careful handling of endpoints.
ogrid: Arrays of evenly spaced numbers in N-dimensions
mgrid: Grid-shaped arrays of evenly spaced numbers in N-dimensions
Examples
--------
>>> np.arange(3)
array([0, 1, 2])
>>> np.arange(3.0)
array([ 0., 1., 2.])
>>> np.arange(3,7)
array([3, 4, 5, 6])
>>> np.arange(3,7,2)
array([3, 5])
- array(...)
- array(object, dtype=None, copy=True, order=None, subok=False, ndmin=True)
Create an array.
Parameters
----------
object : array_like
An array, any object exposing the array interface, an
object whose __array__ method returns an array, or any
(nested) sequence.
dtype : data-type, optional
The desired data-type for the array. If not given, then
the type will be determined as the minimum type required
to hold the objects in the sequence. This argument can only
be used to 'upcast' the array. For downcasting, use the
.astype(t) method.
copy : bool, optional
If true (default), then the object is copied. Otherwise, a copy
will only be made if __array__ returns a copy, if obj is a
nested sequence, or if a copy is needed to satisfy any of the other
requirements (`dtype`, `order`, etc.).
order : {'C', 'F', 'A'}, optional
Specify the order of the array. If order is 'C' (default), then the
array will be in C-contiguous order (last-index varies the
fastest). If order is 'F', then the returned array
will be in Fortran-contiguous order (first-index varies the
fastest). If order is 'A', then the returned array may
be in any order (either C-, Fortran-contiguous, or even
discontiguous).
subok : bool, optional
If True, then sub-classes will be passed-through, otherwise
the returned array will be forced to be a base-class array (default).
ndmin : int, optional
Specifies the minimum number of dimensions that the resulting
array should have. Ones will be pre-pended to the shape as
needed to meet this requirement.
Examples
--------
>>> np.array([1, 2, 3])
array([1, 2, 3])
Upcasting:
>>> np.array([1, 2, 3.0])
array([ 1., 2., 3.])
More than one dimension:
>>> np.array([[1, 2], [3, 4]])
array([[1, 2],
[3, 4]])
Minimum dimensions 2:
>>> np.array([1, 2, 3], ndmin=2)
array([[1, 2, 3]])
Type provided:
>>> np.array([1, 2, 3], dtype=complex)
array([ 1.+0.j, 2.+0.j, 3.+0.j])
Data-type consisting of more than one element:
>>> x = np.array([(1,2),(3,4)],dtype=[('a','<i4'),('b','<i4')])
>>> x['a']
array([1, 3])
Creating an array from sub-classes:
>>> np.array(np.mat('1 2; 3 4'))
array([[1, 2],
[3, 4]])
>>> np.array(np.mat('1 2; 3 4'), subok=True)
matrix([[1, 2],
[3, 4]])
- beta(...)
- beta(a, b, size=None)
The Beta distribution over ``[0, 1]``.
The Beta distribution is a special case of the Dirichlet distribution,
and is related to the Gamma distribution. It has the probability
distribution function
.. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1}
(1 - x)^{\beta - 1},
where the normalisation, B, is the beta function,
.. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1}
(1 - t)^{\beta - 1} dt.
It is often seen in Bayesian inference and order statistics.
Parameters
----------
a : float
Alpha, non-negative.
b : float
Beta, non-negative.
size : tuple of ints, optional
The number of samples to draw. The ouput is packed according to
the size given.
Returns
-------
out : ndarray
Array of the given shape, containing values drawn from a
Beta distribution.
- bincount(...)
- bincount(x, weights=None)
Return the number of occurrences of each value in array of nonnegative
integers.
The output, b[i], represents the number of times that i is found in `x`.
If `weights` is specified, every occurrence of i at a position p
contributes `weights` [p] instead of 1.
Parameters
----------
x : array_like, 1 dimension, nonnegative integers
Input array.
weights : array_like, same shape as `x`, optional
Weights.
See Also
--------
histogram, digitize, unique
- binomial(...)
- 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 : float (but truncated to an integer)
parameter, > 0.
p : float
parameter, >= 0 and <=1.
size : {tuple, int}
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
samples : {ndarray, scalar}
where the values are all integers in [0, n].
See Also
--------
scipy.stats.distributions.binom : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Binomial distribution is
.. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N},
where :math:`n` is the number of trials, :math:`p` is the probability
of success, and :math:`N` is the number of successes.
When estimating the standard error of a proportion in a population by
using a random sample, the normal distribution works well unless the
product p*n <=5, where p = population proportion estimate, and n =
number of samples, in which case the binomial distribution is used
instead. For example, a sample of 15 people shows 4 who are left
handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
so the binomial distribution should be used in this case.
References
----------
.. [1] Dalgaard, Peter, "Introductory Statistics with R",
Springer-Verlag, 2002.
.. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
Fifth Edition, 2002.
.. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden
and Quigley, 1972.
.. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/BinomialDistribution.html
.. [5] Wikipedia, "Binomial-distribution",
http://en.wikipedia.org/wiki/Binomial_distribution
Examples
--------
Draw samples from the distribution:
>>> n, p = 10, .5 # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.
A real world example. A company drills 9 wild-cat oil exploration
wells, each with an estimated probability of success of 0.1. All nine
wells fail. What is the probability of that happening?
Let's do 20,000 trials of the model, and count the number that
generate zero positive results.
>>> sum(np.random.binomial(9,0.1,20000)==0)/20000.
answer = 0.38885, or 38%.
- bytes(...)
- bytes(length)
Return random bytes.
Parameters
----------
length : int
Number of random bytes.
Returns
-------
out : str
String of length `N`.
Examples
--------
>>> np.random.bytes(10)
' eh\x85\x022SZ\xbf\xa4' #random
- can_cast(...)
- can_cast(from=d1, to=d2)
Returns True if cast between data types can occur without losing precision.
Parameters
----------
from: data type code
Data type code to cast from.
to: data type code
Data type code to cast to.
Returns
-------
out : bool
True if cast can occur without losing precision.
- chisquare(...)
- 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 : int
Number of degrees of freedom.
size : tuple of ints, int, optional
Size of the returned array. By default, a scalar is
returned.
Returns
-------
output : ndarray
Samples drawn from the distribution, packed in a `size`-shaped
array.
Raises
------
ValueError
When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
is given.
Notes
-----
The variable obtained by summing the squares of `df` independent,
standard normally distributed random variables:
.. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i
is chi-square distributed, denoted
.. math:: Q \sim \chi^2_k.
The probability density function of the chi-squared distribution is
.. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
x^{k/2 - 1} e^{-x/2},
where :math:`\Gamma` is the gamma function,
.. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.
References
----------
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods,
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
.. [2] Wikipedia, "Chi-square distribution",
http://en.wikipedia.org/wiki/Chi-square_distribution
Examples
--------
>>> np.random.chisquare(2,4)
array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272])
- compare_chararrays(...)
- concatenate(...)
- concatenate((a1, a2, ...), axis=0)
Join a sequence of arrays together.
Parameters
----------
a1, a2, ... : sequence of ndarrays
The arrays must have the same shape, except in the dimension
corresponding to `axis` (the first, by default).
axis : int, optional
The axis along which the arrays will be joined. Default is 0.
Returns
-------
res : ndarray
The concatenated array.
See Also
--------
array_split : Split an array into multiple sub-arrays of equal or
near-equal size.
split : Split array into a list of multiple sub-arrays of equal size.
hsplit : Split array into multiple sub-arrays horizontally (column wise)
vsplit : Split array into multiple sub-arrays vertically (row wise)
dsplit : Split array into multiple sub-arrays along the 3rd axis (depth).
hstack : Stack arrays in sequence horizontally (column wise)
vstack : Stack arrays in sequence vertically (row wise)
dstack : Stack arrays in sequence depth wise (along third dimension)
Examples
--------
>>> a = np.array([[1, 2], [3, 4]])
>>> b = np.array([[5, 6]])
>>> np.concatenate((a, b), axis=0)
array([[1, 2],
[3, 4],
[5, 6]])
>>> np.concatenate((a, b.T), axis=1)
array([[1, 2, 5],
[3, 4, 6]])
- digitize(...)
- digitize(x, bins)
Return the indices of the bins to which each value in input array belongs.
Each index returned is such that `bins[i-1]` <= `x` < `bins[i]` if `bins`
is monotonically increasing, or `bins[i-1]` > `x` >= `bins[i]` if `bins`
is monotonically decreasing. Beyond the bounds of `bins`, 0 or len(`bins`)
is returned as appropriate.
Parameters
----------
x : array_like
Input array to be binned.
bins : array_like
Array of bins.
Returns
-------
out : ndarray
Output array of indices of same shape as `x`.
Examples
--------
>>> x = np.array([0.2, 6.4, 3.0, 1.6])
>>> bins = np.array([0.0, 1.0, 2.5, 4.0, 10.0])
>>> d = np.digitize(x,bins)
>>> d
array([1, 4, 3, 2])
>>> for n in range(len(x)):
... print bins[d[n]-1], "<=", x[n], "<", bins[d[n]]
...
0.0 <= 0.2 < 1.0
4.0 <= 6.4 < 10.0
2.5 <= 3.0 < 4.0
1.0 <= 1.6 < 2.5
- dot(...)
- dot(a, b)
Dot product of two arrays.
For 2-D arrays it is equivalent to matrix multiplication, and for 1-D
arrays to inner product of vectors (without complex conjugation). For
N dimensions it is a sum product over the last axis of `a` and
the second-to-last of `b`::
dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])
Parameters
----------
a : array_like
First argument.
b : array_like
Second argument.
Returns
-------
output : ndarray
Returns the dot product of `a` and `b`. If `a` and `b` are both
scalars or both 1-D arrays then a scalar is returned; otherwise
an array is returned.
Raises
------
ValueError
If the last dimension of `a` is not the same size as
the second-to-last dimension of `b`.
See Also
--------
vdot : Complex-conjugating dot product.
tensordot : Sum products over arbitrary axes.
Examples
--------
>>> np.dot(3, 4)
12
Neither argument is complex-conjugated:
>>> np.dot([2j, 3j], [2j, 3j])
(-13+0j)
For 2-D arrays it's the matrix product:
>>> a = [[1, 0], [0, 1]]
>>> b = [[4, 1], [2, 2]]
>>> np.dot(a, b)
array([[4, 1],
[2, 2]])
>>> a = np.arange(3*4*5*6).reshape((3,4,5,6))
>>> b = np.arange(3*4*5*6)[::-1].reshape((5,4,6,3))
>>> np.dot(a, b)[2,3,2,1,2,2]
499128
>>> sum(a[2,3,2,:] * b[1,2,:,2])
499128
- empty(...)
- empty(shape, dtype=float, order='C')
Return a new array of given shape and type, without initialising entries.
Parameters
----------
shape : {tuple of int, int}
Shape of the empty array
dtype : data-type, optional
Desired output data-type.
order : {'C', 'F'}, optional
Whether to store multi-dimensional data in C (row-major) or
Fortran (column-major) order in memory.
See Also
--------
empty_like, zeros
Notes
-----
`empty`, unlike `zeros`, does not set the array values to zero,
and may therefore be marginally faster. On the other hand, it requires
the user to manually set all the values in the array, and should be
used with caution.
Examples
--------
>>> np.empty([2, 2])
array([[ -9.74499359e+001, 6.69583040e-309], #random data
[ 2.13182611e-314, 3.06959433e-309]])
>>> np.empty([2, 2], dtype=int)
array([[-1073741821, -1067949133], #random data
[ 496041986, 19249760]])
- exponential(...)
- exponential(scale=1.0, size=None)
Exponential distribution.
Its probability density function is
.. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),
for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter,
which is the inverse of the rate parameter :math:`\lambda = 1/\beta`.
The rate parameter is an alternative, widely used parameterization
of the exponential distribution [3]_.
The exponential distribution is a continuous analogue of the
geometric distribution. It describes many common situations, such as
the size of raindrops measured over many rainstorms [1]_, or the time
between page requests to Wikipedia [2]_.
Parameters
----------
scale : float
The scale parameter, :math:`\beta = 1/\lambda`.
size : tuple of ints
Number of samples to draw. The output is shaped
according to `size`.
References
----------
.. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and
Random Signal Principles", 4th ed, 2001, p. 57.
.. [2] "Poisson Process", Wikipedia,
http://en.wikipedia.org/wiki/Poisson_process
.. [3] "Exponential Distribution, Wikipedia,
http://en.wikipedia.org/wiki/Exponential_distribution
- f(...)
- f(dfnum, dfden, size=None)
Draw samples from a 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 should 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
Degrees of freedom in numerator. Should be greater than zero.
dfden : float
Degrees of freedom in denominator. Should be greater than zero.
size : {tuple, int}, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``,
then ``m * n * k`` samples are drawn. By default only one sample
is returned.
Returns
-------
samples : {ndarray, scalar}
Samples from the Fisher distribution.
See Also
--------
scipy.stats.distributions.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",
http://en.wikipedia.org/wiki/F-distribution
Examples
--------
An example from Glantz[1], pp 47-40.
Two groups, children of diabetics (25 people) and children from people
without diabetes (25 controls). Fasting blood glucose was measured,
case group had a mean value of 86.1, controls had a mean value of
82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
data consistent with the null hypothesis that the parents diabetic
status does not affect their children's blood glucose levels?
Calculating the F statistic from the data gives a value of 36.01.
Draw samples from the distribution:
>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)
The lower bound for the top 1% of the samples is :
>>> sort(s)[-10]
7.61988120985
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.
- fastCopyAndTranspose = _fastCopyAndTranspose(...)
- _fastCopyAndTranspose(a)
- frombuffer(...)
- frombuffer(buffer, dtype=float, count=-1, offset=0)
Interpret a buffer as a 1-dimensional array.
Parameters
----------
buffer
An object that exposes the buffer interface.
dtype : data-type, optional
Data type of the returned array.
count : int, optional
Number of items to read. ``-1`` means all data in the buffer.
offset : int, optional
Start reading the buffer from this offset.
Notes
-----
If the buffer has data that is not in machine byte-order, this
should be specified as part of the data-type, e.g.::
>>> dt = np.dtype(int)
>>> dt = dt.newbyteorder('>')
>>> np.frombuffer(buf, dtype=dt)
The data of the resulting array will not be byteswapped,
but will be interpreted correctly.
Examples
--------
>>> s = 'hello world'
>>> np.frombuffer(s, dtype='S1', count=5, offset=6)
array(['w', 'o', 'r', 'l', 'd'],
dtype='|S1')
- fromfile(...)
- fromfile(file, dtype=float, count=-1, sep='')
Construct an array from data in a text or binary file.
A highly efficient way of reading binary data with a known data-type,
as well as parsing simply formatted text files. Data written using the
`tofile` method can be read using this function.
Parameters
----------
file : file or string
Open file object or filename.
dtype : data-type
Data type of the returned array.
For binary files, it is used to determine the size and byte-order
of the items in the file.
count : int
Number of items to read. ``-1`` means all items (i.e., the complete
file).
sep : string
Separator between items if file is a text file.
Empty ("") separator means the file should be treated as binary.
Spaces (" ") in the separator match zero or more whitespace characters.
A separator consisting only of spaces must match at least one
whitespace.
See also
--------
load, save
ndarray.tofile
loadtxt : More flexible way of loading data from a text file.
Notes
-----
Do not rely on the combination of `tofile` and `fromfile` for
data storage, as the binary files generated are are not platform
independent. In particular, no byte-order or data-type information is
saved. Data can be stored in the platform independent ``.npy`` format
using `save` and `load` instead.
Examples
--------
Construct an ndarray:
>>> dt = np.dtype([('time', [('min', int), ('sec', int)]),
... ('temp', float)])
>>> x = np.zeros((1,), dtype=dt)
>>> x['time']['min'] = 10; x['temp'] = 98.25
>>> x
array([((10, 0), 98.25)],
dtype=[('time', [('min', '<i4'), ('sec', '<i4')]), ('temp', '<f8')])
Save the raw data to disk:
>>> import os
>>> fname = os.tmpnam()
>>> x.tofile(fname)
Read the raw data from disk:
>>> np.fromfile(fname, dtype=dt)
array([((10, 0), 98.25)],
dtype=[('time', [('min', '<i4'), ('sec', '<i4')]), ('temp', '<f8')])
The recommended way to store and load data:
>>> np.save(fname, x)
>>> np.load(fname + '.npy')
array([((10, 0), 98.25)],
dtype=[('time', [('min', '<i4'), ('sec', '<i4')]), ('temp', '<f8')])
- fromiter(...)
- fromiter(iterable, dtype, count=-1)
Create a new 1-dimensional array from an iterable object.
Parameters
----------
iterable : iterable object
An iterable object providing data for the array.
dtype : data-type
The data type of the returned array.
count : int, optional
The number of items to read from iterable. The default is -1,
which means all data is read.
Returns
-------
out : ndarray
The output array.
Notes
-----
Specify ``count`` to improve performance. It allows
``fromiter`` to pre-allocate the output array, instead of
resizing it on demand.
Examples
--------
>>> iterable = (x*x for x in range(5))
>>> np.fromiter(iterable, np.float)
array([ 0., 1., 4., 9., 16.])
- frompyfunc(...)
- frompyfunc(func, nin, nout) take an arbitrary python
function that takes nin objects as input and returns
nout objects and return a universal function (ufunc).
This ufunc always returns PyObject arrays
- fromstring(...)
- fromstring(string, dtype=float, count=-1, sep='')
Return a new 1d array initialized from raw binary or text data in
string.
Parameters
----------
string : str
A string containing the data.
dtype : dtype, optional
The data type of the array. For binary input data, the data must be
in exactly this format.
count : int, optional
Read this number of `dtype` elements from the data. If this is
negative, then the size will be determined from the length of the
data.
sep : str, optional
If provided and not empty, then the data will be interpreted as
ASCII text with decimal numbers. This argument is interpreted as the
string separating numbers in the data. Extra whitespace between
elements is also ignored.
Returns
-------
arr : array
The constructed array.
Raises
------
ValueError
If the string is not the correct size to satisfy the requested
`dtype` and `count`.
Examples
--------
>>> np.fromstring('\x01\x02', dtype=np.uint8)
array([1, 2], dtype=uint8)
>>> np.fromstring('1 2', dtype=int, sep=' ')
array([1, 2])
>>> np.fromstring('1, 2', dtype=int, sep=',')
array([1, 2])
>>> np.fromstring('\x01\x02\x03\x04\x05', dtype=np.uint8, count=3)
array([1, 2, 3], dtype=uint8)
Invalid inputs:
>>> np.fromstring('\x01\x02\x03\x04\x05', dtype=np.int32)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: string size must be a multiple of element size
>>> np.fromstring('\x01\x02', dtype=np.uint8, count=3)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: string is smaller than requested size
- gamma(...)
- 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 : scalar > 0
The shape of the gamma distribution.
scale : scalar > 0, optional
The scale of the gamma distribution. Default is equal to 1.
size : shape_tuple, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
out : ndarray, float
Returns one sample unless `size` parameter is specified.
See Also
--------
scipy.stats.distributions.gamma : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Gamma distribution is
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
where :math:`k` is the shape and :math:`\theta` the scale,
and :math:`\Gamma` is the Gamma function.
The Gamma distribution is often used to model the times to failure of
electronic components, and arises naturally in processes for which the
waiting times between Poisson distributed events are relevant.
References
----------
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/GammaDistribution.html
.. [2] Wikipedia, "Gamma-distribution",
http://en.wikipedia.org/wiki/Gamma-distribution
Examples
--------
Draw samples from the distribution:
>>> shape, scale = 2., 2. # mean and dispersion
>>> s = np.random.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, ignored = plt.hist(s, 50, normed=True)
>>> y = bins**(shape-1)*((exp(-bins/scale))/\
(sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()
- geometric(...)
- geometric(p, size=None)
Draw samples from the geometric distribution.
Bernoulli trials are experiments with one of two outcomes:
success or failure (an example of such an experiment is flipping
a coin). The geometric distribution models the number of trials
that must be run in order to achieve success. It is therefore
supported on the positive integers, ``k = 1, 2, ...``.
The probability mass function of the geometric distribution is
.. math:: f(k) = (1 - p)^{k - 1} p
where `p` is the probability of success of an individual trial.
Parameters
----------
p : float
The probability of success of an individual trial.
size : tuple of ints
Number of values to draw from the distribution. The output
is shaped according to `size`.
Returns
-------
out : ndarray
Samples from the geometric distribution, shaped according to
`size`.
Examples
--------
Draw ten thousand values from the geometric distribution,
with the probability of an individual success equal to 0.35:
>>> z = np.random.geometric(p=0.35, size=10000)
How many trials succeeded after a single run?
>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
- get_state(...)
- get_state()
Return a tuple representing the internal state of the generator.
Returns
-------
out : tuple(string, list of 624 integers, int, int, float)
The returned tuple has the following items:
1. the string 'MT19937'
2. a list of 624 integer keys
3. an integer pos
4. an integer has_gauss
5. and a float cached_gaussian
See Also
--------
set_state
- getbuffer(...)
- getbuffer(obj [,offset[, size]])
Create a buffer object from the given object referencing a slice of
length size starting at offset. Default is the entire buffer. A
read-write buffer is attempted followed by a read-only buffer.
- geterrobj(...)
- geterrobj()
Used internally by `geterr`.
Returns
-------
errobj : list
Internal numpy buffer size, error mask, error callback function.
- gumbel(...)
- gumbel(loc=0.0, scale=1.0, size=None)
Gumbel distribution.
Draw samples from a Gumbel distribution with specified location (or mean)
and scale (or standard deviation).
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, it may be derived by
considering a Gaussian process of measurements, and generating the pdf for
the maximum values from that set of measurements (see examples).
Parameters
----------
loc : float
The location of the mode of the distribution.
scale : float
The scale parameter of the distribution.
size : tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
See Also
--------
scipy.stats.gumbel : probability density function,
distribution or cumulative density function, etc.
weibull, scipy.stats.genextreme
Notes
-----
The probability density for the Gumbel distribution is
.. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
\beta}},
where :math:`\mu` is the mode, a location parameter, and :math:`\beta`
is the scale parameter.
The Gumbel (named for German mathematician Emil Julius Gumbel) was used
very early in the hydrology literature, for modeling the occurrence of
flood events. It is also used for modeling maximum wind speed and rainfall
rates. It is a "fat-tailed" distribution - the probability of an event in
the tail of the distribution is larger than if one used a Gaussian, hence
the surprisingly frequent occurrence of 100-year floods. Floods were
initially modeled as a Gaussian process, which underestimated the frequency
of extreme events.
It is one of a class of extreme value distributions, the Generalized
Extreme Value (GEV) distributions, which also includes the Weibull and
Frechet.
The function has a mean of :math:`\mu + 0.57721\beta` and a variance of
:math:`\frac{\pi^2}{6}\beta^2`.
References
----------
.. [1] Gumbel, E.J. (1958). Statistics of Extremes. Columbia University
Press.
.. [2] Reiss, R.-D. and Thomas M. (2001), Statistical Analysis of Extreme
Values, from Insurance, Finance, Hydrology and Other Fields,
Birkhauser Verlag, Basel: Boston : Berlin.
.. [3] Wikipedia, "Gumbel distribution",
http://en.wikipedia.org/wiki/Gumbel_distribution
Examples
--------
Draw samples from the distribution:
>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=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 = np.random.normal(mu, beta, 1000)
... means.append(a.mean())
... maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, normed=True)
>>> beta = np.std(maxima)*np.pi/np.sqrt(6)
>>> 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(...)
- 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 : float (but truncated to an integer)
parameter, > 0.
nbad : float
parameter, >= 0.
nsample : float
parameter, > 0 and <= ngood+nbad
size : {tuple, int}
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
samples : {ndarray, scalar}
where the values are all integers in [0, n].
See Also
--------
scipy.stats.distributions.hypergeom : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Hypergeometric distribution is
.. math:: P(x) = \frac{\binom{m}{n}\binom{N-m}{n-x}}{\binom{N}{n}},
where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n`
for P(x) the probability of x successes, n = ngood, m = nbad, and
N = number of samples.
Consider an urn with black and white marbles in it, ngood of them
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.
References
----------
.. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
and Quigley, 1972.
.. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/HypergeometricDistribution.html
.. [3] Wikipedia, "Hypergeometric-distribution",
http://en.wikipedia.org/wiki/Hypergeometric-distribution
Examples
--------
Draw samples from the distribution:
>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> 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 = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
# answer = 0.003 ... pretty unlikely!
- inner(...)
- inner(a, b)
Inner product of two arrays.
Ordinary inner product of vectors for 1-D arrays (without complex
conjugation), in higher dimensions a sum product over the last axes.
Parameters
----------
a, b : array_like
If `a` and `b` are nonscalar, their last dimensions of must match.
Returns
-------
out : ndarray
`out.shape = a.shape[:-1] + b.shape[:-1]`
Raises
------
ValueError
If the last dimension of `a` and `b` has different size.
See Also
--------
tensordot : Sum products over arbitrary axes.
dot : Generalised matrix product, using second last dimension of `b`.
Notes
-----
For vectors (1-D arrays) it computes the ordinary inner-product::
np.inner(a, b) = sum(a[:]*b[:])
More generally, if `ndim(a) = r > 0` and `ndim(b) = s > 0`::
np.inner(a, b) = np.tensordot(a, b, axes=(-1,-1))
or explicitly::
np.inner(a, b)[i0,...,ir-1,j0,...,js-1]
= sum(a[i0,...,ir-1,:]*b[j0,...,js-1,:])
In addition `a` or `b` may be scalars, in which case::
np.inner(a,b) = a*b
Examples
--------
Ordinary inner product for vectors:
>>> a = np.array([1,2,3])
>>> b = np.array([0,1,0])
>>> np.inner(a, b)
2
A multidimensional example:
>>> a = np.arange(24).reshape((2,3,4))
>>> b = np.arange(4)
>>> np.inner(a, b)
array([[ 14, 38, 62],
[ 86, 110, 134]])
An example where `b` is a scalar:
>>> np.inner(np.eye(2), 7)
array([[ 7., 0.],
[ 0., 7.]])
- int_asbuffer(...)
- laplace(...)
- laplace(loc=0.0, scale=1.0, size=None)
Laplace or double exponential distribution.
It has the probability density function
.. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda}
\exp\left(-\frac{|x - \mu|}{\lambda}\right).
The Laplace distribution is similar to the Gaussian/normal distribution,
but is sharper at the peak and has fatter tails.
Parameters
----------
loc : float
The position, :math:`\mu`, of the distribution peak.
scale : float
:math:`\lambda`, the exponential decay.
- lexsort(...)
- lexsort(keys, axis=-1)
Perform an indirect sort using a sequence of keys.
Given multiple sorting keys, which can be interpreted as columns in a
spreadsheet, lexsort returns an array of integer indices that describes
the sort order by multiple columns. The last key in the sequence is used
for the primary sort order, the second-to-last key for the secondary sort
order, and so on. The keys argument must be a sequence of objects that
can be converted to arrays of the same shape. If a 2D array is provided
for the keys argument, it's rows are interpreted as the sorting keys and
sorting is according to the last row, second last row etc.
Parameters
----------
keys : (k,N) array or tuple containing k (N,)-shaped sequences
The `k` different "columns" to be sorted. The last column (or row if
`keys` is a 2D array) is the primary sort key.
axis : int, optional
Axis to be indirectly sorted. By default, sort over the last axis.
Returns
-------
indices : (N,) ndarray of ints
Array of indices that sort the keys along the specified axis.
See Also
--------
argsort : Indirect sort.
ndarray.sort : In-place sort.
sort : Return a sorted copy of an array.
Examples
--------
Sort names: first by surname, then by name.
>>> surnames = ('Hertz', 'Galilei', 'Hertz')
>>> first_names = ('Heinrich', 'Galileo', 'Gustav')
>>> ind = np.lexsort((first_names, surnames))
>>> ind
array([1, 2, 0])
>>> [surnames[i] + ", " + first_names[i] for i in ind]
['Galilei, Galileo', 'Hertz, Gustav', 'Hertz, Heinrich']
Sort two columns of numbers:
>>> a = [1,5,1,4,3,4,4] # First column
>>> b = [9,4,0,4,0,2,1] # Second column
>>> ind = np.lexsort((b,a)) # Sort by a, then by b
>>> print ind
[2 0 4 6 5 3 1]
>>> [(a[i],b[i]) for i in ind]
[(1, 0), (1, 9), (3, 0), (4, 1), (4, 2), (4, 4), (5, 4)]
Note that sorting is first according to the elements of ``a``.
Secondary sorting is according to the elements of ``b``.
A normal ``argsort`` would have yielded:
>>> [(a[i],b[i]) for i in np.argsort(a)]
[(1, 9), (1, 0), (3, 0), (4, 4), (4, 2), (4, 1), (5, 4)]
Structured arrays are sorted lexically by ``argsort``:
>>> x = np.array([(1,9), (5,4), (1,0), (4,4), (3,0), (4,2), (4,1)],
... dtype=np.dtype([('x', int), ('y', int)]))
>>> np.argsort(x) # or np.argsort(x, order=('x', 'y'))
array([2, 0, 4, 6, 5, 3, 1])
- loads(...)
- loads(string) -- Load a pickle from the given string
- logistic(...)
- 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
scale : float > 0.
size : {tuple, int}
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
samples : {ndarray, scalar}
where the values are all integers in [0, n].
See Also
--------
scipy.stats.distributions.logistic : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Logistic distribution is
.. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},
where :math:`\mu` = location and :math:`s` = scale.
The Logistic distribution is used in Extreme Value problems where it
can act as a mixture of Gumbel distributions, in Epidemiology, and by
the World Chess Federation (FIDE) where it is used in the Elo ranking
system, assuming the performance of each player is a logistically
distributed random variable.
References
----------
.. [1] Reiss, R.-D. and Thomas M. (2001), Statistical Analysis of Extreme
Values, from Insurance, Finance, Hydrology and Other Fields,
Birkhauser Verlag, Basel, pp 132-133.
.. [2] Weisstein, Eric W. "Logistic Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/LogisticDistribution.html
.. [3] Wikipedia, "Logistic-distribution",
http://en.wikipedia.org/wiki/Logistic-distribution
Examples
--------
Draw samples from the distribution:
>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> count, bins, ignored = plt.hist(s, bins=50)
# plot against distribution
>>> def logist(x, loc, scale):
... return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2)
>>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\
... logist(bins, loc, scale).max())
>>> plt.show()
- lognormal(...)
- lognormal(mean=0.0, sigma=1.0, size=None)
Return samples drawn from a log-normal distribution.
Draw samples from a log-normal distribution with specified mean, standard
deviation, and 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
Mean value of the underlying normal distribution
sigma : float, >0.
Standard deviation of the underlying normal distribution
size : tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
See Also
--------
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.
Notes
-----
A variable `x` has a log-normal distribution if `log(x)` is normally
distributed.
The probability density function for the log-normal distribution is
.. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}
where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation
of the normally distributed logarithm of the variable.
A log-normal distribution results if a random variable is the *product* of
a large number of independent, identically-distributed variables in the
same way that a normal distribution results if the variable is the *sum*
of a large number of independent, identically-distributed variables
(see the last example). It is one of the so-called "fat-tailed"
distributions.
The log-normal distribution is commonly used to model the lifespan of units
with fatigue-stress failure modes. Since this includes
most mechanical systems, the log-normal distribution has widespread
application.
It is also commonly used to model oil field sizes, species abundance, and
latent periods of infectious diseases.
References
----------
.. [1] Eckhard Limpert, Werner A. Stahel, and Markus Abbt, "Log-normal
Distributions across the Sciences: Keys and Clues", May 2001
Vol. 51 No. 5 BioScience
http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
.. [2] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
Values, Birkhauser Verlag, Basel, pp 31-32.
.. [3] Wikipedia, "Lognormal distribution",
http://en.wikipedia.org/wiki/Lognormal_distribution
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, normed=True, align='center')
>>> 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.
>>> b = []
>>> for i in range(1000):
... a = 10. + np.random.random(100)
... b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, normed=True, align='center')
>>> 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(...)
- logseries(p, size=None)
Draw samples from a Logarithmic Series distribution.
Samples are drawn from a Log Series distribution with specified
parameter, p (probability, 0 < p < 1).
Parameters
----------
loc : float
scale : float > 0.
size : {tuple, int}
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
samples : {ndarray, scalar}
where the values are all integers in [0, n].
See Also
--------
scipy.stats.distributions.logser : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Log Series distribution is
.. math:: P(k) = \frac{-p^k}{k \ln(1-p)},
where p = probability.
The Log Series distribution is frequently used to represent species
richness and occurrence, first proposed by Fisher, Corbet, and
Williams in 1943 [2]. It may also be used to model the numbers of
occupants seen in cars [3].
References
----------
.. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional
species diversity through the log series distribution of
occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
Volume 5, Number 5, September 1999 , pp. 187-195(9).
.. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
relation between the number of species and the number of
individuals in a random sample of an animal population.
Journal of Animal Ecology, 12:42-58.
.. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
Data Sets, CRC Press, 1994.
.. [4] Wikipedia, "Logarithmic-distribution",
http://en.wikipedia.org/wiki/Logarithmic-distribution
Examples
--------
Draw samples from the distribution:
>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> count, bins, ignored = plt.hist(s)
# plot against distribution
>>> def logseries(k, p):
... return -p**k/(k*log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/\
logseries(bins, a).max(),'r')
>>> plt.show()
- multinomial(...)
- multinomial(n, pvals, size=None)
Draw samples from a multinomial distribution.
The multinomial distribution is a multivariate generalisation 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
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the ``p`` different outcomes. These
should sum to 1 (however, the last element is always assumed to
account for the remaining probability, as long as
``sum(pvals[:-1]) <= 1)``.
size : tuple of ints
Given a `size` of ``(M, N, K)``, then ``M*N*K`` samples are drawn,
and the output shape becomes ``(M, N, K, p)``, since each sample
has shape ``(p,)``.
Examples
--------
Throw a dice 20 times:
>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]])
It landed 4 times on 1, once on 2, etc.
Now, throw the dice 20 times, and 20 times again:
>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3],
[2, 4, 3, 4, 0, 7]])
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.
A loaded dice is more likely to land on number 6:
>>> np.random.multinomial(100, [1/7.]*5)
array([13, 16, 13, 16, 42])
- multivariate_normal(...)
- multivariate_normal(mean, cov[, size])
Draw random samples from a multivariate normal distribution.
The multivariate normal, multinormal or Gaussian distribution is a
generalisation of the one-dimensional normal distribution to higher
dimensions.
Such a distribution is specified by its mean and covariance matrix,
which are analogous to the mean (average or "centre") and variance
(standard deviation squared or "width") of the one-dimensional normal
distribution.
Parameters
----------
mean : (N,) ndarray
Mean of the N-dimensional distribution.
cov : (N,N) ndarray
Covariance matrix of the distribution.
size : 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 sample is returned.
Returns
-------
out : ndarray
The drawn samples, arranged according to `size`. If the
shape given is (m,n,...), then the shape of `out` is is
(m,n,...,N).
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
value drawn from the distribution.
Notes
-----
The mean is a coordinate in N-dimensional space, which represents the
location where samples are most likely to be generated. This is
analogous to the peak of the bell curve for the one-dimensional or
univariate normal distribution.
Covariance indicates the level to which two variables vary together.
From the multivariate normal distribution, we draw N-dimensional
samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix
element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
"spread").
Instead of specifying the full covariance matrix, popular
approximations include:
- Spherical covariance (`cov` is a multiple of the identity matrix)
- Diagonal covariance (`cov` has non-negative elements, and only on
the diagonal)
This geometrical property can be seen in two dimensions by plotting
generated data-points:
>>> mean = [0,0]
>>> cov = [[1,0],[0,100]] # diagonal covariance, points lie on x or y-axis
>>> import matplotlib.pyplot as plt
>>> x,y = np.random.multivariate_normal(mean,cov,5000).T
>>> plt.plot(x,y,'x'); plt.axis('equal'); plt.show()
Note that the covariance matrix must be non-negative definite.
References
----------
.. [1] A. Papoulis, "Probability, Random Variables, and Stochastic
Processes," 3rd ed., McGraw-Hill Companies, 1991
.. [2] R.O. Duda, P.E. Hart, and D.G. Stork, "Pattern Classification,"
2nd ed., Wiley, 2001.
Examples
--------
>>> mean = (1,2)
>>> cov = [[1,0],[1,0]]
>>> x = np.random.multivariate_normal(mean,cov,(3,3))
>>> x.shape
(3, 3, 2)
The following is probably true, given that 0.6 is roughly twice the
standard deviation:
>>> print list( (x[0,0,:] - mean) < 0.6 )
[True, True]
- negative_binomial(...)
- negative_binomial(n, p, size=None)
Negative Binomial distribution.
- newbuffer(...)
- newbuffer(size)
Return a new uninitialized buffer object of size bytes
- noncentral_chisquare(...)
- noncentral_chisquare(df, nonc, size=None)
Draw samples from a noncentral chi-square distribution.
The noncentral :math:`\chi^2` distribution is a generalisation of
the :math:`\chi^2` distribution.
Parameters
----------
df : int
Degrees of freedom.
nonc : float
Non-centrality.
size : tuple of ints
Shape of the output.
- noncentral_f(...)
- noncentral_f(dfnum, dfden, nonc, size=None)
Noncentral F distribution.
- normal(...)
- 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
Mean ("centre") of the distribution.
scale : float
Standard deviation (spread or "width") of the distribution.
size : tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
See Also
--------
scipy.stats.distributions.norm : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Gaussian distribution is
.. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },
where :math:`\mu` is the mean and :math:`\sigma` the standard deviation.
The square of the standard deviation, :math:`\sigma^2`, is called the
variance.
The function has its peak at the mean, and its "spread" increases with
the standard deviation (the function reaches 0.607 times its maximum at
:math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that
`numpy.random.normal` is more likely to return samples lying close to the
mean, rather than those far away.
References
----------
.. [1] Wikipedia, "Normal distribution",
http://en.wikipedia.org/wiki/Normal_distribution
.. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random
Variables and Random Signal Principles", 4th ed., 2001,
pp. 51, 51, 125.
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)
Verify the mean and the variance:
>>> abs(mu - np.mean(s)) < 0.01
True
>>> abs(sigma - np.std(s, ddof=1)) < 0.01
True
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=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()
- packbits(...)
- packbits(myarray, axis=None)
Packs the elements of a binary-valued array into bits in a uint8 array.
The result is padded to full bytes by inserting zero bits at the end.
Parameters
----------
myarray : array_like
An integer type array whose elements should be packed to bits.
axis : int, optional
The dimension over which bit-packing is done.
``None`` implies packing the flattened array.
Returns
-------
packed : ndarray
Array of type uint8 whose elements represent bits corresponding to the
logical (0 or nonzero) value of the input elements. The shape of
`packed` has the same number of dimensions as the input (unless `axis`
is None, in which case the output is 1-D).
See Also
--------
unpackbits: Unpacks elements of a uint8 array into a binary-valued output
array.
Examples
--------
>>> a = np.array([[[1,0,1],
... [0,1,0]],
... [[1,1,0],
... [0,0,1]]])
>>> b = np.packbits(a, axis=-1)
>>> b
array([[[160],[64]],[[192],[32]]], dtype=uint8)
Note that in binary 160 = 1010 0000, 64 = 0100 0000, 192 = 1100 0000,
and 32 = 0010 0000.
- pareto(...)
- pareto(a, size=None)
Draw samples from a Pareto distribution with specified shape.
This is a simplified version of the Generalized Pareto distribution
(available in SciPy), with the scale set to one and the location set to
zero. Most authors default the location to one.
The Pareto distribution must be greater than zero, and is unbounded above.
It is also known as the "80-20 rule". In this distribution, 80 percent of
the weights are in the lowest 20 percent of the range, while the other 20
percent fill the remaining 80 percent of the range.
Parameters
----------
shape : float, > 0.
Shape of the distribution.
size : tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
See Also
--------
scipy.stats.distributions.genpareto.pdf : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Pareto distribution is
.. math:: p(x) = \frac{am^a}{x^{a+1}}
where :math:`a` is the shape and :math:`m` the location
The Pareto distribution, named after the Italian economist Vilfredo Pareto,
is a power law probability distribution useful in many real world problems.
Outside the field of economics it is generally referred to as the Bradford
distribution. Pareto developed the distribution to describe the
distribution of wealth in an economy. It has also found use in insurance,
web page access statistics, oil field sizes, and many other problems,
including the download frequency for projects in Sourceforge [1]. It is
one of the so-called "fat-tailed" distributions.
References
----------
.. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
Sourceforge projects.
.. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
.. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
Values, Birkhauser Verlag, Basel, pp 23-30.
.. [4] Wikipedia, "Pareto distribution",
http://en.wikipedia.org/wiki/Pareto_distribution
Examples
--------
Draw samples from the distribution:
>>> a, m = 3., 1. # shape and mode
>>> s = np.random.pareto(a, 1000) + m
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, normed=True, align='center')
>>> fit = a*m**a/bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit),linewidth=2, color='r')
>>> plt.show()
- permutation(...)
- permutation(x)
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.
Returns
-------
out : ndarray
Permuted sequence or array range.
Examples
--------
>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6])
>>> np.random.permutation([1, 4, 9, 12, 15])
array([15, 1, 9, 4, 12])
- poisson(...)
- poisson(lam=1.0, size=None)
Poisson distribution.
- power(...)
- power(a, size=None)
Power distribution.
- putmask(...)
- putmask(a, mask, values)
Changes elements of an array based on conditional and input values.
Sets ``a.flat[n] = values[n]`` for each n where ``mask.flat[n]==True``.
If `values` is not the same size as `a` and `mask` then it will repeat.
This gives behavior different from ``a[mask] = values``.
Parameters
----------
a : array_like
Target array.
mask : array_like
Boolean mask array. It has to be the same shape as `a`.
values : array_like
Values to put into `a` where `mask` is True. If `values` is smaller
than `a` it will be repeated.
See Also
--------
place, put, take
Examples
--------
>>> x = np.arange(6).reshape(2, 3)
>>> np.putmask(x, x>2, x**2)
>>> x
array([[ 0, 1, 2],
[ 9, 16, 25]])
If `values` is smaller than `a` it is repeated:
>>> x = np.arange(5)
>>> np.putmask(x, x>1, [-33, -44])
>>> x
array([ 0, 1, -33, -44, -33])
- rand(...)
- rand(d0, d1, ..., dn)
Random values in a given shape.
Create an array of the given shape and propagate it with
random samples from a uniform distribution
over ``[0, 1)``.
Parameters
----------
d0, d1, ..., dn : int
Shape of the output.
Returns
-------
out : ndarray, shape ``(d0, d1, ..., dn)``
Random values.
See Also
--------
random
Notes
-----
This is a convenience function. If you want an interface that
takes a shape-tuple as the first argument, refer to
`random`.
Examples
--------
>>> np.random.rand(3,2)
array([[ 0.14022471, 0.96360618], #random
[ 0.37601032, 0.25528411], #random
[ 0.49313049, 0.94909878]]) #random
- randint(...)
- randint(low, high=None, size=None)
Return random integers x such that low <= x < high.
If high is None, then 0 <= x < low.
- randn(...)
- randn(d0, d1, ..., dn)
Returns zero-mean, unit-variance Gaussian random numbers in an
array of shape (d0, d1, ..., dn).
Note: This is a convenience function. If you want an
interface that takes a tuple as the first argument
use numpy.random.standard_normal(shape_tuple).
- random = random_sample(...)
- random_sample(size=None)
Return random floats in the half-open interval [0.0, 1.0).
Parameters
----------
size : shape tuple, optional
Defines the shape of the returned array of random floats.
Returns
-------
out : ndarray, floats
Array of random of floats with shape of `size`.
- random_integers(...)
- random_integers(low, high=None, size=None)
Return random integers x such that low <= x <= high.
If high is None, then 1 <= x <= low.
- random_sample(...)
- random_sample(size=None)
Return random floats in the half-open interval [0.0, 1.0).
Parameters
----------
size : shape tuple, optional
Defines the shape of the returned array of random floats.
Returns
-------
out : ndarray, floats
Array of random of floats with shape of `size`.
- ranf = random_sample(...)
- random_sample(size=None)
Return random floats in the half-open interval [0.0, 1.0).
Parameters
----------
size : shape tuple, optional
Defines the shape of the returned array of random floats.
Returns
-------
out : ndarray, floats
Array of random of floats with shape of `size`.
- rayleigh(...)
- rayleigh(scale=1.0, size=None)
Rayleigh distribution.
- restoredot(...)
- Restore `dot`, `vdot`, and `innerproduct` to the default non-BLAS
implementations.
Typically, the user will only need to call this when troubleshooting and
installation problem, reproducing the conditions of a build without an
accelerated BLAS, or when being very careful about benchmarking linear
algebra operations.
See Also
--------
alterdot : `restoredot` undoes the effects of `alterdot`.
- sample = random_sample(...)
- random_sample(size=None)
Return random floats in the half-open interval [0.0, 1.0).
Parameters
----------
size : shape tuple, optional
Defines the shape of the returned array of random floats.
Returns
-------
out : ndarray, floats
Array of random of floats with shape of `size`.
- seed(...)
- seed(seed=None)
Seed the generator.
seed can be an integer, an array (or other sequence) of integers of any
length, or None. If seed is None, then RandomState will try to read data
from /dev/urandom (or the Windows analogue) if available or seed from
the clock otherwise.
- set_numeric_ops(...)
- set_numeric_ops(op1=func1, op2=func2, ...)
Set numerical operators for array objects.
Parameters
----------
op1, op2, ... : callable
Each ``op = func`` pair describes an operator to be replaced.
For example, ``add = lambda x, y: np.add(x, y) % 5`` would replace
addition by modulus 5 addition.
Returns
-------
saved_ops : list of callables
A list of all operators, stored before making replacements.
Notes
-----
.. WARNING::
Use with care! Incorrect usage may lead to memory errors.
A function replacing an operator cannot make use of that operator.
For example, when replacing add, you may not use ``+``. Instead,
directly call ufuncs:
>>> def add_mod5(x, y):
... return np.add(x, y) % 5
...
>>> old_funcs = np.set_numeric_ops(add=add_mod5)
>>> x = np.arange(12).reshape((3, 4))
>>> x + x
array([[0, 2, 4, 1],
[3, 0, 2, 4],
[1, 3, 0, 2]])
>>> ignore = np.set_numeric_ops(**old_funcs) # restore operators
- set_state(...)
- set_state(state)
Set the state from a tuple.
Parameters
----------
state : tuple(string, list of 624 ints, int, int, float)
The `state` tuple is made up of
1. the string 'MT19937'
2. a list of 624 integer keys
3. an integer pos
4. an integer has_gauss
5. and a float for the cached_gaussian
Returns
-------
out : None
Returns 'None' on success.
See Also
--------
get_state
Notes
-----
For backwards compatibility, the following form is also accepted
although it is missing some information about the cached Gaussian value.
state = ('MT19937', int key[624], int pos)
- set_string_function(...)
- set_string_function(f, repr=1)
Set a Python function to be used when pretty printing arrays.
Parameters
----------
f : Python function
Function to be used to pretty print arrays. The function should expect
a single array argument and return a string of the representation of
the array.
repr : int
Unknown.
Examples
--------
>>> def pprint(arr):
... return 'HA! - What are you going to do now?'
...
>>> np.set_string_function(pprint)
>>> a = np.arange(10)
>>> a
HA! - What are you going to do now?
>>> print a
[0 1 2 3 4 5 6 7 8 9]
- seterrobj(...)
- seterrobj(errobj)
Used internally by `seterr`.
Parameters
----------
errobj : list
[buffer_size, error_mask, callback_func]
See Also
--------
seterrcall
- shuffle(...)
- shuffle(x)
Modify a sequence in-place by shuffling its contents.
- standard_cauchy(...)
- standard_cauchy(size=None)
Standard Cauchy with mode=0.
- standard_exponential(...)
- standard_exponential(size=None)
Standard exponential distribution (scale=1).
- standard_gamma(...)
- standard_gamma(shape, size=None)
Standard Gamma distribution.
- standard_normal(...)
- standard_normal(size=None)
Returns samples from a Standard Normal distribution (mean=0, stdev=1).
Parameters
----------
size : int, shape tuple, optional
Returns the number of samples required to satisfy the `size` parameter.
If not given or 'None' indicates to return one sample.
Returns
-------
out : float, ndarray
Samples the Standard Normal distribution with a shape satisfying the
`size` parameter.
- standard_t(...)
- standard_t(df, size=None)
Standard Student's t distribution with df degrees of freedom.
- strftime(...)
- strftime(format[, tuple]) -> string
Convert a time tuple to a string according to a format specification.
See the library reference manual for formatting codes. When the time tuple
is not present, current time as returned by localtime() is used.
- triangular(...)
- triangular(left, mode, right, size=None)
Triangular distribution starting at left, peaking at mode, and
ending at right (left <= mode <= right).
- uniform(...)
- uniform(low=0.0, high=1.0, size=1)
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, 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
Upper boundary of the output interval. All values generated will be
less than high. The default value is 1.0.
size : tuple of ints, int, optional
Shape of output. If the given size is, for example, (m,n,k),
m*n*k samples are generated. If no shape is specified, a single sample
is returned.
Returns
-------
out : ndarray
Drawn samples, with shape `size`.
See Also
--------
randint : Discrete uniform distribution, yielding integers.
random_integers : Discrete uniform distribution over the closed interval
``[low, high]``.
random_sample : Floats uniformly distributed over ``[0, 1)``.
random : Alias for `random_sample`.
rand : Convenience function that accepts dimensions as input, e.g.,
``rand(2,2)`` would generate a 2-by-2 array of floats, uniformly
distributed over ``[0, 1)``.
Notes
-----
The probability density function of the uniform distribution is
.. math:: p(x) = \frac{1}{b - a}
anywhere within the interval ``[a, b)``, and zero elsewhere.
Examples
--------
Draw samples from the distribution:
>>> s = np.random.uniform(-1,0,1000)
All values are within the given interval:
>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True
Display the histogram of the samples, along with the
probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, normed=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()
- unpackbits(...)
- unpackbits(myarray, axis=None)
Unpacks elements of a uint8 array into a binary-valued output array.
Each element of `myarray` represents a bit-field that should be unpacked
into a binary-valued output array. The shape of the output array is either
1-D (if `axis` is None) or the same shape as the input array with unpacking
done along the axis specified.
Parameters
----------
myarray : ndarray, uint8 type
Input array.
axis : int, optional
Unpacks along this axis.
Returns
-------
unpacked : ndarray, uint8 type
The elements are binary-valued (0 or 1).
See Also
--------
packbits : Packs the elements of a binary-valued array into bits in a uint8
array.
Examples
--------
>>> a = np.array([[2], [7], [23]], dtype=np.uint8)
>>> a
array([[ 2],
[ 7],
[23]], dtype=uint8)
>>> b = np.unpackbits(a, axis=1)
>>> b
array([[0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1, 1, 1],
[0, 0, 0, 1, 0, 1, 1, 1]], dtype=uint8)
- vdot(...)
- Return the dot product of two vectors.
The vdot(`a`, `b`) function handles complex numbers differently than
dot(`a`, `b`). If the first argument is complex the complex conjugate
of the first argument is used for the calculation of the dot product.
For 2-D arrays it is equivalent to matrix multiplication, and for 1-D
arrays to inner product of vectors (with complex conjugation of `a`).
For N dimensions it is a sum product over the last axis of `a` and
the second-to-last of `b`::
dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])
Parameters
----------
a : array_like
If `a` is complex the complex conjugate is taken before calculation
of the dot product.
b : array_like
Second argument to the dot product.
Returns
-------
output : ndarray
Returns dot product of `a` and `b`. Can be an int, float, or
complex depending on the types of `a` and `b`.
See Also
--------
dot : Return the dot product without using the complex conjugate of the
first argument.
Notes
-----
The dot product is the summation of element wise multiplication.
.. math::
a \cdot b = \sum_{i=1}^n a_i^*b_i = a_1^*b_1+a_2^*b_2+\cdots+a_n^*b_n
Examples
--------
>>> a = np.array([1+2j,3+4j])
>>> b = np.array([5+6j,7+8j])
>>> np.vdot(a, b)
(70-8j)
>>> np.vdot(b, a)
(70+8j)
>>> a = np.array([[1, 4], [5, 6]])
>>> b = np.array([[4, 1], [2, 2]])
>>> np.vdot(a, b)
30
>>> np.vdot(b, a)
30
- vonmises(...)
- vonmises(mu=0.0, kappa=1.0, size=None)
Draw samples from a von Mises distribution.
Samples are drawn from a von Mises distribution with specified mode (mu)
and dispersion (kappa), on the interval [-pi, pi].
The von Mises distribution (also known as the circular normal
distribution) is a continuous probability distribution on the circle. It
may be thought of as the circular analogue of the normal distribution.
Parameters
----------
mu : float
Mode ("center") of the distribution.
kappa : float, >= 0.
Dispersion of the distribution.
size : {tuple, int}
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
samples : {ndarray, scalar}
The returned samples live on the unit circle [-\pi, \pi].
See Also
--------
scipy.stats.distributions.vonmises : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the von Mises distribution is
.. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},
where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
and :math:`I_0(\kappa)` is the modified Bessel function of order 0.
The von Mises, named for Richard Edler von Mises, 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. (ed.), Handbook of Mathematical
Functions, National Bureau of Standards, 1964; reprinted Dover
Publications, 1965.
.. [2] von Mises, Richard, 1964, Mathematical Theory of Probability
and Statistics (New York: Academic Press).
.. [3] Wikipedia, "Von Mises distribution",
http://en.wikipedia.org/wiki/Von_Mises_distribution
Examples
--------
Draw samples from the distribution:
>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 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, ignored = plt.hist(s, 50, normed=True)
>>> x = arange(-pi, pi, 2*pi/50.)
>>> y = -np.exp(kappa*np.cos(x-mu))/(2*pi*sps.jn(0,kappa))
>>> plt.plot(x, y/max(y), linewidth=2, color='r')
>>> plt.show()
- wald(...)
- wald(mean, scale, size=None)
Wald (inverse Gaussian) distribution.
- weibull(...)
- weibull(a, size=None)
Weibull distribution.
Draw samples from a 1-parameter Weibull distribution with the given
shape parameter.
.. math:: X = (-ln(U))^{1/a}
Here, U is drawn from the uniform distribution over (0,1].
The more common 2-parameter Weibull, including a scale parameter
:math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.
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.
Parameters
----------
a : float
Shape of the distribution.
size : tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
See Also
--------
scipy.stats.distributions.weibull : probability density function,
distribution or cumulative density function, etc.
gumbel, scipy.stats.distributions.genextreme
Notes
-----
The probability density for the Weibull distribution is
.. math:: p(x) = \frac{a}
{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
where :math:`a` is the shape and :math:`\lambda` the scale.
The function has its peak (the mode) at
:math:`\lambda(\frac{a-1}{a})^{1/a}`.
When ``a = 1``, the Weibull distribution reduces to the exponential
distribution.
References
----------
.. [1] Waloddi Weibull, Professor, 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, 1951 "A Statistical Distribution Function of Wide
Applicability", Journal Of Applied Mechanics ASME Paper.
.. [3] Wikipedia, "Weibull distribution",
http://en.wikipedia.org/wiki/Weibull_distribution
Examples
--------
Draw samples from the distribution:
>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> def weib(x,n,a):
... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()
- where(...)
- where(condition, [x, y])
Return elements, either from `x` or `y`, depending on `condition`.
If only `condition` is given, return ``condition.nonzero()``.
Parameters
----------
condition : array_like, bool
When True, yield `x`, otherwise yield `y`.
x, y : array_like, optional
Values from which to choose.
Returns
-------
out : ndarray or tuple of ndarrays
If both `x` and `y` are specified, the output array, shaped like
`condition`, contains elements of `x` where `condition` is True,
and elements from `y` elsewhere.
If only `condition` is given, return the tuple
``condition.nonzero()``, the indices where `condition` is True.
See Also
--------
nonzero, choose
Notes
-----
If `x` and `y` are given and input arrays are 1-D, `where` is
equivalent to::
[xv if c else yv for (c,xv,yv) in zip(condition,x,y)]
Examples
--------
>>> x = np.arange(9.).reshape(3, 3)
>>> np.where( x > 5 )
(array([2, 2, 2]), array([0, 1, 2]))
>>> x[np.where( x > 3.0 )] # Note: result is 1D.
array([ 4., 5., 6., 7., 8.])
>>> np.where(x < 5, x, -1) # Note: broadcasting.
array([[ 0., 1., 2.],
[ 3., 4., -1.],
[-1., -1., -1.]])
>>> np.where([[True, False], [True, True]],
... [[1, 2], [3, 4]],
... [[9, 8], [7, 6]])
array([[1, 8],
[3, 4]])
>>> np.where([[0, 1], [1, 0]])
(array([0, 1]), array([1, 0]))
- zeros(...)
- zeros(shape, dtype=float, order='C')
Return a new array of given shape and type, filled with zeros.
Parameters
----------
shape : {tuple of ints, int}
Shape of the new array, e.g., ``(2, 3)`` or ``2``.
dtype : data-type, optional
The desired data-type for the array, e.g., `numpy.int8`. Default is
`numpy.float64`.
order : {'C', 'F'}, optional
Whether to store multidimensional data in C- or Fortran-contiguous
(row- or column-wise) order in memory.
Returns
-------
out : ndarray
Array of zeros with the given shape, dtype, and order.
See Also
--------
numpy.zeros_like : Return an array of zeros with shape and type of input.
numpy.ones_like : Return an array of ones with shape and type of input.
numpy.empty_like : Return an empty array with shape and type of input.
numpy.ones : Return a new array setting values to one.
numpy.empty : Return a new uninitialized array.
Examples
--------
>>> np.zeros(5)
array([ 0., 0., 0., 0., 0.])
>>> np.zeros((5,), dtype=numpy.int)
array([0, 0, 0, 0, 0])
>>> np.zeros((2, 1))
array([[ 0.],
[ 0.]])
>>> s = (2,2)
>>> np.zeros(s)
array([[ 0., 0.],
[ 0., 0.]])
>>> np.zeros((2,), dtype=[('x', 'i4'), ('y', 'i4')])
array([(0, 0), (0, 0)],
dtype=[('x', '<i4'), ('y', '<i4')])
- zipf(...)
- zipf(a, size=None)
Draw samples from a Zipf distribution.
Samples are drawn from a Zipf distribution with specified parameter (a),
where a > 1.
The zipf distribution (also known as the zeta
distribution) is a continuous probability distribution that satisfies
Zipf's law, where the frequency of an item is inversely proportional to
its rank in a frequency table.
Parameters
----------
a : float
parameter, > 1.
size : {tuple, int}
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn.
Returns
-------
samples : {ndarray, scalar}
The returned samples are greater than or equal to one.
See Also
--------
scipy.stats.distributions.zipf : probability density function,
distribution or cumulative density function, etc.
Notes
-----
The probability density for the Zipf distribution is
.. math:: p(x) = \frac{x^{-a}}{\zeta(a)},
where :math:`\zeta` is the Riemann Zeta function.
Named after 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] Weisstein, Eric W. "Zipf Distribution." From MathWorld--A Wolfram
Web Resource. http://mathworld.wolfram.com/ZipfDistribution.html
.. [2] Wikipedia, "Zeta distribution",
http://en.wikipedia.org/wiki/Zeta_distribution
.. [3] Wikipedia, "Zipf's Law",
http://en.wikipedia.org/wiki/Zipf%27s_law
.. [4] Zipf, George Kingsley (1932): Selected Studies of the Principle
of Relative Frequency in Language. Cambridge (Mass.).
Examples
--------
Draw samples from the distribution:
>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps
Truncate s values at 50 so plot is interesting
>>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
>>> x = arange(1., 50.)
>>> y = x**(-a)/sps.zetac(a)
>>> plt.plot(x, y/max(y), linewidth=2, color='r')
>>> plt.show()
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