nbodykit.source.catalog.uniform.
MPIRandomState
(comm, seed, N)[source]¶Bases: mtrand.RandomState
A wrapper around numpy.random.RandomState
that can return
random numbers in parallel, independent of the number of ranks.
Parameters: |
---|
Methods
beta (a, b[, size]) |
Draw samples from a Beta distribution. |
binomial (n, p[, size]) |
Draw samples from a binomial distribution. |
bytes (length) |
Return random bytes. |
chisquare (df[, size]) |
Draw samples from a chi-square distribution. |
choice (a[, size, replace, p]) |
Generates a random sample from a given 1-D array |
dirichlet (alpha[, size]) |
Draw samples from the Dirichlet distribution. |
exponential ([scale, size]) |
Draw samples from an exponential distribution. |
f (dfnum, dfden[, size]) |
Draw samples from an F distribution. |
gamma (shape[, scale, size]) |
Draw samples from a Gamma distribution. |
geometric (p[, size]) |
Draw samples from the geometric distribution. |
get_state () |
Return a tuple representing the internal state of the generator. |
gumbel ([loc, scale, size]) |
Draw samples from a Gumbel distribution. |
hypergeometric (ngood, nbad, nsample[, size]) |
Draw samples from a Hypergeometric distribution. |
laplace ([loc, scale, size]) |
Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). |
logistic ([loc, scale, size]) |
Draw samples from a logistic distribution. |
lognormal ([mean, sigma, size]) |
Draw samples from a log-normal distribution. |
logseries (p[, size]) |
Draw samples from a logarithmic series distribution. |
multinomial (n, pvals[, size]) |
Draw samples from a multinomial distribution. |
multivariate_normal (mean, cov[, size, …) |
Draw random samples from a multivariate normal distribution. |
negative_binomial (n, p[, size]) |
Draw samples from a negative binomial distribution. |
noncentral_chisquare (df, nonc[, size]) |
Draw samples from a noncentral chi-square distribution. |
noncentral_f (dfnum, dfden, nonc[, size]) |
Draw samples from the noncentral F distribution. |
normal ([loc, scale, size]) |
Draw random samples from a normal (Gaussian) distribution. |
pareto (a[, size]) |
Draw samples from a Pareto II or Lomax distribution with specified shape. |
permutation (x) |
Randomly permute a sequence, or return a permuted range. |
poisson ([lam, size]) |
Draw samples from a Poisson distribution. |
power (a[, size]) |
Draws samples in [0, 1] from a power distribution with positive exponent a - 1. |
rand (d0, d1, …, dn) |
Random values in a given shape. |
randint (low[, high, size, dtype]) |
Return random integers from low (inclusive) to high (exclusive). |
randn (d0, d1, …, dn) |
Return a sample (or samples) from the “standard normal” distribution. |
random_integers (low[, high, size]) |
Random integers of type np.int between low and high, inclusive. |
random_sample ([size]) |
Return random floats in the half-open interval [0.0, 1.0). |
rayleigh ([scale, size]) |
Draw samples from a Rayleigh distribution. |
seed ([seed]) |
Seed the generator. |
seeded_context (seed) |
A context manager to set and then restore the random seed |
set_state (state) |
Set the internal state of the generator from a tuple. |
shuffle (x) |
Modify a sequence in-place by shuffling its contents. |
standard_cauchy ([size]) |
Draw samples from a standard Cauchy distribution with mode = 0. |
standard_exponential ([size]) |
Draw samples from the standard exponential distribution. |
standard_gamma (shape[, size]) |
Draw samples from a standard Gamma distribution. |
standard_normal ([size]) |
Draw samples from a standard Normal distribution (mean=0, stdev=1). |
standard_t (df[, size]) |
Draw samples from a standard Student’s t distribution with df degrees of freedom. |
tomaxint ([size]) |
Random integers between 0 and sys.maxint , inclusive. |
triangular (left, mode, right[, size]) |
Draw samples from the triangular distribution over the interval [left, right] . |
uniform ([low, high, size]) |
Draw samples from a uniform distribution. |
vonmises (mu, kappa[, size]) |
Draw samples from a von Mises distribution. |
wald (mean, scale[, size]) |
Draw samples from a Wald, or inverse Gaussian, distribution. |
weibull (a[, size]) |
Draw samples from a Weibull distribution. |
zipf (a[, size]) |
Draw samples from a Zipf distribution. |
__getattribute__
(name)[source]¶Decorate callable functions of RandomState such that they return chunks of the total N random numbers generated
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
where the normalisation, B, is the beta function,
It is often seen in Bayesian inference and order statistics.
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized beta distribution. |
Return type: | ndarray or scalar |
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: |
|
---|---|
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()
Notes
The probability density for the binomial distribution is
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
[R676289] | Dalgaard, Peter, “Introductory Statistics with R”, Springer-Verlag, 2002. |
[R776289] | Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002. |
[R876289] | Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972. |
[R976289] | Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html |
[R1076289] | 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
(length)¶Return random bytes.
Parameters: | length (int) – Number of random bytes. |
---|---|
Returns: | out – String of length length. |
Return type: | str |
Examples
>>> np.random.bytes(10)
' eh\x85\x022SZ\xbf\xa4' #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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized chi-square distribution. |
Return type: | ndarray or scalar |
Raises: |
|
Notes
The variable obtained by summing the squares of df independent, standard normally distributed random variables:
is chi-square distributed, denoted
The probability density function of the chi-squared distribution is
where \(\Gamma\) is the gamma function,
References
[R1176294] | NIST “Engineering Statistics Handbook” http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm |
Examples
>>> np.random.chisquare(2,4)
array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272])
choice
(a, size=None, replace=True, p=None)¶Generates a random sample from a given 1-D array
New in version 1.7.0.
Parameters: |
|
---|---|
Returns: | samples – The generated random samples |
Return type: | single item or ndarray |
Raises: |
|
See also
Examples
Generate a uniform random sample from np.arange(5) of size 3:
>>> np.random.choice(5, 3)
array([0, 3, 4])
>>> #This is equivalent to np.random.randint(0,5,3)
Generate a non-uniform random sample from np.arange(5) of size 3:
>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0])
Generate a uniform random sample from np.arange(5) of size 3 without replacement:
>>> np.random.choice(5, 3, replace=False)
array([3,1,0])
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]
Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:
>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0])
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']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'],
dtype='|S11')
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. Dirichlet pdf is the conjugate prior of a multinomial in Bayesian inference.
Parameters: | |
---|---|
Returns: | samples – The drawn samples, of shape (size, alpha.ndim). |
Return type: | ndarray, |
Notes
Uses the following property for computation: for each dimension, draw a random sample y_i from a standard gamma generator of shape alpha_i, then \(X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)\) is Dirichlet distributed.
References
[R1276295] | David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.phy.cam.ac.uk/mackay/ |
[R1376295] | Wikipedia, “Dirichlet distribution”, http://en.wikipedia.org/wiki/Dirichlet_distribution |
Examples
Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.
>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> 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
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 [R1676297].
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 [R1476297], or the time between page requests to Wikipedia [R1576297].
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized exponential distribution. |
Return type: | ndarray or scalar |
References
[R1476297] | Peyton Z. Peebles Jr., “Probability, Random Variables and Random Signal Principles”, 4th ed, 2001, p. 57. |
[R1576297] | Wikipedia, “Poisson process”, http://en.wikipedia.org/wiki/Poisson_process |
[R1676297] | Wikipedia, “Exponential distribution”, http://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 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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Fisher distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.f()
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
[R1776300] | Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill, Fifth Edition, 2002. |
[R1876300] | 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.
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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized gamma distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.gamma()
Notes
The probability density for the Gamma distribution is
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
[R1976302] | Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html |
[R2076302] | Wikipedia, “Gamma distribution”, http://en.wikipedia.org/wiki/Gamma_distribution |
Examples
Draw samples from the distribution:
>>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2)
>>> s = np.random.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)*(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
where p is the probability of success of an individual trial.
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized geometric distribution. |
Return type: | ndarray or scalar |
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
()¶Return a tuple representing the internal state of the generator.
For more details, see set_state.
Returns: | out – The returned tuple has the following items:
|
---|---|
Return type: | tuple(str, ndarray of 624 uints, int, int, float) |
See also
Notes
set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.
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: |
|
---|---|
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
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
[R2176304] | Gumbel, E. J., “Statistics of Extremes,” New York: Columbia University Press, 1958. |
[R2276304] | 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:
>>> 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.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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized hypergeometric distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.hypergeom()
Notes
The probability density for the Hypergeometric distribution is
where \(0 \le x \le m\) and \(n+m-N \le x \le n\)
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
[R2376306] | Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972. |
[R2476306] | Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html |
[R2576306] | 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!
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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Laplace distribution. |
Return type: | ndarray or scalar |
Notes
It has the probability density function
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
[R2676309] | Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972. |
[R2776309] | Kotz, Samuel, et. al. “The Laplace Distribution and Generalizations, ” Birkhauser, 2001. |
[R2876309] | Weisstein, Eric W. “Laplace Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html |
[R2976309] | Wikipedia, “Laplace distribution”, http://en.wikipedia.org/wiki/Laplace_distribution |
Examples
Draw samples from the distribution
>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)
Display the histogram of the samples, along with the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized logistic distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.logistic()
Notes
The probability density for the Logistic distribution is
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
[R3076313] | 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. |
[R3176313] | Weisstein, Eric W. “Logistic Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html |
[R3276313] | 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
(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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized log-normal distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.lognorm()
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:
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
[R3376316] | Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf |
[R3476316] | Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32. |
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='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.
>>> 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='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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized logarithmic series distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.logser()
Notes
The probability density for the Log Series distribution is
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
[R3576318] | 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). |
[R3676318] | 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. |
[R3776318] | D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. |
[R3876318] | 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
(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: |
|
---|---|
Returns: | out – The drawn samples, of shape size, if that was provided. If not,
the shape is In other words, each entry |
Return type: | ndarray |
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 die is more likely to land on number 6:
>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26])
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:
>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT
array([38, 62])
not like:
>>> np.random.multinomial(100, [1.0, 2.0]) # WRONG
array([100, 0])
multivariate_normal
(mean, cov[, size, check_valid, tol])¶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 (standard deviation, or “width,” squared) of the one-dimensional normal distribution.
Parameters: |
|
---|---|
Returns: | out – The drawn samples, of shape size, if that was provided. If not,
the shape is In other words, each entry |
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 points are oriented along 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 positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.
References
[R3976322] | Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991. |
[R4076322] | 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]]
>>> 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:
>>> list((x[0,0,:] - mean) < 0.6)
[True, True]
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 trials and p probability of success where n is an integer > 0 and p is in the interval [0, 1].
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of trials it took to achieve n - 1 successes, N - (n - 1) failures, and a success on the, (N + n)th trial. |
Return type: | ndarray or scalar |
Notes
The probability density for the negative binomial distribution is
where \(n-1\) is the number of successes, \(p\) is the probability of success, and \(N+n-1\) is the number of trials. The negative binomial distribution gives the probability of n-1 successes and N failures in N+n-1 trials, and success on the (N+n)th 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.
References
[R4176324] | Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html |
[R4276324] | Wikipedia, “Negative binomial distribution”, http://en.wikipedia.org/wiki/Negative_binomial_distribution |
Examples
Draw samples from the distribution:
A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?
>>> s = np.random.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 generalisation of the \(\chi^2\) distribution.
Parameters: |
|
---|---|
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
where \(Y_{q}\) is the Chi-square with q degrees of freedom.
In Delhi (2007), it is noted that the noncentral chi-square is useful in bombing and coverage problems, the probability of killing the point target given by the noncentral chi-squared distribution.
References
[R4376326] | Delhi, M.S. Holla, “On a noncentral chi-square distribution in the analysis of weapon systems effectiveness”, Metrika, Volume 15, Number 1 / December, 1970. |
[R4476326] | Wikipedia, “Noncentral chi-square distribution” http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution |
Examples
Draw values from the distribution and plot the histogram
>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
... bins=200, normed=True)
>>> plt.show()
Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.
>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
... bins=np.arange(0., 25, .1), normed=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
... bins=np.arange(0., 25, .1), normed=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(np.random.noncentral_chisquare(3, 20, 100000),
... bins=200, normed=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: |
|
---|---|
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
[R4576328] | Weisstein, Eric W. “Noncentral F-Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html |
[R4676328] | Wikipedia, “Noncentral F-distribution”, http://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.
>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, normed=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, normed=True)
>>> 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 [R4876330], 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 [R4876330].
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized normal distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.norm()
Notes
The probability density for the Gaussian distribution is
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 \(x - \sigma\) [R4876330]). This implies that numpy.random.normal is more likely to return samples lying close to the mean, rather than those far away.
References
[R4776330] | Wikipedia, “Normal distribution”, http://en.wikipedia.org/wiki/Normal_distribution |
[R4876330] | (1, 2, 3) 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()
pareto
(a, size=None)¶Draw samples from a Pareto II or Lomax distribution with specified shape.
The Lomax or Pareto II distribution is a shifted Pareto
distribution. The classical Pareto distribution can be
obtained from the Lomax distribution by adding 1 and
multiplying by the scale parameter m
(see Notes). The
smallest value of the Lomax distribution is zero while for the
classical Pareto distribution it is mu
, where the standard
Pareto distribution has location mu = 1
. Lomax can also
be considered as a simplified version of the Generalized
Pareto distribution (available in SciPy), with the scale set
to one and the location set to zero.
The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the “80-20 rule”. In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range.
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Pareto distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.lomax()
scipy.stats.genpareto()
Notes
The probability density for the Pareto distribution is
where \(a\) is the shape and \(m\) the scale.
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [R4976332]. It is one of the so-called “fat-tailed” distributions.
References
[R4976332] | Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. |
[R5076332] | Pareto, V. (1896). Course of Political Economy. Lausanne. |
[R5176332] | Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. |
[R5276332] | Wikipedia, “Pareto distribution”, http://en.wikipedia.org/wiki/Pareto_distribution |
Examples
Draw samples from the distribution:
>>> a, m = 3., 2. # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m
Display the histogram of the samples, along with the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, normed=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()
permutation
(x)¶Randomly permute a sequence, or return a permuted range.
If x is a multi-dimensional array, it is only shuffled along its first index.
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 – Permuted sequence or array range. |
Return type: | ndarray |
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])
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8],
[0, 1, 2],
[3, 4, 5]])
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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Poisson distribution. |
Return type: | ndarray or scalar |
Notes
The Poisson distribution
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 long type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.
References
[R5376336] | Weisstein, Eric W. “Poisson Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html |
[R5476336] | Wikipedia, “Poisson distribution”, http://en.wikipedia.org/wiki/Poisson_distribution |
Examples
Draw samples from the distribution:
>>> import numpy as np
>>> s = np.random.poisson(5, 10000)
Display histogram of the sample:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, normed=True)
>>> plt.show()
Draw each 100 values for lambda 100 and 500:
>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
poisson_lam_max
= 9.2233720064847708e+18¶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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized power distribution. |
Return type: | ndarray or scalar |
Raises: |
|
Notes
The probability density function is
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
[R5576338] | Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003. |
[R5676338] | 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. http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf |
Examples
Draw samples from the distribution:
>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)
Display the histogram of the samples, along with the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()
Compare the power function distribution to the inverse of the Pareto.
>>> from scipy import stats
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)
>>> plt.figure()
>>> plt.hist(rvs, bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('np.random.power(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of 1 + np.random.pareto(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of stats.pareto(5)')
rand
(d0, d1, ..., dn)¶Random values in a given shape.
Create an array of the given shape and populate it with
random samples from a uniform distribution
over [0, 1)
.
Parameters: | d1, .., dn (d0,) – The dimensions of the returned array, should all be positive. If no argument is given a single Python float is returned. |
---|---|
Returns: | out – Random values. |
Return type: | ndarray, shape (d0, d1, ..., dn) |
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 np.random.random_sample .
Examples
>>> np.random.rand(3,2)
array([[ 0.14022471, 0.96360618], #random
[ 0.37601032, 0.25528411], #random
[ 0.49313049, 0.94909878]]) #random
randint
(low, high=None, size=None, dtype='l')¶Return random integers from low (inclusive) to high (exclusive).
Return random integers from the “discrete uniform” distribution of the specified dtype in the “half-open” interval [low, high). If high is None (the default), then results are from [0, low).
Parameters: |
|
---|---|
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 |
See also
random.random_integers()
Examples
>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
>>> np.random.randint(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:
>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1],
[3, 2, 2, 0]])
randn
(d0, d1, ..., dn)¶Return a sample (or samples) from the “standard normal” distribution.
If positive, int_like or int-convertible arguments are provided,
randn generates an array of shape (d0, d1, ..., dn)
, filled
with random floats sampled from a univariate “normal” (Gaussian)
distribution of mean 0 and variance 1 (if any of the \(d_i\) are
floats, they are first converted to integers by truncation). A single
float randomly sampled from the distribution is returned if no
argument is provided.
This is a convenience function. If you want an interface that takes a tuple as the first argument, use numpy.random.standard_normal instead.
Parameters: | d1, .., dn (d0,) – The dimensions of the returned array, should be all positive. If no argument is given a single Python float is returned. |
---|---|
Returns: | Z – A (d0, d1, ..., dn) -shaped array of floating-point samples from
the standard normal distribution, or a single such float if
no parameters were supplied. |
Return type: | ndarray or float |
See also
random.standard_normal()
Notes
For random samples from \(N(\mu, \sigma^2)\), use:
sigma * np.random.randn(...) + mu
Examples
>>> np.random.randn()
2.1923875335537315 #random
Two-by-four array of samples from N(3, 6.25):
>>> 2.5 * np.random.randn(2, 4) + 3
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], #random
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) #random
random_integers
(low, high=None, size=None)¶Random integers of type np.int between low and high, inclusive.
Return random integers of type np.int from the “discrete uniform” distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int type translates to the C long type used by Python 2 for “short” integers and its precision is platform dependent.
This function has been deprecated. Use randint instead.
Deprecated since version 1.11.0.
Parameters: |
|
---|---|
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 |
See also
random.randint()
Notes
To sample from N evenly spaced floating-point numbers between a and b, use:
a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
Examples
>>> np.random.random_integers(5)
4
>>> type(np.random.random_integers(5))
<type 'int'>
>>> np.random.random_integers(5, size=(3,2))
array([[5, 4],
[3, 3],
[4, 5]])
Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set \({0, 5/8, 10/8, 15/8, 20/8}\)):
>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ])
Roll two six sided dice 1000 times and sum the results:
>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2
Display results as a histogram:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, normed=True)
>>> plt.show()
random_sample
(size=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 \(Unif[a, b), b > a\) multiply the output of random_sample by (b-a) and add a:
(b - a) * random_sample() + 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. |
---|---|
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 |
Examples
>>> np.random.random_sample()
0.47108547995356098
>>> type(np.random.random_sample())
<type 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428])
Three-by-two array of random numbers from [-5, 0):
>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984],
[-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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Rayleigh distribution. |
Return type: | ndarray or scalar |
Notes
The probability density function for the Rayleigh distribution is
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
[R5776340] | Brighton Webs Ltd., “Rayleigh Distribution,” http://www.brighton-webs.co.uk/distributions/rayleigh.asp |
[R5876340] | Wikipedia, “Rayleigh distribution” http://en.wikipedia.org/wiki/Rayleigh_distribution |
Examples
Draw values from the distribution and plot the histogram
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, normed=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 = np.random.rayleigh(modevalue, 1000000)
The percentage of waves larger than 3 meters is:
>>> 100.*sum(s>3)/1000000.
0.087300000000000003
seed
(seed=None)¶Seed the generator.
This method is called when RandomState is initialized. It can be called again to re-seed the generator. For details, see RandomState.
Parameters: | seed (int or array_like, optional) – Seed for RandomState. Must be convertible to 32 bit unsigned integers. |
---|
See also
RandomState()
set_state
(state)¶Set the internal state of the generator from a tuple.
For use if one has reason to manually (re-)set the internal state of the “Mersenne Twister”[R5976342] pseudo-random number generating algorithm.
Parameters: | state (tuple(str, ndarray of 624 uints, int, int, float)) – The state tuple has the following items:
|
---|---|
Returns: | out – Returns ‘None’ on success. |
Return type: | None |
See also
Notes
set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.
For backwards compatibility, the form (str, array of 624 uints, int) is
also accepted although it is missing some information about the cached
Gaussian value: state = ('MT19937', keys, pos)
.
References
[R5976342] | M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998. |
shuffle
(x)¶Modify a sequence in-place by shuffling its contents.
This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.
Parameters: | x (array_like) – The array or list to be shuffled. |
---|---|
Returns: | |
Return type: | None |
Examples
>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8]
Multi-dimensional arrays are only shuffled along the first axis:
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5],
[6, 7, 8],
[0, 1, 2]])
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
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
[R6076343] | NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm |
[R6176343] | Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html |
[R6276343] | Wikipedia, “Cauchy distribution” http://en.wikipedia.org/wiki/Cauchy_distribution |
Examples
Draw samples and plot the distribution:
>>> s = np.random.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)¶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. |
---|---|
Returns: | out – Drawn samples. |
Return type: | float or ndarray |
Examples
Output a 3x8000 array:
>>> n = np.random.standard_exponential((3, 8000))
standard_gamma
(shape, size=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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized standard gamma distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.gamma()
Notes
The probability density for the Gamma distribution is
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
[R6376346] | Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html |
[R6476346] | Wikipedia, “Gamma distribution”, http://en.wikipedia.org/wiki/Gamma_distribution |
Examples
Draw samples from the distribution:
>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.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, ignored = plt.hist(s, 50, normed=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)¶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. |
---|---|
Returns: | out – Drawn samples. |
Return type: | float or ndarray |
Examples
>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, #random
-0.38672696, -0.4685006 ]) #random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)
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: |
|
---|---|
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
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
[R6576348] | Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002. |
[R6676348] | Wikipedia, “Student’s t-distribution” http://en.wikipedia.org/wiki/Student’s_t-distribution |
Examples
From Dalgaard page 83 [R6576348], suppose the daily energy intake for 11 women in 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?
We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value?
>>> s = np.random.standard_t(10, size=100000)
>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727
Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1.
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(s, bins=100, normed=True)
For a one-sided t-test, how far out in the distribution does the t statistic appear?
>>> np.sum(s<t) / float(len(s))
0.0090699999999999999 #random
So the p-value is about 0.009, which says the null hypothesis has a probability of about 99% of being true.
tomaxint
(size=None)¶Random integers between 0 and sys.maxint
, inclusive.
Return a sample of uniformly distributed random integers in the interval
[0, sys.maxint
].
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: | out – Drawn samples, with shape size. |
Return type: | ndarray |
See also
randint()
random_integers()
Examples
>>> RS = np.random.mtrand.RandomState() # need a RandomState object
>>> RS.tomaxint((2,2,2))
array([[[1170048599, 1600360186],
[ 739731006, 1947757578]],
[[1871712945, 752307660],
[1601631370, 1479324245]]])
>>> import sys
>>> sys.maxint
2147483647
>>> RS.tomaxint((2,2,2)) < sys.maxint
array([[[ True, True],
[ True, True]],
[[ True, True],
[ True, True]]], dtype=bool)
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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized triangular distribution. |
Return type: | ndarray or scalar |
Notes
The probability density function for the triangular distribution is
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
[R6776350] | Wikipedia, “Triangular distribution” http://en.wikipedia.org/wiki/Triangular_distribution |
Examples
Draw values from the distribution and plot the histogram:
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
... normed=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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized uniform distribution. |
Return type: | ndarray or scalar |
See also
randint()
random_integers()
[low, high]
.random_sample()
[0, 1)
.random()
rand()
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
anywhere within the interval [a, b)
, and zero elsewhere.
When high
== low
, values of low
will be returned.
If high
< low
, the results are officially undefined
and may eventually raise an error, i.e. do not rely on this
function to behave when passed arguments satisfying that
inequality condition.
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()
vonmises
(mu, kappa, size=None)¶Draw samples from a von Mises distribution.
Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].
The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized von Mises distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.vonmises()
Notes
The probability density for the von Mises distribution is
where \(\mu\) is the mode and \(\kappa\) the dispersion, 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
[R6876351] | Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972. |
[R6976351] | von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964. |
Examples
Draw samples from the distribution:
>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.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, normed=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: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Wald distribution. |
Return type: | ndarray or scalar |
Notes
The probability density function for the Wald distribution is
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
[R7076353] | Brighton Webs Ltd., Wald Distribution, http://www.brighton-webs.co.uk/distributions/wald.asp |
[R7176353] | Chhikara, Raj S., and Folks, J. Leroy, “The Inverse Gaussian Distribution: Theory : Methodology, and Applications”, CRC Press, 1988. |
[R7276353] | Wikipedia, “Wald distribution” http://en.wikipedia.org/wiki/Wald_distribution |
Examples
Draw values from the distribution and plot the histogram:
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=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.
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: |
|
---|---|
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
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
[R7376356] | 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. |
[R7476356] | Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951. |
[R7576356] | 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
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(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()
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 continuous probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.
Parameters: |
|
---|---|
Returns: | out – Drawn samples from the parameterized Zipf distribution. |
Return type: | ndarray or scalar |
See also
scipy.stats.zipf()
Notes
The probability density for the Zipf distribution is
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
[R7676359] | 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 = 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
>>> from scipy import special
Truncate s values at 50 so plot is interesting:
>>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)
>>> plt.plot(x, y/max(y), linewidth=2, color='r')
>>> plt.show()
nbodykit.source.catalog.uniform.
RandomCatalog
(csize, seed=None, comm=None, use_cache=False)[source]¶Bases: nbodykit.base.catalog.CatalogSource
A CatalogSource that can have columns added via a collective random number generator.
The random number generator stored as rng
behaves
as numpy.random.RandomState
but generates random
numbers only on the local rank in a manner independent of
the number of ranks.
Parameters: |
---|
Attributes
attrs |
A dictionary storing relevant meta-data about the CatalogSource. |
columns |
All columns in the CatalogSource, including those hard-coded into the class’s defintion and override columns provided by the user. |
csize |
The total, collective size of the CatalogSource, i.e., summed across all ranks. |
hardcolumns |
A list of the hard-coded columns in the CatalogSource. |
rng |
A MPIRandomState that behaves as numpy.random.RandomState but generates random numbers in a manner independent of the number of ranks. |
size |
The number of particles on the local rank |
use_cache |
If set to True , use the built-in caching features of dask to cache data in memory. |
Methods
Selection () |
A boolean column that selects a subset slice of the CatalogSource. |
Value () |
When interpolating a CatalogSource on to a mesh, the value of this array is used as the Value that each particle contributes to a given mesh cell. |
Weight () |
The column giving the weight to use for each particle on the mesh. |
compute (*args, **kwargs) |
Our version of dask.compute() that computes multiple delayed dask collections at once. |
copy () |
Return a copy of the CatalogSource object |
get_hardcolumn (col) |
Construct and return a hard-coded column. |
make_column (array) |
Utility function to convert a numpy array to a dask.array.Array . |
read (columns) |
Return the requested columns as dask arrays. |
save (output, columns[, datasets, header]) |
Save the CatalogSource to a bigfile.BigFile . |
to_mesh ([Nmesh, BoxSize, dtype, interlaced, …]) |
Convert the CatalogSource to a MeshSource, using the specified parameters. |
update_csize () |
Set the collective size, csize . |
Selection
()¶A boolean column that selects a subset slice of the CatalogSource.
By default, this column is set to True
for all particles.
Value
()¶When interpolating a CatalogSource on to a mesh, the value of this array is used as the Value that each particle contributes to a given mesh cell.
The mesh field is a weighted average of Value
, with the weights
given by Weight
.
By default, this array is set to unity for all particles.
Weight
()¶The column giving the weight to use for each particle on the mesh.
The mesh field is a weighted average of Value
, with the weights
given by Weight
.
By default, this array is set to unity for all particles.
__delitem__
(col)¶Delete a column; cannot delete a “hard-coded” column
__getitem__
(sel)¶The following types of indexing are supported:
__len__
()¶The local size of the CatalogSource on a given rank.
__setitem__
(col, value)¶Add columns to the CatalogSource, overriding any existing columns
with the name col
.
attrs
¶A dictionary storing relevant meta-data about the CatalogSource.
columns
¶All columns in the CatalogSource, including those hard-coded into the class’s defintion and override columns provided by the user.
compute
(*args, **kwargs)¶Our version of dask.compute()
that computes
multiple delayed dask collections at once.
This should be called on the return value of read()
to converts any dask arrays to numpy arrays.
If use_cache
is True
, this internally caches data, using
dask’s built-in cache features.
Parameters: | args (object) – Any number of objects. If the object is a dask collection, it’s computed and the result is returned. Otherwise it’s passed through unchanged. |
---|
Notes
The dask default optimizer induces too many (unnecesarry) IO calls – we turn this off feature off by default. Eventually we want our own optimizer probably.
copy
()¶Return a copy of the CatalogSource object
Returns: | the new CatalogSource object holding the copied data columns |
---|---|
Return type: | CatalogCopy |
csize
¶The total, collective size of the CatalogSource, i.e., summed across all ranks.
It is the sum of size
across all available ranks.
get_hardcolumn
(col)¶Construct and return a hard-coded column.
These are usually produced by calling member functions marked by the @column decorator.
Subclasses may override this method and the hardcolumns attribute to bypass the decorator logic.
hardcolumns
¶A list of the hard-coded columns in the CatalogSource.
These columns are usually member functions marked by @column decorator.
Subclasses may override this method and use get_hardcolumn()
to
bypass the decorator logic.
logger
= <logging.Logger object>¶make_column
(array)¶Utility function to convert a numpy array to a dask.array.Array
.
read
(columns)¶Return the requested columns as dask arrays.
Parameters: | columns (list of str) – the names of the requested columns |
---|---|
Returns: | the list of column data, in the form of dask arrays |
Return type: | list of dask.array.Array |
rng
¶A MPIRandomState
that behaves as
numpy.random.RandomState
but generates random
numbers in a manner independent of the number of ranks.
save
(output, columns, datasets=None, header='Header')¶Save the CatalogSource to a bigfile.BigFile
.
Only the selected columns are saved and attrs
are saved in
header
. The attrs of columns are stored in the datasets.
Parameters: |
|
---|
size
¶The number of particles on the local rank
to_mesh
(Nmesh=None, BoxSize=None, dtype='f4', interlaced=False, compensated=False, window='cic', weight='Weight', value='Value', selection='Selection', position='Position')¶Convert the CatalogSource to a MeshSource, using the specified parameters.
Parameters: |
|
---|---|
Returns: | mesh – a mesh object that provides an interface for gridding particle data onto a specified mesh |
Return type: |
update_csize
()¶Set the collective size, csize
.
This function should be called in __init__()
of a subclass,
after size
has been set to a valid value (not NotImplemented
)
use_cache
¶If set to True
, use the built-in caching features of dask
to cache data in memory.
nbodykit.source.catalog.uniform.
UniformCatalog
(nbar, BoxSize, seed=None, comm=None, use_cache=False)[source]¶Bases: nbodykit.source.catalog.uniform.RandomCatalog
A CatalogSource that has uniformly-distributed Position
and Velocity
columns.
The random numbers generated do not depend on the number of available ranks.
Parameters: |
---|
Attributes
attrs |
A dictionary storing relevant meta-data about the CatalogSource. |
columns |
All columns in the CatalogSource, including those hard-coded into the class’s defintion and override columns provided by the user. |
csize |
The total, collective size of the CatalogSource, i.e., summed across all ranks. |
hardcolumns |
A list of the hard-coded columns in the CatalogSource. |
rng |
A MPIRandomState that behaves as numpy.random.RandomState but generates random numbers in a manner independent of the number of ranks. |
size |
The number of particles on the local rank |
use_cache |
If set to True , use the built-in caching features of dask to cache data in memory. |
Methods
Position () |
The position of particles, uniformly distributed in BoxSize |
Selection () |
A boolean column that selects a subset slice of the CatalogSource. |
Value () |
When interpolating a CatalogSource on to a mesh, the value of this array is used as the Value that each particle contributes to a given mesh cell. |
Velocity () |
The velocity of particles, uniformly distributed in 0.01 x BoxSize |
Weight () |
The column giving the weight to use for each particle on the mesh. |
compute (*args, **kwargs) |
Our version of dask.compute() that computes multiple delayed dask collections at once. |
copy () |
Return a copy of the CatalogSource object |
get_hardcolumn (col) |
Construct and return a hard-coded column. |
make_column (array) |
Utility function to convert a numpy array to a dask.array.Array . |
read (columns) |
Return the requested columns as dask arrays. |
save (output, columns[, datasets, header]) |
Save the CatalogSource to a bigfile.BigFile . |
to_mesh ([Nmesh, BoxSize, dtype, interlaced, …]) |
Convert the CatalogSource to a MeshSource, using the specified parameters. |
update_csize () |
Set the collective size, csize . |
Selection
()¶A boolean column that selects a subset slice of the CatalogSource.
By default, this column is set to True
for all particles.
Value
()¶When interpolating a CatalogSource on to a mesh, the value of this array is used as the Value that each particle contributes to a given mesh cell.
The mesh field is a weighted average of Value
, with the weights
given by Weight
.
By default, this array is set to unity for all particles.
Weight
()¶The column giving the weight to use for each particle on the mesh.
The mesh field is a weighted average of Value
, with the weights
given by Weight
.
By default, this array is set to unity for all particles.
__delitem__
(col)¶Delete a column; cannot delete a “hard-coded” column
__getitem__
(sel)¶The following types of indexing are supported:
__len__
()¶The local size of the CatalogSource on a given rank.
__setitem__
(col, value)¶Add columns to the CatalogSource, overriding any existing columns
with the name col
.
attrs
¶A dictionary storing relevant meta-data about the CatalogSource.
columns
¶All columns in the CatalogSource, including those hard-coded into the class’s defintion and override columns provided by the user.
compute
(*args, **kwargs)¶Our version of dask.compute()
that computes
multiple delayed dask collections at once.
This should be called on the return value of read()
to converts any dask arrays to numpy arrays.
If use_cache
is True
, this internally caches data, using
dask’s built-in cache features.
Parameters: | args (object) – Any number of objects. If the object is a dask collection, it’s computed and the result is returned. Otherwise it’s passed through unchanged. |
---|
Notes
The dask default optimizer induces too many (unnecesarry) IO calls – we turn this off feature off by default. Eventually we want our own optimizer probably.
copy
()¶Return a copy of the CatalogSource object
Returns: | the new CatalogSource object holding the copied data columns |
---|---|
Return type: | CatalogCopy |
csize
¶The total, collective size of the CatalogSource, i.e., summed across all ranks.
It is the sum of size
across all available ranks.
get_hardcolumn
(col)¶Construct and return a hard-coded column.
These are usually produced by calling member functions marked by the @column decorator.
Subclasses may override this method and the hardcolumns attribute to bypass the decorator logic.
hardcolumns
¶A list of the hard-coded columns in the CatalogSource.
These columns are usually member functions marked by @column decorator.
Subclasses may override this method and use get_hardcolumn()
to
bypass the decorator logic.
logger
= <logging.Logger object>¶make_column
(array)¶Utility function to convert a numpy array to a dask.array.Array
.
read
(columns)¶Return the requested columns as dask arrays.
Parameters: | columns (list of str) – the names of the requested columns |
---|---|
Returns: | the list of column data, in the form of dask arrays |
Return type: | list of dask.array.Array |
rng
¶A MPIRandomState
that behaves as
numpy.random.RandomState
but generates random
numbers in a manner independent of the number of ranks.
save
(output, columns, datasets=None, header='Header')¶Save the CatalogSource to a bigfile.BigFile
.
Only the selected columns are saved and attrs
are saved in
header
. The attrs of columns are stored in the datasets.
Parameters: |
|
---|
size
¶The number of particles on the local rank
to_mesh
(Nmesh=None, BoxSize=None, dtype='f4', interlaced=False, compensated=False, window='cic', weight='Weight', value='Value', selection='Selection', position='Position')¶Convert the CatalogSource to a MeshSource, using the specified parameters.
Parameters: |
|
---|---|
Returns: | mesh – a mesh object that provides an interface for gridding particle data onto a specified mesh |
Return type: |
update_csize
()¶Set the collective size, csize
.
This function should be called in __init__()
of a subclass,
after size
has been set to a valid value (not NotImplemented
)
use_cache
¶If set to True
, use the built-in caching features of dask
to cache data in memory.