Home | Trees | Indices | Help |
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thisPath =
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ALLOW_THREADS = 1
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BUFSIZE = 10000
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CLIP = 0
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ERR_CALL = 3
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ERR_DEFAULT = 0
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ERR_DEFAULT2 = 2084
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ERR_IGNORE = 0
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ERR_LOG = 5
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ERR_PRINT = 4
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ERR_RAISE = 2
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ERR_WARN = 1
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FLOATING_POINT_SUPPORT = 1
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FPE_DIVIDEBYZERO = 1
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FPE_INVALID = 8
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FPE_OVERFLOW = 2
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FPE_UNDERFLOW = 4
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False_ = False
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Inf = inf
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Infinity = inf
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MAXDIMS = 32
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NAN = nan
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NINF = -inf
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NZERO = -0.0
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NaN = nan
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PINF = inf
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PZERO = 0.0
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RAISE = 2
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SHIFT_DIVIDEBYZERO = 0
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SHIFT_INVALID = 9
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SHIFT_OVERFLOW = 3
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SHIFT_UNDERFLOW = 6
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ScalarType =
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True_ = True
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UFUNC_BUFSIZE_DEFAULT = 10000
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UFUNC_PYVALS_NAME =
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WRAP = 1
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__package__ =
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absolute = <ufunc 'absolute'>
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add = <ufunc 'add'>
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arccos = <ufunc 'arccos'>
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arccosh = <ufunc 'arccosh'>
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arcsin = <ufunc 'arcsin'>
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arcsinh = <ufunc 'arcsinh'>
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arctan = <ufunc 'arctan'>
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arctan2 = <ufunc 'arctan2'>
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arctanh = <ufunc 'arctanh'>
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bitwise_and = <ufunc 'bitwise_and'>
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bitwise_not = <ufunc 'invert'>
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bitwise_or = <ufunc 'bitwise_or'>
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bitwise_xor = <ufunc 'bitwise_xor'>
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c_ = <numpy.lib.index_tricks.CClass object at 0xaaa824c>
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cast = {<type 'numpy.int64'>: <function <lambda> at 0xa9a741c>
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ceil = <ufunc 'ceil'>
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conj = <ufunc 'conjugate'>
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conjugate = <ufunc 'conjugate'>
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cos = <ufunc 'cos'>
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cosh = <ufunc 'cosh'>
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deg2rad = <ufunc 'deg2rad'>
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degrees = <ufunc 'degrees'>
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divide = <ufunc 'divide'>
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e = 2.71828182846
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equal = <ufunc 'equal'>
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exp = <ufunc 'exp'>
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exp2 = <ufunc 'exp2'>
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expm1 = <ufunc 'expm1'>
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fabs = <ufunc 'fabs'>
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floor = <ufunc 'floor'>
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floor_divide = <ufunc 'floor_divide'>
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fmax = <ufunc 'fmax'>
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fmin = <ufunc 'fmin'>
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fmod = <ufunc 'fmod'>
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frexp = <ufunc 'frexp'>
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greater = <ufunc 'greater'>
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greater_equal = <ufunc 'greater_equal'>
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hypot = <ufunc 'hypot'>
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index_exp = <numpy.lib.index_tricks.IndexExpression object at
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inf = inf
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infty = inf
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invert = <ufunc 'invert'>
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isfinite = <ufunc 'isfinite'>
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isinf = <ufunc 'isinf'>
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isnan = <ufunc 'isnan'>
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ldexp = <ufunc 'ldexp'>
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left_shift = <ufunc 'left_shift'>
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less = <ufunc 'less'>
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less_equal = <ufunc 'less_equal'>
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little_endian = True
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log = <ufunc 'log'>
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log10 = <ufunc 'log10'>
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log1p = <ufunc 'log1p'>
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logaddexp = <ufunc 'logaddexp'>
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logaddexp2 = <ufunc 'logaddexp2'>
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logical_and = <ufunc 'logical_and'>
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logical_not = <ufunc 'logical_not'>
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logical_or = <ufunc 'logical_or'>
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logical_xor = <ufunc 'logical_xor'>
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maximum = <ufunc 'maximum'>
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mgrid = <numpy.lib.index_tricks.nd_grid object at 0xaaa812c>
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minimum = <ufunc 'minimum'>
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mod = <ufunc 'remainder'>
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modf = <ufunc 'modf'>
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multiply = <ufunc 'multiply'>
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nan = nan
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nbytes = {<type 'numpy.int64'>: 8, <type 'numpy.int16'>: 2, <t
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negative = <ufunc 'negative'>
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newaxis = None
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not_equal = <ufunc 'not_equal'>
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ogrid = <numpy.lib.index_tricks.nd_grid object at 0xaaa814c>
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ones_like = <ufunc 'ones_like'>
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pi = 3.14159265359
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r_ = <numpy.lib.index_tricks.RClass object at 0xaaa81cc>
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rad2deg = <ufunc 'rad2deg'>
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radians = <ufunc 'radians'>
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reciprocal = <ufunc 'reciprocal'>
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remainder = <ufunc 'remainder'>
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right_shift = <ufunc 'right_shift'>
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rint = <ufunc 'rint'>
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s_ = <numpy.lib.index_tricks.IndexExpression object at 0xaaa83ec>
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sctypeDict =
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sctypeNA =
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sctypes =
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sign = <ufunc 'sign'>
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signbit = <ufunc 'signbit'>
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sin = <ufunc 'sin'>
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sinh = <ufunc 'sinh'>
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sqrt = <ufunc 'sqrt'>
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square = <ufunc 'square'>
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subtract = <ufunc 'subtract'>
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tan = <ufunc 'tan'>
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tanh = <ufunc 'tanh'>
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true_divide = <ufunc 'true_divide'>
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trunc = <ufunc 'trunc'>
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typeDict =
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typeNA =
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typecodes =
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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. |
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%. |
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 |
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]) |
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 |
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. |
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() |
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 |
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 |
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() |
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! |
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. |
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() |
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() |
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() |
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]) |
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] |
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. |
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() |
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() |
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]) |
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 |
Return random integers x such that low <= x < high. If high is None, then 0 <= x < low. |
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). |
Return random integers x such that low <= x <= high. If high is None, then 1 <= x <= low. |
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 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 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) |
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. |
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() |
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() |
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() |
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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