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Diffstat (limited to 'numpy/random/mtrand/mtrand.pyx')
| -rw-r--r-- | numpy/random/mtrand/mtrand.pyx | 113 |
1 files changed, 67 insertions, 46 deletions
diff --git a/numpy/random/mtrand/mtrand.pyx b/numpy/random/mtrand/mtrand.pyx index 6b054a20f..134bc2e09 100644 --- a/numpy/random/mtrand/mtrand.pyx +++ b/numpy/random/mtrand/mtrand.pyx @@ -1,3 +1,5 @@ +# cython: language_level=3 + # mtrand.pyx -- A Pyrex wrapper of Jean-Sebastien Roy's RandomKit # # Copyright 2005 Robert Kern (robert.kern@gmail.com) @@ -844,16 +846,16 @@ cdef class RandomState: Examples -------- >>> np.random.random_sample() - 0.47108547995356098 + 0.47108547995356098 # random >>> type(np.random.random_sample()) - <type 'float'> + <class 'float'> >>> np.random.random_sample((5,)) - array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) + array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random Three-by-two array of random numbers from [-5, 0): >>> 5 * np.random.random_sample((3, 2)) - 5 - array([[-3.99149989, -0.52338984], + array([[-3.99149989, -0.52338984], # random [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) @@ -954,14 +956,14 @@ cdef class RandomState: Examples -------- >>> np.random.randint(2, size=10) - array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) + array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random >>> 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], + array([[4, 0, 2, 1], # random [3, 2, 2, 0]]) """ @@ -1076,34 +1078,34 @@ cdef class RandomState: Generate a uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3) - array([0, 3, 4]) + array([0, 3, 4]) # random >>> #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]) + array([3, 3, 0]) # random 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]) + array([3,1,0]) # random >>> #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]) + array([2, 3, 0]) # random Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) - array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], - dtype='|S11') + array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random + dtype='<U11') """ @@ -1470,11 +1472,11 @@ cdef class RandomState: Examples -------- >>> np.random.random_integers(5) - 4 + 4 # random >>> type(np.random.random_integers(5)) - <type 'int'> + <class 'numpy.int64'> >>> np.random.random_integers(5, size=(3,2)) - array([[5, 4], + array([[5, 4], # random [3, 3], [4, 5]]) @@ -1483,7 +1485,7 @@ cdef class RandomState: :math:`{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 ]) + array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) # random Roll two six sided dice 1000 times and sum the results: @@ -1679,9 +1681,9 @@ cdef class RandomState: Parameters ---------- a : float or array_like of floats - Alpha, non-negative. + Alpha, positive (>0). b : float or array_like of floats - Beta, non-negative. + Beta, positive (>0). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), @@ -2068,8 +2070,8 @@ cdef class RandomState: The lower bound for the top 1% of the samples is : - >>> sort(s)[-10] - 7.61988120985 + >>> np.sort(s)[-10] + 7.61988120985 # random So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% @@ -2166,6 +2168,7 @@ cdef class RandomState: >>> NF = np.histogram(nc_vals, bins=50, density=True) >>> c_vals = np.random.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, density=True) + >>> import matplotlib.pyplot as plt >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() @@ -2261,7 +2264,7 @@ cdef class RandomState: Examples -------- >>> np.random.chisquare(2,4) - array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) + array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random """ cdef ndarray odf @@ -2443,6 +2446,7 @@ cdef class RandomState: -------- Draw samples and plot the distribution: + >>> import matplotlib.pyplot as plt >>> s = np.random.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) @@ -2785,7 +2789,7 @@ cdef class RandomState: Parameters ---------- a : float or array_like of floats - Shape of the distribution. Should be greater than zero. + Shape parameter of the distribution. Must be nonnegative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), @@ -3279,12 +3283,13 @@ cdef class RandomState: >>> loc, scale = 10, 1 >>> s = np.random.logistic(loc, scale, 10000) + >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=50) # plot against distribution >>> def logist(x, loc, scale): - ... return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2) + ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2) >>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\\ ... logist(bins, loc, scale).max()) >>> plt.show() @@ -3479,6 +3484,7 @@ cdef class RandomState: -------- Draw values from the distribution and plot the histogram + >>> from matplotlib.pyplot import hist >>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave @@ -3492,7 +3498,7 @@ cdef class RandomState: The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. - 0.087300000000000003 + 0.087300000000000003 # random """ cdef ndarray oscale @@ -3873,9 +3879,9 @@ cdef class RandomState: 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): + >>> for i in range(1, 11): # doctest: +SKIP ... probability = sum(s<i) / 100000. - ... print i, "wells drilled, probability of one success =", probability + ... print(i, "wells drilled, probability of one success =", probability) """ cdef ndarray on @@ -4233,6 +4239,7 @@ cdef class RandomState: >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000) + >>> from matplotlib.pyplot import hist >>> hist(s) # note that it is very unlikely to grab both bad items @@ -4342,14 +4349,15 @@ cdef class RandomState: >>> a = .6 >>> s = np.random.logseries(a, 10000) + >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s) # plot against distribution >>> def logseries(k, p): - ... return -p**k/(k*log(1-p)) + ... return -p**k/(k*np.log(1-p)) >>> plt.plot(bins, logseries(bins, a)*count.max()/ - logseries(bins, a).max(), 'r') + ... logseries(bins, a).max(), 'r') >>> plt.show() """ @@ -4474,7 +4482,7 @@ cdef class RandomState: standard deviation: >>> list((x[0,0,:] - mean) < 0.6) - [True, True] + [True, True] # random """ from numpy.dual import svd @@ -4580,14 +4588,14 @@ cdef class RandomState: Throw a dice 20 times: >>> np.random.multinomial(20, [1/6.]*6, size=1) - array([[4, 1, 7, 5, 2, 1]]) + array([[4, 1, 7, 5, 2, 1]]) # random It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> np.random.multinomial(20, [1/6.]*6, size=2) - array([[3, 4, 3, 3, 4, 3], + array([[3, 4, 3, 3, 4, 3], # random [2, 4, 3, 4, 0, 7]]) For the first run, we threw 3 times 1, 4 times 2, etc. For the second, @@ -4596,7 +4604,7 @@ cdef class RandomState: 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]) + array([11, 16, 14, 17, 16, 26]) # random The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take @@ -4605,7 +4613,7 @@ cdef class RandomState: other should be sampled like so: >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT - array([38, 62]) + array([38, 62]) # random not like: @@ -4659,8 +4667,9 @@ cdef class RandomState: 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. + generalization of a Beta distribution. The Dirichlet distribution + is a conjugate prior of a multinomial distribution in Bayesian + inference. Parameters ---------- @@ -4684,13 +4693,24 @@ cdef class RandomState: Notes ----- - .. math:: X \\approx \\prod_{i=1}^{k}{x^{\\alpha_i-1}_i} - 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 - :math:`X = \\frac{1}{\\sum_{i=1}^k{y_i}} (y_1, \\ldots, y_n)` is - Dirichlet distributed. + The Dirichlet distribution is a distribution over vectors + :math:`x` that fulfil the conditions :math:`x_i>0` and + :math:`\\sum_{i=1}^k x_i = 1`. + + The probability density function :math:`p` of a + Dirichlet-distributed random vector :math:`X` is + proportional to + + .. math:: p(x) \\propto \\prod_{i=1}^{k}{x^{\\alpha_i-1}_i}, + + where :math:`\\alpha` is a vector containing the positive + concentration parameters. + + The method uses the following property for computation: let :math:`Y` + be a random vector which has components that follow a standard gamma + distribution, then :math:`X = \\frac{1}{\\sum_{i=1}^k{Y_i}} Y` + is Dirichlet-distributed References ---------- @@ -4710,6 +4730,7 @@ cdef class RandomState: >>> s = np.random.dirichlet((10, 5, 3), 20).transpose() + >>> import matplotlib.pyplot as plt >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') @@ -4798,14 +4819,14 @@ cdef class RandomState: >>> arr = np.arange(10) >>> np.random.shuffle(arr) >>> arr - [1 7 5 2 9 4 3 6 0 8] + [1 7 5 2 9 4 3 6 0 8] # random 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], + array([[3, 4, 5], # random [6, 7, 8], [0, 1, 2]]) @@ -4885,14 +4906,14 @@ cdef class RandomState: Examples -------- >>> np.random.permutation(10) - array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) + array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random >>> np.random.permutation([1, 4, 9, 12, 15]) - array([15, 1, 9, 4, 12]) + array([15, 1, 9, 4, 12]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.permutation(arr) - array([[6, 7, 8], + array([[6, 7, 8], # random [0, 1, 2], [3, 4, 5]]) |