170 lines
6.9 KiB
Python
170 lines
6.9 KiB
Python
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import numpy as np
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from scipy._lib._util import check_random_state
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def rvs_ratio_uniforms(pdf, umax, vmin, vmax, size=1, c=0, random_state=None):
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"""
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Generate random samples from a probability density function using the
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ratio-of-uniforms method.
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Parameters
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----------
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pdf : callable
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A function with signature `pdf(x)` that is proportional to the
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probability density function of the distribution.
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umax : float
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The upper bound of the bounding rectangle in the u-direction.
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vmin : float
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The lower bound of the bounding rectangle in the v-direction.
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vmax : float
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The upper bound of the bounding rectangle in the v-direction.
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size : int or tuple of ints, optional
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Defining number of random variates (default is 1).
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c : float, optional.
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Shift parameter of ratio-of-uniforms method, see Notes. Default is 0.
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random_state : {None, int, `~np.random.RandomState`, `~np.random.Generator`}, optional
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If `random_state` is `None` the `~np.random.RandomState` singleton is
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used.
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If `random_state` is an int, a new ``RandomState`` instance is used,
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seeded with random_state.
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If `random_state` is already a ``RandomState`` or ``Generator``
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instance, then that object is used.
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Default is None.
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Returns
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-------
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rvs : ndarray
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The random variates distributed according to the probability
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distribution defined by the pdf.
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Notes
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-----
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Given a univariate probability density function `pdf` and a constant `c`,
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define the set ``A = {(u, v) : 0 < u <= sqrt(pdf(v/u + c))}``.
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If `(U, V)` is a random vector uniformly distributed over `A`,
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then `V/U + c` follows a distribution according to `pdf`.
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The above result (see [1]_, [2]_) can be used to sample random variables
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using only the pdf, i.e. no inversion of the cdf is required. Typical
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choices of `c` are zero or the mode of `pdf`. The set `A` is a subset of
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the rectangle ``R = [0, umax] x [vmin, vmax]`` where
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- ``umax = sup sqrt(pdf(x))``
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- ``vmin = inf (x - c) sqrt(pdf(x))``
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- ``vmax = sup (x - c) sqrt(pdf(x))``
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In particular, these values are finite if `pdf` is bounded and
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``x**2 * pdf(x)`` is bounded (i.e. subquadratic tails).
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One can generate `(U, V)` uniformly on `R` and return
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`V/U + c` if `(U, V)` are also in `A` which can be directly
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verified.
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The algorithm is not changed if one replaces `pdf` by k * `pdf` for any
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constant k > 0. Thus, it is often convenient to work with a function
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that is proportional to the probability density function by dropping
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unneccessary normalization factors.
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Intuitively, the method works well if `A` fills up most of the
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enclosing rectangle such that the probability is high that `(U, V)`
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lies in `A` whenever it lies in `R` as the number of required
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iterations becomes too large otherwise. To be more precise, note that
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the expected number of iterations to draw `(U, V)` uniformly
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distributed on `R` such that `(U, V)` is also in `A` is given by
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the ratio ``area(R) / area(A) = 2 * umax * (vmax - vmin) / area(pdf)``,
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where `area(pdf)` is the integral of `pdf` (which is equal to one if the
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probability density function is used but can take on other values if a
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function proportional to the density is used). The equality holds since
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the area of `A` is equal to 0.5 * area(pdf) (Theorem 7.1 in [1]_).
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If the sampling fails to generate a single random variate after 50000
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iterations (i.e. not a single draw is in `A`), an exception is raised.
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If the bounding rectangle is not correctly specified (i.e. if it does not
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contain `A`), the algorithm samples from a distribution different from
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the one given by `pdf`. It is therefore recommended to perform a
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test such as `~scipy.stats.kstest` as a check.
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References
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----------
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.. [1] L. Devroye, "Non-Uniform Random Variate Generation",
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Springer-Verlag, 1986.
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.. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian
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random variates", Statistics and Computing, 24(4), p. 547--557, 2014.
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.. [3] A.J. Kinderman and J.F. Monahan, "Computer Generation of Random
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Variables Using the Ratio of Uniform Deviates",
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ACM Transactions on Mathematical Software, 3(3), p. 257--260, 1977.
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Examples
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--------
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>>> from scipy import stats
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Simulate normally distributed random variables. It is easy to compute the
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bounding rectangle explicitly in that case. For simplicity, we drop the
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normalization factor of the density.
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>>> f = lambda x: np.exp(-x**2 / 2)
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>>> v_bound = np.sqrt(f(np.sqrt(2))) * np.sqrt(2)
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>>> umax, vmin, vmax = np.sqrt(f(0)), -v_bound, v_bound
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>>> np.random.seed(12345)
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>>> rvs = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=2500)
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The K-S test confirms that the random variates are indeed normally
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distributed (normality is not rejected at 5% significance level):
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>>> stats.kstest(rvs, 'norm')[1]
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0.33783681428365553
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The exponential distribution provides another example where the bounding
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rectangle can be determined explicitly.
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>>> np.random.seed(12345)
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>>> rvs = stats.rvs_ratio_uniforms(lambda x: np.exp(-x), umax=1,
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... vmin=0, vmax=2*np.exp(-1), size=1000)
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>>> stats.kstest(rvs, 'expon')[1]
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0.928454552559516
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"""
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if vmin >= vmax:
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raise ValueError("vmin must be smaller than vmax.")
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if umax <= 0:
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raise ValueError("umax must be positive.")
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size1d = tuple(np.atleast_1d(size))
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N = np.prod(size1d) # number of rvs needed, reshape upon return
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# start sampling using ratio of uniforms method
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rng = check_random_state(random_state)
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x = np.zeros(N)
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simulated, i = 0, 1
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# loop until N rvs have been generated: expected runtime is finite.
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# to avoid infinite loop, raise exception if not a single rv has been
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# generated after 50000 tries. even if the expected numer of iterations
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# is 1000, the probability of this event is (1-1/1000)**50000
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# which is of order 10e-22
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while simulated < N:
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k = N - simulated
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# simulate uniform rvs on [0, umax] and [vmin, vmax]
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u1 = umax * rng.uniform(size=k)
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v1 = rng.uniform(vmin, vmax, size=k)
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# apply rejection method
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rvs = v1 / u1 + c
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accept = (u1**2 <= pdf(rvs))
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num_accept = np.sum(accept)
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if num_accept > 0:
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x[simulated:(simulated + num_accept)] = rvs[accept]
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simulated += num_accept
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if (simulated == 0) and (i*N >= 50000):
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msg = ("Not a single random variate could be generated in {} "
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"attempts. The ratio of uniforms method does not appear "
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"to work for the provided parameters. Please check the "
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"pdf and the bounds.".format(i*N))
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raise RuntimeError(msg)
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i += 1
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return np.reshape(x, size1d)
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