This is an implementation of Latin Hypercube Sampling with Multi-Dimensional Uniformity (LHS-MDU) from Deutsch and Deutsch, "Latin hypercube sampling with multidimensional uniformity.

## Project description

LHS-MDU
--------

Basics
======
This is a package for generating latin hypercube samples with multi-dimensional uniformity.

To use, simply do::

>>> import lhsmdu
>>> k = lhsmdu.sample(2, 20) # Latin Hypercube Sampling with multi-dimensional uniformity

This will generate a nested list with 2 variables, with 20 samples each.

To plot and see the difference between Monte Carlo and LHS-MDU sampling for a 2 dimensional system::

>>> l = lhsmdu.createRandomStandardUniformMatrix(2, 20) # Monte Carlo sampling
>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> ax = fig.gca()
>>> ax.set_xticks(numpy.arange(0,1,0.1))
>>> ax.set_yticks(numpy.arange(0,1,0.1))
>>> plt.scatter(k, k, col="g", label="LHS-MDU")
>>> plt.scatter(l, l, col="r", label="MC")
>>> plt.grid()
>>> plt.show()

You can use the strata generated by the algorithm to sample again, if you so desire. For this, you can do::

>>> m = lhsmdu.resample()
>>> n = lhsmdu.resample()
>>> o = lhsmdu.resample()

This will again generate the same number of samples as before, a nested list with 2 variables, with 20 samples each.

You can plot these together and see the sampling from the strata::

>>> fig = plt.figure()
>>> ax = fig.gca()
>>> ax.set_xticks(numpy.arange(0,1,0.1))
>>> ax.set_yticks(numpy.arange(0,1,0.1))
>>> plt.title("LHS-MDU")
>>> plt.scatter(k, k, c="g", label="sample 1")
>>> plt.scatter(m, m, c="r", label="resample 2")
>>> plt.scatter(n, n, c="b", label="resample 3")
>>> plt.scatter(o, o, c="y", label="resample 4")
>>> plt.grid()
>>> plt.show()

Alternatively, you can choose to get new strata each time, and see the sampling hence::

>>> p = lhsmdu.sample(2, 20) # Latin Hypercube Sampling with multi-dimensional uniformity
>>> q = lhsmdu.sample(2, 20) # Latin Hypercube Sampling with multi-dimensional uniformity
>>> r = lhsmdu.sample(2, 20) # Latin Hypercube Sampling with multi-dimensional uniformity
>>> fig = plt.figure()
>>> ax = fig.gca()
>>> ax.set_xticks(numpy.arange(0,1,0.1))
>>> ax.set_yticks(numpy.arange(0,1,0.1))
>>> plt.title("LHS-MDU")
>>> plt.scatter(k, k, c="g", label="sample 1")
>>> plt.scatter(p, p, c="r", label="sample 2")
>>> plt.scatter(q, q, c="b", label="sample 3")
>>> plt.scatter(r, r, c="y", label="sample 4")
>>> plt.grid()
>>> plt.show()

===========================================================================================

Sampling from arbitrary CDFs
=======================

After uniformly distributed samples have been generated from LHSMDU, you can convert these to samples from arbitrary distributions using inverse tranform sampling. In this, the CDF [0,1] of the distribution of interest is inverted, and then data points corresponding to the uniformly sampled points are picked up. To do this, you must have a `rv_contiuous` or `rv_discrete` distribution instance taken from scipy.stats. You can also use frozen distributions (after setting loc and scale parameters). Following is an example for normal distribution.::

>>> import scipy.stats.distributions as ssd
>>> p = ssd.norm
>>> new_samples = lhsmdu.inverseTransformSample(p, k)
>>> plt.hist(lhsmdu.inverseTransformSample(p, k))
>>> plt.show()

## Project details

This version 0.1