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

# Installation

You can install lhsmdu using pip::

```
$ pip install lhsmdu
```

Alternatively, you can clone on github and then install the package locally::

```
$ git clone https://github.com/sahilm89/lhsmdu
$ cd lhsmdu
$ python setup.py install --user # for this user only.
```

or::

```
$ pip install git+https://github.com/sahilm89/lhsmdu --user
```

# 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
>>> k = lhsmdu.sample(2, 20) # Latin Hypercube Sampling with multi-dimensional uniformity
>>> k = np.array(k)
>>> l = np.array(l)
>>> 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[0], k[1], color="g", label="LHS-MDU")
>>> plt.scatter(l[0], l[1], color="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:: >>> m = np.array(m) >>> n = np.array(n) >>> o = np.array(o)

```
>>> 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[0], k[1], c="g", label="sample 1")
>>> plt.scatter(m[0], m[1], c="r", label="resample 2")
>>> plt.scatter(n[0], n[1], c="b", label="resample 3")
>>> plt.scatter(o[0], o[1], 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
>>> p = np.array(p)
>>> q = np.array(q)
>>> r = np.array(r)
>>> 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[0], k[1], c="g", label="sample 1")
>>> plt.scatter(p[0], p[1], c="r", label="sample 2")
>>> plt.scatter(q[0], q[1], c="b", label="sample 3")
>>> plt.scatter(r[0], r[1], c="y", label="sample 4")
>>> plt.grid()
>>> plt.show()
```

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

# Changing the random seed

You will notice that the strata generated are the same each time you run the program again. This is because the random seed is a global constant set to a default value by design, so that simulations can be replicated. In order to change this behavior, you can set a new random seed using the following code::

```
>>> randSeed = 11 # random number of choice
>>> lhsmdu.setRandomSeed(randSeed) # Latin Hypercube Sampling with multi-dimensional uniformity
>>> lhsmdu.sample(2, 20) # Latin Hypercube Sampling with multi-dimensional uniformity
```

Alternatively, you can also set the seed by using sample with a new seed::

```
>>> lhsmdu.sample(2, 20, randomSeed=randSeed) # Latin Hypercube Sampling with multi-dimensional uniformity
```

To change the random seed in every run, you can set on top of the program::

```
>>> lhsmdu.setRandomSeed(None)
```

# 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[0])
>>> plt.hist(new_samples[0])
>>> plt.show()
```

# Citing this repository

To cite, please cite both the original paper from Deutsch and Deutsch: http://dx.doi.org/10.1016%2Fj.jspi.2011.09.016. and the repository: https://doi.org/10.5281/zenodo.3929531

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