# Validation by comparing QoI distributions¶

This tutorial shows how to use a Validation Similarities pattern in EasyVVUQ.

We test here two quantities of interest (QoI) represented by two analytical functions with Gaussian uncertainties.

The first function is a parabolic function:

mu1 = (y - 50.)**2 / 500.
sig1 = 0.2
dist1 = chaospy.Normal(mu1, sig1)


The second function a constant but with changing uncertainty on one side:

mu2 = 2.5
sig2 = 0.1 + 0.01 * y
dist2 = chaospy.Normal(mu2, sig2)


In the upper pannel of the figure given below, you see how these functions look when we are varing y in the intervalle [0, 100].

## Validations metrics¶

In EasyVVUQ, we implemented the calculation of three different metrics: Hellinger, Jensen-Shannon and Wasserstein (cf. references below for more details). This allows us to compute distances between two QoI distributions.

## QoI distributions¶

We can use Chaospy to compute the probability densities and the cummulative distributions functions needed for the above-mentioned metrics:

# Probabily densities: for Hellinger and Jensen-Shannon
p1 = dist1.pdf(x)
p2 = dist2.pdf(x)

# Cummulative distributions (with weight): for Wasserstein
dx = x[-1] - x[0]
c1 = dx * dist1.cdf(x)
c2 = dx * dist2.cdf(x)


The sampling values x can be computed using the min/max values of a common large support of QoI distrubtions, for example:

x = np.linspace(min_value, max_value, 1000, endpoint=True)


Note 1: The min/max values can be obtained from lower and upper bound of the distributions. In case of univarainte distribution, we can use: dist.lower[0] and dist.upper[0].

Note 2: Distribution based on samples

To build QoI distribution from list of samples that resutls fron UQ simulations, observations or measurements, we can use:

dist = chaospy.SampleDist(samples)


It estimates a distribution from the given samples by constructing a kernel density estimator (KDE).

## Validate similarities¶

Once probabily densities functions (or Cummulative distributions) are comupted for each QoI, we create a validater, object of EasyVVUQ, and get the distance using compare routine. We can use for example Hellinger metric by comparing two lists of probabily densities, pdf1 and pdf2:

# Validater based on Hellinger metric
validater = easyvvuq.comparison.ValidateSimilarityHellinger()
distance = validater.compare(pdf1, pdf2)


The complete code for this example, using other metrics, can be found here.

Finally, in the lower panel of the the different distances between QoI 1 and Qo 2 are displayed:

The first two are yielding answers between 0 (zero distance: identical distributions) and 1 (very different), Wasserstein instead are unrestricted with a lower limit of zero.