# 3.1.3.1. Forward Propagation¶

The forward propagation provides a probabilistic understanding of output variables by producing sample realizations and statistical moments (mean, standard deviation, skewness, and kurtosis). Currently, only the Monte Carlo Sampling (MCS) method is available in the SimCenterUQ engine and the other sampling methods (Latin hypercube sampling, Surrogate model-based efficient sampling) are available in the Dakota engine.

## Sampling from dataset¶

SimCenterUQ uniquely provides an option to define the distribution of random variables (RVs) directly from the samples of the RVs (See RV: Random Variables). This feature is useful when the user does not know the actual distribution of an RV but has sample realizations. When a dataset is provided, quoFEM either treats the data as a discrete distribution (when `Discrete`

distribution option is selected) or fits a parametric distribution to the data (when another distribution option is selected). For the former case, quoFEM uniformly samples the discrete index of the dataset provided by the user and uses the value corresponding to the index for the forward propagation analysis. The index is sampled either **with** replacement if the data size is smaller than the number of samples to draw or **without** replacement if the data size is larger than the number of samples to draw. Further, when the user provides coupled samples of *correlated* RVs, vector-wise resampling can be performed instead of independent resampling. In particular, a single index is sampled and shared in multiple variables for each realization.

To enable this feature, the user can explicitly define the group of RVs which will share the index samples in the UQ tab using the `Resample RVs from correlated dataset`

:

The actual datasets of the RVs written in this field should be imported in the RV tab, through

`parameters`

-`discrete`

option.The RVs inside each group should be provided with the same length of the samples (e.g. in Fig. 3.1.3.1.1, \(w\) and \(wR\) should have the same sample size \(N_1\) and \(alp\) and \(F_y\) should have the same sample size \(N_2\))

Note

Any correlation values for this coupled datasets **additionally specified in the RV tab** will be **ignored**. Note that the correlation of the data is already reflected in the analysis by introducing the same resampling index.

For example, consider the case where two variables \(w\) and \(wR\) are provided as 10 discrete data points in RV tab as in Fig. 3.1.3.1.2.

Below is the an example 100 realization the two variables when they are considered to be *independent*, i.e. without checking the “Resample RVs from correlated dataset” option.

On the other hand, if the two datasets are considered correlated, i.e. if “Resample RVs from correlated dataset” are checked and the group {w,wR} is reported in the field as shown in Fig. 3.1.3.1.1, 100 realization pairs of the RVs will be stacked on top of the provided 10 dataset.

This feature is especially useful when the user wants to perform a forward UQ analysis directly using the posterior samples obtained from Markov Chain Monte Carlo or other Bayesian sampling approaches.