Problem of Large Dimensional Spaces

A Problem of Probability Density Function Estimation for Large Dimensional Spaces With Many Low-Influential Dimensions

Jergus Suja, Martin Kubicek and Tomas Koutnik

Abstract

All engineering problems consider uncertainties. These range from small production uncertainties to large-scale uncertainties coming from outside, such as variable wind speed or sunlight. Currently, modern methods for uncertainty propagation have large difficulties with estimation of statistics for large-scale problems which considers hundreds of these uncertain parameters. Due to the complexity of the problem and limitations of the modern methods, a common approach for modelling large scale problems is to select a few important parameters and model statistics for these parameters. However, this can lead to an important problem. In this paper, an application of the UptimAI’s UQ propagation algorithm is used to discuss a new problem arising from very high dimensional spaces where a large number of parameters have negligible impact on the final solution. In other words, when a problem consists of a great number of uncertain design parameters, common practice is to focus on the most important ones and neglect the non-influential ones. However, a combination of a great number of noninfluential parameters can lead to completely different results. This is especially a problem for modelling large dimensional statistical models where a common approach is to perform sensitivity analysis and neglect the non-influential variables, i.e. set the non-influential variables to nominal value. Therefore, using a common approach of neglecting the non-influential variables could lead to a dramatic error and hence, we call this problem ”many times nothing killed a horse”. This problem cannot be observed for cases with a small number of design parameters, which are commonly solved in statistical modelling. The reason for this issue is that the combined influence of neglected variables is extremely small and such that has no influence on the final output. Application of the UptimAl’s UQ propagation algorithm to modern engineering problems and the possibilities of mitigation of the cumulative influence of non-influential parameters is discussed in detail. The problem is shown on a case of economic load dispatch (ELD) problem which consist of 140 dimensions [1]. To this problem was applied UptimAI’s UQ propagation algorithm to obtain accurate statistics for the problem and to get deeper insight into the statistics. Using the accurate model obtained by UptimAI’s algorithm, we compare statistics of using only important variables and using all variables. This lead to a significant difference between results and such that put a large question mark on standard approach. The obtained results are validated with the Monte Carlo simulation applied directly to ELD problem. Application of UptimAl’s UQ propagation algorithm to modern engineering problems and the possibilities of mitigation of the cumulative influence of non-influential parameters is discussed in detail.

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