# Applications : a few examples

## Bilateral distributions with fat tails and package *FatTailsR*

## Numerical simulation, NOLH design of experiments and neural networks

### This presentation (Nafems France congress, October 13th, 2010, Paris), prepared with the help of Valéo, describes an original experimental strategy to build and train metamodels (surrogate models) of costly numerical code (about 6 hours per numerical run) of fluid mechanic : the torque and the yield of an automotive fan blade. Here, we try to minimize the number of trials using nonlinear black box models.

### The experimental strategy uses at the first step nearly orthogonal latin hypercubes or NOLH designs (Cioppa, 2002 and 2007), then D-optimal designs for neural networks at the second and third steps.

### The metamodels are a simple second order polynomial model for the torque (a very simple linear model indeed but with a few surprises) and a static neural network with 3 hidden neurons for the yield (due to a severe nonlinearity in the middle of the experimental domain).

### We confirm that NOLH designs are good designs to fill the experimental domain but they are not perfect : they must be completed with some points located at the border of the experimental domain for both polynoms and neural networks. The D-optimal criterion helps find those additional points.

### Such an iterative strategy (sometimes refered as "adaptative designs") returns an adequate model within 2 or 3 steps from an initial starting point with no knowledge at all. The ratio number of trials to number of coefficients of the model at the end of the study is less than 3, which is remarkable for a nonlinear black box model. The study conducted here on 5 inputs will be soon extended to 11 and 16 inputs.

### Let's note that this very economical approach would have been impossible before 2007 : Cioppa's NOLH designs were unknown in France before this date, Valéo had no extensive computation workflow available, the software Neuro Pex dedicated to the design of experiments had just become available since winter 2006.

### Read the presentation InModelia-Valéo (in English) at Nafems conference. The same presentation, in French.

### Contact InModelia to get a copy of the NOLH designs published by Cioppa (2007) and the improved NOLH designs published by De Rainville (2009) at the time of the Valéo study.

## Neural networks, capability index and non-standard distributions

### This presentation (AEC-APC, Advanced Equipment Control - Advanced Process Control Conference, March 29-31, 2006, Aix-en-Provence) made by ST-Microelectronics at the time when I worked at Netral, describes the results obtained with neural networks in modelling cumulative distribution functions and probability density functions of exotic distributions that can be meet sometimes in industry and finance.

### Non-gaussian distributions, asymetric, bimodals, trimodals, quadrimodals, with fat tails : neural networks are able to learn all of them !

### With some reverse calculation tools, one can then estimate in a very realistic manner the quantiles and capability index Cp and Cpk that are of main interest for manufacturing managers and quality managers.

### But there is a trick ! Contact me.

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