## Moon (2010) High-Dimensionality Function

### Description:

*Dimensions: 20*

This is a nonlinear function with many terms, including linear, quadratic and linear-by-linear interaction effects. The active inputs are x

_{1}, x

_{7}, x

_{12}, x

_{18}and x

_{19}(Moon et al., 2012).

Several properties make this a difficult function for input variable screening. First, some of the non-active effects are not much smaller than the active effects, making it difficult to differentiate between them. Also, the four active terms don't involve linear effects of the active variables; they only include interaction or quadratic effects.

All of the terms of the summation are included in the table below. The table entries are coefficients corresponding to the interaction term involving the two variables. Interactions between an x-variable and 1 are linear terms, and interactions between an x-variable and itself are quadratic terms.

### Input Domain:

The function is evaluated on x_{i}∈ [0, 1], for all i = 1, …, 20.

### Modifications and Alternate Forms:

To allow easier differentiation between the active and non-active effects, Moon (2010) and Moon et al. (2012) try three alternate forms of the function.The first form sets all non-active coefficients to 0, keeping the four active coefficients as they were:

The second form keeps all the non-active coefficients as they were, and doubles the active coefficients:

Finally, the third form keeps all the non-active coefficients as they were, and triples the active coefficients:

### Code:

MATLAB Implementation

R Implementation

MATLAB Implementation, C-1

R Implementation, C-1

MATLAB Implementation, C-2

R Implementation, C-2

MATLAB Implementation, C-3

R Implementation, C-3

R Implementation

MATLAB Implementation, C-1

R Implementation, C-1

MATLAB Implementation, C-2

R Implementation, C-2

MATLAB Implementation, C-3

R Implementation, C-3

### References:

Moon, H. (2010). *Design and Analysis of Computer Experiments for Screening Input Variables* (Doctoral dissertation, Ohio State University).

Moon, H., Dean, A. M., & Santner, T. J. (2012). Two-stage sensitivity-based group screening in computer experiments. *Technometrics, 54*(4), 376-387.

For questions or comments, please email Derek Bingham at: dbingham@stat.sfu.ca.

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