phiP {DiceDesign} | R Documentation |

## phiP criterion

### Description

Compute the `\phi_p`

criterion (strongly linked to mindist criterion)

### Usage

`phiP(design, p=50)`

### Arguments

`design` |
a matrix (or a data.frame) corresponding to the design of experiments. |

`p` |
the "p" in the Lp norm which is taken |

### Details

The `\phi_p`

criterion is defined by the `L_p`

norm of the sum of the inverses of the design inter-point euclidean distances:

`\phi_{p}=\left[\sum_{i,j=1\ldots N,i<j}\,\,d_{ij}^{-p}\right]^{\frac{1}{p}}`

A higher value corresponds to a more regular scaterring of design points.

When `p`

tends to infinity, optimizing a design with `\phi_p`

is equivalent to optimizing a design with `mindist`

.

### Value

A real number equal to the value of the `\phi_p`

criterion for the `design`

.

### Author(s)

G. Damblin & B.Iooss

### References

Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of sapce filling designs: optimization of Latin Hypercube Samples and subprojection properties, *Journal of Simulation*, 7:276-289, 2013.

Fang K.-T., Li R. and Sudjianto A. (2006). Design and Modeling for Computer Experiments, *Chapman & Hall*.

Pronzato, L. and Muller, W. (2012). Design of computer experiments: space filling and beyond, *Statistics and Computing*, 22:681-701.

### See Also

geometric criterion (`mindist`

)

### Examples

```
dimension <- 2
n <- 40
X <- matrix(runif(n*dimension), n, dimension)
phiP(X)
```

*DiceDesign*version 1.10 Index]