# The MVCH Algorithm

### Description

The function determines the multivariate mode by iteratively selecting minimum volume convex subsets.

### Usage

1 |

### Arguments

`data` |
At least a two-dimensional data matrix is required. Number of observations needs to be greater than number of dimensions. |

`ps` |
A numeric value between 0 and 1. Fraction of points to be retained in each iteration. Default is set to |

`pf` |
A numeric value between 0 and 1. Fraction of points determining the size of the final subset. Default is set to |

`k` |
The maximum number of iterations. Default is set to |

`a.poi` |
An integer |

`del.poi` |
An integer |

### Details

The algorithm iteratively determines a sequence of subsets of certain size with minimum convex hull volume (i.e. minimum volume subsets) until a certain threshold is reached. In the first iteration a minimum volume subset of size *n_1=floor(n*ps)* is sought. In the second iteration, out of the subset found in iteration 1, a subset of size *n_2=floor(n_1*ps)* is determined. The procedure continues until the threshold is reached: *ceil(n*pf)* where `n`

is the number of observations in `data`

. The mode is calculated as the arithmetic mean of the observations in the final subset. Hence, the combination of `ps`

and `pf`

determines the running time and robustness of the procedure. Highest robustness (in terms of maximum breakdown point) is achieved for *ps=floor((n+d+1)/2)*. Small values of `pf`

guarantee an accurate mode estimation also for asymmetric data sets but running times increase.

To find a minimum volume subset, in each iteration `in.subs`

atomic subsets (consisting of `d+1`

observations) are constructed. Each of these atomic subsets is iteratively expanded by adding the `a.poi`

closest points and deleting `del.poi`

. All three values determine the accuracy of the subset identification (and, hence, the estimate) as well as the running time of the algorithm. Small values of `in.subs`

reduce running time. Choosing similar values for `a.poi`

and `del.poi`

increases running time and algorithm accuracy.

For more details on the algorithm see the reference.

### Value

A list with following entries:

`mode` |
The mode estimate. |

`set` |
The final subset used for mode calculation. |

`vol` |
The convex hull volume of the final subset. |

`set.1` |
The subset identified after the first iteration (outlier-free subset). |

### Author(s)

Thomas Kirschstein <thomas.kirschstein@wiwi.uni-halle.de>

### References

Kirschstein, T., Liebscher, S., Porzio, G., Ragozini, G. (2015): Minimum volume peeling: a robust non-parametric estimator of the multivariate mode, *Computational Statistics and Data Analysis*, DOI: 10.1016/j.csda.2015.04.012.

### Examples

1 2 3 4 5 6 7 8 | ```
# maximum breakdown point estimation
# MVCH(halle, ps = floor((nrow(halle) + ncol(halle) + 1)/2), pf = 0.05)
# slower estimation
# MVCH(halle, ps = 0.75, pf = 0.05)
# quicker estimation
# MVCH(halle, ps = 0.25, pf = 0.05)
``` |