# Computation of the Modified Band Depth

### Description

`MBD`

computes the modified band depth of each observation within a sample which either includes or not the given observation.

### Usage

1 2 |

### Arguments

`x` |
a data matrix containing the observations (samples) by rows and the variables (genes) by columns. |

`xRef` |
an optional data matrix containing the sample of observations with respect to the modified band depth is computed. If unprovided, then all elements in matrix x are taken into account to compute the depth. |

`plotting` |
logical value. If TRUE then the observations in the data matrix x are plotted, in parallel coordinates. |

`grayscale` |
logical value. If TRUE then a different color from a given color palette is assigned to each sample, according to its depth. |

`band` |
logical value. If TRUE then the convex hull (a polygon) of the bands formed by the percentage p of most internal samples are represented. Different values of p can be set with the argument band.limits. |

`band.limits` |
a vector of values in the range 0-1 giving the proportion p of most central curves to be considered to form a band. |

`lty` |
the line type for drawing both the data set and the reference set. |

`lwd` |
the line width for both the data set and the reference set. The thickness of the deepest point is increased by 0.5 with respect to the thickest line drawn. |

`col` |
the color specification for the data set, except for the deepest point. If grayscale is true and no color is specified, then the depth of each point is represented in grayscale colors, with higher intensities corresponding to smaller depths. |

`cold` |
the color used to plot the deepest point. |

`colRef` |
the color specification for the reference data set. |

`ylim` |
numeric vector giving the y coordinates range. |

`cex` |
the magnification used for the legend. |

`...` |
graphical parameters (see 'par') and any further arguments of 'plot'. |

### Details

The modified band depth is the average proportion of components of the considered observation that are between the corresponding components of all possible pairs of elements in the sample with respect to the depth is computed. The depth is efficiently obtained using the multiplicity of each value in the data matrix ordered by columns rather than exhaustively searching for all pairs of samples.

### Value

a list containing:

`ordering ` |
vector giving the ordering of the samples according to their corresponding depths |

`MBD ` |
vector of the computed depths |

### Author(s)

Sara Lopez-Pintado sl2929@columbia.edu and

Aurora Torrente etorrent@est-econ.uc3m.es

### References

Lopez-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. *Journal of the American Statistical Association*, 104, 486-503.

Lopez-Pintado, S. *et al.* (2010). Robust depth-based tools for the analysis of gene expression data. *Biostatistics*, 11 (2), 254-264.

### See Also

scalecurve, R.test

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
## MBD of all elements within a sample
## simulated data
set.seed(0)
x <- matrix(rnorm(1000),100,10)
x.depths.1<-MBD(x,plotting=TRUE)
## real data
data(prostate)
prost.x<-prostate[,1:100]
prost.y<-prostate[,101]
normal.depths<-MBD(prost.x[prost.y==0,],plotting=TRUE,
main="Normal samples")
tumor.depths<-MBD(prost.x[prost.y==1,],plotting=TRUE, band=TRUE,
band.limits=c(.33,.67,1),grayscale=TRUE)
## MBD of a vector with respect to a set of observations
## simulated data
set.seed(0)
v <- matrix(c(2,1,0,3,-2,1,2,1,0,-2,rnorm(3)),3,5)
xR <- matrix(rnorm(100),10,5)
depths<-MBD(v,xR,plotting=TRUE)
# MBD of normal prostate samples with respect to tumoral ones
normal.depths<-MBD(prost.x[prost.y==0,],prost.x[prost.y==1,],
plotting=TRUE)
normal.depths<-MBD(prost.x[prost.y==0,],prost.x[prost.y==1,],plotting=TRUE,
band=TRUE,band.limits=c(.33,.67,1),grayscale=TRUE)
``` |

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