# depth.space.projection: Calculate Depth Space using Projection Depth In ddalpha: Depth-Based Classification and Calculation of Data Depth

## Description

Calculates the representation of the training classes in depth space using projection depth.

## Usage

 ```1 2``` ```depth.space.projection(data, cardinalities, method = "random", num.directions = 1000, seed = 0) ```

## Arguments

 `data` Matrix containing training sample where each row is a d-dimensional object, and objects of each class are kept together so that the matrix can be thought of as containing blocks of objects representing classes. `cardinalities` Numerical vector of cardinalities of each class in `data`, each entry corresponds to one class. `method` to be used in calculations. `"random"` Here the depth is determined as the minimum univariate depth of the data projected on lines in several directions. The directions are distributed uniformly on the (d-1)-sphere; the same direction set is used for all points. `"linearize"` The Nelder-Mead method for function minimization, taken from Olsson, Journal of Quality Technology, 1974, 6, 56. R-codes of this function were written by Subhajit Dutta. `num.directions` Number of random directions to be generated for `method = "random"`. With the growth of n the complexity grows linearly for the same number of directions. `seed` the random seed. The default value `seed=0` makes no changes.

## Details

The depth representation is calculated in the same way as in `depth.projection`, see 'References' for more information and details.

## Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument `cardinalities`.

## References

Donoho, D.L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper. Department of Statistics, Harvard University.

Liu, R.Y. (1992). Data depth and multivariate rank tests. In: Dodge, Y. (ed.), L1-Statistics and Related Methods, North-Holland (Amsterdam), 279–294.

Liu, X. and Zuo, Y. (2014). Computing projection depth and its associated estimators. Statistics and Computing 24 51–63.

Stahel, W.A. (1981). Robust estimation: infinitesimal optimality and covariance matrix estimators. Ph.D. thesis (in German). Eidgenossische Technische Hochschule Zurich.

Zuo, Y.J. and Lai, S.Y. (2011). Exact computation of bivariate projection depth and the Stahel-Donoho estimator. Computational Statistics and Data Analysis 55 1173–1179.

`ddalpha.train` and `ddalpha.classify` for application, `depth.projection` for calculation of projection depth.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# Generate a bivariate normal location-shift classification task # containing 20 training objects class1 <- mvrnorm(10, c(0,0), matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE)) class2 <- mvrnorm(10, c(2,2), matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE)) data <- rbind(class1, class2) # Get depth space using projection depth depth.space.projection(data, c(10, 10), method = "random", num.directions = 1000) depth.space.projection(data, c(10, 10), method = "linearize") data <- getdata("hemophilia") cardinalities = c(sum(data\$gr == "normal"), sum(data\$gr == "carrier")) depth.space.projection(data[,1:2], cardinalities) ```