reduced_mds {bigmds} | R Documentation |

## Reduced MDS

### Description

A data subset is selected and classical MDS is performed on it to obtain the corresponding low dimensional configuration.Then the reaming points are projected onto this initial configuration.

### Usage

```
reduced_mds(x, l, r, n_cores)
```

### Arguments

`x` |
A matrix with |

`l` |
The size for which classical MDS can be computed efficiently
(using |

`r` |
Number of principal coordinates to be extracted. |

`n_cores` |
Number of cores wanted to use to run the algorithm. |

### Details

*Gower's interpolation formula* is the central piece of this algorithm
since it allows to add a new set of points to an existing MDS configuration
so that the new one has the same coordinate system.

Given the matrix `x`

with `n`

points (rows) and
and `k`

variables (columns), a first data subsets (based on a random sample)
of size `l`

is taken and it is used to compute a MDS configuration.

The remaining part of `x`

is divided into `p=({n}-`

`l`

)`/l`

data subsets (randomly). For every data point, it is obtained a MDS
configuration by means of *Gower's interpolation formula* and the first
MDS configuration obtained previously. Every MDS configuration is appended
to the existing one so that, at the end of the process, a global MDS
configuration for `x`

is obtained.

#'This method is similar to `landmark_mds()`

and `interpolation_mds()`

.

### Value

Returns a list containing the following elements:

- points
A matrix that consists of

`n`

individuals (rows) and`r`

variables (columns) corresponding to the principal coordinates. Since we are performing a dimensionality reduction,`r`

`<<k`

- eigen
The first

`r`

largest eigenvalues:`\lambda_i, i \in \{1, \dots, r\}`

, where each`\lambda_i`

is obtained from applying classical MDS to the first data subset.

### References

Delicado P. and C. Pachón-García (2021). *Multidimensional Scaling for Big Data*.
https://arxiv.org/abs/2007.11919.

Paradis E. (2018). *Multidimensional Scaling With Very Large Datasets*. Journal of Computational and Graphical Statistics.

Borg, I. and P. Groenen (2005). *Modern Multidimensional Scaling: Theory and Applications*. Springer.

Gower JC. (1968). *Adding a point to vector diagrams in multivariate analysis*. Biometrika.

### Examples

```
set.seed(42)
x <- matrix(data = rnorm(4 * 10000), nrow = 10000) %*% diag(c(9, 4, 1, 1))
mds <- reduced_mds(x = x, l = 200, r = 2, n_cores = 1)
head(mds$points)
mds$eigen
```

*bigmds*version 3.0.0 Index]