landmark_mds {bigmds} | R Documentation |

## Landmark MDS

### Description

Landmark MDS (LMDS) algorithm applies first classical MDS to a
subset of the data (*landmark points*) and then the remaining individuals are
projected onto the landmark low dimensional configuration using a
distance-based triangulation procedure.

### Usage

```
landmark_mds(x, num_landmarks, r)
```

### Arguments

`x` |
A matrix with |

`num_landmarks` |
Number of landmark points to obtain an initial MDS configuration. It is
equivalent to |

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

### Details

LMDS applies first classical MDS to a subset of the data (*landmark points*). Then,
it uses a distance-based triangulation procedure to project the non-landmark individuals. This
distance-based triangulation procedure coincides with *Gower's interpolation formula*.

This method is similar to `interpolation_mds()`

and `reduced_mds()`

.

### Value

Returns a list containing the following elements:

- points
A matrix that consists of

`n`

points (rows) and`r`

variables (columns) corresponding to the principal coordinates. Since a dimensionality reduction is performed,`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.

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

De Silva V. and JB. Tenenbaum (2004). *Sparse multidimensional scaling using landmark points*. Technical Report, Stanford University.

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 <- landmark_mds(x = x, num_landmarks = 200, r = 2)
head(mds$points)
mds$eigen
```

*bigmds*version 3.0.0 Index]