# PCA for Tensor-Valued Observations

### Description

Computes the tensorial principal components.

### Usage

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### Arguments

`x` |
Numeric array of an order at least three. It is assumed that the last dimension corresponds to the sampling units. |

`p` |
A vector of the percentages of variation per each mode the principal components should explain. |

`d` |
A vector of the exact number of components retained per each mode. At most one of this and the previous argument should be supplied. |

### Details

The observed tensors (array) *X* of size *p_1 x p_2 x ... x p_r* measured on *N* units are projected from each mode on the eigenspaces of the *m*-mode covariance matrices of the corresponding modes. As in regular PCA, by retaining only some subsets of these projections (indices) with respective sizes *d_1, d_2, ... d_r*, a dimension reduction can be carried out, resulting into observations tensors of size *d_1 x d_2 x ... x d_r*. In R the sample of *X* is saved as an `array`

of dimensions
*p_1, p_2, ..., p_r, N*.

### Value

A list containing the following components:

`S ` |
Array of the same size as x containing the principal components. |

`U ` |
List containing the rotation matrices |

`D ` |
List containing the amounts of variance explained by each index in each mode. |

`p_comp` |
The percentages of variation per each mode that the principal components explain. |

`Xmu ` |
The data location. |

### Author(s)

Joni Virta

### References

Virta, J., Taskinen, S. and Nordhausen, K. (2016), Applying fully tensorial ICA to fMRI data, *manuscript under preparation*, **???**, ???–???.

### Examples

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