Description Usage Arguments Details Value Author(s) References Examples

Majorisation-Minimisation based Estimation for Reduced-Rank Regression with a Cauchy Distribution Assumption.
This method is robust in the sense that it assumes a heavy-tailed Cauchy distribution
for the innovations. This method is an iterative optimization algorithm. See `References`

for a similar setting.

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`y` |
Matrix of dimension N*P. The matrix for the response variables. See |

`x` |
Matrix of dimension N*Q. The matrix for the explanatory variables to be projected. See |

`z` |
Matrix of dimension N*R. The matrix for the explanatory variables not to be projected. See |

`mu` |
Logical. Indicating if a constant term is included. |

`r` |
Integer. The rank for the reduced-rank matrix |

`itr` |
Integer. The maximum number of iteration. |

`earlystop` |
Scalar. The criteria to stop the algorithm early. The algorithm will stop if the improvement
on objective function is small than |

`initial_A` |
Matrix of dimension P*r. The initial value for matrix |

`initial_B` |
Matrix of dimension Q*r. The initial value for matrix |

`initial_D` |
Matrix of dimension P*R. The initial value for matrix |

`initial_mu` |
Matrix of dimension P*1. The initial value for the constant |

`initial_Sigma` |
Matrix of dimension P*P. The initial value for matrix Sigma. See |

`return_data` |
Logical. Indicating if the data used is return in the output.
If set to |

The formulation of the reduced-rank regression is as follow:

*y = μ +AB' x + D z+innov,*

where for each realization *y* is a vector of dimension *P* for the *P* response variables,
*x* is a vector of dimension *Q* for the *Q* explanatory variables that will be projected to
reduce the rank,
*z* is a vector of dimension *R* for the *R* explanatory variables
that will not be projected,
*μ* is the constant vector of dimension *P*,
*innov* is the innovation vector of dimension *P*,
*D* is a coefficient matrix for *z* with dimension *P*R*,
*A* is the so called exposure matrix with dimension *P*r*, and
*B* is the so called factor matrix with dimension *Q*r*.
The matrix resulted from *AB'* will be a reduced rank coefficient matrix with rank of *r*.
The function estimates parameters *μ*, *A*, *B*, *D*, and *Sigma*, the covariance matrix of
the innovation's distribution, assuming the innovation has a Cauchy distribution.

A list of the estimated parameters of class `RRRR`

.

- spec
The input specifications.

*N*is the sample size.- history
The path of all the parameters during optimization and the path of the objective value.

- mu
The estimated constant vector. Can be

`NULL`

.- A
The estimated exposure matrix.

- B
The estimated factor matrix.

- D
The estimated coefficient matrix of

`z`

.- Sigma
The estimated covariance matrix of the innovation distribution.

- obj
The final objective value.

- data
The data used in estimation if

`return_data`

is set to`TRUE`

.`NULL`

otherwise.

Yangzhuoran Yang

Z. Zhao and D. P. Palomar, "Robust maximum likelihood estimation of sparse vector error correction model," in2017 IEEE Global Conferenceon Signal and Information Processing (GlobalSIP), pp. 913–917,IEEE, 2017.

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