Simulate data for Reduced-rank regression. See `Detail`

for the formulation
of the simulated data.

RRR_sim( N = 1000, P = 3, Q = 3, R = 1, r = 1, mu = rep(0.1, P), A = matrix(rnorm(P * r), ncol = r), B = matrix(rnorm(Q * r), ncol = r), D = matrix(rnorm(P * R), ncol = R), varcov = diag(P), innov = mvtnorm::rmvt(N, sigma = varcov, df = 3), mean_x = 0, mean_z = 0, x = NULL, z = NULL )

N | Integer. The total number of simulated realizations. |
---|---|

P | Integer. The dimension of the response variable matrix. See |

Q | Integer. The dimension of the explanatory variable matrix to be projected. See |

R | Integer. The dimension of the explanatory variable matrix not to be projected. See |

r | Integer. The rank of the reduced rank coefficient matrix. See |

mu | Vector with length P. The constants. Can be |

A | Matrix with dimension P*r. The exposure matrix. See |

B | Matrix with dimension Q*r. The factor matrix. See |

D | Matrix with dimension P*R. The coefficient matrix for |

varcov | Matrix with dimension P*P. The covariance matrix of the innovation. See |

innov | Matrix with dimension N*P. The innovations. Default to be simulated from a Student t distribution, See |

mean_x | Integer. The mean of the normal distribution \(x\) is simulated from. |

mean_z | Integer. The mean of the normal distribution \(z\) is simulated from. |

x | Matrix with dimension N*Q. Can be used to specify \(x\) instead of simulating form a normal distribution. |

z | Matrix with dimension N*R. Can be used to specify \(z\) instead of simulating form a normal distribution. |

A list of the input specifications and the data \(y\), \(x\), and \(z\), of class `RRR_data`

.

- y
Matrix of dimension N*P

- x
Matrix of dimension N*Q

- z
Matrix of dimension N*R

The data simulated can be used for the standard reduced-rank regression testing
with the following formulation
$$y = \mu +AB' x + D z+innov,$$
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,
\(\mu\) 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 simulates \(x\), \(z\) from multivariate normal distribution and \(y\) by specifying
parameters \(\mu\), \(A\), \(B\), \(D\), and \(varcov\), the covariance matrix of
the innovation's distribution. The constant \(\mu\) and the term \(Dz\) can be
dropped by setting `NULL`

for arguments `mu`

and `D`

. The `innov`

in the argument is
the collection of innovations of all the realizations.