Weighted Procrustes analysis to register landmark configurations into optimal registration using translation, rotation and scaling. Registration without scaling is also an option. Also, obtains principal components, and some summary statistics.

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`x` |
Input k x m x n real array, where k is the number of points, m is the number of dimensions, and n is the sample size. |

`fixcovmatrix` |
If FALSE then the landmark covariance matrix is estimated. If a fixed covariance matrix is desired then the value should be given here, e.g. fixcovmatrix=diag(8) for the identity matrix with 8 landmarks. |

`initial` |
The initial value of the estimated covariance matrix. "Identity" - identity matrix, "Rawdata" - based on sample variance of the raw landmarks. Also, could be a k x k symmetric positive definite matrix. |

`maxiterations` |
The maximum number of iterations for estimating the covariance matrix |

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`scale` |
Logical quantity indicating if scaling is required |

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`reflect` |
Logical quantity indicating if reflection invariance is required |

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`prior` |
Indicates the type of prior. "Exponential" is exponential for the inverse eigenvalues. "Identity" is an inverse Wishart with the identity matrix as parameters. |

`diagonal` |
Logical. Indicates if the diagonal of the landmark covariance matrix (only) should be used. Diagonal matrices can lead to some landmarks having very small variability, which may or may not be desirable. |

`sampleweights` |
Gives the weights of the observations in the sample, rather than the landmarks. This is a fixed quatity. "Equal" indicates that all observations in the sample have equal weight. The weights do not need to sum to 1. |

The factored covariance model is assumed: $Sigma_k x I_m$ with $Sigma_k$ being the covariance matrix of the landmarks, and the cov matrix at each landmark is the identity matrix.

A list with components

`k` |
no of landmarks |

`m` |
no of dimensions (m-D dimension configurations) |

`n` |
sample size |

`mshape` |
Weighted Procrustes mean shape. |

`tan` |
This is the mk x n matrix of Procrustes residuals $X_i^P$ - Xbar. |

`rotated` |
the k x m x n array of weighted Procrustes rotated data |

`pcar` |
the columns are eigenvectors (PCs) of the sample covariance Sv of tan |

`pcasd` |
the square roots of eigenvalues of Sv using tan (s.d.'s of PCs) |

`percent` |
the percentage of variability explained by the PCs using tan. |

`size` |
the centroid sizes of the configurations |

`scores` |
standardised PC scores (each with unit variance) using tan |

`rawscores` |
raw PC scores using tan |

`rho` |
Kendall's Riemannian distance rho to the mean shape |

`rmsrho` |
r.m.s. of rho |

`rmsd1` |
r.m.s. of full Procrustes distances to the mean shape $d_F$ |

`Sigmak` |
Estimate of the sample covariance matrix of the landmarks |

Ian Dryden

Dryden, I.L. and Mardia, K.V. (1998). Statistical Shape Analysis, Wiley, Chichester.

Goodall, C.R. (1991). Procrustes methods in the statistical analysis of shape (with discussion). Journal of the Royal Statistical Society, Series B, 53: 285-339.

procGPA

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Questions? Problems? Suggestions? Tweet to @rdrrHQ or email at ian@mutexlabs.com.

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