Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 76

01 1
10 FM () . 1

1 1 41 . 0

where columns represent landmarks 1, 2, and 3 read from left to right,
and rows represent landmarks 1, 2, and 3 read from top to bottom. The
matrix is square because the number of columns is equal to the number
of rows. Each cell represents the distance between the landmark that
heads a row and the landmark that heads a column. The cell in the
upper left-hand corner represents the linear distance between land-
mark 1 and itself. The second cell in the first row represents the
distance between landmarks 1 and 2, while the third cell in the first row
represents the distance between landmarks 1 and 3. Notice that the
entries on the diagonal are all 0 because the distance between a land-
mark and itself is zero. The matrix is symmetric, meaning that the
entry for the ( i, j )-th cell is equal to the entry for the ( j, i ) - t h cell. The
first cell in the second row represents the distance between landmarks
2 and 1, which is equal to the value entered in the second cell in the first
row, the distance between landmarks 1 and 2.
Since this matrix is symmetric, we need not write it in full matrix
format but can abbreviate it by collecting only the above-diagonal ele-
ments and writing those numbers as a vector.

When written as a vector, the elements to the right of the first diago-
nal element in row 1 are written in order first, followed by the
elements to the right of the diagonal in row 2 and so on. Unless we
specify otherwise, the form matrix is always written as a vector.

3.7 The ability to estimate the mean form and variance

In this section, we provide a verbal description of two key results
regarding the estimation of the mean form and variance. These results
are proven in Part 2 of this chapter. The discussion in this section is