Page 381 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

356 HUMAN PERFORMANCE IN MOTION PLANNING

Let X be the column matrix of the original sample data; its element x ij is
the value of the jth sample variable (the jth column vector of X)for the ith
observation row vector (the ith subject, ith row of X). Denote the covariance
matrix computed from the sample matrix X by R. Let matrix A contain as
its column vectors the eigenvectors of matrix R:The ith column of A is the
ith principal component of R.Both R and A are square matrices of the same
rank (normally equal to the number of sample variables). Matrix A can then
be seen as a transformation matrix that relates the original data to the principal
components: Each original data point (described by a row vector of X) can now
be described in terms of new coordinates (principal components), as a row vector
z; hence
Z= X*A (7.1)

Let matrix be a diagonal matrix of the same rank, with eigenvalues of R in
its diagonal positions, ordered from largest to smallest.
The matrix of principal component loadings, denoted by L,is

L=A* 1/2 (7.2)
An element l ij of the square matrix L is the correlation coefficient between the ith
variable and the jth principal component. It informs us about the “importance,” or
“contribution,” of the ith variable to the jth principal component. Geometrically,
the loading is the projection of ith variable onto the jth component.
If only independent variables represented by the interface and visibility factors
are considered in our arm test, this will include four out of the eight tasks listed
in Section 7.3.1:

virtual–visible
virtual–invisible
physical–visible
physical–invisible

The loading matrix of the corresponding four principal components in the arm test
is shown in Table 7.2. As seen in the table, the first PC (principal component)
accounts for 36.3% of the total variance (top four numbers in column 1) and
is tied primarily to the virtual–visible and physical–invisible tasks (0.679 and
0.695 loads, accordingly). The second PC accounts for the next 25.6% of the
total variance and is tied primarily to the other two tasks, virtual–invisible and
physical–visible (loads 0.666 and 0.739), and so on. Since the contribution of
successive PCs into the total variance falls off rather smoothly, with 85.9% of
the variance being accounted for by the first three PCs, we conclude (somewhat
vaguely) that to a large extent the four tasks measure something different, each
one bringing new information about the subjects’ performance, and hence cannot
be replaced by a smaller number of “hidden factors.”