Morphometric Data - An Invariant Approach to Statistical Analysis of Shapes - page 59

son.” Consider the population of all individuals in which we are inter-
ested. Suppose we calculate the mean of the heights of all these
individuals. This mean is called the “population mean,” usually denot-
ed by ( mu ). Similarly the variance of the heights of all these
individuals is called the “population variance,” denoted by

( sigma
square ). Suppose further that, when we plot the histogram of the
heights of all the individuals in the population, it looks like a “bell
shaped curve.” Then we say that the population distribution is Normal.
Now suppose we select an individual from this population at random.
What can we say about the height of this individual? The height of this
individual is more likely to come from the “middle” of the distribution
near than the tails. In statistical terminology, this is expressed as X
~ N (

2

2 ) and we read it as “ X follows a normal distribution with mean
and variance

,

.” Suppose we select “n” different individuals ran-
domly from this population. Let X i denote the height of the i -th
individual. We say that X i ~ N (

2

2 ) are independent (because the indi-
viduals are selected randomly from the population) and identically
distributed (because they come from the same population) random
variables.
Let us now say that the experimenter used inches as the unit for
measuring height. The population values, also called the parameters,
and

,

, are in the units of inches and inches 2 respectively. Suppose
that by mistake the height of the 10th individual was measured and
reported in units of centimeters. Clearly we cannot say that X 10 ~ N
(

2

2 ) because the units are different. What we can say is that: X 10 ~ N
( 2.5

,

) 2 ) . We simply change the units for the mean parameter and
the variance parameter to match the units of the other measurement.
The moral of the story is that if we transform an observation, the dis-
tribution of that observation changes as well. We next generalize the
idea of transformation to Matrix Normal distribution and landmark
coordinate matrices.
Recall that a matrix Normal distribution is characterized by two
quantities, the mean matrix M and the variance-covariance matrix
V . Suppose that the covariance matrix V can be written in the
Kronecker product form, V

, (2.5

K D . Suppose we observe a landmark
coordinate matrix of an object and denote it by X . Suppose further that
X ~ N ( M,

K D ) . Below we summarize how translation and rotation
of the original object affects the distribution. (This is similar to the
transformation from inches to centimeters described in the previous
paragraph.)