Marti J. Anderson and Angus A. Thompson. 2004. Multivariate control charts for ecological and environmental monitoring. Ecological Applications 14:1921–1935.

Appendix A. Derivation of a multivariate control chart criterion using interpoint distances. A pdf file of this appendix is also available.

Let  be the ith observation in a set of i = 1, …,t points on variable k for k = 1, …, p variables and let  and . We wish to obtain the following criterion:  using only inter-point distances.

First, define two quantities which can be obtained using only inter-point distances: the sum of squared distances from the points to their centroid for the first (t – 1) points:

(A.1)

and for all t points:

.

(A.2)

Now, we can write:

(A.3)

and expanding the square this gives:

.

(A.4)

Recognizing the first term in the rhs of the expression, we have:

.

(A.5)

Next, we can simplify the following expression  to

,

(A.6)

and plugging this quantity where appropriate into Eq. A.5, we have

(A.7)

which, rearranging, gives

(A.8)

Q.E.D.