From:    Jay Bai 
Subject: Vegetation Analysis

Dear Colleagues,

I would like to share the following information with vegetation scientists and
quantitative ecologists, as well as to look for more comments.

T. Jay BAI, Ph.D.
Quantitative Ecologist
MDSM Research
PO Box 272628
Fort Collins, CO 80527
970-490-8345
Bai@gpsr.colostate.edu
http://lamar.colostate.edu/~jbais/dbsearch.html
=============================
Vegetation Science and Vector Analysis

Abstract
Vegetation is a community of plants. Communities are natural phenomena
different from systems and individuals. A community has composition, and the
composition is an essential for vegetation. Vegetation can be described by
multi-component vectors. The magnitude of the vector expresses the mass of the
vegetation, while the direction of the vector expresses the composition of the
vegetation. An m-vector is different from an m*1 matrix. M-vectors have
directions, and the directions are essential for the vectors. M-vector can be
used for vegetation classification and vegetation dynamic analysis.


I posted a commentary entitled "A Critique of Matrix Solutions for Ecology" in
Bulletin of the Ecological Society of America, on April, 1998. Since then, I
have received several responses about the commentary. One response suggested
that a vector is a kind of matrix and that I seemed to be inflating the
importance of the vector.  (One friend wrote that he learned about the vector
in linear algebra, and said that vector is just a kind of matrix. He asked why
I made a big deal about vectors?) My answer is that the distinction between
vectors and matrices is important because the m*1 matrix is NOT an m component
vector (m-vector). Although both m*1 matrix and m-vectors are m component
arrays and an m-vector can be treated as m*1 matrix some times, the m*1 matrix
is NOT an m-vector. In fact, an m-vector is more than an m*1 matrix. Vectors
not only have the magnitude, the same as a matrix, but also have a direction,
and are thus different from a matrix. According to the Grolier Multimedia
Encyclopedia, 1995, a vector is a quantity that has magnitude and direction. In
my official language, Chinese, a vector is called XIANG LIANG, which means
direction ( i.e., XIANG), and magnitude ( i.e., LIANG). It is the direction
that makes vectors different from other quantities.

Here is an example of what m-vector direction means to vegetation science. In
vegetation science, we often consider the composition of the vegetation to mean
more than the total biomass (Bai, 1984). Imagine, for example, a three
dimensional space (3-space), where the three dimensions are trees, shrubs, and
grasses. In this 3-space, point A=(3, 1, 0) is different from point B=(0, 1,
3), although the two have the same magnitude equal to square root of 10. How
does a vector distinguish between those when their magnitudes are the same?
(What makes the difference when the magnitudes of the two are the same?) The
answer is the position of the points in 3-space, in other words, the DIRECTION
of the 3-vectors in the 3-space. The position of point A, or direction of the
vector OA (where O is the origin, and A is the point in space, see Fig 1.) is
closer to the first axis of tree. Consequently, A can be classified as a
woodland. On the other hand, B is closer to the third axis (grass), and may be
classified as grassland. The two 3-vectors, A=(3, 1, 0) and B=(0, 1, 3),
represent different composition and are different types of vegetation. For the
same reason, point 2A=(6, 2, 0) in the same 3-space may be classified as
woodland the same as A=(3, 1, 0), since A and 2A have the same composition
ratio (Fig 1). A and 2A have a difference in magnitude. This difference,
however, is not essential to separate the two. This difference maybe caused by
different sampling areas. For example, 2A might have two times of sampling area
as A did. This is a minor factor, and would not change the vegetation
classification. What is essential to vegetation science is whether A and 2A
have the same composition; or whether A and 2A located in the same ray in
multi-species space(m-space) (Gauch, 1982). In other words, whether OA and O2A
(2A was the point, O was the origin) have the same direction in the 3-space.
Generally speaking, all the points representing the same vegetation in m-space
are located on the same ray. In other words, all m-vectors representing the
same vegetation in m-space have the same direction. Just as composition is
essential to vegetation science, direction is essential to a vector. Different
directions mean different vectors, and the same direction mean the same vector
in our case. Furthermore, any composition change in vegetation can be expressed
by changes in vector direction. Therefore, vegetation dynamics, i.e. changes in
composition, can be expressed by vector rotation in m-space. We can monitor
vegetation change by monitoring the rotation of a state vector in m-space (Bai,
et al. 1997).

The direction of an m-vector in m-space can be expressed by vector's m-cosine
values (Bai, 1982, Gauch, 1982, Jongman, et. al., 1995):

Direction A = cosine A(i)= A(i)/|A(i)|, i=1,2,..m,

where |A(i)| is the magnitude of the vector, the square root of the sum of the
squares. For example, the vector OA and O2A in above can be expressed as

A'=(6/SQRT(40), 2/SQRT(40), 0/ SQRT (40))
= (3/ SQRT (10), 1/ SQRT (10), 0/ SQRT (10)).

The cosine  =cosine <2A> =cosine  is also a 3-vector, when A is a
3-vector. However, the relation between two vectors is expressed by a scalar
which is the cosine value between the two vectors. For example, the cosine = 1, but the cosine = 1/10=0.1.

Using m-vector and m-space, we can build a standardized vegetation
classification system. This systems can accept as many species as we want, and
can handle as many samples as we have. The differences of sampling size, such
as 10 m^2 vs. 100 m^2, sampling shape, such as line vs. square, and
measurement, such as, weight vs. cover, would be filtered out, but only remain
the information of the composition of the vegetation.

As different vectors can have the same magnitude, magnitude by itself tells us
little about the composition of the community. On the other hand, if we were
provided direction, we can determine any individual component and the vector
magnitude given only a single component. For example, if we know the type of
vegetation was woodland, and the components (element) of the vegetation were 3,
1, and 0, as discussed above, then given a shrub value of 2, we can project the
vegetation as (6, 2, 0), and the magnitude of the vegetation would be
calculated as the square root of 40. This is the principle that
Multi-Dimensional Sphere Model (MDSM) uses for community dynamic analysis. For
more detailed information, interested readers can visit the web site:
http://lamar.colostate.edu/~jbais/dbsearch.html

Conclusion:
Vegetation occurs as communities, and communities can be accurately expressed
as m-vectors. An m-vector is fully described by it' magnitude and direction.
The magnitude of a vector expresses the amount of materials, energy, and/or
information that the community contains, while the direction expresses the
distribution of these materials, energy, and/or information among the
components. The direction of a vector can be expressed by cosine values. The
cosine value between two vectors expressing their relation is a scalar.
However, the cosine values associated with each of the m axes expressing
vector's direction in m-space is itself an m-vector. More specifically, it is a
point on the m-unit hypersphere.

Reference:

Bai, T. J., 1998, A Critique of Matrix Solution for Ecology. Bulletin of the
Ecological Society of America, Vol. 79, No.2.

Bai, T. J., T. Cottrell, D.Y. Hao, T. Te, 1997: Multi-Dimensional Sphere Model
and vegetation instantaneous trend analysis. Ecological Modelling, Vol.97
No.1-2, pp75-86

Bai, T. J, 1984, A digital prediction of the succession trend of grassland,
Grassland Research Institute of China, Hohhot, China (in Chinese).

Bai, T., 1982, Exploration of Numerical Classification of Form Leymus Chinensis
in Silingolo River Basin. Inner Mongolia University, Vol.4, pp. 123-132 (in
Chinese with English abstract).

Gauch, H. G., 1982, Multivariate analysis in community ecology. Cambridge
University Press, pp. 117.

Grolier Multimedia Encyclopedia, 1995.

Jongman, R. H. G., C.J.F.Ter Braak & O.F.R. VanTongeren, 1995, Data Analysis in
Community and Landscape Ecology. Cambridge University Press, pp. 178.

Fig 1 (omitted from email)

Acknowledgement:
Thanks to H. S and T. C they prove red and edited the English of an earlier
version.