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A Critique of Matrix Solutions for Ecology
Dear Colleague,
Since I posted my "Re: A Critique for Ecology" on Ecol_l, I received a
few responses, including a few papers using matrix conducting time
series analysis. I am replying my general and simplified comments
explaining more about that "Matrix generally does not have division or
inverse, at least for ecology".
I am considering to post it in Ecol-l or to submit to Bulletin of ESA
to share with more colleagues, as well as seeking for more comments.
As this is so important that I would like to have a few different
opinions before I go any further. Would you please correct me if there
is any mistake in the following messages.
If this is conflict with your interest, please excuse me.
Thank you in advance.
T. Jay Bai
A Critique of Matrix Solutions for Ecology
There is no definite answer for the equation AX=B, where A and B are
two known sequential states of an ecosystem, and X is unknown m*m
transition matrix.
1). When A and B are m-vectors, and X is m*m matrix, since the number
of unknown components, m^2, is greater than the number of equations,
m, there are many answers, i.e. no definite answer. (The definite
solution for AX=B is diagonal matrix, because of |X(i,i)|>|X(i,j)|,
as replicate is the essence of biology. However, a diagonal matrix is
nothing else than a vector.)
2). When A and B are matrices, there is no inverse for matrix A. There
is no solution for X=B/A. Furthermore, if there is no inverse for A,
then there won't be an inverse in the forms of AA', AYA', or YAY',
either. Though these solutions may be mathematically sound, they won't
work for ecology.
The main reason that there is no inverse of a variable*sample matrix
for ecology is that the rank of the matrix is not full. As long as the
samples were collected from a homogeneous vegetation, then even if
they were sampled randomly, there would not be enough independent
vectors in the matrix. All the samples would be varieties of the
centroid vector. If the sampling variances were considered, then there
would be only one independent vector in the matrix. Furthermore, if
the matrix was not full rank, then there would not be a transformation
that could increase the rank.
This suggests that matrix solutions may be a dead end for dynamic
analysis in ecology. I am sorry to be the one delivering this negative
message. I hope some one in this listserve can correct me. Otherwise,
we may have to be very careful in using of matrices for temporal
dynamic analysis for ecosystem/community.
=====================================
The previous mail is attached here for the readers who missed it last
time:
A Critique for Ecology (Posted on Ecol-l, Oct. 20)
Hi, colleagues,
This discussion is interesting and important. Here is my two cents.
One of the reasons that ecologists have not been able to make
predictions is that we did not have a suitable tool. When we handle
every single variable, we can not find the rules behind the phenomena.
When we handle the community as a whole using matrices, the variables
can only be treated as linear (The formal name of matrix algebra is
linear algebra. It has addition and subtraction, but generally does
not have division or inverse, at least for ecology). But behaviors of
most of the variables in biology/ecology may not be linear. Now, the
situation may have changed for ecologists. If we treat our object as a
community/system using multi-component vectors and describe and
analyze them by applying a vector algorithm, then we may discover some
new rules for community/system. Then we can extend these rules to make
projections. This new data synthesis and analysis method based on
m-vectors, named MultiDimensional Sphere Model, was published recently
in Ecological Modelling, 97/1-2, pp.75-86.
In other words, we may adjust the exponential growth equation (as
exponential growth is the essence of biology), by replacing the
intrinsic rate with empirical rate and extending the mono-species
model to a multi-species situation, then we may be able to project
community/ecosystem dynamics. One of the experiments has been
conducting using stock market data (Using stock data can avoid the
sampling error discussion and satisfy the statisticians). The results
shown are promising for a multivariate exponential growth system.
Interested readers can visit the web site,
http://lamar.colostate.edu/~jbai.
************************************
T. Jay BAI, Ph.D.
Quantitative Ecologist
MDSM Research
P.O.Box 272628
Fort Collins, CO 80527
USA
Tel: 970-490-8345
FAX: 970-495-8310
email: jbai@lamar.colostate.edu
http://lamar.colostate.edu/~jbai
************************************