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M-exponential equation and vegetation dynamic analysis
Dear friends,
The multidimensional sphere model (MDSM) has been developed a lot, as well
as my expression (both in English and math), since I posted it in email. I
have learned much more from the email discussion than my presentations
that I made to Pacific Science Congress, ESA, SRM, etc. Thanks for this
new technology and your helps, the MDSM research is in her last phase and
the first paper in English has been published (Ecological Modelling
97/1-2, pp.75-86).
The new development of the MDSM is a multi-variate exponential equation.
My poster "M-exponential equation and vegetation dynamic analysis" is
arranged on Thursday August 14, in ESA 82. From my experience, most of you
may be gone by then. Thus, I am posting it here again and looking for your
comments and help. For the netters who are not interested in this
research, please discard these emails and excuse me for posting.
Thank you for your attention,
T. Jay Bai, Ph.D.
Quantitative Ecologist
MDSM Research
P.O.Box 272628
Fort Collins, CO 80527
http://lamar.colostate.edu/~jbai
M-exponential Equation and Vegetation Dynamic Analysis
-Testing MDSM with NYSE data-
T. Jay Bai, Ph.D.
MDSM Research, March, 1997
Abstract
The Multi-Dimensional Sphere Model (MDSM) is a new data analysis method
based on vector directions. It uses multi-component vectors, m-vectors, or
the points in multi-dimensional space, m-space, to represent vegetation
and uses the angles between vectors to express the relation between the
corresponding vegetation. The MDSM can be applied to vegetation time
serial analysis, and express the vegetation dynamics as an m-exponential
equation:
y(i,k)=(1-alpha)y(i,0)*t(i,0)^k + alpha*d(i,k), i=1,2,...m, k=-1,0,1,..o;
alpha=0, when k<1, 1=>alpha=>0, when k=>1.
The y is the dependent variable of vegetation that made of m plant
species. The first subscript, i, is the index of species. The second, k,
is the index of time and the independent variable. The y, t, and d are
m-vectors. The d is the observation or sampled data. The t, the
Multivariate Instantaneous Trend, is calculated from given and previous
vegetation:
t(i,k)=y'(i,k)/y'(i,k-1) , where y'(i,k)= y(i,k)/sqrt(sum(y(i,k)^2)) is
the projection of the vegetation, y(i,k), from m-space onto the unit
hypersphere, i=1,2,...m.
The model was applied to a sample data of five variables with three time
intervals (alpha=0) from New York Stock Exchange (NYSE) to generate the
Trends. The trends were then tested by a simulated investment. The results
show that a higher trend indicates a higher return. Further more, a trend
of longer-term was obtained by the production of sequential shorter term
trends. The function of the trend, t, can be paralleled as the tangent
vector to the unit hypersphere in a linear model. However, by definition,
instantaneous trend is the ratio of the states, instead of ratio of
differences. Some other potential applications of the m-exponential
equations: trend analysis, prediction, sustainable development, and system
monitoring were also briefly introduced.
Key Words:
Vector analysis, m-vector, m-space, m-exponential equation, instantaneous
trends, m-unit sphere, MDSM, dynamic analysis, multivariate time series.