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?N-exponential equation and vegetation dynamic analysis?

Dear friends,

I am posting the following abstract to this list for comments on
science and corrections on language, grammar or vocabulary.

The n-exponential equation was used on vegetation dynamic analysis. 
Compare to the classic exponential growth equation, i.e.,
Y(x)=Yo*e^rx, the n-exponential equation has three main improvements:
1. The equation can be used on multivariate cases, as y_i,k_, d_i,k_,
and t_i,k, are n-vectors.
2. It uses a multivariate instantaneous changing trend, t_i,k_, to
replace the intrinsic growth rate, e^r.
3. It add an observation, d_i,k+x_, at time k+x to adjust the
projection.  By adding d, the new sampled data was included in the
calculation of the next trend, t_i,k+x_, so that the trend vector can
be updated every time. It also blocks the exponential growth between k
and k+x, to prevent system from collapse. 

The file was in ASCII form: 
_i,k_                     = subscript of i and k
^x^                       = power of x
sigma(d_i,k_), i=1,2,...n = sum of d over i from 1 to n
sqrt(d)                   = square root of (d)

Any comments or corrections are welcome, especially, the comments or
corrections made before 31 will be highly appreciated.

Thank you in advance.

Jay Bai
================================

         N-exponential equation and vegetation dynamic analysis
                               (abstract)
          By Tugjayzhab Jay Bai, MDSM Research, P.O.Box 272628
                       Fort Collins, CO 80527, USA
                                    
The Multi-Dimensional Sphere Model (MDSM) is a new data analysis
method based on multi-variable space, n-space. It uses n-vector, a
point in n-space, to represent vegetation, and uses angles between the
vectors to express the relations between the vegetation. The MDSM can
be applied to vegetation dynamic analysis. The vegetation dynamics,
generally, can be expressed as an n-exponential equation:

     y_i,k+x_=y_i,k_*t_i,k_^x^ + d_i,k+x_, i=1,2,...n.

Where x is the independent variable of time, and y is the dependant
variable of vegetation. The first subscript, i, is index numbering
species from 1 to n, and the second, k, as well as k+x, is index of
time. The x and k are scalars, while y, t, and d are n-vectors.  Thus,
the y_i,k+x_ and y_i,k_, i=1,2,...n, are representing the states of
vegetation at time k+x and k, respectively, while d_i,k+x_,
i=1,2,...n, is the observation, or sampled data, at
time k+x. The key parameter in this equation, t, is the multivariate
instantaneous trend. MDSM determines it's value from existing data:

     t_i,k_=y'_i,k_/y'_i,k-1_ , i=1,2,...n.

Where y'_i,k_=y_i,k_/=sqrt(sigma(y_i,k_^2^)), i=1,2,...n, is the
projection onto the unit hypersphere of the vegetation y_i,k_ from
n-space. The instantaneous trend, t, can be imagined as tangent
vector.

For most long term vegetation monitoring programs, where the
vegetation was sampled every year, x=1, the n-exponential equation can
be used as a step by step vegetation monitoring model:

     y_i,k+1_=y_i,k_*t_i,k_ + d_i,k+1_, i=1,2,...n.

Where, y_i,k_*t_i,k_=p_i,k+1_ is the projection for the next year
based on given information, but d_i,k+1_ is sampled data at the next
year. The t_i,k_=[y_i,k_/sqrt(sigma(y_i,k_^2^))]:
[y_i,k-1_/sqrt(sigma(y_i,k-1_^2^))], i=1,2,...n, can be interpreted as
"changes of relative coverage by species over year" for vegetation
science. Furthermore, the projection and observation can have
different weighing factors based on their reliabilities:
y_i,k+1_=alpha*p_i,k+1_+beta*d_i,k+1_, where alpha+beta=1. If
variances of projection and observation are not available, then
alpha=beta=0.5.

The model was applied to the data from land condition trend analysis
program and it generated a extremely satisfied results.
 
Key words: Multi-Dimensional Sphere Model, n-space, n-vector, n-
exponential equation, multivariate instantaneous trend, tangent
vector, vegetation dynamic analysis, monitoring, ...