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Summary of the MDSM seminar discussion
Dear Friends,
In last annual meeting during the ESA1996, it was discussed that the statistic section would organize a seminar on the theme on "spacial and temporal dynamics" in ESA 1998 meeting. Myself and two other scientists volunteered our time to do some preparing work for the forum. As part of the effort, I start posting some of my research "M-exponential equation and vegetation dynamic analysis" to this list.
As a paper on multidimensional sphere model has been published, the MDSM is not my own idea anymore, but a public available resource. It may be true that not all of the netters in this list are interested in vegetation dynamics, or quantitative methods. In that case, please excuse me and ignore the mails from T. Jay Bai.
Thank you very much.
T. Jay BAI, Ph.D.
Quantitative Ecologist
MDSM Research
POBox 272628
Ft. Collins, CO 80527
Tel: (970)490-8370
http://lamar.colostate.edu/~jbai
Attachment
A few questions that were discussed in the Multi-Dimensional Sphere Model seminar on Monday, June 14, 1997 at Natural Resource Building 100, Colorado State University.
1. How does this model work with climate change? The Multi-Dimensional Sphere Model (MDSM) is designed for discovering vegetation response to climate change. This model by itself can not find climate change. For example, if this year is a wet year, and, consequently, each species increased exactly 2.0-fold from last year, then the model would indicate no change in species relative composition. In stock market example, if the federal government adjusted the interest rate so that all five stocks increased 2.0-fold, then there is no difference between the five stocks. Any of them will give you the 200% return. However, if the species responded to the climate change differently, then this model can discover the differences. In the stock market example, if five stocks increase 220%, 210%, 200%, 190%, and 180% of their initial values, then this model can describe their different responses to the interest adjustment. The MDSM acts as a filter for outside influences and describes only the relative changes inside the systems.
2. How does the model capture changes affected by different treatments? This can be done by trend analysis with blocks. If the data were blocked by two different treatments, and the trends from different blocks were different, then the difference of trends could be attributed to the difference of treatments. For example, if the trend of a land with tank training is different from land with no training, then the effect of tanks on vegetation may be discovered.
3. How does this model handle sampling errors? The model does not handle sampling errors in the case of trend analysis. It assumes that the closing prices ARE representative of the stock market. Whether the closing prices are the true representatives or not is beyond this paper's discussion. (This model, however, can handle variances in clustering analysis.)
4. How can this model help a land manager? The ShangGao Index rate [SGI% (k)=SGI (k)/SGI(k-1)], or trend index will tell the manager the overall condition of the land. When subdivisions are labeled with SGI%, the manager can see what trend happened to which subdivision on his land. A pseudo trend index was applied (Bai and Linn, 1994). Furthermore, this model can be used for clustering analysis. A result of vegetation classification was amazing (1995).
5. Comparison of linear regression, Y(x)=Y(0)+T*X (here T is the slope, or tangent), and exponential equation, Y(x)=Y(0)*T^X (here T is the instantaneous trend). The linear model can fit any curve. The exponential model is most appropriate for biology, as living organisms have a special feature, i.e., replication, offspring producing offspring. Similarly, in economics, interest produces interest. The exponential model can be fit to linear data, provided T is close to one or the time interval is small enough. Therefore, it can fit any curve, too. The two models are mathematically convertible under certain conditions (Ma, 6/14/97). Furthermore, the exponential model can be used to explain species explosion, system crash, etc., but the linear model could not. The most important feature of MDSM which distinguishes it from other models is that, MDSM can handle as many variables as desired at the same time.
6. Sigmoid growth and exponential growth. Sigmoid growth is a variation of exponential growth. A typical formula for sigmoid growth is Y(x+1)=Y(x)*e^{r[(C-Y(x))/C]}, where, C is the environment carrying capacity, but e^r is intrinsic rate (Vandermeer, 1981). While comparing it to an exponential growth, Y(x+1)=Y(x)*T(x), where T is the instantaneous trend, people may find that C>Y(x) is parallel to T(x)>1, C=Y(x) is parallel to T(x)=1, and C<Y(x) is parallel to T(x)<1. As MDSM handles multiple dimensions, the ecosystem described by MDSM is exhibiting sigmoid growth continuously in different dimensions.
7. Validation. This model can be validated by comparison of the two correlation. One is random correlation between sequential samples, but the other is between the prediction and expectation. MDSM uses cos<D(k), D(k-1)> to express the random correlation, and uses cos<P(k),E(k)> to express fitness of prediction [where, P(k) =Y(k-1) *T(k-1) ], and E(k) =[P(k)*alpha + D(k) *beta]/(alpha+beta). The System Monitoring Coefficient (SMC) is defined as cos<PE>/cos<DD>. When SMC is greater or equal to one, then the prediction error is equal or less than random error, indicating that using model is better than no model. The use of SMC is a new feature of MDSM for this year. Previously, the cos<PE> was used for measurement of the prediction fitness as presented at ESA 1996 annual meeting. Also, the SGI% is a new development of this model (Bai, 1997).