Multi-Dimensional Sphere Model and Vegetation Instantaneous Trend Analysis
T. JAY BAI1, TOM COTTRELL2, DUN-YUAN HAO3, TALA TE4, and
ROBERT J. BROZKA5
(Ecological Modelling, 97/1-2, revised by author, for blog)
1 MDSMMT Research LLC, PO Box 80524-3864, Fort. Collins, CO 80524, USA;
2 Dept. of Biology, Luther College, Decorah, IA 52101, USA;
3 Dept. of Mathematics, University of Inner Mongolia, Huhhot, P.R.China;
4 Dept. of Physics, Normal University of Inner Mongolian, Huhhot, P.R.China, 5Research Scientist, Center for Ecological Management of Military Lands, Dept. of Forest Sciences, Colorado State University, Fort Collins, CO 80523, USA.
Telephone, fax, and email address of corresponding author, T.Jay Bai: 970/495-9716, 970/581-0253, mdsm95bai@gmail.com
Permanent mailing address: T. Jay Bai, 615 JoAnne St., Fort Collins, CO 80524, USA.
Abstract
The Multi-Dimensional Sphere Model (MDSM), a new method for multivariate instantaneous trend analysis, is introduced. The model handles three subscript data, Z(i,j,k), e.g., for vegetation analysis, i, j, and k are species, quadrats, and time, respectively. The MDSM uses species, or species groups, as dimensions of a multi-dimensional space, and uses quadrats as points (vectors) in the m-space. The quadrats are standardized to 1.0 by division by their vector length, i.e., the square root of the sum of the squares of the components of a quadrat, q'(i)=q(i) /Ö (å( q(i) 2)), i=1,2,..m. All quadrats are projected onto the unit hypersphere. This maintains the composition information of each species for every quadrat in the data set, and makes all quadrats comparable because their vector lengths equal 1.0. The MDSM groups and combines quadrats on the hypersphere based on the cosine values between the m-vectors to form a state vector, z', representing the vegetation, z'(i) = å q'(i,j) , i=1,2,..m, j=1,2,..n. When performing trend analysis, the MDSM defines the quotient of components of previous (k-1) and present (k) state vectors as an instantaneous trend at a given time. This is referred to as a trend vector, and describes vegetation composition change over time, t(k)=z'(k) /z'(k-1) . The components of a trend vector (here called the t-value of the species) carry information from both previous and present states for species and community. This trend can then be extended to project the future states of the vegetation, p(k+1) = z(k) *t(k) . The MDSM combines correlation analysis, cluster analysis, trend analysis, and prediction of future vegetation states, making it a powerful and promising multivariate analysis method. The model was tested with data from the Land Condition Trend Analysis program at Fort Carson in southeastern Colorado. The model shows promising results for vegetation trend analysis; however, geometric meaning of the vector quotient is not yet clear. To improve our understanding, comparison with an additive model and a validation analysis are needed.
Key Words Multi-dimensional space, m-space, vector analysis, hypersphere, multivariate time series, centralization, centroid vector, vector inverse, standardization, state vector, Importance Value, multivariate instantaneous trend, Trend analysis, trend vector, trend value, similarity coefficient.
https://blog.sciencenet.cn/home.php?mod=space&uid=333331&do=blog&id=449959
