Multi-dimensional sphere model and instantaneous vegetation trend analysis

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 quadrats as points (vectors) in the 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) √[Σ q 2 (i)] . 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 synthesizes the quadrats into state vectors representing the vegetation, z′ ( i) = Σ q′ ( i, j) . 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 predict 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.