许多人也许还记得,或卷入了1997年在美国生态学会网站上展开的所谓生态学批判的讨论。讨论的中心议题是:“为什么生态学还没有能够预测未来”.我的讨论心得被刊登在 Bulletin of the Ecological Society of America , 79卷2期, April, 1998. 后来收录在我的专著趋势分析及其在生态股市中的应用。现翻译成中文，转到科学网，寻求知音的批评，指点和讨论。因为是我自己的文章,所以仍列为原创。转载时有更新。 生态学的矩阵解批判 美国生态学会网站上关于“生态学批判”的讨论很有意义。我以为所谓的“生态学还不能预测”的一个重要原因是我们没有合适的数学工具。当我们处理一个一个单个的变量，我们无法发现现象后面的真实规律；当我们把对象作为系统，群落，作为整体来处理，则只能作为线性变化处理。矩阵代数的正式名称是“线性代数”。它有加法和减法，但通常没有除法，没有逆，至少对生态学是如此。这个致命的弱点限制了它在生态学中，时间/空间动态分析中，的应用。 生态学系统演替的方程 AX = B 没有确定的矩阵解。 上式中， A 和 B 是生态系统/群落两连续已知状态， X 是我们欲求的 m* m 状态转移矩阵 。 下面，从两个方面讨论。 1) A 和 B 都是样本-变量矩阵。 矩阵 A 没有逆 ，所以描述生态系统变化的方程的解 X = B/A 不存在。而且，如果矩阵 A 没有逆，则其它的矩阵形式，如 AA', AYA' , 或 YAY , 也都没有逆。虽然，这种形式的解也许在数学上能成立，但在生态学里不成立。生态学中的样本-变量矩阵没有逆的主要原因是 矩阵不满秩 。 即使取样严格遵守了“随机取样”的原则，只要样本取自“匀质(homogeneous)”的植被，样本-变量矩阵里就不会有足够的“独立向量”。所有的样本都是“中心向量”的变异。换句话说，如果把“取样方差”考虑进去，则，样本-变量矩阵中仅有一个独立向量。而且， 如果矩阵不满秩，则任何变换都无法增加矩阵的秩(Leon 1994:170) 。 2) A 和 B 是由样本-变量矩阵的平均值组成的多元向量( m -vectors)，而 X 仍旧是我们欲求的 m * m 的 状态转移矩阵 。由于未知变量的数目( m*m )多于方程的数目( m )，方程有无数多的解，或说，没有确定的解(Bai 1984)http://www.planta.cn/forum/viewtopic.php?t=11187。 状态转移矩阵 X 有确定解的充分必要条件是：待定的变量的个数是 m ，而其它 m *( m - 1) 个分量为已知，或为零。由于状态转移矩阵是我们要求的，未知的，所以我们先把 m * ( m - 1) 个分量设为零。 这样，下一个问题就是，解出的 m 个非零变量在矩阵中如何摆布。由于存在“生殖隔离”，我们可以假定： 变量内的自相关要大于变量间的互相关 ，也就是： | X ( i, i )| | X ( i, j )|， 其中， i = 1,2, . . . m , j = 1,2,. . . m , 而且 i j . 所以， 转移矩阵 的 m 个非零分量要分布在对角线上: X(i, i) 0，而其它 m * ( m - 1)个零分布在非对角线位置上: X(i, j) = 0， i j。 也就是说， 生态学中的系统状态转移矩阵的解应该是对角矩阵，而且对角元素的值是对应分量的商，非对角元素是零: X(i, i) = b(i) / a(i) ， X (i, j) = 0 。 然而，我们知道，就运算而言，对角矩阵几乎无异于“多元向量”。 也就是说， 定义了乘法和除法的多元向量( m -vector)可能是生态系统时间动态分析的正确工具 。如果我们把我们研究的群落，系统用多元向量来表示，并使用一套向量的计算方法，我们也许可以发现生态系统／群落变化的规律。然后我们就可以延伸这个变化规律来预测未来。这个新的基于多元向量代数的数据分析综合方法，被称为“超球面模型”，最近发表在了《生态模型》97 (l-2):75-86, http://blog.sciencenet.cn/blog-333331-449959.html 从另一个角度，由于生物的增殖实质上是指数增长(Vandermeer 1981)，我们可以用经验增率(empirical rate)来代替内禀增率(intrinsic rate)，把适用于单一种群的标量的指数增长方程扩展到向量，用“多元向量的指数增长方程”来做群落，做生态系统的预测. 文献(略，参考英文原文) http://www.sciencenet.cn/m/user_content.aspx?id=296609 进一步讨论，请访问 http://blog.sciencenet.cn/blog-333331-676833.html
许多人也许还记得,或卷入了1997年在美国生态学会网站上展开的所谓生态学批判的讨论.讨论的中心议题是: 为什么生态学还没有能够预测未来 .我的讨论心得被刊登在Bulletin of the Ecological Society of America, 79卷2期, April, 1998. 后来收录在我的专著趋势分析及其在生态股市中的应用.因为是我自己的文章,所以仍列为原创.转载时有更新。 文章有两重点： 样本-变量矩阵无逆，所以状态转移矩阵无解。如用多元向量代表植被，则状态转移矩阵通解的形式是对角矩阵，进而，通解是同维多元向量。 与博友共同探讨。 A Critique of Matrix Solutions for Ecology The discussion about A Critique of Ecology on Ecol-L attracted my attention. I think one of the reasons that ecologists have not been able to make predictions is that we did not have a suitable tool. When we handle every single variable, we cannot find the rules behind the phenomena. When we handle the community as a whole using matrices, the variables can only be treated as linear. The formal name of matrix algebra is linear algebra. It has addition and subtraction, but generally does not have division or inverse, at least for ecology. This fatal limitation has restricted its application for temporal or spatial dynamic analysis on ecology. There is no definite matrix answer for the equation: AX = B, where A and B are two known sequential states of an ecosystem/community, and X is m* m transition matrix that needs to be determined. We can consider two cases 1) A and B are sample-variable matrices. There is no inverse for matrix A; thus there is no solution for X = B/A. Furthermore, if there is no inverse for A, then there will not be an inverse in the forms of AA', AYA', or YAY, either. Although these solutions may be mathematically sound, they will not work for ecology. The main reason that there is no inverse of a variable-sample matrix for ecology is that the rank of the matrix is not full. As long as the samples were collected from a homogeneous vegetation/community, then even if they were sampled randomly, there would not be enough independent vectors in the matrix. All the samples would be varieties of the centroid vector. In other words, if the sampling variances were considered, then there would be only one independent vector in the matrix. Furthermore, if the matrix was not full rank, then there would not be a transformation that could increase the rank (Leon 1994:170) . 2) A and B are multi-component vectors , m- vectors, which were made of averages of the sample-variable matrix over samples, but X is an m* m matrix. Since the number of unknown components, m* m, is greater than the number of equations, m, there are many answers, i.e., no definite answer (Bai 1984) . The transition matrix X has a definite solution only if the number of components that need to be determined is m, but the other m* (m - 1) components are either known or zeroes. As we know nothing about the transition matrix, to determine these values of the m components, those other m*(m - 1) components should be set to zeroes first. Then the next question would be how to locate these m nonzero components and those m*(m - 1) zeroes in the transition matrix X. As there is reproductive isolation, we can assume that the autocorrelation of a species would be greater than the intercorrelation with other species, i.e., |X(i,i) | | X (i,j) |, where i = 1,2, . . . m, j= 1,2, . . . m, i j. Therefore, these m nonzero components should be placed in the diagonal positions of the transition matrix , X ( i i) 0, but those m*(m - 1) zeroes should be in the offdiagonal positions , X( i,j) = 0, i j. It means that the transition matrix in ecology should be in diagonal form, and X(i,i) = b (i) /a(i) , X(i,j)=0. However, a diagonal matrix is nothing else than a vector. This suggests that the m- vector with division and multiplication may be the correct tool for temporal dynamic analysis in ecology. If we treat our object as a community/ system using m -vectors, and describe and analyze them by applying a vector algorithm, then we may discover some new rules for communities/ systems. Then we can extend these rules to make projections. This new data synthesis and analysis method based on m -vectors, called the Multi-Dimensional Sphere Model, was published recently in Ecological Modelling 97 (l-2):75-86. In other words, as exponential growth is the essence of biology (Vandermeer 1981), we may adjust the exponential growth equation by replacing the intrinsic rate with the empirical rate and extending the monospecies model to a multispecies situation; then we may be able to use the m -exponential equation to make predictions. Literature cited Bai, T. 1984. Numerical prediction of grassland succession trend. (In Chinese.) Grassland Research Institute of the Chinese Academy of Agricultural Sciences (GRIC), Hohhot, China. Leon, S. J. 1994. Linear algebra with applications. Fourth edition. Prentice-Hall, Englewood Cliffs, New Jersey, USA. Vandermeer, J. 1981. Elementary mathematical ecology. John Wiley Sons, New York, New York, USA. T. Jay Bai, Ph. D. MDSM Research Fort Collins, CO 80524 (970) 495-9716 E-mail: mdsm95bai@gmail.com Bulletin of the Ecological Society of America 中文稿点击: http://www.sciencenet.cn/m/user_content.aspx?id=296610 原件附印件请点击: A Critique of Matrix Solutions for Ecology
标签: 动态分析
