Edge selection based on the geometry of dually flat spaces for Gaussian graphical models

We propose a method for selecting edges in undirected Gaussian graphical models. Our algorithm takes after our previous work, an extension of Least Angle Regression (LARS), and it is based on the information geometry of dually flat spaces. Non-diagonal elements of the inverse of the covariance matrix, the concentration matrix, play an important role in edge selection. Our iterative method estimates these elements and selects covariance models simultaneously. A sequence of pairs of estimates of the concentration matrix and an independence graph is generated, whose length is the same as the number of non-diagonal elements of the matrix. In our algorithm, the next estimate of the graph is the nearest graph to the latest estimate of the concentration matrix. The next estimate of the concentration matrix is not just the projection of the latest estimate, and it is shrunk to the origin. We describe the algorithm and show results for some datasets. Furthermore, we give some remarks on model identification and prediction.     

Analysis of changes of insect-plant ratio-improvement of key-factor analysis for evaluating bottom-up effects in insect population dynamics

A revised key-factor analysis was presented for analyzing the temporal changes in the ratio of insect absolute number to plant resource. Ten data sets for 5 insect species were then analyzed. In this key-factor analysis, the key factor is defined as the factor contributing highly to between-year variation inR _r , the log rate of the inter-year change of the insect-plant ratio. The yearly change of plant resource was handled as a separate factor, expressed byr _pl , log ratio of plant resource in yearn to plant resource in yearn+1. The following was revealed: 1) In 7 of the 10 data sets examined,r _pl influenced variations ofR _r ; in particular in 3 casesr _pl was the main key factor. 2) Generation-to-generation fluctuations of absolute insect densities showed density dependence in 4 cases, while those of insect-plant ratios, in 8 cases. 3) The Royama model or a linear model, explained well the relationship between log insect-plant ratio (X _r ) andR _r and the relationship betweenX _r and log yearly change rate of absolute insect density (R _abs ). However, in the 7 cases in whichr _pl was a critical factor for variations ofR _r , with, increase ofX _r ,R _r showed a steeper, decrease around the equilibrium point (the point for whichR _r is 0) thanR _abs . This occurred becauser _pl tended to be negatively correlated withX _r . Consequently, in two casesX _r fluctuated cyclicly or chaotically although without the changes in plant resource, fluctuations ofX _r would be damped oscillations approaching equilibrium.