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COMFAC is employed as a sequential simplification device in empirical analysis. Analysis of the finite‐sample statistical properties checked the power when a COMFAC representation of economic agents’ behaviour was valid. Improved test power response surfaces were proposed. Combined, the findings suggested using other model reduction devices, retaining COMFAC as a destructive testing device for invalid ‘autocorrelation correction’ claims. Diagnostic testing of the initial model was emphasized. Model selection (lag length, choice of simplification, and parameter) is investigated for highly seasonal data.

For almost 30 years, Lande and Arnold's approximation of individual fitness surfaces through multiple regression has provided a common framework for comparing the strength and form of phenotypic selection across traits, fitness components and sexes. This chapter provides an overview of the statistical and geometric approaches available for the multivariate analysis of phenotypic selection that build upon the Lande and Arnold approach. First, it details least squares based approaches for the estimation of multivariate selection in a single population. Second, it shows how these approaches can be extended for the statistical comparison of individual fitness surfaces among groups such as populations or experimental treatments, addressing the inferential differences between analyses of randomly chosen groups versus situations in which groups are experimentally fixed. In each case, it points out known issues and caveats associated with the approaches. Finally, using case studies, the chapter shows how these estimates of multivariate selection can be integrated with quantitative genetic analyses to better understand issues such as the maintenance of genetic variance under selection and how genetic constraints can bias evolutionary responses to selection.

Examines methods of testing for co‐integration in single equations via static regressions, and provides simulation estimates of the percentiles of the distributions of statistics used in these tests. The finite‐sample biases of the estimates of the co‐integrating vectors and powers of the tests based on static regressions are discussed within the framework of extensive Monte Carlo simulations. Dynamic models leading to an error‐correction mechanism based test (ECM test for co‐integration) and non‐parametrically modified estimators are also considered as better ways of estimating the co‐integrating relationships.

This chapter is about practical uses of mathematical models to simplify the task of finding the best conditions under which to crystallize a macromolecule. The models describe a system’s response to changes in the independent variables under experimental control. Such a mathematical description is a surface, whose two-dimensional projections can be plotted, so it is usually called a ‘response surface’. Various methods have been described for navigating an unknown surface. They share important characteristics: experiments performed at different levels of the independent variables are scored quantitatively, and fitted implicitly or explicitly, to some model for system behaviour. Initially, one examines behaviour on a coarse grid, seeking approximate indications for multiple crystal forms and identifying important experimental variables. Later, individual locations on the surface are mapped in greater detail to optimize conditions. Finding ‘winning combinations’ for crystal growth can be approached successively with increasingly well-defined protocols and with greater confidence. Whether it is used explicitly or more intuitively, the idea of a response surface underlies the experimental investigation of all multivariate processes, like crystal growth, where one hopes to find a ‘best’ set of conditions. The optimization process is illustrated schematically in Figure 1. In general, there are three stages to this quantitative approach: (a) Design. One must first induce variation in some desired experimental result by changing the experimental conditions. Experiments are performed according to a plan or design. Decisions must be made concerning the experimental variables and how to sample them. (b) Experiments and scores. Each experiment provides an estimate for how the system behaves at the corresponding point in the experimental space. When these estimates are examined together as a group, patterns often appear. For example, a crystal polymorphism may occur only in restricted regions of the variable space explored by the experiment. (c) Fitting and testing models. Imposing a mathematical model onto such patterns provides a way to predict how the system will behave at points where there were no experiments. The better the predictions, the better the model. Adequate models provide accurate interpolation within the range of experimental variables originally sampled; occasionally a very good model will correctly predict behaviour outside it (1).

The population dynamics of the larch budmoth (LBM), Zeiraphera diniana, in the Swiss Alps are perhaps the best example of periodic oscillations in ecology (figure 7.1). These oscillations are characterized by a remarkably regular periodicity, and by an enormous range of densities experienced during a typical cycle (about 100, 000-fold difference between peak and trough numbers). Furthermore, nonlinear time series analysis of LBM data (e.g., Turchin 1990, Turchin and Taylor 1992) indicates that LBM oscillations are definitely generated by a second-order dynamical process (in other words, there is a strong delayed density dependence—see also chapter 1). Analysis of time series data on LBM dynamics from five valleys in the Alps suggests that around 90% of variance in Rt is explained by the phenomenological time series model employing lagged LBM densities, R, =f(Ni-1, Ni-2, ) (Turchin 2002). As discussed in the influential review by Baltensweiler and Fischlin (1988) about a decade ago, ecological theory suggests a number of candidate mechanisms that can produce the type of dynamics observed in the LBM (see also chapter 1). Baltensweiler and Fischlin concluded that changes in food quality induced by previous budmoth feeding was the most plausible explanation for the population cycles. During the last decade, the issue of larch budmoth oscillations was periodically revisited by various population ecologists looking for general insights about insect population cycles (e.g., Royama 1977, Bowers et al. 1993, Ginzburg and Taneyhill 1994, Den Boer and Reddingius 1996, Hunter and Dwyer 1998, Berryman 1999). These authors generally concurred with the view that budmoth cycles are driven by the interaction with food quality. A recent reanalysis of the rich data set on budmoth population ecology collected by Swiss researchers over a period of several decades, however, suggested that the role of parasitism is underappreciated (Turchin et al. 2002). Before focusing on the roles of food quality and parasitism in LBM dynamics, we briefly review the status of other hypotheses that were discussed in the literature on LBM cycles. First, the natural history of the LBM-larch system is such that food quantity is an unlikely factor to explain LBM oscillations.