Niche Modeling: Predictions From Statistical Distributions - Chapter 9

Chapter 9 Non-linearity In mathematics, non-linear systems are, obviously, not linear. Linear sys-tem and some non-linear systems are easily solvable as they are expressible as a sum of their parts. Desirable assumptions and approximations flow from particular model forms, like linearity and separability, allowing for easier com-putation of results. In the context of niche modeling, an equation describing the response of a species y given descriptors x1,...xn is obviously linear: y = a1x1 +...+anxn (9.1) However this form of equation is unsuitable for niche modeling as it cannot represent the inverted ‘U’ response that is a minimal requirement for repre-senting environmental preferences. The second order polynomial can represent the curved response appropriately and is also linear and easily solvable: y = a11x1 +a12x2 +...+an2x2 However, our concern here is not with solvability. Our concerns are the errors that occur when non-linear (i.e. curved) systems such as equation 9.2 above, are modeled as linear, systems like equation 9.1. Specifically we would like to apply niche modeling methods and determine how types of non-linearity affect reliability of models for reconstructing past temperatures over the last thousand years from measurements of tree ring width. This will demonstrate another potential application of niche modeling to dendroclimatology. Response of an individual and species as a whole to their environment is basic to climate reconstructions from proxies. Simulation, first in one and then two dimensions, can help to understand the potential errors in this methodology from non-linearity. 143 © 2007 by Taylor and Francis Group, LLC 144 Niche Modeling 9.1 Growth niches While we use a one dimension example of reconstructions of temperatures for simplicity, the results are equally applicable in two dimensions. However, we initially use actual reconstructions of past temperature. Reconstructions of past climates using tree-rings have been used in many fields, including climate change and biodiversity [Ker05]. It is believed by many that ”carefully selected tree-ring chronologies ... can preserve such co-herent, large-scale, multicentennial temperature trends if proper methods of analysis are used”[ECS02]. The general methodological approach in dendroclimatology is to normalize across the length and the variance of the raw chronology to reduce extravagant juvenile growth, calibrate a model on the approximately 150 years of instru-ment records (climate principals), and then apply the model to the historic proxy records to recover past temperatures. The attraction of this process is that past climates can then be extrapolated back potentially thousands of years. Non-linearity of response has not been greatly studied. Evidence for nonlin-ear response emerges from detailed latitudinal studies of the responses of single species to multiple climate principals [LLK+00]. Mild forms of non-linearity such as signal saturation were mentioned in a comprehensive review of climate proxy studies [SB03]. These are described variously as a breakdown, ‘insen-sitivity’, a threshold, or growth suppression at higher temperature. Here we explore the consequences of assuming the response is linear when the various forms of growth response to temperature could be: • linear, • sigmoid, • quadratic (or inverted ‘U’), and • cubic. Such non-linear models represent the full range of growth responses based on knowledge of the species physiological and ecological responses, is basic to niche modeling, and a logical necessity of upper and lower limits to organism survival. © 2007 by Taylor and Francis Group, LLC Non-linearity 145 TABLE 9.1: Global temperatures and temperature reconstructions. names 1 year 2 CRU 3 J98 4 MBH99 5 MJ03 6 CL00 7 BJ00 8 BJ01 9 Esp02 10 Mob05 Reference year Climate Research Unit Jones et al. 1998 Holocene Mann et al. 1999 Geophys Res Lett Mann and Jones 2003 Crowley and Lowery 2000 Ambio Briffa 2000 Quat Sci Rev Briffa et al. 2001 J Geophys Res Esper 2002 Science Moberg 2005 Science The series we examine in the linear and sigmoidal sections are listed in Table 9.1. Non-linear models for the quadratic reconstructions were con-structed from an ARMA time series of length 1000 with the addition of a sinusoidal curve to approximate the temperatures from the Medieval Warm Period (MWP) at around 1000AD through the Little Ice Age (LIA), to the period of current relative warmth. The coefficients were determined by fitting an ARIMA(1,0,1) model to the residuals of a linear fit to 150 annual mean temperature anomalies from the Climate Research Unit [Uni] resulting in the following coefficients: AR=0.93, MA = -0.57 and SD = 0.134. 9.1.1 Linear A linear model is fit to the calibration period using r = at + c. The linear equation can be inverted to t = (r − b)/a to predict temperature over the range of the proxy response as shown in Figure 9.1. Some result in better reconstructions of themselves than others, depending largely on the degree of correlation with CRU temperatures over the calibration period. The table 9.2 shows slope and r2 values for each reconstruction. All re-constructions have generally lower slope than one. While a perfect proxy of temperature would be expected to have a slope of one with actual tempera-tures, this loss of sensitivity might result from inevitable loss of information due to noise [vSZJ+04]. 9.1.2 Sigmoidal It is of course not possible for tree growth to increase indefinitely with temperature increases; it has to be limited. The obvious choice for a more © 2007 by Taylor and Francis Group, LLC 146 Niche Modeling 1800 1850 1900 1800 1850 1900 1800 1850 1900 1800 1850 1900 1950 2000 1950 2000 1950 2000 1950 2000 1800 1850 1900 1800 1850 1900 1800 1850 1900 1800 1850 1900 1950 2000 1950 2000 1950 2000 1950 2000 FIGURE 9.1: Reconstructed smoothed temperatures against proxy values for eight major reconstructions. © 2007 by Taylor and Francis Group, LLC Non-linearity 147 TABLE 9.2: Slope and correllation coefficient of temperature reconstructions with temperature. Slope r2 J98 0.78 0.35 MBH99 0.84 0.69 MJ03 0.43 0.47 CL00 0.27 0.23 BJ00 0.36 0.24 BJ01 0.57 0.33 Esp02 0.80 0.41 Mob05 0.28 0.09 accurate model of tree response is a sigmoidal curve. To evaluate the potential of a sigmoidal response I fit a logistic curve to each of the studies and compared the results with a linear fit on the period for which there are values of both temperature and the proxy. The results were as follows (Figure 9.2). The logistic curve did not give a stunning increase in the r2 valuesA´c al-though they were comparable. I had to estimate the maximum and minimum temperatures for each proxy from the maximum value and 0.1 minus the minimum value. Perhaps there is room for improvement in estimating these parameters as well and would improve the r2 statistic. 9.1.3 Quadratic The possibility of inverted U’s in the proxy response is even more critical with possibility that growth suppression at higher temperatures may have happened in the past. Figure 9.3 shows an idealized tree-ring record, with a linear calibration model (C) and the reconstruction resulting from back ex-trapolation. Due to the fit of model to an increasing proxy, smaller rings indi-cate cooler temperatures. A second possible solution (dashed) due to higher temperatures is shown above. Thus the potential for smaller tree-rings due to excess heat, not excess cold, affects the reliability of specific reconstructed temperatures after the first return to the maximum of the chronology. It is obvious that in this case statistical tests on the limited calibration period will not detect nonlinearity outside the calibration period and will not guarantee reliability of the reconstruction. The simple second order quadratic function for tree response to a single © 2007 by Taylor and Francis Group, LLC ... - tailieumienphi.vn