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 sp ecies y given descriptors x 1 , x n is obviously linear: y = a 1 x 1 + + a n x n (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 p olynomial can represent the curved response appropriately and is also linear and easily solvable: y = a 11 x 1 + a 12 x 2 1 + + a n2 x 2 n 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 ab ove, 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. Resp onse 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 sp ecies 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 Reference 1 year year 2 CRU Climate Research Unit 3 J98 Jones et al. 1998 Holocene 4 MBH99 Mann et al. 1999 Geophys Res Lett 5 MJ03 Mann and Jones 2003 6 CL00 Crowley and Lowery 2000 Ambio 7 BJ00 Briffa 2000 Quat Sci Rev 8 BJ01 Briffa et al. 2001 J Geophys Res 9 Esp02 Esper 2002 Science 10 Mob05 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 p eriod 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 r 2 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 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 1800 1850 1900 1950 2000 −1.0 0.0 0.5 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 r 2 values ˜ A´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 r 2 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 mo del (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 148 Niche Modeling −0.4 −0.2 0.0 0.2 0.4 0.2 0.6 1.0 −0.4 −0.2 0.0 0.2 0.4 0.1 0.4 0.7 −0.4 −0.2 0.0 0.2 0.4 0.1 0.3 0.5 −0.4 −0.2 0.0 0.2 0.4 0.1 0.3 0.5 −0.4 −0.2 0.0 0.2 0.4 0.1 0.3 0.5 0.7 −0.4 −0.2 0.0 0.2 0.4 0.2 0.6 1.0 −0.4 −0.2 0.0 0.2 0.4 0.2 0.6 1.0 −0.4 −0.2 0.0 0.2 0.4 0.2 0.6 FIGURE 9.2: Fit of a logistic curve to each of the studies. © 2007 by Taylor and Francis Group, LLC Non-linearity 149 climate principle is: r = f(x) = max[0, ax 2 + bx + c)] Where addition of a second climate principle is necessary, such as precipita- tion, the principle of the maximum limiting factor is a simple and conventional way to incorporate both factors. Here tT is temperature and pP is precipi- tation (Figure 2C): r = min[f(t), f(p)] The inverse function of the quadratic has two solutions, positive and nega- tive: t = −b ±  b 2 −4a(c−r) 2a This formulation of the problem clearly shows the inherent nondeterminism in the solution, producing two distinct reconstructions of past climate. In contrast, the solution to the linear model produces a single solution. Figure 9.4 shows the theoretical quadratic growth response to a periodically fluctuating environmental principle such as temperature. The solid lines are the growth responses for three trees located above, centered and below their optimum temperature range. The response has two peaks, because the opti- mum temperature is visited twice for a single cycle in temperature. Note the p eaks are coincident with the optimal response temperatures, not the maxi- mum temperatures of the principle (dotted lines). The peak size and locations do not match the underlying temperature trend. Figure 9.5 shows the theoretical growth response to two slightly out of phase environmental drivers, e.g. temperature and rainfall) where the response func- tion is the limiting factor. The resulting response now has four peaks, and the non-linear response produces a complex pattern of fluctuations centered on the average of climate principles. The addition of more out of phase drivers would add further complexity. To describe this behavior in niche modeling terms, a tree has a prefer- ence function determined by climatic averages and optimal growth conditions. When the variation in climate is small, e.g. the climate principle varies only within the range of the calibration period, the signal is passed unchanged from principal to proxy. But when the amplitude of variation is large, as in the case for extraction of long time-scale climate variation, the amplitude of the proxy is limited, and the interpretation becomes ambiguous. © 2007 by Taylor and Francis Group, LLC 150 Niche Modeling time temperature ? ? C tree−rings FIGURE 9.3: Idealized chronology showing tree-rings and the two possible solutions due to non-linear response of the principle (solid and dashed line) after calibration on the end region marked C. principal response time principal q=−0.25 q=0.0 q=0.25 FIGURE 9.4: Nonlinear growth response to a simple sinusoidal driver (e.g. temperature) at three optimal response points (dashed lines). © 2007 by Taylor and Francis Group, LLC Non-linearity 151 r t response q=−0.25 q=0.0 q=0.25 FIGURE 9.5: Nonlinear growth response to two out of phase simple sinu- soidal drivers (e.g. temperature and rainfall) at three response points. Solid and dashed lines are climate principles; dotted lines the response of the prox- ies. We now examine the consequences of reconstructing temperatures from non-linear responses calibrated by four different regions of the response curve (mo dels R1-4). The left-hand boxes in Figures 9.7, 9.8, 9.9 and 9.10 illustrate the linear and nonlinear calibration models (lines) from the subset years of the series (circles). The right-hand graphs show the reconstructions (solid line) resulting from inversion of the derived model plotted over the temperatures (dots). 9.1.3.1 R1 The first model is a linear model fit to the portion of the graph from 650 to 700 (Figure 9.7). This corresponds to a reconstruction using the portion of the instrumental record where proxies are responding almost linearly to temperature. While the model shows a good fit to the calibration data, and the reconstruction shows good agreement over the calibration period, the pre- diction becomes rapidly more inaccurate in the future and the past. The amplitude of the reconstructed temperatures is 50% of the actual tempera- tures, and the peaks bear no relation to peaks in temperature. Temporal shift in peaks may be partly responsible for significant offsets in timing of warmth in different regions [CTLT00]. The nonlinear model suggests that timing of peaks should be correlated with regional location of trees, and may be a factor contributing to apparent large latitudinal and regional variations in the magnitude of the MWP [MHCE02]. © 2007 by Taylor and Francis Group, LLC 152 Niche Modeling −1.0 −0.5 0.0 0.5 1.0 −0.6 −0.2 0.2 1000 1200 1400 1600 1800 2000 −2.0 −1.0 0.0 1.0 FIGURE 9.6: Example of fitting a quadratic model of response to a re- construction. As response over the given range is fairly linear, reconstruction does not differ greatly. 0.5 0.7 0.9 −0.7 −0.5 −0.3 −0.1 response temperature −2 −1 0 1 2 0 200 400 600 800 1000 time R1 FIGURE 9.7: Reconstruction from a linear model fit to the portion of the graph from 650 to 700. © 2007 by Taylor and Francis Group, LLC [...]... Group, LLC Niche Modeling −1 0.6 0 0.8 1 1.0 R3 −2 0.4 response 1.2 2 154 −0.6 −0.2 0.0 0.2 0.4 0 200 400 600 800 1000 time temperature FIGURE 9. 9: Reconstruction from a quadratic model derived from data years 700 to 800, the period of ideal nonlinear response to the driving variable 4 2 0.6 −2 0 0.4 0.0 −4 0.2 response 0.8 R4 −0.4 0.0 0.2 0.4 0.6 temperature 0 200 400 600 800 1000 time FIGURE 9. 10: Reconstruction... FIGURE 9. 10: Reconstruction resulting from a quadratic model calibrated from 750 to 850 with two out of phase driving variables, as shown in Figure 9. 5 9. 1.3.4 R4 The fourth model is a quadratic model corresponding to a reconstruction fit to the period of maximum response of the species from 750 to 850 with two out of phase driving variables, as shown in Figure 9. 10 The poor reconstruction shows that... reconstruction is greatly increased In contrast to R1, the inverted function overestimates the amplitude of past temperatures Figure 9. 8 shows variance and may be exaggerated as calibrated slope decreases 9. 1.3.3 R3 The third, Figure 9. 9, is a quadratic model derived from data years 700 to 800, which corresponds to a record of the period of ideal nonlinear response to the driving variable The resulting... principles Thus niche modeling demonstrates its usefulness as a theory for explaining aspects of ecology not previously explained in the linear model 9. 1.4 Cubic A research project to insert salient features of ‘normal’ nonlinear physiological behavior between the proxy and principle could go in a number of © 2007 by Taylor and Francis Group, LLC Non-linearity 155 directions One could draw from plant ecology... nonlinearity 9. 2 Summary These results demonstrate that procedures with linear assumptions are unreliable when applied to the non-linear responses of niche models Reliability of reconstruction of past climates depends, at minimum, on the correct specification of a model of response that holds over the whole range of the proxy, not just the calibration period Use of a linear model of non-linear response... exaggerate variance By visual inspection of Figure 9. 7, it can be seen that ignoring one of the two possible solutions reduces the apparent amplitude of the long time-scale climate reconstructions by half 9. 1.3.2 R2 The second row in Figure 9. 8 is a linear model fit to years 600 to 800 corresponding to reconstruction practice where the proxies show a significant downturn in growth Model R2 simulates the...153 2 Non-linearity 0 0.6 −1 0.2 −2 −0.2 response 1 1.0 R2 −1.0 −0.5 0.0 0.5 0 temperature 200 400 600 800 1000 time FIGURE 9. 8: A linear model fit to years 600 to 800 where the proxies show a significant downturn in growth We have also shown here that linear models can either reduce or exaggerate variance By visual inspection of Figure 9. 7, it can be seen that ignoring one... entering at present, described as increasing ‘insensitivity’ to temperature, as temperatures pass over the peak of response and lower the slope [Bri00] Another study of tree-line species in Alaska attributed a significant inverted U-shaped relationship between the chronology and summer temperatures to a negative growth effect when temperatures warm beyond a physiological threshold [DKD+ 04] The position... model of non-linear response can cause apparent growth decline with higher temperatures, signal degradation with latitudinal variation, temporal shifts in peaks, period doubling, and depressed long time-scale amplitude © 2007 by Taylor and Francis Group, LLC . LLC 146 Niche Modeling 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0. LLC Non-linearity 145 TABLE 9. 1: Global temperatures and temperature reconstructions. names Reference 1 year year 2 CRU Climate Research Unit 3 J98 Jones et al. 199 8 Holocene 4 MBH 99 Mann et al. 199 9. 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 1800 1850 190 0 195 0 2000 −1.0 0.0 0.5 FIGURE 9. 1: Reconstructed smoothed temperatures