Niche Modeling: Predictions From Statistical Distributions - Chapter 9 pdf

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

Chapter 9

Non-linearity

In mathematics, non-linear systems are, obviously, not linear Linear sys-tem and some non-linear syssys-tems 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

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+ a12x21+ + an2x2n

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

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

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

TABLE 9.1: Global temperatures and temperature reconstructions

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

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

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

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

1800 1850 1900 1950 2000

FIGURE 9.1: Reconstructed smoothed temperatures against proxy values for eight major reconstructions

TABLE 9.2: Slope and correllation coefficient of temperature reconstructions with temperature

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

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

The possibility of inverted U’s in the proxy response is even more critical with possibility that growth suppression at higher temperatures may have

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

−0.4 −0.2 0.0 0.2 0.4

FIGURE 9.2: Fit of a logistic curve to each of the studies

climate principle is:

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 tT is temperature and pP 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 ±

q

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 9.4shows 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 peaks 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 9.5shows 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

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

time

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)

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

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)

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]

−1.0 −0.5 0.0 0.5 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

temperature

time R1

FIGURE 9.7: Reconstruction from a linear model fit to the portion of the graph from 650 to 700

temperature

time R2

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

one of the two possible solutions reduces the apparent amplitude of the long time-scale climate reconstructions by half

The second row in Figure 9.8 is a linear model fit to years 600 to 800 cor-responding to reconstruction practice where the proxies show a significant downturn in growth Model R2 simulates the condition we may be entering

at present, described as increasing ‘insensitivity’ to temperature, as temper-atures 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 nega-tive growth effect when temperatures warm beyond a physiological threshold

The position of the peaks is the same as in the previous models, but due

to the inversion of the linear model, the amplitude of the temperature re-construction 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

800, which corresponds to a record of the period of ideal nonlinear response

to the driving variable The resulting reconstructions shows the accurate location and amplitude of the peaks, up to the essential nondeterminism This demonstrates that an accurate reconstruction of temperature could potentially

be recovered if it were possible to choose the correct solution at each inflection point

temperature time

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

temperature

time R4

FIGURE 9.10: Reconstruction resulting from a quadratic model calibrated

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 reconstruc-tion shows that confounding variables together with nondeterminism greatly increase the difficulty of accurately reconstructing climate in greater than one nonlinear dimension

strong positive correlation with temperature, trees at middle latitudes show

a very confused response The nonlinear model developed here is consistent with high latitude trees corresponding more closely to temperature, and trees

in the middle of the range being more disrupted by the two, temperature and rainfall principles Thus niche modeling demonstrates its usefulness as a theory for explaining aspects of ecology not previously explained in the linear model

A research project to insert salient features of ‘normal’ nonlinear phys-iological behavior between the proxy and principle could go in a number of

directions One could draw from plant ecology where probability of occurrence

of species across a temperature ecotone is well modeled by a skew quadratic [Aus87], incorporate the tendency to flattened peaks suggested both by com-parison of ring width and density indices, or even asymptotic behavior sug-gested by physiological models of temperature response of high latitude trees [Loe00]

A second order quadratic model of responses to temperature has the advan-tage of solvability However, the simple quadratic response is not proposed

as an exact theory Rather, it is used to provide insights into the expected behavior of the modeled system under ideal or average conditions, and to develop intuition for the consequences of nonlinearity

9.2 Summary

These results demonstrate that procedures with linear assumptions are un-reliable when applied to the non-linear responses of niche models Reliability

of reconstruction of past climates depends, at minimum, on the correct spec-ification 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 can cause apparent growth decline with higher temperatures, signal degrada-tion with latitudinal variadegrada-tion, temporal shifts in peaks, period doubling, and depressed long time-scale amplitude