Niche Modeling: Predictions From Statistical Distributions - Chapter 10

Chapter 10 Long term persistence Below is an investigation of scaling or long term persistence (LTP) in time series including temperature, precipitation and tree-ring proxies. The recog-nition, quantification and implications for analysis are drawn largely from Koutsoyiannis [Kou02]. They are characterized in many ways, as having long memory, self similarity in distribution, ‘long’ or ‘fat’ tails in the distribution or other properties. There are important distinctions to make between short term persistence (STP), and LTP phenomena. STP occurs for example in Markov or AR(1) process where each value depends only on the previous step. As shown pre-viously, the autocorrelations in an STP series decay much more rapidly than LTP. In addition, LTP are related to a number of properties that are inter-esting in themselves. These properties may not all be present in a particular situation, and definitions of LTP also vary between authors. Some of the properties are [KMF04] : Defn. I Persistent autocorrelation at long lags: Where ρ(k) is the autocorrelation function (ACF) with lag k then a series Xt is LTP if there is a real number α ∈ (0,1) and a constant cp > 0 such that limk→∞ cρ(k) = 1 In other words, the definition states that the ACF decays to zero with a hyperbolic rate of approximately k−α. In contrast the ACF of a STP process decays exponentially. Defn. II Infinite ACF sum: As a consequence of the hyperbolic rate of decay, the ACF of a LTP is usually non-summable: Pk ρ(k) = ∞ Defn. III High standard errors for large samples: The standard error, or variances of the sample mean of a LTP process decay more slowly than the reciprocal of the sample size. 157 © 2007 by Taylor and Francis Group, LLC 158 Niche Modeling VAR[Xm] ∼ a2m−α as m → ∞ where α < 1 Here m refers to the size of the aggregated process, i.e. the sequen-tial sum of m terms of X. Due to this property, classical statistical tests are incorrect, and confidence intervals underestimated. Defn. IV Infinite power at zero wavelength: Thespectralfrequency obeys an increasing power law near the origin, i.e. f(λ) ∼ aλ−α as wavelength λ → 0. In contrast, with STP f(λ) at λ = 0 is positive and finite. Defn. V Constant slope on log-log plot The rescaled adjusted range statistic is characterized with a power exponent H E[R(m)/S(m)] ∼ amH as → ∞ with 0.5 < H < 1. H, called the Hurst exponent, is a measure of the strength of the LTP and H = 1− α Defn. VI Self-similarity: Similar to the above definition, a process is self-similar if a property such as distribution is preserved over large scales of space and/or time Xmt and mHXt have identical distributions for all m > 0 Here m is a scaling factor and H is the constant Hurst exponent. Self-similarity can refer to a number of properties being preserved irrespective of scaling in space and/or time, such as variance or autocorrelation. This can provide very concise descriptions of be-haviour of widely varying scales, such as the ‘burstiness’ of internet traffic [KMF04]. Here we show, and this is far from accepted, is that LTP is a fact of natural phenomena. LTP is seen by some to be an ‘exotic’ phenomenon requiring system with ‘long term memory’. However, if for whatever reason systems do exhibit LTP behavior, it is important to incorporate LTP into our assump-tions. © 2007 by Taylor and Francis Group, LLC Long term persistence 159 Here examine a set of proxy series listed in Table 9.1 of the previous chapter, as well as the temperature and precipitation from a sample of landscape. 10.1 Detecting LTP One of the main operations in examining LTP are aggregates of series. Aggregates are calculated as follows. For example, given a series of numbers X, the aggregated series X1, X2 and X3 is as follows. > x <- seq(0, 1, by = 0.1) > hagg(x, 1:3, sum) [[1]] [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 [[2]] [1] 0.1 0.5 0.9 1.3 1.7 [[3]] [1] 0.3 1.2 2.1 Figure 10.1 compares the two diagnostic tools. The first used previously is the plot of the ACF for successive lags. Note that the series walk, SSS and CRU decay slowly relative to IID and AR. This is consistent with Definition I of LTP in these series with high correlations at long lags. Figure 10.2 shows a similar pattern in a different way, by plotting the cor-relation at lag 1 against the series aggregated at successively long time scales. The persistence of autocorrelation at higher aggregations for series walk, SSS and CRU decay over IID and AR is clear. A plot of the logarithm of standard deviation of the simulated series against the logarithm of the level of aggregation or time scale on Figure 10.3 shows scale invariant series, as per definitions V and VI, as straight lines with a slope greater than 0.5. Random numbers form a straight line of low slope (0.5). The random walk is also a straight line of higher slope as are CRU and SSS. Notably the slope of the AR(1) model declines with higher aggregations, converging towards the slope of the random line. This demonstrates that AR has STP as per definition V but not LTP. © 2007 by Taylor and Francis Group, LLC 160 Niche Modeling walk sss CRU iid 0 5 10 15 20 25 30 lag FIGURE 10.1: One way of plotting autocorrelation in series: the ACF function at lags 1 to k. walk sss iid AR CRU 0 10 20 30 40 k FIGURE 10.2: A second way of plotting autocorrelation in series: the ACF at lag 1 of the aggregated processes at time scales 1 to k. © 2007 by Taylor and Francis Group, LLC Long term persistence 161 walk sss iid 1 2 5 10 20 log.k FIGURE 10.3: The log-log plot of the standard deviation of the aggregated simulated processes vs. scale k. The implications of high self similarity or H value are most apparent in the standard error, or standard deviation of the mean. The standard error of IID series and AR series increase with the square root of aggregations. The aggregation is equivalent to sample size. Thus the usual rule for calculating standard error of the mean applies: s.e. = σ Series such as the SSS, CRU and the random walk would maintain high standard errors of the mean with increasing sample size. This means that where a series has a high H, increasing numbers of data do not decrease our uncertainty in the mean very much. Alternatively, there are few effective points. At the level of the CRU of H = 0.95 the uncertainty in a mean value of 30 points is almost as high as the uncertainty in a mean of a few points. It is this feature of LTP series that is of great concern where accurate estimates of confidence are needed. © 2007 by Taylor and Francis Group, LLC ... - tailieumienphi.vn