Uncertainty, Emergence, and Statistics in Dendrochronology

Abstract

Some fundamental concepts of dendrochronological analysis are reviewed in the context of statistically modeling the climatically related environmental signals in cross-dated tree-ring series. Significant uncertainty exists due to our incomplete mechanistic understanding of radial growth of most tree species in the natural world, one where environmental effects are unobserved, uncontrolled, and steadily changing over time. This biological uncertainty cascades into the realm of statistical uncertainty in ways that are difficult to quantify even though the latter may be well constrained by theory. Therefore, great care must be taken to apply the many well-developed and tested statistical methods of dendrochronology in ways that reduce the probability of making false inferences. This is especially true in the case of biological emergence. This is a special case of uncertainty that arises from the way in which trees as complex organisms can have properties expressed in their ring widths that are impossible to predict from a basic understanding of lower-level physiological processes. Statistical modeling must be conducted in ways that allow for the discovery of such phenomena and, at the same time, protect from the incorrect acceptance of spurious emergent properties. To reduce the probability of the latter, we argue that model verification be an important part of any dendrochronological inquiry based on statistics. Correlation and response function analysis is used to illustrate some of the concepts discussed here. The value of empirical signal strength statistics as predictors of climatic signal strength in tree rings is also investigated.

Appendix

Basic chronology statistics for tree-ring series of length n:

Arithmetic Mean:

$$\mathop x\limits^ - = \frac{1}{n}\sum\limits_{i = 1}^n {x_i }$$

Standard Deviation:

$$s_{} = \sqrt{\frac {\sum\limits_{i = 1}^n {(x_i - \mathop x\limits^\_ )^2 }}{n-1}}$$

Lag-1 Autocorrelation:

$$\mathop r\nolimits_1 = \frac{{\sum\limits_{i = 2}^n {(x_i - \mathop x\limits^ - )(x_{i - 1} - \mathop x\limits^ - )} }}{{(n - 1)s_x^2 }}$$

Mean Sensitivity:

$${\textrm{ms}} = \frac{1}{{n - 1}}\sum\limits_{i = 2}^n {\left| {\frac{{2(x_i - x_{i - 1} )}}{{(x_i + x_{i - 1} )}}} \right|}$$

Empirical signal strength statistics for m tree-ring series from t trees of length n:

Mean chronology:

$$\overline X = \frac{1}{m}\sum\limits_{j = 1}^m {x_{i,j} }$$

for each year, i = 1, n

Chronology MS:

$${\textrm{MS}}_c = \frac{1}{{n - 1}}\sum\limits_{i = 2}^n {\left| {\frac{{2(\overline X _i - \overline X _{i - 1} )}}{{(\overline X _i + \overline X _{i - 1} )}}} \right|}$$

Average ms of the m series:

$$\overline {{\textrm{MS}}} _s = \frac{1}{m}\sum\limits_{j = 1}^m {{\textrm{ms}}_j }$$

Edmund Schulman’s R:

$${\textrm{ESR}} = \frac{{\overline {{\textrm{MS}}} _s }}{{{\textrm{MS}}_c }}$$

For t trees and m total cores, calculate all possible between-series Pearson cross-correlations as

Pearson correlation:

$$\mathop r\nolimits_{xy} = \frac{{\sum\limits_{i = 1}^n {(x_i - \mathop x\limits^ - )(y_i - \mathop y\limits^ - )} }}{{(n - 1)\mathop s\nolimits_x \mathop s\nolimits_y }}$$

where r _xy is the correlation between cores x and y.

From Briffa and Jones (1990), let c _i equal the number of within-tree cores for a given tree and let RBAR be a given average correlation statistic. Then,

RBAR for all series:

$${\textrm{RTOT}} = \frac{1}{{{\textrm{NTOT}}}}\sum\limits_{i = 1}^t {\sum\limits_{\begin{array}{l}\scriptstyle l = 1\\ \scriptstyle i\neq1 \end{array}}^t {\sum\limits_{j = 1}^{c_i } {r_{ilj} } } }$$

where

$${\textrm{NTOT}} = \frac{1}{2}\left[ {\sum\limits_{i = 1}^t {c_i } } \right]\left\{ {\left[ {\sum\limits_{i = 1}^t {c_i } } \right] - 1} \right\}$$

RBAR within trees:

$${\textrm{RWT}} = \frac{1}{{{\textrm{NWT}}}}\sum\limits_{i = 1}^t {\left[ {\sum\limits_{j = 2}^{c_i } {r_{ij} } } \right]}$$

where

$${\textrm{NWT}} = \sum\limits_{i = 1}^t {\frac{1}{2}} c_i (c_i - 1)$$

RBAR between trees:

$${\textrm{RBT}} = \frac{1}{{{\textrm{NBT}}}}({\textrm{RTOT}} \cdot {\textrm{NTOT}}) - {\textrm{RWT}} \cdot {\textrm{NWT}})$$

where ${\textrm{NBT}} = {\textrm{NTOT}} - {\textrm{NWT}}$

The effective RBAR (REFF) is a weighted average of RBT and RWT that includes the added signal strength due to the within-tree correlations.

RBAR effective:

$${\textrm{REFF}} = \frac{{{\textrm{RBT}}}}{{{\textrm{RWT}} + \frac{{1 - {\textrm{RWT}}}}{{{\textrm{CEFF}}}}}}$$

where

$$\frac{1}{{{\textrm{CEFF}}}} = \frac{1}{t}\sum\limits_{i = 1}^t {\frac{1}{{c_i }}}$$

The signal-to-noise ratio and expressed population signal are estimated from REFF now as

Signal-to-noise ratio:

$${\textrm{SNR}} = \frac{{t \cdot {\textrm{REFF}}}}{{1 - {\textrm{REFF}}}}$$

Expressed population signal:

$${\textrm{EPS}} = \frac{{t \cdot {\textrm{REFF}}}}{{t \cdot {\textrm{REFF}} + (1 - {\textrm{REFF}})}}$$