treeclim: an R package for the numerical calibration of proxy‐climate relationships

Abstract

The R package treeclim helps perform numerical calibration of proxy‐climate relationships, with an emphasis on tree‐ring chronologies. The package provides a unified, fast, and public‐domain compilation of established methods while adding novel functionality not implemented in other software. treeclim includes static and moving bootstrapped response and correlation functions, seasonal correlation analysis, a test for spurious temporal changes in proxy‐climate relations, and the evaluation of reconstruction skills. The stationary bootstrap method has been incorporated into the program as a ‘blocks of blocks’ resampling scheme. Applications of treeclim include the calibration of proxy timeseries used in paleoclimatology, forest ecology, and environmental monitoring.

Tree‐ring records, given their abundance and relative ease of collection, have provided fertile ground for the application of numerical methods. Dendrochronological information is used in various areas of science – from forestry to archaeology, from climatology to geochemistry, from ecology to forensics (Speer 2010). Consequently, investigators with widely different backgrounds have contributed to the discipline by proposing an array of data processing tools that date as far back as the origin of modern statistics. For instance, Douglass’ seminal book (1919), in a search for connections between climate, tree growth, and sun spots, contained several numerical methods, including moving averages and accumulated departures from the mean. A compilation of quantitative tools used in dendrochronology up to the late 1980s can be found in Cook and Kairiukstis (1990), but a number of other techniques have been proposed and applied since then.

A common challenge in dendrochronology is the extraction of climatic information from annually resolved tree‐ring chronologies, a process termed ‘calibration’ in dendrochronology and paleoclimatology (Bradley 2014). By correlating climate data, usually at monthly resolution, with tree‐ring chronologies, researchers seek to identify conditions that limit wood growth, and are consequentially reflected in tree‐ring records (Fritts 1976). Response and correlation functions (Blasing et al. 1984) are widely used to calculate dendrochronological relationships with climate variables. While correlation functions consist of Pearson's linear correlation estimates, response functions are another variant of indirect regression techniques (Christiansen 2011) that aim at mitigating the effects of predictor multicollinearity (Cropper 1984) by regressing the proxy record against the principal components of the climate data. This overcomes the shortcoming of potentially inflated variances of the ordinary least squares estimates of the regression coefficients (Jolliffe 2002). Even though principal component regression is not necessarily the best approach to deal with multicollinearity, it still shows competitive performance under moderate multicollinearity (Dormann et al. 2013). Because not all principal components are retained, confidence intervals for the regression coefficients may not be properly estimated using normality assumptions, resulting in over‐fitted response functions. Guiot (1991) proposed bootstrapping (Efron and Tibshirani 1986) as one solution to obtain more robust parameter estimates.

Biondi (2000) introduced the concept of moving and evolving response and correlation functions, which allow for evaluating the stationarity of dendroclimatic responses. This topic has now become widely debated because of the ‘divergence phenomenon’, i.e. the temporal instability of climate‐tree growth relations since the end of the 20th century (D'Arrigo et al. 2008). Consideration of the time‐dependence that may affect dendroclimatic results is increasingly embedded into ecological studies (Schuster and Oberhuber 2013). Time‐varying bootstrapped correlation and response functions were introduced to the public domain by Biondi and Waikul (2004) through the software DENDROCLIM2002 for the Windows operating systems. Although that software has remained functional under the constantly updated Windows versions, its development has now been transferred to the R public‐domain language and environment (R Development Core Team). The package bootRes, a DENDROCLIM2002 clone in R (Zang and Biondi 2013), is not bound to specific operating systems, has better graphics, and can easily be used in scripts to batch‐analyze large data sets. An addition to existing dendroclimatic tools was provided by Meko et al. (2011) with the seascorr software for MATLAB (2014). In their approach, varying combinations of monthly climate variables are bundled into seasonal targets. By allowing one seasonal signal per climatic predictor, this approach facilitates the identification of climate variables to be reconstructed beyond the period of available instrumental climate records (Touchan et al. 2014). It should be emphasized that established methodology provided by freely available software packages has complemented modern efforts to share data between research units, leading to an expansion of spatial coverage in studies of dendroclimatic relations for ecological characterization and risk assessment (Vicente‐Serrano et al. 2014).

Our new package treeclim represents a unified, fast, public‐domain compilation of dendroclimatic tools, including seasonal correlations. Current methodology was expanded by adding a flexible specification model for the design matrix, the possibility to test moving correlation functions for spurious significance, and the evaluation of reconstruction skills. Key features include the possibility to choose between different bootstrapping schemes, and the utilization of a high‐level visualization package for generating publication‐ready figures. Its flexibility and adaptability to various research questions notwithstanding, treeclim was designed with entry‐level users of R in mind, and sensible defaults were provided throughout, so that all functions can be used with a minimal set of parameters. While treeclim's main focus is on dendrochronological applications, it can benefit the analysis of other annually resolved natural archives, such as corals, ice cores, or varve sediments.

Package functionality

Besides monthly climate predictors, treeclim allows variables aggregated from monthly climate data into the design matrix. A flexible variable selection scheme is used for building ranges, means, and sums of different monthly variables, potentially excluding months, such as dormancy periods, which are a priori considered to be of minor importance. Numerically, treeclim uses the algorithm implemented in DENDROCLIM2002 to calculate response and correlation functions; format of input data is the same as for DENDROCLIM2002 and bootRes. In the case of response functions, the design matrix is orthogonalized so that the regression is performed against principal components of the design matrix, retained according to the PVP criterion (Guiot 1991), which corresponds to the determinant of a correlation matrix of uncorrelated variables. Estimated regression coefficients are then transformed back into the original parameter space (Zang and Biondi 2013). Correlation function analysis uses Pearson's linear correlation computed between the response variable and each subvector of the climate design matrix.

Bootstrap resampling (1000 iterations) is used to test for significant correlations. Three options have been included in treeclim: a) regular bootstrapping (Efron and Tibshirani 1986), as in DENDROCLIM2002; b) stationary bootstrapping (Politis and Romano 1994) to account for temporal autocorrelation, with the length of data blocks chosen from a geometric distribution; and c) exact simulation as in seascorr (Meko et al. 2011), where the proxy data is simulated using circulant embedding (Percival and Constantine 2006). The stationary bootstrap mimics the time series properties of the original data by resampling within blocks. The number of observations in each block follows a geometric probability distribution; thus, the choice of the distribution parameter will affect the autocorrelation structure of the resampled time series. In treeclim, an optimal (expected) block length is chosen following Politis and White (2004) and using functionality from the R package np (Hayfield and Racine 2008). In the case of exact simulation, the proxy series is regarded as a realization of a stationary Gaussian process. After estimating the spectrum of the proxy series via direct Fourier transform and periodogram, independent realizations of the Gaussian process are computed, hence only the predictand is simulated in this approach as in Meko et al. (2011). Significance of the coefficients that connect the predictand with the original predictors is estimated using the percentile range method (Dixon 2001) for regular and stationary bootstrapping, and via empirical nonexceedance probabilities computed by the Weibull formula for exact simulation (Stedinger et al. 1993).

Analogous to the DENDROCLIM2002 approach, treeclim allows exploring the time‐dependency of proxy/climate relationships by applying response and correlation functions in moving window intervals. As a new feature of treeclim, we have incorporated the ability to test whether potential low‐frequency modulation in proxy/climate relationships are significantly stronger or weaker than could be expected by chance. This evaluation is accomplished by the multivariate extension of a Monte‐Carlo scheme introduced by Gershunov et al. (2001). A total of 1000 random data sets are generated, where the climate time series are simulated as Gaussian noise, and the proxy series as a linear combination of the climate parameters using the coefficients of the original correlation function, plus an error component with variance equal to the variance unexplained by the individual climate predictors. For each iteration, a moving correlation function is calculated with the same settings as in the original model. The standard deviation over the individual windows for each climate predictor is then compared to the bootstrapped distribution of standard deviations of the simulated data to test for significantly higher or lower low‐frequency modulations.

Seasonal partial correlations are defined using the dendroclimatic window, i.e. the first and last month of the period used to identify climate predictors, together with the season length, which is then overlapped by one month going back in time. For example, using a season of three months and a dendroclimatic window of 14 months that ends in September of the current year, the first season comprises current July to current September, the second season comprises current June to current August, and so forth going progressively back in time, with the last season including the previous June to the previous August. This scheme generates, for any given season length, as many seasonal intervals as there are months in the dendroclimatic window. Considering two climatic variables, a primary and a secondary one, the correlation of the primary variable with tree‐growth is computed as Pearson's linear correlation coefficient, and then this linear correlation is removed to compute the partial correlation of the secondary variable with the predictand. The significance of each (partial) correlation is evaluated using exact simulation via circulant embedding of the data (Percival and Constantine 2006).

After calibration with climate observations, proxy records can be used to reconstruct climatic conditions for periods prior to the instrumental one. treeclim facilitates transitioning from multivariate exploration of climatic signals in a proxy series, to the identification of a meaningful reconstruction target, to testing the reconstruction skill of the proxy series. Such a last step is accomplished by means of split calibration using either ordinary least squares or ranged major axis regression to better represent symmetric relationships using functionality from R package lmodel2 (Legendre 2013). treeclim reports the results of three established tests in the context of reconstruction assessment: the Durbin–Watson test for autocorrelated residuals as implemented in R package lmtest (Zeileis and Hothorn 2002), the reduction of error, and the coefficient of efficiency (Cook et al. 1994).

Example workflow

Public‐domain data were downloaded from the International Tree‐Ring Data Bank (ITRDB) at the National Oceanic and Atmospheric Administration (NOAA) Paleoclimatology Program and World Data Center for Paleoclimatology (<www.ncdc.noaa.gov/paleo/>) and the Tyndall Centre for Climate Change Research (TYN CY 1.1, Mitchell et al. 2004). Data sets used in the following example are included in treeclim: spai020, a tree‐ring chronology from Pinus sylvestris near Penata, Spain, and spain_prec and spain_temp as the corresponding country‐level mean temperatures and precipitation sums.

Correlation and response functions are calculated using the function dcc, which is also used for the moving interval variants. First, monthly temperature and precipitation from June of the previous year to current September, i.e. the default dendroclimatic window, are used as predictors of tree growth.

This will calculate (regular) bootstrapped response functions, the default option. The order of climate variables must be the same in the climate and var_names arguments. The numeric results can be viewed by either issuing the command penota_resp, or summary(penota_resp) for more details, as shown in the following console output (truncations are indicated by ellipses):

The output shows that some months (e.g. current April temperature) are significantly related (default p = 0.05) to the tree‐ring chronology, as indicated by the logical ‘TRUE’ flag. To demonstrate the flexible variable selection model, a second analysis is run where monthly temperature is used from the previous June to the current October, but excluding the winter months, from previous November to current March. For precipitation, we use here two seasonal sums: one from previous June to previous October, and one from current April to current October.

Note how different aggregation modifiers (.range, .sum, .mean) are chained together by the ‘+‘ sign to form an arbitrarily complex design matrix. The default graph (Fig. 1) of this response function analysis is generated by the command plot(penota_resp2).

Figure 1

Plot of a bootsrapped response function analysis relating tree‐ring growth of a pine population near Penota, Spain, to monthly mean temperature from previous June to current October, but excluding the winter months from previous November to current March, and to two seasonal sums of precipitation, one from previous June to previous October, and one from current April to current October. Abbreviated previous‐year months are given in lowercase letters, current year ones in uppercase letters. Only current April temperature is judged significant by the boostrapping procedure.

To analyze the temporal stability of dendroclimatic relations, in this example we demonstrate the use of moving correlation functions. Here, we specify a dendroclimatic window from March to October of the current year and a 35‐yr moving interval with a 5‐yr offset:

Detailed results can be obtained with summary(penota_moving_cor), and a plot (Fig. 2) can be generated by plot(penota_moving_cor). Results highlight temporally unstable dendroclimatic relations, which often change sign. To evaluate which of these fluctuations is significant, i.e. different from fluctuation we could expect from random time series, the object generated from dcc becomes input to the test function g_test, i.e. g_test(penota_moving_cor). Assessing the relevance of such fluctuations helps investigating complex issues related to the ‘divergence phenomenon’.

Figure 2

Plot of moving correlation function relating tree‐ring growth of a pine population near Penota, Spain, to temperature and precipitation of current March to current October. The moving correlation is carried out in windows of 35 yr, offset by 5 yr. Significant correlations are denoted by asterisks. The naming scheme, i.e. abbreviated year (current or previous) and month, for the climate predictors results in a sequence of names taken from the supplied data or explicitly given by the user. It is evident that the majority of coefficients displays temporal fluctuations, as only a few climate variables, such as current April temperature, do not change correlation sign with tree growth over time.

For the last example we look at seasonal correlations at a site near Visdalen, Norway. The example data are also included in treeclim.

By default, the seascorr function calculates seasonal correlations for lengths of 1, 3, and 6 months, but the user may change this option and can supply an arbitrary number of season lengths. For instance, setting option season_length to c(1:7) will give results for seasons from 1 to 7 months in length. The primary climatic variable is by default the one listed first in the ‘climate’ argument by default. The ‘primary’ and ‘secondary’ arguments to the function allow the user to choose primary and secondary predictors by name or position.

A bar plot (Fig. 3) can be obtained by issuing plot(nor_seascorr). Following the approach of Meko et al. (2011), the Gaussian circulant embedding scheme is not a resampling scheme, but a simulation one, where data are generated with the same spectral characteristics as the original observations. Thus, significance is tested using the exceedance probability, and no confidence interval can be computed, since there is no resampling of the original data. To quickly evaluate a summer temperature reconstruction, one can use the skills function:

Figure 3

Plot of a seasonal correlation analysis relating tree‐ring growth from a pine population near Visdalen, Norway, to seasonal temperature (primary variable, upper panel) and precipitation (secondary variable, lower panel) composites with lengths of 1, 3, and 6 months. Previous year months are in lowercase letters, current year ones in uppercase letters. A positive relation can be recognized between tree growth and temperature of previous‐year autumn and current‐year summer.

Here, the reconstruction performs better than using the mean of the reconstructed period (indicated by RE > 0), and contains useful information (CE > 0, Cook et al. 1994). However – to clarify the limits of this example – the Durbin–Watson statistic suggests autocorrelated residuals, which render the estimated reconstruction questionable. A possible solution would be to remove the time‐series autocorrelation in the proxy data prior to the analysis through a process called prewhitening (Biondi and Swetnam 1987). We refer to package dplR (Bunn 2010) and its function chron to obtain prewithened chronologies from tree‐ring series.

Implementation and installation

treeclim is implemented mostly in R. The speed‐critical parts are written in C++ using the Armadillo linear algebra library interfaced to R with RcppArmadillo (Eddelbuettel and Sanderson 2014). The various plots are generated using package ggplot2 (Wickham 2009).

treeclim is free software released under the GPL ver. 3 copyleft license. The latest stable version of the package can be installed from the Contemporary R Archive Network (<http://cran.r‐project.org/web/packages/treeclim>). From within R, type install.packages(‘treeclim’). The code is hosted publicly on GitHub (<http://github.com/cszang/treeclim>), where users can obtain development snapshots, file bug reports or claim pull requests for code changes.

To cite treeclim or acknowledge its use, cite this Software note as follows, substituting the version of the application that you used for ‘version V’:

Zang, C. and Biondi, F. 2014. treeclim: an R package for the numerical calibration of proxy‐climate relationships. – Ecography 38: 431–436 (ver. V).

Acknowledgements

CZ acknowledges funding by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007‐2013/ERC grant agreement no. 282250). FB was supported, in part, by a Charles Bullard Fellowship in Forest Research awarded by Harvard Univ. to visit Harvard Forest in Petersham, Massachusetts, as well as from the US National Science Foundation under grants P2C2‐0823480, AGS‐EAGER‐1256603, and BSC‐1230329.