Concordance: A Measure of Similarity Between Matrices of Time Series with Applications in Dendroclimatology

Abstract

A fundamental assumption in dendroclimatology is that the common signal produced by multiple trees of the same species, growing under similar environmental conditions within the same climate region, relates to changes in the climate within that region in the same way. However, there are concerns that the climate response of young kauri trees may differ to older kauri. As a result, the inclusion of radii from young kauri may weaken the climate signal of the composite chronology. To address this concern, a subset of the data containing tree rings formed when the trees were young was compared to those formed when the trees were old. These subsets contained time series of correlated tree rings aligned by year with start and end years differing for each series. Existing techniques for comparing subsets of time series lack reliability for ragged arrays of dependent non-stationary time series. The concordance method was developed to overcome this. Concordance is a non-parametric method based on bootstrapping that is used to test the hypothesis that two subsets of time series are similar in terms of mean, variance or both. Simulations show that concordance is effective for detecting difference in both the level and scale of two submatrices containing non-stationary and dependent time series. When applied to tree-ring data, the concordance method was able to detect evidence against the subset of young tree rings having the same mean, variance or both than older, more established trees.

Appendix

1.1 Justification of Size adjustment

Definition 2 provides a size adjustment for the bootstrapped means. This size adjustment ensures that the concordance is independent of the number of series within the subsets. Let $\overline{\varvec{X}}^*$ and $\overline{\varvec{Y}}^*$ be the unadjusted matrices of bootstrapped means and $I_X$ and $I_Y$ be the number of series within each of the subsets.

Theorem 2

The distribution of the bootstrap mean converges to the normal distribution by the central limit theorem, thus,

$$\begin{aligned} \sqrt{I_X} \left( \overline{\varvec{X}}^* - \mathbb {E}(\overline{\varvec{X}}^*) \right) \xrightarrow {d} {\text {N}} (0, \varvec{\sigma ^2_X}), \end{aligned}$$

where $\varvec{\sigma ^2_X}$ is the variance–covariance matrix for $X$.

If $\mu _{X_t}$ and $\sigma _{X_t}$ are the population mean and variance for the subset $X$ and $\mu _{\overline{\varvec{X}}_t^*}$ and $\sigma _{\overline{\varvec{X}}_t^*}$ are the mean and variance of the matrix of bootstrapped means at time $t$. Similarly, using the same notation for $Y$, we have $\mu _{Y_t}$, $\sigma _{Y_t}$, $\mu _{\overline{\varvec{Y}}_t^*}$ and $\sigma _{\overline{\varvec{Y}}_t^*}$. Using Theorem 2 the following definition holds:

Definition 4

Under large samples

$$\begin{aligned} \mu _{\overline{\varvec{X}}_t^*}&= \mu _{X_t},&\sigma _{\overline{\varvec{X}}_t^*}&= \frac{\sigma _{X_t}}{\sqrt{I_{X}}}, \\ \mu _{\overline{\varvec{Y}}_t^*}&= \mu _{Y_t},&\sigma _{\overline{\varvec{Y}}_t^*}&= \frac{\sigma _{Y_t}}{\sqrt{I_{Y}}}. \end{aligned}$$

Assuming that $\sqrt{I_{X}} \approx \sqrt{I_{Y}}$ and $\sigma _{X_t} = \sigma _{Y_t}$, for small samples the comparison will be more influenced by the variance in the populations, and a larger difference in the means is required in order for the concordance to detect a dissimilarity. Figure 3a shows that when the concordance is calculated using the unadjusted bootstrapped means, it is dependent on the number of series within each subset. Therefore, a size adjustment is required.

Thus, let $\breve{\varvec{X}}$ and $\breve{\varvec{Y}}$ be the matrices of adjusted bootstrapped means by Definition 2. Then assuming that $B$, the number of bootstrapped samples, is large,

$$\begin{aligned} \text{ Var } \left[ \breve{\varvec{X}} \right] \longrightarrow I_X \text{ Var } \left[ \overline{\varvec{X}}^* \right] . \end{aligned}$$

Therefore, when $\mu _{\breve{\varvec{X}}}$ and $\sigma _{\breve{\varvec{X}}}$ is the mean and variance of the adjusted bootstrapped mean, then

Definition 5

Under large samples

$$\begin{aligned} \mu _{\breve{\varvec{X}}}&= \mu _{\overline{\varvec{X}}^*} = \mu _{X_t},&\sigma _{\breve{\varvec{X}}}&\approxeq \sqrt{I_X} \sigma _{\overline{\varvec{X}}^*} = \sigma _{X_t}, \end{aligned}$$

(9)

$$\begin{aligned} \mu _{\breve{\varvec{Y}}}&= \mu _{\overline{\varvec{Y}}^*} = \mu _{Y_t},&\sigma _{\breve{\varvec{Y}}}&\approxeq \sqrt{I_Y} \sigma _{\overline{\varvec{Y}}^*} = \sigma _{X_t}. \end{aligned}$$

(10)

The size adjustment produced concordance indices that are independent of the size of the samples. Simulated results for the concordance before and after the size adjustment are shown in Fig. 3. The trend in the concordance is similar for the different size of samples, showing that the concordance is not dependent on sample size.

1.2 R Package for Performing Concordance

An R package ‘dplRCon’ has been developed to implement the concordance and contains examples for performing the concordance.