**3. Data**

where *λf*þ*e*þ*<sup>u</sup>* is the weight for the interaction term, *suvui*1*,*0*,*…*,in,*<sup>0</sup> is the transition

and *Yt* are time series that constitute random realizations of two states {0, 1} and the covariates *C*1*,* …*, Ce* are also defined by bivariate states {0, 1}, then the set of all possible states for *Xt* ð Þ *; Yt* is {(0, 0), (0, 1), (1, 0), (1, 1)}. Hence the transition

is a 4 � 4 matrix as specified below.

0, 0 (1) (1, 1) (1, 2) (1, 3) (1, 4) 0, 1 (2) (2, 1) (2, 2) (2, 3) (2, 4) 1, 0 (3) (3, 1) (3, 2) (3, 3) (3, 4) 1, 1 (4) (4, 1) (4, 2) (4, 3) (4, 4)

> 0 0 h i

*Xt* ð Þ *; Yt* **0 (1) 1 (2)** 0, 0 (1) (1, 1) (1, 2) 0, 1 (2) (2, 1) (2, 2) 1, 0 (3) (3, 1) (3, 2) 1, 1 (4) (4, 1) (4, 2)

Moran in [17, 18] suggested that if two series xt and *yt* are synchronous, and if *xt* can be estimated by a model *f*(*x*), the residuals from series *xt* fitted to *f*(*x*), and the residuals from series *yt*, fitted with the same model, but with observations, *yt*, then, *f* (*y*) will be positively correlated. The synchrony of two series can then be examined by testing the significance of the correlation of these two series of residuals (using the same model). Moran used an autoregressive integrated moving average

(ARIMA) model to test synchrony. Moran's theorem suggests that if two (or more) populations sharing a common linear density-dependence (in a so-called renewal process) are disturbed with correlated noise, they will become synchronised with a correlation matching the noise correlation (see details in [4], and also [6, 15, 21]). In this chapter we adopt the *k*th order linear stochastic difference to assess synchrony. Goodness of fit of the second order AR (*k* = 2) model is obtained. The series of residuals can then be found by subtracting the predicted (fitted species) value from the observed series. In summary, synchrony (or otherwise) of two series can be established by performing a test of significance on the correlation coefficient

*<sup>g</sup>*¼<sup>1</sup> *<sup>λ</sup><sup>g</sup>* <sup>¼</sup> 1. For example, if both *Xt*

**0, 0 (1) 0, 1 (2) 1, 0 (3) 1, 1 (4)**

, *h* = 1,…, *e*, are 2 � 4 matrices as below.

**Covariate state**

probability between covariate *h*<sup>1</sup> and covariate *h*<sup>2</sup> interaction term

) and *Xt* ð Þ *; Yt* , and where <sup>P</sup>*f*þ*e*þ*<sup>l</sup>*

**Previous state** ð Þ *Xt*�**<sup>1</sup>***; Yt*�**<sup>1</sup> Current state (***Xt***,** *Yt***)**

(*vu* ¼ *dh*1*<sup>j</sup>*

*h*1 � *dh*2*<sup>j</sup> h*2

matrix *Q* ¼ *qig <sup>i</sup>*

0 *<sup>g</sup> i*0*i* 0 0 h i

*Probability, Combinatorics and Control*

The transition matrices *Dh* ¼ *dhjhi*0*<sup>i</sup>*

**2.3 Synchrony analysis using Moran's approach**

calculated from the two series of residuals as follows:

**56**

Flowering data were sourced from the Box-Ironbark Forest near Maryborough, Victoria, in particular the flowering records of *E. leucoxylon*, *E. microcarpa*, *E. polyanthemos* and *E. tricarpa* (1940 and 1971). Flowering intensity was calculated by using a rank score (from 0 to 5) based on the quantity and distribution of flowering [4, 20, 23].

Flowering intensity scores were dichotomised into two discrete states, namely on and off (1/0) flowering (**Figure 1**) as in [6]. One temperature variant, mean monthly diurnal temperature (MeanT), in addition to the monthly rainfall (Rain) were included as climate covariates in the MTDg models; along with the temperature by rain interaction effect. We used discrete state low/high (lower than median temperature *vs* higher than median temperature) for the temperature variable dichotomies and less/more (less than the median rainfall *vs* more than the median

**Figure 1.** *Flowering of the four eucalypts species.*


**Table 1.**

*Cut-points for climate variables based on medians.*

rainfall) for the rainfall variable. The cut-points for the states or low/high categories of each climate covariate are shown in **Table 1**.
