**4.2 Moran tests on residuals of the MTDg models incorporating climatic covariates**

In this section synchronisation among species pairs is tested using Moran's correlation method on the cross-residuals, based on MTDg models which incorporate both climate covariates and lagged effects of previous flowering. This work is based on [16], where MTDg models allowing interactions were fitted to the same four species. We present here only MTDg models with two covariates, namely, mean temperature and rainfall.

Parameters of the MTDg models are shown in **Table 5**. Significant lag effects of previous flowering states (*lag j*, where *j* = 1, ..., 12 months), and of the climatic covariates (*meanT* and *rain*) and their interaction (*meanT\*rain*) are also given in **Table 5**. The estimated parameters for the MTDg models generally show a (positive) 1 month lag effect and 9, 11 and 12 months lag effects of previous flowering status (**Table 5**).

From **Tables 5** and **6** we observe that mean diurnal temperature (*meanT*) has a significant effect on flowering for all species; *rain* impacts significantly only on *E. tricarpa* (Tri) and an interaction effect between *rain* and *meanT* exists for *E. polyanthemos* (Pol). Overall, flowering increases as temperature (*MeanT*)


increases for *E. microcarpa*; and flowering decreases as temperature increases for both *E. leucoxylon* and *E. tricarpa*. Rainfall positively impacts the flowering of

*E. polyanthemos* exhibits increased flowering at low *meanT* when there is contemporaneous below average rainfall and at high *meanT* with above average rainfall

In what follows we denote the species used to estimate the parameters for the

*E. tricarpa* (i.e. flowering increases with more rainfall). Interestingly

*E. polyanthemos* (i.e. (0.88, 0.12, 0.20, 0.96)) in **Table 6**.

*Synchrony relationships among the four eucalypts species.*

**Species Climate**

*E. poly* Inter-

*1*

*2*

*3*

*4*

*ϕ*

**Table 7.**

**Figure 7.**

**65**

**Table 6**.

**effects**

**(temp/ rain)**

action

*Cut point for less rain: rain 40.44 mm.*

*Cut point for more rain: rain >40.45 mm. Note that '-' indicates cells with zero probabilities.*

*Cut point for low temperature states: MeanT 13.83°C.*

*Cut point for high temperature states: MeanT >13.84°C.*

*Transition probabilities of flowering for the meanT and rain MTDg models.*

*A negative and significant correlation indicates an asynchronous species pair.*

*Significant Moran correlations (in brackets) from the MTDg models.*

**Previous flowering**

*DOI: http://dx.doi.org/10.5772/intechopen.88554*

**Off On Low1 High<sup>2</sup> Less<sup>3</sup> More<sup>4</sup> Low/**

*Mixture Transition Distribution Modelling of Multivariate Time Series of Discrete State…*

*E. mic* (+/) 0.00 1.00 0.00 1.00 0.39 0.28 - - - -

*E. leu* (/+) 0.05 1.00 1.00 0.00 0.88 0.94 - - - - *E. tri* (/+) 0.00 1.00 1.00 0.00 0.00 1.00 - - - -

**Model species mic pol leu tri** Synchronous fitted species tri (0.14) leu (0.14) pol (0.16) mic (0.15)

Asynchronous fitted species mic (0.14<sup>ϕ</sup>

**Temperature Rain Temperature by rain interaction**

**less**

0.01 1.00 0.00 0.34 0.94 0.03 0.88 0.12 0.20 0.96

**Low/ more**

tri (0.11)

)

**High/ less**

**High/ more**

(see the transition probabilities to flowering for the interaction effect of

MTD-based equation as the 'Model species' and the species fitted with these

*ϕ Covariate effects above 0.03 are considered significant.*

*- indicates cells with zero probabilities.*

#### **Table 5.**

*MTDg mixing probabilities of MeanT and rain models.*

*Mixture Transition Distribution Modelling of Multivariate Time Series of Discrete State… DOI: http://dx.doi.org/10.5772/intechopen.88554*


*1 Cut point for low temperature states: MeanT 13.83°C.*

*2 Cut point for high temperature states: MeanT >13.84°C.*

*3 Cut point for less rain: rain 40.44 mm.*

*4 Cut point for more rain: rain >40.45 mm.*

*Note that '-' indicates cells with zero probabilities.*

#### **Table 6**.

• *E. polyanthemos* flowering is synchronous only with *E. leucoxylon*; and

flowering effects and climate covariates (see also **Table 7** and **Figure 7**).

**4.2 Moran tests on residuals of the MTDg models incorporating climatic**

In this section synchronisation among species pairs is tested using Moran's correlation method on the cross-residuals, based on MTDg models which incorporate both climate covariates and lagged effects of previous flowering. This work is based on [16], where MTDg models allowing interactions were fitted to the same four species. We present here only MTDg models with two covariates, namely,

Parameters of the MTDg models are shown in **Table 5**. Significant lag effects of

From **Tables 5** and **6** we observe that mean diurnal temperature (*meanT*) has a significant effect on flowering for all species; *rain* impacts significantly only on *E. tricarpa* (Tri) and an interaction effect between *rain* and *meanT* exists for *E. polyanthemos* (Pol). Overall, flowering increases as temperature (*MeanT*)

**Species lag 1 lag 9 lag 10 lag 11 lag 12 Temp variable Rain Temp rain** *E. mic* 0.534 - - 0.032<sup>ϕ</sup> 0.275 0.136 - - *E. poly* 0.530 0.060 - 0.160 0.105 0.091 0.009 0.045 *E. leu* 0.611 - - 0.124 0.042 0.202 - - *E. tri* 0.617 0.059 0.009 0.096 - 0.157 0.062 -

previous flowering states (*lag j*, where *j* = 1, ..., 12 months), and of the climatic covariates (*meanT* and *rain*) and their interaction (*meanT\*rain*) are also given in **Table 5**. The estimated parameters for the MTDg models generally show a (positive) 1 month lag effect and 9, 11 and 12 months lag effects of previous flowering

• *E. tricarpa* flowering is synchronous only with that of *E. leucoxylon*; and is asynchronous with *E. polyanthemos* (and has no relationship with

We can view **Figure 5** as the transition signatures from past states, where both species flowering is off or both species flowering is on, for synchronous pairings (LeuTri or LeuPol) and the asynchronous species pairs (PolTri, LeuMic and MicPol). **Figure 6** likewise delineates transition signatures from past states, where only one species of the pair is flowering. These signatures (**Figures 5** and **6**) distinctly differ according to whether a species pair is synchronous or asynchronous. For MicTri the associated sum of the probabilities for transitions A and B (both off/on to one off/on) is 0.591 (see **Table 4**), which is close to the threshold for synchrony of 0.65. Note that the more sophisticated MTDg modelling approach in Section 4.2 which incorporates covariates (mean temperature and rainfall) with interactions, shows that indeed *E. microcarpa* and *E. tricarpa* are synchronous (**Tables 6** and **7**), wherein the MTDg model allows for prior lag 1 to lag 12 month

asynchronous with both *E. microcarpa* and *E. tricarpa*.

*E. microcarpa*).

*Probability, Combinatorics and Control*

**covariates**

status (**Table 5**).

*ϕ*

**64**

**Table 5.**

mean temperature and rainfall.

*Covariate effects above 0.03 are considered significant.*

*MTDg mixing probabilities of MeanT and rain models.*

*- indicates cells with zero probabilities.*

*Transition probabilities of flowering for the meanT and rain MTDg models.*


#### **Table 7.**

*Significant Moran correlations (in brackets) from the MTDg models.*

**Figure 7.**

*Synchrony relationships among the four eucalypts species.*

increases for *E. microcarpa*; and flowering decreases as temperature increases for both *E. leucoxylon* and *E. tricarpa*. Rainfall positively impacts the flowering of *E. tricarpa* (i.e. flowering increases with more rainfall). Interestingly *E. polyanthemos* exhibits increased flowering at low *meanT* when there is contemporaneous below average rainfall and at high *meanT* with above average rainfall (see the transition probabilities to flowering for the interaction effect of *E. polyanthemos* (i.e. (0.88, 0.12, 0.20, 0.96)) in **Table 6**.

In what follows we denote the species used to estimate the parameters for the MTD-based equation as the 'Model species' and the species fitted with these

estimated parameters as the 'Fitted species'. **Table 7** gives the resultant significant Moran correlations based on the residual series from the MTDg-based model and fitted species equations. Significant Moran correlations from both the MTDg (and the EKF models show that (a)synchronous pairings found via the MTD and EKF models in [15–19] generally agree (**Tables 7** and **8**); refer also to **Figure 7**, where a solid line indicates synchronous pairs and a dashed line indicates asynchronous pairs of species.

**Table 7** shows significant positive MTDg-based correlations (*P* < 0.006) for the following (model species: fitted species) pairs—(LeuPol), (PolLeu), (LeuTri), (MicTri) and (TriMic), indicating that *E. leucoxylon* is synchronous with *E. polyanthemos*, in agreement with the rules of synchrony described earlier (**Tables 3** and **4**). *E. leucoxylon* is synchronous with *E. tricarpa*; and that *E. microcarpa* and *E. tricarpa* are synchronous. The synchrony of the latter species pair (MicTri) however, contrasts the results of Moran-based results on raw intensity profiles which indicate that *E. microcarpa* and *E. tricarpa* were neither synchronous or asynchronous (**Table 4**). It is noteworthy however, that for this species pairing, *E. microcarpa and E. tricarpa* (i.e. MicTri or TriMic), the associated sum of the probabilities for transitions D and E (one species off/on to both species off/on) is 0.591 (**Table 4**), which is close to the threshold for synchrony of 0.65 (**Tables 3** and **4**).

**Tables 7** and **8** shows significant negative-based correlations (*P* < 0.001) for the following (model species: fitted species) pairs; (LeuMic), (PolMic) and (MicLeu) indicating that that *E. leucoxylon* is asynchronous with *E. microcarpa* and *E. microcarpa* is asynchronous with *E. polyanthemos* (only via the EKF-based residuals) (**Figure 7** RHS); in agreement with the rule for asynchrony (**Table 4**) and Moranbased AR analysis of the flowering intensities.

The resulting parameters estimated from the MTDg models with and without interaction terms can be compared among all four species using **Figure 8**, which shows that the separation of *E. tricarpa* (/+) and *E. microcarpa* (+/) from other species along the horizontal axis 1, is due to the effect of mean temperature. Although *E. leucoxylon* is affected by the similar lag 1 and 11 month flowering terms as *E. polyanthemos*, *E. leucoxylon* (/+) commences flowering at low temperature and shuts down at high temperatures. *E. microcarpa* begins flowering at high temperature (+/). **Figure 8** also displays the similarity (synchronicity) of *E. leucoxylon*

*Distances in the (λ, Q, d, s) parameters among the 4 species—without interaction terms (left) or with*

*Mixture Transition Distribution Modelling of Multivariate Time Series of Discrete State…*

*DOI: http://dx.doi.org/10.5772/intechopen.88554*

The highest degree of synchrony (via the B-MTD rules of synchrony, the MTD models and Moran AR method) occurs between *E. leucoxylon* and *E. tricarpa*; then followed by *E. polyanthemos* and *E. leucoxylon* which indicates the potential for intense competition for potential pollinators, and therefore the prospect for a high level of hybridization. Both these species pairs were shown to be synchronous by Keatley et al., [20]; with *E. leucoxylon* and *E. tricarpa* having 6 years of no overlap (and a long term mean synchrony value of 0.62); and *E. polyanthemos* and *E. leucoxylon* having 5 years of the 31 years (between 1940 and 1970) with no overlap (long term mean synchrony value of 0.51); as quantified in [20]. The degree of synchrony or overlap of flowering was however determined using the method outlined in [22] which measures the extent of overlapping in the flowering periods

*E. leucoxylon* is the only species to synchronise flowering with *E. tricarpa*, as shown by all three methods, namely the B-MTD rules of synchrony, MTD models and Moran's AR method. Synchrony between *E. leucoxylon* and *E. tricarpa*, may be explained in terms of niche/competition and also facilitation may be a factor, due to their different modes of flower production. This agrees with the findings of [20]. Interestingly the MTD models discussed here (see also [6, 16, 24]) show that the climatic drivers or signature of *E. leucoxylon* and *E. tricarpa* is similar with respect to temperature, in that both exhibit decreased flowering with increased temperature. Likewise *E. leucoxylon* is the only species to synchronise flowering with *E. polyanthemos. E. leucoxylon* and *E. polyanthemos* sometimes occur in the same

and *E. polyanthemos*.

*interaction terms (right).*

**Figure 8.**

**67**

**5. Discussion and conclusion**

among pairs of individuals in a population.

Both the MTDg- and EKF-based models show that *E. tricarpa* is not asynchronous with *E. polyanthemos* (**Tables 7** and **8**). Note that for this species pairing *E. tricarpa* and *E. polyanthemos* (i.e. TriPol and PolTri) the associated sum of the probabilities for transitions P(A) and P(B) (both species off/on to one species off/ on) is equal to 0.802 (**Table 4**), which is just above the to the threshold for asynchrony of 0.80.
