**4. Results**

#### **4.1 Bivariate MTD (B-MTD) discrete states results**

Four eucalypts species, *E. leucoxylon*, *E. microcarpa*, *E. polyanthemos* and *E. tricarpa* were modelled using the order 1 B-MTD model discussed in Section 2.2 without the inclusion of covariates (such as temperature (variants) and rainfall). These species were paired as follows: *E. leucoxylon* and *E. microcarpa* (LeuMic); *E. leucoxylon* and *E. polyanthemos* (LeuPol); *E. leucoxylon* and *E. tricarpa* (LeuTri) and so on; hence 6 pairs were modelled via B-MTD (see **Table 2**) for the corresponding bivariate transition probabilities (see also **Figure 2**).

The possible states for any pair of species is the set {(0, 0), (0, 1), (1, 0), (1, 1)}, where no flowering is represented as 0 (state = 0 = no flowering) and flowering is represented by a 1 (state = 1 = flowering). Since lag order 1 B-MTD models were used, the mixing probability *λ* is equal to 1.0.

The corresponding transition matrices for the 6 B-MTD models are given in **Table 2**. These transition profiles are also shown schematically as flow diagrams in **Figures 3**–**4**, and also as transition signatures in **Figures 5**–**6**. These shall be discussed in more detail later. The transitions to differing states (from **Table 2**) are shown as arrows (transitions A to F) in the schematic diagram of **Figure 2**. The exact probabilities of such transitions are given by the off diagonal elements of **Table 2** and also shown above or below the arrows in **Figures 3** and **4**.

The transitions have the following intuitive interpretation and associated probability (sum), which are derived from the subcomponents of the transition matrices *Q* (see **Table 2**).


synchronising. However, transitions that lead towards only one species being on or off (flowering) (A and B) and where within a species pair flowering switches

**Species Previous state Current state**

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

LeuMic (0, 0) **0.6667** 0.2280 0.1053 0.0000

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

LeuPol (0, 0) **0.6970** 0.0303 0.2626 0.0101

LeuTri (0, 0) **0.6947** 0.1263 0.1053 0.0737

MicPol (0, 0) **0.7637** 0.1429 0.0879 0.0055

MicTri (0, 0) **0.7975** 0.0316 0.1329 0.0380

PolTri (0, 0) **0.7464** 0.1739 0.0797 0.0000

(0, 1) 0.0000 **0.6000** 0.1333 0.2667 (1, 0) 0.0845 0.0376 **0.8357** 0.0423 (1, 1) 0.0000 0.0612 0.4490 **0.4898**

(0, 1) 0.4444 **0.3889** 0.0000 0.1667 (1, 0) 0.0562 0.0000 **0.7921** 0.1517 (1, 1) 0.1309 0.0952 0.1429 **0.6310**

(0, 1) 0.0455 **0.3636** 0.0000 0.5909 (1, 0) 0.2203 0.0085 **0.7034** 0.0678 (1, 1) 0.0069 0.0069 0.1736 **0.8125**

(0, 1) 0.1818 **0.6705** 0.1023 0.0455 (1, 0) 0.2737 0.0000 **0.6842** 0.0421 (1, 1) 0.0714 0.2141 0.3572 **0.3573**

(0, 1) 0.2232 **0.7500** 0.0179 0.0089 (1, 0) 0.1090 0.0182 **0.5819** 0.2909 (1, 1) 0.0000 0.4259 0.0000 **0.5741**

(0, 1) 0.0719 **0.7842** 0.0360 0.1079 (1, 0) 0.3067 0.0400 **0.6400** 0.0133 (1, 1) 0.0370 0.1482 0.4074 **0.4074**

**(0, 0) (0, 1) (1, 0) (1, 1)**

Note that the probabilities of staying in the same state; e.g. both species continuing to be in a non-flowering state (a (0, 0) to (0, 0) transition); one species flowering off and the other species in the pair with flowering on, (a (0, 1) to (0, 1)

(transitions C) are considered to be asynchronous.

**Figure 2.**

**59**

**Table 2.**

*Subcomponents of possible transitions.*

*Transition matrices for the 6 B-MTD models.*


In this chapter we shall demonstrate that transitions that lead towards both species being off or both species being on (states D, E or F), are considered to be


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

#### **Table 2.**

rainfall) for the rainfall variable. The cut-points for the states or low/high categories

**Climate variables Low (less) High (more)** Mean diurnal temp (°C) ≤13.84 >13.84 Rain (mm) ≤40.45 >40.45

Four eucalypts species, *E. leucoxylon*, *E. microcarpa*, *E. polyanthemos* and *E. tricarpa* were modelled using the order 1 B-MTD model discussed in Section 2.2 without the inclusion of covariates (such as temperature (variants) and rainfall). These species were paired as follows: *E. leucoxylon* and *E. microcarpa* (LeuMic); *E. leucoxylon* and *E. polyanthemos* (LeuPol); *E. leucoxylon* and *E. tricarpa* (LeuTri) and so on; hence 6 pairs were modelled via B-MTD (see **Table 2**) for the corresponding

The possible states for any pair of species is the set {(0, 0), (0, 1), (1, 0), (1, 1)}, where no flowering is represented as 0 (state = 0 = no flowering) and flowering is represented by a 1 (state = 1 = flowering). Since lag order 1 B-MTD models were

The corresponding transition matrices for the 6 B-MTD models are given in **Table 2**. These transition profiles are also shown schematically as flow diagrams in **Figures 3**–**4**, and also as transition signatures in **Figures 5**–**6**. These shall be

discussed in more detail later. The transitions to differing states (from **Table 2**) are shown as arrows (transitions A to F) in the schematic diagram of **Figure 2**. The exact probabilities of such transitions are given by the off diagonal elements of

The transitions have the following intuitive interpretation and associated probability (sum), which are derived from the subcomponents of the transition matrices

• A: transition of both species off to one species on: q(0, 0;0, 1) + q(0, 0, 1, 0)

• B: transition of both species on to one species off: q(1, 1;0, 1) + q(1, 1, 1, 0)

• D: transition of one species off to both species off: q(0, 1;0, 0) + q(1, 0;0, 0)

• E: transition of one species on to both species on: q(0, 1;1, 1) + q(1, 0;1, 1)

In this chapter we shall demonstrate that transitions that lead towards both species being off or both species being on (states D, E or F), are considered to be

• F: transition of one species on/off to both species off/on: q(0, 0;1, 1) +

**Table 2** and also shown above or below the arrows in **Figures 3** and **4**.

• C: species switching states: q(0, 1;1, 0) + q(1, 0; 0, 1)

of each climate covariate are shown in **Table 1**.

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

*Probability, Combinatorics and Control*

**4.1 Bivariate MTD (B-MTD) discrete states results**

bivariate transition probabilities (see also **Figure 2**).

used, the mixing probability *λ* is equal to 1.0.

**4. Results**

**Table 1.**

*Q* (see **Table 2**).

q(1, 1;0, 0)

**58**

*Transition matrices for the 6 B-MTD models.*

#### **Figure 2.**

*Subcomponents of possible transitions.*

synchronising. However, transitions that lead towards only one species being on or off (flowering) (A and B) and where within a species pair flowering switches (transitions C) are considered to be asynchronous.

Note that the probabilities of staying in the same state; e.g. both species continuing to be in a non-flowering state (a (0, 0) to (0, 0) transition); one species flowering off and the other species in the pair with flowering on, (a (0, 1) to (0, 1)

#### **Figure 3.**

*Diagram of transition probabilities for synchronous pairs: LeuTri and LeuPol.*

same (bivariate) state as the previous state (see highlighted transition probabilities on the diagonals). For synchronous species pairs, such as LeuPol, and LeuTri the likelihood of species switching flowering state (states C), i.e. transition from one species flowering in a pair previous state = (0, 1) to the other species flowering, current state = (1, 0) never occurs (transition probability = 0.0000); or the likelihood of the transition from one species flowering to the other species flowering (i.e. a (1, 0) to (0, 1) transition) is rare (0.0000 ≤ transition probability ≤ 0.0085). For asynchronous species pairs such as LeuMic, MicPol, and PolTri, their switching probabilities are significantly higher in that at least one of the transition probabilities from (0, 1) to (1, 0); or from (1, 0) to (0, 1) is greater than 0.036, with

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

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

*Transition probabilities from (0, 0) and (1, 1) states for 6 species pairs.*

Overall for synchronous pairs the probabilities of one species flowering to both

In summary the transitions that lead to both species being off (no flowering) or

or no species flowering, i.e. one off to both off, or one on to both on are high (>0.30). The latter are delineated by D and E transitions in **Figure 2** and **Table 4**. Overall for asynchronous pairs there are high probabilities of both off (or on) to one off (or on). The latter transitions are delineated by A and B in **Figure 2**, with

both species being on (flowering) (transitions D, E or F), are considered to be

associated probability ≥0.076.

**Figure 5.**

**61**

probabilities given in **Tables 3** and **4**.

**Figure 4.** *Diagram of transition probabilities for asynchronous pairs: PolTri, LeuMic and MicPol.*

transition); one species on the other in the pair off (a (1, 0) to (1, 0) transition); and both species continuing to flower (a (1, 1) to (1, 1) transition) are not shown on **Figure 2**. These to same states transitions, are given for each species, by the diagonal elements in the transition matrices (from previous to current states) in **Table 2**; and are also shown in **Figures 3** and **4** as numbers (positioned next to the 4 states as boxes).

An examination of the transition probabilities for the species pairs in **Table 2** shows that there is a significantly high propensity (probability) to remain in the

**Figure 5.** *Transition probabilities from (0, 0) and (1, 1) states for 6 species pairs.*

same (bivariate) state as the previous state (see highlighted transition probabilities on the diagonals). For synchronous species pairs, such as LeuPol, and LeuTri the likelihood of species switching flowering state (states C), i.e. transition from one species flowering in a pair previous state = (0, 1) to the other species flowering, current state = (1, 0) never occurs (transition probability = 0.0000); or the likelihood of the transition from one species flowering to the other species flowering (i.e. a (1, 0) to (0, 1) transition) is rare (0.0000 ≤ transition probability ≤ 0.0085). For asynchronous species pairs such as LeuMic, MicPol, and PolTri, their switching probabilities are significantly higher in that at least one of the transition probabilities from (0, 1) to (1, 0); or from (1, 0) to (0, 1) is greater than 0.036, with associated probability ≥0.076.

Overall for synchronous pairs the probabilities of one species flowering to both or no species flowering, i.e. one off to both off, or one on to both on are high (>0.30). The latter are delineated by D and E transitions in **Figure 2** and **Table 4**. Overall for asynchronous pairs there are high probabilities of both off (or on) to one off (or on). The latter transitions are delineated by A and B in **Figure 2**, with probabilities given in **Tables 3** and **4**.

In summary the transitions that lead to both species being off (no flowering) or both species being on (flowering) (transitions D, E or F), are considered to be

transition); one species on the other in the pair off (a (1, 0) to (1, 0) transition); and both species continuing to flower (a (1, 1) to (1, 1) transition) are not shown on **Figure 2**. These to same states transitions, are given for each species, by the diagonal elements in the transition matrices (from previous to current states) in **Table 2**; and are also shown in **Figures 3** and **4** as numbers (positioned next to the 4 states as

*Diagram of transition probabilities for asynchronous pairs: PolTri, LeuMic and MicPol.*

An examination of the transition probabilities for the species pairs in **Table 2** shows that there is a significantly high propensity (probability) to remain in the

boxes).

**60**

**Figure 4.**

**Figure 3.**

*Diagram of transition probabilities for synchronous pairs: LeuTri and LeuPol.*

*Probability, Combinatorics and Control*

• Two species are synchronous if P(D or E) > 0.65, i.e. P(one on to both on)

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

• Two species are asynchronous if P(A or B) > 0.80, i.e. P(both off to one

• A: transition of both species off (in the past state) to one species flowering (on)

According to the rules given in **Table 3**, the synchronous pairs are LeuTri and LeuPol (with P(D or E) > 0.65); asynchronous pairs are: PolTri, LeuMic and MicPol

In summary we have a simple rule for (a) synchrony, which in agreement with the work of [6] (see also [25]), using the synchronisation theory of Moran that:

(with P(A or B) > 0.80) and a species pair that is neither synchronous nor

• *E. leucoxylon* flowering is synchronous with both *E. polyanthemos* and

• *E. microcarpa* is synchronous with none of three species; specifically it is asynchronous with both *E. leucoxylon* and *E. polyanthemos* (and has no

**Transition probability sums P(A) P(B) P(A or B) P(C) P(D) P(E) P(D or E) P(F)**

LeuTri 0.232 0.181 0.008 0.266 0.659 0.081 LeuPol 0.293 0.238 0.000 0.501 0.318 0.141

PolTri 0.254 0.556 0.076 0.379 0.121 0.037 LeuMic 0.333 0.510 0.171 0.085 0.309 0.000 MicPol 0.231 0.571 0.102 0.455 0.088 0.077

MicTri 0.165 0.426 0.036 0.332 0.300 0.038

*Transition probabilities of events A to F for each species pair categorised into synchronous and asynchronous (or*

The transitions have the following interpretation and probabilities

+ P(one off to both off) > 0.65,

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

off) + P(both on to one on) > 0.8.

• B: transition of both species on to one species off;

• D: transition of one species off to both species off;

• E: transition of one species on to both species on;

*E. tricarpa*, but asynchronous with *E. microcarpa*.

relationship with *E. tricarpa)*.

• F: transition of one species on/off to both species off/on.

(**Tables 3** and **4**):

in the current state;

asynchronous is MicTri.

**Synchronous (S) pairs**

**Asynchronous (A) pairs**

**Neither S nor A**

*neither) species pairs.*

**Table 4.**

**63**

• C: species switching states;

**Figure 6.** *Transition probabilities from (0, 1) and (1, 0) states for 6 species pairs.*


#### **Table 3.**

*Descriptions and rules of (a) synchrony based on the transitions A-F.*

synchronizing. However, transitions that lead to only one species being on or off (no flowering) (transitions A and B) and where a species pairs' flowering status switches (transitions C) are considered to be asynchronous.

We now provide a rule for synchrony (or asynchrony) based on subcomponent (sums) of the transition probabilities derived from the B-MTD model:

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


The transitions have the following interpretation and probabilities (**Tables 3** and **4**):


According to the rules given in **Table 3**, the synchronous pairs are LeuTri and LeuPol (with P(D or E) > 0.65); asynchronous pairs are: PolTri, LeuMic and MicPol (with P(A or B) > 0.80) and a species pair that is neither synchronous nor asynchronous is MicTri.

In summary we have a simple rule for (a) synchrony, which in agreement with the work of [6] (see also [25]), using the synchronisation theory of Moran that:



#### **Table 4.**

*Transition probabilities of events A to F for each species pair categorised into synchronous and asynchronous (or neither) species pairs.*

synchronizing. However, transitions that lead to only one species being on or off (no flowering) (transitions A and B) and where a species pairs' flowering status

We now provide a rule for synchrony (or asynchrony) based on subcomponent

switches (transitions C) are considered to be asynchronous.

*Descriptions and rules of (a) synchrony based on the transitions A-F.*

*Transition probabilities from (0, 1) and (1, 0) states for 6 species pairs.*

**Description Probability: sum of**

**subcomponents**

<sup>C</sup><sup>ϕ</sup> Switching q(0, 1;1, 0) + q(1,0;0, 1) <sup>&</sup>lt;0.05▼ <sup>≥</sup>0.05▲

<sup>A</sup> Both off to one off q(0, 0;0, 1) + q(0, 0;1, 0) <sup>&</sup>lt;0.30▼ <sup>≥</sup>0.30▲ P(A or B) <sup>&</sup>gt;

asynchrony <sup>B</sup> Both on to one on q(1, 1;0, 1) + q(1, 1;1, 0) <sup>&</sup>lt;0.50▼ <sup>≥</sup>0.50▲

<sup>D</sup><sup>ϕ</sup> One off to both off q(0, 1;0, 0) + q(1, 0;0, 0) <sup>≥</sup>0.40▲ <sup>&</sup>lt;0.40▼ P(D or E) <sup>&</sup>gt;

synchrony <sup>E</sup> One on to both on q(0, 1;1, 1) + q(1, 0;1, 1) <sup>≥</sup>0.40▲ <sup>&</sup>lt;0.40▼

q(0, 0;1, 1) + q(1, 1;0, 0) <sup>≥</sup>0.08▲ <sup>&</sup>lt;0.08▼

**Threshold for synchrony**

**Threshold for asynchrony** **Rules**

0.8 for

0.65 for

**Figure 6.**

*ϕ*

**62**

**Table 3.**

**Transition names**

F Both on (off) to

*Probability, Combinatorics and Control*

both off (on)

*Events or transitions C and F do not occur often.*

(sums) of the transition probabilities derived from the B-MTD model:


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