**4.3 Principal component analysis on the (λ***,* **Q** *,* **d***,* **s) parameters**

In this section a novel approach which invokes a principal component analysis (PCA) of the resultant (*λ*, *Q*, *d*, *s*) parameters (Section 2.2) which details the weight *λ*, *q*, *d* and *s* parameters from the MTD (*n* = 4) models) is performed. The resultant two dimensional PCA axis plots (**Figure 8**) of the rotated (*λ*, *Q*, *d*, *s*)-based PCs provides an informative visualisation of the synchronous and asynchronous species groupings (of *n* > 2 species) allowing for interpretation of the main climate drivers and climatic profiles (e.g.+/ or ( /+)) detailed in **Table 6**.


#### **Table 8.**

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

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

#### **Figure 8.**

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

**Table 7** shows significant positive MTDg-based correlations (*P* < 0.006) for the

*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

**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 Moran-

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

In this section a novel approach which invokes a principal component analysis (PCA) of the resultant (*λ*, *Q*, *d*, *s*) parameters (Section 2.2) which details the weight *λ*, *q*, *d* and *s* parameters from the MTD (*n* = 4) models) is performed. The resultant two dimensional PCA axis plots (**Figure 8**) of the rotated (*λ*, *Q*, *d*, *s*)-based PCs provides an informative visualisation of the synchronous and asynchronous species groupings (of *n* > 2 species) allowing for interpretation of the main climate drivers

**Model species mic pol leu tri** Synchronous fitted species tri (0.12) leu (0.19) pol (0.18) leu (0.26)

) mic (0.10<sup>ϕ</sup>

)

tri (0.33)

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.*

0.591 (**Table 4**), which is close to the threshold for synchrony of 0.65

**4.3 Principal component analysis on the (λ***,* **Q** *,* **d***,* **s) parameters**

and climatic profiles (e.g.+/ or ( /+)) detailed in **Table 6**.

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

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

Asynchronous fitted species leu (0.17<sup>ϕ</sup>

based AR analysis of the flowering intensities.

pairs of species.

*Probability, Combinatorics and Control*

(**Tables 3** and **4**).

asynchrony of 0.80.

*ϕ*

**66**

**Table 8.**

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

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* and *E. polyanthemos*.

## **5. Discussion and conclusion**

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 among pairs of individuals in a population.

*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

geographical area; and earlier studies have shown they overlap significantly [20]. From the flowering behaviour indices of Keatley and Hudson in [23], *E. leucoxylon* and *E. polyanthemos* were shown to have temporally separated months of peak flowering, September and November, respectively; likewise their flowering commencement months May and October, respectively. These two species can occur in the same geographical area and their flowering period. Differentiation of these two species is based on their differing months of peak flowering as well as their separated months of most probable flowering; October and November, respectively. Likewise their flowering commencement months differ, May and October, respectively [23].

peak flowering, and the timing of start and finishing of flowering, as well as possibly specific climate drivers for flowering [4]. The four species studied were shown to be influenced by temperature and rainfall and as a consequence their flowering phenology will change in response to climate change. This in turn will

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

\*, Susan Won Sun Kim<sup>2</sup> and Marie Keatley<sup>3</sup>

3 School of Ecosystem and Forest Sciences, The University of Melbourne,

\*Address all correspondence to: irene.hudson@rmit.edu.au

provided the original work is properly cited.

1 Department of Mathematical Sciences, Royal Melbourne Institute of Technology,

2 South Australian Health and Medical Research Institute, Adelaide, South Australia

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Extensions of the B-MTD models to allow for climate covariates and for the comparison of more than 2 species at a time (a so-called multivariate M-MTD) is the topic of future work. Other forthcoming research is to examine the timing and a/synchronisation of the within species phenostages of both budding and flowering. Refer to earlier work using wavelets [26] and Generalized Additive Model for Location, Scale and Shape (GAMLSS) [27] to model the relationship between climate (mean monthly minimum, maximum temperatures and rainfall) during bud development and the flowering cycles of *Eucalyptus leucoxylon* and *E. tricarpa* from the Maryborough region of Victoria between 1940 and 1962. Monthly behaviour (start, peak, finish, monthly intensity, duration and success) in budding and flowering was assessed using, as in this current chapter, the indices

have an impact on species interactions and community [4].

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

of Keatley in [23].

**Author details**

Melbourne, Australia

Melbourne, Australia

**69**

Irene Hudson<sup>1</sup>

The least degree of synchrony (via the B-MTD rules of synchrony, the MTD models and Moran method) is shown in this chapter to occur between *E. leucoxylon* and *E. microcarpa*; then followed by *E. polyanthemos* and *E. microcarpa*. Our results agree with the findings in [20], which established that a cross between *E. leucoxylon* and *E. microcarpa* is impossible. In terms of climatic signatures: the flowering of *E. microcarpa* behaves differently from *E. leucoxylon* and *E. tricarpa*. *E. microcarpa* flowers at higher temperature and its flowering has a significant and positive relationship with flowering a year ago, refer also to the results reported in [23].

*Eucalyptus tricarpa* and *E. polyanthemos* were shown in this chapter also to be asynchronous (discordant or out of phase). This is in agreement with conclusions reported in [2]. The MTDg model found a significant interaction between two climate variables, mean temperature and rainfall on the flowering of *E. polyanthemos*. As flowering is viewed as either 'off' or 'on' this interaction appears to be delineating *E. polyanthemos'* flowering period. It usually commences flowering in late spring—as mean temperature is increasing and rainfall is decreasing and ceases in early summer; just prior to the warmest mean temperature and lowest rainfall.

Specific temperature thresholds for commencement and for the cessation of flowering for the four species studied here, have been established, see [5, 7, 8]. For example, *E. microcarpa* was shown to flower at high temperatures, and *E. leucoxylon* and *E. tricarpa* both at lower temperatures. The flowering of *E. polyanthemos* was shown to be impacted by both rainfall and temperature, with increased flowering when conditions were either cool and dry, or hot and wet—indicative of a rainfall by temperature interaction.

Moran residual analysis and the B-MTD analysis described in this chapter showed that *E. tricarpa* and *E. microcarpa* did not exhibit a significant synchronous nor an asynchronous relationship. However, for this species pairing, the associated sum of the probabilities for transitions A and B (both off/on to one off/on) is 0.591, which is close to the threshold for synchrony of 0.65. Indeed the more sophisticated MTDg modelling approach which incorporates covariates (mean temperature and rainfall) with interactions, showed that *E. microcarpa* and *E. tricarpa* are synchronous, wherein the MTDg model allows for prior lag 1 to lag 12 month flowering effects and climate covariates.

SOM-based clustering [4] and Moran AR (2) tests also found that *E. polyanthemos* was asynchronous to *E. microcarpa* and *E. tricarpa*, in agreement with the extended Kalman filter (EKF)-based synchrony measures in [15, 21]. Note also it was demonstrated in [20] that *E. polyanthemos* and *E. microcarpa* have 25 years with no overlap (with a long term mean synchrony value of 0.29). Note that the more sophisticated MTDg modelling approach which incorporates covariates (mean temperature and rainfall) with interactions, showed that indeed *E. microcarpa* and *E. tricarpa* are synchronous, wherein the MTDg model allows for prior lag 1 to lag 12 month flowering effects and climate covariates.

Recently synchronisation of eucalypt flowering is shown to be a complex mechanism that incorporates all the flowering elements—flowering duration, timing of

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

peak flowering, and the timing of start and finishing of flowering, as well as possibly specific climate drivers for flowering [4]. The four species studied were shown to be influenced by temperature and rainfall and as a consequence their flowering phenology will change in response to climate change. This in turn will have an impact on species interactions and community [4].

Extensions of the B-MTD models to allow for climate covariates and for the comparison of more than 2 species at a time (a so-called multivariate M-MTD) is the topic of future work. Other forthcoming research is to examine the timing and a/synchronisation of the within species phenostages of both budding and flowering. Refer to earlier work using wavelets [26] and Generalized Additive Model for Location, Scale and Shape (GAMLSS) [27] to model the relationship between climate (mean monthly minimum, maximum temperatures and rainfall) during bud development and the flowering cycles of *Eucalyptus leucoxylon* and *E. tricarpa* from the Maryborough region of Victoria between 1940 and 1962. Monthly behaviour (start, peak, finish, monthly intensity, duration and success) in budding and flowering was assessed using, as in this current chapter, the indices of Keatley in [23].
