**1. Introduction**

Separation or lack of overlap of flowering time in eucalypts has been suggested as a mechanism for maintaining overall 'generic identity' of a plant species. If, for example flowering times and pollinators overlap in sympatric species, hybridization can occur between closely related eucalypts species. Therefore examination of longterm synchrony establishes a baseline of flowering behaviour which may assist in detecting recent or future changes. Although *Eucalyptus* as a genus dominates much of the Australian landscape [1, 2], few studies have quantified eucalypt flowering

overlap within or between species, due to the shortage of phenological data in Australia [3, 4, 7]. This chapter examines flowering synchrony over a 31 year period, 1940–1970, at the population level among four eucalypts species—*Eucalyptus leucoxylon, E. microcarpa, E. polyanthemos* and *E. tricarpa* [3–8].

variants in the case study discussed). The high-order MTD transition probabilities

*P Xt* ¼ *i*0j*Xt*�<sup>1</sup> ¼ *i*1*;* …*Xt*�*<sup>f</sup>* ¼ *if ;C*<sup>1</sup> ¼ *c*1*;* …*Ce* ¼ *ce; M*<sup>1</sup> ¼ *m*1*;* …*Ml* ¼ *ml* � �

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

*l*

*u*¼1

*h*1 � *dh*2*<sup>j</sup> h*2

We refer the reader to [6], and further works in the seminal book by Hudson

Let f g *Xt* and f g *Yt* be sequences of random variables (say two (flowering intensity) time series) taking values in the finite set *N* = {1, …, *k*}. In a *f* th-order Markov

> 0 0 � �, (*i*0*, i*<sup>0</sup>

tion of values taken by *Xt*�*<sup>f</sup> ,* …*, Xt*�<sup>1</sup>*, Yt*�*<sup>f</sup> ,* …*, Yt*�1. In the MTD model, the contributions of the different lags are combined additively. Then a bivariate MTD model,

� �j*Xt*�<sup>1</sup> <sup>¼</sup> *<sup>i</sup>*1*;* …*;Xt*�*<sup>f</sup>* <sup>¼</sup> *if ; Yt*�<sup>1</sup> <sup>¼</sup> *<sup>i</sup>*

<sup>0</sup> ∈ *N*, the probabilities *qig ,i*

(i.e. each row sums to 1 and the elements are nonnegative) and *λ* ¼ *λ<sup>f</sup> ;* …*; λ*<sup>1</sup>

*P*ð *X*1*,t* f g *;* …*;Xn,t* ¼ f g *i*1*,*0*;* …*; in,*<sup>0</sup> ∣*Xt*�<sup>1</sup> ¼ *i*1*,* …*, Xt*�*<sup>f</sup>* ¼ *if , Yt*�<sup>1</sup> ¼ *i*

*h*¼1

vector of lag parameters. To ensure that the results of the model are probabilities,

Covariates and interaction terms can be added to the bivariate MTD (B-MTD) as

� �

*λf*þ*e*þ*usuvui*<sup>0</sup>

) and *Xt*, and where P*f*þ*e*þ*<sup>l</sup>*

<sup>0</sup> ∈ *N*) depends on the combina-

<sup>1</sup>*;* …*; Yt*�*<sup>f</sup>* ¼ *i*

0 *f*

are elements of an *<sup>m</sup>*<sup>2</sup> � *<sup>m</sup>*<sup>2</sup>

0

*λ<sup>f</sup>*þ*e*þ*usuvui*1*,*0*,*…*,in,*<sup>0</sup>

� �<sup>0</sup>

<sup>1</sup>*,* …*, Yt*�*<sup>f</sup>* ¼ *i*

*<sup>g</sup>*¼<sup>1</sup> *<sup>λ</sup><sup>g</sup>* <sup>¼</sup> <sup>1</sup>

0 *f ,*

(3)

(2)

is a

0

, each row of which is a probability distribution

*l*

*u*¼1

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

≤1 the vector *λ* is subject to the constraints P*<sup>f</sup>*

*<sup>λ</sup><sup>f</sup>*þ*hdhjh,i*1*,*0*,*…*,in,*<sup>0</sup> <sup>þ</sup><sup>X</sup>

where *λf*þ*e*þ*<sup>u</sup>* is the weight for the interaction term, *qig <sup>i</sup>*<sup>0</sup> is the transition probability from modality *ig* observed at time *t*�*g* and modality *i*<sup>0</sup> observed at time *t* in the transition matrix *Q*, *suvui*<sup>0</sup> is transition probability between covariate *h*<sup>1</sup> and

(1)

*<sup>g</sup>*¼<sup>1</sup> *<sup>λ</sup><sup>g</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>λ</sup>f*þ*hdhjhi*<sup>0</sup> <sup>þ</sup><sup>X</sup>

are computed as follows:

<sup>¼</sup> <sup>X</sup> *f*

and where *λ<sup>g</sup>* ≥ 0.

*g*¼1

*<sup>λ</sup>gqig <sup>i</sup>*<sup>0</sup> <sup>þ</sup>X*<sup>e</sup>*

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

covariate *h*<sup>2</sup> interaction term (*vu* ¼ *dh*1*<sup>j</sup>*

*h*¼1

and Keatley [7] for further mathematical details.

chain, the probability that *Xt* f g *; Yt* ¼ *i*0*; i*

0 0

*λgqig ,i* 0 *<sup>g</sup> ,i*0*,i* 0 0

> *qig ,i* 0 *<sup>g</sup> ,i*0*,i* 0 0 �

*C*<sup>1</sup> ¼ *c*1*,* …*, Ce* ¼ *ce, M*<sup>1</sup> ¼ *m*1*,* …*, Ml* ¼ *ml*Þ

*<sup>λ</sup>gqi*1*, <sup>g</sup> ,*…*,in, <sup>g</sup> ,i*1*,*0*,*…*,in,*<sup>0</sup> <sup>þ</sup>X*<sup>e</sup>*

*P Xt* f g *; Yt* ¼ *i*0*; i*

where *if ,* …*, i*0*, i*<sup>0</sup>

transition matrix *<sup>Q</sup>* <sup>¼</sup> �

that is, 0≤ P*<sup>f</sup>*

<sup>¼</sup> <sup>X</sup> *f*

*g*¼1

and *λ<sup>g</sup>* ≥0.

follows:

**55**

<sup>¼</sup> <sup>X</sup> *f*

*<sup>g</sup>*¼<sup>1</sup> *<sup>λ</sup>gqig <sup>i</sup>* 0 *<sup>g</sup> i*0*i* 0 0

*g*¼1

*<sup>f</sup> ,* …*, i*<sup>0</sup>

**2.2 The bivariate mixture transition distribution (B-MTD)**

which we denote by B-MTD, has the following formulation:

A new approach to assess synchronicity developed in this chapter is a novel bivariate extension of the of the MTDg model [6, 9] (we coin this B-MTD). The aim of this chapter is to test mixture transition distribution (MTD) and an extended MTDg with interactions; and a novel bivariate extension of MTD (B-MTD) to investigate synchrony in phenological data. The MTDg model [6, 9] was the first approach developed to study the multivariate relationship between the probability of flowering with two states of rain and mean temperature via a mixture transition distribution (MTD), assuming, however a different transition matrix from each lag to the present time (our MTDg analysis), thus generalising the MTD approach in [13], (see also [10]) which led to the development of the MARCH software to perform MTD without covariates [11, 12]. The MTDg model is different to MARCH not only in terms of incorporating interactions between the covariates but also in its minimization process, namely using the AD Model BuilderTM [14], which uses auto-differentiation as a minimisation tool. This is shown to be computationally less intensive than MARCH. The assumption Berchtold's MTD model, namely the assumed equality of the transition matrices among different lags, was a strong assumption, so the idea of the mixture transition distribution model was to consider independently the effect of each lag to the present instead of considering the effect of the combination of lags as in pure Markov chain processes. Specifically, an extended model for MTDg analysis which accommodates interactions was developed in [6], and applied to MTDg modelling of the flowering of four eucalyptus species studied in this chapter, as multivariate time series.

This work extends both MARCH and the work in [15, 16] to allow for differing transition matrices among the lags, i.e. our B-MTD method builds on this approach of the MTDg with interaction model [6, 9]. The MTDg model with interactions showed that the flowering of *E. leucoxylon* and *E. tricarpa* behave similarly with temperature (both flower at low temperature) and both have a positive relationship with flowering intensity 11 months ago. The flowering of *E. microcarpa* behaves differently in that *E. microcarpa* flowers at high temperature.

Our B-MTD approach developed in this chapter allows us the derive rules of thumb for synchrony and asynchrony between pairs of species. The latter B-MTD rules are based on transition probabilities between all possible on and off flowering states from previous to current time. Synchronisation is also tested using residuals from the resultant models via an adaptation of Moran's [17, 18] classical synchrony statistic, incorporating MTDg residuals [17–19].

We also apply the earlier MTDg modelling in [6] using climate covariates and lagged flowering states as predictors to model flowering states (on/off) and thus assess synchronisation using an adaptation of the approach of Moran to the resultant MTDg model and fitted residuals. We compare these MTDg (with covariates) based synchrony measures with our B-MTD results in addition to those using the extended Kalman filter (EKF) [15, 19], based residuals obtained earlier in [21].
