SOM Clustering and Modelling of Australian Railway Drivers’ Sleep, Wake, Duty Profiles

Abstract

Two SOM ANN approaches were used in a study of Australian railway drivers (RDs) to classify RDs’ sleep/wake states and their sleep duration time series profiles over 14 days follow-up. The first approach was a feature-based SOM approach that clustered the most frequently occurring patterns of sleep. The second created RD networks of sleep/wake/duty/break feature parameter vectors of between-states transition probabilities via a multivariate extension of the mixture transition distribution (MTD) model, accommodating covariate interactions. SOM/ANN found 4 clusters of RDs whose sleep profiles differed significantly. Generalised Additive Models for Location, Scale and Shape of the 2 sleep outcomes confirmed that break and sleep onset times, break duration and hours to next duty are significant effects which operate differentially across the groups. Generally sleep increases for next duty onset between 10 am and 4 pm, and when hours since break onset exceeds 1 day. These 2 factors were significant factors determining current sleep, which have differential impacts across the clusters. Some drivers groups catch up sleep after the night shift, while others do so before the night shift. Sleep is governed by the RD’s anticipatory behaviour of next scheduled duty onset and hours since break onset, and driver experience, age and domestic scenario. This has clear health and safety implications for the rail industry.

Appendix A: Mathematics of M-MTDg

1.1 A.1 The Mixture Transition Distribution (MTD) Model

The classic Markov chain is a probabilistic model that represents dependences between successive observations of a random variable (usually over time). In this chapter a discrete state random variable (or multivariate analogues of) taking values in the finite set {1,…, m} is considered in the MTD formulation, which allows for a covariate interaction and modelling of high-order Markov chains (from a time series viewpoint). Markov chains are traditionally used to predict the current value as a function of the previous observations of this same variable (the so-called lagged dependency). The Markov chain was introduced by Andrej A. Markov [126] at the beginning of the twentieth century and has wide applicability in many areas such as: mathematical biology [127], internet applications [128], economics [129], meteorology [130], geography [131], biology [132], chemistry [133], physics [134], behavioural science [135], social sciences [136] and music [137]. For a comprehensive treatment of Markov chains and early applications see Bremaud [138]. Seneta [139] provides an account of Markov’s motivations including an excellent discussion of the early development of the theory.

Raftery [140] introduced the mixture transition distribution (MTD) model to model high-order Markov chains. Berchtold [135, 141–149] subsequently developed software (called Markovian Models Computation and Analysis, MARCH) to model Markov chains using a suite of methods including the MTD and the double chain Markov model. The MTD model has been applied to genomic sequence and time series data [147, 150–153].

The aim of part A of this chapter is to study the multivariate relationship between the probability of sleep with 4-states each of sleep onset times, break onset times, next duty onset times, break duration, and time of sleep via a multivariate mixture transition distribution (M-MTD) which accommodates a different transition matrix from each lag to the present (MTD_g) analysis [154]. The issue of accommodating for interaction terms between covariates itself had not, till the work of Kim [155] and of Hudson et al. [156], been addressed in the MTD [140], MTD_g nor MARCH [149] literature. Kim et al. [154, 157] and Hudson et al. [158] first introduced the concept of interactions based on work in this current chapter, along with GAMLSS [159].

The idea of the original 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. The assumption behind the MTD model, namely the assumed equality of the transition matrices among different lags, is a strong assumption. We further extend the MTD_g model to allow for interactions (between break, duration and next onset times with sleep times) to account for changes in the transition matrices amongst the differing covariates.

This work extends both the MARCH MTD software of Berchtold [149] and the previous work Hudson et al. [156] and of Kim et al. [160–162]. Our model is different to MARCH in terms of incorporating interactions between the covariates and also in its minimisation process [155]. It uses the AD Model Builder^TM [163], [164]. This M-MTD adaptation also utilises auto-differentiation as a minimisation tool, and was shown to be computationally less intensive than MARCH (see the papers of Kim et al. [161, 162] and Kim [155]). The AD Model Builder^TM platform (see Fournier et al. [164]) has great application in fisheries research and recently in computational mathematics and operations management (e.g. electronic systems models [165, 166]).

1.1.1 A.1.1 The MTD Model

Let {Y _i } be a sequence of random variables taking values in the finite set N = {1,…, m}. In a lth-order Markov chain, the probability that i _t , …, i _o ∈ N depends on the combination of values taken by X _t-l , …, X _t-1 . In the MTD model, the contributions of the different lags are combined additively, as follows:

$$ \begin{aligned} & P\left( {X_{t} = i_{0} |X_{0} = i_{t} , \ldots ,X_{t - 1} = i_{1} } \right) \\ & = \sum\limits_{g = 1}^{l} {\lambda_{g} P\left( {X_{t} = i_{0} |X_{t - g} = i_{g} } \right) = \sum\limits_{g = 1}^{l} {\lambda_{g} q_{{i_{g} i_{0} }} } } \\ \end{aligned} $$

(A.1)

where i _t , …, i _o ∈ N, and where the probabilities $q_{{i_{g} i_{0} }}$ are elements of a m × m transition matrix $Q = \left[ {q_{{i_{g} i_{0} }} } \right]$, each row of which is a probability distribution (i.e., each row sums to 1 and the elements are nonnegative) and $\lambda = \left( {\lambda_{l} , \ldots ,\lambda_{1} } \right)^{\prime }$ is a vector of lag parameters, such that

$$ 0 \le \sum\limits_{g = 1}^{l} {\lambda_{g} q_{{i_{g} i_{0} }} \le 1} $$

The vector λ is made subject to the following constraints, $\sum\limits_{g = 1}^{l} {\lambda_{g} = 1} ,$ and $\lambda_{g} \ge 0$. Equation (A.1) gives the probability for each individual combination of i _l , …, i ₀ . The model can also be written in matrix form, giving the whole distribution of X _t [19].

Each row of the transition matrix Q is a probability distribution and as such sums to 1, where the matrix has m(m-1) independent parameters. In addition, a lth-order model has l lag parameters λ₁,…, λ _l , but only (l− 1) of them are independent. Thus a lth order MTD model has m (m – 1) + (l – 1) independent parameters, which is far more parsimonious than the corresponding fully parameterised Markov chain which has m ^l(m – 1) parameters. Moreover, each additional lag in a MTD model adds only one extra parameter. For the basic MTD model of Raftery [15], the same transition matrix Q is used to model the relationship between any of the lags and the present state.

1.1.2 A.1.2 The MTD_g Model

Let {Y _i } be a sequence of random variables taking values in the finite set N = {1,…, m}. In an lth-order Markov chain, the probability that $X_{t} = i_{0} \,,\,i_{0} \in N$, depends on the combination of values taken by $Y_{t - l} , \ldots ,Y_{t - 1}$. In the basic MTD model, the same transition matrix Q is used to model the relation between any of the lags and the present. The idea of the mixture transition distribution (MTD) model is to consider independently the effect of each lag to the present instead of considering the effect of the combination of lags (Fig. A.1), as in the case of the more traditional pure Markov chain process. The constraints imposed by the use of only one transition matrix to represent the relation between each lag and the present is sometimes too strong to allow good modeling of the real high-order transition matrix. In this case, it is possible to replace the basic MTD model by an MTD_g model. The principle of the MTD_g model is to use a different transition matrix of size (k × k) to represent the relationship between each lag and the present. The high-order transition probabilities are then written as follows,

Fig. A.1

Comparison between a 3rd order Markov chain and its MTD model analogue. In a real high-order Markov chain, the combination of all lags influences the probability of the present

$$ P\left( {Y_{t} = i_{0} |Y_{t - 1} = i_{1} , \ldots ,Y_{t - f} = i_{f} } \right) = \sum\limits_{g = 1}^{f} {\lambda_{g} q_{{gi_{g} i_{0} }} } $$

where $q_{{gi_{g} i_{0} }}$ is the transition probability from modality i _g observed at time t-g and modality i ₀ observed at time t in the transition matrix Q _g associated with the gth lag. In addition to the lag weight vector $\left[ {\lambda_{1} , \ldots ,\lambda_{f} } \right]$, the MTD_g model implies the estimation of f transition matrices Q ₁ ,…,Q _f , for a total of f _k (k − 1) + (f − 1) independent parameters. This is much more than was involved in the basic MTD model, but this number of parameters remains small compared to the number of independent parameters of a real fully parameterised fth order Markov chain; thus the MTD_g and its extensions model prove useful in many situations, as shown in this chapter. Here, the contribution of each lag upon the present is considered independently. The MTD model thus approximates high-order Markov chains with far fewer parameters than the fully parameterised model. Though Markov chains are well suited to represent high-order dependencies between successive observations (of a random variable), as the order l of the chain and the number of possible values m increase, the number of independent parameters increases exponentially. The problem then becomes too large to be estimated efficiently, as is often the case for data sets of the size typically encountered in practice [145].

1.1.3 A.1.3 The MTD_g Model with Interactions

The MTD_g model with interactions can also have a different transition matrix of size (k × k) to represent the relationship between each lag and the present [155]. The high-order transition probabilities are then computed as follows

$$ \begin{aligned} & P\left( {Y_{t} = i_{0} |Y_{t - 1} = i_{1} , \ldots ,Y_{t - f} = i_{f} ,C_{1} = c_{1} , \ldots ,C_{e} = c_{e} ,M_{1} = m_{1} , \ldots ,M_{l} = m_{l} } \right) \\ & = \sum\limits_{g = 1}^{f} {\lambda_{g} q_{{gi_{g} i_{0} }} } + \sum\limits_{h = 1}^{e} {\lambda_{f + h} d_{{hj_{h} i_{0} }} } + \sum\limits_{u = 1}^{l} {\lambda_{f + e + u} s_{{uv_{u} i_{0} }} } \\ \end{aligned} $$

where $\lambda_{f + e + u}$ is the weight for the interaction term, $q_{{gi_{g} i_{0} }}$ is the transition probability from modality i _g observed at time t–g and modality i ₀ observed at time t in the transition matrix Q _g associated with the gth lag, $s_{{uv_{u} i_{0} }}$ is transition probability between covariate h ₁ and covariate h ₂ interaction term ($v_{u} = d_{{h_{1} j_{{h_{1} }} }} \times d_{{h_{2} j_{{h_{2} }} }}$) and Y _t , and where $\sum\limits_{g = 1}^{f + e + l} {\lambda_{g} = 1}$ and $\lambda_{g} \ge 0$.

1.1.4 A.1.4 Parameter Estimation

The parameters λ and q of the MTD_g model can be estimated by minimising the negative the log-likelihood (NLL) of the model:

$$ NLL = - \sum\limits_{{i_{l} , \ldots ,i_{0} = 1}}^{m} {n_{{i_{l} , \ldots ,i_{0} }} \log \left( {\sum\limits_{g = 1}^{f} {\lambda_{g} q_{{gi_{g} i_{0} }} } + \sum\limits_{h = 1}^{e} {\lambda_{f + h} d_{{hj_{h} i_{0} }} } + \sum\limits_{u = 1}^{l} {\lambda_{f + e + u} s_{{uv_{u} i_{0} }} } } \right)} $$

where $n_{{i_{l} , \ldots ,i_{0} }}$ is the number of sequences of the form

$$ Y_{t - 1} = i_{1} , \ldots ,Y_{t - f} = i_{f} ,C_{1} = c_{1} , \ldots ,C_{e} = c_{e} ,M_{1} = m_{1} , \ldots ,\,\,M_{l} = m_{l} . $$

To ensure that the model defines a high order Markov chain, the negative log-likelihood is minimised with respect to the constraints delineated above. ADMB^TM was used to minimise the negative the log-likelihood (NLL). This uses auto-differentiation (AUTODIFF) [164] as the minimisation tool, shown to be computationally less intensive than MARCH [155]. Estimation algorithms relevant to this procedure can be found in Fournier [163, 167]. A major advantage of the new model is that its run-time is considerably shorter (less than one minute as compared with two days for the original MTD model [157]) and it can be run from a batch file in DOS. Hence, multiple models can be tested consecutively in remote mode. Outputs can also be appended into one file to allow easy access by any graphical software package [155].