Learning Continuous Time Bayesian Networks in Non-stationary Domains

1 Introduction

The identification of relationships and statistical dependencies between components in multivariate time-series, and the ability of reasoning about whether and how these dependencies change over time is crucial in many research domains such as biology, economics, finance, traffic engineering and neurology, to mention just a few. In biology, for example, knowing the gene regulatory network allows to understand complex biological mechanisms ruling the cell. In such a context, Bayesian networks (BNs) [Segal et al., 2005; Nagarajan and Scutari, 2013], dynamic Bayesian networks (DBNs) [Vinh et al., 2012] and continuous time Bayesian networks (CTBNs) [Acerbi et al., 2014; 2016] have been used to reconstruct transcriptional regulatory networks from gene expression data. The effectiveness of discrete DBNs has been investigated to identify functional correlations among neuroanatomical regions of interest [Burge et al., 2009], while a useful primer on BNs for functional magnetic resonance imaging data analysis has been made available [Mumford and Ramsey, 2014]. However, the mentioned applications require the time-series to be generated from a stationary distribution, i.e. one which does not change over time. While stationarity is a reasonable assumption in many situations, there are cases where the data generating process is clearly non-stationary. Indeed, in recent years, researchers from different disciplines, ranging from economics to computational biology, to sociology and to medicine have become interested in representing relationships and dependencies which change over time. More precisely, researchers have been interested in analyzing the temporal evolution of genetic networks [Lebre et al., 2010], the flow over neural information networks [Smith et al., 2006], heart failure [Liu et al., 2016], complications in type 1 diabetes [Marini et al., 2015] and the dependence structure among financial markets during crisis [Durante and Dunson, 2014]. According to the specialized literature [Robinson and Hartemink, 2010], evolution models can be divided into two main categories: structurally nonstationary, i.e. those models which are allowed to change their structure over time, and parametrically non-stationary, i.e. those models which only allow the parameters' values to change over time.

∗This paper is an extended abstract of an article in the Journal of Artificial Intelligence Research [Villa and Stella, 2016].

In this paper, the structurally non-stationary continuous time Bayesian network model (nsCTBN) is introduced. A nsCTBN consists of a sequence of CTBNs which improves expressiveness over a single CTBN. Indeed, a nsCTBN allows the parents set of each node to change over time at specific transition times and thus it allows to model nonstationary systems. To learn a nsCTBN, the Bayesian score for learning CTBNs [Nodelman et al., 2003] is extended.

The contribution of this paper is twofold. Firstly, it defines the model of structurally non-stationary continuous time Bayesian networks. Secondly, the effectiveness of learning non-stationary continuous time Bayesian networks from realworld data is assessed by analyzing macroeconomic data.

2 Model

In this section, continuous time Bayesian networks and nonstationary continuous time Bayesian networks are introduced together with three knowledge learning settings for structural learning from complete data.

2.1 Continuous Time Bayesian Networks

Continuous time Bayesian networks combine Bayesian networks and homogeneous Markov processes together to efficiently model discrete state continuous time dynamical systems [Nodelman et al., 2002]. They are particularly useful for modeling domains in which variables evolve at different time granularities, such as to model the presence of people at their computers [Nodelman and Horvitz, 2003], to study reliability of dynamical systems [Boudali and Dugan, 2006], to model failures in server farms [Herbrich et al., 2007], and to detect network intrusion [Xu and Shelton, 2008].

The continuous time Bayesian network model is defined as follows.

Definition 1. Continuous time Bayesian network [Nodelman et al., 2002]. Let X be a set of random variables X = {X1,X2, . . . ,XN}. Each X has a finite domain of values V al(X) = {x1, x2, . . . , xI}. A continuous time Bayesian network over X consists of two components: the first is an initial distribution P 0X , specified as a Bayesian network over X, the second is a continuous time transition model specified as: a directed (possibly cyclic) graph G whose nodes are X1, X2, . . . , XN ; a conditional intensity matrix, QPa(X) X , for each variable X ∈ X, where Pa(X) denotes the set of parents of X in the graph G.

The conditional intensity matrix QPa(X) X consists of the set of intensity matrices

where pau ranges over all possible configurations of the parents set Pa(X), while qpau xi = ∑xj̸=xi qpau xixj . Off-diagonal elements of Qpau X , i.e. qpau xixj , are proportional to the probability that the variable X transitions from state xi to state xj given the parents' state pau. The intensity matrix Qpau X can be equivalently summarized with two independent sets: qpau X = {qpau xi : 1 ≤ i ≤ I}, i.e. the set of intensities parameterizing the exponential distributions over when the next transition occurs, and θpau X = {θpau xixj = qpau xixj /qpau xi : 1 ≤ i, j ≤ I, j ̸= i}, i.e. the set of probabilities parameterizing the multinomial distributions over where the state transitions. Note that the CTBN model assumes that only one single variable can change state at any specific instant, while its transition dynamics are specified by its parents via the conditional intensity matrix and they are independent of all other variables given its Markov Blanket 1 .

Continuous time Bayesian networks are both structurally stationary, as the graph does not change over time, and parametrically stationary, as the conditional intensity matrices do not change over time. These stationarity assumptions are reasonable in many situations, but there are cases where the data generating process is intrinsically non-stationary and thus CTBNs can no longer be used. Therefore, the CTBNs have been extended to become structurally non-stationary, i.e. the CTBN's structure is allowed to change over continuous time.

2.2 Non-stationary CTBNs

In the non-stationary continuous time Bayesian network model, the graph of the CTBN is replaced by a graphs sequence G = (G1, G2, . . . , GE), where a graph Ge represents the causal dependency structure of the model for the epoch e ∈ {1, 2, . . . , E}2 . This model is structurally non-stationary because of the introduction of the graphs sequence and it can handle transition times that are common to the whole network and/or node-specific. Following the notations and definitions used for non-stationary dynamic Bayesian networks (nsDBNs), we let T = (t1, . . . , tE-1) be the transition times sequence, i.e. the times at which the causal dependency structure Ge, active at epoch e, is replaced by the causal dependency structure Ge+1, which becomes active at epoch e + 1. An epoch is defined to be the period of time between two consecutive transitions, i.e. the epoch e is active during the period of time starting at te-1 and ending at te. The graph Ge+1, which is active during the epoch e+1, differs from the graph Ge, which is active during the epoch e, in a set of edges that we call the set of edge changes ∆Ge. A non-stationary continuous time Bayesian network is defined as follows.

1 The set of its parents, children, and its children's other parents.

2 It is worthwhile to mention that the first epoch, i.e. the epoch starting at time 0 and ending at time t1 is associated with the graph G1, while the last epoch, i.e. the epoch starting at time tE-1 and ending at time T (the supremum of the considered time interval, i.e. [0,T]) is associated with the graph GE.

Definition 2. (Structurally) non-stationary continuous time Bayesian network. Let X be a set of random variables X1, . . . ,XN. Each X has a finite domain of values V al(X) = {x1, . . . , xI}. A (structurally) non-stationary continuous time Bayesian network Nns = (B,Mns) over X consists of two components:

• an initial distribution P 0X , specified as a Bayesian network B over X,
• a non-stationary continuous time transition model Mns specified as: a sequence of directed (possibly cyclic) graphs G = (Ge)Ee=1 whose nodes are X1, . . . ,XN, where E represents the number of epochs; a conditional intensity matrix, QPaG(X) X,H(X) , ∀X ∈ X, where PaG(X) denotes the parents sets of X in G, and H(X) denotes the intervals associated with X.

The conditional intensity matrix QPaG(X) X,H(X) consists of a set of intensity matrices

one for each configuration pau of each parents set Pa(X) ∈ PaG(X) which is active during the interval hm ∈ H(X). 3

Non-stationary continuous time Bayesian networks allow each node to have its own sequence of parents sets, each parents set being active at a given epoch. Therefore, we introduce the concept of homogeneous interval H(X) = (h1, . . . , hM) associated with node X, which is defined as the union of consecutive epochs during which the same parents set Pa(X) is active for the node X. Note that if each epoch is associated with a different parents set, then M is equal to E. Figure 1 shows the graphs sequence G = (G1, G2, G3, G4) of a nsCTBN consisting of four epochs (E = 4) with transition times T = (t1, t2, t3).

Learning a non-stationary continuous time Bayesian network from a fully observed dataset can be done using the Bayesian learning framework. The following learning settings are considered; known transition times (KTT), where transition times between epochs are given, known number of epochs (KNE), where the number of epochs is given, and unknown number of epochs (UNE) where nothing is known about epochs. These learning settings have increasing difficulty, for each of them the Bayesian score decomposition has been formally derived and the corresponding learning algorithm has been designed [Villa and Stella, 2016].

3 Note that the equation qpau xi,hm = ∑xj̸=xi qpau xixj,hm still holds.

3 Numerical Experiments

Numerical experiments have been performed on both synthetic and real-world datasets. Synthetic datasets were used to compare nsCTBNs to nsDBNs under three knowledge settings in terms of accuracy, precision, recall and F1 measure. The following real-world datasets: drosophila, saccharomyces cerevisiae and songbird, were used to compare nsCTBNs to state-of-the-art algorithms, i.e. TSNI (a method based on ordinary differential equations), nsDBN [Robinson and Hartemink, 2010] and non-homogeneous dynamic Bayesian networks with Bayesian regularization (TVDBN) [Dondelinger et al., 2013], under the UNE knowledge setting. Results show that nsCTBNs are highly competitive when compared to state-of-the-art algorithms for both synthetic and real-world datasets [Villa and Stella, 2016].

In this paper, for the sake of brevity, we illustrate results of nsCTBNs structural learning on the macroeconomics dataset which consists of 17 financial/economic time-series pertaining to the economy of the United States. Time-series have different time granularity and span from 1st January 1986 to 31st March 2015. More specifically, five time-series have daily granularity, namely Crude oil (OIL), USD to EUR spot exchange rate (USDEUR), Gold (GOLD), S&P500 equity index (S&P500) and the 10-years treasury bond yield rate (US10yrsNote). Eleven time-series have monthly granularity, namely production of total industry (PTI), real manufacturing and trade industries sales (RMTIS), personal income (PI), unemployment (UN), consumer price index (CPI), federal funds rate (RATE), producer price index (PPI), non-farm payrolls (NFP), new one-family houses sold (NHSold), new houses for sale (NHSale) and new private house permits (NHPermit). Finally, the gross domestic product (GDP) time-series has quarterly granularity.

The goal of this study is to discover how the financial and economic environment evolve over time. In particular, we focused the attention to detect business cycles 4 and the associated change of relationships among financial and economic variables. Given that the duration of a business cycle is highly variable, the ability to identify the turning point of a cycle (i.e. when a recession starts) is of considerable importance to policymakers, financial companies as well as to individuals. A substantial literature is available about the business cycle turning points detection relying on Markov-switching models [Hamilton and Raj, 2005]. However, these models are not able to represent important features such as the dependence structure among variables in each business cycle. In order to use the nsCTBN model in such a context, we applied a binary discretization to the variable associated with each time-series. Discretization was performed using a look-back period of 1 year for daily data (i.e. if the current value is greater than the past one, then the binary variable is set to 1 otherwise it is set to 0), while a look-back of 1 month for monthly data and 3 months for quarterly data. The approach of looking back into the past is widely used in finance [Moskowitz et al., 2012]. nsCTBNs learning was performed under the UNE setting using the following parameter values: λc = {0.5, 1, 2}, λe = {0.1, 1, 10}, 2 maximum parents per node, 300 iterations and 10 runs.

Figure 2 shows the probability of transition (left side, left axis) versus the S&P500 equity index used as a reference (left side, right axis) and the posterior probability over the number of epochs (right side). The nsCTBN consists of three epochs with transition times close to the end of July 2000 and the end of November 2007.

4 Business cycles are fluctuations in aggregate economic activity, they are recurrent (i.e. it is possible to identify expansion-recession cycles), persistent and not periodic (i.e. they differ in length and severity).

If we compare these dates to the turning points of the US business cycle reported by the National Bureau of Economic Research 5 , then we see that we are not far from the turning point of March 2001 and very close to the one of December 2007, while the turning point occurred in July 1990 is missed, probably because of the limited length of the dataset.

Figure 3 shows the structure of the nsCTBN model corresponding to the most probable number of epochs, i.e. E = 3. An arc is included in the nsCTBN model when it occurs in more than 75% of the performed runs in each epoch. The retrieved networks correspond to the following time periods: from January 1986 to July 2000, from August 2000 to November 2007 and from December 2007 to March 2015.

The novelty of such approach opens the door to many considerations about the economic variables during business cycles, e.g. the learned models show the well-known central role of the personal income and its relation to the unemployment [Mankiw, 2014] and the less known relation of the nonfarm payrolls to the S&P500 equity index [Miao et al., 2014].

5 See the official data at http://www.nber.org/cycles.html

4 Conclusions

We introduced non-stationary continuous time Bayesian networks (nsCTBNs), a probabilistic graphical model generalizing continuous time Bayesian networks to the case where the structure of the probabilistic graphical model is allowed to change over time. nsCTBNs have been shown to be competitive with the state-of-the-art algorithms when both synthetic and real-world datasets are considered. The results of applying the nsCTBN's structural learning algorithm when the number of epochs is unknown show the usefulness of the proposed approach but at the same time warn about the following issues; the variables are assumed to be discrete; specifically each variable of the dataset must take value over a countable number of states and finding the optimal parameter settings of structural learning is extremely difficult as it happens for non-stationary dynamic Bayesian networks. Possible directions for further research include the development of a modeling approach that goes towards the direction of allowing each node to change its parents set asynchronously as well as the objects classification task in a streaming domain.