Generally, prior arrangements to study and/or analyze time series data has focused on identifying groups of similar or co-regulated elements using clustering techniques or finding patterns via data mining. However, there has been a limited amount of research performed to infer and determine causal relationships between the elements of these time series. When attempting to decipher the underlying structure of a system, enumeration of the formulas governing its behavior may be one object of such research. For example, the knowledge of what is responsible for patterns of activity may lead to a greater understanding of systems as well as the ability to better predict future events.
In biologically-related systems, one research goal may be to discover dependencies between genes and genes that influence others. These types of networks can provide a model of biological processes that may then be tested and validated using knock-out or exclusionary experiments, for example. Research in this area has primarily used graph-based methods, such as Bayesian Networks, which can be limited in terms of the relationships they may represent and infer.
Earlier attempts in automating the inference of causal relationships have been described in, e.g., J. Pearl, Causality: Models, Reasoning, and Inference, Cambridge University Press, 2000, and P. Spirtes, C. Glymour, and R. Scheines, Causation, Prediction, and Search. MIT Press, 2000, using graphical models, such as Bayesian networks (BNs). In these approaches, the causal structure of the system may be represented as a graph, where variables can be represented by nodes and the edges between them can represent conditional dependence (and the absence of an edge may imply conditional independence).
A number of assumptions about the data can be used to direct these edges from cause to effect. The result may be a directed acyclic graph (DAG) where a directed edge between two nodes may mean the first causes the second. In these graphical approaches, the edges may be oriented without the use of time course data, as a consequence of the other assumptions. Terminology of SGS may be used their work primarily described though these assumptions and the general procedure are used by many with some variation.
First, it can be assumed that a node in the graph (variable) is independent of every node other than its direct effects conditional on its direct causes (e.g., those that are connected to the node by one edge). This may be referred to as the Causal Markov condition (CMC). The inference of causal structures may rely on two more assumptions: faithfulness and causal sufficiency.
Faithfulness can assume that exactly the independence relations found in the causal graph hold in the probability distribution over the set of variables. This may imply that the independence relations obtained from the causal graph are due to the causal structure generating it. If there are independence relations that are not a result of CMC, then the population may be unfaithful. Faithfulness may be used for determining whether independencies are from some structure, and not from chance coincidence or latent variables.
Causal sufficiency can assume that a set of measured variables includes all of the common causes of pairs on that set. In cases where causal sufficiency does not hold, the inferred graphs may include those with nodes representing unmeasured common causes that can also lead to the observed distribution. Knowledge about temporal ordering may also be used at this point if it is available. In general, the conditional independence conditions can be assumed to be exact conditional independence, though it may be possible to define some threshold to decide when two variables will be considered independent. The result may be a set of graphs that represent the independencies in the data, where the set may contain only one graph in some cases when all assumptions are fulfilled.
However, when using these graphical models there may be no natural way of representing or inferring and/or determining the time between the cause and the effect or a more complex causal relationship than just one node causing another at some future time. An update to Bayesian networks (BNs), dynamic Bayesian networks (DBNs) (see, e.g., N. Friedman, K. Murphy, and S. Russell, Learning the structure of dynamic probabilistic networks, In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence (UAI98), pp. 139-147, 1998) can be introduced to address the temporal component of these relationships. DBNs extend BNs, to show how the system evolves over time. For this purpose, they begin with a prior distribution (described by a DAG structure) as well as two more DAGs: one representing the system at time t and another at t+1, where these hold for any values of t. The connections between these two time slices then may describe the change over time. Similarly to the above, there may generally be one node per variable, with edges representing conditional independence.
This can imply that while the system may start in any state, after that, the structure and dependencies may repeat themselves. For example, the relationships from time 10 to 11 may be exactly the same as those from time 11 to 12. Research by Eichler and Didelez (M. Eichler and V. Didelez, Causal reasoning in graphical time series models. In Proceedings of the 23rd Annual Conference on Uncertainty in Artificial Intelligence, 2007) has largely focused on time series and explicitly capturing the time elapsed between cause and effect. They define that one time series may cause another if an intervention on the first alters the second at some later time. For example, there can be lags of arbitrary length between the series, and these lags may be found to be part of the inference process. While it may be possible to also define the variables in this framework such that they represent a complex causal relationship as well as the timing of the relationship, the resulting framework can still not easily lead to a general method for determining these relationships. Further, while DBNs are a compact representation in the case of sparse structures, it may be difficult to extend them to the case of highly dependent data sets with thousands of variables, none of which can be eliminated.
With respect to one thing causing another, particularly in terms of scientific data, rarely is it as simple as “a causes b”, deterministically, with no other relevant factors. Work by Langmead et al. (2006) (C. Langmead, S. Jha, and E. Clarke, Temporal logics as query languages for dynamic bayesian networks: Application to d. melanogaster embryo development. Technical Report CMU-CS-06-159, Carnegie Mellon University, 2006) describes the use of temporal logic for querying pre-existing DBNs, by translating them into structures that may allow for model checking. This approach may facilitate the use of known DBNs for inference of relationships described by temporal logic formulae. However, only a subset of DBNs can be translated in this way (see, e.g., Langmead, 2008) (C. J. Langmead, Towards inference and learning in dynamic bayesian networks using generalized evidence. Technical Report CMU-CS-08-151, Carnegie Mellon University, 2008.), and thus the benefit of this approach (as opposed to one where the model inferred already allows for model checking) can be limited.
In terms of experimental work, research has been performed in applying notions of causality to the problem of determining relationships among genes (usually from microarray data). Techniques used for inferring and modeling causality amongst genes include, e.g.,: Granger causality, Bayesian networks, mutual information and likelihood-based approaches. Each method can begin with pairwise correlations across the entire time series, connecting them to form graphs of networks. However, it can be difficult to see how the network describing one set of experiments differs from that of another (e.g., between two cancer patients). One method can begin with a correlation network and transform it into one that includes causation. The partially directed network employed may allow for the visualization of multiple relationship types simultaneously, as well as the identification of hub nodes. However it does not easily lead to the probabilistic rules that may be useful in various applications, such as when applied to financial data, for example.
The conventional methods do not appear to facilitate an explicit reasoning about the elapsed time between cause and effect, probabilities of causation as well as relationships more complex than one-to-one, which can be especially useful when attempting to make inferences and/or determinations in time course data, particularly in the case of such data with additional background knowledge.
Thus, there appears to be a need to address at least some of the deficiencies described above. Accordingly, exemplary embodiments of a framework, system, process, computer-accessible medium and software arrangement according to the present disclosure can be provided using which, arbitrarily complex causal relationships can be, e.g., inferred, described and analyzed.