Detecting causality from short time-series data based on prediction of topologically equivalent attractors

Abstract

Background: Detecting causality for short time-series data such as gene regulation data is quite important but it is usually very difficult. This can be used in many fields especially in biological systems. Recently, several powerful methods have been set up to solve this problem. However, it usually needs very long time-series data or much more samples for the existing methods to detect causality among the given or observed data. In our real applications, such as for biological systems, the obtained data or samples are short or small. Since the data or samples are highly depended on experiment or limited resource.

Results: In order to overcome these limitations, here we propose a new method called topologically equivalent position method which can detect causality for very short time-series data or small samples. This method is mainly based on attractor embedding theory in nonlinear dynamical systems. By comparing with inner composition alignment, we use theoretical models and real gene expression data to show the effectiveness of our method.

Conclusions: As a result, it shows our method can be effectively used in biological systems. We hope our method can be useful in many other fields in near future such as complex networks, ecological systems and so on.

Keywords: Causality; Gene regulations; Short time-series; Topologically equivalent position.

Fig. 1

Definition of causality. a. the attractors M _X and M _Y are reconstructed from variables X and Y by lagged-coordinates and they are topological equivalent. b. The predicted time-series (points) $Q_k^{\prime }\left(k=1,2,\cdots \right)$ are located in the nearest neighborhood of Q _k (k = 1, 2, ⋯) which implies causality (see the above part). The predicted time-series (points) $Q_k^{\prime }\left(k=1,2,\cdots \right)$ are outside the nearest neighborhood of Q _k (k = 1, 2, ⋯) which implies no causality (see the below part). Here, the nearest neighborhood is measured by a ball with a small radius r (see the gray district)

Fig. 2

Illustration of topologically equivalent attractors and topologically equivalent position. The two attractors M _X and M _Y are reconstructed from the original system by lagged-coordinates of its components X and Y. These two reconstructed attractors are topological equivalent. There are two short time-series {P _i} and {Q _i}(i = 1, 2, ⋯) on these two topologically equivalent attractors, respectively. And we call two points as topologically equivalent position, taking P ₄ and Q ₄ for example, if and only if the relative distance from P ₄ and Q ₄ to any other points on their corresponding attractors are invariant

Fig. 3

The results of the numerical examples by our method. a. The results of Logistic model by our method (rows → columns). b. The real interaction of a 5-species model. In this model, Y ₁, Y ₂ and Y ₃ are coupled each other, and also Y ₁, Y ₂ , Y ₃ drive Y ₄ and Y ₅ . However, Y ₄ and Y ₅ do not have any effect on Y ₁, Y ₂ and Y ₃ . c. The results of the five species model by our method (rows → columns). Here 0 means that there is no causal relation

Fig. 4

ROC curves of the E. coli network with 100 genes with different noise levels. IOTA represents the method used in [13], and TEP is our method used in this paper

Fig. 5

ROC curves of the Yeast network with 100 genes with different noise levels. IOTA represents the method used in [13], and TEP is our method used in this paper

Fig. 6

ROC curves of the Rat circadian rhythm with 18 key related genes by applying drug at different time points. IOTA represents the method used in [13], and TEP is our method used in this paper

Fig. 7

The real gene regulatory network of E.Coli with 100 genes

Fig. 8

The real gene regulatory network of Yeast with 100 genes