User:Insoulswetrust/sandbox

MuTE toolbox is s freeware software, developed by Alessandro Montalto, that allows to detect the directed dynamical links among time series by means of Granger Causality and Transfer Entropy estimators. Model-free approaches such as binning and nearest neighbor techniques, model-based approaches such as linear method and neural networks approaches are used to evaluate Granger Causality and Transfer Entropy according to both uniform and non-uniform frameworks. MuTE toolbox is implemented in MATLAB and provides both the opportunity to run the methods by means of a script template and by means of a graphical user interface.

Structure
MuTE is structured as follows...

Some Formalism
Before letting the reader go into the details of the embedding approaches and the entropy estimators we would like to introduce some formalism that we will use from now on.

Let us consider a composite physical system described by a set of M interacting dynamical (sub) systems and suppose that, within the composite system, we are interested in evaluating the information flow from the source system $$\mathcal X$$ to the destination system $$\mathcal{Y}$$, collecting the remaining systems in the vector $$\mathbf{\mathcal{Z}} = \left\{Z^k\right\}_{k = 1,\ldots,M-2}$$. We develop our framework under the assumption of stationarity, which allows to perform estimations replacing ensemble averages with time averages (for non-stationary formulations see, e.g., Ledberg (2012), and references therein). Accordingly, we denote X, Y and $$\mathbf{Z}$$ as the stationary stochastic processes describing the state visited by the systems $$\mathcal{X}$$, $$\mathcal{Y}$$ and $$\mathcal{Z}$$ over time, and $$X_n, Y_n$$ and $$\mathbf{Z}_n$$ as the stochastic variables obtained sampling the processes at the present time n. Moreover, we denote $$X_n^-=[X_{n-1}X_{n-2}\ldots], Y_n^-=[Y_{n-1}Y_{n-2}\ldots], \textbf{Z}_n^-=[\textbf{Z}_{n-1}\textbf{Z}_{n-2}\ldots]$$ as the infinite-dimensional vector variables representing the whole past of the processes X, Y and $$\mathbf{Z}$$. Then, the multivariate transfer entropy (TE) from X to Y conditioned to $$\mathbf{Z}$$ is defined as:



TE_{X \rightarrow Y|\mathbf{Z}} = \sum{p\left( Y_n, Y_n^-, X_n^-, \mathbf{Z}_n^- \right) log{\frac{p\left(Y_n | Y_n^-, X_n^-, \mathbf{Z}_n^- \right)}{p\left(Y_n | Y_n^-, \mathbf{Z}_n^- \right)}} } $$

where the sum extends over all the phase space points forming the trajectory of the composite system. $$p(\textbf{a})$$ is then the probability associated with the vector variable $$\textbf{a}$$ while $$p(b|\textbf{a}) = p(\textbf{a},b)/p(\textbf{a})$$ is the probability of observing b given that the variables forming the vector $$\textbf{a}$$ are known. The conditional probabilities used in (\ref{eq:1}) can be interpreted as transition probabilities, in the sense that they describe the dynamics of the transition of the destination system from its past states to its present state, accounting for the past of the other systems. Utilization of the transition probabilities makes the resulting measure able to quantify the extent to which the transition of the destination system $$\mathcal{Y}$$ into its present state is affected by the past states visited by the source system $$\mathcal{X}$$. Specifically, the TE quantifies the information provided by the past of the process X about the present of the process Y that is not already provided by the past of Y or any other process included in $$\mathbf{Z}$$.

The formulation presented in (\ref{eq:1}) is an extension of the original TE measure proposed for pairwise systems, Schreiber (2000), to the case of multiple interacting processes. The conditional TE formulation, also denoted as partial TE, Vakorin (2010), Kugiumtzis (2013), rules out the information shared between $X$ and $Y$ that could be possibly triggered by their common interaction with $\mathbf{Z}$. Note that the TE can be seen as a difference of two conditional entropies (CE), or equivalently as a sum of four Shannon entropies:



TE_{X \rightarrow Y| \mathbf{Z}} = H(Y_n | Y_n^-, \textbf{Z}_n^-) - H(Y_n | Y_n^-, X_n^-, \textbf{Z}_n^-) = H(Y_n, Y_n^-, \textbf{Z}_n^-)-H(Y_n^-, \textbf{Z}_n^-) - H(Y_n,Y_n^-, X_n^-, \textbf{Z}_n^-)+H(Y_n^-, X_n^-, \textbf{Z}_n^-) $$

The TE has a great potential in detecting information transfer because it does not assume any particular model that can describe the interactions governing the system dynamics, it is able to discover purely non linear interactions and to deal with a range of interaction delays, Vicente (2011). Recent research has proven that TE is equivalent to Granger Causality (GC) for Gaussianly distributed data, Barnett (2009), Hlavackova (2011). This establishes a convenient joint framework for both measures. Here we evaluate GC in the TE framework and compare a classical VAR model implemented in both versions, UE and NUE, with two model-free approaches.