Causality in Time Series - ClopiNet

Popescuvalued vector. A Data Generating Process is a quintuple {s a ,p w ,T a ,T w } where T a , T ware finite time Turing machines which perform the following operations: Given an inputof the incompressible string p w the machine T w calculates a rational valued matrix w.The machine T a when given matrices y, a, u, t, w and a positive rational ∆t outputs avector y i+1 which is assigned for future operations to the time t i+1 = max(t) + ∆tThe definition is somewhat unusual in terms of the definition of stochastic systemsas embodiments of Turing machines, but it is quite standard in terms of defining aninnovations term w, a probability distribution thereof p w , a state y, a generating functionp a with parameters a and an exogenous input u. The motivation for using theterminology of algorithmic information theory is to analyse causality assignment as acomputational problem. For reasons of finite description and computability our variablesare rational, rather than real valued. Notice that there is no real restriction onhow the time series is to be generated, recursively or otherwise. The initial conditionin case of recursion is implicit, and time is specified as distinct and increasing but otherwisearbitrarily distributed - it does not necessarily grow in constant increments (itis asynchronous). The slight paradox about describing stochastic dynamical systemsin algorithmic terms is the necessity of postulating a random number generator (an oracle)which in some ways is our main tool for abstracting the complexity of the realworld, but yet is a physical impossibility (since such an oracle would require infinitecomputational time see Li and Vitanyi (1997) for overview). Also, the Turing machineswe consider have finite memory and are time restricted (they implement a predefinedmaximum number of operations before yielding a default output). Otherwise the rulesof algebra (since they perform algebraic operations) apply normally. The cover of aDGP can be defined as:Definition 2 The cover of a Data Generating Process (DGP) class is the cover of theset of all outputs y that a DGP calculates for each member of the set of admissibleparameters a,u,t,w and for each initial condition y 1 . Two DGPs are stochasticallyequivalent if the cover of the set of their possible outputs (for fixed parameters) is thesame.Let us now attempt to define a Granger Causality statistic in algorithmic terms.Allowing for the notation j..k = { j − 1, j − 2..,k + 1,k} if j > k and in reverse order ifj < k1i∑︁K(y 1, j | y 1, j−1..1 ,u j−1..1 ) − K(y 1, j | y 2, j−1..1 ,y 1, j−1..1 ,u j−1..1 ) (4)ij=1This differs from Equation (3) in two elemental ways: it is not a statement of independencebut a number (statistic), namely the average difference (rate) of conditional(or prefix) Kolmogorov complexity of each point in the presumed effect vector whengiven both vector histories or just one, and given the exogenous input history. It is ageneralized conditional entropy rate, and may be reasonably be normalized as such:42