Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable Systems3.2.2. Partitioned Settable SystemsIn elementary settable systems, each single response Y i can freely respond to settingsof all other system variables. We now consider systems where several settable variablesjointly respond to settings of the remaining settable variables, as when responses representthe solution to a joint optimization problem. For this, partitioned settable systemsgroup jointly responding variables into blocks. In elementary settable systems, everyunit i forms a block by itself. We now define general partitioned settable systems.Definition 3.2 (Partitioned Settable System) Let (A,a),(Ω,F , P a ), X 0 , n, and S i , i =1,...,n, be as in Definition 3.1. Let Π = {Π b } be a partition of {1,...,n}, with cardinalityB ∈ ¯N + (B := #Π).For i = 1,2,...,n, let ZiΠZi Π ,i Π b , and taking values in S Π (b) ⊆ Ω 0 × iΠb S i , S Π (b)be settings and let Z Π (b) be the vector containing Z 0 and ∅, b = 1,..., B. For b = 1,..., Band i ∈ Π b , suppose there exist measurable functions r Π i ( · ;a) : SΠ (b) → S i, specific to Πsuch that responses Y Π i(ω) are jointly determined asY Π i:= r Π i (ZΠ (b) ;a).Define the settable variables X Π i: {0,1} × Ω → S i asX Π i (0,ω) := YΠ i (ω) and X Π i (1,ω) := ZΠ i (ω i ) ω ∈ Ω.Put X Π := {X 0 ,X Π 1 ,XΠ 2 ...}. The triple S := {(A,a),(Ω,F ),(Π,XΠ )} is a partitionedsettable system.The settings Z(b) Π may be partition-specific; this is especially relevant when the admissibleset S Π (b) imposes restrictions on the admissible values of ZΠ (b). Crucially, responsefunctions and responses are partition-specific. In Definition 3.2, the joint responsefunction r[b] Π := (rΠ i ,i ∈ Π b) specifies how the settings Z(b) Π outside of block Π bdetermine the joint response Y[b] Π := (YΠ i,i ∈ Π b ), i.e., Y[b] Π = rΠ [b] (ZΠ (b);a). For conveniencebelow, we let Π 0 = {0} represent the block corresponding to X 0 .Example 3.2 makes use of partitioning. Here, we have n = 4 settable variables withB = 2 blocks. Let settable variables 1 and 2 correspond to beer and pizza consumption,respectively, and let settable variables 3 and 4 correspond to price and income. Theagent partition groups together all variables under the control of a given agent. Let theconsumer be agent 2, so Π 2 = {1,2}. Let the rest of the economy, determining price andincome, be agent 1, so Π 1 = {3,4}. The agent partition is Π a = {Π 1 ,Π 2 }. Then for block2,y 1 = Y a 1 (ω) = ra 1 (Z 0(ω 0 ),Z a 3 (ω 3),Z a 4 (ω 4);a) = r a 1 (p,m;a)y 2 = Y a 2 (ω) = ra 2 (Z 0(ω 0 ),Z a 3 (ω 3),Z a 4 (ω 4);a) = r a 2 (p,m;a)represents the joint demand for beer and pizza (belonging to block 2) as a function ofsettings of price and income (belonging to block 1). This joint demand is unique under11