The figure above shows a table of maximum responses for each set of initial conditions, side by side with the graph that gave rise to the set of equations that we integrate to generate the responses. Our naive trading strategy consists of taking the pair of nodes with the highest computed response (the first row in the table), and trading the dependent node (i.e. WTIC futures in the above figure), subject to the action of the independent node (S&P 500 Index). With subsequent recomputations of the graph, which occur once every 100 periods, we switch our trading to target the dependent node in the new max-response pair, and so on.
Why is this so different from using, say, lagged correlations as a basis for a trading strategy? Altaridey’s key insight is that a signals-based systems can be mapped to an isomorphic gene activation network, and the principles of Biochemical kinetics can then be applied to make predictions about the future of this isomorph, and, by extension, of the original signals system. Because we seek to characterize an entire system via a set of pair relationships–instead of focusing narrowly on a single correlation–the resultant insights can be far more powerful and wider-ranging.
The critical assumption underlying this mapping is that a directed edge in our graph expresses not simply a statistical relationship between two signals, but also something like a conserved quantity, with the stimulation-dissipation cycle in one signal feeding into the behavior of the other signal. And this assumption is not unfounded, given that we can make the following observations about our system: 1. the directed edges express causation rather than correlation; 2. the overwhelming majority of the signals we’re tracking are mean-reverting on the time scale of their respective sampling; and 3. unlike chemical kinetics, which enforce the principle of mass conservation, Biochemical (e.g. Hill) kinetics are forgiving of quantities that are not conserved.
In practice, the magnitude of a pairwise response obtained through this strategy is a weighted measure of a few things:
- the centrality of the dependent node (i.e. how many incoming edges, as a % of all edges in the graph)
- the terminality of the dependent node (the balance between incoming and outgoing edges into the node)
- topological closeness between the independent and the dependent nodes
Taking a closer look at the graph in the above figure, there are a few general observations to be made. Notice the sets of bidirected edges (a pair of nodes connected by a pair of edges pointing in the opposite directions) in the upper left corner of the graph, such as between nodes 34299 and 34300. Under the graph discovery scheme in Altaridey, such edges are an indication of zero lag in the signal correlation at the present data collection frequency, and may indicate the need for a higher frequency (with zero lag, causality will be recovered in either direction).
This is also a good opportunity to consider how the dynamic control systems (Level 3) applications are a straightforward extension of the present case study. Rather than finding an absolute maximum response from among the
N^2 - N distinct pairwise responses, in a control system application, we subject the maximization to a constraint: we seek the highest response in the controlled variable. We still perform the same
N integrations, but we’re only looking at the response in our controlled variable, or in a signal that is a proxy for it. Once we’ve ranked the responses, we select those signals that (1.) induce a high positive or negative response in our controlled variable and (2.) can themselves be controlled, and generate our controller according to Control Theory rules.