Time Series Temporality: Rogov-Causality Test
By an analogy with the Granger-causality test, the author has developed the following temporality test (Rogov-causality):
Z = LCS (X, Y) : Zi = 7, = Xti+lagi, i = i, 2 …l (6)
Let us consider the null hypothesis Ho stating that the first time series X from a pair of (X, Y) is not the Rogov-cause of the second time series Y.
This null hypothesis is rejected if there is a long enough (LCSS(X, Y) > 0.5) longest common subsequence of this time series pair, such that the cumulative distribution function (CDF) of time lags lagi at zero (i. e. the probability of a negative time lag between those values of the time series pair that have fallen within their long enough longest common subsequence) is sufficiently high.
This test is designed for the purposes of risk scenario generation, based on such predictors as lagged time series related in terms of the longest common subsequence similarity (LCSS). The test makes it possible to determine which of the two time series, X or Y, with a long-enough longest common subsequence Z= LCS(X, Y), is most likely not the advanced one (i. e., not the Rogov-cause). It is possible to discover the dynamics of which of the two series can be used (if at all) for scenario generation of the other time series of the pair, based on lagged values.
To avoid misinterpreting the term “causality,” one should bear in mind that the presence of Rogov-causality does not mean the existence of a proven cause-effect relationship, but rather characterizes the temporality (the existence of prevailing succession of events in time).
Example: A detective with a limited staff of agents must catch two suspected spies, X and Y, who are visiting different cities around the country, and one of whom is likely to leave reports for the other who follows him. If the detective has an adequate list Z of cities visited sequentially by both spies, he can write out the time delays between the visits of Y and X to the same cities and compose a series of time lags. If the probability of a negative lag (i. e. Y visiting ahead of X) is low, then the visits of Y are unlikely to be the Rogov-cause of the visits of X. If so (i. e. the test has been successfully passed), then, with surveillance of Y alone, one can calculate the confidence interval for the time lag and organize an ambush to catch X in the cities visited by Y an appropriate lag-time ago. Distribution of agents by city should correspond to the lag distribution.