Fig. 3: Hit rates to scalar memory \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})\) for synthetic models.

In each case we compute the percentage of the times within an ensemble of 10³ realisations that the estimated effective memory \({{{\Omega }}}_{{{{{\mathrm{eff}}}}}}({{{{{{{\mathcal{G}}}}}}}})\) and estimated pair memory \({{{\Omega }}}_{{{{{\mathrm{pair}}}}}}({{{{{{{\mathcal{G}}}}}}}})\) exactly match the scalar memory \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})\) (the memory parameter p is randomly sampled from UNIFORM{1,...,10} for each realisation). Models depend on parameters q, y and (where applicable) c (see SI section IV for details), so each curve scans hit rates for the whole range of a given parameter and fix the values of the other parameters to q = 0.9, y = 0.1, c = 0.1 (in every case, time series size is T = 10⁶). In DARN(p) and eDARN(p) models (panels a and b) where virtual loops are by construction absent, \({{{\Omega }}}_{{{{{\mathrm{eff}}}}}}({{{{{{{\mathcal{G}}}}}}}})={{{\Omega }}}_{{{{{\mathrm{pair}}}}}}({{{{{{{\mathcal{G}}}}}}}})\) and their estimation typically coincide with \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})\) for a large range of model parameters, as expected. In CDARN(p) and eCDARN(p) models (panels c and d), (probabilistic) virtual loops are expected to kick in, inducing a mismatch between \({{{\Omega }}}_{{{{{\mathrm{eff}}}}}}({{{{{{{\mathcal{G}}}}}}}})\) and \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})\) (the mismatch is notably smaller for \({{{\Omega }}}_{{{{{\mathrm{pair}}}}}}({{{{{{{\mathcal{G}}}}}}}})\) as this quantity disregards diagonal entries of the co-memory matrix and thus cannot account for first-order virtual loops).