Fig 1.
Summary of criteria for detecting causality.
(A) Schematic of cross-map algorithm for testing Y → X. Delay vectors in X, mapped to values in Y with lag ℓ, are bootstrap-sampled to construct a prediction library. For each delay vector in X, reconstructed values are calculated from a distance-weighted sum of Y values from nearest neighbors in the library. Many sampled libraries yield a distribution of cross-map correlations between actual Y and reconstructed
. (B) Criterion 1 (cross-map increase). Bootstrap distributions of cross-map correlation are calculated at minimum and maximum library sizes with ℓ = 0; causality is inferred if the correlation at Lmax is significantly greater than the correlation at Lmin. (C) Criterion 2 (negative cross-map lag). Cross-map correlations are calculated across different values of ℓ. Causality is inferred if the highest cross-map correlation for negative ℓ is positive and significantly greater than the highest value for nonnegative ℓ.
Fig 2.
Interactions detected as a function of process noise and the strength of interaction (C2 → C1) and representative time series.
(A) Heat maps show the fraction of 100 replicates significant for each inferred interaction for different parameter combinations. A significant increase in cross-map correlation ρ with library length L indicated a causal interaction. The time series consisted of 1000 years of monthly data. (B) Representative 25-year sample of the time series for which mutual interactions were inferred (σ12 = 0.25, η = 0.01). (C) Representative sample of the time series for which C2 is inferred to drive C1 but not vice-versa (σ12 = 0.25, η = 0.05).
Fig 3.
Shared frequency spectra predict probability of inferred interaction.
Points show the maximum cross-spectral densities of strains 1 and 2 plotted against the p-values for C1 → C2 for 1000 years of annual data. In all replicates, C1 never actually drives C2. Point color indicates the strength of C2 → C1 (σ12), and point size indicates the standard deviation of the process noise (η) on transmission rates.
Fig 4.
Interactions detected as a function of process noise and the strength of interaction (C2 → C1) and representative time series.
Heat maps show the fraction of 100 replicates significant for each inferred interaction for different parameter combinations. A maximum, positive cross-map correlation ρ at a negative lag indicated a causal interaction. Each replicate used 100 years of monthly incidence.
Fig 5.
Incorrect inference with changing transmission rate.
Example time series for testing transient dynamics. Each time series contained 100 years of monthly incidence data. The transmission rate β1 for the driven strain C1 was fixed at β1 = 0.30 (A) and β1 = 0.32 (B), and varied linearly over time between the two values (C). The transient time series yields high false positive and false negative rates under CCM. Interaction strength was σ21 = 0.5, process noise was η = 0.01, and seasonal forcing was ϵ = 0.
Fig 6.
Historical childhood infections in New York City and Chicago and inferred interactions from two reconstruction methods.
Time series show weekly incidence of infections per 1000 inhabitants of New York City (A) and Chicago (B). Delay-embeddings were constructed by maximizing the univariate correlation (C) or through a random projection method (D) Arrows indicate the inferred interactions from the New York (blue) and Chicago (red) time series under Criterion 2 (negative cross-map lag).
Fig 7.
Compartmental representation of strain-competition model.
Hosts are susceptible (S), infected/infective (I), or recovered (R) with respect to each strain. Hosts move from S to I based on a seasonally varying transmission rate, and from I to R at a constant recovery rate. Competition takes place through cross-immunity, which is implemented by having hosts skip the infected state for one strain with some probability if they are already infected with another strain.
Table 1.
Default parameter values.