Sensitivity of Convergent Cross Mapping to temporal discontinuities: A case study of seasonal influenza in Hong Kong

Abstract

Convergent cross mapping (CCM) method has been widely applied to investigate environmental drivers of infectious disease dynamics, particularly for seasonal influenza. However, its robustness to temporal gaps and missing observations—common features in surveillance data—remains largely unexplored. Using seasonal influenza surveillance data from Hong Kong, we systematically assessed the sensitivity of inferred environment–disease relationships to different data-exclusion scenarios, including the removal of low-activity periods and targeted time points. We compared CCM with quasi-binomial generalized linear models (GLM) and the Peter–Clark Momentary Conditional Independence (PCMCI⁺) framework.

Across all scenarios, CCM-based inference exhibited pronounced sensitivity to data gaps, with both causal strength and inferred relationships varying substantially across gap configurations. In contrast, GLM and PCMCI⁺ estimates remained stable, consistently indicating a negative association between ozone and influenza transmission. These findings highlight a critical limitation of CCM when applied to incomplete time series and underscore the need for caution in interpreting causality from gap-affected epidemiological data.

1. Introduction

Seasonal influenza is a highly transmissible acute respiratory infection that poses a substantial and recurring burden on global public health. There are around a billion cases of seasonal influenza annually, including 3 to 5 million cases of severe illness. In addition, seasonal influenza contributes to approximately 290,000 to 650,000 respiratory-related deaths annually, affecting nearly all countries worldwide (World Health Organization, 2025).

Across different climatic regions, observed influenza activity exhibits distinct seasonal patterns. In temperate regions, influenza epidemics display pronounced seasonality, typically occurring during the winter months of the respective hemispheres (Finkelman et al., 2007; Hope-Simpson, 1981; Tamerius et al., 2011). In contrast to temperate settings, influenza activity in many tropical regions persists throughout the year with less well-defined seasonal structure. In some tropical and subtropical areas, two distinct epidemic peaks may be observed within a single year, as documented in regions such as Singapore and Hong Kong (Shaman & Kohn, 2009; Viboud, Alonso, & Simonsen, 2006).

Numerous studies have sought to identify the key drivers underlying the seasonality of influenza outbreaks. Broadly, three major mechanisms have been proposed to drive seasonal influenza dynamics: variation in host contact rates, environmental modulation of viral survival and transmissibility, and seasonal fluctuations in host immune function (Tamerius et al., 2011). Environmental variables, particularly absolute humidity and temperature, have been shown to substantially influence the transmission efficiency of influenza viruses. Influenza epidemics tend to occur during periods characterized by low absolute humidity and colder temperatures (Tamerius et al., 2013). Under these conditions, viral stability in the environment is enhanced and airborne transmission is facilitated, thereby creating favorable conditions for influenza spread (Lowen et al., 2007). Host-related factors, including age, underlying health conditions, workflows, and immunological history, also play an important role in shaping outbreak dynamics (Cauchemez et al., 2014; Viboud, Bjornstad, et al., 2006).

For environmental modulation of viral survival and transmissibility, CCM has been widely applied to investigate the causal effects of climate factors on seasonal influenza transmission (Chen et al., 2024; Deyle et al., 2016; Guo et al., 2024). However, despite its widespread use, an important feature of influenza surveillance data—namely, the presence of temporal discontinuities or gaps—has received limited attention. For example, Guo et al. (Guo et al., 2024) restricted their analysis to influenza seasons from October through May, thereby excluding low-transmission months. While such season-restricted analyses capture periods of heightened influenza activity, they may conflate seasonal co-variation with causal effects and limit the ability to assess whether the ozone–influenza relationship persists outside peak transmission periods.

To explicitly examine the impact of temporal discontinuities, we utilize year-round influenza surveillance data from Hong Kong and systematically evaluate the stability of CCM-based causal inference under different data exclusion schemes. Specifically, we consider four scenarios: (i) the unaltered continuous series spanning 2014–2019; (ii) annual removal of each year's lowest-activity month; (iii) exclusion of the single lowest-activity month over the entire period; and (iv) truncation by omitting only the final month (November 2019).

In addition to convergent cross mapping method, we applied quasi-binomial generalized linear models (GLM) and the Peter–Clark Momentary Conditional Independence (PCMCI⁺) algorithm as robustness checks. GLM characterizes conditional mean relationships between influenza activity and environmental covariates through likelihood-based formulations, while PCMCI⁺ is grounded in time-series graphical models and infers directed dependencies via conditional independence testing. The choice of GLM and PCMCI⁺ as benchmark methods is theoretically motivated by their inferential contrast with CCM. Accordingly, this study explicitly examines whether causal inference methods relying on state-space reconstruction (CCM) exhibit greater sensitivity to data continuity than methods based on statistical association (GLM) or conditional independence within graphical models (PCMCI⁺).

2. Results

2.1. Dynamics of influenza activity and meteorological drivers

We analyzed a complete and continuous weekly time series from Hong Kong spanning 2014 to 2019. Fig. 1 summarizes the temporal evolution of FluP alongside key meteorological variables, including air temperature, absolute humidity (AH), ozone concentration, relative humidity (RH), and solar radiation. FluP is an influenza activity proxy defined as the product of the influenza-like illness (ILI) consultation rate (per 1000 consultations) and the laboratory-confirmed influenza positivity rate among those consultations (Guo et al., 2024; Shaman et al., 2013). Across the study period, all five meteorological drivers exhibited pronounced seasonal patterns, with clear annual periodicity.

Fig. 1.

Panels A–E show dual-axis time series of FluP against (A) temperature (°C), (B) absolute humidity (kg/kg), (C) ozone concentration (ppb), (D) relative humidity (%), and (E) solar radiation (W/m²). Each panel's left y-axis corresponds to the raw meteorological variable, while the right y-axis shows FluP on its original scale.

We further assessed the stationarity of each time series using the Augmented Dickey–Fuller (ADF) test. All variables consistently rejected the null hypothesis of a unit root across the different data-exclusion scenarios (all p < 0.05; Supplementary Table S1), indicating no evidence of non-stationarity. This suggests that the stationarity properties of the time series remained stable under different data-removal schemes and therefore support the validity of the subsequent sensitivity analyses.

2.2. Sensitivity of CCM method

Convergent cross-mapping (CCM) was applied to evaluate the causal relationships between environmental variables and influenza activity under four data-exclusion protocols: (1) the uninterrupted 2014–2019 time series (“No Gap”); (2) exclusion of each year's lowest-activity month; (3) removal of the single month with the lowest overall incidence (June 2018); and (4) truncation omitting only the final month (November 2019).

As shown in Fig. 2, under the uninterrupted time series, most environmental variables did not exhibit significant CCM signals at lag 0 or lag −1. A significant effect was observed only for ozone (O₃) at lag −2, with ρ = 0.223 and p = 0.011.

Fig. 2.

CCM effect sizes under different data-exclusion scenarios. Each panel shows the change in cross–map skill Δρ_CCM for ozone predicting influenza in Hong Kong. Panels are arranged in a 4 × 3 grid: rows correspond to (from top to bottom) the full time series, exclusion of the annual minimum-flu month, exclusion of June 2018, and exclusion of November 2019; columns correspond to lags of 0, 1, and 2 weeks. In each panel, gray horizontal lines indicate the 95% confidence interval of Δρ_CCM under 1000 seasonal surrogate time series, and the vertical dashed line marks Δρ_CCM = 0. Red circles show the observed Δρ_CCM at each lag with open symbols represent non-significant in the surrogate test, p > 0.05 and filled symbols represent significant, p ≤ 0.05).

In contrast, after excluding the annual minimum influenza months, a greater number of environmental variables exhibited statistically significant CCM signals. Specifically, at lag 0, both air temperature (ρ = 0.187, p = 0.011) and absolute humidity (ρ = 0.161, p = 0.047) showed significant positive CCM signals. At lag −1, ozone (ρ = 0.205, p = 0.001), absolute humidity (ρ = 0.151, p = 0.028), and temperature (ρ = 0.143, p = 0.020) all displayed statistically significant CCM effects, with ozone showing the largest and most consistent signal. At lag −2, ozone remained the only variable with a statistically significant CCM signal (ρ = 0.151, p = 0.015).

To assess sensitivity to individual time points, we further examined the impact of excluding specific months. Removing June 2018—the month with the lowest influenza incidence over 2014–2019—produced results broadly similar to those obtained when excluding annual minimum months. For example, ozone at lag −1 increased from ρ = 0.178(p = 0.056) to ρ = 0.215(p = 0.001), indicating enhanced statistical significance, while temperature at lag 0 increased from ρ = 0.169(p = 0.123) to ρ = 0.187(p = 0.011). This indicates that excluding a single low-activity month can substantially alter CCM inference.

By contrast, excluding the final month of the series (November 2019) yielded CCM results highly consistent with those from the uninterrupted time series. Although ozone at lag −2 showed a modest decrease from ρ = 0.223(p = 0.011) to ρ = 0.165(p = 0.093), most other variables exhibited only minor changes in both ρ and p values. This suggests that CCM inference remains relatively stable when temporal continuity is preserved, and that the observed sensitivity is not driven by end-of-series artifacts.

Across all data-exclusion protocols and lag specifications, solar radiation did not exhibit stable or significant CCM signals (ρ ranging from −0.014 to 0.059, with all p ≥ 0.234), indicating that it is unlikely to act as a direct causal driver of influenza activity in this system.

To further assess the robustness of the model estimates, we conducted a sensitivity analysis by removing the single lowest observation in the time series (dated 2014–11–15; Fig. 3). This test was designed to evaluate whether CCM-based inference is disproportionately influenced by observations associated with specific dynamical regimes. After excluding this extreme low-value point, the estimated associations changed notably. In particular, absolute humidity (AH) exhibited a statistically significant positive association with influenza positivity (FluP) at lag 0 (ρ = 0.246, p = 0.028), whereas this relationship was not statistically significant in the full dataset. These results suggest that the inferred exposure–response relationships may be sensitive to influential observations, with even a single low-activity time point capable of altering the strength and significance of CCM signals.

Fig. 3.

Sensitivity analysis after removing the global minimum of the time series (2014–11–15).

2.3. Sensitivity of GLM method

Using quasi-binomial generalized linear models, we quantified associations between environmental variables and influenza activity in Hong Kong (Fig. 4). Associations were evaluated across multiple lag structures (lag 0–2) and under four data-exclusion schemes to assess sensitivity to influential time points. Overall, at lag 0, absolute humidity (AH) exhibited a statistically significant association with influenza activity. In contrast, ozone (O₃) showed statistically significant associations across all examined lag structures (lag 0, 1, and 2).

Fig. 4.

State-specific generalized linear model (GLM) estimates of environmental effects on influenza activity under different data exclusion schemes. Points denote the estimated regression coefficients β scaled per one standard deviation increase in each environmental variable, while horizontal bars represent the corresponding 95% confidence intervals. Results are shown separately for lag 0, lag 1, and lag 2 (from left to right). Filled symbols indicate statistically significant associations (p < 0.05), whereas open symbols denote non-significant effects.

Importantly, the estimated GLM effects were highly consistent across alternative data-exclusion strategies. Both the effect sizes, expressed as beta estimates per standard deviation increase, and their corresponding statistical significance remained stable regardless of whether annual minimum influenza months were excluded, a single extremely low-incidence month (June 2018) was removed, or the terminal month of the series (November 2019) was truncated. These results indicate that the inferred associations from the GLM analysis are robust to data handling choices and are not driven by specific influential observations.

2.4. Sensitivity of PCMCI ⁺ method

As shown in Fig. 5, using the PCMCI ⁺ framework with a significance level of α_PC = 0.05, we investigated the directed dependency structure among meteorological variables and influenza activity under four data-exclusion schemes. In the full time series without data gaps, PCMCI⁺ identified a coherent network structure linking temperature (T), absolute humidity (AH), relative humidity (RH), solar radiation (RAD), and ozone (O₃), with O₃ and temperature exhibiting a direct influence on influenza activity at a one-week lag.

Fig. 5.

PCMCI ⁺ results under different data-exclusion schemes.

Across different data-exclusion strategies, the relationships among meteorological variables showed modest changes, whereas the inferred dependencies associated with influenza activity were largely preserved.

2.5. Comparison of CCM, PCMCI⁺, and GLM results

Table 1 summarizes the statistical significance of environmental effects on influenza activity inferred by CCM, PCMCI⁺, and GLM under gap and no-gap scenarios across multiple lag structures, where the gap scenario corresponds to the exclusion of the minimum-influenza month in each year.

Table 1.

Comparison of CCM, PCMCI⁺ and GLM p-values under gap and no-gap scenarios.

	CCMpvalues		PCMCI+pvalues		GLMpvalues
	Gap	No-gap	Gap	No-gap	Gap	No-gap
Ozone
Lag 0	5.60 × 10−2	3.13 × 10−1	7.70 × 10−2	7.06 × 10−2	2.58 × 10−3∗∗	2.57 × 10−3∗∗
Lag 1	5.00 × 10−4∗∗∗	5.60 × 10−2	1.72 × 10−4∗∗∗	2.58 × 10−4∗∗∗	6.06 × 10−9∗∗∗	5.39 × 10−9∗∗∗
Lag 2	1.50 × 10−2∗	1.10 × 10−2∗	9.39 × 10−1	9.99 × 10−1	4.52 × 10−3∗∗	4.06 × 10−3∗∗
Absolute humidity
Lag 0	2.70 × 10−2∗	1.14 × 10−1	6.40 × 10−1	6.12 × 10−1	3.00 × 10−2	2.94 × 10−2
Lag 1	2.80 × 10−2∗	1.13 × 10−1	7.10 × 10−2	6.75 × 10−2	2.15 × 10−1	2.13 × 10−1
Lag 2	5.80 × 10−2	1.18 × 10−1	5.85 × 10−1	6.70 × 10−1	8.37 × 10−1	8.50 × 10−1
Temperature
Lag 0	1.40 × 10−2∗	1.23 × 10−1	3.55 × 10−1	3.65 × 10−1	7.23 × 10−1	7.20 × 10−1
Lag 1	2.00 × 10−2∗	1.34 × 10−1	7.33 × 10−3∗∗	8.69 × 10−3∗∗	4.13 × 10−1	4.14 × 10−1
Lag 2	7.50 × 10−2	2.19 × 10−1	3.18 × 10−1	3.45 × 10−1	5.98 × 10−2	6.04 × 10−2
Relative humidity
Lag 0	2.49 × 10−1	1.79 × 10−1	5.37 × 10−1	6.06 × 10−1	8.91 × 10−1	8.89 × 10−1
Lag 1	2.90 × 10−2∗	6.70 × 10−2	1.99 × 10−1	2.14 × 10−1	1.59 × 10−1	1.58 × 10−1
Lag 2	1.00	2.22 × 10−1	9.32 × 10−1	9.88 × 10−1	2.72 × 10−1	2.71 × 10−1
Solar radiation
Lag 0	1.00	1.00	2.28 × 10−1	2.35 × 10−1	5.25 × 10−2	5.12 × 10−2
Lag 1	2.34 × 10−1	3.76 × 10−1	7.85 × 10−2	6.39 × 10−2	6.34 × 10−1	6.42 × 10−1
Lag 2	1.00	4.67 × 10−1	4.53 × 10−1	5.26 × 10−1	2.23 × 10−1	2.21 × 10−1

Note: Here, “gap” refers to the exclusion of the minimum month in each year. Boldface indicates p < 5 × 10⁻². CCM = convergent cross mapping; PCMCI⁺ = Peter–Clark Momentary Conditional Independence algorithm; GLM = generalized linear model.

It should be noted that the reported p-values are not directly comparable across methods, as they are associated with different null hypotheses and statistical interpretations. Specifically, GLM p-values assess the significance of regression coefficients deviating from zero; PCMCI ⁺ p-values are derived from conditional independence tests and reflect the existence of edges in a causal graph given other variables and lags; whereas CCM significance is not based on a single correlation coefficient, but on the convergence property of cross-map skill with increasing library size. In particular, we quantify CCM convergence strength as the difference between cross-map skill at the maximum and minimum library sizes (Δρ = ρ_max − ρ_min), and evaluate its empirical significance by comparison with a null distribution generated from seasonal surrogate data (p_CCM).

Accordingly, Table 1 is used to assess the internal consistency and robustness of each method under data-exclusion (gap vs. no-gap) scenarios, focusing on whether each method yields consistent binary significance calls under a common threshold.

Across all three analytical frameworks, ozone (O₃) exhibits the most consistent and robust association with influenza activity, with statistically significant effects detected across multiple lags, particularly in the GLM and under the gap scenario in CCM and PCMCI⁺. Absolute humidity (AH) shows significant associations primarily at lag 0 in both CCM and GLM, while evidence at longer lags is weaker or non-significant across methods.

In contrast, temperature and relative humidity display more lag-specific and method-dependent patterns. Temperature shows statistically significant effects at lag 1 in PCMCI⁺ and GLM, whereas CCM detects significant signals only under the gap scenario. Relative humidity exhibits intermittent significance, with evidence concentrated at lag 1. Solar radiation does not show associations with influenza activity across any method or lag structure, suggesting a limited role in driving the observed influenza activity in this setting.

Importantly, the inferred CCM results differ substantially between the gap and no-gap scenarios, indicating a high sensitivity of CCM to data exclusion. In contrast, the results obtained from PCMCI⁺ and GLM remain largely stable across both scenarios.

3. Methods

3.1. Influenza data

Weekly influenza surveillance data were obtained from the Hong Kong Government's Open Data portal (“Flu Express”, https://data.gov.hk/en-data/dataset/hk-dh-chpsebcddr-flu-express) for the period January 2014 to December 2019. This dataset provides multiple surveillance indicators, including the influenza-like illness (ILI) consultation rate per 1000 visits, the proportion of respiratory specimens testing positive for influenza, the count of seasonal influenza virus detections, and the number of severe influenza cases, among others. For this analysis, we selected a subset of these indicators.

To address under-reporting and incomplete laboratory confirmation, we followed established practices in influenza epidemiology by integrating syndromic surveillance data (ILI consultation rates) and virological confirmation data (influenza-positive test results) into a unified proxy measure for influenza incidence (Reed et al., 2009; Shaman et al., 2013). Specifically, we defined the weekly influenza proxy incidence (FluP_t) as:

$${\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{P}}_t=({\mathrm{I}\mathrm{L}\mathrm{I}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}}_t)\times ({\mathrm{I}\mathrm{n}\mathrm{fl}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{z}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}}_t),$$

an approach previously validated to effectively account for under-ascertainment and reliably represent influenza activity within the community (Viboud, Alonso, & Simonsen, 2006).

3.2. Environmental and air pollution data

Environmental data were obtained from two official sources in Hong Kong. Meteorological variables—including daily mean temperature (°C), absolute and relative humidity (%), and solar radiation (MJ/m²)—were retrieved from the Hong Kong Observatory. Ozone data were obtained from the Hong Kong Environmental Protection Department (EPD), which provides publicly accessible monitoring data through its online portal (https://www.epd.gov.hk/epd/english/environmentinhk/air/data/air_data.html).

All variables were originally reported at a daily resolution and were aggregated to weekly values using the median within each epidemiological week, to align with the temporal resolution of the influenza surveillance data.

3.3. Augmented Dickey–Fuller test

The Augmented Dickey–Fuller (ADF) test is a widely used statistical procedure for detecting unit roots in time series (Dickey & Fuller, 1981). The presence of a unit root formally indicates non-stationarity. The ADF test evaluates the following hypotheses:

• •${\mathcal{H}}_0$: The time series contains a unit root and is therefore non-stationary.
• •${\mathcal{H}}_1$: The time series does not contain a unit root and is stationary.

In this study, the ADF test was applied to assess the stationarity of each analyzed time series, providing a basis for the later sensitivity analyses. Statistical significance was evaluated at the 5% level.

3.4. CCM methods

Many scientific disciplines rely on models—typically sets of hypothesized equations—to approximate complex systems for hypothesis testing, explanation of past observations, or forecasting future behavior. When underlying mechanisms are unknown or too intricate for simple parametric forms, explicit equations may become impractical. Empirical models offer a flexible alternative by inferring system structure directly from data rather than imposing a rigid functional form(Casdagli et al., 1991; Deyle & Sugihara, 2011; Sauer et al., 1991; Takens, 1981).

Empirical Dynamic Modeling (EDM) is a non-parametric framework designed for nonlinear, state-dependent systems that often operate far from equilibrium (Casdagli et al., 1991; Deyle & Sugihara, 2011; Sauer et al., 1991; Takens, 1981). EDM begins by reconstructing the system's attractor manifold via time-delay embedding of one or more observed variables:

$${\mathbf{x}}_t=\left(x_t,x_{t-\tau },\dots ,x_{t-(E-1)\tau }\right),$$

where E is the embedding dimension and τ the time lag. According to Takens' theorem, this “shadow attractor” is diffeomorphic to the true attractor under mild conditions, enabling recovery of the system dynamics (Sauer et al., 1991; Takens, 1981).

Convergent Cross Mapping (CCM) is a nonlinear causality detection method based on state-space reconstruction (Sugihara et al., 2012). Once the time series is partitioned, the complete CCM procedure—including the selection of delay τ and embedding dimension E, any Simplex/S-map diagnostics, and the final cross-mapping—is carried out independently within each segment; no neighbour searches are permitted across segment boundaries. For both the complete dataset and datasets with temporal discontinuities, we followed the CCM analysis workflow described in Guo et al. (2024) and Deyle et al. (2016).

In the CCM framework, causal strength is quantified by the cross-map skill (ρ), defined as the Pearson correlation between predicted and observed values, i.e., ρ = cor(Y_pred, Y_true). This measure reflects the ability of one variable to reconstruct another via the reconstructed state space. Following Sugihara et al. (2012), causality is inferred based on the convergence of ρ with increasing library size, together with statistical significance assessed using surrogate testing. In line with previous studies, we evaluate statistical significance at the p < 0.05 level. To facilitate interpretation, we further classify causal strength based on the magnitude of ρ.

Implementation details. Firstly, we do embedding dimension selection (simplex projection). For each univariate series X, we reconstructed its shadow attractor in ${\mathbb{R}}^E$. We varied E from 2 up to $\lfloor \sqrt{N}\rfloor$ (where N is the length of the influenza time series) and performed out-of-sample forecasting via simplex projection: for each x_t, we identified its k = E+ 1 nearest neighbors and predicted x_t+1 as the distance-weighted average of their successors. Forecast skill was quantified by the Pearson correlation ρ(E) between predicted and observed values. The optimal embedding E∗ was chosen as

$$E^{\ast }=\arg {\max }_E\rho (E).$$

Using this E∗, we then applied Convergent Cross Mapping (CCM) to assess directional causality between the reconstructed attractors of the putative driver X (environmental factor) and response Y (influenza activity index). To test whether X drives Y, we performed cross mapping from Y to X: for each library size L, we used Y's embedding to locate the k = E∗ +1 nearest neighbors of each point y_t, and predicted x_t as the weighted average of their corresponding $x_{t_i}$. Cross-map skill ρ_Y→X(L) was computed as the Pearson correlation between predicted and observed x_t.

To formally assess convergence, we quantified the increase in cross-map skill as

$$\mathrm{\Delta }{\rho }_{\text{CCM}}=\rho (L_{\max })-\rho (L_{\min }),$$

where the minimum and maximum library sizes were defined as L_min = E∗ +2 and L_max = m − E∗ −2, respectively, with m denoting the length of the available time series.

To control for spurious causality driven by shared seasonality or stochastic noise, we generated seasonal surrogate time series for each environmental variable by extracting seasonal anomalies and estimating additive noise using a smoothing spline approach. Statistical significance was assessed by comparing the observed Δρ_CCM against the surrogate-based null distribution, with p < 0.05 indicating a statistically significant causal relationship (Deyle et al., 2016; Guo et al., 2024).

We implemented all analyses using the R package rEDM (Ye et al., 2019).

3.5. GLM method

We analyzed daily influenza counts (FluP) with quasi-binomial generalized linear model (GL) fitted via mgcv::gam. The quasi-binomial family retains the usual logit link but inflates the variance by a data-driven dispersion factor ϕ, thereby correcting the standard errors for over-dispersion that is ubiquitous in infectious-disease counts.

For each state and for lags ℓ = 0, 1, 2 we regressed FluP_t on one lagged environmental predictor—ozone (O₃), absolute humidity (AH), temperature (T), relative humidity (RH), or solar radiation (RAD)—while adjusting for year-fixed and month-fixed effects and for short-term autocorrelation through log(FFluP_t−1) and log(FluP_t−2). Formally,

$$\text{logit}\left[\mathrm{Pr}({\mathrm{fl}\mathrm{u}\mathrm{P}}_t)\right]={\beta }_0+{\beta }_1{X}_{t-\ell }+{\gamma }_{\text{year}}+{\gamma }_{\text{month}}+{\varphi }_1\log {\mathrm{fl}\mathrm{u}\mathrm{P}}_{t-1}+{\varphi }_2\log {\mathrm{fl}\mathrm{u}\mathrm{P}}_{t-2}.$$

The analysis was carried out twice: (i) on the complete series and (ii) on a truncated series in which, for every year, the month with the lowest influenza burden was removed. Coefficients were subsequently re-scaled to represent the effect of a one-standard-deviation increase in exposure, and lag-1 estimates are shown in the main text. For consistency across analytical frameworks, statistical significance was evaluated at the p < 0.05 level. In the GLM framework, effect size is represented by the regression coefficient, with p-values obtained from standard hypothesis tests assessing whether coefficients differ significantly from zero.

3.6. PCMCI ⁺ method

Lagged causal relationships among influenza incidence proxy (FluP) and five meteorological variables (O₃, T, AH, RH, and RAD) were inferred using the PCMCI⁺ algorithm implemented in Tigramite (version 6.0). PCMCI⁺ combines the constraint-based Peter–Clark (PC) algorithm (Spirtes et al., 2001) with the Momentary Conditional Independence (MCI) test (Runge, 2018), enabling efficient removal of spurious associations followed by unbiased estimation of remaining causal links.

Causal inference using PCMCI⁺ relies on several standard assumptions, including causal sufficiency, the causal Markov condition, and faithfulness (Runge, 2018). In the classical time-series causal graph framework, satisfaction of the causal Markov condition for lagged parent sets typically requires the additional assumption that no instantaneous (contemporaneous, τ = 0) causal effects are present, such that causal influences act only through lagged pathways (Spirtes et al., 2000). In practical observational settings as we discussed here, however, limited temporal sampling resolution may cause fast interactions occurring within a sampling interval to manifest statistically as contemporaneous dependencies. Accordingly, in this study we allow τ = 0 associations to be identified within the PCMCI⁺ framework in order to capture potential instantaneous statistical dependencies. These contemporaneous links are interpreted solely as instantaneous dependence rather than as directed or mechanistic causal effects. Causal interpretation in this study therefore focuses primarily on lagged relationships with explicit temporal ordering (τ > 0).

Then all time series were ℓ₂-normalised prior to analysis, and candidate time lags were restricted to τ = 0, 1, 2. The procedure begins with a PC-based screening step: for each target variable, candidate parents are iteratively tested, and any link whose partial correlation fails to meet the significance threshold α_PC = 0.05—conditional on the current parent set—is pruned. Given the relatively limited sample size, conditional independence was assessed using linear partial correlation (ParCorr), which assumes approximately linear and Gaussian dependencies.

All surviving candidate links X_t−τ → Y_t are then evaluated using the MCI test, which conditions on the selected parents of both the source variable X and the target variable Y. The resulting signed partial correlation coefficient from the MCI test is retained as the edge weight (cross-MCI), representing the strength and direction of the lagged causal association. Statistical significance was assessed based on p-values derived from the conditional independence tests (ParCorr). No additional multiple testing correction was applied, and results were evaluated at a nominal significance level of α = 0.05.

Finally, temporal ordering (τ > 0) was used to assign edge directions, yielding the weighted causal network presented in this study.

4. Discussion

4.1. Environmental drivers of influenza

In this study, we examined the robustness of the ozone–influenza relationship in a subtropical setting using year-round influenza surveillance data from Hong Kong between 2014 and 2019. By explicitly incorporating complete year-long observations and implementing targeted data-exclusion schemes, our analysis extends prior influenza research that primarily focused on peak transmission seasons and provides a more comprehensive assessment of the temporal stability and plausibility of environmental drivers of observed influenza activity.

Across multiple complementary causal inference frameworks, including convergent cross-mapping (CCM), quasi-binomial generalized linear model (GLM), and the PCMCI⁺ algorithm, ambient ozone (O₃) consistently emerged as an influential factor associated with influenza activity. In particular, regression-based analyses indicated a negative association between ozone concentration and observed influenza activity intensity. This population-level finding is consistent with previous epidemiological studies conducted in Hong Kong and the United States, which reported inverse relationships between ambient ozone levels and influenza activity (Ali et al., 2018; Guo et al., 2024).

Ozone is most likely to influence observed influenza activity through environmental modulation of viral survival and transmissibility. Laboratory studies have shown that ozone exposure can substantially reduce the infectivity of aerosolized influenza A virus, with the virucidal effect strongly modulated by relative humidity. These findings provide mechanistic support for an inverse association between ambient ozone levels and observed influenza activity (Dubuis et al., 2021).

In addition to ozone, we observed that the effects of traditional climatic drivers such as absolute humidity and temperature were less consistent in this subtropical setting. In the GLM analyses, absolute humidity showed a significant association with influenza activity primarily at lag 0, whereas temperature exhibited a detectable effect only at lag 1 in the PCMCI⁺ framework. However, these associations were not consistently identified across methods or lag structures, suggesting that their influence on observed influenza activity in Hong Kong may be weaker or more context-dependent.

This pattern is consistent with previous findings that the role of temperature in observed influenza activity differs by climatic zone. While influenza epidemics in many temperate regions are strongly associated with cooler temperatures, such relationships are often less pronounced or less stable in tropical and subtropical areas (Tamerius et al., 2011).

4.2. Sensitivity of CCM-method

Previous studies by Cobey and Baskerville (Baskerville & Cobey, 2017; Cobey & Baskerville, 2016) have demonstrated that CCM is sensitive to the strongly periodic, noisy, and transient dynamics commonly observed in ecological systems, highlighting fundamental challenges in state-space reconstruction under real-world conditions. Building on these insights, we explicitly accounted for shared seasonality and stochastic noise by constructing seasonal surrogate time series based on anomaly extraction and smoothing spline–derived noise components.

Despite these controls, we find that temporal discontinuities—specifically, data gaps and the selective removal of time points—can substantially alter CCM inference, leading to inconsistent causal conclusions across analytical settings. This indicates that CCM sensitivity arises not only from intrinsic system dynamics, as previously recognized, but also from the temporal structure and continuity of the observed data. Importantly, while CCM has been widely applied in epidemiological and ecological studies, particularly in high-impact literature, the effects of temporal gaps and data discontinuities have received limited attention. Our findings therefore identify a previously underexplored source of instability in CCM-based causal inference and underscore the need for caution when applying CCM to incomplete or discontinuous time series.

Our analyses showed that applying CCM to truncated time series segments yielded markedly different effect sizes and significance patterns from those obtained using uninterrupted time series. Although this finding is context-specific, it has important methodological implications for the use of CCM in influenza surveillance research. In many epidemiological studies, influenza data are restricted to predefined epidemic seasons because of study design or reporting practices, particularly in temperate settings where transmission is strongly seasonal. By contrast, in tropical and subtropical regions, influenza circulation often persists throughout the year. Our results suggest that, under such conditions, restricting analyses to truncated seasonal segments may compromise the validity or interpretability of CCM-based inference. While a broadly related concern has been noted in earlier work more than a decade ago, our findings extend this issue in a distinct epidemiological context and indicate that the limitations of CCM may be especially relevant when continuous transmission dynamics are artificially segmented.

4.3. Sensitivity of GLM and PCMCI ⁺ methods

In contrast, the results obtained from the GLM and PCMCI⁺ analyses were comparatively more stable across different data-exclusion schemes. Unlike CCM, these methods do not rely on local geometric properties of a reconstructed state space. Removal of individual observations therefore primarily affects sample size or statistical power, rather than altering neighborhood structure or underlying model geometry, leading to greater apparent robustness in point-wise sensitivity tests.

This apparent stability, however, should be interpreted in light of the underlying assumptions of each method. GLM-based inference depends on logit-linear relationship, which may fail to capture nonlinear or state-dependent dynamics inherent in infectious disease systems. Similarly, PCMCI⁺ relies on strong causal assumptions—including causal sufficiency, the causal Markov condition, and faithfulness—that are difficult to verify empirically in observational epidemiological data.

As a result, the relative stability of GLM and PCMCI⁺ across exclusion schemes reflects robustness of estimated statistical associations or conditional dependencies under their respective modeling assumptions, rather than necessarily implying more accurate or complete causal reconstruction.

4.4. Limitations

A major limitation of this study is that, although we found CCM to be sensitive to truncation and temporal discontinuities in time series, it remains difficult to determine more systematically the conditions under which this sensitivity is most severe, the mechanisms through which bias is introduced, and whether robust correction strategies are possible. This issue is particularly relevant in influenza research, where many surveillance datasets cover only predefined epidemic seasons. In such settings, artificial temporal breakpoints may disrupt state-space reconstruction and thereby affect the stability of CCM-based causal inference. Although we attempted to minimize this problem by using year-round data and evaluating multiple data-exclusion schemes, the resulting bias cannot be fully ruled out.

Ideally, these issues should be evaluated in simulation systems with known causal structures. However, constructing simulation models that faithfully reproduce real-world influenza dynamics is inherently difficult, particularly because the mechanisms by which climatic and environmental factors influence influenza transmission remain incompletely understood. As a result, simulation studies are themselves difficult to regard as fully reliable benchmarks. As an alternative empirical check, we additionally conducted a sensitivity analysis in the Supplementary Materials using 2010–2015 U.S. sentinel influenza data, which showed a consistent pattern with main-text results based on Hong Kong influenza data: temporal gaps again tended to produce more significant CCM results.

CRediT authorship contribution statement

Linxi He: Writing – original draft, Visualization, Methodology, Formal analysis, Data curation. Daihai He: Writing – review & editing, Validation, Supervision, Formal analysis.

Conflict of interest

D He is an associate editor of this journal and D He is not involved in the editorial process. The authors have declared no other conflict of interest.

Handling Editor: Dr Yiming Shao

Appendix A. Supplementary data

The following is the Supplementary data to this article:

Data availability

Weekly influenza surveillance data were obtained from the Hong Kong Government's Open Data portal (https://data.gov.hk/en-data/dataset/hk-dh-chpsebcddr-flu-express). Environmental and air pollution data were obtained from the Hong Kong Observatory and the Hong Kong Environmental Protection Department (EPD) (https://www.epd.gov.hk/epd/english/environmentinhk/air/data/air_data.html).