Descriptive statistics included the mean, maximum, standard deviation, and coefficient of variation (CV), providing an initial quantification of streamflow magnitude and variability. Structural changes were identified using Pettitt's test (Pettitt, 1979), which revealed a significant breakpoint in the time-series mean. Sen's slope estimator (Sen, 1968) quantified long-term trends, while the Mann–Kendall test (Mann, 1945) evaluated trend significance.

Stationarity was explored through additive time series decomposition using classical seasonal decomposition based on moving averages, which separates the series into trend, seasonal, and residual components, enabling visual inspection of long-term behavior, seasonal structure, and residual stability. Subsequently, stationarity was formally evaluated using the Phillips–Perron (PP) test, which tests the null hypothesis of a unit root and is robust to autocorrelation and heteroskedasticity in the residuals.

Temporal dependencies were examined using autocorrelation function (ACF) and partial autocorrelation function (PACF) analyses (Chatfield, 1989). The ACF quantifies the linear correlation at lag k and is defined as: 1rk=∑t=1n-k(xt-x‾)(xt+k-x‾)∑t=1n-k(xt-x‾)2, where xt is the series value at time t, x‾ is the mean, and n is the number of observations.

PACF measures correlation at lag k controlling for intermediate lags, informing autoregressive and moving average term selection in SARIMA models (Box et al., 2015). ACF and PACF plots were examined for seasonality and persistence, using 95 % confidence intervals to identify significant lags.

The hydrological relationship between precipitation and streamflow in the Cravo Sur basin has been previously demonstrated (Ocampo-Marulanda et al., 2025), supporting the use of CHIRPS precipitation data as potential predictors. In this study, 81 precipitation time series corresponding to CHIRPS grid cells within the Tocaría sub-basin were analyzed and reduced in dimension to extract the main information contained in this dataset (Xn) using an NLPCA approach. The nonlinear principal components (NLPCs) obtained in the bottleneck layer of the NLPCA model (ym) represent the main modes of precipitation variability in the region. For this purpose, a Multi-Layer Perceptron (MLP) neural network was used, with an input layer, one or more hidden layers, and an output layer. During training, the back-propagation algorithm was used to identify the optimal combination of synaptic weights by minimizing reconstruction error through iterative updates of weights and bias terms. Several network topologies were evaluated, and the [81-2-81] architecture produced the best results; the components were estimated in hierarchical order. Linear activation functions were used for the hidden and output layers, and the best performance was obtained after 100 training epochs.

The nonlinear principal components were estimated using only the 70 % subset of precipitation data corresponding to the SARIMAX model training period (1985–2008), as defined in Sect. 2.5. Implemented through an autoencoder with bottleneck architecture, NLPCA minimizes the reconstruction error between inputs and outputs, thereby capturing nonlinear relationships beyond the capabilities of traditional PCA (Canchala et al., 2019; Scholz et al., 2007). Validation of the extracted components included reconstruction accuracy and latent-space visualization, confirming their effectiveness in summarizing precipitation variability.

MVs comprising sea surface temperature anomalies, sea-level pressure, and wind indices known to influence Colombian hydrology (Canchala et al., 2020a, b; Cerón et al., 2020; Poveda et al., 2001) were evaluated.

Spearman rank correlations between each MV and streamflow were calculated for lags of up to 14 months to capture delayed effects. Variables exhibiting statistically significant correlations at lags ≥ 6 months were retained, reflecting hydroclimatic memory. Only predictors with demonstrated hydrological relevance and minimal intercorrelation were preserved to ensure model parsimony and stability.

A two-step screening procedure was applied to select the exogenous predictors used in the forecasting models. First, pairwise correlation matrices visualized as heatmaps were used to identify groups of highly correlated MVs, thereby detecting potential redundancy among predictors and identifying those most strongly associated with streamflow (Dormann et al., 2013). Second, the Variance Inflation Factor (VIF) was computed to quantify multivariate collinearity among the candidate predictors (Katrutsa and Strijov, 2017).

Predictors with VIF values greater than 10 were considered excessively collinear and were iteratively removed. When several variables conveyed similar large-scale climatic information, selection was guided not only by the reduction of VIF values but also by the strength of their absolute correlation with streamflow, in order to retain the predictor with the most hydrologically relevant signal while preserving model parsimony. After each elimination step, VIF values were recalculated until the reduced predictor set showed acceptable multicollinearity levels.

Model performance was first assessed using R2 and RMSE for the training, validation, and test periods, alongside AIC and BIC to evaluate goodness-of-fit. These metrics provide a quantitative overview of how well each model reproduces the general behavior of the streamflow series.

Forecasts were then compared against simple baselines: monthly climatology, computed as the average flow for each month in the training period, and persistence, which assumes that the flow at time t equals the flow at time t-1. This comparison allows us to contextualize the added value of the SARIMAX model relative to straightforward, intuitive approaches.

Special attention was given to the evaluation of high flows, defined as events above the 90th percentile, by computing RMSE for these peaks. Since the SARIMAX model is designed to capture extreme events, this peak-focused analysis was prioritized to assess its skill in predicting the most critical conditions for water management.

Finally, 95 % confidence intervals were estimated for all forecasts. After fitting the model on log-transformed data with exogenous variables, predictions and confidence intervals were obtained in log scale and back-transformed to the original units. These intervals illustrate the uncertainty around the forecasted values, providing a clear sense of the expected range for future streamflow.

Forecasting accuracy typically decreases as the prediction horizon extends, a well-documented phenomenon referred to as forecast error degradation (Xiang et al., 2024). To assess this behavior, the best-performing SARIMAX model was evaluated over multiple lead times using RMSE and R2 as performance metrics. Forecasts were produced for 2 to 24 months ahead in increments of 2 months, enabling the quantification of predictive accuracy across short, medium, and long term horizons.

To capture the evolution of forecast quality, RMSE and R2 values were analyzed in pairs of consecutive horizons. This pairing facilitated the visualization of degradation trends through line plots, offering insights into the temporal limits of model effectiveness.

Residual diagnostics are essential for evaluating model adequacy by verifying whether prediction errors conform to standard statistical assumptions. Residuals were computed as the difference between observed and forecasted streamflow values, forming a time series that was subjected to rigorous statistical and graphical analysis.

Normality was tested using the Jarque–Bera test (Jarque and Bera, 1980), assessing whether residuals follow a Gaussian distribution, a prerequisite for valid inference and uncertainty quantification. Autocorrelation was examined through the Durbin–Watson statistic (Durbin and Watson, 1950), where values near 2 suggest the absence of significant serial dependence, thus confirming that temporal structures have been effectively captured. Heteroscedasticity was assessed via the Breusch–Pagan test (Breusch and Pagan, 1979); significant p values indicated non-constant residual variance, potentially undermining the model's generalizability.

Complementing these tests, graphical diagnostics were employed to detect patterns indicative of model misspecification. Histograms with kernel density estimation (Silverman, 1986) were used to visually assess normality, while residuals-versus-fitted-values plots revealed heteroscedastic patterns or structural biases. Quantile-quantile plots (Wilk and Gnanadesikan, 1968) further evaluated the agreement between empirical and theoretical residual quantiles. Finally, ACF plots were used to identify lagged dependencies; the absence of significant peaks outside the confidence bounds supported the white-noise assumption.

The streamflow time series at El Playón station exhibits pronounced interannual and seasonal variability, with values ranging from 2.5 to 280 m3s-1 and a mean of 92.4 m3s-1. This range is consistent with the monomodal rainfall regime characteristic of the region (Urrea et al., 2016). A standard deviation of 66.4 m3s-1 and a coefficient of variation of 71.8 % confirm the high dispersion and temporal heterogeneity of the data, which pose a challenge for accurate modeling and prediction.

Trend-detection tests were also applied. Visual inspection suggested a potential breakpoint in January 2019; however, the Pettitt test yielded a p value of 1.00, indicating that this change is not statistically significant. Likewise, Sen's slope estimator and the Mann–Kendall test indicated a slight upward trend in discharge, but without statistical significance (p = 0.3445), suggesting the absence of a persistent monotonic trend (see Table 1 and Fig. 4).

These characteristics highlight the complexity of the time series and justify the need for rigorous preprocessing, including stationarity and autocorrelation analysis.

Stationarity was assessed through a combination of exploratory decomposition and formal statistical testing. Figure 5 presents the seasonal decomposition results. The trend component does not exhibit a persistent increasing or decreasing pattern over the analyzed period; instead, it shows multi-year fluctuations characterized by alternating phases of higher and lower streamflow. These variations suggest low-frequency hydroclimatic variability rather than a clear long-term trend, indicating that mean discharge remained broadly stable over time, although modulated by interannual and decadal variability.

The seasonal component displays a clear and regular periodicity, reflecting a 12-month seasonal cycle in streamflow. This pronounced seasonality is consistent with the regional hydroclimatic regime, dominated by the alternation between wet and dry seasons, and confirms that streamflow variability in the Tocaría River is largely controlled by seasonal precipitation patterns.

The residual component exhibits a random distribution centered around zero, with no discernible systematic patterns or trends. It captures short-term variability, extreme events, and noise not explained by the trend or seasonal components, supporting the adequacy of the seasonal decomposition and the assumption of weak stationarity after seasonal removal.

The Phillips–Perron test further confirmed stationarity, yielding a test statistic of -6.5005 and a p value < 0.0001, allowing rejection of the null hypothesis of a unit root. These results justify modeling without differencing and the selection of d=0 in the SARIMA model.

The ACF displays significant periodic peaks at regular intervals, indicative of strong seasonal periodicity and persistent temporal dependencies beyond short lags (Fig. 6). This cyclicality aligns with hydrological patterns driven by seasonal precipitation and runoff dynamics, underscoring the appropriateness of seasonal forecasting models such as SARIMA.

The PACF shows a prominent spike at lag 1 followed by rapid decay to values within the confidence bounds (Fig. 7), consistent with an autoregressive process of order 1 (AR(1)); however, the selected SARIMA model incorporates higher-order dynamics to capture the longer memory and seasonal structure identified in the ACF. The lack of significant PACF values at higher lags indicates limited benefit from additional autoregressive terms.

Based on combined ACF and PACF analysis, a SARIMA (4,0,4)(0,0,3)12 model was selected. This model configuration incorporates the identified seasonal periodicity (season length s=12), confirms stationarity (d=0), and captures short-term autoregressive and moving average dynamics. Residual diagnostics validated the model's adequacy, demonstrating no significant autocorrelation.

The 81 CHIRPS precipitation time series for the Tocaría River basin were compressed into two nonlinear principal components (NLPC1 and NLPC2) using an autoencoder-based NLPCA. The resulting latent scores showed variances of 17.64 (NLPC1) and 6.30 (NLPC2), corresponding to 73.69 % and 26.31 % of the variance within the two-dimensional latent space, respectively. These percentages describe the relative contribution of each latent coordinate. The two NLPCs were subsequently used as exogenous covariates in the streamflow forecasting model, providing a parsimonious representation of basin-scale precipitation variability. This approach is consistent with Ocampo-Marulanda et al. (2025), who used CHIRPS-derived nonlinear precipitation components as compact predictors for streamflow reconstruction in data-scarce basins.

The influence of 21 MVs on streamflow variability in the Tocaría River was assessed using a lagged Spearman correlation analysis, considering time lags from 0 to 14 months. This asynchronous framework allowed the identification of delayed relationships between climate anomalies and hydrological responses, which are crucial for predictive modeling. This lagged-correlation framework enables the detection of delayed hydroclimatic teleconnections, which are essential for long-lead streamflow forecasting.

Statistical significance was evaluated using two-tailed tests with confidence levels between 90 % and 99 %. Figure 8 displays the lagged correlations between the MVs and streamflow at El Playón station. Several variables showed statistically significant correlations across multiple lags and exhibited a quasi-periodic structure that reflects the hydroclimatic seasonality modulated by the Intertropical Convergence Zone (ITCZ).

Key Pacific-related indices – namely NINO12 (ρ=0.75) and NINO3 (ρ=0.60) – emerged as the strongest predictors, underscoring the robust influence of ENSO-related sea surface temperature (SST) anomalies on the Tocaría River streamflow. These findings are consistent with previous studies across Colombia (e.g., Cadavid and Salazar, 2008; Canchala et al., 2020c; Poveda et al., 2002), which reported comparable lagged correlations between ENSO indices and regional streamflow, often with cyclic behavior.

In addition, Atlantic-related variables such as AMM (ρ=0.32) and TSA (ρ=0.25) showed statistically significant, albeit weaker, correlations. These results corroborate the relevance of Atlantic SST variability in eastern Colombia. The Llanos low-level jet and moisture transport from the tropical Atlantic – modulated by indices such as AMM and TSA – have been highlighted as key controls of precipitation and streamflow in the Orinoquía region (Correa et al., 2024; Builes-Jaramillo et al., 2022; Labat et al., 2012; Nieto et al., 2008).

The set of significantly correlated predictors spans both Pacific and Atlantic basins: NINO3, NINO4, NINO12, NINO34, PDO, TNI, NP, QBO (Pacific-related), and AMO, NTA, TNA, NAO, AMM, TSA (Atlantic-related). This reinforces the complexity of the climate–hydrology interactions in the Tocaría River basin and emphasizes the need to account for both direct and lagged macroclimatic signals in predictive modeling.

The results of the correlation heatmap (Fig. 9) reveal substantial interdependence among several MVs, indicating that the inclusion of multiple closely related predictors may introduce redundancy and model instability. For instance, the ENSO indices NINO3, NINO4, NINO12, and NINO34 exhibit strong mutual correlations, suggesting that a single representative index is sufficient to capture ENSO-related variability. Similarly, the high correlations observed among the TNA, NTA, and TSA indices indicate overlapping information content; therefore, only one of these indices was retained in the final predictor set to mitigate multicollinearity. In contrast, variables such as NAO, PDO, TNI, and QBO showed weaker absolute correlations with streamflow and were therefore considered less informative for forecasting purposes than the predictors retained in the final subset.

Similarly, the high correlations observed among the TNA, NTA, and TSA indices indicate overlapping information content; therefore, only one of these indices was retained in the final predictor set to mitigate multicollinearity. In contrast, variables such as NAO, PDO, TNI, and QBO showed weaker absolute correlations with streamflow and, therefore, were considered less informative for forecasting purposes than the predictors retained in the final subset.

To ensure model robustness and reduce the risk of overfitting, multicollinearity among the candidate predictors was quantified using VIF. Table 2 summarizes the VIF values for all variables considered. While most predictors exhibited acceptable multicollinearity levels (VIF < 10), several variables, particularly those associated with ENSO, showed extreme multicollinearity. Specifically, the NINO3, NINO4, NINO12, and NINO34 indices presented VIF values ranging from 2492 to 81 825. In addition, the TNA and TSA indices yielded infinite VIF values, indicating perfect collinearity with other predictors, most likely due to structural redundancy among Atlantic SST indices. This behavior is consistent with the strong interdependencies observed in the correlation heatmap.

To address this issue, various combinations of variables were tested through an iterative model-refinement process that prioritized predictors with strong individual relationships to streamflow and minimal mutual collinearity. This selection strategy balances predictive power and model parsimony, mitigating multicollinearity-induced variance inflation and enhancing generalizability.

Ultimately, the optimal subset of exogenous predictors was determined to be PNLPCA_PC1, PNLPCA_PC2, NINO12, TNA, AMO, and AMM.

This final configuration minimized multicollinearity, preserved critical information from both precipitation and climate variability, and improved the interpretability and reliability of the streamflow forecasting model. The final predictor pool, comprising key macroclimatic indices and the first two NLPCs, provided the input for the forecasting models evaluated in the subsequent section.

Figure 10 shows the streamflow prediction results for El Playón station using the LOG-SARIMAX-MVNL model, which integrates exogenous variables, including MVs and NLPCs derived from local precipitation data. The figure includes observed streamflow, model predictions for the training period, and forecasts for the validation and test periods.

The observed streamflow series reveals pronounced seasonality, characterized by recurrent peaks and troughs consistent with the hydroclimatic regime of the basin. During the training phase, the model reproduces the seasonal dynamics and intra-annual variability with high accuracy, showing minimal deviations. In the validation and test period, although some differences appear; particularly underestimations of extreme highs or delays in response; the general flow patterns, including the timing and intensity of seasonal peaks, are well preserved. These results highlight the model's capacity to capture both historical and prospective streamflow behaviors based on relevant exogenous predictors.

Figure 11 shows the performance of the log-transformed SARIMAX model for river-flow forecasting. Observed flows are shown in black, the model fit during training in blue, and the forecast mean in red. Shaded areas indicate uncertainty: the lighter band represents the 95 % prediction interval, and the darker band represents the 95 % confidence interval. The model captures the overall trend and seasonal patterns well, although extreme peaks tend to be underestimated and uncertainty increases for higher flows, reflecting their inherent variability. Overall, the figure highlights both the model's accuracy and the reliability of its forecasts.

To assess the model's temporal generalization capability, streamflow forecasts were generated for lead times of up to 24 months (Fig. 12). RMSE values remained below 30 m3s-1 during the first 8 forecast months, which is low relative to the standard deviation of the observed series (σ≈ 66 m3s-1). Beyond month 14, RMSE increased progressively before stabilizing below 50 m3s-1. The coefficient of determination (R2) remained above 0.92 up to month 12, after which it gradually declined, reaching 0.82 by month 18 and remaining above 0.80 throughout the 24-month forecast horizon.

These results confirm the robustness of the LOG-SARIMAX-MVNL model for medium- and long-term forecasting, maintaining a favorable balance between accuracy and lead time. The performance is suitable for practical decision-making applications in water resource planning, including drought early warning and integrated watershed management in the Tocaría River basin.

The residuals are not perfectly normal (Shapiro–Wilk W = 0.913, p = 0.0004; Jarque–Bera JB = 58.951, p < 0.001), show some positive autocorrelation (Durbin–Watson = 1.243), and exhibit slight heteroscedasticity over time (Breusch–Pagan BP = 3.987, p = 0.0459). These patterns provide useful insights into the dynamics of the series and support a more nuanced interpretation of its temporal behavior.

The diagnostic plots show that the residuals generally follow the expected distribution (see Fig. 13), with slight deviations in the tails (Q–Q plot). Residuals tend to spread more at higher fitted values, indicating mild heteroscedasticity (Residuals vs Fitted). The ACF shows mostly no significant autocorrelation, though the first lags suggest weak positive dependence. Overall, the model captures the main dynamics, with residuals showing only minor deviations.

In summary, the statistical analysis indicates that the LOG-SARIMAX-MVNL model provides reliable streamflow forecasts under peak-flow conditions; however, its ability to accurately reproduce high-magnitude extreme flows remains limited, likely due to complex hydrometeorological processes that are not fully captured by the selected predictors.