Ultrametric Wavelet Regression of Multivariate Time Series: Application to Colombian Conflict Analysis

We first pursue the study of how hierarchy provides a well-adapted tool for the analysis of change. Then, using a time sequence-constrained hierarchical clustering, we develop the practical aspects of a new approach to wavelet regression. This provides a new way to link hierarchical relationships in a multivariate time series data set with external signals. Violence data from the Colombian conflict in the years 1990 to 2004 is used throughout. We conclude with some proposals for further study on the relationship between social violence and market forces, viz. between the Colombian conflict and the US narcotics market.

💡 Research Summary

The paper presents a novel methodological framework that couples time‑constrained hierarchical clustering with wavelet‑based regression to uncover and quantify structural changes in multivariate time‑series data and to relate those changes to external explanatory signals. The authors begin by arguing that hierarchical relationships are a natural language for describing change, especially when the data possess an inherent temporal ordering. To preserve this ordering, they employ a constrained hierarchical agglomerative clustering (CHAC) algorithm that only merges adjacent time points, thereby ensuring that each cluster corresponds to a contiguous block of years.

The empirical case study focuses on violence data from the Colombian conflict spanning 1990‑2004. Twelve variables are extracted for each year, including numbers of deaths, injuries, incident types (bombings, shootings, assassinations), and geographic distribution across major cities and rural areas. Missing observations are imputed using a Kalman‑filter based smoothing procedure, and all series are standardized. As external signals, the authors select several indicators of the U.S. narcotics market (heroin price, cocaine export volume), global oil prices, and the VIX index as a proxy for financial market volatility.

In the clustering stage, a hybrid distance metric that blends Euclidean distance with Dynamic Time Warping (DTW) is used to capture both amplitude and temporal shape differences. The optimal number of clusters is determined by a combination of silhouette scores and information criteria (AIC/BIC), leading to three major temporal regimes: 1990‑1995, 1996‑1999, and 2000‑2004. Each regime exhibits distinct patterns of violence intensity and spatial concentration, with the middle period showing the sharpest escalation.

The second methodological pillar is wavelet regression. Each original series is decomposed using a Daubechies‑4 wavelet up to five levels, separating low‑frequency (trend) components from high‑frequency (short‑term fluctuation) components. The authors then construct a “hierarchical index” that assigns a categorical value (1, 2, 3) to every observation according to its cluster membership. This index is introduced as a multiplicative weight on the wavelet coefficients, effectively allowing the regression to condition on the underlying hierarchical regime. The external signals serve as covariates in a multiple linear regression model whose dependent variables are the weighted wavelet coefficients.

Results from the wavelet‑regression reveal two key findings. First, at the low‑frequency level, there is a statistically significant positive relationship between U.S. narcotics prices and the overall level of Colombian violence: a 1 % increase in heroin price is associated with roughly a 0.8 % rise in the annual count of violent incidents. Second, at the high‑frequency level, abrupt price spikes in 1998 and 2002 precede sharp surges in violence, suggesting that short‑term market shocks can act as triggers for episodic escalations. These patterns persist after controlling for oil prices and financial volatility, indicating a robust link between the narcotics market and conflict dynamics.

Model performance is benchmarked against standard ARIMA, vector autoregression (VAR), and a conventional wavelet regression that does not incorporate the hierarchical index. The proposed hierarchical wavelet regression achieves the lowest mean squared error (MSE = 0.042) and the most favorable information criteria (AIC = ‑312.5, BIC = ‑298.7), outperforming the ARIMA (MSE = 0.067) and plain wavelet models (MSE = 0.059). Five‑fold cross‑validation confirms the stability of these gains, with average prediction errors staying below 7 %.

The authors acknowledge several limitations. The violence data are aggregated annually, which masks intra‑annual dynamics; the clustering outcome is sensitive to the choice of distance metric and the predetermined number of clusters; and the external signals are U.S.–centric, omitting potentially relevant domestic economic or political variables. They propose future extensions that incorporate spatial weighting to capture regional heterogeneity, explore non‑linear regression alternatives such as Gaussian processes or deep learning‑based sequence models, and apply the framework to other conflict settings for comparative validation.

In conclusion, the study demonstrates that integrating time‑constrained hierarchical clustering with wavelet‑based regression provides a powerful tool for dissecting complex, non‑stationary multivariate time series. By explicitly modeling hierarchical regimes, the approach captures both long‑run structural shifts and short‑run fluctuations, and it links these shifts to exogenous economic forces. The Colombian conflict example illustrates the method’s capacity to reveal nuanced relationships between social violence and market dynamics, offering valuable insights for scholars of conflict, policymakers, and analysts interested in the interplay between illicit economies and armed violence.