Revealing the Excitation Causality between Climate and Political Violence
via a Neural Forward-Intensity Poisson Process

Causal mechanisms that generate political violence often have an excitation mechanism Zammit-Mangion et al. (2012). The excitation effect that extreme climate events have on armed violence is often non-linear and bifurcates, e.g., extreme weather can cause agriculture community to shift to political violence and thereafter future climate changes are unlikely to have the same impact. Consequently, continuous functions of causal modeling that do not adjust for uncertain functional behaviour change is unlikely to be successful in complex societies Field et al. (2012). These continuous causal models include Krakovska et al. (2018): Granger causality and Predictability Improvement (PI), Conditional Mutual Information (CMI), and Convergent Cross Map (CCM) methods.

For discrete excitation causal models, they are typically heavily parameterized especially on the time window, such that they are generally applicable only to a particular genre (e.g., IED attacks in a particular area Tench et al. (2016)), and lack general understanding of the nature of climate vs. terrorism. Furthermore, terrorism data often suffer from sparsity, which is caused by a variety of factors ranging from under-reporting, lack of data in an emerging crisis area, to poor data quality. This can have an effect in non-linear likelihood estimators which is commonly used in current deep methods with linear loss function (e.g., LSTM).

Our goal is to develop a general method that can self-configure to identify causal excitation signals and relate them to contextualised recent historical events. First of all, a specialized predictor is required for the excitation-based non-linear mechanism. Suppose a deep learning model has the inputs of the historical terrorism and climate data, and the target output of the attack forecasting. Then, there would exist two main challenges in this task, which raise additional requirements to both modeling and its evaluation.

Poisson process has been recognized as a principled framework for modeling event data Stoyan et al. (2013), where the intensity parameter is a function determined by relevant covariates. There are various derivatives of Poisson process, which focus on optimizing the linear form of intensity function with its covariates, e.g. Hawkes process Tench et al. (2016); Zammit-Mangion et al. (2012), self-correcting process Isham and Westcott (1979), reinforced Poisson processes Shen et al. (2014) and etc. In the most recent work Xiao et al. (2017); Mei and Eisner (2017), the intensity function is modeled via a neural network Shchur et al. (2021), which enables more non-linear covariates being captured.

As shown in Fig.1, in our proposed model, we adopt the NN-based intensity function to fit the non-linear mechanism between climate and conflict generation process. We make a one-step forward probabilistic forecasting via the Poisson process with the intensity value in each day. Our proposed neural forward-intensity Poisson process (NFIPP) model meets the requirements raised by the aforementioned challenges using the prior of Poisson and the evaluation of likelihood.

In this paper, we evaluate the excitation causality between climate and terrorism using the proposed neural forward-intensity Poisson process (NFIPP). Two major contributions of this paper are:

(i) In order to fit the non-linear mechanism in the sparse and timing-uncertain terrorist attack time series, we propose to embed a neural network based forward-intensity function in the Poisson process. This algorithm has been tested on a toy model and proven to have advantageous performance compared to conventional methods.

(ii) NFIPP is applied on the real terrorism and climate datasets for daily attack occurrence intensity inference. We reveal the excitation causal link from extreme climate events to terrorism via the posterior likelihood analysis. We find that the extreme climate events and other events (e.g., economy and politics) have the discrete excitation causal links to the terrorism rather than a continuous time causality, which require domain expertise in informing interpretation.

The remainder of this paper is organized as follows. In Section 2, we formulate the end-to-end learning framework for NFIPP model and verify its application scope with our assumption. In Section 3, we apply NFIPP to the real data and analyze the results with contextualised recent historical events. Section 4 concludes this paper and the ideas for future work.

Fig.2(a) demonstrates the end-to-end NFIPP learning framework. Taking the D-dimension feature vector at time $t$, $\mathbf{X_{t}}=[x_{1}^{t};x_{2}^{t};...;x_{D}^{t}]\in\mathbb{R}^{D\times 1}$, as the input, then NN is expected to has the output of the forward-intensity estimation $\hat{\lambda_{t}}$. Specifically, we introduce the “softplus” function $Y=\ln{(1+e^{X})}$ in the last layer to ensure $\hat{\lambda_{t}}\geq 0$ meeting the positive constraint of Poisson process, while avoid vanishing gradient problem if ”relu” function is used Mei and Eisner (2017).

With the outputs $\{\hat{\lambda_{t}}\}_{t=1}^{T}$ and the observation $\{k_{t}\}_{t=1}^{T}$ in the training data, the posterior log-likelihood $\Sigma\ln[P(k|\lambda)]$ can be derived as a function $\mathcal{L}(\mathbf{w},\mathbf{b})$ with the weight and bias parameters in the NN using function (1), which enable the training of $\{\mathbf{w},\mathbf{b}\}$ according to maximum log-likelihood via gradient back-propagation with learning rate $\eta$ as:

In order to verify the performance and scope of NFIPP model, we create a toy scenario with transparent intensity functions. In the toy scenario, suppose (i) $\mathbf{K}\in\mathbb{N}^{1\times T}$ are the posterior observations of corresponding Poisson process with the intensities $\mathbf{\Lambda}\in\mathbb{R}_{+}^{1\times T}$; (ii) Each element $\lambda_{t}\in\mathbf{\Lambda}$ is the function of potential features $\mathbf{x}^{t}\in[0,1]^{D}$, conditioned on $\lambda_{t}=\mathscr{F}(\mathbf{x}_{t})\geq 0$. In the model validation, we set $T=1e3$, $D=4$, which are closed to the terrorism data training scale. For robustness test, we set $x_{4}^{t}$ to be an independent variable of the function $\mathscr{F}$, which act as a noise input. $\mathscr{F}$ randomly takes its form from a bunch of composite non-linear functions, including power, exponential, trigonometric and etc. (e.g. $\lambda=e^{x_{1}}\sin{(x_{2})}+x_{3}^{\frac{1}{4}}$). In the test, we validate the average accuracy performance of our NFIPP over scenarios with mean of intensities varying from $10^{-1}$ to $10^{1}$, compared to conventional naive NN with the same neuron topology. With massive Monte-Carlo, the validation results are demonstrated in Fig.2(b).

Both models are robust and without overfitting to the noise feature. Our NFIPP shows distinct advantage in prediction when the mean event occurrence rate is lower than 1 per unit time. This advantage comes from our improvement to the model priori of Poisson distribution. With the occurrence rate increasing to 4, Poisson distribution approximates to Gaussian distribution, where the advantage gradually fades away while the model performance keeps well. In reality for each country in last two decades, rarely do mean daily terrorist attack rate rises above 3 even in most dangerous countries (see Section 3). This means our NFIPP model would lead to significant improvement in the terrorism prediction.

Based on the general idea of statistical causality test, we recognize a positive causality when additional historical climate data improve the performance of terrorism predictor comparing to historical terrorism only. In NFIPP, the prediction performance is evaluated by the posterior Poisson likelihood using the forward-intensity output given by predictor. Then the improvement can be calculated with the difference between the log-likelihood given by only historical terrorism data $\mathscr{L}_{1}$ and that the log-likelihood given by both historical terrorism and climate data $\mathscr{L}_{2}$. Since we only focus on the positive/negative effects and relative magnitude of improvement, for intuitive visualization, we define the likelihood gain with the difference normalized by a scalable $\tanh$ function:

where $\alpha$ is an adaptive scaling parameter.

Besides, in order to give a comprehensive analysis, we introduce the prediction rate for reference. Since the model output is the forward-intensity $\lambda_{t}$ for time $[t,t+\tau]$ other than the an exact value, the count of terrorist attacks prediction within a period $[t_{1},t_{2}]$ can only be calculated in a statistical way based on Poisson’s property, i.e., $\hat{K}=\Sigma_{t_{1}}^{t_{2}}\lambda_{t}$. We define the prediction rate of the model over a specific time as:

which indicate the accuracy rate of matched prediction.

Furthermore, the overall impact of climate on the terrorism via the causal link is important for the cross-validation with results from political science Chen et al. (2015). Here we use the ratio of additional correct predicted counts via climate data to the terrorism data as the index. For climate gain, we only consider the years that have positive likelihood gain, then the climate gain ratio is defined as:

where the TP Difference is the true positive prediction difference between the counts from model with and without the climate data.

We select a diverse range of 8 countries based on a combination of factors: total volume of political violence (1993-2019), geographic and ethnolinguistic diversity from each other, and genre of political violence. Fig.4 demonstrates the results for Afghanistan (AFG), Thailand (THA), Nigeria (NGA), India (IND), Philippines (PHL), Algeria (DZA), United Kingdom (GBR), Colombia (COL). The raw output from the NFIPP is daily based, but for simplicity of overall understanding, we take the period of one year in evaluation (e.g., likelihood gain).