Spatial point process models are widely applied to point pattern data from various fields in the social and environmental sciences. However, a serious hurdle in fitting point process models is the presence of duplicated points, wherein multiple observations share identical spatial coordinates. This often occurs because of decisions made in the geo-coding process, such as assigning representative locations (e.g., aggregate-level centroids) to observations when data producers lack exact location information. Because spatial point process models like the Log-Gaussian Cox Process (LGCP) assume unique locations, researchers often employ {\it ad hoc} solutions (e.g., jittering) to address duplicated data before analysis. As an alternative, this study proposes a Modified Minimum Contrast (MMC) method that adapts the inference procedure to account for the effect of duplicates without needing to alter the data. The proposed MMC method is applied to LGCP models, with simulation results demonstrating the gains of our method relative to existing approaches in terms of parameter estimation. Interestingly, simulation results also show the effect of the geo-coding process on parameter estimates, which can be utilized in the implementation of the MMC method. The MMC approach is then used to infer the spatial clustering characteristics of conflict events in Afghanistan (2008-2009).

翻译：空间点过程模型广泛应用于社会科学和环境科学各领域的点模式数据分析。然而，拟合点过程模型时面临的一个严重障碍是重复点的存在，即多个观测点具有完全相同的空间坐标。这种情况通常源于地理编码过程中的决策，例如当数据生产者缺乏精确位置信息时，会将代表性位置（如聚合层面的质心）分配给观测点。由于像对数高斯柯克斯过程（LGCP）这样的空间点过程模型假设位置具有唯一性，研究者常在分析前采用临时解决方案（如坐标微扰）来处理重复数据。作为替代方案，本研究提出了一种改进的最小对比度方法，该方法通过调整推断程序来考虑重复点的影响，而无需修改原始数据。所提出的MMC方法应用于LGCP模型，仿真结果表明该方法在参数估计方面相较于现有方法具有优势。值得注意的是，仿真结果还揭示了地理编码过程对参数估计的影响，这一发现可用于MMC方法的实际应用。最后，我们运用MMC方法推断了阿富汗（2008-2009年）冲突事件的空间聚类特征。