Modeling noisy time-series data of crime with stochastic differential equations

Abstract

We develop and calibrate stochastic continuous models that capture crime dynamics in the city of Valencia, Spain. From the emergency phone, data corresponding to three crime events, aggressions, stealing and women alarms, are available from the year 2010 until 2020. As the resulting time series, with monthly counts, are highly noisy, we decompose them into trend and seasonality parts. The former is modeled by geometric Brownian motions, both uncorrelated and correlated, and the latter is accommodated by randomly perturbed sine-cosine waves. Albeit simple, the models exhibit high ability to simulate the real data and show promising for crimes-interaction identification and short-term predictive policing.

Spatio-temporal stochastic differential equations for crime incidence modeling

“…These types of models have certainly been considered in environmental sciences (Calatayud et al. 2022 , 2023b ; Chowell et al. 2016 ; Nafidi et al.…”

“…Further, it includes spatial correlations in the formulation, in contrast to similar stochastic models (Calatayud et to assess the evolution of social behaviors. These types of models have certainly been considered in environmental sciences (Calatayud et al 2022(Calatayud et al , 2023bChowell et al 2016;Nafidi et al 2022;Pell et al 2018) (Zika, Ebola, COVID-19 and CO 2 emissions, with certain exponential growths) and turn out to be successful in our context of offenses. Nevertheless, phenomenological forecasting models are limited by the assumption that future incidence will follow the patterns of incidence observed in the past.…”

“…However, the solution stochastic process x t ¼ x 0 e lt would not be irregular in such a formulation, but smooth (Neckel and Rupp 2013). This issue could be fixed by incorporating a certain random model error E t , with resulting response process x t ¼ x 0 e lt þ E t , and then applying Bayesian or maximum-likelihood inference for parameter estimation (Calatayud and Jornet 2020;Calatayud et al 2022Calatayud et al , 2023bSmith 2013;Xiu 2010). Nonetheless, a certain structure of the residuals of the deterministic exponential model would then be required, for example, symmetry around zero, homoscedasticity, etc.…”

Improving crime prediction through ensembles

“…A seguir, Calatayud et al (2022) propuseram a modelagem de ruído a partir de stochastic differential equations. Identificando a grande quantidade de ruídos em séries temporais relacionadas a crimes, se dispuseram a decompor uma série temporal relacionada a cidade de Valência na Espanha e observar em que medida o modelo replicaria com qualidade os dados reais, a partir da modelagem de ruídos [Calatayud et al 2023].…”