## 1. Introduction

Recent work on statistical methods for spatial point pattern data has made it easy to fit a wide range of models to real data in applications. Parametric inference, model selection and goodness-of-fit testing are also feasible with Markov chain Monte Carlo methods.

However, tools for checking or criticizing the fitted model are quite limited. There is currently no analogue for spatial point patterns of the comprehensive strategy for model criticism in the linear model, which uses tools such as residual plots and influence diagnostics to identify unusual or influential observations, to assess model assumptions one by one and to recognize the form of departures from the model. Indeed it is widespread practice in the statistical analysis of spatial point pattern data to focus primarily on comparing the data with a homogeneous Poisson process (‘complete spatial randomness’), which is generally the *null* model in applications, rather than the *fitted* model. The paucity of model criticism in spatial statistics is a weakness in applications, especially in areas such as spatial epidemiology where fitted models may invite very close scrutiny.

Accordingly, this paper sets out to develop residuals and residual plots for models that are fitted to spatial point patterns. Our goal is a coherent strategy for model criticism in spatial point process models, resembling the existing methods for the linear model. This depends crucially on finding the right definition of residuals for a spatial point process model fitted to point pattern data. Additionally we must develop appropriate plots and transformations of the residuals for assessing each component (‘assumption’) of the fitted model, with a statistical rationale for each plot.

Our definition of residuals is a natural generalization of the well-known residuals for point processes in time, which are used routinely in survival analysis. It had been thought that no such generalization exists for spatial point processes, because of the lack of a natural ordering in two-dimensional space, and that the analysis of spatial point patterns necessitated quite a different approach (Cox and Isham (1980), section 6.1, and Ripley (1988), introduction). Nevertheless the generalization from temporal to spatial point processes is straightforward after one crucial change. The key is to replace the usual conditional intensity of the process (or hazard rate of the lifetime distribution) by the Papangelou conditional intensity of the spatial process. Antecedents of this approach are to be found in the work of Stoyan and Grabarnik (1991).

Next, diagnostic plots are developed systematically, by exploiting an analogy between point process models and generalized linear models (GLMs). The Papangelou conditional intensity *λ* of the spatial point process corresponds, under this analogy, to the mean response in a GLM. The spatial point process residuals that are introduced in this paper correspond to the usual residuals for Poisson log-linear regression. The components of a point process model (spatial trend, dependence on spatial covariates and interaction between points of the pattern) cor-respond to model terms in a GLM. Thus the well-understood diagnostic plots for assessing each term in a GLM can be carried across to spatial point processes.

Section 2 presents motivating examples. Section 3 offers a review and critique of current techniques. Section 4 reviews existing theory of residuals for point processes in time and space–time. Section 5 introduces spatial point process models and the essential background for our definition of residuals. Section 6 describes the diagnostic of Stoyan and Grabarnik (1991). Our new residuals for spatial point processes are defined in Sections 7 and 8. Properties of the residuals are studied in Section 9. Sections 10–12 develop diagnostic plots for assessing each component of a spatial point process model. Sections 13 and 14 discuss practical implementation and scope of the techniques.