Rolf Turner

PRACTICAL MAXIMUM PSEUDOLIKELIHOOD FOR SPATIAL POINT PATTERNS (with Discussion)

Summary This paper describes a technique for computing approximate maximum pseudolikelihood estimates of the parameters of a spatial point process. The method is an extension of Berman & Turner’s (1992) device for maximizing the likelihoods of inhomogeneous spatial Poisson processes. For a very wide class of spatial point process models the likelihood is intractable, while the pseudolikelihood is known explicitly, except for the computation of an integral over the sampling region. Approximation of this integral by a finite sum in a special way yields an approximate pseudolikelihood which is formally equivalent to the (weighted) likelihood of a loglinear model with Poisson responses. This can be maximized using standard statistical software for generalized linear or additive models, provided the conditional intensity of the process takes an ‘exponential family’ form. Using this approach a wide variety of spatial point process models of Gibbs type can be fitted rapidly, incorporating spatial trends, interaction between points, dependence on spatial covariates, and mark information.

Sustained Acceleration of Achievement in Reading Comprehension: The New Zealand Experience

Schools with primarily indigenous and ethnic minorities in low socioeconomic areas have long been associated with low levels of achievement, particularly in literacy. This is true for New Zealand despite high levels of reading comprehension by international comparisons (e.g., PISA). Recent reviews of schooling improvement suggest small gains over the short term are possible with well-designed interventions, but for children in the middle primary school years, the criterion against which effective interventions need to be judged is sustained and systematic acceleration across levels of achievement in order to achieve equitable distributions of achievement. Plotting gains across time is also needed to examine whether “summer effects” can be overcome. The present quasi-experimental design study was a three-year research and development collaboration among schools, government, and researchers to raise reading comprehension through critical discussions of achievement and teacher observation data and linking research on effective comprehension practices to specific needs. The collaboration resulted in increased rates of achievement that were variable but sustained across three years. The growth model showed a step-like pattern with rapid gains over school months and a plateau over summer. Over three years this represented an average achievement gain across cohorts followed longitudinally of one year’s progress in addition to expected progress over that period with stanine effect sizes of d = 0.62. The results show the significance of testing effects against the criterion of sustained and systematic achievement and the need to examine growth over multiple calendar years to better represent the pattern of gains.

Modelling spatial point patterns in R

We describe practical techniques for fitting stochastic models to spatial point pattern data in the statistical package R. The techniques have been implemented in our package spatstat in R. They are demonstrated on two example datasets.

Direct maximization of the likelihood of a hidden Markov model

Ever since the introduction of hidden Markov models by Baum and his co-workers, the method of choice for fitting such models has been maximum likelihood via the EM algorithm. In recent years it has been noticed that the gradient and Hessian of the log likelihood of hidden Markov and related models may be calculated in parallel with a filtering process by which the likelihood may be calculated. Various authors have used, or suggested the use of, this idea in order to maximize the likelihood directly, without using the EM algorithm. In this paper we discuss an implementation of such an approach. We have found that a straightforward implementation of Newtons method sometimes works but is unreliable. A form of the Levenberg-Marquardt algorithm appears to provide excellent reliability. Two rather complex examples are given for applying this algorithm to the fitting of hidden Markov models. In the first a better than 6-fold increase in speed over the EM algorithm was achieved. The second example turned out to be problematic (somewhat interestingly) in that the maximum likelihood estimator appears to be inconsistent. Whatever its merit, this estimator is calculated much faster by Levenberg-Marquardt than by EM. We also compared the Levenberg-Marquardt algorithm, applied to the first example, with a generic numerical maximization procedure. The Levenberg-Marquardt algorithm appeared to perform almost three times better than the generic procedure, even when analytic derivatives were provided, and 19 times better when they were not provided.

Point patterns of forest fire locations

In this paper I demonstrate some of the techniques for the analysis of spatial point patterns that have become available due to recent developments in point process modelling software. These developments permit convenient exploratory data analysis, model fitting, and model assessment. Efficient model fitting, in particular, makes possible the testing of statistical hypotheses of genuine interest, even when interaction between points is present, via Monte Carlo methods. The discussion of these techniques is conducted jointly with and in the context of some preliminary analyses of a collection of data sets which are of considerable interest in their own right. These data sets (which were kindly provided to me by the New Brunswick Department of Natural Resources) consist of the complete records of wildfires which occurred in New Brunswick during the years 1987 through 2003. In treating these data sets I deal with data-cleaning problems, methods of exploratory data analysis, means of detecting interaction, fitting of statistical models, and residual analysis and diagnostics. In addition to demonstrating modelling techniques, I include a discussion on the nature of statistical models for point patterns. This is given with a view to providing an understanding of why, in particular, the Strauss model fails as a model for interpoint attraction and how it has been modified to overcome this difficulty. All actual modelling of the New Brunswick fire data is done only with the intent of illustrating techniques. No substantive conclusions are or can be drawn at this stage. Realistic modelling of these data sets would require incorporation of covariate information which I do not so far have available.

Residual Diagnostics for Covariate Effects in Spatial Point Process Models

For a spatial point process model in which the intensity depends on spatial covariates, we develop graphical diagnostics for validating the covariate effect term in the model, and for assessing whether another covariate should be added to the model. The diagnostics are point-process counterparts of the well-known partial residual plots (component-plus-residual plots) and added variable plots for generalized linear models. The new diagnostics can be derived as limits of these classical techniques under increasingly fine discretization, which leads to efficient numerical approximations. The diagnostics can also be recognized as integrals of the point process residuals, enabling us to prove asymptotic results. The diagnostics perform correctly in a simulation experiment. We demonstrate their utility in an application to geological exploration, in which a point pattern of gold deposits is modeled as a point process with intensity depending on the distance to the nearest geological fault. Online supplementary materials include technical proofs, computer code, and results of a simulation study.

University Admission Models that Address Quality and Equity.

University entrance criteria can be a contentious topic, particularly in respect of equity. In this paper we discuss studies which demonstrate that revisions of entrance criteria which are designed with no explicit reference to equity issues can have a surprisingly positive impact on the fractions of disadvantaged subgroups admitted. We demonstrate this impact in the New Zealand context with special reference to a “dual admission model” that we believe could have broad applicability. An assessment of the variability of the estimates of the impact of changed entry criteria (calculated using a simulation method) demonstrates the statistical significance of our results, thus providing confirmation that the apparent impact is not illusory.

Adjusted composite likelihood ratio test for spatial Gibbs point processes

We investigate an analogue of the likelihood ratio test for spatial Gibbs point process models fitted by maximum pseudolikelihood or maximum composite likelihood. The test statistic must be adjusted in order to obtain an asymptotic distribution under the null hypothesis. Adjustments developed for composite likelihoods of finite systems of random variables are adapted to the point process setting. Recent results in point process theory are used to estimate the composite information J and sensitivity H from the point pattern data. In a large simulation experiment we find that the proposed test is exact if J and H are known exactly; it is slightly conservative when J and H are estimated from the data.

Bias correction for parameter estimates of spatial point process models

When a spatial point process model is fitted to spatial point pattern data using standard software, the parameter estimates are typically biased. Contrary to folklore, the bias does not reflect weaknesses of the underlying mathematical methods, but is mainly due to the effects of discretization of the spatial domain. We investigate two approaches to correcting the bias: a Newton–Raphson-type correction and Richardson extrapolation. In simulation experiments, Richardson extrapolation performs best.

A note on the Berman condition

It is established that if a time series satisfies the Berman condition, and another related (summability) condition, the result of filtering that series through a certain type of filter also satisfies the two conditions. In particular it follows that if