Analysis of Spatial Count Data using Kalman Smoothing

Claus Dethlefsen Dept. of Mathemati al S ien es Aalborg University Fr. Bajers Vej 7G 9220 Aalborg, Denmark Abstra t This paper onsiders spatial ount data from an agri ultural eld experiment. Counts of weed plants in a eld have been re orded in a proje t on pre ision farming. Interest is in mapping the weed intensity so that the dose of herbi ide applied at any lo ation an be adjusted to the amount of weed present at the lo ation. We elaborate on a link between state spa e models and Markov random elds. The observations are modelled as independent Poisson ounts onditional on a Gaussian Markov random eld. We show that the model may be written as a state spa e model whi h may be analysed by ombining approximate Kalman lter te hniques with importan e sampling. 1 Introdu tion We analyse a data set kindly put to our disposal by Danish Institute of Agri ultural S ien es. The data were olle ted in onne tion with a proje t in pre ision farming at the Danish Institute of Agri ultural S ien es. Counts of weed plants on a eld were re orded in 1993, 1994 and 1995. Interest is in mapping the weed intensity so that the dose of herbi ide applied at any lo ation an be adjusted to the amount of weed present at the lo ation. Along with the weed ounts, 11 explanatory variables were also measured. Among these variables, Christensen et al. [2000℄ found that the intensity of weed was related to the per entage of organi matter in the soil and that there is a north-south de reasing trend in the data. Here, we model the relation between ounts of the spe ies Viola arvensis in year 1994 and the two explanatory variables. The weed ounts are displayed in Figure 1, using the * * * * * * * * * * 1 7 6 2 6 2 16 15 3 3 3 1 1 1 4 1 2 7 4 6 4 * 8 3 1 * 1 2 5 1 2 6 23 3 6 2 1 1 2 4 2 3 * 1 2 7 * 10 * * 10 2 3 1 2 2 6 5 * * 8 8 1 2 1 1 1 1 2 1 * 7 2 * 1 2 1 6 * 1 1 1 1 1 3 3 1 1 4 1 2 1 1 1 1 1 1 1 4 1 1 * 1 * * * 1 1 1 1 2 1