# Estimation of the Space-Time Inhomogeneous Pair Correlation function Compute an estimate of the space-time pair correlation function. ```r PCFhat(xyt, s.region, t.region, dist, times, lambda, ks="box", hs, kt="box", ht, correction = "isotropic") ``` ## Arguments - `xyt`: Coordinates and times $(x,y,t)$ of the point pattern. - `s.region`: Two-column matrix specifying polygonal region containing all data locations. If `s.region` is missing, the bounding box of `xyt[,1:2]` is considered. - `t.region`: Vector containing the minimum and maximum values of the time interval. If `t.region` is missing, the range of `xyt[,3]` is considered. - `dist`: Vector of distances $u$ at which $g(u,v)$ is computed. If missing, the maximum of `dist` is given by $min(S_x, S_y)/4$, where $S_x$ and $S_y$ represent the maximum width and height of the bounding box of `s.region`. - `times`: Vector of times $v$ at which $g(u,v)$ is computed. If missing, the maximum of `times` is given by $(T_max - T_min)/4$, where $T_min$ and $T_max$ are the minimum and maximum of the time interval $T$. - `lambda`: Vector of values of the space-time intensity function evaluated at the points $(x,y,t)$ in $S x T$. If `lambda` is missing, the estimate of the space-time pair correlation function is computed as for the homogeneous case, i.e. considering $n/|S x T|$ as an estimate of the space-time intensity. - `ks`: Kernel function for the spatial distances. Default is the `"box"` kernel. Can also be `"epanech"` for the Epanechnikov kernel or `"gaussian"` or `"biweight"`. - `hs`: Bandwidth of the kernel function `ks`. - `kt`: Kernel function for the temporal distances. Default is the `"box"` kernel. Can also be `"epanech"` for the Epanechnikov kernel or `"gaussian"` or `"biweight"`. - `ht`: Bandwidth of the kernel function `kt`. - `correction`: A character vector specifying the edge correction(s) to be applied among `"isotropic"`, `"border"`, `"modified.border"`, `"translate"` and `"none"` (see Details). The default is `"isotropic"`. ## Details An approximately unbiased estimator for the space-time pair correlation function, based on data giving the locations of events $x_i: i = 1,...,n$ on a spatio-temporal region $SxT$, where $S$ is an arbitrary polygon and $T$ a time interval: $$ \widehat{g}(u,v)=\frac{1}{4\pi u}\sum_{i=1}^{n}\sum_{j \neq i} \frac{1}{w_{ij}}\frac{k_{s}(u-\|s_i-s_j\|)k_{t}(v-|t_i-t_j|)}{\lambda(x_i) \lambda(x_j)},g(u,v) = 1/|SxT| 1/(4 pi u) sum_{i = 1,...,n} sum_{j = 1,...,n; j != i} 1/w_ij k_s(u -||s_i - s_j||) k_t(v - |t_i - t_j|)/(lambda(x_i) lambda(x_j)), $$ where $lambda(x_i)$ is the intensity at $x_i = (s_i, t_i)$ and $w_ij$ is an edge correction factor to deal with spatial-temporal edge effects. The edge correction methods implemented are: `isotropic`: $w_ij = |S x T| w_ij^(s) w_ij^(t)$, where the temporal edge correction factor $w_ij^(t) = 1$ if both ends of the interval of length $2|t_i - t_j|$ centred at $t_i$ lie within $T$ and $w_ij^(t) = 1/2$ otherwise and $w_ij^(s)$ is the proportion of the circumference of a circle centred at the location $s_i$ with radius $||s_i - s_j||$ lying in $S$ (also called Ripley's edge correction factor). `border`: c("$w_ij = (sum_{j = 1,...,n} 1{d(s_j, S) > u ; d(t_j, T) > v}/\n$", "$lambda(x_j)) / 1{d(s_i, S) > u ; d(t_i, T) > v}$"), where $d(s_i, S)$ denotes the distance between $s_i$ and the boundary of $S$ and $d(t_i, T)$ the distance between $t_i$ and the boundary of $T$. `modified.border`: $w_ij = |S_(-u) x T_(-v)| / 1{d(s_i, S) > u ; d(t_i, T) > v}$, where $S_(-u)$ and $T_(-v)$ are the eroded spatial and temporal region respectively, obtained by trimming off a margin of width $u$ and $v$ from the border of the original region. `translate`: c("$w_ij = |S intersect S_(s_i - s_j)\n$", "$ x T intersect T_(t_i - t_j)|$"), where $S_(s_i - s_j)$ and $T_(t_ i - t_j)$ are the translated spatial and temporal regions. `none`: No edge correction is performed and $w_ij = |S x T|$. $k_s()$ and $k_t()$ denotes kernel functions with bandwidth $h_s$ and $h_t$. Experience with pair correlation function estimation recommends box kernels (the default), see Illian et al. (2008). Epanechnikov, Gaussian and biweight kernels are also implemented. Whatever the kernel function, if the bandwidth is missing, a value is obtain from the function `dpik` of the package KernSmooth. Note that the bandwidths play an important role and their choice is crucial in the quality of the estimators as they heavily influence their variance. ## Returns A list containing: - **pcf**: `ndist` x `ntimes` matrix containing values of $g(u,v)$. - **pcftheo**: `ndist` x `ntimes` matrix containing theoretical values for a Poisson process. - **dist, times**: Parameters passed in argument. - **kernel**: A vector of names and bandwidths of the spatial and temporal kernels. - **correction**: The name(s) of the edge correction method(s) passed in argument. ## References Baddeley, A., Rubak, E., Turner, R., (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton. Gabriel E., Diggle P. (2009). Second-order analysis of inhomogeneous spatio-temporal point process data. Statistica Neerlandica, 63, 43--51. Gabriel E., Rowlingson B., Diggle P. (2013). stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software, 53(2), 1--29. Gabriel E. (2014). Estimating second-order characteristics of inhomogeneous spatio-temporal point processes: influence of edge correction methods and intensity estimates. Methodology and computing in Applied Probabillity, 16(2), 411--431. Illian JB, Penttinen A, Stoyan H and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, London. ## Author(s) Edith Gabriel ## Examples ```r # First example data(fmd) data(northcumbria) FMD<-as.3dpoints(fmd[,1]/1000,fmd[,2]/1000,fmd[,3]) Northcumbria=northcumbria/1000 # estimation of the temporal intensity Mt<-density(FMD[,3],n=1000) mut<-Mt$y[findInterval(FMD[,3],Mt$x)]*dim(FMD)[1] # estimation of the spatial intensity h<-mse2d(as.points(FMD[,1:2]), Northcumbria, nsmse=50, range=4) h<-h$h[which.min(h$mse)] Ms<-kernel2d(as.points(FMD[,1:2]), Northcumbria, h, nx=500, ny=500) atx<-findInterval(x=FMD[,1],vec=Ms$x) aty<-findInterval(x=FMD[,2],vec=Ms$y) mhat<-NULL for(i in 1:length(atx)) mhat<-c(mhat,Ms$z[atx[i],aty[i]]) # estimation of the pair correlation function g1 <- PCFhat(xyt=FMD, dist=1:15, times=1:15, lambda=mhat*mut/dim(FMD)[1], s.region=northcumbria/1000,t.region=c(1,200)) # plotting the estimation plotPCF(g1) plotPCF(g1,type="persp",theta=-65,phi=35) # Second example xyt=rpp(lambda=200) g2=PCFhat(xyt$xyt,dist=seq(0,0.16,by=0.02), times=seq(0,0.16,by=0.02),correction=c("border","translate")) plotPCF(g2,type="contour",which="border") ```

PCFhat function