aij.theta | R Documentation |
This function computes the A=a_{ij}(θ) matrix useful in calculations for Tango's test T(θ) for spatial (disease) clustering (see Eqn (2) of \insertCitetango:2007;textualnnspat. Here, A=a_{ij}(θ) is any matrix of a measure of the closeness between two points i and j with aii = 0 for all i = 1, …,n, and θ = (θ_1,…,θ_p)^t denotes the unknown parameter vector related to cluster size and δ = (δ_1,…,δ_n)^t, where δ_i=1 if z_i is a case and 0 otherwise. The test is then
T(θ)=∑_{i=1}^n∑_{j=1}^nδ_i δ_j a_{ij}(θ)=δ^t A(θ) δ
where A=a_{ij}(θ).
T(θ) becomes Cuzick and Edwards T_k tests statistic (\insertCitecuzick:1990;textualnnspat),
if a_{ij}=1 if z_j is among the k
NNs of z_i and 0 otherwise.
In this case θ=k and aij.theta
becomes aij.mat
(more specifically,
aij.mat(dat,k)
and aij.theta(dat,k,model="NN")
.
In Tango's exponential clinal model (\insertCitetango:2000;textualnnspat), a_{ij}=\exp≤ft(-4 ≤ft(\frac{d_{ij}}{θ}\right)^2\right) if i \ne j and 0 otherwise, where θ is a predetermined scale of cluster such that any pair of cases far apart beyond the distance θ cannot be considered as a cluster and d_{ij} denote the Euclidean distance between two points i and j.
In the exponential model (\insertCitetango:2007;textualnnspat), a_{ij}=\exp≤ft(-\frac{d_{ij}}{θ}\right) if i \ne j and 0 otherwise, where θ and d_{ij} are as above.
In the hot-spot model (\insertCitetango:2007;textualnnspat), a_{ij}=1 if d_{ij} ≤ θ and i \ne j and 0 otherwise, where θ and d_{ij} are as above.
The argument model
has four options, NN
, exp.clinal
, exponential
, and
hot.spot
, with exp.clinal
being the default.
And the theta
argument specifies the scale of clustering or the clustering parameter in the particular
spatial disease clustering model.
See also (\insertCitetango:2007;textualnnspat) and the references therein.
aij.theta(dat, theta, model = "exp.clinal", ...)
dat |
The data set in one or higher dimensions, each row corresponds to a data point. |
theta |
A predetermined cluster scale so that any pair of cases farther apart then the distance θ is unlikely to be cluster. |
model |
Type of Tango's spatial clustering model with four options:
|
... |
are for further arguments, such as |
The A=a_{ij}(θ) matrix useful in calculations for Tango's test T(θ).
Elvan Ceyhan
aij.mat
, aij.nonzero
and ceTk
n<-20 #or try sample(1:20,1) Y<-matrix(runif(3*n),ncol=3) k<-3#1 #try also 2,3 #aij for CE's Tk Aij<-aij.theta(Y,k,model = "NN") Aij2<-aij.mat(Y,k) sum(abs(Aij-Aij2)) #check equivalence of aij.theta and aij.mat with model="NN" Aij<-aij.theta(Y,k,method="max") Aij2<-aij.mat(Y,k) range(Aij-Aij2) theta=.2 aij.theta(Y,theta,model = "exp.clinal") aij.theta(Y,theta,model = "exponential") aij.theta(Y,theta,model = "hot.spot")
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