An object of class "SpatPatterns".
Given the set of \(n\) points, dat, in a region, this function assigns \(n_1=\)round(n*ult.prop,0) of them as cases,
and the rest as controls with first selecting \(k_0=\)round(n*init.prop,0) as cases initially, then selecting
a contagious case and then assigning the label case to the remaining points with infection probabilities
inversely proportional to their position among the kNNs.
The initial and ultimate number of cases will be \(k_0\) and \(n_1\) on the average if the argument poisson=TRUE
(i.e., \(k_0=\)rpois(1,round(n*init.prop,0)) and \(n_1=\)rpois(1,round(n*ult.prop,0)) ), otherwise
they will be exactly equal to \(n_1=\)round(n*ult.prop,0) and \(k_0=\)round(n*init.prop,0).
More specifically, let \(z_1,\ldots,z_{k_0}\) be the initial cases. Then one of the cases is selected as a
contagious case, say \(z_j\) and then its kNNs (among the non-cases) are found.
Then label these kNN non-case points as cases with infection probabilities prob equal to the value
of the rho*(1/(1:k))^pow values at these points, where rho is a scaling parameter for
the infection probabilities and pow is a parameter in the power adjusting the kNN dependence.
We stop when we first exceed \(n_1\) cases. rho has to be in \((0,1)\) for prob to be a vector of probabilities,
and for a given rho, pow must be \(> \ln(rho)/\ln(k)\).
If rand.init=TRUE, first \(k_0\) entries are chosen as the initial cases in the data set,
dat, otherwise, \(k_0\) initial cases are selected randomly among the data points.
Algorithmically, first all dat points are treated as non-cases (i.e., controls or healthy subjects). Then the function follows the following steps for labeling of the points:
step 0: \(n_1\) is generated randomly from a Poisson distribution with mean = round(n*ult.prop,0), so that the
average number of ultimate cases will be round(n*ult.prop,0) if the argument poisson=TRUE, else \(n_1=\)round(n*ult.prop,0).
And \(k_0\) is generated randomly from a Poisson distribution with mean = round(n*init.prop,0), so that the
average number of initial cases will be round(n*init.prop,0) if the argument poisson=TRUE, else \(k_0=\)round(n*init.prop,0).
step 1: Initially, \(k_0\) many points from dat are selected as cases.
The selection of initial cases are determined based on the argument rand.init (with default=TRUE)
where if rand.init=TRUE then the initial cases are selected randomly from the data points, and if rand.init=
FALSE, the first \(k_0\) entries in the data set, dat, are selected as the cases.
step 2: Then it selects a contagious case among the cases, and randomly labels its k control NNs as cases with
decreasing infection probabilities prob=rho*(1/(1:k))^pow. See the description for the details
of the parameters in the prob.
step 3: The procedure ends when number of cases \(n_c\) exceeds \(n_1\), and \(n_c-n_1\) of the cases (other than the initial cases) are randomly selected and relabeled as controls, i.e., 0s, so that the number of cases is exactly \(n_1\).
Note that the infection probabilities of the kNNs of each initial case increase
with increasing rho; and probability of infection decreases as further NNs are considered from
a contagious case (i.e., as k increases in the kNNs).
See ceyhan:SiM-seg-ind2014;textualnnspat for more detail where type II non-RL pattern is the
case 2 of non-RL pattern considered in Section 6 with \(n_1\) is
fixed as a parameter rather than being generated from a Poisson distribution and pow=1.
Although the non-RL pattern is described for the case-control setting, it can be adapted for any two-class setting when it is appropriate to treat one of the classes as cases or one of the classes behave like cases and other class as controls.
rnonRLII(
dat,
k,
rho,
pow,
init.prop,
ult.prop,
rand.init = TRUE,
poisson = FALSE
)A list with the elements
="cc" for the case-control patterns for RL or non-RL of the given data points, dat
The type of the point pattern
Number of NNs, k, a scaling parameter for the infection probabilities of kNNs, rho,
a parameter in the power adjusting the kNN dependence of the infection probabilities, initial proportion
of cases, init.prop, and the ultimate proportion of cases, ult.prop.
The set of points non-RL procedure is applied to obtain cases and controls randomly in the type II fashion
The labels of the points as 1 for cases and 0 for controls after the type II nonRL procedure is
applied to the data set, dat. Cases are denoted as red dots and controls as black circles in the plot.
The initial cases in the data set, dat. Denoted as red crosses in the plot of the points.
The contagious cases in the data set, dat. Denoted as blue points in the plot of the points.
Both are NULL for this function, as initial set of points, dat, are provided
for the non-RL procedure.
Description of the point pattern
The "main" title for the plot of the point pattern
The vector of two numbers, which are the number of cases and controls.
The possible ranges of the \(x\)- and \(y\)-coordinates of the generated and the reference points
A set of points the non-RL procedure is applied to obtain cases and controls randomly in the type II fashion (see the description).
An integer representing the number of NNs considered for each contagious case, i.e.,
kNNs of each contagious case are candidates to be infected to become cases.
A scaling parameter for the probabilities of labeling the points as cases (see the description).
A parameter in the power adjusting the kNN dependence in the probabilities of labeling the
points as cases (see the description).
A real number between 0 and 1 representing the initial proportion of cases in the data set,
dat. The selection of the initial cases depends on the parameter rand.init (see the description).
A real number between 0 and 1 representing the ultimate proportion of cases in the data set,
dat after the non-RL assignment.
A logical argument (default is TRUE) to determine the choice of the initial cases in the data set, dat.
If rand.init=TRUE then the initial cases are selected randomly from the data points, and if rand.init=
FALSE, the first init.prop*n entries in the data set, dat, are labeled as the cases.
A logical argument (default is FALSE) to determine whether the number of initial and ultimate
cases, \(k_0\) and \(n_1\), will be random or fixed. If poisson=TRUE then the \(k_0\) and \(n_1\) are from a Poisson distribution,
\(k_0=\)rpois(1,round(n*init.prop,0)) and \(n_1=\)rpois(1,round(n*ult.prop,0))
otherwise they are fixed, \(k_0=\)round(n*init.prop,0) and \(n_1=\)round(n*ult.prop,0).
Elvan Ceyhan
rnonRLI, rnonRLIII, rnonRLIV, and rnonRL
n<-40; #try also n<-20; n<-100;
#data generation
dat<-cbind(runif(n,0,1),runif(n,0,1))
rho<-.8
pow<-2
knn<-5 #try 2 or 3
ip<-.3 #initial proportion
up<-.5 #ultimate proportion
Xdat<-rnonRLII(dat,knn,rho,pow,ip,up,poisson=FALSE) #labeled data, try poisson=TRUE
Xdat
table(Xdat$lab)
summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)
#normal original data
n<-40; #try also n<-20; n<-100;
#data generation
dat<-cbind(rnorm(n,0,1),rnorm(n,0,1))
rho<-0.8
pow<-2
knn<-5 #try 2 or 3
ip<-.3 #initial proportion
up<-.5 #ultimate proportion
Xdat<-rnonRLII(dat,knn,rho,pow,ip,up,poisson=FALSE) #labeled data, try poisson=TRUE
Xdat
table(Xdat$lab)
summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)
Run the code above in your browser using DataLab