The normal scan statistic evaluates the statistic which compares the node attribute within
the subgraph with that outside the subgraph while the node attribute follows the normal distribution.
Usage
norm.stat(obs, pop = 1, zloc)
Arguments
obs
Numeric vector of observation values.
pop
Numeric vector of population values ; default = 1.
zloc
Numeric vector of selected nodes.
Value
Three values will be returned. The first value is test statistic.
The second is the estimated means which estimated outside the selected nodes.
The third is the estimated means estimated within the selected nodes.
Details
A network with interested attributes is denoted as $G=(V,E,X)$,
where $X=(x_1,\ldots,x_{|V|})$ follows a defined distribution. Suppose a subgraph, $Z$, is selected.
$$\lambda_A(Z)=n\ln (\sqrt{(\hat{\sigma}^2))}-n\ln (\sqrt{(2 \hat{\sigma}_z^2)}),$$
where $\hat{\sigma}^2=\sum_{i=1}^n(x_i-\bar{x})^2/n$, and
$\hat{\sigma}_z^2=[\sum_{i \in Z}(x_i-\bar{x}_z)^2-\sum_{j \notin Z}(x_j-\bar{x}_x)^2]/n$,
in which $n$ is the number of nodes, and $\bar{x}_z=\sum_{i \in Z} x_i/n_z$ and
$\bar{x}_c=\sum_{j \notin Z} x_j/(n-n_z)$.
It is equivalent to minimize the variance within the subgraph $Z$.
References
Kulldorff, M., Huang, L., & Konty, K. (2009).
A scan statistic for continuous data based on the normal probability model.
International journal of health geographics, 8(1), 58.