The WD-LI measurements in the population can be put in a behavior-function context, in which the neighboring pixels of the WD-LI space form behavior information while SER acts as function information. The core target of the analysis is to quantify the WD-LI behavior in relationship with the function measure (e.g., SER). For measurement that is continuous in both behavior and function space, simple regression analysis between the two would provide a good estimate of such dynamics. However, the WD-LI behavior measurements for each individual are sparse in WD-LI space. For example,
Figure 2 shows the WD-LI behaviors of all subjects in the space. The empty pixels indicate that there is no behavior in the pixels (i.e., there is no PoT). To address the sparsity in the space, for each pixel, we need to “borrow” the information from the neighboring behavior. We use the two-dimensional Gaussian kernel function
16 (1) to delimit the neighbors and assign the weight for every neighbor of the pixel located at (
i,
j).
\begin{equation}N\left\{ {\left( {i\# ,j\# } \right)|\left( {i,j} \right),\sigma } \right\} = exp\left\{ {\frac{{ - {{\left| {\left| {\left( {i\# ,j\# } \right),\left( {i,j} \right)} \right|} \right|}^2}}}{{2{\sigma ^2}}}} \right\},\end{equation}
where
i = {0, 1, …,
N − 1} represents the index on the horizontal axis and
j = {0, 1, …,
N − 1} represents the index on the vertical axis,
N = 40. The hyperparameter σ controls the width of the Gaussian kernel. The pixels in the circle with the radius 2σ are considered the neighboring pixels of the pixel. (
i*,
j*) represents the index of neighboring pixels of the pixel (
i,
j) in the circle, that is, (
i*,
j*) = {(0, 0), (0, 1), (1, 0), (1, 1), …}, where (
i*)
2 + (
j*)
2 ≤ (2σ)
2.
\(( {i\# ,j\# } )\) represents one element of (
i*,
j*). || .|| is the Euclidean distance. The weighted values of all the neighboring pixels of the pixel (
i,
j) are represented by (
2):
\begin{eqnarray} {{\boldsymbol w}_{ij}} &=& \{ N\left\{ {\left( {i\# ,j\# } \right)|\left( {i,j} \right),\sigma } \right\}{\rm{\;}}|{\rm{\;}}\left( {i\# ,j\# } \right){\rm{\;}} \in {\rm{\;}}\left( {i{\rm{*}},j{\rm{*}}} \right),\nonumber\\
&&{\rm{\;}}{\left( {i{\rm{*}}} \right)^2} + {\left( {j{\rm{*}}} \right)^2} \le {\left( {2\sigma } \right)^2}\} .\end{eqnarray}