This function calculates the kurtosis distance (Vidal, 2020), which is an heuristic measure to select the number of components to be computed in kffobi and pspline.kffobi.
kd(fdx, hm = fdPar(fdx), rho = NULL, r = 2, centerfd = FALSE, qmin = 2, qmax = 5)a functional data object obtained from the fda package.
a functional parameter object, obtained from the fda package, that defines the independent component functions to be estimated in kffobi.
the smoothing parameter to perform kd on pspline.kffobi.
a number indicating the order of the penalty to perform kd on pspline.kffobi
a logical value indicating whether the mean function has to be subtracted from each functional observation.
the minimum allowable \(q\) degree.
the maximum allowable \(q\) degree.
A vector of \(\mathtt{KD}\) values.
The kurtosis distance (\(\mathtt{KD}\)) measures the degree of extremeness in an independent component coordinate space of degree \(q\). For a fixed \(q\), the kurtosis distance is defined as $$\mathtt{KD}(z_{j})=\mathrm{max}\left[\sum_{j=1}^{q}\mathrm{kurt}(z_{j})\right]-\mathrm{min}\left[\sum_{j=1}^{q}\mathrm{kurt}(z_{j})\right],$$ where \(z_{j}\) is the vector of independent components calculated from \(\int_{T}x_{i}^{st}(t)h_{j}(t)dt\). Thus, \(\mathtt{KD}\) is calculated from the the standardized original sample instead of the standardized principal component expansion. Then, the kd function calculates \(\mathtt{KD}\) for different values of \(q=2,\ldots,n-1\). The user can then consider choosing a value from those amongst the first relative maxima of the vector of \(\mathtt{KD}\) values. If the value of \(q\) is increased too much, there is a risk of losing accurate estimates, as more of the independent part (noise) of the model is induced.
Vidal, M. (2020). Functional Independent Component Analysis in Bioelectrical Signal Processing. MA thesis. Universidad de Granada.