A notion of depth for sparse functional data

Sparse Functional Boxplots for Multivariate Curves

“…All of them show a bit of weaker association when the curve is sparser on average. The Spearman coefficient may reach closer to 0.95 when α in ( 4) is increased from 0.05 to above 0.2 for all RMFHDs; RMFHD aw in the univariate setting follows this pattern (Sguera & López-Pintado 2020). While seeking the best α requires more computation, when α is above 0.05, the result usually is not statistically convincing.…”

“…If there is only one variable, then the data reduce to univariate functional data, and MFPCA becomes univariate functional principal component analysis (UFPCA, Yao et al 2005). The depth notions we consider are either from the direct generalization of the univariate revised functional depth (Sguera & López-Pintado 2020) to multivariate functional data, or our new idea described in Section 2.2.2.…”

“…In the second approach, we revise the notions of depth to handle sparse multivariate functional data by incorporating the fitted data and its confidence interval into the revised depth (see the proposal by Sguera & López-Pintado (2020) in the univariate case). We evaluate these two approaches in the following subsections and select the best depth for sparse multivariate functional depth in a simulation study (Section 4.2).…”

“…, where the number of observations is three times the original one; see Sguera & López-Pintado (2020) for the univariate functional case. The third only considers the confidence intervals and combines upper and lower confidence levels into one observation, where X f upd,2 := {( X ub,1 , X lb,1 ) , .…”

“…Other available methods for fitting sparse functional data include functional principal component analysis (James et al 2000, Liu et al 2017), B-spline models (Thompson & Rosen 2008), and covariance estimation (Xiao et al 2018). One example of the second approach is in Sguera & López-Pintado (2020),…”