  This paper introduces the sparse functional boxplot and the intensity sparse
functional boxplot as practical exploratory tools. Besides being available for
complete functional data, they can be used in sparse univariate and
multivariate functional data. The sparse functional boxplot, based on the
functional boxplot, displays sparseness proportions within the 50\% central
region. The intensity sparse functional boxplot indicates the relative
intensity of fitted sparse point patterns in the central region. The two-stage
functional boxplot, which derives from the functional boxplot to detect
outliers, is furthermore extended to its sparse form. We also contribute to
sparse data fitting improvement and sparse multivariate functional data depth.
In a simulation study, we evaluate the goodness of data fitting, several depth
proposals for sparse multivariate functional data, and compare the results of
outlier detection between the sparse functional boxplot and its two-stage
version. The practical applications of the sparse functional boxplot and
intensity sparse functional boxplot are illustrated with two public health
datasets. Supplementary materials and codes are available for readers to apply
our visualization tools and replicate the analysis.
