A geometric approach to visualization of variability in functional data

We propose a new method for the construction and visualization of boxplot-type displays for functional data. We use a recent functional data analysis framework, based on a representation of functions called square-root slope functions, to decompose observed variation in functional data into three main components: amplitude, phase, and vertical translation. We then construct separate displays for each component, using the geometry and metric of each representation space, based on a novel definition of the median, the two quartiles, and extreme observations. The outlyingness of functional data is a very complex concept. Thus, we propose to identify outliers based on any of the three main components after decomposition. We provide a variety of visualization tools for the proposed boxplot-type displays including surface plots. We evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface temperature functions, electrocardiogram functions and growth curves

Dataset of manually measured QT intervals in the electrocardiogram

BACKGROUND: The QT interval and the QT dispersion are currently a subject of considerable interest. Cardiac repolarization delay is known to favor the development of arrhythmias. The QT dispersion, defined as the difference between the longest and the shortest QT intervals or as the standard deviation of the QT duration in the 12-lead ECG is assumed to be reliable predictor of cardiovascular mortality. The seventh annual PhysioNet/Computers in Cardiology Challenge, 2006 addresses a question of high clinical interest: Can the QT interval be measured by fully automated methods with accuracy acceptable for clinical evaluations? METHOD: The PTB Diagnostic ECG Database was given to 4 cardiologists and 1 biomedical engineer for manual marking of QRS onsets and T-wave ends in 458 recordings. Each recording consisted of one selected beat in lead II, chosen visually to have minimum baseline shift, noise, and artifact. In cases where no T wave could be observed or its amplitude was very small, the referees were instructed to mark a 'group-T-wave end' taking into consideration leads with better manifested T wave. A modified Delphi approach was used, which included up to three rounds of measurements to obtain results closer to the median. RESULTS: A total amount of 2*5*548 Q-onsets and T-wave ends were manually marked during round 1. To obtain closer to the median results, 8.58 % of Q-onsets and 3.21 % of the T-wave ends had to be reviewed during round 2, and 1.50 % Q-onsets and 1.17 % T-wave ends in round 3. The mean and standard deviation of the differences between the values of the referees and the median after round 3 were 2.43 ± 0.96 ms for the Q-onset, and 7.43 ± 3.44 ms for the T-wave end. CONCLUSION: A fully accessible, on the Internet, dataset of manually measured Q-onsets and T-wave ends was created and presented in additional file: 1 (Table 4) with this article. Thus, an available standard can be used for the development of automated methods for the detection of Q-onsets, T-wave ends and for QT interval measurements