In the last decade, the analysis of data that are not vectors but functional objects is having an increasing spread in modern statistics, which has to face data objects characterised by increasingly complexity. In my talk I will discuss three case studies.
The first case study is motivated by a multiple profile monitoring problem: the health monitoring of a steam steriliser during its life cycle. Indeed, each sterilisation run gives several profiles related to machine health, and degradation of the steam steriliser during its life cycle modifies the profile curvature unpredictably. Hence the need for a control chart capable of monitoring multiple sterilisation profiles during the steriliser life cycle.
For covering this kind of problems, we introduce general functional EWMA control charts. When functional data to be monitored are smooth enough to be representable by a finite dimensional basis, a particular version of these functional EWMA’s is shown to be a multivariate EWMA applied to basis coefficients. Hence it is called f-EWMA for monitoring single profiles and f-MEWMA for multiple profiles. Control limits and control chart performance are assessed on Monte Carlo simulations.
The second case study is related to geographic gaps of radiosonde monitoring networks. Since radiosondes measure atmospheric profiles for temperature and other thermodynamic variables, the natural data representation is to consider these profiles as functional objects.
In this frame, a gap region is defined as an atmospheric region where the spatial prediction uncertainty is high. To do this global bi-daily radiosonde profiles are modelled as a spatiotemporal process on the sphere × time with functional values. In particular, we use splines with random coefficients given by spatiotemporal processes and estimate the model parameter using the EM algorithm implemented in DSTEM package. After that, the functional kriging variance is used to identify the gaps. Techniques for large data sets are considered.
The third case study is related to the measurement uncertainty of atmospheric profiles obtained by remote sensing and radiosoundings, which is crucial in climate change studies. In this frame, some modelling issues related to functional data representation of temperature profiles are discussed.
In particular, co-location mismatch of a satellite profile and a radiosonde profile is discussed. The objective is the assessment of the vertical smoothing mismatch uncertainty related to this profile comparison. To see this, radiosondes are harmonised to match the satellite data in a two steps procedure, which is based on the maximum likelihood approach and exploits the measurement uncertainties in a natural way. At the first step, radiosonde profiles are transformed into continuous functions using splines. At the second step, radiosonde profiles are harmonised by considering weighting functions based on the generalised extreme values probability density function with parameters depending on altitude. The variation between harmonised and non-harmonised radiosonde is then informative on vertical smoothing mismatch.