The authors would like to thank the discussants for their interesting contributions, their insightful comments and suggestions. We believe that the discussions provide evidence that the field of statistical process control (SPC) for high dimensional and / or nonstationary processes deserves great attention and is key to keeping modern processes in-control.
Indeed, nowadays process data are gathered at high sampling rates, providing multivariate (and often repeated) information about each and every sample or batch. Those data acquisition characteristics have their repercussion on two main aspects of classical univariate SPC. First, modern process sensors such as (near infrared) spectrophotometers, high frequency vibration analyzers or digital cameras result in data streams where often the number of variables is higher than the number of observations leading to ill-conditioned settings. This issue is mainly tackled in our paper and the discussion by Ferrer by considering latent variables multivariate SPC (MSPC). Second, the fact that sampling intervals are often much smaller than the natural process dynamics results in data that display a high degree of autocorrelation. The issue of autocorrelation and nonstationarity also comes in from a totally different perspective, namely that of monitoring living organisms whose behavior is time dependent. Incorporating such autocorrelation and time dependency in the SPC setting is discussed from a time series perspective in our paper, and is further elaborated in the discussion by De Vries.
The issue raised by Ferrer concerning in-control processes indeed is an important one -in-control should mean in any case that the process behaves as expected given its inherent natural variability. In a sense, this is not restricted to a fixed distribution (e.g., a normal distribution) nor autocorrelation structure. As stated both in our paper as in the discussion, 'the type of data and the way they are collected are subject to change', with high sampling rates and high dimensional data being key characteristics. Seeing the peculiarities of most multivariate data (often generated by spectral devices) with correlations amongst variables > 0.99, latent variables-based MSPC seem to be the natural choice to avoid an ill-conditioned covariance matrix. Publications in this field have spurred during the last 15 years, and we feel that in the coming years this research field will further develop. As Ferrer suggested, the field of Functional Data Analysis (FDA) is an interesting one when considering spectral data. Saeys et al. [1] already showed the potential of FDA in the field of chemometrics (i.e., the field of using spectral data for classification / prediction purposes). They showed that the use of latent variables model such as PCA or partial least squares is prone to important risks because they do not require an understanding of the physical problem. Using a simple example of spectral data (transmissions as a function of the wavelength ), Saeys et al. [1] showed that altering the dataset by random permutation of the covariates (i.e., the data vector x i D x i . / is altered to x i D x i .Ä/ with › being a random permutation of , so destroying the spectral information) has no influence on the predictive power of a PLS model. As far as the authors are aware of, the use of FDA as an alternative to the latent variables MSPC is un(der)developed and requires attention as Ferrer suggested.
In his discussion, De Vries focuses on monitoring animals (individually, or group wise), a discipline often referred to as precision livestock farming (PLF). As De Vries addresses, the concepts of (advanced) SPC are still scarcely used in daily practice in those industries. We believe this is related to several aspects: