## 1 Introduction  
For a time-varying system, detection of unexpected or unwanted change in its evolution can be of paramount importance. Examples include environmental monitoring, process control, or, referring to the examples considered in this article, identification of potentially malignant changes in skin moles or the onset of food quality deterioration (see, for example, [1; 2; 3; 4]). The first changes may be small and manifest themselves in different scales and it may be crucial to detect them as early as possible. Statistical scale-space methodologies (see Section 2) can be very useful in such situations since exploring the measurements in multiple resolutions can help identify subtle changes. Examples of scale-space methods designed for change detection are the SiNos technique for capturing non-stationarities in a time series [5] and the iBiSeiZer method for detecting changes in images [6]. Our goal was to develop a method that can detect minor change while at the same time keeping the number of false alarms to a minimum. This is important in practical applications as a successful method must have both high sensitivity and high specificity.  
Recently, [PERSON] et al. proposed a scale-space method for testing whether \(k\) multivariate data sets of same dimension originate from the same distribution [7]. Thus, the proposed method solves the classical \(k\)-sample problem using scale-space analysis and the method has proven successful in many applications. In the applications considered here the observed data consist of multivariate vectors obtained from spectral signatures and therefore changes in their characteristics can also be analyzed with this method. Unfortunately, it turns out that in this context the method suffers from two serious shortcomings: failing to detect very small changes and producing unacceptably high rates of false alarms in some situations (see Section 4). Our goal therefore is to design a scale-space method that would suffer less from these shortcomings.  
As an illustration of the difficulty of detecting very small changes, consider the example in Figure 1 which is discussed in more detail in Sections 2 and 4. The original data set consists of a number of spectral signatures acquired by a push-broom hyperspectral camera, each signature corresponding to a particular spot in a skin mole. Several acquisitions of the mole are taken at the same time, and an example of one acquisition is given in Figure 1a where each curve corresponds to a specific spectral signature. To simulate a situation where the mole might begin to turn malignant, we manually distorted just one spectral signature (thus corresponding to a very small local change in the mole) in another acquisition of the same mole at spectral channel 80 on the horizontal axis in Figure 1a. In case of real moles, the first changes may be extremely hard to detect and a method with high sensitivity and specificity is therefore crucial. In our test, the distorted set of signatures in Figure 1b was compared with 14 other acquisitions and the goal was to detect the small change we manually introduced. It turned out that such a small change is indeed detected by our new methodology but not by the method suggested in [7] nor by a standard approach such as principal components analysis (PCA). We will return in more detail to this example in Section 4.  
Figure 1: (**a**) The original undistorted curve families for the artificial example. (**b**) An example where a small artificial change has been introduced to the data set in Figure 1a at spectral channel 80.