## 6 Discussion and Conclusions  
We demonstrated that it is possible to formulate an objective function with a closed-form solution for a clustering problem that avoids the otherwise computationally intensive iteration scheme. This is especially advantageous in applications where the cluster analysis itself is part of an iterative scheme as in the example of indirect reference interval estimation. The purpose of this paper was not to introduce a new method for indirect reference interval estimation but to demonstrate that a suitable formulation of the objective function for clustering can lead to a closed-form solution when the clustering problem is simplified. Of course, with the consideration of one-dimensional data with one main cluster and a noise  
Figure 3: Mean computation time of three methods for indirect reference estimation for different sample sizes on a standard PC. The \(x\)-axis is shown on a logarithmic scale. cluster, our assumptions are very restrictive but still have an application potential as the example from laboratory medicine shows. Our approach is of practical use whenever a dataset to be clustered consists of a main cluster with a relatively symmetric distribution and an unknown number of small clusters with unknown distributions. For example, when estimating reference intervals from \"impure\" values, most of which were collected from a population of healthy reference persons and a smaller proportion from persons with different diseases. While conventional methods for distribution deconvolution require statements about the distributions of the pathological fractions ([PERSON] and [PERSON], 2003), our method allows the entirety of these sick populations to be defined as noise and the cluster of healthy individuals to be found. Medical laboratories must evaluate their reference intervals periodically. With hundreds of analytes with different reference intervals for women, men, and various age groups, altogether thousands of reference intervals must be computed so that fast computation plays an important role. Although the different algorithms usually yield coherent reference intervals as in Table 1, in some cases one or the other algorithm fails with an implausible result. Because there is no \"best\" solution, it is recommended to apply different algorithms simultaneously and not to rely on a single one. With its reasonable computation time, our algorithm can therefore be seen as an additional module for computing and evaluating reference intervals. We see another useful application in the evaluation of single-cell analyses, e.g., using flow cytometry. Here, thousands of measurement data are collected per experiment, from which the population of the cells of interest is to be recognised against the background noise of unknown other cells or particles. These can be, for example, clonal tumour cells against the background of connective tissue cells or microvesicles against the background of preparative impurities. Although flow cytometry data are in principle multidimensional, there are one-dimensional representations ([PERSON] and [PERSON], 2003) and applications where such one-dimensional representations are the basis for the identification of the cells of interest ([PERSON] and [PERSON], 2003). Our results may also encourage consideration of reformulations of the objective functions in other clustering approaches, making them amenable to closed-form solutions. The results of cluster analysis depend heavily on selecting a suitable distance measure. Here, we used the squared Euclidean distance, which, while common, is not always the best choice for all clustering problems. Replacing the Euclidean distance with a suitable other distance or dissimilarity measure might also open a path to closed-form solutions. In this way, the restrictions of our approach to one-dimensional data and one main cluster with an additional noise cluster could be overcome. Implementations of all algorithms developed in this paper are available as R-code in the Supplementary Material.