## 1 Introduction  
A common task in today's research is the identification of specific markers, as predictors of a classification yielded in clustering analysis of the data. For instance, this approach is particularly useful after high-throughput experiments to compare gene expression or methylation profiles among different cell lines [1]. This task is used in the nascent field of single-cell sequencing, leading to the important step of clustering cells for further classification or as a qualifying metric of the sequencing process [2]. Regarding the vastly used gene expression assays, the vector of profiles for each marker across different cell lines is recorded using hierarchical clustering algorithms. These algorithms yield a dendrogram and a heat map representing the vector of marker profiles, illustrating the arrangement of the clusters. To assess how well the clustering is segregating different cell lines, a class stating the desired partitioning of each cell line is provided a posteriori. Then, a simple visual inspection of the vector of classes is used to estimate whether the clustering is providing a good partition. Such a partition vector is colored according to the classification that each item is associated with, and it is expected that similar items will be contiguous, so the groups formed are assessed qualitatively against the biological background of each item. This procedure should not be confused with \"supervised clustering\", which provides a vector of classes starting the desired partitioning a priori. This is then used to guide the clustering algorithms by allowing the learning of the metric distances that optimizes the partitioning [3]. Additionally, it may become confused with the metric assessment of the clustering algorithms, especially with the external cluster evaluation. For this, various metrics have been developed to qualify the clustering algorithm itself, such as intrinsic and extrinsic measures. The extrinsic validation compares the clustering to a goal to say whether it is good clustering or not. The internal validation compares the elements within the cluster and their differences [4]. Partition quantitative assessment (_PQA_) involves characteristics of both kinds of validation, through using both the crafted goal standard and the yielded signal itself (clustered vector). However, _PQA_ gathers these elements not qualifying the clustering algorithm itself but to quantify the noise embedded in the cluster, this noise may be due to the intrinsic metric or marker used to order the data set. A possible caveat of the qualitative assessment discussed above is that humans tend to perceive meaningful patterns within random data leading to a cognitive bias known as apphenia [5]. While interpreting the partitions obtained from unsupervised clustering analysis, researchers attempt to visually assess how close the classifications are to each other, finding patterns that are not well supported by the data. Such an effect is created because the adjacency between items may give a notion of the dissimilarity distance in the dendrogram leaves. Unfortunately, as far as we know, there is no method to quantitatively assess the quality of the groups of classifications from the clustering or, at least, there is no attempt to quantify whether certain configuration or order of the items may be due to randomness. This is a serious caveat, since the insertion of noise can lead to a false conclusion or misleading results. Furthermore, the purging of this noise can lead to more efficient descriptions of markers and its phenomena, accelerating the advance in many fields. In statistics, serial correlation (SC) is a term used to describe the relationship between observations of the same variable over specific periods. It was originally used in engineering to determine how a signal, for instance, a radio wave, varies with itself over time. Later, SC was adapted to econometrics to analyze economic data over time principally to predict stock prices and, in other fields, to model-independent random variables [6].