# Bootstrap-CURE: A Novel Clustering Approach for Sensor Data--An Application to 3D Printing Industry
## 3 State-of-the-Art
### Clustering Strategies
#### 3.2.3 Detection of the Number of Clusters  
The most common strategies used in deciding the number of clusters in a hierarchical setting are cross-validation, resampling, and finding the _knee_ or _elbow_ of an error curve. The approach of cross-validation aims at computing the regression coefficients on the \(v-1\) portion of the dataset and validating the same on the \(v\)th portion, which is not used in building the regression model. The approach, however, becomes time intensive for large data and is not recommended. In [65], [PERSON] et al. proposed a criterion to detect the number of clusters in a modular network, using a leave-one-out cross-validation approach. [PERSON] et al. [66] proposed a cross-validation-based approach to determine the number of clusters in a k-means type of clustering. Hence, the idea of using cross validation in clustering is to iteratively test which cluster solution yields the lowest error rate and consider that the best clustering solution. However, as pointed out by [PERSON] et al. in [67], cluster analysis has no such linear coefficients which can be applied to multiple random samples and true cluster membership is rather unknown. Consequently, the cross-validation strategy cannot be used globally for all cases. This was also highlighted by [PERSON] et al. in [68], where the authors demonstrated how the cross-validation technique fails in hierarchical clustering by doing an _Monte-Carlo_ simulation-based study of cross-validation techniques under different conditions. Several works related to using resampling methods to decide the number of clusters in data can be found in the literature. In [69], [PERSON] and [PERSON] presented a replication-based stopping rule in which a replication defined by higher-order clustering helps identify the distinct underlying populations (clusters) in a multidimensional space. An improved version of this criterion was presented in [70], where [PERSON] et al. used the bootstrapping procedure along with the increase in the size of the resampled dataset with respect to the primary dataset and thus showed an increase in accuracy of the clustering solution. Related work is found in [71], where [PERSON] et al. discussed a bootstrapping-based approach to estimate the clustering instability and then select the best number of clusters. Another popular method of finding the optimal number of clusters is to seek the local maxima (or minima) corresponding to the _knee_ or _elbow_ of a curve that plots values of some clustering evaluation criteria on a range of numbers. [PERSON] et al. in [72] reviewed several CVIs and their association with the type of data. [PERSON] et al. [73] also provided a measure called the _gap statistic_. It tests the hypothesis that the model has a single cluster (\(K=1\)) and tries to reject it with an alternative hypothesis (\(K>1\)). This method compares the total within intra-cluster variation for different values of k with their expected values under null reference distribution of the data, i.e., when there is no underlying clustering. However, the rejection of the null hypothesis only indicates insufficient evidence in support of the null hypothesis and does not really make it true. The underlying methodology in _elbow_ or _knee_ is, in fact, agnostic to the cluster validity index being used. In [74], [PERSON] et al. proposed _clustering gain_ to find the right number of clusters in hierarchical clustering. Clustering gain is designed to have a maximum value when the intra-cluster similarity is maximized and inter-cluster similarity is minimized. The optimal number of clusters is then chosen based on the maximum point in the clustering gain curve. Similarly, in [75], [PERSON] et al. proposed another criterion, called _CSP_ (compact separation proportion) wherein the optimal number of clusters is estimated corresponding to the maximum average value of the _CSP_ index.