How do you validate and compare clustering results with missing values?

In some cases, experts may be able to provide additional information or insights that can help handle missing values. For example, domain experts may have knowledge of common patterns in missing data for a particular dataset and can provide guidance on how to handle them.

Listwise deletion or complete case analysis involves removing any observations that have missing values in any of the variables. This method is commonly used for MCAR, but it can also be used for MAR if the missingness is low. This method can lead to a loss of information, and the remaining sample may not be representative of the original sample.

Imputation involves filling in the missing values with plausible values based on the observed values. There are different imputation methods available such as mean imputation, mode imputation, regression imputation, and multiple imputation. Imputation is commonly used for MAR and MNAR, but the quality of the imputation can be affected by the degree of missing values and the pattern of missing values.

Categorical imputation is similar to imputation but is specific to categorical variables. It involves filling in the missing categorical values with the mode (most frequent) value of that variable.

Missing indicator involves creating a new binary variable that indicates whether a value is missing or not for each variable with missing values. This method can be used to retain all the observations in the analysis while accounting for the missing values.

Other methods that can handle missing values include: K-Medoids: K-Medoids is a variant of K-Means that replaces the mean with a medoid (the most representative point) as the center of each cluster. This method is robust to outliers and can handle missing values by computing the distance between each pair of data points using only the observed values. Fuzzy C-Means with imputation: Fuzzy C-Means is a soft clustering algorithm that assigns each data point to multiple clusters based on the degree of membership. This method can handle missing values by using imputation techniques to replace the missing values with plausible values based on the observed data.

Two more methods that can handle missing values: Multiple Imputation with Expectation-Maximization (EM) algorithm: This method involves imputing missing values multiple times and using the EM algorithm to estimate the parameters of the clustering model. This method can handle missing values that are MAR or MCAR. DBSCAN with imputation: DBSCAN is a density-based clustering algorithm that groups together data points that are in dense regions. This method can handle missing values by using imputation techniques to replace the missing values with plausible values based on the observed data.

Hierarchical clustering is a popular clustering algorithm that creates a tree-like structure called a dendrogram by recursively merging clusters based on the similarity between them. Hierarchical clustering is sensitive to missing values as the distance metric used to compute the similarity between the data points can be affected by the missing values. However, some variations of hierarchical clustering, such as the weighted pair-group method with arithmetic mean (WPGMA), can handle missing values by assigning a weight of zero to the missing values.

Gaussian Mixture Models (GMM) is a soft clustering algorithm that models the data distribution as a combination of Gaussian distributions. GMM assumes that the data is complete and normally distributed, and therefore, cannot handle missing values directly. However, one approach to handling missing values in GMM is to use imputation techniques to replace the missing values with plausible values based on the observed data.

Self-Organizing Maps (SOM) is an unsupervised learning algorithm that maps high-dimensional data onto a low-dimensional grid while preserving the topological structure of the data. SOM is sensitive to missing values as the distance metric used to compute the similarity between the data points can be affected by the missing values. However, one approach to handling missing values in SOM is to use imputation techniques to replace the missing values with plausible values based on the observed data.

Normalized Mutual Information (NMI) measures the mutual information between two clusterings, taking into account the size of the clusters and the number of clusters in each clustering.

Silhouette Coefficient measures how well each data point fits into its assigned cluster, and ranges from -1 to 1, where a value of 1 indicates a good clustering result.

Homogeneity, Completeness, and V-Measure are three metrics that evaluate the quality of a clustering result in terms of the extent to which each cluster contains only data points from a single class (homogeneity), the extent to which all data points belonging to a given class are assigned to the same cluster (completeness), and a harmonic mean of both metrics (V-Measure).

Fowlkes-Mallows Index (FMI) measures the similarity between two clusterings based on the geometric mean of the precision and recall of each cluster.