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Choose six illustrative values $y=(-2.2,-1.8,-1,1,1.8,2.2)$, fixed mixture weights ${\pi }_1={\pi }_2=1/2$, known standard deviation $\sigma =1$, and initial means $({\mu }_1^{(0)},{\mu }_2^{(0)})=(-1,1)$. These are constructed data for a Gaussian mixture distribution, not measurements. With $\varphi$ the standard normal distribution probability density function, the fitted log-likelihood isAt the initial means, the component density ratio is $\varphi (y_i-1)/\varphi (y_i+1)=e^{2y_i}$. The first E-step therefore gives

The paired values $y$ and $-y$ have complementary mixture responsibilities, so $N_1^{(0)}=N_2^{(0)}=3$ exactly. The first M-step givesThe hyperbolic tangent expression follows by pairing the contributions at $y$ and $-y$. Assigning every negative value wholly to the first mixture component and every positive value wholly to the second would instead give means $\pm 5/3$. Those hard assignments discard the overlap: the expectation-maximization algorithm is not the same update as K-means clustering.

The observed-data log-likelihood rises from $-11.420338905$ initially to $-10.368745063$ after one update and $-10.340655142$ after two. Further updates approach means $\pm 1.638023833$, with log-likelihood approximately $-10.340509617$. These values describe this initialization; EM likelihood monotonicity supplies no certificate of a global maximum.

For stable numerical evaluation, put $a_{ij}=\log (1/2)-\frac{1}{2}\log (2\pi )-\frac{1}{2}(y_i-{\mu }_j{)}^2$ and $b_i={\max }_j{a}_{ij}$. The log-sum-exp function givesSubtracting $b_i$ keeps at least one exponential equal to one, avoiding the all-zero sum that direct evaluation of tiny component densities can produce.

Cluster analysis groups observations without given class labels. It starts by deciding what similarity means, usually through a dissimilarity matrix. Euclidean distance is natural for comparable continuous coordinates; a Mahalanobis distance can account for differing scales and correlations when a meaningful nonsingular covariance estimate is available. Scaling and variable selection can change the groups. Binary, ordinal or categorical data may need a different dissimilarity, rather than arbitrary numerical codes and Euclidean distance.

Agglomerative hierarchical clustering begins with singleton groups, repeatedly merges the pair with smallest intergroup dissimilarity, and records the merges in a dendrogram. The height represents the algorithm's merge criterion. Cutting the dendrogram gives partitions at different resolutions, and merges cannot subsequently be reversed. In single-linkage clustering, the criterion is the nearest cross-pair distance; this detects chains and nonconvex groups, but a thin bridge can connect otherwise separated clouds. Complete-linkage clustering uses the farthest cross-pair distance and tends to produce compact groups, at the cost of sensitivity to extreme points. Average-linkage clustering uses the mean cross-pair distance, lying between these two criteria for a given pair of clusters.

For squared Euclidean geometry, Ward clustering merges the pair that least increases the within-group sum of squares. Expanding deviations about the merged mean givesThis favors low within-group dispersion, but interpreting it as a sum-of-squares increase requires Euclidean data. Different software may plot a rescaled height, so the vertical units of a dendrogram from Ward clustering must be stated.

A nonhierarchical alternative is k-means clustering. For a prescribed $K$, minimizeAt fixed assignments, expanding squared distances or differentiating shows that the minimizing $m_k$ is the group mean. At fixed centroids, each point is assigned to a nearest centroid. Alternating these operations does not increase $W_K$; with a consistent tie rule and nonempty groups it reaches a locally stable partition. Initialization matters, so use multiple starts, and give an explicit rule for empty groups. It favors roughly spherical clusters and is sensitive to outliers and scaling. It cannot be expected to reproduce a nonconvex single-linkage partition.

The sketch displays a constructed point configuration and the complete-linkage merges computed from its actual Euclidean distances. A cut between the within-pair and between-pair merge heights yields three groups. Such visual separation is useful evidence, not proof of a uniquely correct number of populations. Select a resolution using subject knowledge, stability under resampling, separation diagnostics or the change of $W_K$ with $K$. Clustering always gives a partition even for unstructured data, and different distance or linkage choices can yield different sensible summaries. Clustering is an exploratory construction whose conclusions depend on the geometry and the chosen resolution.