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• Method of single linkage or nearest neighbour. Proximity between two clusters is the proximity between their two closest objects. This value is one of values of the input matrix. The conceptual metaphor of this built of cluster, its archetype, is spectrum or chain. Chains could be straight or curvilinear, or could be like "snowflake" or "amoeba" view. Two most dissimilar cluster members can happen to be very much dissimilar in comparison to two most similar. Single linkage method controls only nearest neighbours similarity.

• Method of complete linkage or farthest neighbour. Proximity between two clusters is the proximity between their two most distant objects. This value is one of values of the input matrix. The metaphor of this built of cluster is circle (in the sense, by hobby or plot) where two most distant from each other members cannot be much more dissimilar than other quite dissimilar pairs (as in circle). Such clusters are "compact" contours by their borders, but they are not necessarily compact inside.

• Method of between-group average linkage (UPGMA). Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side. The metaphor of this built of cluster is quite generic, just united class or close-knit collective; and the method is frequently set the default one in hierarhical clustering packages. Clusters of miscellaneous shapes and outlines can be produced.

• Simple average, or method of equilibrious between-group average linkage (WPGMA) is the modified previous. Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side; while the subclusters of which each of these two clusters were merged recently have equalized influence on that proximity – even if the subclusters differed in the number of objects.

• Method of within-group average linkage (MNDIS). Proximity between two clusters is the arithmetic mean of all the proximities in their joint cluster. This method is an alternative to UPGMA. It usually will lose to it in terms of cluster density, but sometimes will uncover cluster shapes which UPGMA will not.

• Centroid method (UPGMC). Proximity between two clusters is the proximity between their geometric centroids: [squared] euclidean distance between those. The metaphor of this built of cluster is proximity of platforms (politics). Like in political parties, such clusters can have fractions or "factions", but unless their central figures are apart from each other the union is consistent. Clusters can be various by outline.

• Median, or equilibrious centroid method (WPGMC) is the modified previous. Proximity between two clusters is the proximity between their geometric centroids ([squared] euclidean distance between those); while the centroids are defined so that the subclusters of which each of these two clusters were merged recently have equalized influence on its centroid – even if the subclusters differed in the number of objects. Name "median" is partly misleading because the method doesn't use medians of data distributions, it is still based on centroids (the means).

• Ward’s method, or minimal increase of sum-of-squares (MISSQ), sometimes incorrectly called "minimum variance" method. Proximity between two clusters is the magnitude by which the summed square in their joint cluster will be greater than the combined summed square in these two clusters: $SS_{12}-(SS_1+SS_2)$. (Between two singleton objects this quantity = squared euclidean distance / $2$.) The metaphor of this built of cluster is type. Intuitively, a type is a cloud more dense and more concentric towards its middle, whereas marginal points are few and could be scattered relatively freely.

• Method of single linkage or nearest neighbour. Proximity between two clusters is the proximity between their two closest objects. This value is one of values of the input matrix. The conceptual metaphor of this build of cluster, its archetype, is spectrum or chain. Chains could be straight or curvilinear, or could be like "snowflake" or "amoeba" view. Two most dissimilar cluster members can happen to be very much dissimilar in comparison to two most similar. Single linkage method controls only nearest neighbours similarity.

• Method of complete linkage or farthest neighbour. Proximity between two clusters is the proximity between their two most distant objects. This value is one of values of the input matrix. The metaphor of this build of cluster is circle (in the sense, by hobby or plot) where two most distant from each other members cannot be much more dissimilar than other quite dissimilar pairs (as in circle). Such clusters are "compact" contours by their borders, but they are not necessarily compact inside.

• Method of between-group average linkage (UPGMA). Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side. The metaphor of this build of cluster is quite generic, just united class or close-knit collective; and the method is frequently set the default one in hierarhical clustering packages. Clusters of miscellaneous shapes and outlines can be produced.

• Simple average, or method of equilibrious between-group average linkage (WPGMA) is the modified previous. Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side; while the subclusters of which each of these two clusters were merged recently have equalized influence on that proximity – even if the subclusters differed in the number of objects.

• Method of within-group average linkage (MNDIS). Proximity between two clusters is the arithmetic mean of all the proximities in their joint cluster. This method is an alternative to UPGMA. It usually will lose to it in terms of cluster density, but sometimes will uncover cluster shapes which UPGMA will not.

• Centroid method (UPGMC). Proximity between two clusters is the proximity between their geometric centroids: [squared] euclidean distance between those. The metaphor of this build of cluster is proximity of platforms (politics). Like in political parties, such clusters can have fractions or "factions", but unless their central figures are apart from each other the union is consistent. Clusters can be various by outline.

• Median, or equilibrious centroid method (WPGMC) is the modified previous. Proximity between two clusters is the proximity between their geometric centroids ([squared] euclidean distance between those); while the centroids are defined so that the subclusters of which each of these two clusters were merged recently have equalized influence on its centroid – even if the subclusters differed in the number of objects. Name "median" is partly misleading because the method doesn't use medians of data distributions, it is still based on centroids (the means).

• Ward’s method, or minimal increase of sum-of-squares (MISSQ), sometimes incorrectly called "minimum variance" method. Proximity between two clusters is the magnitude by which the summed square in their joint cluster will be greater than the combined summed square in these two clusters: $SS_{12}-(SS_1+SS_2)$. (Between two singleton objects this quantity = squared euclidean distance / $2$.) The metaphor of this build of cluster is type. Intuitively, a type is a cloud more dense and more concentric towards its middle, whereas marginal points are few and could be scattered relatively freely.

First 5 methods permit any proximity measures (any similarities or distances) and results will, naturally, depend on the measure chosen.

Last 6 methods require distances; and fully correct will be to use only squared euclidean distances with them, because these methods compute centroids in euclidean space. Therefore distances should be euclidean for the sake of geometric correctness (these 6 methods are called together geometric linkage methods). At worst case, you might input other metric distances at admitting more heuristic, less rigorous analysis. Now about that "squared". Computation of centroids and deviations from them are most convenient mathematically/programmically to perform on squared distances, that's why HAC packages usually require to input and are tuned to process the squared ones. However, there exist implementations - fully equivalent yet a bit slower - based on nonsquared distances input and requiring those; see for example "Ward-2" implementation for Ward's method. You should consult with the documentation of you clustering program to know which - squared or not - distances it expects at input to a "geometric method" in order to do it right.

Still other methods represent some specialized set distances. HAC algorithm can be based on them, only not on the generic Lance-Williams formula; such distances include, among other: Hausdorff distance and Point-centroid cross-distance (I've implemented a HAC program for SPSS based on those.)

First 5 methods described permit any proximity measures (any similarities or distances) and results will, naturally, depend on the measure chosen.

Next 6 methods described require distances; and fully correct will be to use only squared euclidean distances with them, because these methods compute centroids in euclidean space. Therefore distances should be euclidean for the sake of geometric correctness (these 6 methods are called together geometric linkage methods). At worst case, you might input other metric distances at admitting more heuristic, less rigorous analysis. Now about that "squared". Computation of centroids and deviations from them are most convenient mathematically/programmically to perform on squared distances, that's why HAC packages usually require to input and are tuned to process the squared ones. However, there exist implementations - fully equivalent yet a bit slower - based on nonsquared distances input and requiring those; see for example "Ward-2" implementation for Ward's method. You should consult with the documentation of you clustering program to know which - squared or not - distances it expects at input to a "geometric method" in order to do it right.

• Method of single linkage or nearest neighbour. Proximity between two clusters is the proximity between their two closest objects. This value is one of values of the input matrix. The conceptual metaphor of this built of cluster, its archetype, is spectrum or chain. Chains could be straight or curvilinear, or could be like "snowflake" or "amoeba" view. Two most dissimilar cluster members can happen to be very much dissimilar in comparison to two most similar. Single linkage method controls only nearest neighbours similarity.

• Method of complete linkage or farthest neighbour. Proximity between two clusters is the proximity between their two most distant objects. This value is one of values of the input matrix. The metaphor of this built of cluster is circle (in the sense, by hobby or plot) where two most distant from each other members cannot be much more dissimilar than other quite dissimilar pairs (as in circle). Such clusters are "compact" contours by their borders, but they are not necessarily compact inside.

• Method of between-group average linkage (UPGMA). Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side. The metaphor of this built of cluster is quite generic, just united class or close-knit collective; and the method is frequently set the default one in hierarhical clustering packages. Clusters of miscellaneous shapes and outlines can be produced.

• Simple average, or method of equilibrious between-group average linkage (WPGMA) is the modified previous. Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side; while the subclusters of which each of these two clusters were merged recently have equalized influence on that proximity – even if the subclusters differed in the number of objects.

• Method of within-group average linkage (MNDIS). Proximity between two clusters is the arithmetic mean of all the proximities in their joint cluster. This method is an alternative to UPGMA. It usually will lose to it in terms of cluster density, but sometimes will uncover cluster shapes which UPGMA will not.

• Centroid method (UPGMC). Proximity between two clusters is the proximity between their geometric centroids: [squared] euclidean distance between those. The metaphor of this built of cluster is proximity of platforms (politics). Like in political parties, such clusters can have fractions or "factions", but unless their central figures are apart from each other the union is consistent. Clusters can be various by outline.

• Median, or equilibrious centroid method (WPGMC) is the modified previous. Proximity between two clusters is the proximity between their geometric centroids ([squared] euclidean distance between those); while the centroids are defined so that the subclusters of which each of these two clusters were merged recently have equalized influence on its centroid – even if the subclusters differed in the number of objects.

• Ward’s method, or minimal increase of sum-of-squares (MISSQ), sometimes incorrectly called "minimum variance" method. Proximity between two clusters is the magnitude by which the summed square in their joint cluster will be greater than the combined summed square in these two clusters: $SS_{12}-(SS_1+SS_2)$. (Between two singleton objects this quantity = squared euclidean distance / $2$.) The metaphor of this built of cluster is type. Intuitively, a type is a cloud more dense and more concentric towards its middle, whereas marginal points are few and could be scattered relatively freely.

• Method of single linkage or nearest neighbour. Proximity between two clusters is the proximity between their two closest objects. This value is one of values of the input matrix. The conceptual metaphor of this built of cluster, its archetype, is spectrum or chain. Chains could be straight or curvilinear, or could be like "snowflake" or "amoeba" view. Two most dissimilar cluster members can happen to be very much dissimilar in comparison to two most similar. Single linkage method controls only nearest neighbours similarity.

• Method of complete linkage or farthest neighbour. Proximity between two clusters is the proximity between their two most distant objects. This value is one of values of the input matrix. The metaphor of this built of cluster is circle (in the sense, by hobby or plot) where two most distant from each other members cannot be much more dissimilar than other quite dissimilar pairs (as in circle). Such clusters are "compact" contours by their borders, but they are not necessarily compact inside.

• Method of between-group average linkage (UPGMA). Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side. The metaphor of this built of cluster is quite generic, just united class or close-knit collective; and the method is frequently set the default one in hierarhical clustering packages. Clusters of miscellaneous shapes and outlines can be produced.

• Simple average, or method of equilibrious between-group average linkage (WPGMA) is the modified previous. Proximity between two clusters is the arithmetic mean of all the proximities between the objects of one, on one side, and the objects of the other, on the other side; while the subclusters of which each of these two clusters were merged recently have equalized influence on that proximity – even if the subclusters differed in the number of objects.

• Method of within-group average linkage (MNDIS). Proximity between two clusters is the arithmetic mean of all the proximities in their joint cluster. This method is an alternative to UPGMA. It usually will lose to it in terms of cluster density, but sometimes will uncover cluster shapes which UPGMA will not.

• Centroid method (UPGMC). Proximity between two clusters is the proximity between their geometric centroids: [squared] euclidean distance between those. The metaphor of this built of cluster is proximity of platforms (politics). Like in political parties, such clusters can have fractions or "factions", but unless their central figures are apart from each other the union is consistent. Clusters can be various by outline.

• Median, or equilibrious centroid method (WPGMC) is the modified previous. Proximity between two clusters is the proximity between their geometric centroids ([squared] euclidean distance between those); while the centroids are defined so that the subclusters of which each of these two clusters were merged recently have equalized influence on its centroid – even if the subclusters differed in the number of objects. Name "median" is partly misleading because the method doesn't use medians of data distributions, it is still based on centroids (the means).

• Ward’s method, or minimal increase of sum-of-squares (MISSQ), sometimes incorrectly called "minimum variance" method. Proximity between two clusters is the magnitude by which the summed square in their joint cluster will be greater than the combined summed square in these two clusters: $SS_{12}-(SS_1+SS_2)$. (Between two singleton objects this quantity = squared euclidean distance / $2$.) The metaphor of this built of cluster is type. Intuitively, a type is a cloud more dense and more concentric towards its middle, whereas marginal points are few and could be scattered relatively freely.