@@ -728,7 +728,7 @@ In the `impetuous.clustering` module you will find several codes for assessing i
The "Link" codes are more efficient at creating a link hierarchy of the data but can be thought of as throwing away information at every linking step. The lost information is deemed unuseful by the heuristic. The full link algorithm determines the new cluster distance to the rest of the points in a self consistent fashion by employing the same heuristic. Using simple linkage, or `min` value distance assignment, will produce an equivalent [hierarchy](https://online.stat.psu.edu/stat555/node/86/) as compared to the one deduced by a connection algorithm. Except for some of the cases when there are distance ties in the link evaluation. This is a computational quirk that does not affect "connection" based hierarchy construction.
The "Link" method is thereby not useful for the deterministic treatment of a particle system where all the true connections in it are important, such as in a water bulk system when you want all your quantum-mechanical waters to be treated at the same level of theory based on their connectivity. This is indeed why my connectivity algorithm was invented by me in 2009. If you are only doing black box statistics then this distinction is not important and computational efficiency is probably what you care about. You can construct hierarchies from both algorithm types but the connection algorithm will always produce a unique and well-determined structure while the link algorithms will be unique but structurally dependent on how ties are resolved and which heuristic is employed for construction. The connection hierarchy is exact and deterministic, but slow to construct, while the link hierarchies are heuristic dependent and non-deterministic, but fast to construct. We will study this more in the following code example as well as the case when they are equivalent.
The "Link" method is thereby not useful for the deterministic treatment of a particle system where all the true connections in it are important, such as in a water bulk system when you want all your quantum-mechanical waters to be treated at the same level of theory based on their connectivity at a specific level or distance. This is indeed why my connectivity algorithm was invented by me in 2009. If you are only doing black box statistics on a complete hierarchy then this distinction is not important and computational efficiency is probably what you care about. You can construct hierarchies from both algorithm types but the connection algorithm will always produce a unique and well-determined structure while the link algorithms will be unique but structurally dependent on how ties are resolved and which heuristic is employed for construction. The connection hierarchy is exact and deterministic, but slow to construct, while the link hierarchies are heuristic dependent, but fast to construct. We will study this more in the following code example as well as the case when they are equivalent.
## 14.1 Link hierarchy construction
The following code produces two distance matrices. One has distance ties and the other one does not. The second matrix is well known and the correct minimal linkage hierarchy is well known. Lets see compare the results between scipy and our method.
