ward#
- scipy.cluster.hierarchy.ward(y)[source]#
- Perform Ward’s linkage on a condensed distance matrix. - See - linkagefor more information on the return structure and algorithm.- The following are common calling conventions: - Z = ward(y)Performs Ward’s linkage on the condensed distance matrix- y.
- Z = ward(X)Performs Ward’s linkage on the observation matrix- Xusing Euclidean distance as the distance metric.
 - Parameters:
- yndarray
- A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that - pdistreturns. Alternatively, a collection of m observation vectors in n dimensions may be passed as an m by n array.
 
- Returns:
- Zndarray
- The hierarchical clustering encoded as a linkage matrix. See - linkagefor more information on the return structure and algorithm.
 
 - See also - linkage
- for advanced creation of hierarchical clusterings. 
- scipy.spatial.distance.pdist
- pairwise distance metrics 
 - Notes - wardhas experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable- SCIPY_ARRAY_API=1and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.- Library - CPU - GPU - NumPy - ✅ - n/a - CuPy - n/a - ⛔ - PyTorch - ✅ - ⛔ - JAX - ✅ - ⛔ - Dask - ⚠️ merges chunks - n/a - See Support for the array API standard for more information. - Examples - >>> from scipy.cluster.hierarchy import ward, fcluster >>> from scipy.spatial.distance import pdist - First, we need a toy dataset to play with: - x x x x x x x x x x x x - >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] - Then, we get a condensed distance matrix from this dataset: - >>> y = pdist(X) - Finally, we can perform the clustering: - >>> Z = ward(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) - The linkage matrix - Zrepresents a dendrogram - see- scipy.cluster.hierarchy.linkagefor a detailed explanation of its contents.- We can use - scipy.cluster.hierarchy.fclusterto see to which cluster each initial point would belong given a distance threshold:- >>> fcluster(Z, 0.9, criterion='distance') array([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], dtype=int32) >>> fcluster(Z, 1.1, criterion='distance') array([1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8], dtype=int32) >>> fcluster(Z, 3, criterion='distance') array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) >>> fcluster(Z, 9, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) - Also, - scipy.cluster.hierarchy.dendrogramcan be used to generate a plot of the dendrogram.