# Similarity analysis#

The design of a product family requires the creation of a basic product architecture from which all product family members can be derived. By analyzing the similarity of products within an existing product portfolio one can get an indication of which and how many product variants have to be supported by the architecture.

## Input#

The input for a similarity analysis is a domain mapping matrix (or bipartite graph) that maps individual products from the portfolio to attributes that characterize the functionality and designs of the products. The datasets module contains an example data set for performing a similarity analysis. So let’s get started and load the ‘similarity’ dataset:

>>> from ragraph import datasets
>>> graph = datasets.get("similarity")


and subsequently visualize it using dmm:

>>> import ragraph.plot
>>> fig = ragraph.plot.dmm(
...     rows=[n for n in graph.nodes if n.kind == "attribute"],
...     cols=[n for n in graph.nodes if n.kind == "product"],
...     edges=graph.edges,
...     sort=False,
... )
>>> fig.write_image(dmm_path)  # e.g.: "dmm.svg"


which results in Figure 15, which shows the resulting domain mapping matrix (DMM), in which twelve products (columns) are mapped to ten attributes (rows). A mark at position $$i,j$$ indicates that product $$j$$ is characterized by attribute $$i$$.

In selecting attributes for performing a similarity analysis you should balance the attribute granularity level. That is, very fine grained (detailed) attributes will yield a very sparse mapping matrix, while very coarse (high level) attributes will yield a very dense mapping matrix. Moreover, you should take care in ensuring that the attributes are non-overlapping. In general, it is advised to use attributes that define functionality, working-principles, and embodiments of product(s) (modules).

Once you have defined your graph, mapping products to attributes, you can use the SimilarityAnalysis object to perform the similarity analysis. First, you have to instantiate the SimilarityAnalysis object for which a minimal example is shown below:

>>> from ragraph.analysis.similarity import SimilarityAnalysis
>>> sa = SimilarityAnalysis(
...     rows=[n for n in graph.nodes if n.kind == "attribute"],
...     cols=[n for n in graph.nodes if n.kind == "product"],
...     edges=graph.edges,
... )


The SimilarityAnalysis object requires three parameters: rows which is a list of Node objects which are the row elements of a DMM (in this case attributes), cols which is a list of Node objects which are the column elements of a DMM (in this case products), and edges which is a list of Edge objects that map column elements to row elements.

Internally, the jaccard_matrix is calculated for both the columns elements as the row elements, which are stored within the row_similarity_matrix and col_similarity_matrix attributes:

>>> print(sa.row_similarity_matrix)
[[1.   0.   0.   0.25 0.   0.25 0.17 0.2  0.   0.  ]
[0.   1.   0.   0.   0.   0.   0.2  0.   0.   0.33]
[0.   0.   1.   0.   0.5  0.   0.17 0.   0.25 0.  ]
[0.25 0.   0.   1.   0.   0.   0.   0.25 0.   0.  ]
[0.   0.   0.5  0.   1.   0.   0.17 0.   0.25 0.  ]
[0.25 0.   0.   0.   0.   1.   0.2  0.25 0.   0.  ]
[0.17 0.2  0.17 0.   0.17 0.2  1.   0.   0.   0.2 ]
[0.2  0.   0.   0.25 0.   0.25 0.   1.   0.   0.  ]
[0.   0.   0.25 0.   0.25 0.   0.   0.   1.   0.  ]
[0.   0.33 0.   0.   0.   0.   0.2  0.   0.   1.  ]]
>>> print(sa.col_similarity_matrix)
[[1.   0.25 0.33 0.   0.33 0.   0.   0.   0.   0.25 0.   0.  ]
[0.25 1.   0.   0.25 0.25 0.25 0.25 0.   0.   0.2  0.   0.  ]
[0.33 0.   1.   0.   0.33 0.   0.   0.   0.   0.   0.   0.  ]
[0.   0.25 0.   1.   0.   0.   0.33 0.   0.   0.   0.33 0.  ]
[0.33 0.25 0.33 0.   1.   0.   0.   0.   0.   0.25 0.   0.  ]
[0.   0.25 0.   0.   0.   1.   0.33 0.   0.   0.   0.33 0.  ]
[0.   0.25 0.   0.33 0.   0.33 1.   0.   0.   0.   0.33 0.  ]
[0.   0.   0.   0.   0.   0.   0.   1.   0.33 0.25 0.   0.33]
[0.   0.   0.   0.   0.   0.   0.   0.33 1.   0.67 0.   0.33]
[0.25 0.2  0.   0.   0.25 0.   0.   0.25 0.67 1.   0.   0.25]
[0.   0.   0.   0.33 0.   0.33 0.33 0.   0.   0.   1.   0.  ]
[0.   0.   0.   0.   0.   0.   0.   0.33 0.33 0.25 0.   1.  ]]


The data from these matrices are stored within the internal graph property. This graph is a regular Graph object which you can be visualized using the mdm module:

>>> import ragraph.plot
>>> fig = ragraph.plot.mdm(
...     leafs=sa.graph.leafs,
...     edges=sa.graph.edges,
...     style=ragraph.plot.Style(
...         piemap=dict(
...             display="weights",
...             fields=["similarity"]
...         )
...     ),
... )
>>> fig.write_image(smdm1_path)


Figure 16 shows the resulting product - attribute multi-domain-matrix (MDM) in which similarity weights are displayed.

## Pruning#

If a similarity matrix is very dense, you may want to prune the matrix by removing all values below a certain threshold. You can prune the matrices by setting the values of the col_sim_threshold and row_sim_threshold attributes:

>>> sa.col_sim_threshold = 0.30
>>> sa.row_sim_threshold = 0.20


By changing the value of these attributes, the graph attribute will automatically update, as show in Figure 17 in which all edges with a similarity weight below the respective threshold are removed.

>>> import ragraph.plot
>>> fig = ragraph.plot.mdm(
...     leafs=sa.graph.leafs,
...     edges=sa.graph.edges,
...     style=ragraph.plot.Style(
...         piemap=dict(
...             display="weights",
...             fields=["similarity"]
...         )
...     ),
... )
>>> fig.write_image(smdm2_path)


## Clustering#

The aim of a similarity analysis is to find groups (clusters) of products that are similar and could therefore be standardized. Similarly, one could search for groups (clusters) of attributes that have high similarity, which implies that they are often found together within products. To highlight these clusters you can use the cluster_rows and cluster_cols methods:

>>> sa.cluster_rows(alpha=2.0, beta=2.0, mu=2.0)
>>> sa.cluster_cols(alpha=2.0, beta=2.0, mu=2.0)


By default the markov algorithm. You could, however, provide a different algorithm by setting the algo argument if desired.

After clustering, you can re-visualize the product - attribute MDM:

>>> import ragraph.plot
>>> sa.row_sim_threshold = 0.0
>>> sa.col_sim_threshold = 0.0
>>> fig = ragraph.plot.mdm(
...     leafs=sa.graph.leafs,
...     edges=sa.graph.edges,
...     style=ragraph.plot.Style(
...         piemap=dict(
...             display="weights",
...             fields=["similarity"]
...         )
...     ),
... )
>>> fig.write_image(smdm3_path)  # e.g.: "dmm.svg"


Note that we reset the similarity thresholds to 0.0 to ensure that all data is displayed. Figure 18 shows the resulting MDM in which one can observe three product clusters and three attribute clusters. The first product cluster maps to the first attribute cluster, the second product cluster primarily maps to the third attribute cluster, and the third product cluster primarily maps to the second attribute clusters. This indicates the presences of three product families within the product portfolio, each with a distinct set of attributes that characterize their functionality and design.