NIMAA-vignette

Introduction

Many numerical analyses are invalid when working with nominal data because the mode is the only way to measure central tendency for nominal data, and frequency testing, like Chi-square tests, is the most common statistical analysis that makes sense. NIMAA package (Jafari and Chen 2022) proposes a comprehensive set of pipeline to perform nominal data mining, which can effectively find label relationships of each nominal variable according to pairwise association with other nominal data. You can also check for updates, here NIMAA.

It uses bipartite graphs to show how two different types of data are linked together, and it puts them in the incidence matrix to continue with network analysis. Finding large submatrices with non-missing values and edge prediction are other applications of NIMAA to explore local and global similarities within the labels of nominal variables.

Then, using a variety of different network projection methods, two unipartite graphs are constructed on the given submatrix. NIMAA provides several options for clustering projected networks and selecting the best one based on internal measures and external prior knowledge (ground truth). When weighted bipartite networks are considered, the best clustering results are used as the benchmark for edge prediction analysis. This benchmark is used to figure out which imputation method is the best one to predict weight of edges in bipartite network. It looks at how similar the clustering results are before and after the imputations. By using edge prediction analysis, we tried to get more information from the whole dataset even though there were some missing values.

library(NIMAA)

1 Exploring the data

In this section, we demonstrate how to do a NIMAA analysis on a weighted bipartite network using the beatAML dataset. beatAML is one of four datasets that can be found in the NIMAA package (Jafari and Chen 2022). This dataset has three columns: the first two contain nominal variables, while the third contains numerical variables.

Read the data from the package:

# read the data
beatAML_data <- NIMAA::beatAML

1.1 Plotting the original data

The plotIncMatrix() function prints some information about the incidence matrix derived from input data, such as its dimensions and the proportion of missing values, as well as the image of the matrix. It also returns the incidence matrix object.

NB: To keep the size of vignette small enough for CRAN rules, we won’t output the interactive figure here.

beatAML_incidence_matrix <- plotIncMatrix(
  x = beatAML_data, # original data with 3 columns
  index_nominal = c(2,1), # the first two columns are nominal data
  index_numeric = 3,  # the third column is numeric data
  print_skim = FALSE, # if you want to check the skim output, set this as TRUE
  plot_weight = TRUE, # when plotting the weighted incidence matrix
  verbose = FALSE # NOT save the figures to local folder
  )

Na/missing values Proportion: 0.2603

The beatAML dataset as an incidence matrix

1.2 Plotting the bipartite network of the original data

Given that we have the incidence matrix, we can easily reconstruct the corresponding bipartite network. In the NIMAA package, we have two options for visualizing the bipartite network: static or interactive plots.

1.2.1 Static plot

The plotBipartite() function customizes the corresponding bipartite network visualization based on the igraph package (Csardi and Nepusz 2006) and returns the igraph object.

bipartGraph <- plotBipartite(inc_mat = beatAML_incidence_matrix, vertex.label.display = T)

# show the igraph object
bipartGraph
#> IGRAPH bd183bf UNWB 650 47636 -- 
#> + attr: name (v/c), type (v/l), shape (v/c), color (v/c), weight (e/n)
#> + edges from bd183bf (vertex names):
#>  [1] Alisertib (MLN8237)      --11-00261 Barasertib (AZD1152-HQPA)--11-00261
#>  [3] Bortezomib (Velcade)     --11-00261 Canertinib (CI-1033)     --11-00261
#>  [5] Crenolanib               --11-00261 CYT387                   --11-00261
#>  [7] Dasatinib                --11-00261 Doramapimod (BIRB 796)   --11-00261
#>  [9] Dovitinib (CHIR-258)     --11-00261 Erlotinib                --11-00261
#> [11] Flavopiridol             --11-00261 GDC-0941                 --11-00261
#> [13] Gefitinib                --11-00261 Go6976                   --11-00261
#> [15] GW-2580                  --11-00261 Idelalisib               --11-00261
#> + ... omitted several edges

1.2.2 Interactive plot

The plotBipartiteInteractive() function generates a customized interactive bipartite network visualization based on the visNetwork package (Almende B.V. and Contributors, Thieurmel, and Robert 2021).

NB: To keep the size of vignette small enough, we do not output the interactive figure here. Instead, we show a screenshot of part of the beatAML dataset.

plotBipartiteInteractive(inc_mat = beatAML_incidence_matrix)

beatAML dataset as incidence matrix

analysis_reuslt <- analyseNetwork(bipartGraph)

# showing the general measures for network topology
analysis_reuslt$general_stats
#> $vertices_amount
#> [1] 650
#> 
#> $edges_amount
#> [1] 47636
#> 
#> $edge_density
#> [1] 0.2258433
#> 
#> $components_number
#> [1] 1
#> 
#> $eigen_centrality_value
#> [1] 15721.82
#> 
#> $hub_score_value
#> [1] 247175684

2 Extracting large submatrices without missing values

In the case of a weighted bipartite network, the dataset with the fewest missing values should be used for the next steps. This is to avoid the sensitivity problems of clustering-based methods. The extractSubMatrix() function extracts the submatrices that have non-missing values or have a certain percentage of missing values inside (not for elements-max matrix), depending on the argument’s input. The result will also be shown as a plotly plot (Sievert 2020), so you can see the screenshots of beatAML dataset below.

The extraction process is performed and visualized in two ways, which can be chosen depending on the user’s preference: using the original input matrix (row-wise) and using the transposed matrix (column-wise). NIMAA extracts the largest submatrices with non-missing values or with a specified proportion of missing values (using the bar argument) in four ways predefined in the shape argument:

• Square, to extract square matrix with the equal number of rows and columns,
• Rectangular_row, to extract matrix with the maximum number of rows,
• Rectangular_col, to extract matrix with the maximum number of columns,
  
• Rectangular_element_max, to extract matrix with the maximum number of elements.

Here we extract two shapes of submatrix from the beatAML_incidence_matrix including square and rectangular, with the maximum number of elements:

sub_matrices <- extractSubMatrix(
  x = beatAML_incidence_matrix,
  shape = c("Square", "Rectangular_element_max"), # the selected shapes of submatrices
  row.vars = "patient_id",
  col.vars = "inhibitor",
  plot_weight = TRUE,
  print_skim = FALSE
  )
#> binmatnest2.temperature 
#>                20.12122 
#> Size of Square:   96 rows x  96 columns 
#> Size of Rectangular_element_max:      87 rows x  140 columns

Row-wise arrangement

Column-wise arrangement

A value called binmatnest2.temperature is also printed by this function that indicates the nestedness measure of the input matrix. If the input matrix is highly nested (binmatnest2.temperature >= 1), then splitting the input matrix into parts is recommended to follow the analysis in each part independently.

3 Cluster finding analysis of projected unipartite networks

The findCluster() function performs seven commonly used network clustering algorithms, with the option of preprocessing the input incidence matrix following the projecting of the bipartite network into unipartite networks. Preprocessing options such as using normalization and defining a threshold help us remove edges that have low weights or are weak edges. Also, it is possible to continue using binarized or original edge weights before performing community detection or clustering. Finally, all of the clustering methods that were used can be compared and the best one chosen based on internal or external cluster validity measures such as average of silhouette width and Jaccard similarity index. Additionally to displaying in consul, the findings are provided as bar plots.

To begin, let us examine how to utilize findCluster to find clusters of patient_ids based on the Rectangular_element_max submatrix of the beatAML bipartite network projection to a patient’s unipartite similarity network.

cls1 <- findCluster(
  sub_matrices$Rectangular_element_max,
  part = 1,
  method = "all", # all available clustering methods
  normalization = TRUE, # normalize the input matrix
  rm_weak_edges = TRUE, # remove the weak edges in graph
  rm_method = 'delete', # delete the weak edges instead of lowering their weights to 0.
  threshold = 'median', # Use median of edges' weights as threshold
  set_remaining_to_1 = TRUE, # set the weights of remaining edges to 1
  )

Additionally, if external features (prior knowledge) are provided, the function compares the clustering results obtained with the external features in terms of similarity by setting extra_feature argument.

For instance, let’s generate some random features for the patients in the beatAML dataset. The following is an example:

external_feature <- data.frame(row.names = cls1$walktrap$names)
external_feature[,'membership'] <- paste('group',
                                         findCluster(sub_matrices$Rectangular_element_max,
                                                     method = c('walktrap'),
                                                     rm_weak_edges = T,
                                                     set_remaining_to_1 = F,
                                                     threshold = 'median',
                                                     comparison = F)$walktrap$membership)
dset1 <- utils::head(external_feature)
knitr::kable(dset1, format = "html")

Clustering is done on the patient similarity network again, but this time by setting the extra_feature argument to provide external validation for the cluster finding analysis.

cls <- findCluster(
  sub_matrices$Rectangular_element_max,
  part = 1,
  method = "all", 
  normalization = TRUE, 
  rm_weak_edges = TRUE, 
  rm_method = 'delete', 
  threshold = 'median', 
  set_remaining_to_1 = TRUE, 
  extra_feature = external_feature # add an extra feature reference here
  )
#> Warning in findCluster(sub_matrices$Rectangular_element_max, part = 1, method =
#> "all", : cluster_spinglass cannot work with unconnected graph
#> 
#> 
#> |                   |  walktrap|   louvain|   infomap| label_prop| leading_eigen| fast_greedy|
#> |:------------------|---------:|---------:|---------:|----------:|-------------:|-----------:|
#> |modularity         | 0.0125994| 0.0825865| 0.0000000|  0.0000000|     0.0806766|   0.0825865|
#> |avg.silwidth       | 0.2109092| 0.1134990| 0.9785714|  0.9785714|     0.1001961|   0.1134990|
#> |coverage           | 0.9200411| 0.5866393| 1.0000000|  1.0000000|     0.5806783|   0.5866393|
#> |corrected.rand     | 0.8617545| 0.0603934| 0.2518265|  0.2518265|     0.0430117|   0.0603934|
#> |jaccard_similarity | 0.9310160| 0.4431195| 0.7981967|  0.7981967|     0.4409188|   0.4431195|

As you can see, two new indices are included (jaccard_similarity and corrected_rand) as external cluster validity measures which represent the similarity between clustering and the prior knowledge used as ground truth.

The following sections show how to perform clustering analysis on the other part of the beatAML bipartite network, namely, inhibitor. In this case, we simply need to modify the part to 2.

cls2 <- findCluster(
  sub_matrices$Rectangular_element_max, # the same submatrix
  part = 2 # set to 2 to use the other part of bipartite network
  )
#> Warning in findCluster(sub_matrices$Rectangular_element_max, part = 2):
#> cluster_spinglass cannot work with unconnected graph
#> 
#> 
#> |             |  walktrap|   louvain|  infomap| label_prop| leading_eigen| fast_greedy|
#> |:------------|---------:|---------:|--------:|----------:|-------------:|-----------:|
#> |modularity   | 0.0211412| 0.0794066| 0.000000|   0.000000|     0.0794650|   0.0785273|
#> |avg.silwidth | 0.1723698| 0.1349404| 0.954023|   0.954023|     0.1317522|   0.1270078|
#> |coverage     | 0.8433993| 0.5542491| 1.000000|   1.000000|     0.5403528|   0.5938001|

4 Exploring the clusters

In this section, we look at the clustering results, with a particular emphasis on cluster visualization and scoring measures.

4.1 Displaying Clusters inside the unipartite network

The plotCluster() function makes an interactive network figure that shows nodes that belong to the same cluster in the same color and nodes that belong to different clusters in separate colors.

plotCluster(graph=cls2$graph,cluster = cls2$louvain)

NB: To keep the size of vignette small enough, we do not output the interactive figure here. Instead, we show a screenshot.

Interactive plot for function plotCluster()

4.2 Displaying Clusters’ relationship inside the bipartite network

The Sankey diagram is used to depict the connections between clusters within each part of the bipartite network. The display is also interactive, and by grouping nodes within each cluster as “summary” nodes, the visualClusterInBipartite function emphasizes how clusters of each part are connected together.

visualClusterInBipartite(
  data = beatAML_data,
  community_left = cls1$leading_eigen,
  community_right = cls2$fast_greedy,
  name_left = 'patient_id',
  name_right = 'inhibitor')

NB: To keep the size of vignette small enough, we do not output the interactive figure here. Instead, we show a screenshot.

Interactive plot for function visualClusterInBipartite()

scoreCluster(community = cls2$infomap,
             graph = cls2$graph,
             dist_mat = cls2$distance_matrix)

validateCluster(community = cls$leading_eigen,
                extra_feature = external_feature,
                dist_mat = cls$distance_matrix)
#> Warning in fpc::cluster.stats(d = dist_mat, clustering = community$membership, :
#> alt.clustering renumbered because maximum != number of clusters
#> $corrected.rand
#> [1] 0.04301166
#> 
#> $jaccard_similarity
#> [1] 0.4409188

5 Edge prediction

imputations <- predictEdge(inc_mat = beatAML_incidence_matrix,
                                  method = c('svd','median','als','CA'))
# show the result format
summary(imputations)
#>        Length Class  Mode   
#> median 64416  -none- numeric
#> svd    64416  -none- numeric
#> als    64416  -none- numeric
#> CA     64416  -none- numeric

validateEdgePrediction(imputation = imputations,
                   refer_community = cls1$fast_greedy,
                   clustering_args = cls1$clustering_args)
#> 
#> 
#> |       | Jaccard_similarity| Dice_similarity_coefficient| Rand_index| Minkowski (inversed)| Fowlkes_Mallows_index|
#> |:------|------------------:|---------------------------:|----------:|--------------------:|---------------------:|
#> |median |          0.7476353|                   0.8555964|  0.8628983|             1.870228|             0.8556407|
#> |svd    |          0.7224792|                   0.8388829|  0.8458376|             1.763708|             0.8388853|
#> |als    |          0.7476353|                   0.8555964|  0.8628983|             1.870228|             0.8556407|
#> |CA     |          0.6935897|                   0.8190765|  0.8280576|             1.670030|             0.8191111|

#>   imputation_method Jaccard_similarity Dice_similarity_coefficient Rand_index
#> 1            median          0.7476353                   0.8555964  0.8628983
#> 2               svd          0.7224792                   0.8388829  0.8458376
#> 3               als          0.7476353                   0.8555964  0.8628983
#> 4                CA          0.6935897                   0.8190765  0.8280576
#>   Minkowski (inversed) Fowlkes_Mallows_index
#> 1             1.870228             0.8556407
#> 2             1.763708             0.8388853
#> 3             1.870228             0.8556407
#> 4             1.670030             0.8191111