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Clustering through the epigraph and hypograph indices

The EHyClus function

One of the main utilities of the epigraph and the hypograph index is to be able to measure ``extremality’’ of a curve with respect to a bunch of curves and to provide and ordination of the curves from top to bottom or vice-versa.

EHyClus is a methodology for clustering functional data which is based in four steps:

Smooth the functional dataset.
Apply the indices to data, first and second derivatives.
Apply classical multivariate clustering technique.
Obtain a clustering partition in \(k\) groups (\(k\) fixed in advanced).

In ehymet, the function that allows us to perform this process is EHyClus.

EHyClus function is designed for clustering functional datasets. The input data can be a one-dimensional dataset, \(n \times p\) matrix, or a multidimensional one, \(n \times p \times q\) array. The function transforms the initial dataset by first smoothing the data and then applying different indices analyzed on the first vignette such as the epigraph, hypograph, and their modified versions, and then applies various clustering algorithms to this dataset. Indices for one or multiple dimension are applied depending on the size of the input data.

It supports multiple clustering methods: hierarchical clustering (“hierarch”), k-means (“kmeans”), kernel k-means (“kkmeans”), and spectral clustering (“spc”). Also, it allows customization of hierarchical clustering methods, distance metrics, and kernels using parameters like l_dist_hierarch.

For smoothing, it uses B-splines with a specified or automatically selected number of basis functions.

To check the quality of the results, if true labels are provided, the function can validate the clustering results and compute performance metrics such as purity, F-measure, and Rand Index (RI). Also, it records the time taken for each clustering method, in case you want to measure it in case of wanting to know the trade-off between results and execution time

Parameters

All the parameters and its functionality can be found on the function documentation, but arguably the most important ones are:

curves: The dataset containing the curves to be clustered.
vars_combinations: List that determines the combination of variables to use. If not provided, generic combinations of variables will be used.
clustering_methods: A vector specifying which clustering methods to apply.
n_clusters Number of clusters to generate.
k: Number of basis functions for B-splines. Not to be confused with n_clusters.
bs: Type of penalized smoothing basis to use.
true_labels: If provided, evaluation metrics are provided along the result.

Example of usage and validation

In this subsection, we are going to display how to obtain the results of the EHyClus function and validate them (if true_labels are present). First, we are going to generate multidimensional data using sim_model_ex2:

n <- 50
curves <- sim_model_ex2(n = n, i_sim = 4)

And, as we are generating \(n = 50\) curves per group and there are two groups, the true labels are:

true_labels <- c(rep(1, n), rep(2, n))

We can run the algorithm with or without the true labels. For the combinations of variables, we will use “d2dtaMEI” with “d2dtaMHI”.

res <- EHyClus(curves, vars_combinations = list(c("d2dtaMEI", "d2dtaMHI")))
res_with_labels <- EHyClus(curves,
  vars_combinations = list(c("d2dtaMEI", "d2dtaMHI")),
  true_labels = true_labels
)

For the result without labels, it can be seen that we only have the generated clusters:

names(res)
#> [1] "cluster"

Whereas the one we generated with labels has both cluster and metrics:

names(res_with_labels)
#> [1] "cluster" "metrics"

Let’s explore both components of the latter. We’re going to start with cluster.

#>                                               levelName
#> 1  cluster                                             
#> 2   ¦--hierarch                                        
#> 3   ¦   ¦--hierarch_single_euclidean_d2dtaMEId2dtaMHI  
#> 4   ¦   ¦--hierarch_complete_euclidean_d2dtaMEId2dtaMHI
#> 5   ¦   ¦--hierarch_average_euclidean_d2dtaMEId2dtaMHI 
#> 6   ¦   ¦--hierarch_centroid_euclidean_d2dtaMEId2dtaMHI
#> 7   ¦   ¦--hierarch_ward.D2_euclidean_d2dtaMEId2dtaMHI 
#> 8   ¦   ¦--hierarch_single_manhattan_d2dtaMEId2dtaMHI  
#> 9   ¦   ¦--hierarch_complete_manhattan_d2dtaMEId2dtaMHI
#> 10  ¦   ¦--hierarch_average_manhattan_d2dtaMEId2dtaMHI 
#> 11  ¦   ¦--hierarch_centroid_manhattan_d2dtaMEId2dtaMHI
#> 12  ¦   °--hierarch_ward.D2_manhattan_d2dtaMEId2dtaMHI 
#> 13  ¦--kmeans                                          
#> 14  ¦   ¦--kmeans_euclidean_d2dtaMEId2dtaMHI           
#> 15  ¦   °--kmeans_mahalanobis_d2dtaMEId2dtaMHI         
#> 16  ¦--kkmeans                                         
#> 17  ¦   ¦--kkmeans_rbfdot_d2dtaMEId2dtaMHI             
#> 18  ¦   °--kkmeans_polydot_d2dtaMEId2dtaMHI            
#> 19  °--spc                                             
#> 20      ¦--spc_rbfdot_d2dtaMEId2dtaMHI                 
#> 21      °--spc_polydot_d2dtaMEId2dtaMHI

In the suffix of the names we can see the combination of variables used, that is “d2dtaMEI” with “d2dtaMHI”. We can also see the different parameters used. For example, for the kmeans it is easy to see that it has been performed with both the Euclidean and the Mahalanobis distances.

Looking in particular at some of the elements, we see that it contains the following:

str(res_with_labels$cluster$hierarch$hierarch_ward.D2_euclidean_d2dtaMEId2dtaMHI)
#> List of 3
#>  $ cluster: int [1:100] 1 1 1 1 1 1 1 1 1 1 ...
#>  $ valid  : 'table' num [1:3(1d)] 0.93 0.868 0.869
#>   ..- attr(*, "dimnames")=List of 1
#>   .. ..$ : chr [1:3] "Purity" "Fmeasure" "RI"
#>  $ time   : num 0.000973

On the one hand we have cluster which is the vector that assigns each of the curves to a cluster. Then we have valid, which is the validation data. And finally time, which is the time that this particular method has taken to be executed. Let’s take a look at valid:

head(res_with_labels$cluster$hierarch$hierarch_ward.D2_euclidean_d2dtaMEId2dtaMHI$valid)
#>   Purity Fmeasure       RI 
#>   0.9300   0.8678   0.8685

It gives us 3 metrics: Purity, F-measure and the Rand Index (RI). However, we can obtain this information in another way. Going back to the second element of res_with_labels: metrics give us a summary of all metrics:

head(res_with_labels$metrics, 3)
#>                                             Purity Fmeasure     RI         Time
#> kmeans_euclidean_d2dtaMEId2dtaMHI             1.00   1.0000 1.0000 0.0084872246
#> kmeans_mahalanobis_d2dtaMEId2dtaMHI           1.00   1.0000 1.0000 0.0089440346
#> hierarch_ward.D2_euclidean_d2dtaMEId2dtaMHI   0.93   0.8678 0.8685 0.0009729862

It gives us the Purity, F-measure, RI and Time for every clustering method with every combination of parameters that has been executed. We can search for “hierarch_single_euclidean_d2dtaMEId2dtaMHI” and see that it yields the same results as seen previously:

res_with_labels$metrics["hierarch_single_euclidean_d2dtaMEId2dtaMHI", ]
#>                                            Purity Fmeasure     RI        Time
#> hierarch_single_euclidean_d2dtaMEId2dtaMHI   0.52   0.6535 0.4958 0.001051903

But now imagine that we executed EHyClus without giving the true_labels parameter but we want to compute the metrics. For that purpose, we can use the clustering_validation function, using the true labels as the first parameter and the ones generated by the clustering method as the second one:

clustering_validation(true_labels, res$cluster$hierarch$hierarch_single_euclidean_d2dtaMEId2dtaMHI$cluster)
#>   Purity Fmeasure       RI 
#>   0.5200   0.6535   0.4958

Note that we are only taking a look at some of the result. Let’s see if some of them have given us better metrics. Results are sorted based on the RI:

head(res_with_labels$metrics, 5)
#>                                             Purity Fmeasure     RI         Time
#> kmeans_euclidean_d2dtaMEId2dtaMHI             1.00   1.0000 1.0000 0.0084872246
#> kmeans_mahalanobis_d2dtaMEId2dtaMHI           1.00   1.0000 1.0000 0.0089440346
#> hierarch_ward.D2_euclidean_d2dtaMEId2dtaMHI   0.93   0.8678 0.8685 0.0009729862
#> hierarch_ward.D2_manhattan_d2dtaMEId2dtaMHI   0.93   0.8685 0.8685 0.0010530949
#> kkmeans_rbfdot_d2dtaMEId2dtaMHI               0.77   0.6684 0.6422 0.1124079227

Indeed, we can see that some methods have given us excellent results, performing an almost perfect clustering.

In addition, if we only want to obtain the results for the best clustering method, we can use the only_best parameter. Note that this parameter only works when true_labels is provided.

res_only_best <- EHyClus(curves,
  vars_combinations = list(c("d2dtaMEI", "d2dtaMHI")),
  true_labels = true_labels,
  only_best = TRUE
)

If we inspect the object, we see that in fact it only contains the results of the clustering method that has obtained the best results.

res_only_best$cluster
#> $kmeans_euclidean_d2dtaMEId2dtaMHI
#> $kmeans_euclidean_d2dtaMEId2dtaMHI$cluster
#>   [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1
#>  [61] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> 
#> $kmeans_euclidean_d2dtaMEId2dtaMHI$valid
#>   Purity Fmeasure       RI 
#>        1        1        1 
#> 
#> $kmeans_euclidean_d2dtaMEId2dtaMHI$time
#> [1] 0.011976

res_only_best$metrics
#>                                   Purity Fmeasure RI     Time
#> kmeans_euclidean_d2dtaMEId2dtaMHI      1        1  1 0.011976

This can be useful if you want to use the function as if it were a typical clustering method.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.