Binarisation

The fuzzy CoPaM matrix includes membership values for all of the considered genes (data objects) in each of the K clusters. A membership value of unity indicates full belongingness of the given gene to the given cluster, a zero membership indicates absolutely no belongingness, and a fractional membership value indicates respective partial belongingness. If all of the R individual partitions have consensually assigned a given gene to the same cluster, the membership of this gene in that cluster will be unity while being nil in all of the other clusters. However, if the individual partitions have disagreement in assignment of that gene, its membership value is distributed over all of the clusters in which some partitions included it. Indeed, the membership of the gene in any of these partitions is set to be proportionate with the number of individual partitions which assigned it to it.

The next step is to binarise the membership values of the genes in the fuzzy CoPaM. It is clear that a given gene should be assigned to the cluster to which it has been assigned consensually by all partitions. However, in the case of disagreement, should this gene be simultaneously assigned to all of the clusters to which some partitions assign it, or should it be left unassigned from all of the clusters given the dispute? Below are six proposed binarisation techniques addressing this issue in different ways.

3.2.4.1. Maximum value binarisation (MVB)

MVB assigns the gene exclusively to the cluster in which it has its largest membership value, and therefore it generates complementary clusters. Let the resulting binary CoPaM be B* _{with K rows and N columns, where 𝑏𝑏}

𝑘𝑘,𝑖𝑖∗ ∈ {0, 1} is an element in this matrix

representing the binary membership of the ith _{gene in the k}th _{cluster. Similarly, the} corresponding fuzzy CoPaM is U* _{with the elements 𝑢𝑢}

𝑘𝑘,𝑖𝑖

∗ _{∈ [0, 1]. Given that, the MVB} technique can be expressed as:

𝑏𝑏_{𝑘𝑘,𝑖𝑖}∗ _{= �}1, 𝑢𝑢𝑘𝑘,𝑖𝑖∗ = max_{1≤𝑗𝑗≤𝐾𝐾}𝑢𝑢𝑗𝑗,𝑖𝑖∗

0, 𝑙𝑙𝑜𝑜ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 (3.7)

3.2.4.2. Top binarisation (TB)

The TB technique moves from the MVB technique towards producing wider clusters. This is done by assigning the given gene to multiple clusters simultaneously if its membership values in them are not farer than the value of the tuning parameter δ below its maximum membership value. The TB technique is expressed as:

𝑏𝑏𝑘𝑘,𝑖𝑖∗ = �

1, 𝑢𝑢𝑘𝑘,𝑖𝑖∗ ≥ max_{1≤𝑗𝑗≤𝐾𝐾}𝑢𝑢𝑗𝑗,𝑖𝑖∗ − 𝛿𝛿

0, 𝑙𝑙𝑜𝑜ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 (3.8)

3.2.4.3. Difference threshold binarisation (DTB)

In contrast to TB, and in a symmetric manner, the DTB technique moves from the MVB technique towards producing tighter clusters. This is performed by assigning a gene to the cluster in which it has its maximum membership value only if this value is far from the closest competitive cluster at least by the value of the tuning parameter δ; it is not assigned to any of the clusters otherwise. The DTB technique is expressed as:

𝑏𝑏𝑘𝑘,𝑖𝑖∗ = � 1, 𝑢𝑢𝑘𝑘,𝑖𝑖∗ ≥ max_{1≤𝑗𝑗≤𝐾𝐾,} 𝑗𝑗≠𝑘𝑘 𝑢𝑢𝑗𝑗,𝑖𝑖∗ + 𝛿𝛿 0, 𝑙𝑙𝑜𝑜ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 (3.9) We group the aforementioned three techniques in a track of binarisation and we name it as the TB-MVB-DTB track (Figure 3.4). When δ is equal to zero in TB or DTB, they become identical to the MVB technique. When δ increases, TB or DTB start widening or tightening the clusters, respectively. The maximum value of δ is unity. When this value is reached, the TB technique reaches the extreme case of wide clusters in which each one of the clusters includes all of the genes. Also, at the δ value of unity, the DTB produces the tightest clusters in which a gene is assigned to a cluster only if its fuzzy membership value is equal to unity in that cluster and is equal to zero in all of the other clusters, i.e. if all of the R individual partitions have consensually assigned that gene to that cluster. Although DTB may generate many empty clusters at δ = 1.0, this result would not be trivial if some of the clusters still preserved some genes up to this tightest level, as opposed to the TB technique’s results at such δ value.

Figure 3.4. Binarisation tracks

The (left) TB-MVB-DTB track and the (right) UB-VTB-IB track can produce clusters which range from very wide to very tight. Also, the MVB technique of the TB-MVB-DTB track can also provide complementary clusters.

3.2.4.4. Union binarisation (UB)

UB assigns each gene to all of the clusters in which it has non-zero fuzzy membership values, i.e. to all of the clusters in which at least one of the R individual partitions has assigned it. This generates wide and overlapping clusters. UB is expressed as:

𝑏𝑏𝑘𝑘,𝑖𝑖∗ = �1, 𝑢𝑢𝑘𝑘,𝑖𝑖

∗ _{> 0}

0, 𝑙𝑙𝑜𝑜ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 (3.10)

3.2.4.5. Intersection binarisation (IB)

Contrary to the UB technique, IB assigns a gene to a cluster only if all of the R individual partitions have consensually assigned that gene to it, i.e. if its fuzzy membership value in it is unity while being zero elsewhere. This technique generates the tightest and most focused clusters, and is equivalent to the tightest clusters generated by the TB-MVB-DTB track, namely by the DTB technique at δ = 1.0. The IB technique is expressed as:

𝑏𝑏𝑘𝑘,𝑖𝑖∗ = �1, 𝑢𝑢𝑘𝑘,𝑖𝑖

∗ _{= 1.0}

0, 𝑙𝑙𝑜𝑜ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 (3.11)

3.2.4.6. Value threshold binarisation (VTB)

VTB assigns a gene to a cluster if its membership in it is larger than or equal to the value of the tuning parameter α. The VTB technique is expressed as:

𝑏𝑏𝑘𝑘,𝑖𝑖∗ = �1, 𝑢𝑢𝑘𝑘,𝑖𝑖

∗ _{≥ 𝛼𝛼}

0, 𝑙𝑙𝑜𝑜ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒 (3.12)

We group the latter three techniques into a second track of binarisation, namely the UB- VTB-IB track (Figure 3.4). When α is equal to zero, the VTB assigns each gene to all of the clusters, which is a trivial and useless result. At α = ε, where ε is an arbitrarily small real positive number, the VTB technique becomes identical to the UB technique, and at α = 1.0,

it becomes identical to the IB technique. As the value of α increases from ε to unity, the clusters are tightened.

The main semantic difference between the two tracks is that the TB-MVB-DTB track considers comparative criteria based on the competition amongst the clusters over the genes. On the other hand, the UB-VTB-IB track considers the absolute membership values of genes in individual clusters. Nonetheless, the fact that fuzzy membership values are normalised such that their sum for a single gene over all of the clusters is unity implies that those values implicitly consider certain levels of competition between the clusters, even when considered by the UB-VTB-IB track. However, the TB-MVB-DTB track is more explicit in basing gene-cluster assignments on such competitions.

3.3. UNCLES

The objective of the Bi-CoPaM method, while using tightening binarisation techniques, can be summarised as: it aims at identifying the subsets of genes (or any other types of objects) which are consistently co-expressed (highly correlated in profiles) over all of the given datasets and when analysed by all of the adopted clustering methods and setups. Indeed, binarisation parameters control how much tolerance is accepted.

However, other research questions can be answered by other ways of unifying individual clustering results. We therefore propose a more general paradigm of multiple- dataset mining which we call the unification of clustering results from multiple datasets

using external specifications (UNCLES) (Abu-Jamous, et al., 2015c). Bi-CoPaM serves as

a special case of the UNCLES method as it unifies clustering results from multiple datasets while considering consistency in co-expression over all/most of the datasets as external specifications. We name this type of external specifications as “type A”.

Here we propose another type of external specifications, labelled “type B”, which aims at identifying the subsets of genes which are consistently co-expressed in a subset of datasets (S+_{) while being poorly consistently co-expressed in another subset of datasets (S}-_).

To apply UNCLES type B, the type A (Bi-CoPaM) is applied to each of the two subsets of datasets S+ _{and S}- _{separately while adopting DTB binarisation with the δ values of δ}+ _and δ- _{respectively. After that, the genes that are included in the results of processing the S}+ datasets and not included in the results of processing S- _{datasets, indeed after binarisation,} are included in the final result. Therefore, the UNCLES type B method utilises a pair of parameters (δ+_{, δ}-_{) in order to achieve its results. The parameter δ}+ _{controls how well co-} expressed the genes should be in the S+ _{datasets to be included in the final result, while the}

parameter δ- _{controls how well co-expressed the genes should be in the S}- _{datasets to be} excluded from the final result. Note that at the pair (δ+_{, 0) empty clusters are generated} because at δ- _{= 0 all of the genes will be excluded from the final result.}