parallelpam
In parallelpam: Parallel Partitioning-Around-Medoids (PAM) for Big Sets of Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Load package

library(parallelpam)

Purpose

The parallelpam package (@R-parallelpam) is meant as a way to apply the PAM algorithm to quite (or very) big sets of data, such as the results of single-cell sequencing, but can be generally used for any type of data, as long as a distance/dissimilarity matrix can be calculated.

Differently to other packages, its main strength is its ability to perform clustering based on Partitioning Around Medoids (PAM) using a large number of items and doing it in parallel. Memory and speed limitations are reduced by extensive use of C++ programming which allows use of alternative data types (for instance, float vs. double to represent distance/dissimilarity), intermediate disk-storage to avoid memory copy operations whenever possible and use of threads.

Both phases of PAM (initialization with BUILD and optimization) have been parallelized so it you have a multi-core machine, many threads will be launched with a great acceleration of the calculations. This is done automatically, even you are allowed to choose the number of threads yourself for comparison purposes or to allow your machine to execute other things simultaneously.

Also, calculation of the matrix of distances/dissimilarities from the initial data and calculation of silhouette of the resulting clusterization are available, too, and are calculated in parallel in a multi-core machine.

The data are stored in memory reading them from binary files created with the package jmatrix (@R-jmatrix). To be familiar with them please read the vignette jmatrixpp which is included with this package.

WARNING: you must NOT load jmatrix explicitly. Indeed, you do not need even to install it. All its functions have been included here, too, so calling to library(parallelpam) is enough.`

Workflow

Debug messages

First of all, the package can show quite informative (but sometimes verbose) messages in the console. To turn on/off such messages you can use

# Initially, state of debug is FALSE. Turn it on exclusively for the
# parallelpam part with
ParallelpamSetDebug(TRUE)
# There is another parameter, debjmat, to turn on messages about
# binary matrix creation/manipulation. By default is FALSE but turn it on
# if you like with
# ParallelpamSetDebug(TRUE,debjmat=TRUE)

Data load/storage

The first step is to load raw data coming from external sources like the main formats used in single cell experiments which should have been stored as a binary matrix file in jmatrix format. Since this is a separate package, and for purposes of illustration, we will create an artificial matrix for a small problem that fits in R memory with 5000 vectors with 500 dimensions each. Then we will calculate the dissimilarity matrix and finally we will apply the PAM algorithm to it.

# Create the matrix with row names V1 to V5000 and column names d1 to d500
nvec<-5000
ndim<-500
P<-matrix(runif(nvec*ndim),nrow=nvec)
rownames(P)<-paste0("V",1:nvec)
colnames(P)<-paste0("d",1:ndim)
# Write it to disk as a binary file in jmatrix format. Please,
# see vignette jmatrixpp.
JWriteBin(P,"datatest.bin",dtype="float",dmtype="full",
          comment="Synthetic problem data to test PAM")

For your real problem, the input format can be a .csv file. See function CsvToJMat in package scellpam (@R-scellpam).

To know details about the generated files do

JMatInfo("datatest.bin")

Calculating the distance/dissimilarity matrix

This is the most computationally intensive part of the process (particularly, for samples with a high number of points and/or high dimensionality) and therefore has been programmed in parallel, taking advantage of the multiple cores of the machine, if available. The funcion is called CalcAndWriteDissimilarityMatrix. Its input and output files (first and second parameters) are of course compulsory. Input file can be a sparse of full binary jmatrix (but obviously, not a symmetric matrix).

WARNING: notice that the vectors to calculate dissimilarities amongst them MUST be stored BY ROWS. This is due to efficiency reasons.

Output dissimilarity matrix will always be a binary symmetric (obviously square) matrix with a number of rows (and columns) equal to the number of rows (in this case, vectors) of the input file. The type of distance/dissimilarity can be L1 (Manhattan distance), L2 (Euclidean distance) or Pearson (Pearson dissimilarity coefficient). The resulting matrix stores only names for the rows, which are the names of the vectors stored as rows in file datatest.bin. If the number of vectors is $N$, only $N(N+1)/2$ dissimilarity values are really stored.

A note on the number of threads, valid also for other algorithms that will be explained later:

Possible values for the number of threads are:

-1 (or any negative number) to indicate you do not want to use threads (strictly sequential computation).
0 to allow the program to choose the number of threads according to the problem size and the number of available cores.
Any positive number to force the use of such number of threads.

Choosing explicitly a number of threads bigger than the number of available cores is allowed, but discouraged and the program emits a warning about it.

With respect to option 0, for small problems (in this case, less than 1000 vectors) the function makes the choice of not using threads, since the overhead of opening and waiting termination is not worth. For bigger problems the number of chosen threads is the number of available cores, or twice this number if the processor is capable of hyperthreading. Nevertheless, this choice may not be the best, depending on your machine, possibly due (I guess) to the memory access conflicts created by the need of keep processor cache coherence. You may have to do some trials with your data in your machine.

Now, let us try it with this small dataset.

CalcAndWriteDissimilarityMatrix("datatest.bin","datatestL2.bin",
                                distype="L2",restype="float",
                                comment="L2 distance for vectors in
 jmatrix file datatest.bin",nthreads=0)

The resulting matrix is stored as a binary symmetric matrix of float values, as we can check.

JMatInfo("datatestL2.bin")

Applying PAM

The last step is to take the previously calculated matrix and apply the Partitioning Around Medoids classifier. Function is ApplyPAM. First parameter (name of the file containing the dissimilarity matrix in jmatrix format) and second parameter (k, number of medoids) are compulsory. The names and default values for the rest of parameters are as in this example.

L=ApplyPAM("datatestL2.bin",k=5,init_method="BUILD",max_iter=1000,
           nthreads=0)

Parameters init_method (and another optional parameter, initial_med) deserve special comment. The first is the method to initialize the medoids. Its possible values are BUILD, LAB and PREV. The rest of the algorithm make medoid swapping between the points of the initial set made with BUILD or LAB and the rest of points until no swap can reduce the objective function, which is the sum of distances of each point to its closest medoid. But this may fall (and indeed falls) in local minima. If you initialize with BUILD or LAB the optional parameter initial_med cannot be used.

The initialization methods BUILD and LAB are described in the paper from Schubert at al. (@Schubert2019). BUILD is deterministic. LAB uses a sample of the total points to initialize. Obviously, you can run LAB to get different initializations and compare the results.

The returned object is a list with two fields: med and clasif. This will be explained later.

From now on, typical calls to obtain only the initial medoids would be

Lbuild=ApplyPAM("datatestL2.bin",k=5,init_method="BUILD",max_iter=0)
Llab1=ApplyPAM("datatestL2.bin",k=5,init_method="LAB",max_iter=0)
Llab2=ApplyPAM("datatestL2.bin",k=5,init_method="LAB",max_iter=0)

As it can be seen, to get and compare different initializations you must set the parameter max_iter to the value 0. In this case no iterations of objective function reduction are performed, and the returned object contains the initial medoids and the classification induced by them. Notice that even looking equal, the results of the latter two calls are different since LAB initializes with a random component (the sample to choose initial medoids is chosen randomly).

You can check that the medoids, stored in Llab1$med and Llab2$med (see more on this below) are in general different.

Now, these results can be used to initialize PAM if you find that any of them contains a specially good set of medoids. This is the role of method PREV that we have mentioned before. A typical call would be

Llab2Final=ApplyPAM("datatestL2.bin",k=5,init_method="PREV",
                    initial_med=Llab2$med)

The initial set of medoids is taken from the object returned by the former calls.

With respect to that object, as we said it is a list with two vectors. The first one, L$med, has as many components as requested medoids and the second, L$clasif, has as many components as instances.

Medoids are expressed in L$med by its number in the array of vectors (row number in the dissimilarity matrix) starting at 1 (R convention).

L$clasif contains the number of the medoid (i.e.: the cluster) to which each instance has been assigned, according to their order in L$med (also from 1).

This means that if L$clasif[p] is m, the point p belongs to the class grouped around medoid L$med[m]. Let us see it.

# Which are the indexes of the points chosen as medoids?
L$med
#
# In which class has point 147 been classified?
L$clasif[147]
#
# And which is the index (row in the dissimilarity matrix)
# of the medoid closest to point 147?
L$med[L$clasif[147]]

In this way, values in L$clasif are between 1 and the number of medoids, as we can see:

min(L$clasif)
max(L$clasif)

They can be used as factors.

Calculating silhouette

It is interesting to filter points based on the degree in which they belong to a cluster. Indeed, cluster refinement can be done getting rid of points far away from any cluster center, or which are at a similar distance of two or more of them.

This is characterized by the silhouette of each point. Three functions deal with this: CalculateSilhouette, FilterBySilhouetteQuantile and FilterBySilhouetteThreshold.

S=CalculateSilhouette(Llab2$clasif,"datatestL2.bin",nthreads=0)

The parameters to function CalculateSilhouette are the array of class membership, as returned by ApplyPAM in its clasif field, and the file with the matrix of dissimilarities.

A parallel implementation has been programmed, being nthreads as explained before.

Silhouette is a number in $[-1,1]$; the higher its value, the most centered a point is in its cluster.

The returned object S is a numeric vector with the value of the silhouette for each point, which will be a named vector if the classification vector was named.

This vector can be converted to an object of the class cluster:silhouette with the function NumSilToClusterSil (which needs the vector of classifications, too). This is done so that, if you load the package cluster (@R-cluster), plot will generate the kind of silhouette plots included in such package.

If the package cluster is installed you can try to execute this: (Sorry, we can't try ourselves since we don't know if cluster is installed in your system and the CRAN check does not allow the use of installed.packages to test it)

Sclus <- NumSilToClusterSil(Llab2$clasif,S)
library(cluster)
plot(Sclus)

Probably the plot does not look very nice with this random data which yields a very bad clustering (since they are not, by its own nature, organized in clusters) but with real data you should see significant things (see package scellpam (@R-scellpam)).

Once the silhouette is calculated we can filter it by quantile or by threshold. All points under this quantile or threshold will be discarded, except if they are medoids. Parameters are:

The silhouette, as returned by CalculateSilhouette.
The list of medoids/clasif, as returned by ApplyPAM.
The file with matrix of counts for the whole set of individuals, which was our first input.
The file that will contain the matrix of counts of the remaining individuals.
The file with the dissimilarity matrix for the whole set, as calculated by CalcAndWriteDissimilarityMatrix.
The file that will contain the dissimilarity for the remaining individuals.
And (depending on the function used) the quantile in $[0,1]$ or the silhouette threshold in $[-1,1]$.

As an example,

Lfilt=FilterBySilhouetteQuantile(S,Llab2,"datatest.bin",
                                 "datatestFilt.bin","datatestL2.bin",
                                 "datatestL2Filt.bin",0.2)

If the original matrix contained row and column names, the column names are copied and the row names are transported for those rows that remain. The same happens with respect to rows of the dissimilarity matrix.

Notice that the new dissimilarity matrix could have been calculated from the matrix of filtered counts with CalcAndWriteDissimilarityMatrix but creating it here, simply getting rid of the filtered rows and columns is much faster.

Also, if a medoid is below the silhouette quantile, it will not be filtered out, but a warning message will be shown, since this is a strange situation that may indicate that some of your clusters are not real but artifacts due to a few outliers that are close to each other.

But remember that this was the result of the first step of the PAM algorithm, so probably you will want to make them iterate.

Lfinal=ApplyPAM("datatestL2Filt.bin",k=length(Lfilt$med),
                init_method="PREV",initial_med=Lfilt$med)

Of course, we might have used simply 5 as number of medoids, k, since this does not change by filtering, but this is to emphasize the fact that ApplyPAM with method PREV requires both parameters to be consistent.

Comparison with other PAM implementations

The user might want to compare this PAM implementation with others provided for instance by packages cluster (@R-cluster) or ClusterR (@R-ClusterR). In cluster the input is either the data matrix (so the distance matrix is calculated inside the pam function) or directly the distance matrix but as a R vector with the lower-diagonal part of the symmetric matrix ordered by columns (i.e.: column 1 from M(2,1) to M(n,1), followed by column 2 from M(3,2) to M(n,2) and so on, up to M(n,n-1). This is a vector of $n(n-1)/2$ components. To facilitate such comparison the function GetSubdiag is provided which takes as input the jmatrix file with the distance matrix and returns the vector to be passed to pam in the aforementioned packages.

d = GetSubdiag("datatestL2.bin")

Then, explicit comparison could be done with:

(Sorry, we can't try ourselves since we don't know if cluster is installed in your system and the CRAN check does not allow the use of installed.packages to test it)

library(cluster)
clusterpam = pam(d,diss=TRUE,k=5)
print(sort(clusterpam$id.med))
print(sort(L$med))

Similarly, you can check against the ClusterR package. In this package you need the complete dissimilarity matrix to be passed so we have to get it:

# Be patient, this may take some time...
Dm = GetJManyRows("datatestL2.bin",seq(1:nvec))

and then

library(ClusterR)
ClusterRpam = Cluster_Medoids(Dm,clusters=5)
print(sort(ClusterRpam$medoid_indices))
print(sort(L$med))

In all cases we tried with this simple (but random) example results were the same. In other cases with a large number of points some medoids were different in the different implementations but the value of the function to minimize (sum of distances) was always the same, indicating that they were equivalent minima. You can test this with function GetTD as follows:

TDparallelpam = GetTD(L,"datatestL2.bin")

# This is to adapt cluster package output format to ours, since this is what our GetTD function expects...
Lcl = list()
Lcl$med = clusterpam$id.med
Lcl$clasif = clusterpam$clustering
TDcluster = GetTD(Lcl,"datatestL2.bin")

# The same with ClusterR package:
LclR = list()
LclR$med = ClusterRpam$medoid_indices
LclR$clasif = ClusterRpam$clusters
TDClusterR = GetTD(LclR,"datatestL2.bin")