clustering.sc.dp: Optimal Clustering with Sequential Constraint by Using Dynamic Programming
Tibor Szkaliczki
eLearning Department, Institute for Computer Science and Control, Hungarian Academy of Sciences2016-05-01
Abstract
The general clustering algorithms do not guarantee optimality because of the hardness of the problem. Polynomial-time methods can find the clustering corresponding to the exact optimum only in special cases. For example, the dynamic programming algorithm can solve the one-dimensional clustering problem, i.e., when the items to be clustered can be characterised by only one scalar number. Optimal one-dimensional clustering is provided by package Ckmeans.1d.dp in R. The paper shows a possible generalisation of the method implemented in this package to multidimensional data: the dynamic programming method can be applied to find the optimum clustering of vectors when only subsequent items may form a cluster. Sequential data are common in various fields including telecommunication, bioinformatics, marketing, transportation etc. The proposed algorithm can determine the optima for a range of cluster numbers in order to support the case when the number of clusters is not known in advance.Introduction
Clustering plays a key role in various areas including data mining, character recognition, information retrieval, machine learning applied in diverse fields such as marketing, medicine, engineering, computer science, etc. A clustering algorithm forms groups of similar items in a data set which is a crucial step in analysing complex data. Clustering can be formulated as an optimisation problem assigning items to clusters while minimising the distances among the cluster members. The normally used clustering algorithms do usually not find the optimal solution because the clustering problem is NP-complete in the general case. This paper introduces a package implementing an optimisation method for clustering in a special case when a sequential constraint should be met, i.e., when the items to be clustered are sorted and only subsequent items may form a cluster. This constraint is common when clustering data streams, e.g., audio and video streams, trajectories, motion tracks, click-streams etc. The good news is that the exact optimum can be found in polynomial time in this case.
The algorithm recommended for clustering sequential data is based on the dynamic programing approach developed by Wang and Song (2011). They gave a polynomial-time algorithm for one-dimensional clustering, i.e., when the items can be characterised by only one scalar number. Similarly to the heuristic \(k\)-means algorithm, it divides data into \(k\) groups and it minimises the within-cluster sum of squared distances (WCSS or \(\mathit{withinss}\) for short). The algorithm guarantees a solution minimising the optimisation goal. The source code of the algorithm is available in the R package Ckmeans.1d.dp (Song and Wang 2011). The generalisation of the algorithm to the multiple dimensional space has been open so far. We extended the dynamic programming approach from one-dimensional clustering to multidimensional clustering with sequential constraint (i.e., only subsequent elements of the input may form a cluster). The method finds the exact optimum in this case as well. We implemented the algorithm in the R package clustering.sc.dp (Szkaliczki and Song 2015).
Although the original algorithm has been developed to find the optimal solution with exactly \(k\) clusters it can determine the optimal value for all numbers of clusters less than or equal to \(k\) in a single run. For this reason, we implemented two variants of the algorithm. The first one finds the optimal solution for a specific \(k\) which can be used if the number of clusters is known in advance. The second variant returns the vector containing the minimal \(\mathit{withinss}\) for all cluster numbers less than or equal to \(k\). This extension of the algorithm is useful if the number of clusters is not known in advance which is a common case.
The remainder of this paper is organized as follows: In the next section, a brief overview of the related work is presented. Then the optimization problem is formally described and the developed optimization algorithm is introduced in detail. Some evaluation results are also presented and the usage of the implemented package is introduced. A brief summary concludes the paper.
The problem specification
Clustering methods divide a dataset \(X = \left\{\overline x_1,\overline x_2, \ldots,\overline x_n\right\}\) of \(d\)-dimensional vectors into a set \(\Pi = \left\{C_1, C_2,\ldots, C_k\right\}\) of disjoint clusters where \(n\) and \(k\) denote the number of items to be clustered and the number of clusters, respectively. Throughout the paper, vectors are distinguished from scalars by a bar over their symbol. In case of clustering with sequential constraint, the items are sorted and the clusters are formed only by subsequent items: \(C_j = \left\{\overline x_{b_j}, \overline x_{b_j+1},\ldots,\overline x_{b_j+n_j-1}\right\}\) where \(b_j\) and \(n_j\) denote the first item and the number of items in cluster \(C_j\), respectively. The optimisation goal is to minimise the within-cluster sum of squared distances (\(\mathit{withinss}\)) also called sum of squared error (SSE), sum of quadratic errors (SQE) or distortion which is a common measurement of quality in clustering. It is formally defined as follows:
\[\begin{equation} \label{eq:wcss} \mathit{withinss} = \sum_{j=1}^k \sum_{\overline x_i\in C_j}\left\|\overline x_i-\overline \mu_j\right\|^2, \end{equation} (\#eq:eqwcss) \]
where \(\left\|\overline x \right\|\) denotes the Euclidean norm of vector \(\overline x\) and \(\overline \mu_j\) is the cluster mean:
\[\overline \mu_j = \frac{1}{n_j} \sum_{\overline x_i\in C_j}\overline x_i.\]
Now, we can formulate our problem as follows:
Input:
Items to be clustered: \(X = \left\{\overline x_1, \overline x_2, \ldots,\overline x_n\right\}\),
Number of clusters: \(k\).
Output:
Optimal clustering: \(\Pi = \left\{C_1, C_2,\ldots, C_k\right\}\).
Minimise
within-cluster sum of squared distances (\(\mathit{withinss}\)): Eq. @ref(eq:eqwcss).
Subject to
sequential constraint: \[\left(\overline x_{i_1}\in C_j\right) \land \left(\overline x_{i_2}\in C_j\right) \land \left(i_1\le i_3\le i_2\right) \Rightarrow \left(\overline x_{i_3}\in C_j\right).\]
general clustering conditions:
each item is clustered:
\[\forall \overline x_i \exists C_j: \overline x_i\in C_j.\]
one cluster is assigned to each item
\[\left(\overline x_i\in C_{j_1}\right) \land \left(\overline x_i\in C_{j_2}\right)\Rightarrow \left(j_1 = j_2\right).\]
The dynamic programming algorithm
The recursive formula used in the dynamic programming formulation is based on the fact, that if clustering for the first \(i\) items in \(m\) clusters is optimal then after dropping the last cluster the resulting clustering is optimal with \(m-1\) clusters for the remaining items. Let \(D[i,m]\) denote the value of the minimal within-cluster sum of squared distances (\(\mathit{withinss}\)) of the clustering for the first \(i\) items by using \(m\) clusters. If \(j\) denotes the first item of the \(m\)th cluster then the optimality of \(D[i,m]\) implies the optimality of \(D[j-1,m-1]\) as well. \(D[n,k]\) gives the minimal \(\mathit{withinss}\) for clustering all items in \(k\) clusters where \(n\) denotes the total number of items.
The recursive formula applied in the dynamic programming approach is defined as follows (Wang and Song 2011):
\[D[i,m]=\min_{m\le j\le i} \left\{D[j-1,m-1]+d\left(\overline x_j,\ldots,\overline x_i\right)\right\}, 1\le i\le n, 1\le m\le k\]
where \(d\left(\overline x_j,\ldots\overline x_i\right)\) is the sum of squared Euclidean distances from \(\overline x_j,\ldots\overline x_i\) to their mean.
The optimal solution can be determined by dynamic programming in two steps. First, the recursive formula is used to find the minimal \(\mathit{withinss}\). Then backtracking finds the optimal clustering.
\(B[i,m]\) stores the index of the first item \(b_m\) of the last cluster in the partial solution belonging to \(D[i,m]\) which is used for backtracking the optimal solution after determining the minimal \(\mathit{withinss}\). We apply the dynamic programming method to solve optimal clustering for a range of cluster numbers and \(k\) denotes the maximum number of clusters in our algorithm (\(1\le k \le n\)). The steps of calculating \(D[i,m]\) can be implemented as follows:
for i := 0 to n
D[i, 0] := 0 // initialisations
for i := 1 to n
for m := 1 to min(i, k)
D[i, m] := MAX_DOUBLE;
for j := i downto m // calculating the recursive formula
if D[i, m] > D[j - 1, m - 1] + d(xj, ..., xi)
D[i, m] := D[j - 1, m - 1] + d(xj, ..., xi)
B[i, m] := j
Return D[n, m] for m = 1, ..., k \(D[n, m], m = 1,\ldots,k\) gives the minimal distances for different number of clusters. If the number of clusters is known in advance, it is enough to return \(D[n,k]\). Otherwise, \(D[n, m]\) for all \(m\le k\) can be returned for further processing by the user in order to select the proper number of clusters.
The algorithm finds the exact optimum in polynomial time. It runs \(O\left(n^2k\right)\) iterations in which \(D[i, m]\) is checked and, if necessary, updated. Each iteration can be performed in time proportional to the dimensions of the vectors (\(O\left(d\right)\)) independently from the number of items and clusters if \(d\left(\overline x_j, \ldots,\overline x_i\right)\) is computed progressively based on \(d\left(\overline x_{j+1}, \ldots,\overline x_i\right)\) and the average of the items. This can be done similarly to the one-dimensional case in the following way. Let \(\overline \mu_{j,i}\) denote the mean of the items with index between \(j\) and \(i\). If \(j = i\) \(d\left(\overline x_j, \ldots,\overline x_i\right) = 0, \overline \mu_{j,i} = \overline x_i\). For index \(j\) from \(i-1\) down to \(m\), the algorithm iteratively computes \[\begin{aligned} d\left(\overline x_j, \ldots,\overline x_i\right) &= d\left(\overline x_{j+1}, \ldots,\overline x_i\right)+\frac{ i-j}{ i-j+1}\left(\overline x_j-\overline \mu_{j,i-1}\right)^2\\ \overline \mu_{j,i} &= \frac{\overline x_j +\left(i-j\right)\overline \mu_{j+1,i}}{ i-j+1} \end{aligned}\] Using the above iterative computation, the overall running time is quadratic in the number of items and linear in the number of clusters and the dimensions of the vectors: \(O\left(n^2kd\right)\).
The optimal solution with cluster number \(m\) can be backtracked by the help of \(B[i,m]\) (Wang and Song 2011):
\[B[i,m] = \mathrm{argmin}_{m\le j\le i}\left\{D[j-1,m-1]+ d\left(\overline x_j,\ldots, \overline x_i \right)\right\}, 1\le i\le n, 1\le m\le k\]
\(B[n,m]\) is equal to the first item of the last cluster in clustering with \(m\) clusters. The last cluster contains items \(\overline x_{B[n,m]},\ldots,\overline x_n\). The further clusters can be determined by using backtracking as follows: if \(j\) is the first item of the \(l\)th cluster then the preceding cluster is formed by the items \(\overline x_{B[j,l-1]},\ldots,\overline x_{j-1}\). The steps of backtracking to the find optimal clustering using \(m\) number of clusters are as follows:
the mth cluster is B[n, m], ..., n
j := B[n, m]
for l := m to 2
the (l - 1)th cluster is B[j, l - 1], ..., j - 1
j = B[j, l - 1]Backtracking can be performed in linear time in the number of the clusters (\(O\left(k\right)\)).
We would like to mention how to combine the dynamic programming method with methods finding the proper number of clusters. The cluster numbers are typically determined by using a measure of validity (e.g., \(\mathit{withinss}\)) indicating the goodness of clustering. The “best” \(k\) is chosen based on the analysis of the values of the measure for each \(k\) within the range of the possible number of clusters. A huge variety of methods are available in the literature to determine the cluster numbers (Milligan and Cooper 1985; Dimitriadou, Dolnicar, and Weingessel 2002). Typically, they are applied on the output generated by hierarchical clustering methods. Although our dynamic programming approach does not belong to the hierarchical clustering similar methods can be used on the result of our algorithm for finding the proper number of clusters.
Backtracking can be performed only once if the number of clusters is known in advance or it can be selected by analysing \(\mathit{withinss}\) contained in the last column of matrix \(D\). Otherwise, backtracking should be executed for each possible cluster numbers for further analysis. The method can efficiently determine the optimal clustering for all numbers of clusters less than or equal to \(k\) essentially because the most time-consuming part is determining \(D\) and \(B\) which should be executed only once.
Implementation
We implemented this dynamic programming algorithm in C++. The implementation was built on source code from the R package Ckmeans.1d.dp and we created a new R package clustering.sc.dp. In the name of the package, sc and dp refer to sequential constraint and dynamic programming, respectively. The open-source approach made it possible to reuse the code from package Ckmeans.1d.dp but it was beneficial for the original package as well: we suggested a minor change in the code to speed up the code which was incorporated into Ckmeans.1d.dp (\(\geq\) version 3.3.0).
Evaluation
Optimality
If the sequential constraint is considered in clustering than the optimal \(\mathit{withinss}\) is usually larger than in the general case without the constraint since the constraint excludes many possible solutions. In order to compare our algorithm with general clustering methods such as the \(k\)-means method, we generated a dataset where the optima without and with sequential constraint are equal. For this purpose, we created a totally ordered vector set where one vector is simultaneously larger or smaller in each coordinate than another vector (\(\forall i,j \in \{1,2,\ldots,n\} \forall k,l \in \{1,2,\ldots,d\} x_{ik} < x_{jk} \implies x_{il} < x_{jl}\) where \(x_{yz}\) denote the \(z\)th coordinate of item \(x_y\)). We used a random walk as a totally ordered vector set where the steps between subsequent items were generated by using the exponential distribution. The generated dataset consisted of 10,000 two-dimensional vectors.
We compared the dynamic programming algorithm with the
kmeans() function in R which provides the (Hatigan and Wong 1979) implementation of \(k\)-means. We ran the algorithms with
different cluster numbers from 2 to 50. The minimal \(\mathit{withinss}\) found by \(k\)-means was always greater or equal to
the optimum value found by clustering.sc.dp().
We used the relative difference in \(\mathit{withinss}\) from the
kmeans() result to the optimal value produced by
clustering.sc.dp() for measuring deviation of the \(k\)-means result from the optimum.
Figure 1 shows the relative difference as a
function of the cluster numbers. It can be seen that \(k\)-means is able to find the optimum if
the cluster number is at most 10. For larger cluster numbers, its error
starts increasing and the relative difference is more than 20% if the
cluster number is 50.
kmeans() to the optimal value returned by
clustering.sc.dp(). The input data set of size 10 000 were
generated as a random walk having a step size that varies according to
an exponential distribution with rate 1.0 in each coordinate.Runtime
We tested the dynamic programming algorithm on inputs with different sizes, dimensions and cluster numbers in order to find its performance bounds and examine experimentally how the running time depends on the input sizes. The simulations were run on a desktop computer with a Pentium Dual-Core 2.93 GHz processor and 4 GB memory, running Windows 10 operation system. We generated multidimensional Gaussian random walks as data sets for performance tests. The steps between subsequent items were generated independently for each coordinate by using the Gaussian random distribution with zero mean and standard deviation of 0.1.
In the first setting (Figure 2),
runtime is obtained as a function of input data size for running
clustering.sc.dp(). The size of the input varies from 1,000
to 30,000 with a step size of 1,000. The input data consists of
two-dimensional vectors. The number of clusters is set to 2. The runtime
increases quadratically in the number of items to be clustered.
Table 1 presents some runtime data for a different magnitude of the number of items in order to find performance bounds of the method. One can see that the optimal clustering was found very quickly if the number of items is 1,000, it took more than a second, almost two and a half minutes, about four hours for 10,000, 100,000 and one million items, respectively.
| Size | 1000 | 10,000 | 100,000 | 1,000,000 |
|---|---|---|---|---|
| Runtime (sec) | 0.03 | 1.19 | 144.00 | 14,479.88 |
In the second performance test we examine the dependency of the runtime on the dimensions. All input data sets are of the same size 10,000 and the dimensions of the vectors varies from 1 to 512. The dimensions are doubled in each run. Figure 3 shows the results. The algorithm runs less than a second if the input contains one-dimensional vectors (i.e., scalars). The runtime increases linearly with the dimension and it can be solved within two and a half minutes if the dimension of the processed vectors is 512.
In the third performance test, the runtime is examined as a function of the number of clusters. The number of clusters is between 1 and 25. The input data set consist of 10,000 two-dimensional vectors. The black line with circles in Figure 4 shows the result. The runtime increases linearly with the number of clusters.
Finally, we compared the runtime of different functions within the
package. The input data were the same as in the previous setting. One
can see in Figure 4 that the
runtime of findwithinss.sc.dp() is slightly larger than the
one of clustering.sc.dp(). The runtime of
backtracking.sc.dp() remains small even for large cluster
numbers. The package has three typical ways of usage:
clustering.sc.dp()can be called when the number of clusters is known in advance.findwithinss.sc.dp()can be called first and thenbacktracking.sc.dp()for a selected number of clusters.findwithinss.sc.dp()can be called first and thenbacktracking.sc.dp()for all possible number of clusters.
The subsequent call of findwithinss.sc.dp() and
backtracking.sc.dp() is only slightly slower than calling
clustering.sc.dp(). Furthermore, the total runtime of
calling findwithinss.sc.dp() for cluster number \(k\) and backtracking.sc.dp()
for all cluster numbers less than or equal to \(k\) is less than the double of the runtime
of clustering.sc.dp() called for the same cluster number.
This is much faster than calling the clustering function \(k\) times which would be required without
splitting clustering.sc.dp() into phases of finding the
optimal \(\mathit{withinss}\)
(findwithinss.sc.dp()) and backtracking
(backtracking.sc.dp()).
clustering.sc.dp(),
findwithinss.sc.dp(), backtracking.sc.dp(). It
also contains the total running time of
backtracking.sc.dp() when it is called for all cluster
numbers less than or equal to the specific cluster number.Introduction to the R package clustering.sc.dp
R package clustering.sc.dp offers functions to perform
optimal clustering on multidimensional data with sequential constraint.
Method clustering.sc.dp() can find the optimal clustering
if the number of clusters is known. Otherwise, methods
findwithinss.sc.dp() and backtracking.sc.dp()
can be used.
The following examples illustrate how to use the package. Function
clustering.sc.dp() outputs the same fields describing
clustering as Ckmeans.1d.dp() does in the original
package:
\(cluster\): a vector of cluster indices assigned to each element in \(x\). Each cluster is indexed by an integer from 1 to \(k\).
\(centers\): a matrix whose rows represent the vectors of cluster centres (the average of the points within the cluster).
\(\mathit{withinss}\): the within-cluster sum of squared distances for each cluster.
\(size\): a vector containing the number of points in each cluster.
Figure 5 visualizes the input data
and the clusters created by clustering.sc.dp(). See below
for the R source code of the example.
# Example1: clustering data generated from a random walk
x <- rbind(0, matrix(rnorm(99 * 2, 0, 0.1), nrow = 99, ncol = 2))
x <- apply(x, 2, cumsum)
k <- 2
result <- clustering.sc.dp(x, k)
plot(x, type = "b", col = result$cluster)
points(result$centers, pch = 24, bg = 1:k)
The next example demonstrates the usage of functions
findwithinss.sc.dp() and backtracking.sc.dp().
Similarly to the previous example, it also processes data of a random
walk. Function findwithinss.sc.dp() finds optimal \(\mathit{withinss}\) for a range of cluster
numbers. It returns a list with two components:
\(\mathit{withinss}\): a vector of total within-cluster sum of squared distances of the optimal clusterings for each number of clusters less than or equal to
k.\(\mathit{backtrack}\): backtrack data used by
backtracking.sc.dp().
In our example, the first cluster number where \(\mathit{withinss}\) drops below a threshold
is selected as the number of clusters. Function
backtracking.sc.dp() outputs the optimal clustering for the
selected cluster number in the same format as
clustering.sc.dp() does without running the whole
clustering process again (Figure 6).
# Example2: clustering data generated from a random walk with small withinss
x <- rbind(0, matrix(rnorm(99 * 2, 0, 0.1), nrow = 99, ncol = 2))
x <- apply(x, 2, cumsum)
k <- 10
r <- findwithinss.sc.dp(x, k)
# select the first cluster number where withinss drops below a threshold
k_th <- which(r$twithinss <= 5.0)[1]
# backtrack
result <- backtracking.sc.dp(x, k_th, r$backtrack)
plot(x, type = "b", col = result$cluster)
points(result$centers, pch = 24, bg = 1:k_th)
Summary
Clustering data with sequential constraint is a polynomial time solvable variant of the clustering problem. The paper introduced a package implementing a dynamic programming approach that finds the exact optimum of the problem. The algorithm represents an extension of the one-dimensional dynamic programming strategy of Ckmeans.1d.dp to multiple dimensional spaces which has been an open problem in the paper of (Wang and Song 2011). The package supports both cases when the exact number of clusters is given and when the number of clusters is not known in advance. It can also be used to evaluate approximation algorithms for clustering with sequential constraint due to its optimality. The runtime evaluations indicate how fast the algorithm can solve problems with different sizes and parameters. Our future plan is to use the dynamic programming method in video summarisation.
Acknowledgements
Research is supported by the Hungarian National Development Agency under grant HUMAN_MB08-1-2011-0010.