 Research article
 Open Access
 Published:
An efficient clustering algorithm for partitioning Yshort tandem repeats data
BMC Research Notes volume 5, Article number: 557 (2012)
Abstract
Background
YShort Tandem Repeats (YSTR) data consist of many similar and almost similar objects. This characteristic of YSTR data causes two problems with partitioning: nonunique centroids and local minima problems. As a result, the existing partitioning algorithms produce poor clustering results.
Results
Our new algorithm, called kApproximate Modal Haplotypes (kAMH), obtains the highest clustering accuracy scores for five out of six datasets, and produces an equal performance for the remaining dataset. Furthermore, clustering accuracy scores of 100% are achieved for two of the datasets. The kAMH algorithm records the highest mean accuracy score of 0.93 overall, compared to that of other algorithms: kPopulation (0.91), kModesRVF (0.81), New Fuzzy kModes (0.80), kModes (0.76), kModesHybrid 1 (0.76), kModesHybrid 2 (0.75), Fuzzy kModes (0.74), and kModesUAVM (0.70).
Conclusions
The partitioning performance of the kAMH algorithm for YSTR data is superior to that of other algorithms, owing to its ability to solve the nonunique centroids and local minima problems. Our algorithm is also efficient in terms of time complexity, which is recorded as O(km(nk)) and considered to be linear.
Background
YShort Tandem Repeats (YSTR) data represent the number of times an STR motif repeats on the Ychromosome. It is often called the allele value of a marker. For example, if there are eight allele values for the DYS391 marker, the STR would look like the following fragments: [TCTA] [TCTA] [TCTA] [TCTA] [TCTA] [TCTA] [TCTA] [TCTA]. The number of tandem repeats has effectively been used to characterize and differentiate between two people.
In modern kinship analyses, the YSTR is very useful for distinguishing lineages and providing information about lineage relationships [1]. Many areas of study, including genetic genealogy, forensic genetics, anthropological genetics, and medical genetics, have taken advantage of the YSTR method. For example, it has been used to trace a similar group of Ysurname projects to support traditional genealogical studies, e.g., [2–4]. Further, in forensic genetics, the YSTR is one of the primary concerns in human identification for sexual assault cases [5], paternity testing [6], missing persons [7], human migration patterns [8], and the reexamination of ancient cases [9].
From a clustering perspective, the goal of partitioning YSTR data is to group a set of YSTR objects into clusters that represent similar genetic distances. The genetic distance of two YSTR objects is based on the mismatch results from comparing the YSTR objects and their modal haplotypes. For Ysurname applications, if two people share 0, 1, 2, and 3 allele value mismatches for each marker, they are considered to be the most familially related. Furthermore, for Yhaplogroup applications, the number of mismatches is variant and greater than that typically found in Ysurname applications. This is because the haplogroup application is based on larger family groups branched out from the same ancestor, covering certain geographical areas and ethnicities throughout the world. The established YDNA haplogroups named by the letters A to T, with further subdivisions using numbers and lower case letters, are now available for reference (see [10] and [11] for details).
Efforts to group YSTR data based on genetic distances have recently been reported. For example, Schlecht et al. [12] used machine learning techniques to classify YSTR fragments into related groups. Furthermore, Seman et al. [13–19] used partitional clustering techniques to group YSTR data by the number of repeats, a method used in genetic genealogy applications. In this study, we continue efforts to partition the YSTR data based on the partitional clustering approaches carried out in [13–19]. Recently, we have also evaluated eight partitional clustering algorithms over six YSTR datasets [19]. As a result, we found that there is scope to propose a new partitioning algorithm to improve the overall clustering results for the same datasets.
A new partitioning algorithm is required to handle the characteristics of YSTR data, thus producing better clustering results. YSTR data are slightly unique compared to the common categorical data used in [20–25]. The YSTR data contain a higher degree of similarity of YSTR objects in their intraclasses and interclasses. (Note that the degree of similarity is based on the mismatch results when comparing the objects and their modal haplotypes.) For example, many YSTR surname objects are found to be similar (zero mismatches) and almost similar (1, 2, and 3 mismatches) in their intraclasses. In some cases, the mismatch values of interclass objects are not obviously far apart. YSTR haplogroup data contain similar, almost similar, and also quite distant objects. Occasionally, the YSTR haplogroup data may include subclasses that are sparse in their intraclasses.
Partitional clustering algorithms
Classically, clustering has been divided into hierarchical and partitional methods. The main difference between the two is that the hierarchical method breaks the data up into hierarchical clusters, whereas the partitional method divides the data into mutually disjoint partitions. The pillar of the partitional algorithms is the kMeans algorithm [26], introduced almost four decades ago. As a consequence, the kMeans paradigm has been extended to various versions, including the kModes algorithm [25] for categorical data.
The kModes algorithm owes its existence to the ineffectiveness of the kMeans algorithm for handling categorical data. Ralambondrainy [27] attempted to rectify this using a hybrid numeric–symbolic method based on the binary characters 0 and 1. However, this approach suffered from an unacceptable computational cost, particularly when the categorical attributes had many categories. Since then, a variety of kModestype algorithms have been introduced, such as kModes with new dissimilarity measures [21, 22], kPopulation [23], and a new Fuzzy kModes [20].
Partitional algorithms use an objective function in their optimization process, and the determination of this function was described as the P problem by Bobrowski and Bezdek [28] and Salim and Ismail [29]. When he proposed the kModes clustering algorithm, Huang [25] split P into P_{ 1 } and P_{ 2 }. P_{ 1 } denotes the minimization problem of obtaining values for the partition matrix w_{ li } of 0 or 1 (for the hard clustering approach) or 0 to 1 (for the fuzzy clustering approach); see Eq. (1b) as an example. Furthermore, P_{ 2 } denotes the minimization problem of obtaining the value that occurs most often (or the mode of a categorical data set) to represent the center of a cluster (often called the centroid). The minimization of P_{ 2 } by obtaining the appropriate mode essentially causes the minimization of problem P_{ 2 }, and vice versa. As an example of the optimization process for problem P in the Fuzzy kModes algorithm, we wish to solve Eq. (1) subject to Eqs. (1a), (1b), and (1c).
subject to:
And
where:

w_{ li } is a (k × n) partition matrix that denotes the degree of membership of object i in the l th cluster that contains a value of 0 to 1,

k (≤ n) is a known number of clusters,

Z is the centroid such that [Z_{ 1 }, Z_{ 2 },…,Z_{ k }] ∈ R^{mk},

α [1, ∞) is a weighting exponent,

d(X_{ i }, Z_{ l }) is the distance measure between the object X_{ i } and the centroid Z_{ l }, as described in Eqs. (2) and (2a).
$$d\left(x,z\right)={\displaystyle {\sum}_{j=1}^{n}\delta}\left({x}_{j},{z}_{j}\right)$$(2)
where:
Huang and Ng [24] described the optimization process of P_{ 1 } and P_{ 2 } as follows:

Problem P_{ 1 }: Fix Z = $\widehat{Z}$ and solve the reduced problem P(W,$\widehat{Z}$) as in Eq. (3). This process obtains the minimized values of 0–1 of the partition matrix w_{ li }.
$${w}_{\mathit{li}}=\{\begin{array}{ll}1,& If\phantom{\rule{0.12em}{0ex}}{X}_{i}={\widehat{Z}}_{l}\\ 0,& If\phantom{\rule{0.12em}{0ex}}{X}_{i}={\widehat{Z}}_{h},h\ne l\hfill \\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sum _{h=1}^{k}{\left[\frac{d\left({X}_{i,}{\widehat{Z}}_{l}\right)}{d\left({X}_{i,}{\widehat{Z}}_{h}\right)}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\left(\alpha 1\right)$}\right.}}$}\right.,\hfill & If\phantom{\rule{0.12em}{0ex}}{X}_{i}\ne {\widehat{Z}}_{l},\phantom{\rule{0.12em}{0ex}}and\phantom{\rule{0.25em}{0ex}}{X}_{i}\ne {\widehat{X}}_{h,}1\le h\le k\end{array}$$(3) 
Problem P_{ 2 }: Fix W = Ŵ and solve the reduced problem P(Ŵ, Z) as in Eq. (4) subject to Eq. (4a). This process obtains the most frequent attributes, or the modes, which give the centroids.
$${Z}_{\mathit{li}}={a}_{j}^{\left(p\right)}\in \text{DOM}\left({A}_{j}\right)$$(4)
where:
and α ∈ [1, ∞) is a weighting exponent.
Problem of partitioning YSTR data
Due to the characteristics of YSTR data, there are two optimization problems for existing partitional algorithms: nonunique centroids and local minima problems. These two problems are caused by the drawback of the modes mechanism of determining the centroids. Nonunique centroids would result in empty clusters, whereas the local minima problem leads to poorer clustering results. Both problems are a result of the obtained centroids, which are not sufficient to represent their classes.
Therefore, problems will occur for the following two cases:
i)The total number of objects in a dataset is small while the number of classes is large. To illustrate this case, consider the following example.
Example I: Figure 1 shows an artificial example of a dataset consisting of nine objects in three classes: Class A = {A_{ 1 }, A_{ 2 }, A_{ 3 }}, Class B = {B_{ 1 }, B_{ 2 }, B_{ 3 }}, and Class C = {C_{ 1 }, C_{ 2 }, C_{ 3 }}. Each object is composed of three attributes, represented in lower case; e.g., for object A_{ 1 }, the attributes are a_{ 1 }, a_{ 2 }, and a_{ 3 }. The dataset is considered to have a higher degree of similarity between objects in intraclasses, while the number of objects is small and number of classes is large. Thus, the appropriate modes for representing the classes are: Class A – [a_{ 1 }, a_{ 2 }, a_{ 3 }], lass B – [a_{ 1 }, b_{ 2 }, c_{ 3 }], and Class C – [b_{ 1 }, c_{ 2 }, d_{ 4 }]. However, attribute a_{ 1 } in DOMAIN (A_{ 1 }), a_{ 2 } in DOMAIN (A_{ 2 }), and c_{ 3 } in DOMAIN (A_{ 3 }) are too dominant, and would therefore dominate the process of updating P_{ 2 }. Figure 2 shows the possibility that each cluster is formed by the dominant attributes.
As a result, the mode that consists of [a_{ 1 }, a_{ 2 }, c_{ 3 }] would be obtained twice. Thus, P_{ 2 } would not be minimized due to this nonunique centroid. Another possibility is that the two modes are different, but are not distinctive enough to represent their clusters, such as modes [a_{ 1 }, a_{ 2 }, a_{ 3 }] or [a_{ 1 }, a_{ 2 }, b_{ 3 }] for Cluster 2. As a consequence, this case would fall into a local minima problem.
ii)An extreme distribution of objects in a class. To illustrate this case, consider the following example.
Example II: Figure 3 shows a dataset consisting of eight objects in two classes: Class A = {A_{ 1 }, A_{ 2 }, A_{ 3 }, A_{ 4 }, A_{ 5 }, A_{ 6 }} and Class B = {B_{ 1 }, B_{ 2 }}. Each object consists of three attributes, again represented in lower case. The appropriate modes to represent the classes are: Class A – [a_{ 1 }, a_{ 2 }, b_{ 3 }] and Class B – [a_{ 1 }, b_{ 2 }, c_{ 3 }] or [a_{ 1 }, b_{ 2 }, d_{ 3 }]. The distribution of objects in Class A is considerably larger than in Class B, covering approximately 75% of the total set of objects. This characteristic of the data is found to be problematic for P_{ 2 }, particularly for the fuzzy approach. The problem is actually caused by the initial centroid selection. Figure 4 shows the objects in Class A would be equally distributed into clusters 1 and 2.
As a result, object A becomes dominant in both clusters, and so the obtained modes might be represented solely by objects in Class A, e.g., [a_{ 1 }, a_{ 2 }, a_{ 3 }] and [a_{ 1 }, a_{ 2 }, b_{ 3 }].
The above situations cause P not to be fully optimized, thus producing poor clustering results. Therefore, a new algorithm with a new concept of P_{ 2 } is proposed in order to overcome these problems and improve the clustering accuracy results of YSTR data.
Methods
The center of a cluster
The mode mechanism for the center of a cluster (problem P_{ 2 }) is not appropriate for handling the characteristics of YSTR data, and therefore, it cannot be used as a mechanism to represent the center of a cluster (centroid). Instead, the center of YSTR data should be the modal haplotypes, which are required to calculate the distance of YSTR objects. The distance between a YSTR object and its modal haplotype can be formalized as in Eq. (5) subject to Eq. (5a).
subject to:
where m is the number of markers.
The modal haplotype is controlled by groups of objects that are similar or almost similar in YSTR data. The similar and almost similar objects have a lower distance, or a higher degree of membership values in a fuzzy sense. Thus, these two groups are considerably the most dominant objects required to find the Approximate Modal Haplotype. Consider four objects x_{ 1 }, x_{ 2 }, x_{ 3 }, and x_{ 4 } and two clusters c_{ 1 } and c_{ 2 }. The membership value for each object and its cluster are as shown in Table 1, whereby objects x_{ 1 } and x_{ 3 } have a 100% chance of being the most dominant object in cluster c_{ 1 }, but only a 50% chance of being the dominant object in cluster c_{ 2 }, and so on. A dominant weighting value of 1.0 is given to any dominant object and a weight of 0.5 is given to the remaining objects.
The kAMH algorithm
Let X ={X_{ 1 }, X_{ 2 },…, X_{ n }} be a set of n YSTR objects and A ={A_{ 1 },A_{ 2 },…, A_{ m }} be a set of markers (attributes) of a YSTR object. Let H = {H_{ 1 }, H_{ 2 },.,H_{ k }} ∈ X be the set of Approximate Modal Haplotypes for k clusters. Suppose k is known a priori. Let H_{ l } be the Approximate Modal Haplotype, represented as [h_{ l,1 }, h_{ l,2 },…,h_{ l,m }], and therefore, H_{ l,j } = X_{ i,j } for 1≤ j ≤ m and 1≤ i ≤ n. The objective of the algorithm is to partition the categorical objects X into k clusters. Thus, the H_{ l } can be replaced by X_{ i } until n provided they satisfy the condition described in Eq. (6).
Here, P(Á) is the cost function described in Eq. (7), which is subject to Eqs. (7a), (8), (8a), (8b), (9), (9a), (9b), and (9c).
subject to:

W_{ li }^{∝} is a (k × n) partition matrix that denotes the degree of membership of YSTR object i in the l th cluster that contains a value of 0 to 1 as described in Eq. (8), subject to Eqs. (8a) and (8b).
$${W}_{\mathit{li}}^{\propto}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\{{\left(\begin{array}{cc}1,& If,\phantom{\rule{0.5em}{0ex}}{X}_{i}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{H}_{i}\\ 0,& If,\phantom{\rule{0.5em}{0ex}}{X}_{i}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{H}_{z},z\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}l\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sum _{z=1}^{k}$}\right.{\left[\frac{{d}_{\mathit{\text{ystr}}}\left({X}_{i,}{H}_{l}\right)}{{d}_{\mathit{\text{ystr}}}\left({X}_{i},{H}_{z}\right)}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\left(\propto 1\right)$}\right.}\phantom{\rule{0.5em}{0ex}}\hfill & ,If\phantom{\rule{0.5em}{0ex}}{H}_{i}\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}{X}_{j}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}{X}_{i}\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}{H}_{z},\phantom{\rule{0.5em}{0ex}}1\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}z\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}k\end{array},\phantom{\rule{6em}{0ex}}\right)}^{\propto}$$(8) 
subject to:
$${w}_{\mathit{li}}^{\propto}\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\left[0,1\right],\phantom{\rule{0.5em}{0ex}}1\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}i\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}n,\phantom{\rule{0.5em}{0ex}}1\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}l\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}k\text{,}$$(8a)
and
where,

k (≤ n) is a known number of clusters.

H is the Approximate Modal Haplotype (centroid) such that [H_{ 1 }, H_{ 2 },…,H_{ k }] ∈ X.

α ∈ [1, ∞) is a weighting exponent and used to increase the precision of the membership degrees. Note that this alpha is typical based on 1.1 until 2.0 as introduced by Huang and Ng [24].

d_{ ystr }(X_{i,}H_{ l }) is the distance measure between the YSTR object X_{ i } and the Approximate Modal Haplotype H_{ l } as described in Eq. (5) and subject to Eq.(5a).

D_{ li } is another (k × n) partition matrix which contains a dominant weighting value of 1.0 or 0.5, as explained above (See Table 1). The dominant weighting values are based on the value of W_{ li }^{∝} above. D_{ li } is described in Eq. (9), subject to Eqs. (9a), (9b), and (9c).
$${d}_{\mathit{li}}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\{\begin{array}{c}\hfill 1.0,\phantom{\rule{0.5em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}{w}_{\mathit{li}}^{\propto}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\text{max}}^{{w}_{\mathit{li}}^{\propto},1\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}l\phantom{\rule{0.5em}{0ex}}\le \phantom{\rule{0.5em}{0ex}}k}\hfill \\ \hfill 0.5,\phantom{\rule{0.5em}{0ex}}\text{otherwise}\hfill \end{array}$$(9)
subject to:
The basic idea of the kAMH algorithm is to find k clusters in n objects by first randomly selecting an object to be the Approximate Modal Haplotype h for each cluster. The next step is to iteratively replace the objects x onebyone towards the Approximate Modal Haplotype h. The replacement is based on Eq. (6) if the cost function as described in Eq. (7) and subject to (7a), (8), (8a), (8b), (9), (9a), (9b) and, (9c) is maximized. Thus, the differences between the kAMH algorithm and the other kModetype algorithms are as follows.

i.
The objects (the data themselves) are used as the centroids instead of modes. Since the distance of YSTR objects is measured by comparing the objects and their modal haplotypes, we need to approximately find the objects that can represent the modal haplotypes. In finding the final Approximate Modal Haplotype for a particular group (cluster), each object needs to be tested onebyone and replaced on a maximization of a cost function.

ii.
A maximization process of the cost function is required instead of minimizing it as in the kmodetype algorithms.
A detailed description of the kAMH algorithm is given below.
Step 1 – Select k initial objects randomly as Approximate Modal Haplotype (centroids). E.g. if k = 4, then choose randomly 4 objects as the initial Approximate Modal Haplotype.
Step 2 – Calculate distance d_{ ystr }(X_{i,}H_{ l }) according to Eq. (5) and subject to (5a).
Step 3 – Calculate partition matrix w_{ li }^{∝} according to Eq. (8), subject to Eqs. (8a) and (8b). Note that the w_{ li }^{∝} is based on the distance calculated in Step 2.
Step 4 – Assign a weighting dominant of 1.0 or 0.5 for partition matrix D_{ li } according to Eqs. (9), (9a), (9b) and (9c).
Step 5 – Calculate cost function P(Á) based on W_{ li }^{∝}D_{ li } according to Eqs (7) and (7a).
Step 6 – Test for each initial modal haplotype by the other objects onebyone. If current cost function is greater than previous cost function according to Eq. (6), then replace it.
Step 7 – Repeat Step 2 until Step 6 for each x and h
Step 8 – Once the final Approximate Modal Haplotypes are obtained for all clusters, assign the objects to their corresponding crisp clusters C_{ li } according to Eq. (10).
Furthermore, the implementation of the steps above of the algorithm is formalized in the form of pseudocode as follows.
INPUT: Dataset, X, the number of cluster, k , the number of dimensional, d and the fuzziness index,
OUTPUT: A set of clusters, k

01
: Select H_{ l } randomly from X such that 1≤l≤ k

02
: for each H_{ l } an Approximate Modal Haplotype do

03
: for each X_{ i } do

04
: Calculate P(À) = ∑ _{l = 1}^{k} ∑ _{i = 1}^{n}À_{ li }

05
: if P(À) = ∑ _{l = 1}^{k} ∑ _{i = 1}^{n}À_{ li } is maximized, then

06
: Replace H_{ l } by X_{ i }

07
: end if end for

09
: end for

10
: Assign X_{ i } to C_{ l } for all l, 1≤ l ≤ k; 1≤i≤ n as Eq. (10)

11
: Output Results
Optimization of the problem P
In optimizing the problem P, the kAMH algorithm uses a maximization process instead of the minimization process imposed by the kModetype algorithms. This process is formalized in the kAMH algorithm as follows.
Step 1  Choose an Approximate Modal Haplotype, H^{(t)}∈ X. Calculate P(Á); Set t=1
Step 2  Choose X^{(t+1)} such that P(Á)^{t+1} is maximized; Replace H^{1} by X^{(t+1)}
Step 3  Set t=t+1 ; Stop when t=n; otherwise go to Step 2.
*Note: n is the number of objects
The convergence of the algorithm is proven as P_{ 1 } and P_{ 2 } are maximized accordingly. The function P(Á) incorporates the P(W, H) function imposed by the Fuzzy kModes algorithm, where W is a partition matrix and H is the approximate modal haplotype that defines the center of a cluster. Thus, P_{ 1 } and P_{ 2 } are solved by Theorems 1 and 2, respectively.
Theorem 1 – Let Ĥ be fixed. P(W, Ĥ) is maximized if and only if
Proof
Let X= {X_{ 1 },X_{ 2 },.,X_{ n }} be a set of n YSTR categorical objects and H= {H_{ 1 },H_{ 2 },.,H_{ k }} be a set of centroids (Approximate Modal Haplotypes) for k clusters. Suppose that P= {P_{ 1 },P_{ 2 },.,P_{ k }} is a set of dissimilarity measures based on d_{ ystr }(X_{i,}H_{ l }), as described in Eqs. (5) and subject to (5a), ∀ i and l 1 ≤ i ≤ n; 1 ≤ l ≤ k
Definition 1  For X_{ i } = H_{ l } and X_{ i } = H_{ z }, where z ≠ l, the membership value for all i is
For any P that is obtained from d_{ ystr }(X_{i,}H_{ l }) where X_{ i } = H_{ l }, the maximum value of w_{ li }^{∝} is 1 and X_{ i } = H_{ z }, z ≠ l the value of w_{ li }^{∝} is 0. Therefore, because H_{ l } is fixed, w_{ li }^{∝} is maximized.
Definition 2 – For the case of H_{ i } ≠ X_{ i }and X_{ i } ≠ H_{ z }, ∀ z, 1 ≤ z ≤ k, the membership value for all i is
Suppose that p_{ li } ∈ P is the minimum value, we write as
Therefore,
where z ≠ l
Thus, ${{\displaystyle {\sum}_{z=1}^{k}\left[\frac{{P}_{\mathit{li}}}{zi}\right]}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\left(\propto 1\right)$}\right.}\phantom{\rule{1em}{0ex}}<\phantom{\rule{0.5em}{0ex}}{{\displaystyle {\sum}_{z=1}^{k}\left[\frac{{P}_{\mathit{ti}}}{{P}_{\mathit{zi}}}\right]}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\left(\propto 1\right)$}\right.}$ where
t ≠ l and ∀ z and t, 1 ≤ z ≤ k; 1 ≤ t ≤ k It follows that
where t ≠ l
Therefore, based on definitions 1 and 2, w_{ li }^{∝} is maximal. Because Ĥ is fixed, $P\left(W,\widehat{H}\right)$ is maximized.
Theorem 2 – Let h_{ l } ∈ X be the initial center of a cluster for 1 ≤ l ≤ k. h_{ l } is replaced by x_{ i } as the Approximate Modal Haplotype if and only if
Proof
Let D= {D_{ 1 },D_{ 2 },.,D_{ k }} be a set of dominant weighting values. For any maximum value of w_{ li }^{∝} as proved by Theorem 1, we assign an optimum value of 1.0 as a dominant weighting value, otherwise 0.5 as described in Eq, (9) and subject to Eqs. (9a), (9b) and (9c). We write
Because w_{ li }^{∝} and D_{ li } are nonnegative, the product W_{ li }^{∝}D_{ li } must be maximal. It follows that the sum of all quantities ∑ _{l = 1}^{k} ∑ _{i = 1}^{n}Á_{ li } is also maximal. Hence, the result follows.
YSTR Datasets
The YSTR data were mostly obtained from a database called worldfamilies.net [30]. The first, second, and third datasets represent YSTR data for haplogroup applications, whereas the fourth, fifth, and sixth datasets represent YSTR data for Ysurname applications. All datasets were filtered for standardization on 25 similar attributes (25 markers). The chosen markers include DYS393, DYS390, DYS19 (394), DYS391, DYS385a, DYS385b, DYS426, DYS388, DYS439, DYS389I, DYS392, DYS389II, DYS458, DYS459a, DYS459b, DYS455, DYS454, DYS447, DYS437, DYS448, DYS449, DYS464a, DYS464b, DYS464c, and DYS464b. These markers are more than sufficient for determining a genetic connection between two people. According to Fitzpatrick [31], 12 markers (YDNA12 test) are already sufficient to determine who does or does not have a relationship to the core group of a family.
All datasets were retrieved from the respective websites in April 2010, and can be described as follows:

1)
The first dataset consists of 751 objects of the YSTR haplogroup belonging to the Ireland yDNA project [32]. The data contain only 5 haplogroups, namely E (24), G (20), L (200), J (32), and R (475). Thus, k = 5.

2)
The second dataset consists of 267 objects of the YSTR haplogroup obtained from the Finland DNA Project [33]. The data are composed of only 4 haplogroups: L (92), J (6), N (141), and R (28). Thus, k = 4.

3)
The third dataset consists of 263 objects obtained from the Yhaplogroup project [34]. The data contain Groups G (37), N (68), and T (158). Thus, k = 3.

4)
The fourth dataset consists of 236 objects combining four surnames: Donald [35], Flannery [36], Mumma [37], and William [38]. Thus, k = 4.

5)
The fifth dataset consists of 112 objects belonging to the Philips DNA Project [39]. The data consist of eight family groups: Group 2 (30), Group 4 (8), Group 5 (10), Group 8 (18), Group 10 (17), Group 16 (10), Group 17 (12), and Group 29 (7). Thus, k = 8.

6)
The sixth dataset consists of 112 objects belonging to the Brown Surname Project [40]. The data consist of 14 family groups: Group 2 (9), Group 10 (17), Group 15 (6), Group 18 (6), Group 20 (7), Group 23 (8), Group 26 (8), Group 28 (8), Group 34 (7), Group 44 (6), Group 35 (7), Group 46 (7), Group 49 (10), and Group 91 (6). Thus, k = 14.
The values in parentheses indicate the number of objects belonging to that particular group. Datasets 1–3 represent YSTR haplogroups and datasets 4–6 represent YSTR surnames.
Results and discussion
The following results compare the performance of the kAMH algorithm with eight other partitional algorithms: the kModes algorithm [25], kModes with RVF [21, 22, 41], kModes with UAVM [21], kModes with Hybrid 1 [21], kModes with Hybrid 2 [21], the Fuzzy kModes algorithm [24], the kPopulation algorithm [23], and the New Fuzzy kModes algorithm [20].
Our analysis was based on the average accuracy scores obtained from 100 runs for each algorithm and dataset. During the experiments, the objects in the datasets were randomly reordered from the preceding run. The misclassification matrix proposed by Huang [25] was used to obtain the clustering accuracy scores for evaluating the performance of each algorithm. The clustering accuracy r defined by Huang [25] is given by Eq. (11):
where k is the number of clusters, a_{ i } is the number of instances occurring in both cluster i and its corresponding haplogroup or surname, and n is the number of instances in the dataset.
Clustering performance
Table 2 shows the clustering accuracy scores for all datasets (boldface indicates the highest clustering accuracy). Based on these results, the performance of the kAMH algorithm was very promising. Out of six datasets, our algorithm obtained the highest clustering accuracy scores for datasets 1, 2, 4, 5, and 6. In fact, the algorithm also achieved the optimal clustering accuracy for two datasets (4 and 5). However, for dataset 3, the results show that the accuracy of the kAMH algorithm was 0.01 lower than that of the kPopulation algorithm. A statistical ttest was carried out for further verification. This indicated that t(101.39) = 0.65, and p = 0.51. Thus, there was no significant difference at the 5% level between the accuracy score of our kAMH algorithm and the kPopulation algorithm. This means that both algorithms displayed an equal performance for this dataset.
During the experiments, the kAMH algorithm did not encounter any difficulties. However, the Fuzzy kModes and the New Fuzzy kModes algorithms faced problems with datasets 1, 5, and 6. For dataset 1, the problem was caused by the extreme number of objects in Class R (475), which covered about 63% of the total objects. Further, for datasets 5 and 6, the problem was caused by many similar objects in a larger number of classes. In particular, both algorithms faced the problem P_{ 2 } caused by the initial centroid selections. Note also that the results for both algorithms were based on the diverse method, an initial centroid selection proposed by Huang [25].
For an overall comparison, Table 3 shows the results of all YSTR datasets. It clearly indicates that the kAMH algorithm obtained the highest accuracy score of 0.93. The closest score of 0.91 belongs to the kPopulation algorithm. Furthermore, the kAMH algorithm also recorded the best results in terms of standard deviation (0.07), the lower bound (0.93), the upper bound (0.94), and the minimum accuracy score (0.79).
For further verification, a oneway ANOVA test was carried out. This indicated that the assumption of homogeneity of variance was violated; therefore, the Welch Fratio is reported. There was a significant variance in the clustering accuracy scores among the nine algorithms, in which F(8, 2230) = 378, p < 0.001, and ω^{2} = 0.25. Thus, the Games–Howell procedure was used for a multiple comparison among the nine algorithms. Table 4 shows the result of this comparison with regard to the kAMH algorithm against the other eight algorithms. At the 5% level of significance, it is clearly shown that the kAMH algorithm (M = 0.93, 95% CI [0.93, 0.94]) differed from the other eight algorithms (all P values < 0.001). Thus, the performance of kAMH algorithm exhibited a very significant difference compared to the other algorithms.
Efficiency
We now consider the time efficiency of the kAMH algorithm. The computational cost of the algorithm depends on the nested loop for k(nk), where k is the number of clusters and n is the number of data required to obtain the cost function, P(À). The function P(À) involves the number of attributes m in calculating the distances and the membership values for its partition matrix w_{ li }. Thus, the overall time complexity is O(km(nk)). However, the time efficiency of the kAMH algorithm will not reach O(n^{2}) because the value of k in the outer loop will not become equivalent to the value of nk in the inner loop. See pseudocode for a detailed implementation of these loops.
A scalability test was also carried out for the kAMH algorithm. These experiments were based on a dataset called Connect [42]. This dataset consisted of 65,000 data, 42 attributes, and three classes. Two scalability tests were conducted: (a) scalability against the number of objects, when the number of clusters was three, and (b) scalability against the number of clusters, when the number of objects was 65,000. The test was performed on a personal computer with an Intel® Core™ 2 DUO Processor with 2.93 GHz and 2.00 GB memory. Figure 5(a) and (b) illustrate the results of the tests. In conclusion, the runtime of the kAMH algorithm increased linearly with the number of clusters and data.
Conclusions
Our experimental results indicate that the performance of the proposed kAMH algorithm for partitioning YSTR data was significantly better than that of the other algorithms. Our algorithm handled all problems, as described previously, and was not too sensitive to P_{ 0 }, the initial centroid selection, even though the datasets contained a lot of similar objects. Moreover, the concept of P_{ 2 } in using the object (the data itself) as the approximate center of a cluster has significantly improved the overall performance of the algorithm. In fact, our algorithm is the most consistent of those tested because the difference between the minimum and maximum scores is smaller. The kAMH algorithm always produces the highest minimum score for each dataset. In conclusion, the kAMH algorithm is an efficient method of partitioning YSTR categorical data.
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Acknowledgements
This research is supported by Fundamental Research Grant Scheme, Ministry of Higher Eduction Malaysia. We would like to thank RMI, UiTM for their support for this research. We extend our gratitude to many contributors toward the completion of this paper, including Prof. Dr. Daud Mohamed, En. Azizian Mohd Sapawi, Puan Nuru'l'Izzah Othman, Puan Ida Rosmini, and our research assistants: Syahrul, Azhari, Kamal, Hasmarina, Nurin, Soleha, Mastura, Fadzila, Suhaida, and Shukriah.
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Authors' contributions
AS carried out the algorithm development and experiments. ZAB verified the algorithm and the results. MNI verified the YSTR data and also the results. All authors read and approved the final manuscript.
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Keywords
 Algorithms
 Bioinformatics
 Clustering
 Optimization
 Data mining