A partial order framework for incomplete data clustering

Abstract

We propose in this paper a partial order framework for clustering incomplete data. The paramount feature of this framework is that it spans over a partial order that can be leveraged to establish data similarity. We present the underlying theoretical foundations and study the convergence of clustering algorithms in this framework. In addition, we present a partial order-based clustering algorithm (POK-means) that illustrates the embedding of K-means clustering algorithm in our framework. The first contribution of our method is that unlike methods based on imputation of the missing values, our method does not make any assumptions about missing data. Another important contribution is that it alleviates false dismissals caused by other interval-based similarity measures. The experimental results show that, although our method do not assume any prior knowledge of (or assumptions about) missing data, it is competitive to most of published incomplete data clustering methods that are based on assumptions about input data or imputation (e.g. methods based on partial or interval kernel distances) in accuracy and performance.

Appendix: A

Proposition 1

≼ is a partial order.

Proof

• Reflexivity: [a_L,a_R] ≼ [a_L,a_R] since a_L ≤ a_L ∧ a_R ≤ a_R.

• Asymmetry: Let us assume that [a_L,a_R] ≼ [$a_{L}^{\prime },a_{R}^{\prime }$] and [$a_{L}^{\prime },a_{R}^{\prime }$] ≼ [a_L,a_R]. This means that $a_{L} \leq a_{L}^{\prime }$ and $a_{L}^{\prime } \leq a_{L}$. Hence, $a_{L} = a_{L}^{\prime }$. By similar reasoning, we have $a_{R} = a_{R}^{\prime }$. So [a_L,a_R] = [$a_{L}^{\prime },a_{R}^{\prime }$].

• Transitivity: Let us assume that [a_L,a_R] ≼ [$a_{L}^{\prime },a_{R}^{\prime }$] and [$a_{L}^{\prime },a_{R}^{\prime }$] ≼ [$a_{L}^{\prime \prime },a_{R}^{\prime \prime }$]. This means that $a_{L} \leq a_{L}^{\prime }$ and $a_{L}^{\prime } \leq a_{L}^{\prime \prime }$. So $a_{L} \leq a_{L}^{\prime \prime }$. By similar reasoning, we have $a_{R} \leq a_{R}^{\prime \prime }$. Therefore, [a_L,a_R] ≼ [$a_{L}^{\prime \prime },a_{R}^{\prime \prime }$].

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Proposition 2

(${{\mathscr{L}}_{\mathcal {I}}},\preceq$) is a complete lattice.

Proof

Let $S \subseteq {\mathscr{L}}_{\mathcal {I}}$ such that S = {[a₁,b₁], [a₂,b₂], …, [a_n,b_b]}. Then, an upper bound of the elements in S is $u=[\max \limits (a_{i}), \max \limits (b_{i})]$. Let [x,y] another upper bound of S. Based on the partial order ≼, we can prove easily that $\max \limits (a_{i}) \leq x$ and $\max \limits (b_{i}) \leq y$, which means that $[\max \limits (a_{i}), \max \limits (b_{i})] \preceq [x,y]$. Therefore, u is the least upper bound of S. By similar reasoning, S has a greatest lower bound $l = [\min \limits (a_{i}$), $\min \limits (b_{i})]$. □

Theorem 1

Any approximation based on the weighted distance POWD satisfies the lower bounding constraint.

Proof

Let d_i be an approximation value of a distance between two multidimensional data x and y for the i^th feature. We assume that d_i belongs to the interval distance [$d_{\min \limits } = {\sum }_{i} d_{\min \limits }^{i},d_{\max \limits } = {\sum }_{i} d_{\max \limits }^{i}$] in POF. This means that d_i ≤ $d_{\max \limits }$. So, w_i d_i ≤ w_i $d_{\max \limits }$. Thus, $\sum$ w_i d_i ≤ ($\sum$ w_i) $d_{\max \limits }$. Since by the definition of POWD, we have $\sum$ w_i = 1, we conclude that POWD satisfies the lower bounding constraint. □

Theorem 2

POK-Means in its strict version converges to a local minimum of the lattice (${\mathscr{L}}_{\mathcal {I}},\preceq$).

Proof

The convergence proof is based on the following facts:

• The set of (${\mathscr{L}}_{\mathcal {I}},\preceq$) is a complete lattice.

• In each iteration the SSE is minimized. So the sequence of iterations leads to a decreasing chain in term of SSE. Let S be the set of I₀, I₁, …, I_n. S is finite since the number of configurations is finite. Since S is a subset of I has a greatest lower bound.

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