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Hierarchical clustering of subpopulations with a dissimilarity based on the likelihood ratio statistic: application to clustering massive data sets

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Abstract

The problem of clustering subpopulations on the basis of samples is considered within a statistical framework: a distribution for the variables is assumed for each subpopulation and the dissimilarity between any two populations is defined as the likelihood ratio statistic which compares the hypothesis that the two subpopulations differ in the parameter of their distributions to the hypothesis that they do not. A general algorithm for the construction of a hierarchical classification is described which has the important property of not having inversions in the dendrogram. The essential elements of the algorithm are specified for the case of well-known distributions (normal, multinomial and Poisson) and an outline of the general parametric case is also discussed. Several applications are discussed, the main one being a novel approach to dealing with massive data in the context of a two-step approach. After clustering the data in a reasonable number of ‘bins’ by a fast algorithm such as k-Means, we apply a version of our algorithm to the resulting bins. Multivariate normality for the means calculated on each bin is assumed: this is justified by the central limit theorem and the assumption that each bin contains a large number of units, an assumption generally justified when dealing with truly massive data such as currently found in modern data analysis. However, no assumption is made about the data generating distribution.

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Notes

  1. The proof of this and the other propositions of this paper are given in Sect. 7.

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Acknowledgements

We wish to thank Dr. F. Clavel for having shared the EPIC French data with us and Dr. E. Riboli for general assistance in becoming familiar with EPIC. The authors gratefully acknowledge the hospitality of the Department of Epidemiology and Statistics members during M. Castejón and A. González visits to McGill University. M. Castejón also thanks the hospitality of INRIA members during his visit. We gratefully acknowledge support from the Ministerio de Educación y Ciencia de España, Dirección General de Investigación, by means of the DPI2006-14784 and DPI2007-61090 research projects.

Author information

Authors and Affiliations

  1. Department of Epidemiology and Biostatistics, McGill University, Montreal, P.Q., Canada

    Antonio Ciampi

  2. INRIA—Rocquencourt, 87153, Le Chesnay Cedex, France

    Yves Lechevallier

  3. Department of Mechanical, Informatical and Aerospace Engineering, Universidad de León, 24007, León, Spain

    Manuel Castejón Limas & Ana González Marcos

Authors
  1. Antonio Ciampi
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  2. Yves Lechevallier
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  3. Manuel Castejón Limas
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  4. Ana González Marcos
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Corresponding author

Correspondence to Antonio Ciampi.

Appendix: Proofs

Appendix: Proofs

Proof of Proposition 1

Since d(A i , A j ) >  0 we have that f(A r ) > 0, r = 1, 2, ..., M. Thus f(A i ∪ A j ) >  f(A i ) +  f(A j ) and obviously f(A i ∪ A j ) >  max {f(A i ), f(A j )}

Proof of Proposition 2

$$ d\left(B_s,B_t\right)=\left({\overline{x}}_s-{\overline{x}}_{s \cup t}\right)^{\rm T} V_{s}^{-1} \left({\overline{x}}_s-{\overline{x}}_{s \cup t}\right)+ \left({\overline{x}}_t-{\overline{x}}_{s \cup t}\right)^{\rm T} V_{t}^{-1} \left({\overline{x}}_t-{\overline{x}}_{s \cup t}\right) $$
(41)

Now:

$$ {\overline{x}}_s-{\overline{x}}_{s \cup t} = \left(V_{s}^{-1}+V_{t}^{-1}\right)^{-1} \left[\left(V_{s}^{-1}+V_{t}^{-1}\right){\overline{x}}_s - V_{s}^{-1} {\overline{x}}_s - V_{t}^{-1} {\overline{x}}_t \right] $$
(42)
$$ {\overline{x}}_s-{\overline{x}}_{s \cup t} = \left(V_{s}^{-1}+V_{t}^{-1}\right)^{-1} V_t^{-1} \left({\overline{x}}_s-{\overline{x}}_t\right) $$
(43)
$$ {\overline{x}}_t-{\overline{x}}_{s \cup t} = \left(V_{s}^{-1}+V_{t}^{-1}\right)^{-1} V_s^{-1} \left({\overline{x}}_t-{\overline{x}}_s\right) $$
(44)

Therefore:

$$ d\left(B_s,B_t\right) = \left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T} V_t^{-1} \left(V_s^{-1}+V_t^{-1}\right)^{-1}V_s^{-1} \left(V_s^{-1}+V_t^{-1}\right)^{-1}V_t^{-1}\left({\overline{x}}_s- {\overline{x}}_t\right) + \left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T} V_s^{-1} \left(V_s^{-1}+V_t^{-1}\right)^{-1}V_t^{-1} \left(V_s^{-1}+V_t^{-1}\right)^{-1}V_s^{-1}\left({\overline{x}}_s- {\overline{x}}_t\right)= \left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T} \left(V_s+V_t\right)^{-1} \left(V_s^{-1}+V_t^{-1}\right)^{-1} V_t^{-1} \left({\overline{x}}_s-{\overline{x}}_t\right) + \left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T} \left(V_s+V_t\right)^{-1} \left(V_s^{-1}+V_t^{-1}\right)^{-1} V_s^{-1} \left({\overline{x}}_s-{\overline{x}}_t\right)= \left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T}\left(V_s+V_t\right)^{- 1}\left(V_s^{-1}+V_t^{-1}\right)\left(V_s^{-1}+V_t^{-1}\right)^{- 1}\left({\overline{x}}_s-{\overline{x}}_t\right) = \left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T} \left(V_s+V_t\right)^{-1} \left({\overline{x}}_s-{\overline{x}}_t\right) $$
(45)

where we have used

$$ V_s^{-1}\left(V_s^{-1}+V_t^{-1}\right)^{-1}V_t^{-1}= \left(V_s+V_t\right)^{-1} $$
(46)

Notice also that we can write

$$ d\left(B_s,B_t\right)=\left({\overline{x}}_s-{\overline{x}}_t\right)^{\rm T} \left(V_s^{-1}+V_t^{-1}\right)^{-1} V_s^{-1}V_t^{-1}\left({\overline{x}}_s-{\overline{x}}_t\right) $$
(47)

owing to the conmutativity of symmetric matrices.

Proof of Proposition 3

$$ d\left(A_i, A_j \right) = \sum_{r \in A_i \cup A_j} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) - \sum_{i=1}^2 \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right) $$
(48)
$$ d\left(A_i, A_j \right) = \sum_{i=1}^2 \left\{\sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) - \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)\right\} $$
(49)

Let us develop the first term within the curly bracket above:

$$ \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1}\left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) = \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} + {\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} + {\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) = \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right) + \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) + 2 \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) $$
(50)

Now notice that the last term is zero. This follows from:

$$ \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) = \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} \sum_{r \in A_i} V_r^{-1} \left({\overline{x}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right) $$
(51)

and

$$ \sum_{r \in A_i} V_r^{-1}\left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)= \sum_{r \in A_i} V_r^{-1} {\overline{{\mathbf{x}}}}_{r} - \sum_{r \in A_i} V_r^{-1} {\overline{{\mathbf{x}}}}_{A_i}=V_{A_i}^{-1} {\overline{{\mathbf{x}}}}_{A_i} - \left(V_{A_i} \right)^{-1} {\overline{{\mathbf{x}}}}_{A_i} = 0 $$
(52)

since by definition

$$ V_{A_i}^{-1}= \sum_{r \in A_i} V_r^{-1}, \quad {\overline{{\mathbf{x}}}}_{A_i}=V_{A_i} \sum_{r \in A_i} V_r^{-1} {\overline{{\mathbf{x}}}}_{r} $$
(53)

We now have,

$$ d\left(A_i, A_j \right) = \sum_{i=1}^2 \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right) + \sum_{i=1}^2 \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) - \sum_{i=1}^2 \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{r} - {\overline{{\mathbf{x}}}}_{A_i} \right) = \sum_{i=1}^2 \left\{\sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) \right\} $$
(54)

and we can proceed as in proposition 2. We write to avoid confusion

$$ {{\mathcal{U}}}_r= V_r^{-1}, {{\mathcal{U}}}_{A_i}=\sum_{r \in A_i} V_r^{-1} = \sum_{r \in A_i} {{\mathcal{U}}}_r, {{\mathcal{U}}}_{A_i \cup A_j}={{\mathcal{U}}}_{A_i} + {{\mathcal{U}}}_{A_j} $$

whence

$$ {\overline{{\mathbf{x}}}}_{A_i \cup A_j} = {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left[{{\mathcal{U}}}_{A_i} {\overline{{\mathbf{x}}}}_{A_i} + {{\mathcal{U}}}_{A_j} {\overline{{\mathbf{x}}}}_{A_j}\right] $$
(55)

and therefore

$$ {\overline{{\mathbf{x}}}}_{A_i}-{\overline{{\mathbf{x}}}}_{A_i \cup A_j} = {{\mathcal{U}}}_{A_i}^{-1} \sum_{r \in A_i} {{\mathcal{U}}}_r {\overline{{\mathbf{x}}}}_{r} - {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left[{{\mathcal{U}}}_{A_i}{\overline{{\mathbf{x}}}}_{A_i}+{{\mathcal{U}}}_{A_ j}{\overline{{\mathbf{x}}}}_{A_j}\right] = {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left[{{\mathcal{U}}}_{A_i \cup A_j} {\overline{{\mathbf{x}}}}_{A_i} - {{\mathcal{U}}}_{A_i} {\overline{{\mathbf{x}}}}_{A_i} - {{\mathcal{U}}}_{A_j} {\overline{{\mathbf{x}}}}_{A_j} \right] = {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left[{{\mathcal{U}}}_{A_i} {\overline{{\mathbf{x}}}}_{A_i}+ {{\mathcal{U}}}_{A_j}{\overline{{\mathbf{x}}}}_{A1} - {{\mathcal{U}}}_{A_i} {\overline{{\mathbf{x}}}}_{A_i} - {{\mathcal{U}}}_{A_j} {\overline{{\mathbf{x}}}}_{A_j} \right] = {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_j} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right) $$
(56)

and similarly

$$ {\overline{{\mathbf{x}}}}_{A_j} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j}= {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_i} \left({\overline{{\mathbf{x}}}}_{A_j} - {\overline{{\mathbf{x}}}}_{A_i} \right) $$
(57)

Substituting in the expression for d(A i , A j ) above (Eq. 54, p. 36):

$$ d\left(A_i, A_j \right) = \sum_{r \in A_i} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) + \sum_{r \in A_j} \left({\overline{{\mathbf{x}}}}_{A_j} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right)^{\rm T} V_r^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_i \cup A_j} \right) = \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)^{\rm T} {{\mathcal{U}}}_{A_j} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left(\sum_{r \in A_i} V_r^{-1} \right) {{\mathcal{U}}}_{A_j} {{\mathcal{U}}}_{A_i \cup A2}^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right) + \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right) {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left(\sum_{r \in A_j} V_r^{-1} \right) {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)= \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)^{\rm T} {{\mathcal{U}}}_{A_j} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_j} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right) + \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)^{\rm T} {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_j} {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)= \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)^{\rm T} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_j} \left({{\mathcal{U}}}_{A_i} + {{\mathcal{U}}}_{A_j} \right) {{\mathcal{U}}}_{A_i \cup A_j}^{-1} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)= \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right)^{\rm T} {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_j} \left({\overline{{\mathbf{x}}}}_{A_i} - {\overline{{\mathbf{x}}}}_{A_j} \right) $$
(58)

where we have used the commutativity of symmetric matrices, the definition \({{\mathcal{U}}}_{A_i}=\sum_{r \in A_i} V_r^{-1},\) and \({{\mathcal{U}}}_{A_i \cup A_j}={{\mathcal{U}}}_{A_i} + {{\mathcal{U}}}_{A_j}.\) Finally noting that:

$$ {{\mathcal{U}}}_{A_i \cup A_j}^{-1} {{\mathcal{U}}}_{A_i} {{\mathcal{U}}}_{A_j}= \left(V_{A_i}^{-1} + V_{A_j}^{-1} \right)^{-1} V_{A_i}^{-1} V_{A_j}^{-1}=\left(V_{A_i} + V_{A_j} \right)^{-1} $$
(59)

we have

$$ d\left(A_i,A_j\right)=\left({\overline{{\mathbf{x}}}}_{A_i}- {\overline{{\mathbf{x}}}}_{A_j}\right)^{\rm T} \left(V_{A_i} + V_{A_j} \right)^{-1} \left({\overline{{\mathbf{x}}}}_{A_i}-{\overline{{\mathbf{x}}}}_{A_j}\right) $$
(60)

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Ciampi, A., Lechevallier, Y., Limas, M.C. et al. Hierarchical clustering of subpopulations with a dissimilarity based on the likelihood ratio statistic: application to clustering massive data sets. Pattern Anal Applic 11, 199–220 (2008). https://doi.org/10.1007/s10044-007-0088-4

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