Clustering and estimation of finite mixture models under bivariate ranked set sampling with application to a breast cancer study

Abstract

In the literature on modeling heterogeneous data via mixture models, it is generally assumed that the samples are drawn from the underlying population using the simple random sampling (SRS) technique. This study exploits the bivariate ranked set sampling (BVRSS) technique to learn finite mixture models. We generalize the expectation-maximization (EM) algorithm under univariate RSS to the bivariate case. Computationally, through a simulation study under a noisy setting, we compare the performance of the proposed rank-based estimators with that of the SRS-based competitors in estimating unknown parameters and cluster assignments. The proposed methodology is applied to a breast cancer data set to diagnose malignant or benign tumors in patients. The results showed that the extra rank information in BVRSS samples leads to a better inference about the unknown features of mixture models.

Appendix

1.1 Appendix A.1.: Proof of Lemma 3.1

The conditional pdf’s of the latent variables are

$$\begin{aligned}&f\big (\left. {z_{sk}^{(ij)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) = {{f\big (\left. {{x_{[i](j)}},{y_{(i)[j]s}}} \right| z_{sk}^{(ij)},\varvec{\Psi } \big ) f\big (z_{sk}^{(ij)}\big )} \over {{f^{[i](j),(i)[j]}} \big ({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\Psi } \big )}}\\&\quad = {{{c_1}\prod \limits _{k = 1}^K {{{\{ {f_k}({x_{[i](j)s}},{y_{(i)[j]s}};{\varvec{\xi }_k})\} }^{z_{sk}^{(ij)}}}} {{\{ F({x_{[i](j)s}};\varvec{\Psi })\} }^{j - 1}}{{\{ {\bar{F}}({x_{[i](j)s}};\varvec{\Psi })\} }^{m - j}}{{\{ {F^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{i - 1}}{{\{ {{\bar{F}}^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{m - i}} } \over {{c_1}f({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\Psi }){{\{ F({x_{[i](j)s}};\varvec{\Psi })\} }^{j - 1}}{{\{ \bar{F}({x_{[i](j)s}};\varvec{\Psi })\} }^{m - j}}{{\{ {F^{[j]}}({y_{(i)[j]s}};\Psi )\} }^{i - 1}}{{\{ {{\bar{F}}^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{m - i}}}} \\&\qquad \times \prod \limits _{k = 1}^K {\pi _k^{z_{sk}^{(ij)}}} = \prod \limits _{k = 1}^K {{{ \left( {{{\pi _k}{f_k}\big ({x_{[i](j)s}},{y_{(i)[j]s}};{\varvec{\xi }_k}\big )} \over {f\big ({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\Psi } \big )}} \right) }^{z_{sk}^{(ij)}}}},\\&f\big (\left. {w_{sk}^{(j)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi }\big ) = {{f\big (\left. {{x_{[i](j)}},{y_{(i)[j]s}}} \right| w_{sk}^{(j)},\varvec{\Psi }\big ) f\big (w_{sk}^{(j)}\big )} \over {{f^{[i](j),(i)[j]}}\big ({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\Psi }\varvec{)}}}\\&\quad = {{{c_1}{c_2}f({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\Psi })\prod \limits _{k = 1}^K {{{\{ {F_k}({x_{[i](j)s}};{\varvec{\xi }_k})\} }^{w_{sk}^{(j)}}}} {{\{ {\bar{F}}({x_{[i](j)s}};\varvec{\Psi })\} }^{m - j}}{{\{ {F^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{i - 1}}{{\{ {{\bar{F}}^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{m - i}}} \over {{c_1}f({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\Psi }){{\{ F({x_{[i](j)s}};\varvec{\Psi })\} }^{j - 1}}{{\{ \bar{F}({x_{[i](j)s}};\varvec{\Psi })\} }^{m - j}}{{\{ {F^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{i - 1}}{{\{ {{\bar{F}}^{[j]}}({y_{(i)[j]s}};\varvec{\Psi })\} }^{m - i}}}}\\&\qquad \times \prod \limits _{k = 1}^K {\pi _k^{w_{sk}^{(j)}}} = {c_2}\prod \limits _{k = 1}^K {{{\left( {{{\pi _k}{F_k}\big ({x_{[i](j)s}};{\varvec{\xi }_k}\big )} \over {F\big ({x_{[i](j)s}};\varvec{\Psi }\big )}}\right) }^{w_{sk}^{(j)}}}}, \end{aligned}$$

In such a way, $f\big (\left. {v_{sk}^{(j)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) = {c_3}\prod \limits \nolimits _{k = 1}^K {{{\left( {{{\pi _k}{{\bar{F}}_k}({x_{[i](j)s}};{\varvec{\xi }_k})} \over {\bar{F}({x_{[i](j)s}};\varvec{\Psi }) }}\right) }^{v_{sk}^{(j)}}}}$, $f\big (\left. {u_{sk}^{(i)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big )={c_4}\prod \limits \nolimits _{k = 1}^K {{{\left( {{{\pi _k}F_k^{[j]} ({y_{(i)[j]s}};{\varvec{\xi }_k})} \over {{F^{[j]}}({y_{(i)[j]s}};\varvec{\Psi }) }}\right) }^{u_{sk}^{(i)}}}}$, and $f\big (\left. {d_{sk}^{(i)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) = {c_5}\prod \limits \nolimits _{k = 1}^K {{{\left( {{{\pi _k}{\bar{F}}_k^{[j]} ({y_{(i)[j]s}};{\varvec{\xi }_k})} \over {{{\bar{F}}^{[j]}}({y_{(i)[j]s}};\varvec{\Psi }) }}\right) }^{d_{sk}^{(i)}}}}$.

Since the latent variables are independent variables, we have

$$\begin{aligned}&f\big ({x_{[i](j)s}},{y_{(i)[j]s}}, \varvec{z}_s^{(ij)}, \varvec{w}_s^{(j)}, \varvec{v}_s^{(j)}, \varvec{u}_s^{(i)}, \varvec{d}_s^{(i)}; \varvec{\Psi }\big ) \\&\quad = f \big (\left. {z_{sk}^{(ij)},w_{sk}^{(j)},v_{sk}^{(j)},u_{sk}^{(i)} ,d_{sk}^{(i)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) {f^{[i](j),(i)[j]}}\big ({x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big )\\&\quad = f \big (\left. {z_{sk}^{(ij)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) f \big (\left. {w_{sk}^{(j)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) f \big (\left. {v_{sk}^{(j)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big )\\&\qquad \times f\big (\left. {u_{sk}^{(i)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) f\big (\left. {d_{sk}^{(i)}} \right| {x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ){f^{[i](j),(i)[j]}} \big ({x_{[i](j)s}},{y_{(i)[j]s}},\varvec{\Psi } \big ) \\&\quad = c_1 c_2 c_3 c_4 c_5 \Bigg \{ \prod \limits _{k = 1}^K{ {\pi _k^{ z_{sk}^{(ij)} + w_{sk}^{(j)} + v_{sk}^{(j)} +u_{sk}^{(i)} + d_{sk}^{(i)} } } {\big \{ {f_k}\big ({x_{[i](j)s}},{y_{(i)[j]s}};\varvec{\xi }_k \big )\big \} ^{z_{sk}^{(ij)}}}} \\&\qquad \times {{\big \{ {F_k}\big ({x_{[i](j)s}};\varvec{\xi }_k \big )\big \} ^{w_{sk}^{(j)}}}{\big \{ {{\bar{F}}_k}\big ({x_{[i](j)s}};\varvec{\xi }_k \big )\big \} ^{v_{sk}^{(j)}}}{\big \{ F_k^{[j]}\big ({y_{(i)[j]s}};\varvec{\xi }_k \big )\big \} ^{u_{sk}^{(i)}}}{\big \{ \bar{F}_k^{[j]}\big ({y_{(i)[j]s}};\varvec{\xi }_k \big )\big \} ^{d_{sk}^{(i)}}} } \Bigg \}. \end{aligned}$$

1.2 Appendix A.2.: Proof of equation (22)

For maximization (21) with respect to $\pi _k$, we eliminate the terms that do not depend on $\pi _k$. Therefore, the quantity ${Q_{{M_1}}}(\left. \varvec{\Psi } \right| {\varvec{\Psi }^{(t)}})$ reduces to $Q_{M_1}(\pi _k)$ as follows

$$\begin{aligned} Q_{M_1}({\pi _k}) = (2m-1)\sum \limits _{s = 1}^r {\sum \limits _{j = 1}^m {\sum \limits _{i = 1}^m {\sum \limits _{k = 1}^K {{\eta _{k,{M_1}}}({x_{[i](j)s}},{y_{(i)[j]s}};{\varvec{\Psi }^{(t)}}) \log {\pi _k} } } } }, \end{aligned}$$

where ${\eta _{k,{M_1}}}({x_{[i](j)s}},{y_{(i)[j]s}};{\varvec{\Psi }^{(t)}})$ is defined in (23). We use the Lagrangian multiplier method under the constraint $\sum \nolimits _{k=1}^K {\pi _k} = 1$. The Lagrangian function is

$$\begin{aligned} {\mathcal {L}}({\pi _k},\lambda ) = {Q_{{M_1}}}({\pi _k}) - \lambda (\sum \limits _{k = 1}^K {{\pi _k}} - 1). \end{aligned}$$

By setting the differential of ${\mathcal {L}}({\pi _k},\lambda )$ with respect to $\pi _k$ equal to zero, we have

$$\begin{aligned} \pi _{k,{M_1}}^{(t + 1)} = {1 \over \lambda }\sum \limits _{s = 1}^r {\sum \limits _{j = 1}^m {\sum \limits _{i = 1}^m {{\eta _{k,{M_1}}}({x_{[i](j)s}},{y_{(i)[j]s}};{\varvec{\Psi }^{(t)}})} } } , \end{aligned}$$

where from (16)–(20), we have

$$\begin{aligned} \lambda&= \sum \limits _{s = 1}^r {\sum \limits _{j = 1}^m {\sum \limits _{i = 1}^m {\sum \limits _{k = 1}^K {{\eta _{k,{M_1}}}({x_{[i](j)s}},{y_{(i)[j]s}};{\varvec{\Psi }^{(t)}})} } } }\\&= \sum \limits _{s = 1}^r {\sum \limits _{j = 1}^m {\sum \limits _{i = 1}^m {1 \over {(2m-1)}} \big \{1+ (j-1) + (m-j) + (i-1) + (m-i)\big \} }} \\&= rm^2. \end{aligned}$$