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Topic-tracking-based dynamic user modeling with TV recommendation applications

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Abstract

One of the challenging issues in TV recommendation applications based on implicit rating data is how to make robust recommendation for the users who irregularly watch TV programs and for the users who have their time-varying preferences on watching TV programs. To achieve the robust recommendation for such users, it is important to capture dynamic behaviors of user preference on watched TV programs over time. In this paper, we propose a topic tracking based dynamic user model (TDUM) that extends the previous multi-scale dynamic topic model (MDTM) by incorporating topic-tracking into dynamic user modeling. In the proposed TDUM, the prior of the current user preference is estimated as a weighted combination of the previously learned preferences of a TV user in multi-time spans where the optimal weight set is found in the sense of the evidence maximization of the Bayesian probability. So, the proposed TDUM supports the dynamics of public users’ preferences on TV programs for collaborative filtering based TV program recommendation and the highly ranked TV programs by similar watching taste user group (topic) can be traced with the same topic labels epoch by epoch. We also propose a rank model for TV program recommendation. In order to verify the effectiveness of the proposed TDUM and rank model, we use a real data set of the TV programs watched by 1,999 TV users for 7 months. The experiment results demonstrate that the proposed TDUM outperforms the Latent Dirichlet Allocation (LDA) model and the MDTM in log-likelihood for the topic modeling performance, and also shows its superiority compared to LDA, MDTM and Bayesian Personalized Rank Matrix Factorization (BPRMF) for TV program recommendation performance in terms of top-N precision-recall.

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Authors and Affiliations

  1. School of Electrical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Eunhui Kim & Munchurl Kim

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  1. Eunhui Kim
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  2. Munchurl Kim
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Correspondence to Munchurl Kim.

Appendices

Appendix A

In order to derive (6) from (5), we start with the following relation:

$$\begin{array}{@{}rcl@{}} P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) &=&{\int}_{\theta_{t}}\prod\limits_{u=1}^{U}\prod\limits_{s=1}^{S} P\left (\theta_{t,u}|\pi_{t-1,u,s},\eta_{t,u,s}\right )\\ &&\times\prod\limits_{w=1}^{W} P\left (z_{t,u,w}|\theta_{t,u}\right )d_{\theta_{t}} \end{array} $$
(A1)

If we assume that the topic distributions per user are independent of one another, (A1) can be rewritten as

$$\begin{array}{@{}rcl@{}} P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) &=&\prod\limits_{u=1}^{U}{\int}_{\theta_{t,u}}\prod\limits_{s=1}^{S} P\left (\theta_{t,u}|\pi_{t-1,u,s},\eta_{t,u,s}\right )\\ &&\times\prod\limits_{w=1}^{W} P\left (z_{t,u,w}|\theta_{t,u}\right )d_{\theta_{t,u}} \end{array} $$
(A2)

If the probability product in (A2) is expressed for the entire time span S as (.), then, we have

$$\begin{array}{@{}rcl@{}} P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) &=&\prod\limits_{u=1}^{U}{\int}_{\theta_{t,u}} P\left (\theta_{t,u}|\pi_{t-1,u,\left (. \right )},\eta_{t,u,\left (. \right )}\right)\\ &&\times\prod\limits_{w=1}^{W} P\left (z_{t,u,w}|\theta_{t,u}\right )d_{\theta_{t,u}} \end{array} $$
(A3)

Since 𝜃 t, u is a multinomial distribution with Dirichlet prior, (A3) can be rewritten as

$$\begin{array}{@{}rcl@{}} &&P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) =\prod\limits_{u=1}^{U}\\ &&\times{\int}_{\theta_{t,u}} \left [ \frac{\Gamma \left (\sum\limits_{k=1}^{K} \kappa_{k}^{t,u} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\kappa_{k}^{t,u}-1} \prod\limits_{w=1}^{W}P\left (z_{t,u,w}|\theta_{t,u}\right ) \right ]d_{\theta_{t,u}}\\ \end{array} $$
(A4)

where \(\kappa _{k}^{t,u}\equiv \pi _{t-1,u,\left (. \right )},\eta _{t,u,\left (. \right )}\). To further simplify (A4), we introduce the assumption for \({\Gamma } \left ({\sum }_{k=1}^{K}\kappa _{k}^{t,u} \right )\equiv Gamma \left (\eta _{t,u,\left (. \right )}{\sum }_{k=1}^{K}\pi _{t-1,u,\left (. \right )}\right )\) in (8) is applied for (A4), resulting in

$$\begin{array}{@{}rcl@{}} &&P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) =\prod\limits_{u=1}^{U}\\ &&\times{\int}_{\theta_{t,u}} \left [ \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\kappa_{k}^{t,u}-1} \prod\limits_{w=1}^{W}P\left (z_{t,u,w}|\theta_{t,u}\right ) \right ]d_{\theta_{t,u}}\\ \end{array} $$
(A5)

For the term \({\prod }_{w=1}^{W}P\left (z_{t,u,w}|\theta _{t,u}\right )\) in (A5), only one topic label z t, u, w is allocated for one word w. Thus, P(z t, u, w |𝜃 t, u ) can be substituted with 𝜃 t, u, k for a topic label k, which results in

$$\begin{array}{@{}rcl@{}} P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) & =&\prod\limits_{u=1}^{U}{\int}_{\theta_{t,u}} \left [ \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\kappa_{k}^{t,u}-1} \prod\limits_{k=1}^{K}\theta^{N_{t,u,k}} \right ]d_{\theta_{t,u}} \\ & =&\prod\limits_{u=1}^{U}{\int}_{\theta_{t,u}} \left [ \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\left (\kappa_{k}^{t,u}+N_{t,u,k}-1 \right )} \right ]d_{\theta_{t,u}} \\ \end{array} $$
(A6)

Equation (A6) can further be simplified by considering the Dirichlet distribution for observing the exponential part \(\kappa _{k}^{t,u}+N_{t,u,k}-1\) of 𝜃 t, u, k as

$$\begin{array}{@{}rcl@{}} &&P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) \\ && =\prod\limits_{u=1}^{U}{\int}_{\theta_{t,u}} \left [ \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\left (\kappa_{k}^{t,u}+N_{t,u,k}-1 \right )} \right ]d_{\theta_{t,u}} \\ && =\prod\limits_{u=1}^{U}{\int}_{\theta_{t,u}} \left [ \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\left (\kappa_{k}^{t,u}+N_{t,u,k}-1 \right )} \right ] \\ && \cdot \left [ \frac{\Gamma \left (\sum\limits_{k=1}^{K}\kappa_{k}^{t,u}+N_{t,u,k} \right )\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u}+N_{t,u,k} \right )} {\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u}+N_{t,u,k} \right ){\Gamma} \left (\sum\limits_{k=1}^{K}\kappa_{k}^{t,u}+N_{t,u,k} \right )} \right ]d_{\theta_{t,u}}\\ && =\prod\limits_{u=1}^{U}\left [ \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \frac{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u}+N_{t,u,k} \right )} {\Gamma \left (\sum\limits_{k=1}^{K}\kappa_{k}^{t,u}+N_{t,u,k} \right )}\right ]\\ &&\cdot \left [{\int}_{\theta_{t,u}} \frac{\Gamma \left (\sum\limits_{k=1}^{K}\kappa_{k}^{t,u}+N_{t,u,k} \right )} {\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u}+N_{t,u,k} \right ){\Gamma}} \prod\limits_{k=1}^{K}\theta_{t,u,k}^{\left (\kappa_{k}^{t,u}+N_{t,u,k}-1 \right )} \right ]d_{\theta_{t,u}} \\ \end{array} $$
(A7)

Since the integration of Dirichlet distribution in (A7) is equal to one, (A7) can be simplified to

$$ P\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t}\right ) =\prod\limits_{u=1}^{U} \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \frac{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u}+N_{t,u,k} \right )} {\Gamma \left (\sum\limits_{k=1}^{K}\kappa_{k}^{t,u}+N_{t,u,k} \right )} $$
(A8)

Similarly, (7) can be derived as (6). Based on (A8) and (7), the total evidence in (5) is expressed as

$$\begin{array}{@{}rcl@{}} && P(\mathbf{w}_{t},\mathbf{z}_{t}|{\Pi}_{t},H_{t},{\Xi}_{t},{\Lambda}_{t}) \,=\,\prod\limits_{u=1}^{U} \frac{\Gamma \left (\eta_{t,u,\left (. \right )} \right )}{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u} \right )} \frac{\prod\limits_{k=1}^{K}{\Gamma} \left (\kappa_{k}^{t,u}+N_{t,u,k} \right )} {\Gamma \left (\eta_{t,u,\left (. \right )}+N_{t,u,k} \right )} \\ &&\cdot \prod\limits_{k=1}^{K} \frac{\Gamma \left (\lambda_{t,k,\left (. \right )} \right )}{\Gamma \left (\xi_{t,k,w,\left (. \right )}\lambda_{t,k,\left (. \right )} \right )} \frac{\prod\limits_{k=1}^{K}{\Gamma} \left (\xi_{t,k,w,\left (. \right )}\lambda_{t,k,\left (. \right )}+N_{t,k,w} \right )} {\Gamma \left (\lambda_{t,k,\left (. \right )}+N_{t,k,.} \right )}\\ \end{array} $$
(A9)

Appendix B

To obtain (9) from (A9), we simulate (B1) by Gibbs Sampling [10, 21–23] such as

$$\begin{array}{@{}rcl@{}} &&P(z_{t,j}=k|\mathbf{z}_{t,-j},\mathbf{w}_{t},{\Pi}_{t-1},H_{t},{\Xi}_{t-1},{\Lambda}_{t})\\ && \propto P(w_{t,j}|\mathbf{z}_{t,-j},\mathbf{w}_{t,-j},{\Xi}_{t-1},{\Lambda}_{t})\\ &&\cdot P(z_{t,j}=k|\mathbf{z}_{t,-j}{\Pi}_{t-1},H_{t})\\ && = \left [\frac{N_{t,k,w_{t,j},-j}+\sum\limits_{s=0}^{S}\xi_{t-1,k,w_{t,j},s}\lambda_{t,k,s}}{N_{t,k,.-j}+\sum\limits_{s=0}^{S}\lambda_{t,k,s}}\right ]\\ &&\cdot \left [ \frac{N_{t,u,k,-j}+\sum\limits_{s=0}^{S}\pi_{t-1,u,k,s}\eta_{t,u,s}}{N_{t,u,.-j}+\sum\limits_{s=0}^{S}\lambda_{t,u,s}}\right ] \end{array} $$
(B1)

Appendix C

Equation (11) is derived from [14]. Similarly, (10) can also be derived on the basis of the total probability evidence in (A9). For this, we only consider the η related terms for formulation and treat the other terms as a constant C with log probability of

$$\begin{array}{@{}rcl@{}} &&logP\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t} \right ) + C \\ &&=\sum\limits_{u=1}^{U}\left [ log{\Gamma} \left (\sum\limits_{s=0}^{S}\eta_{t,u,s}\right )- log{\Gamma} \left (N_{t,u,.}+\sum\limits_{s=0}^{S}\eta_{t,u,s}\right ) \right ] \\ &&+\sum\limits_{u=1}^{U}\sum\limits_{k=1}^{K}\left [ log{\Gamma} \left (N_{t,u,k}+\sum\limits_{s=0}^{S}\pi_{t-1,u,k,s}\eta_{t,u,s}\right )\right.\\ &&\qquad\qquad\left.-log{\Gamma} \left (\sum\limits_{s=0}^{S}\pi_{t-1,u,k,s}\eta_{t,u,s}\right ) \right ]\\ &&+C \end{array} $$
(C1)

Using the following two inequalities in (C2) and (C3) [26], we get (C4) from (C1) as follows:

$$ log\frac{\Gamma \left (x \right )}{\Gamma \left (n + x \right )} \geq log\frac{\Gamma \left (\hat{x} \right )}{\Gamma \left (n + \hat{x} \right )} + \left ({\Psi}\left (\hat{x} \right )-{\Psi}\left (n + \hat{x} \right ) \right )\left (x-\hat{x} \right ) $$
(C2)
$$ \renewcommand{\theequation}{C\arabic{equation}} log\frac{\Gamma \left (n + x \right )}{\Gamma \left (x \right )} \geq log\frac{\Gamma \left (n + \hat{x} \right )}{\Gamma \left (\hat{x} \right )} + \hat{x}\left( {\Psi}\left (n + \hat{x} \right )-{\Psi}\left (\hat{x} \right ) \right )log\frac{x}{\hat{x}} $$
(C3)
$$\begin{array}{@{}rcl@{}} &&logP\left (\mathbf{z}_{t}|{\Pi}_{t-1},H_{t} \right )+C \\ && \geqslant -\sum\limits_{u=1}^{U}\left [ {\Psi} \left (N_{t,u,.} + \sum\limits_{s=0}^{S} \eta_{t,u,s}^{old}\right ) - {\Psi} \left (\sum\limits_{s=0}^{S} \eta_{t,u,s}^{old}\right ) \right ]\left (\sum\limits_{s=0}^{S} \eta_{t,u,s}\right )\\ && +\sum\limits_{u=1}^{U}\sum\limits_{k=1}^{K}\left [ {\Psi} \left (N_{t,u,k} + \sum\limits_{s=0}^{S} \pi_{t-1,u,k,s}\eta_{t,u,s}^{old}\right )\right.\\ &&~~\qquad\qquad\left. - {\Psi} \left (\sum\limits_{s=0}^{S} \pi_{t-1,u,k,s}\eta_{t,u,s}^{old}\right ) \right ]\\ && \cdot \left (\sum\limits_{s=0}^{S} \pi_{t-1,u,k,s}\eta_{t,u,s}^{old}\right ) \cdot log\left (\sum\limits_{s=0}^{S} \pi_{t-1,u,k,s}\eta_{t,u,s}\right ) + C^{\prime}\\ && = F\left (\eta_{t,u,s} \right ) \end{array} $$
(C4)

Taking the derivative on (C4) with respect to η t, u, s and setting it to zero, then we obtain (10).

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Kim, E., Kim, M. Topic-tracking-based dynamic user modeling with TV recommendation applications. Appl Intell 44, 771–792 (2016). https://doi.org/10.1007/s10489-015-0720-8

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