A Heterogeneous Clustering Approach for Human Activity Recognition

Abstract

Human Activity Recognition (HAR) has a growing research interest due to the widespread presence of motion sensors on user’s personal devices. The performance of HAR system deployed on large-scale is often significantly lower than reported due to the sensor-, device-, and person-specific heterogeneities. In this work, we develop a new approach for clustering such heterogeneous data, represented as a time series, which incorporates different level of heterogeneities in the data within the model. Our method is to represent the heterogeneities as a hierarchy where each level in the hierarchy overcomes a specific heterogeneity (e.g., a sensor-specific heterogeneity). Experimental evaluation on Electromyography (EMG) sensor dataset with heterogeneities shows that our method performs favourably compared to other time series clustering approaches.

Appendix A Posterior Characterization

• $\alpha _i$: The posterior for $\alpha _i$ is:

  $$\begin{aligned} \begin{aligned} f(\alpha _i | rest) \propto N_P(\mu _a, \varSigma _a)\\ \varSigma _a = (\varSigma _{\alpha }^{-1} + Z^T\varSigma _{y}^{-1}Z)^{-1}\\ \mu _a = \varSigma _a Z^T \varSigma _y^{-1} (y_i - X \beta _i - \theta _i)\\ f(\sigma _{\alpha _j}^2 | rest) = IGa(c_0^{\alpha } + \frac{n}{2}, c_1^{\alpha } + \frac{1}{2} \sum _{i=1}^n \alpha _{ij}^2), j = 1,..,p \end{aligned} \end{aligned}$$

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• $\beta _i$: The posterior for $\beta _i$ (or $\beta _{s,r,k}$) is:

  $$\begin{aligned} \begin{aligned} f(\beta _i | rest) \propto N_D (\mu _b, \varSigma _b)\\ \varSigma _b = (\varSigma _{\beta ,s,r,k}^{-1} + X^T \varSigma _{y}^{-1}X )^{-1}\\ \mu _b = \varSigma _b [ X^T \varSigma _{y}^{-1} (y_i - Z \alpha _i - \theta _i) + \varSigma _{\beta ,s,r,k} \overline{\beta _{s,r,k}} ]\\ f(\sigma _{\beta _{s,r,k,i}}^2 | rest) = IGa(c_0^{\beta _{s,r,k,i}} + \frac{m}{2}, c_1^{\beta _{s,r,k,i}} + \frac{1}{2} \sum _{j=1}^m \beta _{s,r,k,i}^2)\\ i = 1,..,p \end{aligned} \end{aligned}$$

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  where m is the number of data points belonging to that cluster.

• $\theta _i$: The posterior for $\theta _i$ (or $\theta _{s,r,k}$) is:

  $$\begin{aligned} \begin{aligned} f(\theta _i | rest) \propto N_T (\mu _c, \varSigma _c)\\ \varSigma _c = (\varSigma _{\theta ,s,r,k}^{-1} + \varSigma _{y}^{-1})^{-1}\\ \mu _c = \varSigma _c [\varSigma _{y}^{-1} (y_i - Z \alpha _i - X \beta _i) + \varSigma _{\theta ,s,r,k} \overline{\theta _{s,r,k}}]\\ f(\sigma _{\theta _{s,r,k,i}}^2 | rest) = IGa(c_0^{\theta _{s,r,k,i}} + \frac{m}{2}, c_1^{\theta _{s,r,k,i}} + \frac{1}{2} \sum _{j=1}^m \theta _{s,r,k,i}^2)\\ i = 1,..,T \end{aligned} \end{aligned}$$

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  where m is the number of data points belonging to that cluster.

• $\sigma _{\epsilon _i}^2$: The posterior for $\sigma _{\epsilon _i}^2$ is:

  $$\begin{aligned} \begin{aligned} f(\sigma _{\epsilon _i}^2 | rest) \propto IGa(c_0^{\epsilon } + \frac{T}{2}, c_1^{\epsilon } + \frac{1}{2} M_i^{'}M_i)\\ M_i = (y_i - Z \alpha _i - X \beta _i - \theta _i) \end{aligned} \end{aligned}$$

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• Level k posterior: The posterior for any level of hierarchy except for top-most level consists of following updates:

  $$\begin{aligned} \begin{aligned} f(\beta _k | rest) \propto N_D(\mu _g, \varSigma _g)\\ \varSigma _g = (\varSigma _{\beta ,r,k}^{-1} + \varSigma _{\beta ,k}^{-1} )^{-1}\\ \mu _g = \varSigma _g (\varSigma _{\beta ,k}\overline{\beta _{k}} + \varSigma _{\beta ,r,k}^{-1} \beta _{r,k})\\ f(\sigma _{\beta _{k,i}}^2 | rest) = IGa(c_0^{\beta _{k,i}} \frac{R}{2}, c_1^{\beta _{k,i}} \sum _{j=1}^S \beta _{k,i}^2)\\ f(\theta _k | rest) \propto N_D(\mu _h, \varSigma _h)\\ \varSigma _h = (\varSigma _{\theta ,r,k}^{-1} + \varSigma _{\theta ,k}^{-1} )^{-1}\\ \mu _h = \varSigma _h (\varSigma _{\theta ,k}\overline{\theta _{k}} + \varSigma _{\theta ,r,k}^{-1} \theta _{r,k})\\ f(\sigma _{\theta _{k,i}}^2 | rest) = IGa(c_0^{\theta _{k,i}} \frac{R}{2}, c_1^{\theta _{k,i}} \sum _{j=1}^R \beta _{k,i}^2) \end{aligned} \end{aligned}$$

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• Top level posterior: The posterior at top-most level is:

  $$\begin{aligned} \begin{aligned} f(\beta | rest) \propto N_D(\mu _e, \varSigma _e)\\ \varSigma _e = (\varSigma _{\beta ,k}^{-1} + \varSigma _{\beta }^{-1} )^{-1}\\ \mu _e = \varSigma _e (\varSigma _{\beta ,k}^{-1} \beta _{k})\\ f(\sigma _{\beta _{i}}^2 | rest) = IGa(c_0^{\beta _{i}} \frac{K}{2}, c_1^{\beta _{i}} \sum _{j=1}^K \beta _{i}^2)\\ f(\theta | rest) \propto N_D(\mu _f, \varSigma _f)\\ \varSigma _f = (\varSigma _{\theta ,k}^{-1} + \varSigma _{\theta }^{-1} )^{-1}\\ \mu _f = \varSigma _f (\varSigma _{\theta ,k}^{-1} \theta _{k})\\ f(\sigma _{\theta }^2 | rest) = IGa(\frac{KT}{2}, \frac{1}{2} \sum _{j=1}^K \theta _j^{'} Q^{-1}\theta _j)\\ f(\rho | rest) \propto |Q|^{-K/2} \exp {\frac{-1}{2 \sigma _{\theta }^2} \sum _{j=1}^K \theta _j^{'} Q^{-1}\theta _j} \frac{\sqrt{1 + \rho ^2}}{1 - \rho ^2} \end{aligned} \end{aligned}$$

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  where $Q_{ij} = \rho ^{|i-j|}$ for $i,j = 1,..,T$.

• Posterior for GDD and p: The posterior for GDD is conjugate with multinomial sampling. The probability p is updated based on the fit of the data with respect to the individual clusters lowest level mean using the likelihood function. The complete detail for GDD posterior characterization can be found in [6].

This completes the posterior characterization of our approach.