A Kernel to Exploit Informative Missingness in Multivariate Time Series from EHRs

Abstract

A large fraction of the electronic health records (EHRs) consists of clinical measurements collected over time, such as lab tests and vital signs, which provide important information about a patient’s health status. These sequences of clinical measurements are naturally represented as time series, characterized by multiple variables and large amounts of missing data, which complicate the analysis. In this work, we propose a novel kernel which is capable of exploiting both the information from the observed values as well the information hidden in the missing patterns in multivariate time series (MTS) originating e.g. from EHRs. The kernel, called TCK$_{IM}$, is designed using an ensemble learning strategy in which the base models are novel mixed mode Bayesian mixture models which can effectively exploit informative missingness without having to resort to imputation methods. Moreover, the ensemble approach ensures robustness to hyperparameters and therefore TCK$_{IM}$ is particularly well suited if there is a lack of labels—a known challenge in medical applications. Experiments on three real-world clinical datasets demonstrate the effectiveness of the proposed kernel.

Appendix—Synthetic benchmark datasets

To test how well TCK$_{IM}$ performs for a varying degree of informative missingness, we generated in total 16 synthetic datasets by randomly injecting missing data into 4 MTS benchmark datasets. The characteristics of the datasets are described in Table 4. We transformed all MTS in each dataset to the same length, T, where T is given by $T = \left\lceil T_{max} / \left\lceil T_{max}/25 \right\rceil \right\rceil .$ Here, $\lceil \, \rceil$ is the ceiling operator and $T_{max}$ is the length of the longest MTS in the original dataset.

Table 4 Characteristics of benchmark datasets. Attr is number of attributes, Train and Test number of training and test samples. $N_c$ is the number of classes, $T_{min}$ and $T_{max}$ length of shortest and longest MTS, and T is the MTS length after the transformation

Datasets. The following procedure was used to create 8 synthetic datasets with missing data from the Wafer and Japanese vowels datasets. We randomly sampled a number $c_v \in \{-1, 1\}$ for each attribute $v \in \{1,\dots , V\}$, where $c_v =1$ indicates that the attribute and the labels are positively correlated and $c_v =-1$ negatively correlated. Thereafter, we sampled a missing rate $\gamma _{nv}$ from $\mathcal {U}[ 0.3 + E \cdot c_v \cdot (y^{(n)}-1), 0.7 + E \cdot c_v \cdot (y^{(n)}-1)]$ for each MTS $X^{(n)}$ and attribute. The parameter E was tuned such that the Pearson correlation (absolute value) between the missing rates for the attributes $\gamma _v$ and the labels $y^{(n)}$ took the values $\{0.2, \, 0.4, \, 0.6, \, 0.8\}$, respectively. By doing so, we could control the amount of informative missingness and because of the way we sampled $\gamma _{nv}$, the missing rate in each dataset was around 50% independently of the Pearson correlation. Further, the following procedure was used to create 8 synthetic datasets from the uWave and Character trajectories datasets, which both consist of only 3 attributes. We randomly sampled a number $c_v \in \{-1, 1\}$ for each attribute $v \in \{1,\dots , V\}$. Attribute(s) with $c_v = -1$ became negatively correlated with the labels by sampling $\gamma _{nv}$ from $\mathcal {U}[ 0.7 - E \cdot (y^{(n)}-1), 1 - E \cdot (y^{(n)}-1)]$, whereas the attribute(s) with $c_v = 1$ became positively correlated with the labels by sampling $\gamma _{nv}$ from $\mathcal {U}[ 0.3 + E \cdot (y^{(n)}-1), 0.6 + E \cdot (y^{(n)}-1)]$. The parameter E was computed in the same way as above. Then, we computed the mean of each attribute $\mu _v$ over the complete dataset and let each element with $x^{(n)}_v(t) > \mu _v$ be missing with probability $\gamma _{nv}$. This means that the probability of being missing is dependent on the value of the missing element, i.e. the missingness mechanism is MNAR within each class. Hence, this type of informative missingness is not the same as the one we created for the Wafer and Japanese vowels datasets.

Baselines. Three baseline models were created. The first baseline, namely ordinary TCK, ignores the missingness mechanism. In the second one, refered to as TCK$_B$, we modeled the missing patterns naively by concatenating the binary missing indicator MTS R to the MTS X and creating a new MTS U with 2V attributes. Then, ordinary TCK was trained on the datasets consisting of $\{U^{(n)}\}$. In the third baseline, TCK$_0$, we investigated how well informative missingness can be captured by imputing zeros for the missing values and then training the TCK on the imputed data.

Table 5 Performance (accuracy) of TCK$_{IM}$ and three baselines

Results. Table 5 shows the performance of the proposed TCK$_{IM}$ and the three baselines for all of the 16 synthetic datasets. We see that the proposed TCK$_{IM}$ achieves the best accuracy for 14 out of 16 datasets and is the only method which consistently has the expected behaviour, namely that the accuracy increases as the correlation between missing values and class labels increases. It can also be seen that the performance of TCK$_{IM}$ is similar to TCK when the amount of information in the missing patterns is low, whereas TCK is clearly outperformed when the informative missingness is high. This demonstrates that TCK$_{IM}$ can effectively exploit informative missingness.