unsupervised-multimodal-trajectory-modeling 2024.2.1

trains mixtures of state space models with expectation maximization

Unsupervised Multimodal Trajectory Modeling

We propose and validate a mixture of state space models to perform unsupervised clustering of short trajectories. Within the state space framework, we let expensive-to-gather biomarkers correspond to hidden states and readily obtainable cognitive metrics correspond to measurements. Upon training with expectation maximization, we find that our clusters stratify persons according to clinical outcome. Furthermore, we can effectively predict on held-out trajectories using cognitive metrics alone. Our approach accommodates missing data through model marginalization and generalizes across research and clinical cohorts.

Data format

We consider a training dataset

$$ \mathcal{D} = {(x_{1:T}^{i}, z_{1:T}^{i}) }_{1 \leq i \leq n_d} $$

consisting of $n_d$ sequences of states and observations paired in time. We denote the states $z_{1:T}^{i} = (z_1^i, z_2^i, \dotsc, z_T^i)$ where $z_t^i \in \mathbb{R}^d$ corresponds to the state at time $t$ for the $i$ th instance and measurements $x_{1:T}^{i} = (x_1^i, x_2^i, \dotsc, x_T^i)$ where $x_t^i \in \mathbb{R}^\ell$ corresponds to the observation at time $t$ for the $i$ th instance. For the purposes of this code, we adopt the convention that collections of time-delineated sequences of vectors will be stored as 3-tensors, where the first dimension spans time $1\leq t \leq T$, the second dimension spans instances $1\leq i \leq n_d$ (these will almost always correspond to an individual or participant), and the third dimension spans the components of each state or observation vector (and so will have dimension either $d$ or $\ell$). We accommodate trajectories of differing lengths by standardising to the longest available trajectory in a dataset and appending np.nan's to shorter trajectories.

Model specification

We adopt a mixture of state space models for the data:

given explicitly by:

$$ p(z^i_{1:T}, x^i_{1:T}) = \sum_{c=1}^{n_c} \pi_{c} \delta_{ \{c=c^i \} } \bigg( p(z_1^i| c) \prod_{t=2}^T p(z_t^i | z_{t-1}^i, c) \prod_{t=1}^T p(x_t^i | z_t^i, c) \bigg) $$

Each individual $i$ is independently assigned to some cluster $c^i$ with probability $\pi_{c}$, and then conditional on this cluster assignment, their initial state $z_1^i$ is drawn according to $p(z_1^i| c)$, with each subsequent state $z_t^i, 2\leq t \leq T$ being drawn in turn using the cluster-specific state model $p(z_t^i | z_{t-1}^i, c)$, depending on the previous state. At each point in time, we obtain an observation $x_t^i$ from the cluster-specific measurement model $p(x_t^i | z_t^i, c)$, depending on the current state. In what follows, we assume both the state and measurement models are stationary for each cluster, i.e. they are independent of $t$. In particular, for a given individual, the relationship between the state and measurement should not change over time.

In our main framework, inspired by the work of Chiappa and Barber[^1], we additionally assume that the cluster-specific state initialisation is Gaussian, i.e. $p(z_1^i| c) = \eta_d(z_1^i; m_c, S_c)$, and the cluster-specific state and measurement models are linear Gaussian, i.e. $p(z_t^i | z_{t-1}^i, c) = \eta_d(z_t^i; z_{t-1}^iA_c, \Gamma_c)$ and $p(x_t^i | z_t^i, c) = \eta_\ell(x_t^i; z_t^iH_c, \Lambda_c)$, where $\eta_d(\cdot, \mu, \Sigma)$ denotes the multivariate $d$-dimensional Gaussian density with mean $\mu$ and covariance $\Sigma$, yielding:

$$ p(z^i_{1:T}, x^i_{1:T}) = \sum_{c=1}^{n_c} \pi_{c} \delta_{ \{c=c^i \} } \bigg( \eta_d(z_1^i; m_c, S_c) \prod_{t=2}^T \eta_d(z_t^i; z_{t-1}^iA_c, \Gamma_c) \prod_{t=1}^T \eta_\ell(x_t^i; z_t^iH_c, \Lambda_c) \bigg). $$

In particular, we assume that the variables we are modeling are continuous and changing over time. When we train a model like the above, we take a dataset $\mathcal{D}$ and an arbitrary set of cluster assignments $c^i$ (as these are also latent/ hidden from us) and iteratively perform M and E steps (from which EM[^2] gets its name):

• [E] Expectation step: given the current model, we assign each data instance $(z^i_{1:T}, x^i_{1:T})$ to the cluster to which it is mostly likely to belong under the current model
• [M] Maximization step: given the current cluster assignments, we compute the sample-level cluster assignment probabilities (the $\pi_c$) and optimal cluster-specific parameters

Optimization completes after a fixed (large) number of steps or when no data instances change their cluster assignment at a given iteration.

Adapting the code for your own use

A typical workflow is described at: https://github.com/burkh4rt/Unsupervised-Trajectory-Clustering-Starter

Caveats & Troubleshooting

Some efforts have been made to automatically handle edge cases. For a given training run, if any cluster becomes too small (fewer than 3 members), training terminates. In order to learn a model, we make assumptions about our training data as described above. While our approach seems to be robust to some types of model misspecification, we have encountered training issues with the following problems:

1. Extreme outliers. An extreme outlier tends to want to form its own cluster (and that's problematic). In many cases this may be due to a typo or failed data-cleaning (i.e. an upstream problem). Generating histograms of each feature is one way to recognise this problem.
2. Discrete / static features. Including discrete data violates our Gaussian assumptions. If we learn a cluster where each trajectory has the same value for one of the states or observations at a given time step, then we are prone to estimating a singular covariance structure for this cluster which yields numerical instabilities. Adding a small bit of noise to discrete features may remediate numerical instability to some extent.

Another assumption that is easy-to-violate is our stationarity assumption for the measurement model.

[^1]: S. Chiappa and D. Barber. Dirichlet Mixtures of Bayesian Linear Gaussian State-Space Models: a Variational Approach. Tech. rep. 161. Max Planck Institute for Biological Cybernetics, 2007.

[^2]: A. Dempster, N. Laird, and D. B. Rubin. Maximum Likelihood from
Incomplete Data via the EM Algorithm. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 39.1 (1977), pp. 1–38.