Package: unsupervised-multimodal-trajectory-modeling
Unsupervised Multimodal Trajectory ModelingWe 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 formatWe 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 appendingnp.nan's to shorter trajectories.Model specificationWe adopt a mixture of state space models for the data:$$
p(z^i_{1:T}, x^i_{1:T})
= \sum_{c=1}^{n_c} \pi_{c} \delta_{ \{c=c^i \} }
\bigg( p(z_1^i| c)  \prod_{t=2}^T p(z_t^i | z_{t-1}^i, c)
\prod_{t=1}^T p(x_t^i | z_t^i, c) \bigg)
$$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-specificstate 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-specificmeasurement 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 parametersOptimization 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 useA typical workflow is described at:https://github.com/burkh4rt/Unsupervised-Trajectory-Clustering-StarterCaveats & TroubleshootingSome 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: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.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 fromIncomplete Data via the EM Algorithm.J. Roy. Stat. Soc. Ser. B (Stat.
Methodol.) 39.1 (1977), pp. 1–38.