Add support for inhomogeneous parameters

Author: Hylke Donker

dynamax/linear_gaussian_ssm/models.py

@@ -7,12 +7,12 @@
 
 from fastprogress.fastprogress import progress_bar
 from functools import partial
-from jax import jit, vmap
+from jax import jit, tree, vmap
 from jax.tree_util import tree_map
 from jaxtyping import Array, Float
 from tensorflow_probability.substrates.jax.distributions import MultivariateNormalFullCovariance as MVN
 from typing import Any, Optional, Tuple, Union, runtime_checkable
-from typing_extensions import Protocol 
+from typing_extensions import Protocol
 
 from dynamax.ssm import SSM
 from dynamax.linear_gaussian_ssm.inference import lgssm_joint_sample, lgssm_filter, lgssm_smoother, lgssm_posterior_sample
@@ -206,7 +206,7 @@ def sample(self,
                key: PRNGKeyT,
                num_timesteps: int,
                inputs: Optional[Float[Array, "num_timesteps input_dim"]] = None) \
-                -> Tuple[Float[Array, "num_timesteps state_dim"], 
+                -> Tuple[Float[Array, "num_timesteps state_dim"],
                          Float[Array, "num_timesteps emission_dim"]]:
         """Sample from the model.
         
@@ -357,7 +357,7 @@ def forecast(self,
                 input_weights=params.emissions.input_weights,
                 cov=1e8 * jnp.ones(self.emission_dim)) # ignore dummy observatiosn
             )
-        
+
         dummy_emissions = jnp.zeros((num_forecast_timesteps, self.emission_dim))
         forecast_inputs = forecast_inputs if forecast_inputs is not None else \
             jnp.zeros((num_forecast_timesteps, 0))
@@ -367,7 +367,7 @@ def forecast(self,
         H = params.emissions.weights
         b = params.emissions.bias
         R = params.emissions.cov if params.emissions.cov.ndim == 2 else jnp.diag(params.emissions.cov)
-        
+
         forecast_emissions = forecast_states.filtered_means @ H.T + b
         forecast_emissions_cov = H @ forecast_states.filtered_covariances @ H.T + R
         return forecast_states.filtered_means, \
@@ -643,6 +643,47 @@ def m_step(self,
         )
         return params, m_step_state
 
+    def _check_params(self, params: ParamsLGSSM, num_timesteps: int) -> ParamsLGSSM:
+        """Replace None parameters with zeros."""
+        dynamics, emissions = params.dynamics, params.emissions
+        is_inhomogeneous = dynamics.weights.ndim == 3
+
+        def _zeros_if_none(x, shape):
+            if x is None:
+                return jnp.zeros(shape)
+            return x
+
+        shape_prefix = ()
+        if is_inhomogeneous:
+            shape_prefix = (num_timesteps - 1,)
+
+        clean_dynamics = ParamsLGSSMDynamics(
+                weights=dynamics.weights,
+                bias=_zeros_if_none(dynamics.bias, shape=shape_prefix + (self.state_dim,)),
+                input_weights=_zeros_if_none(
+                    dynamics.input_weights, shape=shape_prefix + (self.state_dim, self.input_dim)
+                ),
+                cov=dynamics.cov
+            )
+        shape_prefix = ()
+        if is_inhomogeneous:
+            shape_prefix = (num_timesteps,)
+
+        clean_emissions = ParamsLGSSMEmissions(
+            weights=emissions.weights,
+            bias=_zeros_if_none(emissions.bias, shape=shape_prefix + (self.emission_dim,)),
+            input_weights=_zeros_if_none(
+                emissions.input_weights, shape=shape_prefix + (self.emission_dim, self.input_dim)
+            ),
+            cov=emissions.cov
+        )
+        return ParamsLGSSM(
+            initial=params.initial,
+            dynamics=clean_dynamics,
+            emissions=clean_emissions,
+        )
+
+
     def fit_blocked_gibbs(self,
                           key: PRNGKeyT,
                           initial_params: ParamsLGSSM,
@@ -654,7 +695,8 @@ def fit_blocked_gibbs(self,
 
         Args:
             key: random number key.
-            initial_params: starting parameters.
+            initial_params: starting parameters. Include a leading time axis for
+                the dynamics and emissions parameters in inhomogeneous models.
             sample_size: how many samples to draw.
             emissions: set of observation sequences.
             inputs: optional set of input sequences.
@@ -664,67 +706,97 @@ def fit_blocked_gibbs(self,
         """
         num_timesteps = len(emissions)
 
+        # Inhomogeneous models have a leading time dimension.
+        is_inhomogeneous = initial_params.dynamics.weights.ndim == 3
+
         if inputs is None:
             inputs = jnp.zeros((num_timesteps, 0))
 
+        initial_params = self._check_params(initial_params, num_timesteps)
+
         def sufficient_stats_from_sample(states):
             """Convert samples of states to sufficient statistics."""
             inputs_joint = jnp.concatenate((inputs, jnp.ones((num_timesteps, 1))), axis=1)
             # Let xn[t] = x[t+1]          for t = 0...T-2
-            x, xp, xn = states, states[:-1], states[1:]
-            u, up = inputs_joint, inputs_joint[:-1]
+            x, xn = states, states[1:]
+            u = inputs_joint
+            # Let z[t] = [x[t], u[t]] for t = 0...T-1
+            z = jnp.concatenate([x, u], axis=-1)
+            # Let zp[t] = [x[t], u[t]] for t = 0...T-2
+            zp = z[:-1]
             y = emissions
 
             init_stats = (x[0], jnp.outer(x[0], x[0]), 1)
 
             # Quantities for the dynamics distribution
-            # Let zp[t] = [x[t], u[t]] for t = 0...T-2
-            sum_zpzpT = jnp.block([[xp.T @ xp, xp.T @ up], [up.T @ xp, up.T @ up]])
-            sum_zpxnT = jnp.block([[xp.T @ xn], [up.T @ xn]])
-            sum_xnxnT = xn.T @ xn
-            dynamics_stats = (sum_zpzpT, sum_zpxnT, sum_xnxnT, num_timesteps - 1)
+            sum_zpzpT = jnp.einsum('ti,tj->tij', zp, zp)
+            sum_zpxnT = jnp.einsum('ti,tj->tij', zp, xn)
+            sum_xnxnT = jnp.einsum('ti,tj->tij', xn, xn)
+            z_is_observed = jnp.ones(num_timesteps - 1)
+            # The dynamics stats have a leading time dimension.
+            dynamics_stats = (sum_zpzpT, sum_zpxnT, sum_xnxnT, z_is_observed)
             if not self.has_dynamics_bias:
-                dynamics_stats = (sum_zpzpT[:-1, :-1], sum_zpxnT[:-1, :], sum_xnxnT,
-                                  num_timesteps - 1)
+                dynamics_stats = (sum_zpzpT[:, :-1, :-1], sum_zpxnT[:, :-1, :], sum_xnxnT,
+                                  z_is_observed)
 
             # Quantities for the emissions
-            # Let z[t] = [x[t], u[t]] for t = 0...T-1
-            sum_zzT = jnp.block([[x.T @ x, x.T @ u], [u.T @ x, u.T @ u]])
-            sum_zyT = jnp.block([[x.T @ y], [u.T @ y]])
-            sum_yyT = y.T @ y
-            emission_stats = (sum_zzT, sum_zyT, sum_yyT, num_timesteps)
+            sum_zzT = jnp.einsum('ti,tj->tij', z, z)
+            sum_zyT = jnp.einsum('ti,tj->tij', z, y)
+            sum_yyT = jnp.einsum('ti,tj->tij', y, y)
+            y_is_observed = jnp.ones(num_timesteps)
+            # The emissions stats have a leading time dimension.
+            emission_stats = (sum_zzT, sum_zyT, sum_yyT, y_is_observed)
             if not self.has_emissions_bias:
-                emission_stats = (sum_zzT[:-1, :-1], sum_zyT[:-1, :], sum_yyT, num_timesteps)
+                emission_stats = (sum_zzT[:, :-1, :-1], sum_zyT[:, :-1, :], sum_yyT, y_is_observed)
 
             return init_stats, dynamics_stats, emission_stats
 
-        def lgssm_params_sample(rng, stats):
-            """Sample parameters of the model given sufficient statistics from observed states and emissions."""
-            init_stats, dynamics_stats, emission_stats = stats
-            rngs = iter(jr.split(rng, 3))
-
-            # Sample the initial params
+        def _sample_initial_params(rng, init_stats):
             initial_posterior = niw_posterior_update(self.initial_prior, init_stats)
-            S, m = initial_posterior.sample(seed=next(rngs))
+            S, m = initial_posterior.sample(seed=rng)
+            return ParamsLGSSMInitial(mean=m, cov=S)
 
-            # Sample the dynamics params
+        def _sample_dynamics_params(rng, dynamics_stats):
             dynamics_posterior = mniw_posterior_update(self.dynamics_prior, dynamics_stats)
-            Q, FB = dynamics_posterior.sample(seed=next(rngs))
+            Q, FB = dynamics_posterior.sample(seed=rng)
             F = FB[:, :self.state_dim]
             B, b = (FB[:, self.state_dim:-1], FB[:, -1]) if self.has_dynamics_bias \
                 else (FB[:, self.state_dim:], jnp.zeros(self.state_dim))
+            return ParamsLGSSMDynamics(weights=F, bias=b, input_weights=B, cov=Q)
 
-            # Sample the emission params
+        def _sample_emission_params(rng, emission_stats):
             emission_posterior = mniw_posterior_update(self.emission_prior, emission_stats)
-            R, HD = emission_posterior.sample(seed=next(rngs))
+            R, HD = emission_posterior.sample(seed=rng)
             H = HD[:, :self.state_dim]
             D, d = (HD[:, self.state_dim:-1], HD[:, -1]) if self.has_emissions_bias \
                 else (HD[:, self.state_dim:], jnp.zeros(self.emission_dim))
+            return ParamsLGSSMEmissions(weights=H, bias=d, input_weights=D, cov=R)
+
+        def lgssm_params_sample(rng, stats):
+            """Sample parameters of the model given sufficient statistics from observed states and emissions."""
+            init_stats, dynamics_stats, emission_stats = stats
+            rngs = iter(jr.split(rng, 3))
+
+            # Sample the initial params
+            initial_params = _sample_initial_params(next(rngs), init_stats)
+
+            # Sample the dynamics and emission params.
+            if not is_inhomogeneous:
+                # Aggregate summary statistics across time for homogeneous model.
+                dynamics_stats = tree.map(lambda x: jnp.sum(x, axis=0), dynamics_stats)
+                emission_stats = tree.map(lambda x: jnp.sum(x, axis=0), emission_stats)
+                dynamics_params = _sample_dynamics_params(next(rngs), dynamics_stats)
+                emission_params = _sample_emission_params(next(rngs), emission_stats)
+            else:
+                keys_dynamics = jr.split(next(rngs), num_timesteps - 1)
+                keys_emission = jr.split(next(rngs), num_timesteps)
+                dynamics_params = vmap(_sample_dynamics_params)(keys_dynamics, dynamics_stats)
+                emission_params = vmap(_sample_emission_params)(keys_emission, emission_stats)
 
             params = ParamsLGSSM(
-                initial=ParamsLGSSMInitial(mean=m, cov=S),
-                dynamics=ParamsLGSSMDynamics(weights=F, bias=b, input_weights=B, cov=Q),
-                emissions=ParamsLGSSMEmissions(weights=H, bias=d, input_weights=D, cov=R)
+                initial=initial_params,
+                dynamics=dynamics_params,
+                emissions=emission_params,
             )
             return params
 

dynamax/linear_gaussian_ssm/models_test.py

@@ -1,12 +1,16 @@
 """
 Tests for the linear Gaussian SSM models.
 """
+from itertools import count, product
 
-import pytest
+import jax.numpy as jnp
 import jax.random as jr
+from jax import tree
+import pytest
 
 from dynamax.linear_gaussian_ssm import LinearGaussianSSM
 from dynamax.linear_gaussian_ssm import LinearGaussianConjugateSSM
+from dynamax.linear_gaussian_ssm.inference import ParamsLGSSM
 from dynamax.utils.utils import monotonically_increasing
 
 NUM_TIMESTEPS = 100
@@ -29,3 +33,52 @@ def test_sample_and_fit(cls, kwargs, inputs):
     fitted_params, lps = model.fit_em(params, param_props, emissions, inputs=inputs, num_iters=3)
     assert monotonically_increasing(lps)
     fitted_params, lps = model.fit_sgd(params, param_props, emissions, inputs=inputs, num_epochs=3)
+
+@pytest.mark.parametrize(["has_dynamics_bias", "has_emissions_bias"], product([True, False], repeat=2))
+def test_inhomogeneous_lgcssm(has_dynamics_bias, has_emissions_bias):
+    """
+    Test a LinearGaussianConjugateSSM with time-varying dynamics and emission model.
+    """
+    state_dim = 2
+    emission_dim = 3
+    num_timesteps = 4
+    keys = map(jr.PRNGKey, count())
+    kwargs = {
+        "state_dim": state_dim,
+        "emission_dim": emission_dim,
+        "has_dynamics_bias": has_dynamics_bias,
+        "has_emissions_bias": has_emissions_bias,
+    }
+    model = LinearGaussianConjugateSSM(**kwargs)
+    params, param_props = model.initialize(jr.PRNGKey(0))
+    # Repeat the parameters for each timestep.
+    inhomogeneous_dynamics = tree.map(
+        lambda x: jnp.repeat(x[None], num_timesteps - 1, axis=0), params.dynamics,
+    )
+    inhomogeneous_emissions = tree.map(
+        lambda x: jnp.repeat(x[None], num_timesteps, axis=0), params.emissions,
+    )
+
+    _, emissions = model.sample(params, next(keys), num_timesteps=num_timesteps)
+    inhomogeneous_params = ParamsLGSSM(
+        initial=params.initial,
+        dynamics=inhomogeneous_dynamics,
+        emissions=inhomogeneous_emissions,
+    )
+    params_trace = model.fit_blocked_gibbs(
+        next(keys),
+        inhomogeneous_params,
+        sample_size=5,
+        emissions=emissions,
+    )
+
+    # Arbitrarily check the last set of parameters from the Markov chain.
+    last_params = tree.map(lambda x: x[-1], params_trace)
+    assert last_params.initial.mean.shape == (state_dim,)
+    assert last_params.initial.cov.shape == (state_dim, state_dim)
+    assert last_params.dynamics.weights.shape == (num_timesteps - 1, state_dim, state_dim)
+    assert last_params.emissions.weights.shape == (num_timesteps, emission_dim, state_dim)
+    assert last_params.dynamics.bias.shape == (num_timesteps - 1, state_dim)
+    assert last_params.emissions.bias.shape == (num_timesteps, emission_dim)
+    assert last_params.dynamics.cov.shape == (num_timesteps - 1, state_dim, state_dim)
+    assert last_params.emissions.cov.shape == (num_timesteps, emission_dim, emission_dim)