import jax.numpy as jnp
import jax.random as jr
from jax import vmap
from jaxtyping import Float, Array
from dynamax.hidden_markov_model.models.abstractions import HMM, HMMEmissions, HMMParameterSet, HMMPropertySet
from dynamax.hidden_markov_model.models.initial import StandardHMMInitialState, ParamsStandardHMMInitialState
from dynamax.hidden_markov_model.models.transitions import StandardHMMTransitions, ParamsStandardHMMTransitions
from dynamax.parameters import ParameterProperties
from dynamax.types import Scalar
from dynamax.utils.utils import pytree_sum
from dynamax.utils.bijectors import RealToPSDBijector
from tensorflow_probability.substrates import jax as tfp
from typing import NamedTuple, Optional, Tuple, Union
tfd = tfp.distributions
tfb = tfp.bijectors
class ParamsLinearRegressionHMMEmissions(NamedTuple):
weights: Union[Float[Array, "state_dim emission_dim input_dim"], ParameterProperties]
biases: Union[Float[Array, "state_dim emission_dim"], ParameterProperties]
covs: Union[Float[Array, "state_dim emission_dim emission_dim"], ParameterProperties]
class ParamsLinearRegressionHMM(NamedTuple):
initial: ParamsStandardHMMInitialState
transitions: ParamsStandardHMMTransitions
emissions: ParamsLinearRegressionHMMEmissions
class LinearRegressionHMMEmissions(HMMEmissions):
def __init__(self,
num_states,
input_dim,
emission_dim):
"""_summary_
Args:
initial_probabilities (_type_): _description_
transition_matrix (_type_): _description_
emission_matrices (_type_): _description_
emission_biases (_type_): _description_
emission_covariance_matrices (_type_): _description_
"""
self.num_states = num_states
self.input_dim = input_dim
self.emission_dim = emission_dim
@property
def emission_shape(self):
return (self.emission_dim,)
def initialize(self,
key=jr.PRNGKey(0),
method="prior",
emission_weights=None,
emission_biases=None,
emission_covariances=None,
emissions=None):
if method.lower() == "kmeans":
assert emissions is not None, "Need emissions to initialize the model with K-Means!"
from sklearn.cluster import KMeans
key, subkey = jr.split(key) # Create a random seed for SKLearn.
sklearn_key = jr.randint(subkey, shape=(), minval=0, maxval=2147483647) # Max int32 value.
km = KMeans(self.num_states, random_state=int(sklearn_key)).fit(emissions.reshape(-1, self.emission_dim))
_emission_weights = jnp.zeros((self.num_states, self.emission_dim, self.input_dim))
_emission_biases = jnp.array(km.cluster_centers_)
_emission_covs = jnp.tile(jnp.eye(self.emission_dim)[None, :, :], (self.num_states, 1, 1))
elif method.lower() == "prior":
# TODO: Use an MNIW prior
key1, key2, key = jr.split(key, 3)
_emission_weights = 0.01 * jr.normal(key1, (self.num_states, self.emission_dim, self.input_dim))
_emission_biases = jr.normal(key2, (self.num_states, self.emission_dim))
_emission_covs = jnp.tile(jnp.eye(self.emission_dim), (self.num_states, 1, 1))
else:
raise Exception("Invalid initialization method: {}".format(method))
# Only use the values above if the user hasn't specified their own
default = lambda x, x0: x if x is not None else x0
params = ParamsLinearRegressionHMMEmissions(
weights=default(emission_weights, _emission_weights),
biases=default(emission_biases, _emission_biases),
covs=default(emission_covariances, _emission_covs))
props = ParamsLinearRegressionHMMEmissions(
weights=ParameterProperties(),
biases=ParameterProperties(),
covs=ParameterProperties(constrainer=RealToPSDBijector()))
return params, props
def distribution(self, params, state, inputs):
prediction = params.weights[state] @ inputs
prediction += params.biases[state]
return tfd.MultivariateNormalFullCovariance(prediction, params.covs[state])
def log_prior(self, params):
return 0.0
# Expectation-maximization (EM) code
def collect_suff_stats(self, params, posterior, emissions, inputs=None):
expected_states = posterior.smoothed_probs
sum_w = jnp.einsum("tk->k", expected_states)
sum_x = jnp.einsum("tk,ti->ki", expected_states, inputs)
sum_y = jnp.einsum("tk,ti->ki", expected_states, emissions)
sum_xxT = jnp.einsum("tk,ti,tj->kij", expected_states, inputs, inputs)
sum_xyT = jnp.einsum("tk,ti,tj->kij", expected_states, inputs, emissions)
sum_yyT = jnp.einsum("tk,ti,tj->kij", expected_states, emissions, emissions)
return dict(sum_w=sum_w, sum_x=sum_x, sum_y=sum_y, sum_xxT=sum_xxT, sum_xyT=sum_xyT, sum_yyT=sum_yyT)
def initialize_m_step_state(self, params, props):
return None
def m_step(self, params, props, batch_stats, m_step_state):
def _single_m_step(stats):
sum_w = stats['sum_w']
sum_x = stats['sum_x']
sum_y = stats['sum_y']
sum_xxT = stats['sum_xxT']
sum_xyT = stats['sum_xyT']
sum_yyT = stats['sum_yyT']
# Make block matrices for stacking features (x) and bias (1)
sum_x1x1T = jnp.block(
[[sum_xxT, jnp.expand_dims(sum_x, 1)],
[jnp.expand_dims(sum_x, 0), jnp.expand_dims(sum_w, (0, 1))]]
)
sum_x1yT = jnp.vstack([sum_xyT, sum_y])
# Solve for the optimal A, b, and Sigma
Ab = jnp.linalg.solve(sum_x1x1T, sum_x1yT).T
Sigma = 1 / sum_w * (sum_yyT - Ab @ sum_x1yT)
Sigma = 0.5 * (Sigma + Sigma.T) # for numerical stability
return Ab[:, :-1], Ab[:, -1], Sigma
emission_stats = pytree_sum(batch_stats, axis=0)
As, bs, Sigmas = vmap(_single_m_step)(emission_stats)
params = params._replace(weights=As, biases=bs, covs=Sigmas)
return params, m_step_state
[docs]
class LinearRegressionHMM(HMM):
r"""An HMM whose emissions come from a linear regression with state-dependent weights.
This is also known as a *switching linear regression* model.
Let $y_t \in \mathbb{R}^N$ and $u_t \in \mathbb{R}^M$ denote vector-valued emissions
and inputs at time $t$, respectively. In this model, the emission distribution is,
$$p(y_t \mid z_t, u_t, \theta) = \mathcal{N}(y_{t} \mid W_{z_t} u_t + b_{z_t}, \Sigma_{z_t})$$
with *emission weights* $W_k \in \mathbb{R}^{N \times M}$, *emission biases* $b_k \in \mathbb{R}^N$,
and *emission covariances* $\Sigma_k \in \mathbb{R}_{\succeq 0}^{N \times N}$.
The emissions parameters are $\theta = \{W_k, b_k, \Sigma_k\}_{k=1}^K$.
We do not place a prior on the emission parameters.
*Note: in the future we add a* matrix-normal-inverse-Wishart_ *prior (see pg 576).*
.. _matrix-normal-inverse-Wishart: https://github.com/probml/pml2-book
:param num_states: number of discrete states $K$
:param input_dim: input dimension $M$
:param emission_dim: emission dimension $N$
:param initial_probs_concentration: $\alpha$
:param transition_matrix_concentration: $\beta$
:param transition_matrix_stickiness: optional hyperparameter to boost the concentration on the diagonal of the transition matrix.
"""
def __init__(self,
num_states: int,
input_dim: int,
emission_dim: int,
initial_probs_concentration: Union[Scalar, Float[Array, "num_states"]]=1.1,
transition_matrix_concentration: Union[Scalar, Float[Array, "num_states"]]=1.1,
transition_matrix_stickiness: Scalar=0.0):
self.emission_dim = emission_dim
self.input_dim = input_dim
initial_component = StandardHMMInitialState(num_states, initial_probs_concentration=initial_probs_concentration)
transition_component = StandardHMMTransitions(num_states, concentration=transition_matrix_concentration, stickiness=transition_matrix_stickiness)
emission_component = LinearRegressionHMMEmissions(num_states, input_dim, emission_dim)
super().__init__(num_states, initial_component, transition_component, emission_component)
@property
def inputs_shape(self):
return (self.input_dim,)
[docs]
def initialize(self,
key: jr.PRNGKey=jr.PRNGKey(0),
method: str="prior",
initial_probs: Optional[Float[Array, "num_states"]]=None,
transition_matrix: Optional[Float[Array, "num_states num_states"]]=None,
emission_weights: Optional[Float[Array, "num_states emission_dim input_dim"]]=None,
emission_biases: Optional[Float[Array, "num_states emission_dim"]]=None,
emission_covariances: Optional[Float[Array, "num_states emission_dim emission_dim"]]=None,
emissions: Optional[Float[Array, "num_timesteps emission_dim"]]=None
) -> Tuple[HMMParameterSet, HMMPropertySet]:
"""Initialize the model parameters and their corresponding properties.
You can either specify parameters manually via the keyword arguments, or you can have
them set automatically. If any parameters are not specified, you must supply a PRNGKey.
Parameters will then be sampled from the prior (if `method==prior`).
Args:
key: random number generator for unspecified parameters. Must not be None if there are any unspecified parameters.
method: method for initializing unspecified parameters. Both "prior" and "kmeans" are supported.
initial_probs: manually specified initial state probabilities.
transition_matrix: manually specified transition matrix.
emission_weights: manually specified emission weights.
emission_biases: manually specified emission biases.
emission_covariances: manually specified emission covariances.
emissions: emissions for initializing the parameters with kmeans.
Returns:
Model parameters and their properties.
"""
key1, key2, key3 = jr.split(key , 3)
params, props = dict(), dict()
params["initial"], props["initial"] = self.initial_component.initialize(key1, method=method, initial_probs=initial_probs)
params["transitions"], props["transitions"] = self.transition_component.initialize(key2, method=method, transition_matrix=transition_matrix)
params["emissions"], props["emissions"] = self.emission_component.initialize(key3, method=method, emission_weights=emission_weights, emission_biases=emission_biases, emission_covariances=emission_covariances, emissions=emissions)
return ParamsLinearRegressionHMM(**params), ParamsLinearRegressionHMM(**props)
