Tracking an object using the Kalman filter#
Consider an object moving in \(R^2\). We assume that we observe a noisy version of its location at each time step. We want to track the object and possibly forecast its future motion. We now show how to do this using a simple linear Gaussian SSM, combined with the Kalman filter algorithm.
Let the hidden state represent the position and velocity of the object, \(z_t =\begin{pmatrix} u_t & v_t & \dot{u}_t & \dot{v}_t \end{pmatrix}\). (We use \(u\) and \(v\) for the two coordinates, to avoid confusion with the state and observation variables.) The process evolves in continuous time, but if we discretize it with step size \(\Delta\), we can write the dynamics as the following linear system:
\[\begin{align*}
\underbrace{\begin{pmatrix} u_t\\ v_t \\ \dot{u}_t \\ \dot{v}_t \end{pmatrix}}_{z_t}
=
\underbrace{
\begin{pmatrix}
1 & 0 & \Delta & 0 \\
0 & 1 & 0 & \Delta\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
}_{F}
\underbrace{\begin{pmatrix} u_{t-1} \\ v_{t-1} \\ \dot{u}_{t-1} \\ \dot{v}_{t-1} \end{pmatrix}}_{z_{t-1}}
+ q_t
\end{align*}\]
where \(q_t \in R^4\) is the process noise, which we assume is Gaussian, so \(q_t \sim N(0,Q)\).
Now suppose that at each discrete time point we observe the location (but not the velocity). We assume the observation is corrupted by Gaussian noise. Thus the observation model becomes
\[\begin{align*}
\underbrace{\begin{pmatrix} y_{1,t} \\ y_{2,t} \end{pmatrix}}_{y_t}
&=
\underbrace{
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix}
}_{H}
\;
\underbrace{\begin{pmatrix} u_t\\ v_t \\ \dot{u}_t \\ \dot{v}_t \end{pmatrix}}_{z_t}
+ r_t
\end{align*}\]
where \(r_t \sim N(0,R)\) is the observation noise. We see that the observation matrix \(H\) simply ``extracts’’ the relevant parts of the state vector.
Setup#
Show code cell content
%%capture
try:
import dynamax
except ModuleNotFoundError:
print('installing dynamax')
%pip install -q dynamax[notebooks]
import dynamax
from jax import numpy as jnp
from jax import random as jr
from jax import vmap
from matplotlib import pyplot as plt
from dynamax.utils.plotting import plot_uncertainty_ellipses
from dynamax.linear_gaussian_ssm import LinearGaussianSSM
from dynamax.linear_gaussian_ssm import lgssm_smoother, lgssm_filter
Create the model#
state_dim = 4
emission_dim = 2
delta = 1.0
# Create object
lgssm = LinearGaussianSSM(state_dim, emission_dim)
# Manually chosen parameters
initial_mean = jnp.array([8.0, 10.0, 1.0, 0.0])
initial_covariance = jnp.eye(state_dim) * 0.1
dynamics_weights = jnp.array([[1, 0, delta, 0],
[0, 1, 0, delta],
[0, 0, 1, 0],
[0, 0, 0, 1]])
dynamics_covariance = jnp.eye(state_dim) * 0.001
emission_weights = jnp.array([[1.0, 0, 0, 0],
[0, 1.0, 0, 0]])
emission_covariance = jnp.eye(emission_dim) * 1.0
# Initialize model
params, _ = lgssm.initialize(jr.PRNGKey(0),
initial_mean=initial_mean,
initial_covariance=initial_covariance,
dynamics_weights=dynamics_weights,
dynamics_covariance=dynamics_covariance,
emission_weights=emission_weights,
emission_covariance=emission_covariance)
Sample some data from the model#
num_timesteps = 15
key = jr.PRNGKey(310)
x, y = lgssm.sample(params, key, num_timesteps)
# Plot Data
observation_marker_kwargs = {"marker": "o", "markerfacecolor": "none", "markeredgewidth": 2, "markersize": 8}
fig1, ax1 = plt.subplots()
ax1.plot(*x[:, :2].T, marker="s", color="C0", label="true state")
ax1.plot(*y.T, ls="", **observation_marker_kwargs, color="tab:green", label="emissions")
ax1.legend(loc="upper left")
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
Cell In[4], line 3
1 num_timesteps = 15
2 key = jr.PRNGKey(310)
----> 3 x, y = lgssm.sample(params, key, num_timesteps)
5 # Plot Data
6 observation_marker_kwargs = {"marker": "o", "markerfacecolor": "none", "markeredgewidth": 2, "markersize": 8}
File ~/work/dynamax/dynamax/dynamax/linear_gaussian_ssm/models.py:209, in LinearGaussianSSM.sample(self, params, key, num_timesteps, inputs)
202 def sample(
203 self,
204 params: ParamsLGSSM,
(...)
207 inputs: Optional[Float[Array, "ntime input_dim"]] = None
208 ) -> PosteriorGSSMFiltered:
--> 209 return lgssm_joint_sample(params, key, num_timesteps, inputs)
File ~/work/dynamax/dynamax/dynamax/linear_gaussian_ssm/inference.py:421, in lgssm_joint_sample(params, key, num_timesteps, inputs)
418 # Sample the initial state
419 key1, key2 = jr.split(key)
--> 421 initial_state, initial_emission = _sample_initial(key1, params, inputs)
423 # Sample the remaining emissions and states
424 next_keys = jr.split(key2, num_timesteps - 1)
File ~/work/dynamax/dynamax/dynamax/linear_gaussian_ssm/inference.py:397, in lgssm_joint_sample.<locals>._sample_initial(key, params, inputs)
394 def _sample_initial(key, params, inputs):
395 key1, key2 = jr.split(key)
--> 397 initial_state = MVN(params.initial.mean, params.initial.cov).sample(seed=key1)
399 H0, D0, d0, R0 = _get_params(params, num_timesteps, 0)[4:]
400 u0 = tree_map(lambda x: x[0], inputs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/mvn_full_covariance.py:191, in MultivariateNormalFullCovariance.__init__(self, loc, covariance_matrix, validate_args, allow_nan_stats, name)
185 # No need to validate that covariance_matrix is non-singular.
186 # LinearOperatorLowerTriangular has an assert_non_singular method that
187 # is called by the Bijector.
188 # However, cholesky() ignores the upper triangular part, so we do need
189 # to separately assert symmetric.
190 scale_tril = tf.linalg.cholesky(covariance_matrix)
--> 191 super(MultivariateNormalFullCovariance, self).__init__(
192 loc=loc,
193 scale_tril=scale_tril,
194 validate_args=validate_args,
195 allow_nan_stats=allow_nan_stats,
196 name=name)
197 self._parameters = parameters
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/mvn_tril.py:228, in MultivariateNormalTriL.__init__(self, loc, scale_tril, validate_args, allow_nan_stats, experimental_use_kahan_sum, name)
221 linop_cls = (KahanLogDetLinOpTriL if experimental_use_kahan_sum else
222 tf.linalg.LinearOperatorLowerTriangular)
223 scale = linop_cls(
224 scale_tril,
225 is_non_singular=True,
226 is_self_adjoint=False,
227 is_positive_definite=False)
--> 228 super(MultivariateNormalTriL, self).__init__(
229 loc=loc,
230 scale=scale,
231 validate_args=validate_args,
232 allow_nan_stats=allow_nan_stats,
233 experimental_use_kahan_sum=experimental_use_kahan_sum,
234 name=name)
235 self._parameters = parameters
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/mvn_linear_operator.py:205, in MultivariateNormalLinearOperator.__init__(self, loc, scale, validate_args, allow_nan_stats, experimental_use_kahan_sum, name)
202 if loc is not None:
203 bijector = shift_bijector.Shift(
204 shift=loc, validate_args=validate_args)(bijector)
--> 205 super(MultivariateNormalLinearOperator, self).__init__(
206 # TODO(b/137665504): Use batch-adding meta-distribution to set the batch
207 # shape instead of tf.zeros.
208 # We use `Sample` instead of `Independent` because `Independent`
209 # requires concatenating `batch_shape` and `event_shape`, which loses
210 # static `batch_shape` information when `event_shape` is not statically
211 # known.
212 distribution=sample.Sample(
213 normal.Normal(
214 loc=tf.zeros(batch_shape, dtype=dtype),
215 scale=tf.ones([], dtype=dtype)),
216 event_shape,
217 experimental_use_kahan_sum=experimental_use_kahan_sum),
218 bijector=bijector,
219 validate_args=validate_args,
220 name=name)
221 self._parameters = parameters
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/transformed_distribution.py:244, in _TransformedDistribution.__init__(self, distribution, bijector, kwargs_split_fn, validate_args, parameters, name)
238 self._zero = tf.constant(0, dtype=tf.int32, name='zero')
240 # We don't just want to check isinstance(JointDistribution) because
241 # TransformedDistributions with multipart bijectors are effectively
242 # joint but don't inherit from JD. The 'duck-type' test is that
243 # JDs have a structured dtype.
--> 244 dtype = self.bijector.forward_dtype(self.distribution.dtype)
245 self._is_joint = tf.nest.is_nested(dtype)
247 super(_TransformedDistribution, self).__init__(
248 dtype=dtype,
249 reparameterization_type=self._distribution.reparameterization_type,
(...)
252 parameters=parameters,
253 name=name)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/bijectors/bijector.py:1705, in Bijector.forward_dtype(self, dtype, name, **kwargs)
1701 input_dtype = nest_util.broadcast_structure(
1702 self.forward_min_event_ndims, self.dtype)
1703 else:
1704 # Make sure inputs are compatible with statically-known dtype.
-> 1705 input_dtype = nest.map_structure_up_to(
1706 self.forward_min_event_ndims,
1707 lambda x: dtype_util.convert_to_dtype(x, dtype=self.dtype),
1708 nest_util.coerce_structure(self.forward_min_event_ndims, dtype),
1709 check_types=False)
1711 output_dtype = self._forward_dtype(input_dtype, **kwargs)
1712 try:
1713 # kwargs may alter dtypes themselves, but we currently require
1714 # structure to be statically known.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/python/internal/backend/jax/nest.py:324, in map_structure_up_to(shallow_structure, func, *structures, **kwargs)
323 def map_structure_up_to(shallow_structure, func, *structures, **kwargs):
--> 324 return map_structure_with_tuple_paths_up_to(
325 shallow_structure,
326 lambda _, *args: func(*args), # Discards path.
327 *structures,
328 **kwargs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/python/internal/backend/jax/nest.py:353, in map_structure_with_tuple_paths_up_to(shallow_structure, func, expand_composites, *structures, **kwargs)
350 for input_tree in structures:
351 assert_shallow_structure(
352 shallow_structure, input_tree, check_types=check_types)
--> 353 return dm_tree.map_structure_with_path_up_to(
354 shallow_structure, func, *structures, **kwargs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tree/__init__.py:778, in map_structure_with_path_up_to(***failed resolving arguments***)
776 results = []
777 for path_and_values in _multiyield_flat_up_to(shallow_structure, *structures):
--> 778 results.append(func(*path_and_values))
779 return unflatten_as(shallow_structure, results)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/python/internal/backend/jax/nest.py:326, in map_structure_up_to.<locals>.<lambda>(_, *args)
323 def map_structure_up_to(shallow_structure, func, *structures, **kwargs):
324 return map_structure_with_tuple_paths_up_to(
325 shallow_structure,
--> 326 lambda _, *args: func(*args), # Discards path.
327 *structures,
328 **kwargs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/bijectors/bijector.py:1707, in Bijector.forward_dtype.<locals>.<lambda>(x)
1701 input_dtype = nest_util.broadcast_structure(
1702 self.forward_min_event_ndims, self.dtype)
1703 else:
1704 # Make sure inputs are compatible with statically-known dtype.
1705 input_dtype = nest.map_structure_up_to(
1706 self.forward_min_event_ndims,
-> 1707 lambda x: dtype_util.convert_to_dtype(x, dtype=self.dtype),
1708 nest_util.coerce_structure(self.forward_min_event_ndims, dtype),
1709 check_types=False)
1711 output_dtype = self._forward_dtype(input_dtype, **kwargs)
1712 try:
1713 # kwargs may alter dtypes themselves, but we currently require
1714 # structure to be statically known.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/internal/dtype_util.py:247, in convert_to_dtype(tensor_or_dtype, dtype, dtype_hint)
245 elif isinstance(tensor_or_dtype, np.ndarray):
246 dt = base_dtype(dtype or dtype_hint or tensor_or_dtype.dtype)
--> 247 elif np.issctype(tensor_or_dtype):
248 dt = base_dtype(dtype or dtype_hint or tensor_or_dtype)
249 else:
250 # If this is a Python object, call `convert_to_tensor` and grab the dtype.
251 # Note that this will add ops in graph-mode; we may want to consider
252 # other ways to handle this case.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/numpy/__init__.py:397, in __getattr__(attr)
394 raise AttributeError(__former_attrs__[attr])
396 if attr in __expired_attributes__:
--> 397 raise AttributeError(
398 f"`np.{attr}` was removed in the NumPy 2.0 release. "
399 f"{__expired_attributes__[attr]}"
400 )
402 if attr == "chararray":
403 warnings.warn(
404 "`np.chararray` is deprecated and will be removed from "
405 "the main namespace in the future. Use an array with a string "
406 "or bytes dtype instead.", DeprecationWarning, stacklevel=2)
AttributeError: `np.issctype` was removed in the NumPy 2.0 release. Use `issubclass(rep, np.generic)` instead.
Show code cell source
def plot_lgssm_posterior(post_means, post_covs, ax=None, ellipse_kwargs={}, legend_kwargs={}, **kwargs):
"""Plot posterior means and covariances for the first two dimensions of
the latent state of a LGSSM.
Args:
post_means: array(T, D).
post_covs: array(T, D, D).
ax: matplotlib axis.
ellipse_kwargs: keyword arguments passed to matplotlib.patches.Ellipse().
**kwargs: passed to ax.plot().
"""
if ax is None:
fig, ax = plt.subplots()
# This is to stop some weird behaviour where running the function multiple
# # times with an empty argument wouldn't reset the dictionary.
# if ellipse_kwargs is None:
# ellipse_kwargs = dict()
# if 'edgecolor' not in ellipse_kwargs:
# if 'color' in kwargs:
# ellipse_kwargs['edgecolor'] = kwargs['color']
# Select the first two dimensions of the latent space.
post_means = post_means[:, :2]
post_covs = post_covs[:, :2, :2]
# Plot the mean trajectory
ax.plot(post_means[:, 0], post_means[:, 1], **kwargs)
# Plot covariance at each time point.
plot_uncertainty_ellipses(post_means, post_covs, ax, **ellipse_kwargs)
ax.axis("equal")
if "label" in kwargs:
ax.legend(**legend_kwargs)
return ax
Perform online filtering#
lgssm_posterior = lgssm.filter(params, y)
print(lgssm_posterior.filtered_means.shape)
print(lgssm_posterior.filtered_covariances.shape)
print(lgssm_posterior.marginal_loglik)
(15, 4)
(15, 4, 4)
-43.13845
fig2, ax2 = plt.subplots()
ax2.plot(*y.T, ls="", **observation_marker_kwargs, color="tab:green", label="observed")
ax2.plot(*x[:, :2].T, ls="--", color="darkgrey", label="true state")
plot_lgssm_posterior(
lgssm_posterior.filtered_means,
lgssm_posterior.filtered_covariances,
ax2,
color="tab:red",
label="filtered means",
ellipse_kwargs={"edgecolor": "k", "linewidth": 0.5},
legend_kwargs={"loc":"upper left"}
)
<AxesSubplot: >
Perform offline smoothing#
lgssm_posterior = lgssm.smoother(params, y)
fig3, ax3 = plt.subplots()
ax3.plot(*y.T, ls="", **observation_marker_kwargs, color="tab:green", label="observed")
ax3.plot(*x[:, :2].T, ls="--", color="darkgrey", label="true state")
plot_lgssm_posterior(
lgssm_posterior.smoothed_means,
lgssm_posterior.smoothed_covariances,
ax3,
color="tab:red",
label="smoothed means",
ellipse_kwargs={"edgecolor": "k", "linewidth": 0.5},
legend_kwargs={"loc":"upper left"}
)
<AxesSubplot: >
Low-level interface to the underlying inference algorithms#
We can also call the inference code directly, without having to make an LG-SSM object. We just pass the model parameters directly to the function.
filtered_posterior = lgssm_filter(params, y) # Kalman filter
smoothed_posterior = lgssm_smoother(params, y) # Kalman filter + smoother
assert jnp.allclose(lgssm_posterior.filtered_means, filtered_posterior.filtered_means)
assert jnp.allclose(lgssm_posterior.filtered_means, smoothed_posterior.filtered_means)
assert jnp.allclose(lgssm_posterior.smoothed_means, smoothed_posterior.smoothed_means)
Tracking multiple objects in parallel#
# Generate 4 sample trajectories
num_timesteps = 15
num_samples = 4
key = jr.PRNGKey(310)
keys = jr.split(key, num_samples)
xs, ys = vmap(lambda key: lgssm.sample(params, key, num_timesteps))(keys)
# vmap the inference
lgssm_posteriors = vmap(lambda y: lgssm.smoother(params, y))(ys)
def plot_kf_parallel(xs, ys, lgssm_posteriors):
num_samples = len(xs)
dict_figures = {}
# Plot Data
fig, ax = plt.subplots()
for n in range(num_samples):
ax.plot(*xs[n, :, :2].T, ls="--", color=f"C{n}")
ax.plot(*ys[n, ...].T, ".", color=f"C{n}", label=f"Trajectory {n+1}")
ax.set_title("Data")
ax.legend()
dict_figures["missiles_latent"] = fig
# Plot Filtering
fig, ax = plt.subplots()
for n in range(num_samples):
ax.plot(*ys[n, ...].T, ".")
filt_means = lgssm_posteriors.filtered_means[n, ...]
filt_covs = lgssm_posteriors.filtered_covariances[n, ...]
plot_lgssm_posterior(
filt_means,
filt_covs,
ax,
color=f"C{n}",
ellipse_kwargs={"edgecolor": f"C{n}", "linewidth": 0.5},
label=f"Trajectory {n+1}",
)
ax.legend(fontsize=10)
ax.set_title("Filtered Posterior")
dict_figures["missiles_filtered"] = fig
# Plot Smoothing
fig, ax = plt.subplots()
for n in range(num_samples):
ax.plot(*ys[n, ...].T, ".")
filt_means = lgssm_posteriors.smoothed_means[n, ...]
filt_covs = lgssm_posteriors.smoothed_covariances[n, ...]
plot_lgssm_posterior(
filt_means,
filt_covs,
ax,
color=f"C{n}",
ellipse_kwargs={"edgecolor": f"C{n}", "linewidth": 0.5},
label=f"Trajectory {n+1}",
)
ax.legend(fontsize=10)
ax.set_title("Smoothed Posterior")
dict_figures["missiles_smoothed"] = fig
return dict_figures
dict_figures = plot_kf_parallel(xs, ys, lgssm_posteriors)
