# Kalman Filter¶

Kalman filters are linear models for state estimation of dynamic systems [1]. They have been the de facto standard in many robotics and tracking/prediction applications because they are well suited for systems with uncertainty about an observable dynamic process. They use a “observe, predict, correct” paradigm to extract information from an otherwise noisy signal. In Pyro, we can build differentiable Kalman filters with learnable parameters using the `pyro.contrib.tracking`

library

## Dynamic process¶

To start, consider this simple motion model:

where \(k\) is the state, \(X\) is the signal estimate, \(Z_k\) is the observed value at timestep \(k\), \(\mathbf{W}_k\) and \(\mathbf{V}_k\) are independent noise processes (ie \(\mathbb{E}[w_k v_j^T] = 0\) for all \(j, k\)) which we’ll approximate as Gaussians. Note that the state transitions are linear.

## Kalman Update¶

At each time step, we perform a prediction for the mean and covariance:

and a correction for the measurement:

where \(X\) is the position estimate, \(P\) is the covariance matrix, \(K\) is the Kalman Gain, and \(Q\) and \(R\) are covariance matrices.

For an in-depth derivation, see [2]

## Nonlinear Estimation: Extended Kalman Filter¶

What if our system is non-linear, eg in GPS navigation? Consider the following non-linear system:

Notice that \(\mathbf{f}\) and \(\mathbf{h}\) are now (smooth) non-linear functions.

The Extended Kalman Filter (EKF) attacks this problem by using a local linearization of the Kalman filter via a Taylors Series expansion.

where \(\mathbf{H}_k\) is the Jacobian matrix at time \(k\), \(x_k^R\) is the previous optimal estimate, and we ignore the higher order terms. At each time step, we compute a Jacobian conditioned the previous predictions (this computation is handled by Pyro under the hood), and use the result to perform a prediction and update.

Omitting the derivations, the modification to the above predictions are now:

and the updates are now:

In Pyro, all we need to do is create an `EKFState`

object and use its `predict`

and `update`

methods. Pyro will do exact inference to compute the innovations and we will use SVI to learn a MAP estimate of the position and measurement covariances.

As an example, let’s look at an object moving at near-constant velocity in 2-D in a discrete time space over 100 time steps.

```
[ ]:
```

```
import os
import math
import torch
import pyro
import pyro.distributions as dist
from pyro.infer.autoguide import AutoDelta
from pyro.optim import Adam
from pyro.infer import SVI, Trace_ELBO, config_enumerate
from pyro.contrib.tracking.extended_kalman_filter import EKFState
from pyro.contrib.tracking.distributions import EKFDistribution
from pyro.contrib.tracking.dynamic_models import NcvContinuous
from pyro.contrib.tracking.measurements import PositionMeasurement
smoke_test = ('CI' in os.environ)
assert pyro.__version__.startswith('1.7.0')
```

```
[ ]:
```

```
dt = 1e-2
num_frames = 10
dim = 4
# Continuous model
ncv = NcvContinuous(dim, 2.0)
# Truth trajectory
xs_truth = torch.zeros(num_frames, dim)
# initial direction
theta0_truth = 0.0
# initial state
with torch.no_grad():
xs_truth[0, :] = torch.tensor([0.0, 0.0, math.cos(theta0_truth), math.sin(theta0_truth)])
for frame_num in range(1, num_frames):
# sample independent process noise
dx = pyro.sample('process_noise_{}'.format(frame_num), ncv.process_noise_dist(dt))
xs_truth[frame_num, :] = ncv(xs_truth[frame_num-1, :], dt=dt) + dx
```

Next, let’s specify the measurements. Notice that we only measure the positions of the particle.

```
[ ]:
```

```
# Measurements
measurements = []
mean = torch.zeros(2)
# no correlations
cov = 1e-5 * torch.eye(2)
with torch.no_grad():
# sample independent measurement noise
dzs = pyro.sample('dzs', dist.MultivariateNormal(mean, cov).expand((num_frames,)))
# compute measurement means
zs = xs_truth[:, :2] + dzs
```

We’ll use a Delta autoguide to learn MAP estimates of the position and measurement covariances. The `EKFDistribution`

computes the joint log density of all of the EKF states given a tensor of sequential measurements.

```
[ ]:
```

```
def model(data):
# a HalfNormal can be used here as well
R = pyro.sample('pv_cov', dist.HalfCauchy(2e-6)) * torch.eye(4)
Q = pyro.sample('measurement_cov', dist.HalfCauchy(1e-6)) * torch.eye(2)
# observe the measurements
pyro.sample('track_{}'.format(i), EKFDistribution(xs_truth[0], R, ncv,
Q, time_steps=num_frames),
obs=data)
guide = AutoDelta(model) # MAP estimation
```

```
[ ]:
```

```
optim = pyro.optim.Adam({'lr': 2e-2})
svi = SVI(model, guide, optim, loss=Trace_ELBO(retain_graph=True))
pyro.set_rng_seed(0)
pyro.clear_param_store()
for i in range(250 if not smoke_test else 2):
loss = svi.step(zs)
if not i % 10:
print('loss: ', loss)
```

```
[ ]:
```

```
# retrieve states for visualization
R = guide()['pv_cov'] * torch.eye(4)
Q = guide()['measurement_cov'] * torch.eye(2)
ekf_dist = EKFDistribution(xs_truth[0], R, ncv, Q, time_steps=num_frames)
states= ekf_dist.filter_states(zs)
```

**Figure 1:**True track and EKF prediction with error.

## References¶

[1] Kalman, R. E. *A New Approach to Linear Filtering and Prediction Problems.* 1960

[2] Welch, Greg, and Bishop, Gary. *An Introduction to the Kalman Filter.* 2006.