
KF Basics - Part 2
------------------

Probabilistic Generative Laws
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1st Law:
^^^^^^^^

The belief representing the state :math:`x_{t}`, is conditioned on all
past states, measurements and controls. This can be shown mathematically
by the conditional probability shown below:

.. math:: p(x_{t} | x_{0:t-1},z_{1:t-1},u_{1:t})

1) :math:`z_{t}` represents the **measurement**

2) :math:`u_{t}` the **motion command**

3) :math:`x_{t}` the **state** (can be the position, velocity, etc) of
   the robot or its environment at time t.

‘If we know the state :math:`x_{t-1}` and :math:`u_{t}`, then knowing
the states :math:`x_{0:t-2}`, :math:`z_{1:t-1}` becomes immaterial
through the property of **conditional independence**’. The state
:math:`x_{t-1}` introduces a conditional independence between
:math:`x_{t}` and :math:`z_{1:t-1}`, :math:`u_{1:t-1}`

Therefore the law now holds as:

.. math:: p(x_{t} | x_{0:t-1},z_{1:t-1},u_{1:t})=p(x_{t} | x_{t-1},u_{t})

2nd Law:
^^^^^^^^

If :math:`x_{t}` is complete, then:

.. math:: p(z_{t} | x-_{0:t},z_{1:t-1},u_{1:t})=p(z_{t} | x_{t})

:math:`x_{t}` is **complete** means that the past states, controls or
measurements carry no additional information to predict future.

:math:`x_{0:t-1}`, :math:`z_{1:t-1}` and :math:`u_{1:t}` are
**conditionally independent** of :math:`z_{t}` given :math:`x_{t}` of
complete.

The filter works in two parts:

:math:`p(x_{t} | x_{t-1},u_{t})` -> **State Transition Probability**

:math:`p(z_{t} | x_{t})` -> **Measurement Probability**

Conditional dependence and independence example:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`\bullet`\ **Independent but conditionally dependent**

Let’s say you flip two fair coins

A - Your first coin flip is heads

B - Your second coin flip is heads

C - Your first two flips were the same

A and B here are independent. However, A and B are conditionally
dependent given C, since if you know C then your first coin flip will
inform the other one.

:math:`\bullet`\ **Dependent but conditionally independent**

A box contains two coins: a regular coin and one fake two-headed coin
((P(H)=1). I choose a coin at random and toss it twice. Define the
following events.

A= First coin toss results in an H.

B= Second coin toss results in an H.

C= Coin 1 (regular) has been selected.

If we know A has occurred (i.e., the first coin toss has resulted in
heads), we would guess that it is more likely that we have chosen Coin 2
than Coin 1. This in turn increases the conditional probability that B
occurs. This suggests that A and B are not independent. On the other
hand, given C (Coin 1 is selected), A and B are independent.

Bayes Rule:
~~~~~~~~~~~

Posterior =

.. math:: \frac{Likelihood*Prior}{Marginal} 

Here,

**Posterior** = Probability of an event occurring based on certain
evidence.

**Likelihood** = How probable is the evidence given the event.

**Prior** = Probability of the just the event occurring without having
any evidence.

**Marginal** = Probability of the evidence given all the instances of
events possible.

Example:

1% of women have breast cancer (and therefore 99% do not). 80% of
mammograms detect breast cancer when it is there (and therefore 20% miss
it). 9.6% of mammograms detect breast cancer when its not there (and
therefore 90.4% correctly return a negative result).

We can turn the process above into an equation, which is Bayes Theorem.
Here is the equation:

:math:`\displaystyle{\Pr(\mathrm{A}|\mathrm{X}) = \frac{\Pr(\mathrm{X}|\mathrm{A})\Pr(\mathrm{A})}{\Pr(\mathrm{X|A})\Pr(\mathrm{A})+ \Pr(\mathrm{X | not \ A})\Pr(\mathrm{not \ A})}}`

:math:`\bullet`\ Pr(A|X) = Chance of having cancer (A) given a positive
test (X). This is what we want to know: How likely is it to have cancer
with a positive result? In our case it was 7.8%.

:math:`\bullet`\ Pr(X|A) = Chance of a positive test (X) given that you
had cancer (A). This is the chance of a true positive, 80% in our case.

:math:`\bullet`\ Pr(A) = Chance of having cancer (1%).

:math:`\bullet`\ Pr(not A) = Chance of not having cancer (99%).

:math:`\bullet`\ Pr(X|not A) = Chance of a positive test (X) given that
you didn’t have cancer (~A). This is a false positive, 9.6% in our case.

Bayes Filter Algorithm
~~~~~~~~~~~~~~~~~~~~~~

The basic filter algorithm is:

for all :math:`x_{t}`:

1. :math:`\overline{bel}(x_t) = \int p(x_t | u_t, x_{t-1}) bel(x_{t-1})dx`

2. :math:`bel(x_t) = \eta p(z_t | x_t) \overline{bel}(x_t)`

end.

:math:`\rightarrow`\ The first step in filter is to calculate the prior
for the next step that uses the bayes theorem. This is the
**Prediction** step. The belief, :math:`\overline{bel}(x_t)`, is
**before** incorporating measurement(\ :math:`z_{t}`) at time t=t. This
is the step where the motion occurs and information is lost because the
means and covariances of the gaussians are added. The RHS of the
equation incorporates the law of total probability for prior
calculation.

:math:`\rightarrow` This is the **Correction** or update step that
calculates the belief of the robot **after** taking into account the
measurement(\ :math:`z_{t}`) at time t=t. This is where we incorporate
the sensor information for the whereabouts of the robot. We gain
information here as the gaussians get multiplied here. (Multiplication
of gaussian values allows the resultant to lie in between these numbers
and the resultant covariance is smaller.

Bayes filter localization example:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: ipython3

    from IPython.display import Image
    Image(filename="bayes_filter.png",width=400)




.. image:: Kalmanfilter_basics_2_files/Kalmanfilter_basics_2_5_0.png
   :width: 400px



Given - A robot with a sensor to detect doorways along a hallway. Also,
the robot knows how the hallway looks like but doesn’t know where it is
in the map.

1. Initially(first scenario), it doesn’t know where it is with respect
   to the map and hence the belief assigns equal probability to each
   location in the map.

2. The first sensor reading is incorporated and it shows the presence of
   a door. Now the robot knows how the map looks like but cannot
   localize yet as map has 3 doors present. Therefore it assigns equal
   probability to each door present.

3. The robot now moves forward. This is the prediction step and the
   motion causes the robot to lose some of the information and hence the
   variance of the gaussians increase (diagram 4.). The final belief is
   **convolution** of posterior from previous step and the current state
   after motion. Also, the means shift on the right due to the motion.

4. Again, incorporating the measurement, the sensor senses a door and
   this time too the possibility of door is equal for the three door.
   This is where the filter’s magic kicks in. For the final belief
   (diagram 5.), the posterior calculated after sensing is mixed or
   **convolution** of previous posterior and measurement. It improves
   the robot’s belief at location near to the second door. The variance
   **decreases** and **peaks**.

5. Finally after series of iterations of motion and correction, the
   robot is able to localize itself with respect to the
   environment.(diagram 6.)

Do note that the robot knows the map but doesn’t know where exactly it
is on the map.

Bayes and Kalman filter structure
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The basic structure and the concept remains the same as bayes filter for
Kalman. The only key difference is the mathematical representation of
Kalman filter. The Kalman filter is nothing but a bayesian filter that
uses Gaussians.

For a bayes filter to be a Kalman filter, **each term of belief is now a
gaussian**, unlike histograms. The basic formulation for the **bayes
filter** algorithm is:

.. math::

   \begin{aligned} 
   \bar {\mathbf x} &= \mathbf x \ast f_{\mathbf x}(\bullet)\, \, &\text{Prediction} \\
   \mathbf x &= \mathcal L \cdot \bar{\mathbf x}\, \, &\text{Correction}
   \end{aligned}

:math:`\bar{\mathbf x}` is the *prior*

:math:`\mathcal L` is the *likelihood* of a measurement given the prior
:math:`\bar{\mathbf x}`

:math:`f_{\mathbf x}(\bullet)` is the *process model* or the gaussian
term that helps predict the next state like velocity to track position
or acceleration.

:math:`\ast` denotes *convolution*.

Kalman Gain
~~~~~~~~~~~

.. math::  x = (\mathcal L \bar x)

Where x is posterior and :math:`\mathcal L` and :math:`\bar x` are
gaussians.

Therefore the mean of the posterior is given by:

.. math::


   \mu=\frac{\bar\sigma^2\, \mu_z + \sigma_z^2 \, \bar\mu} {\bar\sigma^2 + \sigma_z^2}

.. math:: \mu = \left( \frac{\bar\sigma^2}{\bar\sigma^2 + \sigma_z^2}\right) \mu_z + \left(\frac{\sigma_z^2}{\bar\sigma^2 + \sigma_z^2}\right)\bar\mu

In this form it is easy to see that we are scaling the measurement and
the prior by weights:

.. math:: \mu = W_1 \mu_z + W_2 \bar\mu

The weights sum to one because the denominator is a normalization term.
We introduce a new term, :math:`K=W_1`, giving us:

.. math::

   \begin{aligned}
   \mu &= K \mu_z + (1-K) \bar\mu\\
   &= \bar\mu + K(\mu_z - \bar\mu)
   \end{aligned}

where

.. math:: K = \frac {\bar\sigma^2}{\bar\sigma^2 + \sigma_z^2}

The variance in terms of the Kalman gain:

.. math::

   \begin{aligned}
   \sigma^2 &= \frac{\bar\sigma^2 \sigma_z^2 } {\bar\sigma^2 + \sigma_z^2} \\
   &= K\sigma_z^2 \\
   &= (1-K)\bar\sigma^2 
   \end{aligned}

:math:`K` is the *Kalman gain*. It’s the crux of the Kalman filter. It
is a scaling term that chooses a value partway between :math:`\mu_z` and
:math:`\bar\mu`.

Kalman Filter - Univariate and Multivariate
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

\ **Prediction**\ 

:math:`\begin{array}{|l|l|l|} \hline \text{Univariate} & \text{Univariate} & \text{Multivariate}\\ & \text{(Kalman form)} & \\ \hline \bar \mu = \mu + \mu_{f_x} & \bar x = x + dx & \bar{\mathbf x} = \mathbf{Fx} + \mathbf{Bu}\\ \bar\sigma^2 = \sigma_x^2 + \sigma_{f_x}^2 & \bar P = P + Q & \bar{\mathbf P} = \mathbf{FPF}^\mathsf T + \mathbf Q \\ \hline \end{array}`

:math:`\mathbf x,\, \mathbf P` are the state mean and covariance. They
correspond to :math:`x` and :math:`\sigma^2`.

:math:`\mathbf F` is the *state transition function*. When multiplied by
:math:`\bf x` it computes the prior.

:math:`\mathbf Q` is the process covariance. It corresponds to
:math:`\sigma^2_{f_x}`.

:math:`\mathbf B` and :math:`\mathbf u` are model control inputs to the
system.

\ **Correction**\ 

:math:`\begin{array}{|l|l|l|} \hline \text{Univariate} & \text{Univariate} & \text{Multivariate}\\ & \text{(Kalman form)} & \\ \hline & y = z - \bar x & \mathbf y = \mathbf z - \mathbf{H\bar x} \\ & K = \frac{\bar P}{\bar P+R}& \mathbf K = \mathbf{\bar{P}H}^\mathsf T (\mathbf{H\bar{P}H}^\mathsf T + \mathbf R)^{-1} \\ \mu=\frac{\bar\sigma^2\, \mu_z + \sigma_z^2 \, \bar\mu} {\bar\sigma^2 + \sigma_z^2} & x = \bar x + Ky & \mathbf x = \bar{\mathbf x} + \mathbf{Ky} \\ \sigma^2 = \frac{\sigma_1^2\sigma_2^2}{\sigma_1^2+\sigma_2^2} & P = (1-K)\bar P & \mathbf P = (\mathbf I - \mathbf{KH})\mathbf{\bar{P}} \\ \hline \end{array}`

:math:`\mathbf H` is the measurement function.

:math:`\mathbf z,\, \mathbf R` are the measurement mean and noise
covariance. They correspond to :math:`z` and :math:`\sigma_z^2` in the
univariate filter. :math:`\mathbf y` and :math:`\mathbf K` are the
residual and Kalman gain.

The details will be different than the univariate filter because these
are vectors and matrices, but the concepts are exactly the same:

-  Use a Gaussian to represent our estimate of the state and error
-  Use a Gaussian to represent the measurement and its error
-  Use a Gaussian to represent the process model
-  Use the process model to predict the next state (the prior)
-  Form an estimate part way between the measurement and the prior

Reference
~~~~~~~~~~~

1. Roger Labbe’s
   `repo <https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python>`__
   on Kalman Filters. (Majority of text in the notes are from this)

2. Probabilistic Robotics by Sebastian Thrun, Wolfram Burgard and Dieter
   Fox, MIT Press.
