# Kalman Filter

## **Design of Autonomous Systems**
### csci 6907/4907-Section 86
### Prof. **Sibin Mohan**

---

**Kalman Filter** express state and uncertainty → **Gaussians** distributions

---

### kalman filters and gaussians

---

### Gaussian Distribution

---

### Gaussian Distribution

**continuous probability distribution**

---

### Gaussian Distribution

**continuous probability distribution** → **real-valued random variable**

---

### Gaussian Distribution

**continuous probability distribution** → **real-valued random variable**

<br>

$$
\mathcal{N}\left(\mu, \sigma^{2}\right)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} 
$$

---

### Gaussian Distribution

**continuous probability distribution** → **real-valued random variable**

---

### Gaussian Distribution

a very common Gaussian:

---

### Gaussian Distribution

a very common Gaussian:

---

### Gaussian Distribution

**variance** ($\sigma^2$) → captures the **uncertainty** in system

---

### Gaussian Distribution

**variance** ($\sigma^2$) → captures the **uncertainty** in system

the **square** of standard deviation ($\sigma$)

---

various types of "uncertainty"

---

more specific examples:

---

### some properties of Gaussians

---

### some properties of Gaussians

1. **area** under curve = <scb>1</scb>

Note:
- any guesses why?

---

### some properties of Gaussians

1. **area** under curve = <scb>1</scb>

(since it is the sum of all the probabilities)

---

### some properties of Gaussians

1. **area** under curve = <scb>1</scb>

$$
area = \int_{-\infty}^{+\infty} \mathcal{N}\left(\mu, \sigma^{2}\right)
$$

---

$$
area = \int_{-\infty}^{+\infty} \mathcal{N}\left(\mu, \sigma^{2}\right)
$$

- curve approaches **infinity** ($\infty$) on **either side** 
- the probability of certain events is **never** <scb>0</scb> → no matter how small!

---

### some properties of Gaussians

1. **area** under curve = <scb>1</scb>

$$
area = \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} = \textbf{1}
$$

---

### some properties of Gaussians

2. probability that a value, $x$ → is in a **range**, $(a,b)$:

$$
\operatorname{Pr}(a \leq x \leq b)=\int_{a}^{b} \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} d x
$$

---

### some properties of Gaussians

3. probability **within one standard deviation** (on either side),

$$
\operatorname{Pr}(\mu-\sigma \leq x \leq \mu+\sigma)=\int_{\mu-\sigma}^{\mu+\sigma} \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} d x \approx \textbf{68 \%}
$$

---

### some properties of Gaussians

4. **sum** and **product** of two Gaussian distributions → **easy** to calculate

---

given, two Gaussians,

$X_{1} \sim \mathcal{N}\left(\mu_{1}, \sigma_{1}^{2}\right)$

$X_{2} \sim \mathcal{N}\left(\mu_{2}, \sigma_{2}^{2}\right)
$

---

### sum and product of Gaussians

|result| sum | |
|------|-----|--------|
| **new** Gaussian, $Z \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$ | $Z=X_{1}+X_{2}$ | |

---

### sum and product of Gaussians

|result| sum | |
|------|-----|--------|
| **new** Gaussian, $Z \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$ | $Z=X_{1}+X_{2}$ | |
| new **mean**, $\mu$ | $\mu_{1}+\mu_{2}$ | |

---

### sum and product of Gaussians

|result| sum | |
|------|-----|--------|
| **new** Gaussian, $Z \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$ | $Z=X_{1}+X_{2}$ | |
| new **mean**, $\mu$ | $\mu_{1}+\mu_{2}$ | |
| new **variance**, $\sigma^{2}$ | $\sigma_{1}^{2}+\sigma_{2}^{2}$ | |

---

### sum and product of Gaussians

|result| sum | product|
|------|-----|--------|
| **new** Gaussian, $Z \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$ | $Z=X_{1}+X_{2}$ | |
| new **mean**, $\mu$ | $\mu_{1}+\mu_{2}$ | |
| new **variance**, $\sigma^{2}$ | $\sigma_{1}^{2}+\sigma_{2}^{2}$ | |
||

---

### sum and product of Gaussians

---

### sum and product of Gaussians

|result| sum | product|
|------|-----|--------|
| **new** Gaussian, $Z \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$ | $Z=X_{1}+X_{2}$ | $Z=X_{1} X_{2}$ |
| new **mean**, $\mu$ | $\mu_{1}+\mu_{2}$ | $\mu=\frac{\sigma_{2}^{2} \mu_{1}+\sigma_{1}^{2} \mu_{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}$ |
| new **variance**, $\sigma^{2}$ | $\sigma_{1}^{2}+\sigma_{2}^{2}$ | |
||

---

### sum and product of Gaussians

---

### some properties of Gaussians

5. **closure** under **linear transformations**

---

### some properties of Gaussians

5. **closure** under **linear transformations**

linear transformation on Gaussian → **result remains a Gaussian**!

---

### some properties of Gaussians

5. **closure** under **linear transformations**

ensures that Kalman Filter equations remain,

- elegant and 
- manageable

---

### **State** in Kalman Filter
---

### **State** in Kalman Filters

---

**prior** ↔ **measurement** ↔ **update** process

---

**prior** ↔ **measurement** ↔ **update** process

---

prediction and measurements → not single points → **distributions**

---

### Kalman Filter | **transitions**

similar to Bayes Filter → each edge is now **probabilistic Gaussian value**

---

### Kalman Filter | **Prediction**

---

**process model** * is also a Gaussian!

[*_i.e.,_ model of how system evolves over time → physics model]

---

**process model** * is also a Gaussian!

equations of motion → _velocity_ follows a Gaussian distribution

---

we want to track motion of car

---

**predict** next state using following **process model**:

$$
\overline{x_{k+1}}=x_{k} +v_{k} \Delta t
$$

[Newton's laws of motion]

---

$$
\overline{x_{k+1}}=x_{k} +v_{k} \Delta t
$$

|variable | description
|---------|-----------|
| $x_{k}$ | current state |

---

$$
\overline{x_{k+1}}=x_{k} +v_{k} \Delta t
$$

|variable | description
|---------|-----------|
| $x_{k}$ | current state |
| $x_{k+1}$ | **predicted** next state |

---

$$
\overline{x_{k+1}}=x_{k} +v_{k} \Delta t
$$

|variable | description
|---------|-----------|
| $x_{k}$ | current state |
| $x_{k+1}$ | **predicted** next state |
| **$v_{k}$** | velocity|

---

$$
\overline{x_{k+1}}=x_{k} +v_{k} \Delta t
$$

|variable | description
|---------|-----------|
| $x_{k}$ | current state |
| $x_{k+1}$ | **predicted** next state |
| **$v_{k}$** | velocity|
| $\Delta t$ | time difference |
||

---

velocity → Gaussian distribution

---

velocity → Gaussian distribution

$$
v_{k} \sim \mathcal{N} \left(3 m / \mathrm{s}, 1^{2} m^2/s^2 \right)
$$

---

**current** state, $x_k$ → Gaussian

---

**predicted** state, $x_{k+1}$ → Gaussian *

(* both current state **and** velocity are Gaussians)

---

$$
\overline{x_{k+1}} \sim \mathcal{N}\left(\pmb{?}\ \mathrm{m}, \pmb{?}\ \mathrm{m}^{2}\right) 
$$

---

recall, that **sum** of two Gaussians is,

$$
\mu =\mu_{1}+\mu_{2} 
$$
$$
\sigma^{2} =\sigma_{1}^{2}+\sigma_{2}^{2}
$$

---

assume $\Delta t = 1s$,

---

assume $\Delta t = 1s$,

$$
\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, 2 \mathrm{m}^{2}\right)
$$

---

assume $\Delta t = 1s$,

$$
\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, 2 \mathrm{m}^{2}\right)
$$

---

---

Kalman Filter → **unimodel** *

---

Kalman Filter → **unimodel** *

(* a **single peak** each time)

---

Kalman Filter → **unimodel** *

(* a **single peak** each time)

---

Kalman Filter → **unimodel** *

---

Kalman Filter → **unimodel** *

an obstacle is

---

Kalman Filter → **unimodel** *

an obstacle is

<div>

**not** both,

- $10m$ away (<scb>90%</scb> prob.) **and** 
- $8m$ away (<scb>70%</scb> prob.)

</div>

<div>

</div>
</div>

---

Kalman Filter → **unimodel** *

an obstacle is

<div>

**not** both,

- $10m$ away (<scb>90%</scb> prob.) **and** 
- $8m$ away (<scb>70%</scb> prob.)

</div>

<div>

<br>

- *$9.7m$* away, <scb>98%</scb> prob. or 
- **nothing**

</div>
</div>

Note:
- this is from the viewpoint of a sngle sensor, state model

---

### Kalman Filter | **Prediction**

---

### Kalman Filter | **Prediction**

some examples → consider following velocity models:

|||
|----|----|
| **A** | $$v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 0^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$$
| **B** | $$v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 1^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$$
| **C** | $$v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 2^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$$
||

---

the following **predicted** states:

---

|prediction| velocity model|
|-----|----|
| <img src="img/ekf/kalman/kalman.predict.7.png" width="700"> | <br> **A**, **B** or **C** ? |
| <img src="img/ekf/kalman/kalman.predict.8.png" width="700"> | **A**, **B** or **C** ? |
| <img src="img/ekf/kalman/kalman.predict.9.png" width="700"> | **A**, **B** or **C** ? |
||

---

<br>

</div>

<div>

</div>
</div>

---

|prediction| velocity model|
|-----|----|
| <img src="img/ekf/kalman/kalman.predict.7.png" width="500"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 1^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ [**B**] |
| <img src="img/ekf/kalman/kalman.predict.8.png" width="500"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 0^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ [**A**]|
| <img src="img/ekf/kalman/kalman.predict.9.png" width="500"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 2^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ [**C**]|
||

---

what does **predicted** state look like?

---

|prediction| velocity model| predicted state |
|-----|----|------|
| <img src="img/ekf/kalman/kalman.predict.7.png" width="500"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 1^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | $\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, ? \mathrm{m}^{2}\right)$|
| <img src="img/ekf/kalman/kalman.predict.8.png" width="500"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 0^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | $\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, ? \mathrm{m}^{2}\right)$|
| <img src="img/ekf/kalman/kalman.predict.9.png" width="500"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 2^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | $\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, ? \mathrm{m}^{2}\right)$|
||

---

what does **predicted** state look like?

|prediction| velocity model| predicted state |
|-----|----|------|
| <img src="img/ekf/kalman/kalman.predict.7.png" width="300"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 1^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | |
| <img src="img/ekf/kalman/kalman.predict.8.png" width="300"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 0^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | |
| <img src="img/ekf/kalman/kalman.predict.9.png" width="300"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 2^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | |
||

---

**recall**, $\overline{x_{k}} \sim \mathcal{N}\left(3.5, 1^{2}\right)$ <br>
$\mu =\mu_{1}+\mu_{2}$ and $\sigma^{2} =\sigma_{1}^{2}+\sigma_{2}^{2}$

---

|prediction| velocity model| predicted state |
|-----|----|------|
| <img src="img/ekf/kalman/kalman.predict.7.png" width="300"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 1^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | $\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, 2 \mathrm{m}^{2}\right)$|
| <img src="img/ekf/kalman/kalman.predict.8.png" width="300"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 0^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | $\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, 1 \mathrm{m}^{2}\right)$|
| <img src="img/ekf/kalman/kalman.predict.9.png" width="300"> | $v_{k} \sim \mathcal{N}\left(3 \mathrm{m} / \mathrm{s}, 2^{2} \mathrm{m}^{2} / \mathrm{s}^{2}\right)$ | $\overline{x_{k+1}} \sim \mathcal{N}\left(6.5 \mathrm{m}, 5 \mathrm{m}^{2}\right)$|
||

---

depending on **noise** in intial Gaussian → **uncertainty increases**

even when velocity model had <scb>0</scb> noise!

---

**problem** → don't know how **close** our prediction is to reality

---

**problem** → don't know how **close** our prediction is to reality

we need an **update** step → measurements!

---

### Kalman Filter | **Update**

---

### Kalman Filter | **Update**

**measurement uncertainty** (aka "likelihood") → also Gaussian
---

### Kalman Filter | **Update**

${z_{k}} \sim \mathcal{N}\left(\mu, \sigma^2\right)$

---

### Kalman Filter | **Update**

---

### Kalman Filter | **Update**

**example**: current temperature is $72^{\circ} \mathrm{F} \pm 1^{\circ} \mathrm{F}$

---

recall that,

$$
posterior = prior \times likelihood
$$

---

recall that,

$$
posterior = prior \times likelihood
$$
$$
posterior = \overline x_k \times z_k
$$

---

what does this **actually mean**?

---

consider a few examples...

---

### Example **1**

| prediction ($\overline x_{k}$) | measurement ($z_k$) | posterior ($\overline x_{k} \times z_k$) |
|----------|----------|----------|
|$\overline{x_{k}} \sim \mathcal{n}\left(5,1^{2}\right)$ | $z_{k} \sim \mathcal{n}\left(5,1^{2}\right)$ | **?** |
||

---

let's plot these two Gaussians,

---

perfectly aligned → **same** distribution!

---

| prediction ($\overline x_{k}$) | measurement ($z_k$) | posterior ($\overline x_{k} \times z_k$) |
|----------|----------|----------|
|$\overline{x_{k}} \sim \mathcal{N}\left(5,1^{2}\right)$ | $z_{k} \sim \mathcal{N}\left(5,1^{2}\right)$ | **?** |
||

what is $\overline{x_{k}} \times z_{k}$?

---

recall **product** of Gaussians,

$$
\mu =\frac{\sigma_{2}^{2} \mu_{1}+\sigma_{1}^{2} \mu_{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}} \\
$$
$$
\sigma^{2} =\frac{\sigma_{1}^{2} \sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}
$$

---

hence, the **posterior**:

| prediction ($\overline x_{k}$) | measurement ($z_k$) | posterior ($\overline x_{k} \times z_k$) |
|----------|----------|----------|
|$\overline{x_{k}} \sim \mathcal{N}\left(5,1^{2}\right)$ | $z_{k} \sim \mathcal{N}\left(5,1^{2}\right)$ | $\overline{x_k} \times z_k \sim \mathcal{N}(5,0.5)$ |
||

---

graphically,

---

- an **increase** in the probability around <scb>5</scb> (**taller** Gaussian)

---

- an **increase** in the probability around <scb>5</scb> (**taller** Gaussian)
- a **decrease** in the uncertainty (**narrower** Gaussian)

---

### Example **2**

---

### Example **2** 
this time the initial Guassians differ slightly,

---

### Example **2** 
this time the initial Guassians differ slightly,

| prediction ($\overline x_{k}$) | measurement ($z_k$) | posterior ($\overline x_{k} \times z_k$) |
|----------|----------|----------|
|$\overline{x_{k}} \sim \mathcal{N}\left(5,1^{2}\right)$ | $z_{k} \sim \mathcal{N}\left(5.4,1^{2}\right)$ | $\overline{x_k} \times {z_{k}} \sim \mathcal{N}(5.2,0.5)$ |
||

---

result:

---

result → in the **midddle** of two original Gaussians

---

### Example **3**

the Gaussian's differ **significantly**, _i.e.,_

| prediction ($\overline x_{k}$) | measurement ($z_k$) | posterior ($\overline x_{k} \times z_k$) |
|----------|----------|----------|
|$\overline{x_{k}} \sim \mathcal{N}\left(4,1^{2}\right)$ | $z_{k} \sim \mathcal{N}\left(6,0.5^{2}\right)$ | $\overline{x_{k}} \times z_{k} \sim \mathcal{N}(?,?)$ |
||

---

original Gaussians:

---

post update → resultant distribution,

| prediction ($\overline x_{k}$) | measurement ($z_k$) | posterior ($\overline x_{k} \times z_k$) |
|----------|----------|----------|
|$\overline{x_{k}} \sim \mathcal{N}\left(4,1^{2}\right)$ | $z_{k} \sim \mathcal{N}\left(6,0.5^{2}\right)$ | $\overline{x_{k}} \times z_{k}\sim \mathcal{N}(5.6,0.2)$ |
||

---

**before** update
<img src="img/ekf/kalman/kalman.update.6.png" width="900">
</div>
<div>

**after** update
<img src="img/ekf/kalman/kalman.update.7.png" width="900">
</div>
</div>

---

- **more weight → more ‘certain’ information**

---

- more weight → more ‘certain’ information
- **measurement** (in this case) → has a higher+narrower Gaussian

---

### **posterior**

- **between** prediction ($\overline x_{k}$) and measurement ($z_k$)

---

### **posterior**

- between prediction ($\overline x_{k}$) and measurement ($z_k$)
- closer to **more certain side** → based on variances

---

### **posterior**

- between prediction ($\overline x_{k}$) and measurement ($z_k$)
- closer to **more certain side** → based on the variances
- so, a ‘**weighted average**’

---

### **posterior**

- between prediction ($\overline x_{k}$) and measurement ($z_k$)
- closer to **more certain side** → based on the variances
- so, a ‘**weighted average**’

</div>

---

a more **detailed example**

---

actual position vs measurements

---

**after Kalman Filter is applied**

---

**after Kalman Filter is applied**

1. posterior → **always between** prior and measurement
2. posterior is **closer to** measurement → sensor noise is small

---

look at [textbook](https://autonomy-course.github.io/textbook/autonomy-textbook.html#kalman-filter) for more detailed examples

---

### kalman filter | **update** step

---

### kalman filter | **update** step

---

### kalman filter | **update** step

---

recall,

$$
\begin{aligned}
\mu & =\frac{\sigma_{z}^{2} \bar{\mu}+\bar{\sigma}^{2} \mu_{z}}{\bar{\sigma}^{2}+\sigma_{z}^{2}} \newline
\end{aligned}
$$

---

let's **rewrite** it as,

$$
\begin{aligned}
\mu & =\frac{\sigma_{z}^{2} \bar{\mu}+\bar{\sigma}^{2} \mu_{z}}{\bar{\sigma}^{2}+\sigma_{z}^{2}} \newline
& =K_{1} \mu_{z}+K_{2} \bar{\mu} \newline
\end{aligned}
$$

---

to **simplify** further,

$$
\begin{aligned}
\mu & =\frac{\sigma_{z}^{2} \bar{\mu}+\bar{\sigma}^{2} \mu_{z}}{\bar{\sigma}^{2}+\sigma_{z}^{2}} \newline
& =K_{1} \mu_{z}+K_{2} \bar{\mu} \newline
& =K \mu_{z}+(1-K) \bar{\mu}
\end{aligned}
$$

---

<br>

$K_2 = 1-K$ → a simplification

---

$K_2 = 1-K$ → a simplification

whether posterior → **closer to the prior or the measurement**?

---

whether posterior → **closer to the prior or the measurement**?

<br>

where does it fall on this line?

---

### **Kalman Gain** ($K$)

---

### **Kalman Gain** ($K$)

$$
K = \frac{\bar{\sigma}^{2}}{\bar{\sigma}^{2} + \sigma_{z}^{2}}
$$

---

### **Kalman Gain** ($K$)

$$
K = \frac{\bar{\sigma}^{2}}{\bar{\sigma}^{2} + \sigma_{z}^{2}}
$$

- **how much** measurement is **trusted**

---

### **Kalman Gain** ($K$)

$$
K = \frac{\bar{\sigma}^{2}}{\bar{\sigma}^{2} + \sigma_{z}^{2}}
$$

- **how much** measurement is **trusted**
- **mixing ratio** between measurement and prior

---

**example**

---

**example**

measurement → **three times** more accurate than prior

---

**example**

measurement → **three times** more accurate than prior

what is the Kalman Gain?

---

what is the Kalman Gain?

$$
\begin{gathered}
\bar{\sigma}^{2}=3 \sigma_{z}^{2} \newline
\end{gathered}
$$

---

what is the Kalman Gain?

$$
\begin{gathered}
\bar{\sigma}^{2}=3 \sigma_{z}^{2} \newline
K=\frac{3 \sigma_{z}^{2}}{3 \sigma_{Z}^{2}+\sigma_{Z}^{2}}
\end{gathered}
$$

---

what is the Kalman Gain?

$$
\begin{gathered}
\bar{\sigma}^{2}=3 \sigma_{z}^{2} \newline
K=\frac{3 \sigma_{z}^{2}}{3 \sigma_{Z}^{2}+\sigma_{Z}^{2}}=\frac{3}{4}
\end{gathered}
$$

---

$$
K=\frac{3}{4}
$$

---

### posterior

---

### posterior

rewritten as

---

### posterior

rewritten as

|||
|----|----|
| **residual** | $y = \mu_z - \bar{\mu}$|

---

### posterior

rewritten as

|||
|----|----|
| **residual** | $y = \mu_z - \bar{\mu}$|
| posterior **mean** | $\mu = \bar{\mu} + Ky$|

---

### posterior

rewritten as

|||
|----|----|
| **residual** | $y = \mu_z - \bar{\mu}$|
| posterior **mean** | $\mu = \bar{\mu} + Ky$|
| posterior **noise** | $\sigma^2 = (1-K) \bar{\sigma^2}$|
||

---

### posterior

rewritten as

|||
|----|----|
| **residual** | $y = \mu_z - \bar{\mu}$|
| posterior **mean** | $\mu = \bar{\mu} + Ky$|
| posterior **noise** | $\sigma^2 = (1-K) \bar{\sigma^2}$|
||

(since $\sigma^{2} =\frac{\sigma_{1}^{2} \sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}$)

---

### **higher** K → **more** certainty!

---

### Kalman Filter | **Summary**

---

### Kalman Filter | **Summary**

---

### Kalman Filter | **Summary**

<div>

<br>

1. **step 0**: **initialize** $x_{0} \sim \mathcal{N}\left(\mu_{0}, \sigma_{0}^{2}\right)$
    - initial **belief**
    - reasonable random values/initial measurement
</div>
</div>

---

### Kalman Filter | **Summary**

<div>

2. **step 1**: **predict**
    - calculate **prior**, $\overline{x_{k}} \sim \mathcal{N}\left(\overline{\mu_{k}},{\overline{\sigma_{k}}}^{2}\right)$
    - from previous posterior, $x_{k-1} \sim \mathcal{N}\left(\mu_{k-1}, \sigma_{k-1}^{2}\right)$
    - incorporating process model, $f_{x} \sim \mathcal{N}\left(\mu_{f}, \sigma_{f}^{2}\right)$
</div>
</div>

---

### Kalman Filter | **Summary**

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3. **step 2**: **update**
    - given measurement, $z_{k}$
    - compute **residual** and **Kalman gain**
    - set posterior ($x_{k}$) between prior ($\overline{x_k}$) and measurement ($z_{k}$)
    - based on residual and Kalman gain
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(see textbook for more details/equations/etc.)