Filters¶

Algorithms that use uncertain measurements from sensors to predict unknown variable with acceptable accuracy, to estimate the current state.

Help identify underlying trends, by smoothing time series and hence filtering out noise

They are not for prediction

Working¶

where

• \(\tilde y_t=\) smoothed/filtered value
• \(y_t=\) observed value
• \(y^*_t=\) true value
• \(u_{t, \text{PN}} =\) Process noise
• \(u_{t, \text{MN}} =\) Measurement noise

Filter Design¶

1. Define problem which consists of the state
2. Define the motion model
3. Define how the state will be measured
4. Define uncertainty in system’s dynamic model
5. Implement & test the filter in controlled environment with known inputs & outputs
6. Tune the filter

Notes¶

• Window size = Duration of Seasonality will remove it
• Be wary of over-smoothing
• Never go live without tuning filter

Types wrt Time Dependence¶

• Centered
• Lagging

Filters vs Rolling Statistics¶

The Kalman filter is better suited for estimating things that change over time. The Kalman Filter lets you add more information about how the system you're filtering works. In other words, you can use a signal model to improve the output of the filter.

Sure, a moving average filter can give very good results when you're expecting a close-to-constant output. But as soon as the signal you're modelling is dynamic (think speech or position measurements), then the simple moving average filter will not change quickly enough (or at all) compared with what the Kalman Filter will do.

The Kalman filter uses the signal model, which captures your knowledge of how the signal changes, to improve its output in terms of the variance from "truth".

Concepts¶

Applications¶

These can be used in any field, but a few examples

• Guidance
• Navigation
• Control of vehicles

Batch vs Recursive¶

Uncertainty¶

Variance of measurement errors provided by scale vendor/derived through calibration; Even though we can’t accurately know the estimate error, we can estimate the uncertainty in the estimates

Most modern systems are equipped with multiple sensors that provide estimation of hidden/unknown variables based on series of measurements

One of the biggest challenges of tracking and control systems is to provide accurate and precise estimation of the hidden variables in the presence of uncertainty.

Bias & Variance¶

Very similar to Machine Learning Prediction Bias & Variance

Measurement Filters¶

Alpha Filter¶

Average Filter¶

The \(\alpha\) factor is called as Gain, and is taken as \(\alpha_t = \dfrac{1}{t}\). As number of measurements increase, each successive measurement has less weight in estimation, as \(t \uparrow \implies \alpha \downarrow\)

Gives the less weight to recent data compared to past data

Kalman filter requires an initial guess as a preset; it may be approximate.

\((y_{t, t} - \hat y_{t, t-1})\) is called the measurement residual

Beta Filter¶

where

• \(s=\) seasonality duration

Complex¶

Parameters¶

Dynamic System¶

State(s) change over time

Let’s take the case of 2 states: position & velocity $$ \begin{align} \hat y_{t, t} &= \hat y_{t, t-1} + \Delta t \hat {\dot x}{t, t-1} \tag{1} \ \implies \hat {\dot x}} &= \hat {\dot x{t, t-1} + \beta \left( \dfrac{y_t - \hat y \ \hat y_{1, 0} &= 0; \hat {\dot x}_{1, 0} = 0 \end{align} $$ However, if we assume constant velocity, but measurement residual in }}{\Delta t} \right) \tag{2\(x \ne 0,\) then it could be due to 2 reasons

• Value of \(\beta = \text{const}\), unlike \(\alpha_t\)
• \(\beta \propto \text{Precision} \propto \dfrac{1}{\sigma_\text{measurement}}\)

Adaptive Filter¶

Kalman Filter¶

A low-pass filter with dynamically-changing \(\alpha\)

Assumes that the following are normally-distributed

• measurements
• current state estimates
• next state estimates

Also quantifies the uncertainties associated with the estimates

Optimal filter that combines the prior state estimate with the measurement, such that uncertainty of current state estimate is minimized

flowchart LR

subgraph i[Inputs]
    direction LR
    mp[Measured<br/>Parameter]
    mu[Measurement<br/>Uncertainty]
end

subgraph kf[Kalman Filter]
    direction LR
    u[Update] --> p[Predict] --> ud[Unit<br/>Delay] --> u
end

subgraph o[Outputs]
    direction LR
    sse[System State<br />Estimate]
    eu[Estimate<br />Uncertainty]
end

subgraph ii[Initial Inputs]
    direction LR
    is[Initial State]
    isu[Initial State<br />Uncertainty]
end

ii --> p
i ---> sf[Sensor<br />Fusion] --> u --> o

where

• \(r\) is the measurement uncertainty in variance
• \(p\) is the estimate uncertainty in variance

Equations¶

Kalman Gain
\(\approx 0\)
\(\approx 1\)

Advantages¶

• Handles noise in initial estimate, state transitions, and observations
• Removes need to store historical data
• Can handle asynchronous measurements, ie measurements recorded by multiple sensors at different time points
• Computationally-efficient
• Suitable for real-time applications

Disadvantages¶

• Only for linear gaussian state space models.
• In practice, state transitions may be non-linear/noise and may be non-gaussian.
• Use other types of Kalman filters for this

Kalman Smoother¶

Extended Kalman Filter¶

Developed for non-linear dynamics

Performs analytic linearization of the model at each point

Unscented Kalman Filter¶

Without Model-Based Approach¶

	\(\mu'\)	\({\sigma^2}'\)	Estimation uncertainty
Update Parameter/Measurement Uses Bayes’ rule	\(\left( \dfrac{\mu}{\sigma^2} + \dfrac{\nu}{r^2} \right) {\sigma^2}'\)	\(\dfrac{1}{\dfrac{1}{r^2} + \dfrac{1}{\sigma^2}}\)	decreases
Predict Motion	\(\mu + u\)	\({\sigma^2} + r^2\)	increases

Comments