Subhasis Sanyal — October 30, 2021
Advanced Computer Vision Maths Python

This article was published as a part of the Data Science Blogathon

Overview

  • The importance of the Kalman filter relies on removing the noise from input data. It is one of the great tools to have during the data capturing process.
  • It is based on a linear dynamical system, which is based on the Continuous Markov Chain model.
  • The applicability of the Kalman filter is very vast. Particular use can be seen with sensor-based systems.

Content

  1. Introduction
  2. Linear Dynamical System
  3. Kalman Filter and Hidden Markov Model
  4. Measurement System Analysis and Kalman filter
  5. Why Kalman filters
  6. The internal process of a Kalman Filter
  7. Where to use The Kalman filter
  8. Coding Examples
  9. Disadvantages
  10. Conclusion
  11. Reference
Kalman Filter image

 

Introduction to Kalman Filter

I always believe most of us are fitness freaks. We always seem to find some time for a walk or jog during our busy daily schedule, or we try for it. I feel the advent of the smartwatch has just fueled the situation more because now we can measure and track the data of our physical work, calorie burns, Heart Rate, BP rate, etc. But how much can you trust these sensory measurement data?

I have seen skepticism among people regarding the measurement data. Let me give an example. Each morning I have a personal goal to run 3 KMs. I am 5 feet 11 inches. Each footstep of mine is 3 feet approximately while I am jogging. I can cover about 250 feet in a minute. Ideally, it should take 3300 steps to cover the distance. But, I have seen the normal range is between 3500 Steps to 3200 Steps. But the question arises when I see data like 6000 Steps or 2000 Steps. So, is it a problem related to the GPS or the inbuilt accelerometer sensor? Since I am a health data freak, I want to measure the actual value of my step counts each day when I jog. Well,

I have physically measured my footstep(3 feet), and I know the distance between my home and destination is fixed(3 KMs). Now I have a dilemma in the case of measurement. Do I need to trust sensory or physical data measurement techniques to find the step counts or combine them to estimate? The most practical method is to use both of the assessments to measure the step counts. Based on how much you trust your sensors and your physical estimate, you can combine these two values to best estimate your step counts. Just what I have written about the combination thing is the basic concept of  Kalman Filter.

Linear Dynamical System for Kalman Filter

To know the usability of the Kalman Filter, we need to have a little bit of intuition about the Linear Dynamical System. A dynamical system evolves throughout time, so it’s a function of time. Most of the systems that we see in real life are dynamical systems. We have a particular branch of mathematics known as Calculus which describes a dynamic system very well. It works in the area where we deal with change rates overtime or any other parameters. We call it a differential equation, right? So, if a set of differential equations describes a system, we call it a Linear Dynamic System. We must develop an algorithm to check an autonomous car’s position (latitude and longitude), direction, velocity, and other parameters. Now the position is nothing but a distance. And distance and velocity can be described with the help of differential equations, like rate of change in position or rate of change in direction or rate of change in velocity, etc.

 Linear Dynamical System for Kalman Filter

 

Kalman Filter and Hidden Markov Model

 Kalman Filter and Hidden Markov Model

The Kalman Filter is a method for solving the continuous version of Hidden Markov Models. Hidden Markov deals with latent variables. Let me give a little bit of intuition. When you see a happy face in a crowd, you don’t know the reason for their happiness. There can be many factors like the guy got a job, or meet someone, etc. these factors are latent or hidden because you see only the outcome(happiness). Markov chain says that only your present determines your future, not the past. For our happiness example with Hidden Markov Model, we want to determine the factors or variables(hidden) that cause feelings of happiness before the happy expression(outcome). I have written an article about Markov chain you can check that for more. Below I have given the link for the same. In Hidden Markov Model, we assume the hidden state is one of a few classes, and the movement among these states uses a discrete Markov Chain. Whereas, in Kalman filters, we consider that the unobserved state distribution is Gaussian and moves continuously according to linear dynamics. So obviously, the precision will increase. In case we try to measure a value.

Measurement System Analysis and Kalman Filter

The core objective of a machine learning model is to predict or estimate a value or class, which will be closure to an accurate value or class. When we evaluate the hidden state of the system, we call it estimation. Like inbuilt accelerometer sensor in our smartwatch estimate our step counts or velocity. GPS sensor can calculate our position during the walk. This estimation has to be accurate means close to the actual value. Also, it needs to be precise, which speaks about the variability in measurement. Higher precision means lower variance meaning low uncertainty, while lower precision systems have higher variance meaning high uncertainty. The random measurement error produces the variance. By averaging or smoothing measurements, we can remove the variance. To get a closer estimation, we need more measurements data. Mainly continuous hidden data. Kalman filter advocates about it.

 Measurement System Analysis

Why use Kalman filters?

Let’s consider our first example. Our smartwatches have GPS sensors to track our physical movement during walking or running based on position or location coordinates. The most challenging part is giving you accurate and precise data—many external hidden factors like thermal noise, receiver clock precession, materials, GPS satellite location, etc., create problems in accuracy and precession. The Kalman filter estimates these hidden variables based on inaccurate and uncertain measurement, as it also provides a future state prediction based on the previous estimation. The below image gives some intuition of the factors that the Kalman filter considers.

 Why use Kalman filters?

Take another example of an Autonomous car. In an autonomous vehicle, inbuilt sensors determine the position of objects surrounding the vehicle. Also, a model that predicts the future positions of the objects. The issue with it in real-time system predictive model and sensors may not work well it leads to uncertainty. But with The Kalman filter, we can curb the uncertainty impact with the help of a distribution. More often, a gaussian distribution.

 Why use Kalman filters tree

The internal process of a Kalman Filter

Kalman filter has two stages. It is a recursive process based on the continuous Markov chain. The two steps of it are two predicts and update. “prediction” and “update” are called “propagation” and “correlation.” Before we describe the process, lets us express one statistical term,’ Covariance.’ When we try to find the relation between two variables, we use two words Covariance and correlation. In the case of Covariance, we try to determine how much movement in one variable predicts the movement in corresponding variables. Covariance focuses on direction, not on strength. You can get some intuitive ideas from the below images about variance.

The internal process of a Kalman Filter
The internal process of a Kalman Filter
The internal process of a Kalman Filter 2
bias The internal process

The Kalman Filter process model defines the evolution of the state from time t−1 to time t as:

Kalman Filter process 1

Here F is the state transition matrix that has been applied to the previous state vector xt−1 , B is the control-input matrix used to the control vector ut−1 , and wt−1 is the process noise vector considered zero-mean Gaussian with the covariance Q.

 process 2

I won’t confuse you with the equations since the article is an intuitive picture of the Kalman filter. But one more important term I need to define without any equation. Calculating the Kalman gain(K) involved calculating the covariance matrix for the observation errors and comparing it with the process covariance matrix. In the case of Kalman Gain(K)–> 1, the measurements are accurate, but the estimates are unstable. A smaller value of Kalman gain means estimates are more stable, and the measurements are inaccurate.

Where to use The Kalman filter

It is an exciting question that where the Kalman filter can be used. Everywhere. Any data capturing process can use the Kalman filter. Like Sensors, as we are not sure if the sensor’s measurement data is right or wrong. Data capturing during speech and image processing. Most of the uses case can be found in Space related activities.

Coding Examples

#Install(in Windows) :
pip install filterpy
#Considering a filter that tracks position and velocity using a sensor. only reads position:
from filterpy.kalman import KalmanFilter
f = KalmanFilter (dim_x=2, dim_y=1)
#Assign some initial value for the state (position and velocity):
f.x = np.array([[2.],    # position
                [0.]])   # velocity
#Define the state transition matrix:
f.F = np.array([[1.,1.],
                [0.,1.]])
#Define the measurement function:
f.H = np.array([[1.,0.]])
#Define the covariance matrix.
f.P = np.array([[1000.,    0.],
                [   0., 1000.] ])
#Assign the measurement noise. 
f.R = np.array([[5.]])
#Assign the process noise. 
from filterpy.common import Q_discrete_white_noise
f.Q = Q_discrete_white_noise(dim=2, dt=0.1, var=0.13)
#perform the predict/update loop:
z = get_sensor_reading()
f.predict()
f.update(z)
do_something_with_estimate (f.x)

Conclusion

Kalman filter, also known as linear quadratic estimation (LQE), estimates the underlying states based on Noisy data. Kalman filter tries to filter out the noise. The applicability of the Kalman filter is very vast. Recently Apple Reveals Adding the Kalman Filter to advance ‘Maps’ accuracy for iDevices & especially moving vehicles with CarPlay. With the increased usage of sensory IoT devices, we will use the Kalman filter lot more than what we are using now.

Reference

Material Resources :

https://medium.com/@jaems33/understanding-kalman-filters-with-python-2310e87b8f48

https://www.kalmanfilter.net/default.aspx

https://towardsdatascience.com/kalman-filter-an-algorithm-for-making-sense-from-the-insights-of-various-sensors-fused-together-ddf67597f35e

https://en.wikipedia.org/wiki/Kalman_filter

https://editor.analyticsvidhya.com/preview/index?article_id=18980

Image Resource :

https://qualitytrainingportal.com/resources/measurement-system-analysis/what-is-msa/https://programmathically.com/covariance-and-correlation/

https://towardsdatascience.com/kalman-filter-an-algorithm-for-making-sense-from-the-insights-of-various-sensors-fused-together-ddf67597f35e

Code Resources 

https://filterpy.readthedocs.io/en/latest/kalman/KalmanFilter.html

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