 kalman-filter

Package Overview
Dependencies
3
Maintainers
1
Versions
26
Issues
File Explorer

# kalman-filter

Kalman filter (and Extended Kalman Filter) Multi-dimensional implementation in Javascript

1.13.0latest
GitHub Version published
Maintainers
1
806
decreased by-17.16%

Changelog

### .css-1z04cui{margin-bottom:var(--chakra-space-4);font-size:var(--chakra-fontSizes-md);}v1.13.0

#### Bug Fixes

• package.json (764b713)

#### Features

• control parameter using dynamic.constant (90b3333) ## kalman-filter

Kalman Filter in JavaScript (for both node.js and the browser)

This library implements following features:

• N-dimensional Kalman Filter (for multivariate Gaussian)
• Forward Kalman Filter (Online)
• Forward-Backward Smoothing Kalman Filter
• Split Prediction/Correction steps
• Extended Kalman Filter (when using functions for dynamics and observation matrixes)
• Correlation Matrix

### Demos/Examples

Open an issue to add more examples in this section explaining how you use this library !

### Installation

#### Npm

npm install kalman-filter  const {KalmanFilter} = require('kalman-filter'); 

#### Browser usage

Download the file kalman-filter.min.js from Releases page

<script src="dist/kalman-filter.min.js"></script> <script> var {KalmanFilter} = kalmanFilter; // ... do whatever you want with KalmanFilter </script> 

### Simple Example

#### 1D Smoothing Usage

const {KalmanFilter} = require('kalman-filter'); const observations = [0, 0.1, 0.5, 0.2, 3, 4, 2, 1, 2, 3, 5, 6]; // this is creating a smoothing const kFilter = new KalmanFilter(); const res = kFilter.filterAll(observations) // res is a list of list (for multidimensional filters) // [ // [ 0 ], // [ 0.06666665555510715 ], // [ 0.3374999890620582 ], // [ 0.25238094852592136 ], // [ 1.9509090885288296 ], // [ 3.2173611101031616 ], // [ 2.4649867370240965 ], // [ 1.5595744679428254 ], // [ 1.831772445766021 ], // [ 2.5537767922925685 ], // [ 4.065625882212133 ], // [ 5.26113483436549 ] // ] 

Result is : #### 2D Smoothing Usage

const {KalmanFilter} = require('kalman-filter'); const observations = [[0, 1], [0.1, 0.5], [0.2, 3], [4, 2], [1, 2]]; const kFilter = new KalmanFilter({observation: 2}); // equivalent to // new KalmanFilter({ // observation: { // name: 'sensor', // sensorDimension: 2 // } // }); const res = kFilter.filterAll(observations) 

#### 2D Smoothing with constant-speed model

const {KalmanFilter} = require('kalman-filter'); const observations = [[0, 1], [0.1, 0.5], [0.2, 3], [4, 2], [1, 2]]; const kFilter = new KalmanFilter({ observation: 2, dynamic: 'constant-speed' }); // equivalent to // new KalmanFilter({ // observation: { // name: 'sensor', // sensorDimension: 2 // }, // dynamic: { // name: 'constant-speed' // }, // }); const res = kFilter.filterAll(observations) 

### How to instantiate your kalman filter

This library gives you the ability to fully configure your kalman-filter.

For advanced usage, here is the correspondance table with the matrix name of the wikipedia article

Wikipedia articlekalman-filter js lib
$F_k$, the state-transition modeldynamic.transition
$H_k$, the observation modelobservation.stateProjection
$Q_k$, the covariance of the process noisedynamic.covariance
$R_k$, the covariance of the observation noiseobservation.covariance
$B_k u_k$, the control-input model multiplied by the control vectordynamic.constant
$\mathbf{P}_{0\mid 0}$dynamic.init.covariance
$\mathbf{x}_{0\mid 0}$dynamic.init.mean

#### Configure the dynamic with dynamic.name

dynamic.name is a shortcut to give you access to preconfigured dynamic models, you can also register your own shortcust see Register models shortcuts

Available default models as :

• constant-position
• constant-speed
• constant-acceleration

This will automatically configure the dynamic.transition matrix.

###### constant-position
\begin{align} State :& \begin{bmatrix} x_t \end{bmatrix}\\ Transition Equation :& x_t \sim x_{t-1} \\ dynamic.transition :& \begin{bmatrix} 1 \end{bmatrix} \end{align} 
###### constant-speed
\begin{align} State :& \begin{bmatrix} x_t \\ speed_t \end{bmatrix} \\ Transition Equation :& \begin{split} x_t &\sim x_{t-1} + speed_{t-1},\\ speed_t &\sim speed_{t-1} \end{split} \\ dynamic.transition :& \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \end{align} 
###### constant-acceleration
\begin{align} State :& \begin{bmatrix} x_t \\ speed_t \\ acc_t \end{bmatrix} \\ Transition Equation :& \begin{split} x_t &\sim x_{t-1} + speed_{t-1} \\ speed_t &\sim speed_{t-1} + acc_{t-1} \\ acc_t &\sim acc_{t-1} \end{split} \\ dynamic.transition :& \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{bmatrix} \end{align} 
##### 'constant-position' on 2D data

This is the default behavior

const {KalmanFilter} = require('kalman-filter'); const kFilter = new KalmanFilter({ observation: { sensorDimension: 2, name: 'sensor' }, dynamic: { name: 'constant-position',// observation.sensorDimension == dynamic.dimension covariance: [3, 4]// equivalent to diag([3, 4]) } }); 
##### 'constant-speed' on 3D data
const {KalmanFilter} = require('kalman-filter'); const kFilter = new KalmanFilter({ observation: { sensorDimension: 3, name: 'sensor' }, dynamic: { name: 'constant-speed',// observation.sensorDimension * 2 == state.dimension timeStep: 0.1, covariance: [3, 3, 3, 4, 4, 4]// equivalent to diag([3, 3, 3, 4, 4, 4]) } }); 
##### 'constant-acceleration' on 2D data
const {KalmanFilter} = require('kalman-filter'); const kFilter = new KalmanFilter({ observation: { sensorDimension: 2, name: 'sensor' }, dynamic: { name: 'constant-acceleration',// observation.sensorDimension * 3 == state.dimension timeStep: 0.1, covariance: [3, 3, 4, 4, 5, 5]// equivalent to diag([3, 3, 4, 4, 5, 5]) } }); 

#### Instanciation of a generic linear model

This is an example of how to build a constant speed model, in 3D without dynamic.name, using detailed api.

• dynamic.dimension is the size of the state
• dynamic.transition is the state transition model that defines the dynamic of the system
• dynamic.covariance is the covariance matrix of the transition model
• dynamic.init is used for initial state (we generally set a big covariance on it)
const {KalmanFilter} = require('kalman-filter'); const timeStep = 0.1; const huge = 1e8; const kFilter = new KalmanFilter({ observation: { dimension: 3 }, dynamic: { init: { // We just use random-guessed values here that seems reasonable mean: [, , , , , ], // We init the dynamic model with a huge covariance cause we don't // have any idea where my modeled object before the first observation is located covariance: [ [huge, 0, 0, 0, 0, 0], [0, huge, 0, 0, 0, 0], [0, 0, huge, 0, 0, 0], [0, 0, 0, huge, 0, 0], [0, 0, 0, 0, huge, 0], [0, 0, 0, 0, 0, huge], ], }, // Corresponds to (x, y, z, vx, vy, vz) dimension: 6, // This is a constant-speed model on 3D : [ [Id , timeStep*Id], [0, Id]] transition: [ [1, 0, 0, timeStep, 0, 0], [0, 1, 0, 0, timeStep, 0], [0, 0, 1, 0, 0, timeStep], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1] ], // Diagonal covariance for independant variables // since timeStep = 0.1, // it makes sense to consider speed variance to be ~ timeStep^2 * positionVariance covariance: [1, 1, 1, 0.01, 0.01, 0.01]// equivalent to diag([1, 1, 1, 0.01, 0.01, 0.01]) } }); 

#### Configure the observation

##### Using sensor observation

The observation is made from 2 different sensors with identical properties (i.e. same covariances) , the input measure will be [<sensor0-dim0>, <sensor0-dim1>, <sensor1-dim0>, <sensor1-dim1>].

const {KalmanFilter} = require('kalman-filter'); const timeStep = 0.1; const kFilter = new KalmanFilter({ observation: { sensorDimension: 2,// observation.dimension == observation.sensorDimension * observation.nSensors nSensors: 2, sensorCovariance: [3, 4], // equivalent to diag([3, 4]) name: 'sensor' }, dynamic: { name: 'constant-speed',// observation.sensorDimension * 2 == state.dimension covariance: [3, 3, 4, 4]// equivalent to diag([3, 3, 4, 4]) } }); 
##### Custom Observation matrix

The observation is made from 2 different sensors with different properties (i.e. different covariances), the input measure will be [<sensor0-dim0>, <sensor0-dim1>, <sensor1-dim0>, <sensor1-dim1>].

This can be achived manually by using the detailed API :

• observation.dimension is the size of the observation
• observation.stateProjection is the matrix that transforms state into observation, also called observation model
• observation.covariance is the covariance matrix of the observation model
const {KalmanFilter} = require('kalman-filter'); const timeStep = 0.1; const kFilter = new KalmanFilter({ observation: { dimension: 4, stateProjection: [ [1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0] ], covariance: [3, 4, 0.3, 0.4] }, dynamic: { name: 'constant-speed',// observation.sensorDimension * 2 == state.dimension covariance: [3, 3, 4, 4]// equivalent to diag([3, 3, 4, 4]) } }); 

### Play with Kalman Filter

In order to use the Kalman-Filter with a dynamic or observation model which is not strictly a General linear model, it is possible to use function in following parameters :

• observation.stateProjection
• observation.covariance
• dynamic.transition
• dynamic.covariance
• dynamic.constant

In this situation this function will return the value of the matrix at each step of the kalman-filter.

In this example, we create a constant-speed filter with non-uniform intervals;

const {KalmanFilter} = require('kalman-filter'); const intervals = [1,1,1,1,2,1,1,1]; const kFilter = new KalmanFilter({ observation: { dimension: 2, /** * @param {State} opts.predicted * @param {Array.<Number>} opts.observation * @param {Number} opts.index */ stateProjection: function(opts){ return [ [1, 0, 0, 0], [0, 1, 0, 0] ] }, /** * @param {State} opts.predicted * @param {Array.<Number>} opts.observation * @param {Number} opts.index */ covariance: function(opts){ return [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] ] } }, dynamic: { dimension: 4, //(x, y, vx, vy) /** * @param {State} opts.previousCorrected * @param {Number} opts.index */ transition: function(opts){ const dT = intervals[opts.index]; if(typeof(dT) !== 'number' || isNaN(dT) || dT <= 0){ throw(new Error('dT should be positive number')) } return [ [1, 0, dT, 0], [0, 1, 0, dT] [0, 0, 1, 0] [0, 0, 0, 1] ] }, /** * @param {State} opts.previousCorrected * @param {Number} opts.index */ covariance: function(opts){ const dT = intervals[opts.index]; if(typeof(dT) !== 'number' || isNaN(dT) || dT <= 0){ throw(new Error('dT should be positive number')) } return [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1*dT, 0], [0, 0, 0, 1*dT] ] } } }); 

#### Extended

If you want to implement an extended kalman filter

You will need to put your non-linear functions in the following parameters

• observation.fn
• dynamic.fn

See an example in Sinusoidale Extended Kalman-Filter

#### Using Control model

If you want to add a constant parameter in the dynamic model (also called control input), you can use dynamic.constant function

See an example code in demo/bouncing-ball or the result in Bouncing Ball example

#### Simple Batch usage (run it once for the whole dataset)

const observations = [[0, 2], [0.1, 4], [0.5, 9], [0.2, 12]]; // batch kalman filter const results = kFilter.filterAll(observations); 

#### Online filter

When using online usage (only the forward step), the output of the filter method is an instance of the "State" class.

// online kalman filter let previousCorrected = null; const results = []; observations.forEach(observation => { previousCorrected = kFilter.filter({previousCorrected, observation}); results.push(previousCorrected.mean); }); 

#### Predict/Correct detailed usage (run it online)

If you want to use KalmanFilter in more advanced usage, you might want to dissociate the predict and the correct functions

// online kalman filter let previousCorrected = null; const results = []; observations.forEach(observation => { const predicted = kFilter.predict({ previousCorrected }); const correctedState = kFilter.correct({ predicted, observation }); results.push(correctedState.mean); // update the previousCorrected for next loop iteration previousCorrected = correctedState }); console.log(results); 

#### Batch Forward - Backward smoothing usage

The Forward - Backward process

// batch kalman filter const results = kFilter.filterAll({observations, passMode: 'forward-backward'}); 

### Register models shortcuts

To get more information on how to build a dynamic model, check in the code lib/dynamic/ (or lib/observation for observation models).

If you feel your model can be used by other, do not hesitate to create a Pull Request.

const {registerDynamic, KalmanFilter, registerObservation} = require('kalman-filter'); registerObservation('custom-sensor', function(opts1){ // do your stuff return { dimension, stateProjection, covariance } }) registerDynamic('custom-dynamic', function(opts2, observation){ // do your stuff // here you can use the parameter of observation (like observation.dimension) // to build the parameters for dynamic return { dimension, transition, covariance } }) const kFilter = new KalmanFilter({ observation: { name: 'custom-sensor', // ... fields of opts1 }, dynamic: { name: 'custom-dynamic', // ... fields of opts2 } }); 

### Set your model parameters from the ground truths state values

In order to find the proper values for covariance matrix, we use following approach :

 const {getCovariance, KalmanFilter} = require('kalman-filter'); // Ground truth values in the dynamic model hidden state const groundTruthStates = [ // here this is (x, vx) [[0, 1.1], [1.1, 1], [2.1, 0.9], [3, 1], [4, 1.2]], // example 1 [[8, 1.1], [9.1, 1], [10.1, 0.9], [11, 1], [12, 1.2]] // example 2 ] // Observations of this values const measures = [ // here this is x only [[0.1], [1.3], [2.4], [2.6], [3.8]], // example 1 [[8.1], [9.3], [10.4], [10.6], [11.8]] // example 2 ]; const kFilter = new KalmanFilter({ observation: { name: 'sensor', sensorDimension: 1 }, dynamic: { name: 'constant-speed' } }) const dynamicCovariance = getCovariance({ measures: groundTruthStates.map(ex => return ex.slice(1) ).reduce((a,b) => a.concat(b)), averages: groundTruthStates.map(ex => return ex.slice(1).map((_, index) => { return kFilter.predict({previousCorrected: ex[index - 1]}).mean; }) ).reduce((a,b) => a.concat(b)) }); const observationCovariance = getCovariance({ measures: measures.reduce((a,b) => a.concat(b)), averages: groundTruthStates.map((a) => a).reduce((a,b) => a.concat(b)) }); 

### How to measure how good does a specific model fits with data

There are different ways to measure the performance of a model against some measures :

#### Model fits with a specific measurements

We use Mahalanobis distance

const observations = [[0, 2], [0.1, 4], [0.5, 9], [0.2, 12]]; // online kalman filter let previousCorrected = null; const results = []; observations.forEach(observation => { const predicted = kFilter.predict({ previousCorrected }); const dist = predicted.mahalanobis(observation) previousCorrected = kFilter.correct({ predicted, observation }); distances.push(dist); }); const distance = distances.reduce((d1, d2) => d1 + d2, 0); 

#### How precise is this Model

We compare the model with random generated numbers sequence.

const h = require('hasard') const observationHasard = h.array({value: h.number({type: 'normal'}), size: 2}) const observations = observationHasard.run(200); // online kalman filter let previousCorrected = null; const results = []; observations.forEach(observation => { const predicted = kFilter.predict({ previousCorrected }); const dist = predicted.mahalanobis(measure) previousCorrected = kFilter.correct({ predicted, observation }); distances.push(dist); }); const distance = distances.reduce((d1, d2) => d1 + d2, 0); 

#### Credits

Thanks to Adrien Pellissier for his hard work on this library.

#### Similar Project

For a simple 1D Kalman filter in javascript see https://github.com/wouterbulten/kalmanjs

## Keywords

FAQs

Last updated on 28 Mar 2023

Did you know? Socket installs a Github app to automatically flag issues on every pull request and report the health of your dependencies. Find out what is inside your node modules and prevent malicious activity before you update the dependencies. support@socket.dev Product

• Package Issues
• Integrations
• Docs
• Pricing
• FAQ