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
Correlation Matrix

Demos/Examples

Link	Description	Image
Bikes	4D Constant Acceleration boxes
Bouncing Ball	2D constant acceleration with bounces
Sinusoidale Extended Kalman-Filter	1D Extended KF Sinus
Code pen GPS Data smoothing with constant speed	2D constant speed
Partial Observation	1D / 2 sensor with missing values
Smooth 3x3 rotation matrix	4d smoothing

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

Add it to your project like :

<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

Advanced usage

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 article	kalman-filter js lib
$F_k$, the state-transition model	`dynamic.transition`
$H_k$, the observation model	`observation.stateProjection`
$Q_k$, the covariance of the process noise	`dynamic.covariance`
$R_k$, the covariance of the observation noise	`observation.covariance`
$B_k u_k$, the control-input model multiplied by the control vector	`dynamic.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: [[500], [500], [500], [0], [0], [0]],
			// 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

Use your kalman filter

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[0]).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

Files

README.md

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