!Kalman Filter Bike Gif

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 | !Kalman Filter Bike Gif |
| Bouncing Ball | 2D constant acceleration with bounces | !3x3 rotation matrix |
| Sinusoidale Extended Kalman-Filter | 1D Extended KF Sinus | !Kalman Filter Sin Extended |
| Code pen GPS Data smoothing with constant speed | 2D constant speed | !Kalman Filter GPS |
| Partial Observation | 1D / 2 sensor with missing values | !partial |
| Smooth 3x3 rotation matrix | 4d smoothing | !3x3 rotation matrix |

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

Installation

$3

``sh
npm install kalman-filter
`

`js
const {KalmanFilter} = require('kalman-filter');
`

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

Add it to your project like :

`html


`

Simple Example

$3

`js
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 :

!Kalman Filter 1d example

$3

`js
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)
`

$3

`js
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

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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 |

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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

`math
\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

`math
\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

`math
\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

`js
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


`js
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


`js
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])
}
});

`

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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)

`js
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])
}
});

`

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#### Using sensor observation

The observation is made from 2 different sensors with identical properties (i.e. same covariances) , the input measure will be [, , , ].

`js
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 [, , , ].

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*

`js
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;

`js
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.} opts.observation
* @param {Number} opts.index
*/
stateProjection: function(opts){
return [
[1, 0, 0, 0],
[0, 1, 0, 0]
]
},
/**
* @param {State} opts.predicted
* @param {Array.} 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]
]
}
}
});
`


$3

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

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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

$3

`js
const observations = [[0, 2], [0.1, 4], [0.5, 9], [0.2, 12]];

// batch kalman filter
const results = kFilter.filterAll(observations);
`


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

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

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

`js
// 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);
`

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The Forward - Backward process

`js
// 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.

`js
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 :

`js

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 :

$3

We use Mahalanobis distance

`js
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);
`


$3

We compare the model with random generated numbers sequence.

`js
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);

``

$3

Thanks to _Adrien Pellissier_ for his hard work on this library.

$3

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