# Transition Models¶

class stonesoup.models.transition.base.TransitionModel[source]

Transition Model base class

property ndim: int

Number of dimensions of model

abstract property ndim_state: int

Number of state dimensions

## Linear¶

class stonesoup.models.transition.linear.LinearGaussianTransitionModel(seed: Optional[int] = None)[source]
Parameters

seed (Union[int, NoneType], optional) – Seed for random number generation

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

abstract covar(**kwargs)stonesoup.types.array.CovarianceMatrix

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

abstract matrix(**kwargs)numpy.ndarray

Model matrix

property ndim: int

Number of dimensions of model

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.CombinedLinearGaussianTransitionModel(model_list: Sequence[GaussianModel], seed: Optional[int] = None)[source]

Bases: stonesoup.models.base.LinearModel, stonesoup.models.transition.base._CombinedGaussianTransitionModel

Combine multiple models into a single model by stacking them.

The assumption is that all models are Linear and Gaussian. Time Variant, and Time Invariant models can be combined together. If any of the models are time variant the keyword argument “time_interval” must be supplied to all methods

Parameters
• model_list (Sequence[GaussianModel]) – List of Transition Models.

• seed (Union[int, NoneType], optional) – Seed for random number generation

matrix(**kwargs)[source]

Model matrix $$F$$

Returns

Return type

numpy.ndarray of shape (ndim_state, ndim_state)

covar(**kwargs)

Returns the transition model noise covariance matrix.

Returns

The process noise covariance.

Return type

stonesoup.types.state.CovarianceMatrix of shape (ndim_state, ndim_state)

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

model_list: Sequence[stonesoup.models.base.GaussianModel]

List of Transition Models.

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of combined model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.LinearGaussianTimeInvariantTransitionModel(transition_matrix: numpy.ndarray, covariance_matrix: CovarianceMatrix, seed: Optional[int] = None, control_matrix: numpy.ndarray = None)[source]

Generic Linear Gaussian Time Invariant Transition Model.

Parameters
transition_matrix: numpy.ndarray

Transition matrix $$\mathbf{F}$$.

control_matrix: numpy.ndarray

Control matrix $$\mathbf{B}$$.

covariance_matrix: stonesoup.types.array.CovarianceMatrix

Transition noise covariance matrix $$\mathbf{Q}$$.

matrix(**kwargs)[source]

Model matrix $$F$$

Returns

The model matrix evaluated given the provided time interval.

Return type

numpy.ndarray of shape (ndim_state, ndim_state)

covar(**kwargs)[source]

Returns the transition model noise covariance matrix.

Returns

The process noise covariance.

Return type

stonesoup.types.state.CovarianceMatrix of shape (ndim_state, ndim_state)

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.ConstantNthDerivative(constant_derivative: int, noise_diff_coeff: float, seed: Optional[int] = None)[source]

Discrete model based on the Nth derivative with respect to time being constant, to set derivative use keyword argument constant_derivative

The model is described by the following SDEs:

\begin{eqnarray} dx^{(N-1)} & = & x^{(N)} dt & | {(N-1)th \ derivative \ on \ X-axis (m)} \\ dx^{(N)} & = & q\cdot dW_t,\ W_t \sim \mathcal{N}(0,q^2) & | \ Nth\ derivative\ on\ X-axis (m/s^{N}) \end{eqnarray}

It is hard to represent the matrix form of these due to the fact that they vary with N, examples for N=1 and N=2 can be found in the ConstantVelocity and ConstantAcceleration models respectively. To aid visualisation of $$F_t$$ the elements are calculated as the terms of the taylor expansion of each state variable.

Parameters
• constant_derivative (int) – The order of the derivative with respect to time to be kept constant, eg if 2 identical to constant acceleration

• noise_diff_coeff (float) – The Nth derivative noise diffusion coefficient (Variance) $$q$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

constant_derivative: int

The order of the derivative with respect to time to be kept constant, eg if 2 identical to constant acceleration

noise_diff_coeff: float

The Nth derivative noise diffusion coefficient (Variance) $$q$$

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

matrix(time_interval, **kwargs)[source]

Model matrix

covar(time_interval, **kwargs)[source]

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

property ndim: int

Number of dimensions of model

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.RandomWalk(noise_diff_coeff: float, seed: Optional[int] = None)[source]

This is a class implementation of a discrete, time-variant 1D Linear-Gaussian Random Walk Transition Model.

The target is assumed to be (almost) stationary, where target velocity is modelled as white noise.

Parameters
• noise_diff_coeff (float) – The position noise diffusion coefficient $$q$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

noise_diff_coeff: float

The position noise diffusion coefficient $$q$$

property constant_derivative

For random walk, this is 0.

covar(time_interval, **kwargs)

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.ConstantVelocity(noise_diff_coeff: float, seed: Optional[int] = None)[source]

This is a class implementation of a discrete, time-variant 1D Linear-Gaussian Constant Velocity Transition Model.

The target is assumed to move with (nearly) constant velocity, where target acceleration is modelled as white noise.

The model is described by the following SDEs:

\begin{eqnarray} dx_{pos} & = & x_{vel} d & | {Position \ on \ X-axis (m)} \\ dx_{vel} & = & q\cdot dW_t,\ W_t \sim \mathcal{N}(0,q^2) & | \ Speed on\ X-axis (m/s) \end{eqnarray}

Or equivalently:

$x_t = F_t x_{t-1} + w_t,\ w_t \sim \mathcal{N}(0,Q_t)$

where:

$\begin{split}x & = & \begin{bmatrix} x_{pos} \\ x_{vel} \end{bmatrix}\end{split}$
$\begin{split}F_t & = & \begin{bmatrix} 1 & dt\\ 0 & 1 \end{bmatrix}\end{split}$
$\begin{split}Q_t & = & \begin{bmatrix} \frac{dt^3}{3} & \frac{dt^2}{2} \\ \frac{dt^2}{2} & dt \end{bmatrix} q\end{split}$
Parameters
• noise_diff_coeff (float) – The velocity noise diffusion coefficient $$q$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

noise_diff_coeff: float

The velocity noise diffusion coefficient $$q$$

property constant_derivative

For constant velocity, this is 1.

covar(time_interval, **kwargs)

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.ConstantAcceleration(noise_diff_coeff: float, seed: Optional[int] = None)[source]

This is a class implementation of a discrete, time-variant 1D Constant Acceleration Transition Model.

The target acceleration is modeled as a zero-mean white noise random process.

The model is described by the following SDEs:

\begin{eqnarray} dx_{pos} & = & x_{vel} d & | {Position \ on \ X-axis (m)} \\ dx_{vel} & = & x_{acc} d & | {Speed \ on\ X-axis (m/s)} \\ dx_{acc} & = & q W_t,\ W_t \sim \mathcal{N}(0,q^2) & | {Acceleration \ on \ X-axis (m^2/s)} \end{eqnarray}

Or equivalently:

$x_t = F_t x_{t-1} + w_t,\ w_t \sim \mathcal{N}(0,Q_t)$

where:

$\begin{split}x & = & \begin{bmatrix} x_{pos} \\ x_{vel} \\ x_{acc} \end{bmatrix}\end{split}$
$\begin{split}F_t & = & \begin{bmatrix} 1 & dt & \frac{dt^2}{2} \\ 0 & 1 & dt \\ 0 & 0 & 1 \end{bmatrix}\end{split}$
$\begin{split}Q_t & = & \begin{bmatrix} \frac{dt^5}{20} & \frac{dt^4}{8} & \frac{dt^3}{6} \\ \frac{dt^4}{8} & \frac{dt^3}{3} & \frac{dt^2}{2} \\ \frac{dt^3}{6} & \frac{dt^2}{2} & dt \end{bmatrix} q\end{split}$
Parameters
• noise_diff_coeff (float) – The acceleration noise diffusion coefficient $$q$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

noise_diff_coeff: float

The acceleration noise diffusion coefficient $$q$$

property constant_derivative

For constant acceleration, this is 2.

covar(time_interval, **kwargs)

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.NthDerivativeDecay(decay_derivative: int, noise_diff_coeff: float, damping_coeff: float, seed: Optional[int] = None)[source]

Discrete model based on the Nth derivative with respect to time decaying to 0 exponentially, to set derivative use keyword argument decay_derivative

The model is described by the following SDEs:

\begin{eqnarray} dx^{(N-1)} & = & x^{(N)} dt & | {(N-1)th derivative \ on \ X-axis (m)} \\ dx^{(N)} & = & -K x^{N} dt + q\cdot dW_t,\ W_t \sim \mathcal{N}(0,q^2) & | \ Nth\ derivative\ on\ X-axis (m/s^{N}) \end{eqnarray}

The transition and covariance matrices are very difficult to express simply, but examples for N=1 and N=2 are given in OrnsteinUhlenbeck and Singer respectively.

Parameters
• decay_derivative (int) – The derivative with respect to time to decay exponentially, eg if 2 identical to singer

• noise_diff_coeff (float) – The noise diffusion coefficient $$q$$

• damping_coeff (float) – The Nth derivative damping coefficient $$K$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

decay_derivative: int

The derivative with respect to time to decay exponentially, eg if 2 identical to singer

noise_diff_coeff: float

The noise diffusion coefficient $$q$$

damping_coeff: float

The Nth derivative damping coefficient $$K$$

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

matrix(time_interval, **kwargs)[source]

Model matrix

covar(time_interval, **kwargs)[source]

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

property ndim: int

Number of dimensions of model

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.OrnsteinUhlenbeck(noise_diff_coeff: float, damping_coeff: float, seed: Optional[int] = None)[source]

This is a class implementation of a discrete, time-variant 1D Linear-Gaussian Ornstein Uhlenbeck Transition Model.

The target is assumed to move with (nearly) constant velocity, which exponentially decays to zero over time, and target acceleration is modeled as white noise.

The model is described by the following SDEs:

\begin{eqnarray} dx_{pos} & = & x_{vel} dt & | {Position \ on \ X-axis (m)} \\ dx_{vel} & = & -K x_{vel} dt + q dW_t, W_t \sim \mathcal{N}(0,q) & | {Speed\ on \ X-axis (m/s)} \end{eqnarray}

Or equivalently:

$x_t = F_t x_{t-1} + w_t,\ w_t \sim \mathcal{N}(0,Q_t)$

where:

$\begin{split}x & = & \begin{bmatrix} x_{pos} \\ x_{vel} \end{bmatrix}\end{split}$
$\begin{split}F_t & = & \begin{bmatrix} 1 & \frac{1}{K}(1 - e^{-Kdt})\\ 0 & e^{-Kdt} \end{bmatrix}\end{split}$
$\begin{split}Q_t & = & \begin{bmatrix} \frac{dt - \frac{2}{K}(1 - e^{-Kdt}) + \frac{1}{2K}(1 - e^{-2Kdt})}{K^2} & \frac{\frac{1}{K}(1 - e^{-Kdt}) - \frac{1}{2K}(1 - e^{-2Kdt})}{K} \\ \frac{\frac{1}{K}(1 - e^{-Kdt}) - \frac{1}{2K}(1 - e^{-2Kdt})}{K} & \frac{1 - e^{-2Kdt}}{2K} \end{bmatrix} q\end{split}$
Parameters
• noise_diff_coeff (float) – The velocity noise diffusion coefficient $$q$$

• damping_coeff (float) – The velocity damping coefficient $$K$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

noise_diff_coeff: float

The velocity noise diffusion coefficient $$q$$

damping_coeff: float

The velocity damping coefficient $$K$$

property decay_derivative

The derivative with respect to time to decay exponentially, eg if 2 identical to singer

covar(time_interval, **kwargs)

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.Singer(noise_diff_coeff: float, damping_coeff: float, seed: Optional[int] = None)[source]

This is a class implementation of a discrete, time-variant 1D Singer Transition Model.

The target acceleration is modeled as a zero-mean Gauss-Markov random process.

The model is described by the following SDEs:

\begin{eqnarray} dx_{pos} & = & x_{vel} dt & | {Position \ on \ X-axis (m)} \\ dx_{vel} & = & x_{acc} dt & | {Speed \ on\ X-axis (m/s)} \\ dx_{acc} & = & -K x_{acc} dt + q W_t,\ W_t \sim \mathcal{N}(0,q^2) & | {Acceleration \ on \ X-axis (m^2/s)} \end{eqnarray}

Or equivalently:

$x_t = F_t x_{t-1} + w_t,\ w_t \sim \mathcal{N}(0,Q_t)$

where:

$\begin{split}x & = & \begin{bmatrix} x_{pos} \\ x_{vel} \\ x_{acc} \end{bmatrix}\end{split}$
$\begin{split}F_t & = & \begin{bmatrix} 1 & dt & (K dt-1+e^{-K dt})/K^2 \\ 0 & 1 & (1-e^{-K dt})/K \\ 0 & 0 & e^{-K t} \end{bmatrix}\end{split}$
$\begin{split}Q_t & = & q \begin{bmatrix} \frac{[1-e^{-2K dt}] + 2K dt + \frac{2K^3 dt^3}{3}- 2K^2 dt^2 - 4K dt e^{-K dt} }{2K^5} & \frac{(K dt - [1-e^{-K dt}])^2}{2K^4} & \frac{[1-e^{-2K dt}]-2K dt e^{-K dt}} {2K^3} \\ \frac{(K dt - [1 - e^{-K dt}])^2}{2K^4} & \frac{2K dt - 4[1-e^{-K dt}] + [1-e^{-2K dt}]}{2K^3} & \frac{[1-e^{-K dt}]^2}{2K^2} \\ \frac{[1- e^{-2K dt}]-2K dt e^{-K dt}} {2K^3} & \frac{[1-e^{-K dt}]^2}{2K^2} & \frac{1-e^{-2K dt}}{2K} \end{bmatrix}\end{split}$
Parameters
• noise_diff_coeff (float) – The acceleration noise diffusion coefficient $$q$$

• damping_coeff (float) – The reciprocal of the decorrelation time $$\alpha$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

noise_diff_coeff: float

The acceleration noise diffusion coefficient $$q$$

damping_coeff: float

The reciprocal of the decorrelation time $$\alpha$$

property decay_derivative

The derivative with respect to time to decay exponentially, eg if 2 identical to singer

covar(time_interval, **kwargs)

Model covariance

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.SingerApproximate(noise_diff_coeff: float, damping_coeff: float, seed: Optional[int] = None)[source]
Parameters
• noise_diff_coeff (float) – The acceleration noise diffusion coefficient $$q$$

• damping_coeff (float) – The reciprocal of the decorrelation time $$\alpha$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

property decay_derivative

The derivative with respect to time to decay exponentially, eg if 2 identical to singer

covar(time_interval, **kwargs)[source]

Returns the transition model noise covariance matrix.

Parameters

time_interval (datetime.timedelta) – A time interval $$dt$$

Returns

The process noise covariance.

Return type

stonesoup.types.state.CovarianceMatrix of shape (ndim_state, ndim_state)

damping_coeff: float

The reciprocal of the decorrelation time $$\alpha$$

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of model state dimensions.

Return type

int

noise_diff_coeff: float

The acceleration noise diffusion coefficient $$q$$

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.ConstantTurnSandwich(turn_noise_diff_coeffs: numpy.ndarray, turn_rate: float, model_list: Sequence[LinearGaussianTransitionModel], seed: Optional[int] = None)[source]

This is a class implementation of a time-variant 2D Constant Turn Model. This model is used, as opposed to the normal ConstantTurn model, when the turn occurs in 2 dimensions that are not adjacent in the state vector, eg if the turn occurs in the x-z plane but the state vector is of the form $$(x,y,z)$$. The list of transition models are to be applied to any state variables that lie in between, eg if for the above example you wanted the y component to move with constant velocity, you would put a ConstantVelocity model in the list.

The target is assumed to move with (nearly) constant velocity and also known (nearly) constant turn rate.

Parameters
• turn_noise_diff_coeffs (numpy.ndarray) – The acceleration noise diffusion coefficients $$q$$

• turn_rate (float) – The turn rate $$\omega$$

• model_list (Sequence[LinearGaussianTransitionModel]) – List of Transition Models.

• seed (Union[int, NoneType], optional) – Seed for random number generation

turn_noise_diff_coeffs: numpy.ndarray

The acceleration noise diffusion coefficients $$q$$

turn_rate: float

The turn rate $$\omega$$

model_list: Sequence[stonesoup.models.transition.linear.LinearGaussianTransitionModel]

List of Transition Models.

property ndim_state

ndim_state getter method

Returns

The number of combined model state dimensions.

Return type

int

matrix(time_interval, **kwargs)[source]

Model matrix $$F$$

Returns

Return type

numpy.ndarray of shape (ndim_state, ndim_state)

covar(time_interval, **kwargs)[source]

Returns the transition model noise covariance matrix.

Returns

The process noise covariance.

Return type

stonesoup.types.state.CovarianceMatrix of shape (ndim_state, ndim_state)

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

property ndim: int

Number of dimensions of model

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

class stonesoup.models.transition.linear.ConstantTurn(turn_noise_diff_coeffs: numpy.ndarray, turn_rate: float, seed: Optional[int] = None)[source]

This is a class implementation of a discrete, time-variant 2D Constant Turn Model.

The target is assumed to move with (nearly) constant velocity and also known (nearly) constant turn rate.

The model is described by the following SDEs:

\begin{eqnarray} dx_{pos} & = & x_{vel} d & | {Position \ on \ X-axis (m)} \\ dx_{vel} & = &-\omega y_{pos} d & | {Speed \ on\ X-axis (m/s)} \\ dy_{pos} & = & y_{vel} d & | {Position \ on \ Y-axis (m)} \\ dy_{vel} & = & \omega x_{pos} d & | {Speed \ on\ Y-axis (m/s)} \end{eqnarray}

Or equivalently:

$x_t = F_t x_{t-1} + w_t,\ w_t \sim \mathcal{N}(0,Q_t)$

where:

$\begin{split}x & = & \begin{bmatrix} x_{pos} \\ x_{vel} \\ y_{pos} \\ y_{vel} \end{bmatrix}\end{split}$
$\begin{split}F_t & = & \begin{bmatrix} 1 & \frac{\sin\omega dt}{\omega} & 0 &-\frac{1-\cos\omega dt}{\omega} \\ 0 & \cos\omega dt & 0 & -\sin\omega dt \\ 0 & \frac{1-\cos\omega dt}{\omega} & 1 & \frac{\sin\omega dt}{\omega}\\ 0 & \sin\omega dt & 0 & \cos\omega dt \end{bmatrix}\end{split}$
$\begin{split}Q_t & = & \begin{bmatrix} q_x^2 \frac{dt^3}{3} & q_x^2 \frac{dt^2}{2} & 0 & 0 \\ q_x^2 \frac{dt^2}{2} & q_x^2 dt & 0 & 0 \\ 0 & 0 & q_y^2 \frac{dt^3}{3} & q_y^2 \frac{dt^2}{2}\\ 0 & 0 & q_y^2 \frac{dt^2}{2} & q_y^2 dt \end{bmatrix}\end{split}$
Parameters
• turn_noise_diff_coeffs (numpy.ndarray) – The acceleration noise diffusion coefficients $$q$$

• turn_rate (float) – The turn rate $$\omega$$

• seed (Union[int, NoneType], optional) – Seed for random number generation

property model_list

For a turn in adjacent state vectors, no transition models go in between

covar(time_interval, **kwargs)

Returns the transition model noise covariance matrix.

Returns

The process noise covariance.

Return type

stonesoup.types.state.CovarianceMatrix of shape (ndim_state, ndim_state)

function(state: State, noise: Union[bool, numpy.ndarray] = False, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model linear function $$f_k(x(k),w(k)) = F_k(x_k) + w_k$$

Parameters
Returns

The StateVector(s) with the model function evaluated.

Return type

StateVector or StateVectors

matrix(time_interval, **kwargs)

Model matrix $$F$$

Returns

Return type

numpy.ndarray of shape (ndim_state, ndim_state)

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of combined model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation

turn_noise_diff_coeffs: numpy.ndarray

The acceleration noise diffusion coefficients $$q$$

turn_rate: float

The turn rate $$\omega$$

## NonLinear¶

class stonesoup.models.transition.nonlinear.CombinedNonlinearGaussianTransitionModel(model_list: Sequence[GaussianModel], seed: Optional[int] = None)[source]

Bases: stonesoup.models.transition.base._CombinedGaussianTransitionModel

Combine multiple models into a single model by stacking them.

The assumption is that all models are Gaussian. Time Variant, and Time Invariant models can be combined together. If any of the models are time variant the keyword argument “time_interval” must be supplied to all methods

Parameters
• model_list (Sequence[GaussianModel]) – List of Transition Models.

• seed (Union[int, NoneType], optional) – Seed for random number generation

function(state, **kwargs)stonesoup.types.array.StateVector[source]

Applies each transition model in model_list in turn to the state’s corresponding state vector components. For example, in a 3D state space, with model_list = [modelA(ndim_state=2), modelB(ndim_state=1)], this would apply modelA to the state vector’s 1st and 2nd elements, then modelB to the remaining 3rd element.

Parameters

state (stonesoup.state.State) – The state to be transitioned according to the models in model_list.

Returns

state_vector – of shape (ndim_state, 1). The resultant state vector of the transition.

Return type

stonesoup.types.array.StateVector

covar(**kwargs)

Returns the transition model noise covariance matrix.

Returns

The process noise covariance.

Return type

stonesoup.types.state.CovarianceMatrix of shape (ndim_state, ndim_state)

model_list: Sequence[stonesoup.models.base.GaussianModel]

List of Transition Models.

property ndim: int

Number of dimensions of model

property ndim_state

ndim_state getter method

Returns

The number of combined model state dimensions.

Return type

int

pdf(state1: State, state2: State, **kwargs)Union[stonesoup.types.numeric.Probability, numpy.ndarray]

Model pdf/likelihood evaluation function

Evaluates the pdf/likelihood of state1, given the state state2 which is passed to function().

In mathematical terms, this can be written as:

$p = p(y_t | x_t) = \mathcal{N}(y_t; x_t, Q)$

where $$y_t$$ = state_vector1, $$x_t$$ = state_vector2 and $$Q$$ = covar.

Parameters
Returns

The likelihood of state1, given state2

Return type
rvs(num_samples: int = 1, random_state=None, **kwargs)Union[stonesoup.types.array.StateVector, stonesoup.types.array.StateVectors]

Model noise/sample generation function

Generates noise samples from the model.

In mathematical terms, this can be written as:

$v_t \sim \mathcal{N}(0,Q)$

where $$v_t =$$ noise and $$Q$$ = covar.

Parameters

num_samples (scalar, optional) – The number of samples to be generated (the default is 1)

Returns

noise – A set of Np samples, generated from the model’s noise distribution.

Return type

2-D array of shape (ndim, num_samples)

seed: Optional[int]

Seed for random number generation