source_localization.obs module

class source_localization.obs.Obs(t_obs, X_obs, d_obs, msh, dt_obs_operator=0.0)[source]

Bases: object

Class containing all the features related to the sensors and the observations. This includes:

the observations \(m_{obs}\), the observation times \(t_{obs}\) and the observation positions \(X_{obs}=(x_{obs}, y_{obs})\),

the estimation of the observed variable \(m(c(s))\),

the concentration of pheromone \(c(s)\) at the time and positions required to compute the observed variable,

the observation operator \(c\mapsto m(c)\)

the adjoint operator of its derivative with respect of the state variable \(\phi\mapsto (d_cm(c))^*\cdot\phi\),

the covariance matrix of the observation error \(\mathbf{R}\).

In this class, the observations are the accumulation of pheromone over a given time window. Therefore, the observation operator is the integral of the pheromone concentration over this time window:

\[m(c)_{i}=\int_{t_{obs, i}-\delta t}^{t_{obs, i}}c(x_{obs, i}, y_{obs, i}, \tau)d\tau\]

msh

The geometry of the domain.

Type:: MeshRect2D

t_obs

The observation times \(t_{obs}\).

Type:: ndarray

X_obs

The locations of the observations \(X_{obs}=(x_{obs}, y_{obs})\).

Type:: ndarray

d_obs

The observations data \(m_{obs}\).

Type:: ndarray

N_obs

The number of observations.

Type:: int

nb_sensors

The number of sensors.

Type:: int

X_sensors

The positions of the sensors.

Type:: ndarray

dt_obs_operator

Length of the time window of accumulation \(\delta t\). If the observation is instantaneous, set to 0.

Type:: float

nb_dt_in_obs_time_window

Number of time steps covering the time window of accumulation \([t_{obs, i}-\delta t; t_{obs, i}]\).

Type:: int

index_obs_to_index_time_est

The indexes of the times at which the state variable is required to compute the observation variable given the index of an observation.

Type:: ndarray

index_time_est_to_index_obs

Dictionary where each key is a time index, and the value is an array of indexes of the observations that require the estimate of the state variable at this time to estimate the observed variable.

Type:: dict

c_est

The estimation the concentration of pheromone \(c(s;x,y,t)\) at the times \(t\) and positions \((x,y)\) required to compute the observed variable \(m(c(s))\).

Type:: ndarray

d_est

The estimation of the accumulation of pheromone (observed variable) computed using the pheromone propagation model \(m(c(s))\).

Type:: ndarray

R_inv

The inverse of the covariance matrix of the observation error \(\mathbf{R}^{-1}\).

Type:: sparse matrix, default: identity() matrix

__init__(t_obs, X_obs, d_obs, msh, dt_obs_operator=0.0)[source]

Constructor method.

Parameters:

t_obs (ndarray) – The observation times \(t_{obs}\).
X_obs (ndarray) – The locations of the observations \(X_{obs}=(x_{obs}, y_{obs})\).
d_obs (ndarray) – The observations data \(m_{obs}\).
msh (MeshRect2D) – The geometry of the domain.
dt_obs_operator (float, default: 0) – Length of the time window of accumulation \(\delta t\). If set to 0, the observation is instantaneous.

Raises:

ValueError – if the time array has not been initialized using the methods calc_dt_explicit_solver() or calc_dt_implicit_solver().
ValueError – if the sizes of the arrays containing the observation times \(t_{obs}\), the observation positions \(X_{obs}\) and the observations data \(m_{obs}\) do not coincide.
ValueError – if one observation time \(t_{obs}\) or one observation position \(X_{obs}\) is out of the time window \([0;T]\) or domain \(\Omega\).

adjoint_derivative_obs_operator(t, phi)[source]

compute the spatial map of the adjoint of the derivative of the observation operator with respect to the state variable \(\phi\mapsto (d_cm(c))^*\cdot\phi\), at a given time \(t\) and for an element of the observation space \(\phi\).

Parameters:

t (float or integer) – The current time \(t\).
phi (ndarray) – The element of the observation space \(\phi\) at current time \(t\).

Returns:

The spatial map of the image of the adjoint of the derivative of the observation operator \((d_cm(c))^*\cdot\phi(x,y)\) at the given time raveled into a vector.

Return type:

ndarray

obs_operator()[source]

The observation operator \(c\mapsto m(c)\).

In this class, the observation operator is, for the \(i^{th}\) observation time and position, \(m(c)_{i}=\int_{t_{obs, i}-\delta t}^{t_{obs, i}}c(x_{obs, i}, y_{obs, i}, \tau)d\tau\). Moreover, it is assumed that the integration time window for different data of a same sensor are not overlapping

Notes

This method computes the observed variable and store the values in the attribute d_est using the attribute c_est that contains the estimation the concentration of pheromone \(c(s;x,y,t)\) computed by the pheromone propagation model at the times \(t\) and positions \((x,y)\) required to compute the observed variable. To get these \(c(s;x,y,t)\), the attribute c_est should be computed in a first time using the method solver_est_at_obs_times() of the class DiffusionConvectionReaction2DEquation().

Raises:: ValueError – if the attribute c_est has not been computed in a first time.

onesensor_adjoint_derivative_obs_operator(t, phi, index_sensor)[source]

compute the spatial map of the adjoint of the derivative of the one-sensor observation operator with respect to the state variable \(\phi\mapsto (d_cm_i(c))^*\cdot\phi\), at a given time \(t\), for the \(i^{th}\) sensor and for an element of the observation space \(\phi\).

Parameters:

t (float or integer) – The current time \(t\).
phi (ndarray) – The element of the observation space \(\phi\) at current time \(t\).
index_sensor (int) – The index of the sensor \(i\).

Returns:

The spatial map of the image of the adjoint of the derivative of the one-sensor observation operator \((d_cm_i(c))^*\cdot\phi(x,y)\) at the given time raveled into a vector.

Return type:

ndarray