# ivp2ekf0¶

probnum.diffeq.ivp2ekf0(ivp, prior, evlvar)[source]

Computes measurement model and initial distribution for KF based on IVP and prior.

Initialdistribution:

Conditions the initial distribution of the Gaussian filter onto the initial values.

• If preconditioning is set to False, it conditions the initial distribution $$\mathcal{N}(0, I)$$ on the initial values $$(x_0, f(t_0, x_0), ...)$$ using as many available deri vatives as possible.
• If preconditioning is set to True, it conditions the initial distribution $$\mathcal{N}(0, P P^\top)$$ on the initial values $$(x_0, f(t_0, x_0), ...)$$ using as many available derivatives as possible. Note that the projection matrices $$H_0$$ and $$H_1$$ become $$H_0 P^{-1}$$ and $$H_1 P^{-1}$$ which has to be taken into account during the preconditioning.

Measurement model:

Returns a measurement model $$\mathcal{N}(g(m), R)$$ involving computing the discrepancy

$g(m) = H_1 m(t) - f(t, H_0 m(t)).$

Then it returns either type of Gaussian filter, each with a different interpretation of the Jacobian $$J_g$$:

• EKF0 thinks $$J_g(m) = H_1$$
• EKF1 thinks $$J_g(m) = H_1 - J_f(t, H_0 m(t)) H_0^\top$$
• UKF thinks: ‘’What is a Jacobian?’‘

Note that, again, in the case of a preconditioned state space model, $$H_0$$ and $$H_1$$ become $$H_0 P^{-1}$$ and $$H_1 P^{-1}$$ which has to be taken into account. In this case,

• EKF0 thinks $$J_g(m) = H_1 P^{-1}$$
• EKF1 thinks $$J_g(m) = H_1 P^{-1} - J_f(t, H_0 P^{-1} m(t)) (H_0 P^{-1})^\top$$
• UKF again thinks: ‘’What is a Jacobian?’‘

Note: The choice between $$H_i$$ and $$H_i P^{-1}$$ is taken care of within the Prior.

Returns ExtendedKalmanFilter object that is compatible with the GaussianIVPFilter.

evlvar : float,
measurement variance; in the literature, this is “R”