This package implements some common routines for state-space models.

The provided algorithms are:

- Kalman filter (
`kalman_filter`

) - Tempered particle filter (
`tempered_particle_filter`

): "Tempered Particle Filtering" (2016) - Kalman smoothers:
`hamilton_smoother`

: James Hamilton,*Time Series Analysis*(1994)`koopman_smoother`

: S.J. Koopman, "Disturbance Smoother for State Space Models" (*Biometrika*, 1993)

- Simulation smoothers:
`carter_kohn_smoother`

: C.K. Carter and R. Kohn, "On Gibbs Sampling for State Space Models" (*Biometrika*, 1994)`durbin_koopman_smoother`

: J. Durbin and S.J. Koopman, "A Simple and Efficient Simulation Smoother for State Space Time Series Analysis" (*Biometrika*, 2002)

The tempered particle filter is a particle filtering method which can approximate the log-likelihood value implied by a general (potentially non-linear) state space system with the following representation:

```
s_{t+1} = Φ(s_t, ϵ_t) (transition equation)
y_t = Ψ(s_t) + u_t (measurement equation)
ϵ_t ∼ F_ϵ(∙; θ)
u_t ∼ N(0, E)
Cov(ϵ_t, u_t) = 0
```

- The documentation and code are located in src/filters/tempered_particle_filter.
- The example is located in docs/examples/tempered_particle_filter

```
s_{t+1} = C + T*s_t + R*ϵ_t (transition equation)
y_t = D + Z*s_t + u_t (measurement equation)
ϵ_t ∼ N(0, Q)
u_t ∼ N(0, E)
Cov(ϵ_t, u_t) = 0
```

```
kalman_filter(y, T, R, C, Q, Z, D, E, s_0 = Vector(), P_0 = Matrix(); outputs = [:loglh, :pred, :filt], Nt0 = 0)
tempered_particle_filter(y, Φ, Ψ, F_ϵ, F_u, s_init; verbose = :high, include_presample = true, fixed_sched = [], r_star = 2, c = 0.3, accept_rate = 0.4, target = 0.4, xtol = 0, resampling_method = :systematic, N_MH = 1, n_particles = 1000, Nt0 = 0, adaptive = true, allout = true, parallel = false)
hamilton_smoother(y, T, R, C, Q, Z, D, E, s_0, P_0; Nt0 = 0)
koopman_smoother(y, T, R, C, Q, Z, D, s_0, P_0, s_pred, P_pred; Nt0 = 0)
carter_kohn_smoother(y, T, R, C, Q, Z, D, E, s_0, P_0; Nt0 = 0, draw_states = true)
durbin_koopman_smoother(y, T, R, C, Q, Z, D, E, s_0, P_0; Nt0 = 0, draw_states = true)
```

For more information, see the docstring for each function (e.g. enter `?kalman_filter`

in the REPL).

All of the provided algorithms can handle time-varying state-space systems. To do this, define `regime_indices`

, a `Vector{Range{Int64}}`

of length `n_regimes`

, where `regime_indices[i]`

indicates the range of periods `t`

in regime `i`

. Let `T_i`

, `R_i`

, etc. denote the state-space matrices in regime `i`

. Then the state space is given by:

```
s_{t+1} = C_i + T_i*s_t + R_i*ϵ_t (transition equation)
y_t = D_i + Z_i*s_t + u_t (measurement equation)
ϵ_t ∼ N(0, Q_i)
u_t ∼ N(0, E_i)
```

Letting `Ts = [T_1, ..., T_{n_regimes}]`

, etc., we can then call the time-varying methods of the algorithms:

```
kalman_filter(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0 = Vector(), P_0 = Matrix(); outputs = [:loglh, :pred, :filt], Nt0 = 0)
hamilton_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0, P_0; Nt0 = 0)
koopman_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, s_0, P_0, s_pred, P_pred; Nt0 = 0)
carter_kohn_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0, P_0; Nt0 = 0, draw_states = true)
durbin_koopman_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0, P_0; Nt0 = 0, draw_states = true)
```

02/06/2017

5 days ago

86 commits