class StochVolModel(Model):
name = "Stochastic volatility model"
b = local_param()
# λ = global_param(prior=Normal(0, 1e-4))
σ = global_param(prior=LogNormalPrior(0, 1), rename="α", transform="log")
φ = global_param(prior=BetaPrior(2, 2), rename="ψ", transform="logit")
def ln_joint(self, y, ζ):
# b, λ, (σ, α), (φ, ψ) = self.unpack(ζ)
b, (σ, α), (φ, ψ) = self.unpack(ζ)
ar1_sd = torch.pow(1 - torch.pow(φ, 2), -0.5)
llikelihood = (
Normal(0, torch.exp(.5 * (σ * b))).log_prob(y).sum()
# Normal(0, torch.exp(.5 * (λ + σ * b))).log_prob(y).sum()
+ Normal(φ * b[:-1], 1).log_prob(b[1:]).sum() +
Normal(0., ar1_sd).log_prob(b[0]))
lprior = (
self.ψ_prior.log_prob(ψ) + self.α_prior.log_prob(α)
# + self.λ_prior.log_prob(λ)
)
return llikelihood + lprior
def simulate(self, λ=0., σ=0.5, φ=0.95):
assert σ > 0 and 0 < φ < 1
b = torch.zeros(self.input_length)
φ = torch.tensor(φ)
ar1_sd = torch.pow(1 - torch.pow(φ, 2), -0.5)
b[0] = Normal(0, ar1_sd).sample()
for t in range(1, self.input_length):
b[t] = Normal(φ * b[t - 1], 1).sample()
y = Normal(loc=0., scale=torch.exp(0.5 * (λ + σ * b))).sample()
return y, b
def sample_observed(self, ζ, y, fc_steps=0):
b, (σ, α), (φ, ψ) = self.unpack(ζ)
λ = 0.
if fc_steps > 0:
b = torch.cat([b, torch.zeros(fc_steps)])
for t in range(self.input_length, self.input_length + fc_steps):
b[t] = b[t - 1] * φ + Normal(0, 1).sample()
return Normal(loc=0., scale=torch.exp(0.5 * (λ + σ * b))).sample()
class UnivariateGaussian(Model):
"""Simple univariate Gaussian model.
For the optimization, we transform σ -> ln(σ) = η to ensure σ > 0.
"""
name = "Univariate Gaussian model"
μ = global_param(prior=NormalPrior(0., 10.))
σ = global_param(prior=LogNormalPrior(0., 10.),
rename="η",
transform="log")
def simulate(self, N: int, μ: float, σ: float):
assert N > 2 and σ > 0
return Normal(μ, σ).sample((N, )).type(self.dtype).to(self.device)
def ln_joint(self, y, ζ):
μ, (σ, η) = self.unpack(ζ)
ll = Normal(μ, σ).log_prob(y).sum()
lp = self.μ_prior.log_prob(μ) + self.η_prior.log_prob(η)
return ll + lp
class LocalLevelModel(Model):
name = "Local level model"
z = local_param()
γ = global_param(prior=NormalPrior(1, 3))
η = global_param(prior=LogNormalPrior(0, 3), transform="log", rename="ψ")
σ = global_param(prior=InvGammaPrior(1, 5), transform="log", rename="ς")
ρ = global_param(prior=BetaPrior(2, 2), transform="logit", rename="φ")
def ln_joint(self, y, ζ):
"""Computes the log likelihood plus the log prior at ζ."""
z, γ, (η, ψ), (σ, ς), (ρ, φ) = self.unpack(ζ)
ar1_uncond_var = torch.pow((1 - torch.pow(ρ, 2)), -0.5)
llikelihood = (Normal(γ + η * z, σ).log_prob(y).sum() +
Normal(ρ * z[:-1], 1).log_prob(z[1:]).sum() +
Normal(0., ar1_uncond_var).log_prob(z[0]))
lprior = (self.γ_prior.log_prob(γ) + self.ψ_prior.log_prob(ψ) +
self.ς_prior.log_prob(ς) + self.φ_prior.log_prob(φ))
return llikelihood + lprior
def simulate(self, γ: float, η: float, σ: float, ρ: float):
z = torch.empty([self.input_length])
z[0] = Normal(0, 1 / (1 - ρ**2)**0.5).sample()
for i in range(1, self.input_length):
z[i] = ρ * z[i - 1] + Normal(0, 1).sample()
y = Normal(γ + η * z, σ).sample()
return y.type(self.dtype).to(self.device), z.type(self.dtype).to(
self.device)
def sample_observed(self, ζ, y, fc_steps=0):
z, γ, (η, ψ), (σ, ς), (ρ, φ) = self.unpack(ζ)
if fc_steps > 0:
z = torch.cat([z, torch.zeros(fc_steps)])
# iteratively project states forward
for t in range(self.input_length, self.input_length + fc_steps):
z[t] = z[t - 1] * ρ + Normal(0, 1).sample()
return Normal(γ + η * z, σ).sample()
class FilteredSVModelDualOpt(FilteredStateSpaceModelFreeProposal):
""" A simple stochastic volatility model for estimating with FIVO.
.. math::
x_t = exp(a)exp(z_t/2) ε_t ε_t ~ Ν(0,1)
z_t = b + c * z_{t-1} + ν_t ν_t ~ Ν(0,1)
The proposal density is
.. math::
z_t = d + e * z_{t-1} + η_t η_t ~ Ν(0,1)
The model parameter ζ covers the parameters used in the SV model, ζ={a, b, c}.
The alternative parameter η covers the parameters η={d, e}.
"""
name = "Particle filtered stochastic volatility model"
a = global_param(prior=LogNormalPrior(0, 1), transform="log", rename="α")
b = global_param(prior=NormalPrior(0, 1))
c = global_param(prior=BetaPrior(1, 1), transform="logit", rename="ψ")
d = global_param(prior=NormalPrior(0, 1))
e = global_param(prior=BetaPrior(1, 1), transform="logit", rename="ρ")
def __init__(
self,
input_length: int,
num_particles: int = 50,
resample=True,
dtype=None,
device=None,
):
super().__init__(input_length, num_particles, resample, dtype, device)
self._md = 3
self._pd = 2 # no σ in proposal yet
def simulate(self, a, b, c):
"""Simulate from p(y, z | θ)"""
a, b, c = map(torch.tensor, (a, b, c))
z_true = torch.empty((self.input_length,))
z_true[0] = Normal(b, (1 - c ** 2) ** (-.5)).sample()
for t in range(1, self.input_length):
z_true[t] = b + c * z_true[t - 1] + Normal(0, 1).sample()
y = Normal(0, torch.exp(a) * torch.exp(z_true / 2)).sample()
return (
y.type(self.dtype).to(self.device),
z_true.type(self.dtype).to(self.device),
)
def conditional_log_prob(self, t, y, z, ζ):
"""Compute log p(x_t, z_t | y_{0:t-1}, z_{0:t-1}, ζ).
Args:
t: time index (zero-based)
y: y_{0:t} vector of points observed up to this point (which may
actually be longer, but should only be indexed up to t)
z: z_{0:t} vector of unobserved variables to condition on (ditto,
array may be longer)
ζ: parameter to condition on; should be unpacked with self.unpack
"""
a, b, c, = self.unpack_natural_model_parameters(ζ)
if t == 0:
log_pzt = Normal(b, (1 - c ** 2) ** (-.5)).log_prob(z[t])
else:
log_pzt = Normal(b + c * z[t - 1], 1).log_prob(z[t])
log_pxt = Normal(0, torch.exp(a) * torch.exp(z[t] / 2)).log_prob(y[t])
return log_pzt + log_pxt
def ln_prior(self, ζ: torch.Tensor) -> float:
a, b, c = self.unpack_natural_model_parameters(ζ)
return (
self.a_prior.log_prob(a)
+ self.b_prior.log_prob(b)
+ self.c_prior.log_prob(c)
)
def model_parameters(self):
return [self.a, self.b, self.c]
def proposal_parameters(self):
return [self.d, self.e]
def unpack_natural_model_parameters(self, ζ: torch.Tensor):
α, b, ψ = ζ[0], ζ[1], ζ[2]
return self.a_to_α.inv(α), b, self.c_to_ψ.inv(ψ)
def unpack_natural_proposal_parameters(self, η: torch.Tensor):
d, ρ = η[0], η[1]
return d, self.e_to_ρ.inv(ρ)
def simulate_log_phatN(
self,
y: torch.Tensor,
ζ: torch.Tensor,
η: torch.Tensor,
sample: torch.Tensor = None,
):
"""Apply particle filter to estimate marginal likelihood log p^(y | ζ)
This algorithm is subtly different than the one in fivo.py, because it
also takes η as a parameter.
"""
log_phatN = 0.
log_N = math.log(self.num_particles)
log_w = torch.full(
(self.num_particles,), -log_N, dtype=self.dtype, device=self.device
)
Z = None
proposal = self.proposal_for(y, η)
for t in range(self.input_length):
zt = proposal.conditional_sample(t, Z, self.num_particles).unsqueeze(0)
Z = torch.cat([Z, zt]) if Z is not None else zt
log_αt = self.conditional_log_prob(
t, y, Z, ζ
) - proposal.conditional_log_prob(t, Z)
log_phatt = torch.logsumexp(log_w + log_αt, dim=0)
log_phatN += log_phatt
log_w += log_αt - log_phatt
with torch.no_grad():
ESS = 1. / torch.exp(2 * log_w).sum()
if self.resample and ESS < self.num_particles:
a = Categorical(torch.exp(log_w)).sample((self.num_particles,))
Z = (Z[:, a]).clone()
log_w = torch.full(
(self.num_particles,),
-log_N,
dtype=self.dtype,
device=self.device,
)
if sample is not None:
with torch.no_grad():
# samples should be M * T, where M is the number of samples
assert sample.shape[0] >= self.input_length
idxs = Categorical(torch.exp(log_w)).sample()
sample[: self.input_length] = Z[:, idxs]
return log_phatN
def proposal_for(self, y: torch.Tensor, η: torch.Tensor) -> PFProposal:
"""Return the proposal distribution for the given parameters.
Args:
y: data vector
η: proposal parameter vector
"""
d, e = self.unpack_natural_proposal_parameters(η)
return AR1Proposal(μ=d, ρ=e, σ=1.)
@property
def md(self) -> int:
"""Dimension of the model."""
return self._md
@property
def pd(self) -> int:
"""Dimension of the proposal."""
return self._pd
def sample_observed(self, ζ, y, fc_steps=0):
a, b, c = self.unpack_natural_model_parameters(ζ[:3])
z = self.sample_unobserved(ζ, y, fc_steps)
return Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
def sample_unobserved(self, ζ, y, fc_steps=0):
assert y is not None
a, b, c = self.unpack_natural_model_parameters(ζ[:3])
# get a sample of states by filtering wrt y
z = torch.empty((len(y) + fc_steps,))
self.simulate_log_phatN(y=y, ζ=ζ[:3], η=ζ[3:], sample=z)
# now project states forward fc_steps
if fc_steps > 0:
for t in range(self.input_length, self.input_length + fc_steps):
z[t] = b + c * z[t - 1] + Normal(0, 1).sample()
return Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
def __repr__(self):
return (
f"Stochastic volatility model for dual optimization of model and proposal:\n"
f"\tx_t = exp(a * z_t/2) ε_t t=1, …, {self.input_length}\n"
f"\tz_t = b + c * z_{{t-1}} + ν_t, t=2, …, {self.input_length}\n"
f"\tz_1 = b + 1/√(1 - c^2) ν_1\n"
f"\twhere ε_t, ν_t ~ Ν(0,1)\n\n"
f"Particle filter with {self.num_particles} particles, AR(1) proposal:\n"
f"\tz_t = d + e * z_{{t-1}} + η_t, t=2, …, {self.input_length}\n"
f"\tz_1 = d + 1/√(1 - e^2) η_1\n"
f"\twhere η_t ~ Ν(0,1)\n"
)
class FilteredStochasticVolatilityModelFreeProposal(FilteredStateSpaceModel):
""" A simple stochastic volatility model for estimating with FIVO.
.. math::
x_t = exp(a)exp(z_t/2) ε_t ε_t ~ Ν(0,1)
z_t = b + c * z_{t-1} + ν_t ν_t ~ Ν(0,1)
The proposal density is also an AR(1):
.. math::
z_t = d + e * z_{t-1} + η_t η_t ~ Ν(0,1)
"""
name = "Particle filtered stochastic volatility model"
a = global_param(prior=LogNormalPrior(0, 1), transform="log", rename="α")
b = global_param(prior=NormalPrior(0, 1))
c = global_param(prior=BetaPrior(1, 1), transform="logit", rename="ψ")
d = global_param(prior=NormalPrior(0, 1))
e = global_param(prior=BetaPrior(1, 1), transform="logit", rename="ρ")
f = global_param(prior=LogNormalPrior(0, 1), transform="log", rename="ι")
def simulate(self, a, b, c):
"""Simulate from p(x, z | θ)"""
a, b, c = map(torch.tensor, (a, b, c))
z = torch.empty((self.input_length,))
z[0] = Normal(b, (1 - c ** 2) ** (-.5)).sample()
for t in range(1, self.input_length):
z[t] = b + c * z[t - 1] + Normal(0, 1).sample()
x = Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
return x.type(self.dtype).to(self.device), z.type(self.dtype).to(self.device)
def conditional_log_prob(self, t, y, z, ζ):
"""Compute log p(x_t, z_t | y_{0:t-1}, z_{0:t-1}, ζ).
Args:
t: time index (zero-based)
y: y_{0:t} vector of points observed up to this point (which may
actually be longer, but should only be indexed up to t)
z: z_{0:t} vector of unobserved variables to condition on (ditto,
array may be longer)
ζ: parameter to condition on; should be unpacked with self.unpack
"""
a, b, c, _, _, _ = self.unpack_natural(ζ)
if t == 0:
log_pzt = Normal(b, (1 - c ** 2) ** (-.5)).log_prob(z[t])
else:
log_pzt = Normal(b + c * z[t - 1], 1).log_prob(z[t])
log_pxt = Normal(0, torch.exp(a) * torch.exp(z[t] / 2)).log_prob(y[t])
return log_pzt + log_pxt
def sample_observed(self, ζ, y, fc_steps=0):
a, _, _, _, _, _ = self.unpack_natural(ζ)
z = self.sample_unobserved(ζ, y, fc_steps)
return Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
def sample_unobserved(self, ζ, y, fc_steps=0):
assert y is not None
a, b, c, _, _, _ = self.unpack_natural(ζ)
# get a sample of states by filtering wrt y
z = torch.empty((len(y) + fc_steps,))
self.simulate_log_phatN(y=y, ζ=ζ, sample=z)
# now project states forward fc_steps
if fc_steps > 0:
for t in range(self.input_length, self.input_length + fc_steps):
z[t] = b + c * z[t - 1] + Normal(0, 1).sample()
return Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
def proposal_for(self, y: torch.Tensor, ζ: torch.Tensor) -> PFProposal:
_, _, _, d, e, f = self.unpack_natural(ζ)
return AR1Proposal(μ=d, ρ=e, σ=f)
def __repr__(self):
return (
f"Stochastic volatility model with parameters {{a, b, c}}:\n"
f"\tx_t = exp(a * z_t/2) ε_t t=1,…,{self.input_length}\n"
f"\tz_t = b + c * z_{{t-1}} + ν_t, t=2,…,{self.input_length}\n"
f"\tz_1 = b + 1/√(1 - c^2) ν_1\n"
f"\twhere ε_t, ν_t ~ Ν(0,1)\n\n"
f"Filter with {self.num_particles} particles; AR(1) proposal params {{d, e, f}}:\n"
f"\tz_t = d + e * z_{{t-1}} + f η_t, t=2,…,{self.input_length}\n"
f"\tz_1 = d + f/√(1 - e^2) η_1\n"
f"\twhere η_t ~ Ν(0,1)\n"
)
class FilteredStochasticVolatilityModelFixedParams(FilteredStateSpaceModel):
""" A simple stochastic volatility model for estimating with FIVO.
.. math::
x_t = exp(a)exp(z_t/2) ε_t ε_t ~ Ν(0,1)
z_t = b + c * z_{t-1} + f ν_t ν_t ~ Ν(0,1)
"""
name = "Particle filtered stochastic volatility model"
d = global_param(prior=NormalPrior(0, 1))
e = global_param(prior=BetaPrior(1, 1), transform="logit", rename="ρ")
f = global_param(prior=LogNormalPrior(0, 1), transform="log")
def __init__(
self,
input_length,
num_particles,
resample,
a=0.5,
b=1.,
c=0.95,
dtype=None,
device=None,
):
super().__init__(
input_length=input_length,
num_particles=num_particles,
resample=resample,
dtype=dtype,
device=device,
)
self.a, self.b, self.c = (
torch.tensor(x, dtype=self.dtype, device=self.device) for x in (a, b, c)
)
def simulate(self):
"""Simulate from p(x, z | θ)"""
z_true = torch.empty((self.input_length,), dtype=self.dtype, device=self.device)
z_true[0] = Normal(self.b, (1 - self.c ** 2) ** (-.5)).sample()
for t in range(1, self.input_length):
z_true[t] = (
self.b
+ self.c * z_true[t - 1]
+ torch.randn(1, dtype=self.dtype, device=self.device)
)
x = Normal(0, torch.exp(self.a) * torch.exp(z_true / 2)).sample()
return (
x.type(self.dtype).to(self.device),
z_true.type(self.dtype).to(self.device),
)
def conditional_log_prob(self, t, y, z, ζ):
"""Compute log p(x_t, z_t | y_{0:t-1}, z_{0:t-1}, ζ).
Args:
t: time index (zero-based)
y: y_{0:t} vector of points observed up to this point (which may
actually be longer, but should only be indexed up to t)
z: z_{0:t} vector of unobserved variables to condition on (ditto,
array may be longer)
ζ: parameter to condition on; should be unpacked with self.unpack
"""
if t == 0:
log_pzt = Normal(self.b, (1 - self.c ** 2) ** (-.5)).log_prob(z[t])
else:
log_pzt = Normal(self.b + self.c * z[t - 1], 1).log_prob(z[t])
log_pxt = Normal(0, torch.exp(self.a) * torch.exp(z[t] / 2)).log_prob(y[t])
return log_pzt + log_pxt
def sample_observed(self, ζ, y, fc_steps=0):
z = self.sample_unobserved(ζ, y, fc_steps)
return Normal(0, torch.exp(self.a) * torch.exp(z / 2)).sample()
def sample_unobserved(self, ζ, y, fc_steps=0):
assert y is not None
# get a sample of states by filtering wrt y
z = torch.empty((len(y) + fc_steps,))
self.simulate_log_phatN(y=y, ζ=ζ, sample=z)
# now project states forward fc_steps
if fc_steps > 0:
for t in range(self.input_length, self.input_length + fc_steps):
z[t] = self.b + self.c * z[t - 1] + Normal(0, 1).sample()
return Normal(0, torch.exp(self.a) * torch.exp(z / 2)).sample()
def proposal_for(self, y: torch.Tensor, ζ: torch.Tensor) -> PFProposal:
d, e, f = self.unpack_natural(ζ)
return AR1Proposal(μ=d, ρ=torch.tensor(.95, dtype=self.dtype), σ=f)
def __repr__(self):
return (
f"Stochastic volatility model:\n"
f"\tx_t = exp(a * z_t/2) ε_t t=1, …, {self.input_length}\n"
f"\tz_t = b + c * z_{{t-1}} + ν_t, t=2, …, {self.input_length}\n"
f"\tz_1 = b + 1/√(1 - c^2) ν_1\n"
f"\twhere ε_t, ν_t ~ Ν(0,1)\n\n"
f"Particle filter with {self.num_particles} particles, AR(1) proposal:\n"
f"\tz_t = d + e * z_{{t-1}} + f η_t, t=2, …, {self.input_length}\n"
f"\tz_1 = d + f/√(1 - e^2) η_1\n"
f"\twhere η_t ~ Ν(0,1)\n"
)
class FilteredLocalLevelModel(Model):
"""Local level (linear gaussian) model that exploits the Kalman filter."""
name = "Filtered local level model"
z0 = global_param(prior=NormalPrior(0, 10))
σz0 = global_param(prior=LogNormalPrior(1, 1),
transform="log",
rename="ςz0")
γ = global_param(prior=NormalPrior(1, 3))
η = global_param(prior=LogNormalPrior(0, 1), transform="log", rename="ψ")
σ = global_param(prior=InvGammaPrior(1, 5), transform="log", rename="ς")
ρ = global_param(prior=BetaPrior(1, 1), transform="logit", rename="φ")
def ln_joint(self, y, ζ):
"""Computes the log likelihood plus the log prior at ζ."""
z0, (σz0, ςz0), γ, (η, ψ), (σ, ς), (ρ, φ) = self.unpack(ζ)
# unroll first iteration of loop to set initial conditions
# prediction step
z_pred = ρ * z0
Σz_pred = ρ**2 * σz0**2 + 1
y_pred = γ + η * z_pred
Σy_pred = η**2 * Σz_pred + σ**2
# correction step
gain = Σz_pred * η / Σy_pred
z_upd = z_pred + gain * (y[0] - y_pred)
Σz_upd = Σz_pred - gain**2 * Σy_pred
llik = Normal(y_pred, torch.sqrt(Σy_pred)).log_prob(y[0])
for t in range(2, y.shape[0] + 1):
i = t - 1
# prediction step
z_pred = ρ * z_upd
Σz_pred = ρ**2 * Σz_upd + 1
y_pred = γ + η * z_pred
Σy_pred = η**2 * Σz_pred + σ**2
# correction step
gain = Σz_pred * η / Σy_pred
z_upd = z_pred + gain * (y[i] - y_pred)
Σz_upd = Σz_pred - gain**2 * Σy_pred
llik += Normal(y_pred, torch.sqrt(Σy_pred)).log_prob(y[i])
return llik + self.ln_prior(ζ)
def simulate(
self,
γ: float,
η: float,
σ: float,
ρ: float,
z0: float = None,
σz0: float = None,
):
if z0 is None:
z0 = 0
if σz0 is None:
σz0 = 1. / (1 - ρ**2)**0.5
z = torch.empty([self.input_length])
z[0] = Normal(z0, σz0).sample()
for i in range(1, self.input_length):
z[i] = ρ * z[i - 1] + Normal(0, 1).sample()
y = Normal(γ + η * z, σ).sample()
return y, z
def kalman_smoother(self, y, ζ):
z0, (σz0, ςz0), γ, (η, ψ), (σ, ς), (ρ, φ) = self.unpack(ζ)
z_pred = torch.zeros((self.input_length, )) # z_{t|t-1}
z_upd = torch.zeros((self.input_length, )) # z_{t|t}
Σz_pred = torch.zeros((self.input_length, )) # Σ_{z_{t|t-1}}
Σz_upd = torch.zeros((self.input_length, )) # Σ_{z_{t|t}}
y_pred = torch.zeros((self.input_length, )) # y_{t|t-1}
Σy_pred = torch.zeros((self.input_length, )) # Σ_{y_{t|t-1}}
z_smooth = torch.zeros((self.input_length, )) # z_{t|T}
Σz_smooth = torch.zeros((self.input_length, )) # Σ_{z_{t|T}}
# prediction step
z_pred[0] = ρ * z0
Σz_pred[0] = ρ**2 * σz0**2 + 1
y_pred[0] = η * z_pred[0]
Σy_pred[0] = η**2 * Σz_pred[0] + σ**2
# correction step
gain = Σz_pred[0] * η / Σy_pred[0]
z_upd[0] = z_pred[0] + gain * (y[0] - y_pred[0])
Σz_upd[0] = Σz_pred[0] - gain**2 * Σy_pred[0]
for i in range(1, y.shape[0]):
# prediction step
z_pred[i] = ρ * z_upd[i - 1]
Σz_pred[i] = ρ**2 * Σz_upd[i - 1] + 1
y_pred[i] = η * z_pred[i]
Σy_pred[i] = η**2 * Σz_pred[i] + σ**2
# correction step
gain = Σz_pred[i] * η / Σy_pred[i]
z_upd[i] = z_pred[i] + gain * (y[i] - y_pred[i])
Σz_upd[i] = Σz_pred[i] - gain**2 * Σy_pred[i]
# smoothing step
z_smooth[self.input_length - 1] = z_upd[self.input_length - 1]
Σz_smooth[self.input_length - 1] = Σz_upd[self.input_length - 1]
for i in range(self.input_length - 2, -1, -1):
smooth = Σz_upd[i] * ρ / Σz_pred[i]
z_smooth[i] = z_upd[i] + smooth**2 * (Σz_pred[i + 1] -
Σz_smooth[i])
Σz_smooth[i] = Σz_upd[i] - smooth**2 * (Σz_pred[i + 1] -
Σz_smooth[i + 1])
return {
"z_upd": z_upd,
"Σz_upd": Σz_upd,
"z_smooth": z_smooth,
"Σz_smooth": Σz_smooth,
"y_pred": y_pred,
"Σy_pred": Σy_pred,
}
def filtered_path(self, y, params):
"""Filter path, return final obs."""
z0, σz0, γ, η, σ, ρ, = params
z = torch.zeros_like(y)
z[0] = z0
# unroll first iteration of loop to set initial conditions
# prediction step
z_pred = ρ * z0
Σz_pred = ρ**2 * σz0**2 + 1
y_pred = γ + η * z_pred
Σy_pred = η**2 * Σz_pred + σ**2
# correction step
gain = Σz_pred * η / Σy_pred
z_upd = z_pred + gain * (y[0] - y_pred)
Σz_upd = Σz_pred - gain**2 * Σy_pred
z[1] = z_upd
for t in range(2, y.shape[0] + 1):
i = t - 1
# prediction step
z_pred = ρ * z_upd
Σz_pred = ρ**2 * Σz_upd + 1
y_pred = γ + η * z_pred
Σy_pred = η**2 * Σz_pred + σ**2
# correction step
gain = Σz_pred * η / Σy_pred
z_upd = z_pred + gain * (y[i] - y_pred)
Σz_upd = Σz_pred - gain**2 * Σy_pred
z[t - 1] = z_upd
return z
def forecast_paths(self, y, post, nsteps=10, ndraws=10_000):
# loop over posterior draws
# conditional on draw, obtain x_T draw
# project x_T+1, x_T+2, ..., x_T+h
N = len(y)
z_ext = torch.zeros([N + nsteps])
y_ext = torch.zeros([N + nsteps])
y_ext[:N] = y
fc_paths = torch.zeros([nsteps, ndraws])
is_mcmc_posterior = getattr(post, 'flatnames', None)
if is_mcmc_posterior:
mcmc_draws = post.extract()
flatnames = ['z0', 'sigma_z0', 'gamma', 'eta', 'sigma', 'rho']
param_draws = torch.stack(
[torch.tensor(mcmc_draws[n]) for n in flatnames])
for i in range(ndraws):
if is_mcmc_posterior:
# posterior is matrix of samples
params = param_draws[:, i % param_draws.shape[1]]
else:
params = post.q.sample()
z_ext[:N] = self.filtered_path(y, params)
# allow latent states z to evolve
z0, σz0, γ, η, σ, ρ, = params
# print([γ, η, σ, ρ])
for t in range(N, N + nsteps):
# draw z[t] | z[t-1]
z_ext[t] = z_ext[t - 1] * ρ + torch.normal(torch.zeros(1))
# sample conditionally indepedent ys
y_ext[N:] = torch.normal(γ + η * z_ext[N:], σ * torch.ones(nsteps))
fc_paths[:, i] = y_ext[N:]
return fc_paths
class SVModel(FilteredStateSpaceModel):
""" A simple stochastic volatility model for estimating with FIVO.
.. math::
x_t = exp(a)exp(z_t/2) ε_t ε_t ~ Ν(0,1)
z_t = b + c * z_{t-1} + ν_t ν_t ~ Ν(0,1)
The proposal density is also an AR(1):
.. math::
z_t = d + e * z_{t-1} + η_t η_t ~ Ν(0,1)
"""
class SVModelResult(FilteredStateSpaceModel.FIVOResult):
def forecast(self, steps=1, n=100):
"""Produce forecasts of the specified number of steps ahead.
Procedure: sample ζ, filter to get p(z_T | y, θ), project the state chain
forward, then compute y.
"""
z_T_draws = torch.zeros(n)
z_proj_draws = torch.zeros((n, steps))
y_proj_draws = torch.zeros((n, steps))
sample = torch.zeros((1, n))
self.model.num_particles = n # dodgy hack to simulate more particles
phatns = torch.zeros((n,))
for i in range(n):
ζ = self.q.sample()
a, b, c = ζ[0], ζ[1], ζ[2]
phatns[i] = self.model.simulate_log_phatN(self.y, ζ, sample)
# just take a single particle's z_T
z_T_draws[i] = sample[0, random.randint(0, n - 1)]
z_proj_draws[i, 0] = b + c * z_T_draws[i] + torch.randn(1)
for j in range(1, steps):
z_proj_draws[i, j] = b + c * z_proj_draws[i, j - 1] + torch.randn(1)
y_proj_draws[i, :] = Normal(
0, torch.exp(a) * torch.exp(z_proj_draws[i, :] / 2)
).sample()
kde = stats.gaussian_kde(y_proj_draws[:, -1].cpu().numpy())
return kde, y_proj_draws.cpu().numpy()
name = "Particle filtered stochastic volatility model"
a = global_param(prior=LogNormalPrior(0, 1), transform="log", rename="α")
b = global_param(prior=NormalPrior(0, 1))
c = global_param(prior=ModifiedBetaPrior(0.5, 1.5), transform="logit", rename="ψ")
d = global_param(prior=NormalPrior(0, 1))
e = global_param(prior=BetaPrior(1, 1), transform="logit", rename="ρ")
f = global_param(prior=LogNormalPrior(0, 1), transform="log", rename="ι")
result_type = SVModelResult
def simulate(self, a, b, c):
"""Simulate from p(x, z | θ)"""
a, b, c = map(torch.tensor, (a, b, c))
z = torch.empty((self.input_length,))
z[0] = Normal(b, (1 - c ** 2) ** (-.5)).sample()
for t in range(1, self.input_length):
z[t] = b + c * z[t - 1] + Normal(0, 1).sample()
x = Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
return x.type(self.dtype).to(self.device), z.type(self.dtype).to(self.device)
def conditional_log_prob(self, t, y, z, ζ):
"""Compute log p(x_t, z_t | y_{0:t-1}, z_{0:t-1}, ζ).
Args:
t: time index (zero-based)
y: y_{0:t} vector of points observed up to this point (which may
actually be longer, but should only be indexed up to t)
z: z_{0:t} vector of unobserved variables to condition on (ditto,
array may be longer)
ζ: parameter to condition on; should be unpacked with self.unpack
"""
a, b, c, _, _, _ = self.unpack_natural(ζ)
if t == 0:
log_pzt = Normal(b, (1 - c ** 2) ** (-.5)).log_prob(z[t])
else:
log_pzt = Normal(b + c * z[t - 1], 1).log_prob(z[t])
log_pxt = Normal(0, torch.exp(a) * torch.exp(z[t] / 2)).log_prob(y[t])
return log_pzt + log_pxt
def sample_observed(self, ζ, y, fc_steps=0):
a, _, _, _, _, _ = self.unpack_natural(ζ)
z = self.sample_unobserved(ζ, y, fc_steps)
return Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
def sample_unobserved(self, ζ, y, fc_steps=0):
assert y is not None
a, b, c, _, _, _ = self.unpack_natural(ζ)
# get a sample of states by filtering wrt y
z = torch.empty((len(y) + fc_steps,))
self.simulate_log_phatN(y=y, ζ=ζ, sample=z)
# now project states forward fc_steps
if fc_steps > 0:
for t in range(self.input_length, self.input_length + fc_steps):
z[t] = b + c * z[t - 1] + Normal(0, 1).sample()
return Normal(0, torch.exp(a) * torch.exp(z / 2)).sample()
def proposal_for(self, y: torch.Tensor, ζ: torch.Tensor) -> PFProposal:
_, _, _, d, e, f = self.unpack_natural(ζ)
return AR1Proposal(μ=d, ρ=e, σ=f)
def __repr__(self):
return (
f"Stochastic volatility model with parameters {{a, b, c}}:\n"
f"\ty_t = exp(a * z_t/2) ε_t t=1,…,{self.input_length}\n"
f"\tz_t = b + c * z_{{t-1}} + ν_t, t=2,…,{self.input_length}\n"
f"\tz_1 = b + 1/√(1 - c^2) ν_1\n"
f"\twhere ε_t, ν_t ~ Ν(0,1)\n\n"
f"Filter with {self.num_particles} particles; AR(1) proposal params {{d, e, f}}:\n"
f"\tz_t = d + e * z_{{t-1}} + f η_t, t=2,…,{self.input_length}\n"
f"\tz_1 = d + f/√(1 - e^2) η_1\n"
f"\twhere η_t ~ Ν(0,1)\n"
)