def vonmises_logpdf(data, mus, kappas, mask=None):
"""
Compute the log probability density of a von Mises distribution.
This will broadcast as long as data, mus, and kappas have the same
(or at least compatible) leading dimensions.
Parameters
----------
data : array_like (..., D)
The points at which to evaluate the log density
mus : array_like (..., D)
The means of the von Mises distribution(s)
kappas : array_like (..., D)
The concentration of the von Mises distribution(s)
mask : array_like (..., D) bool
Optional mask indicating which entries in the data are observed
Returns
-------
lps : array_like (...,)
Log probabilities under the von Mises distribution(s).
"""
D = data.shape[-1]
assert mus.shape[-1] == D
assert kappas.shape[-1] == D
# Check mask
mask = mask if mask is not None else np.ones_like(data, dtype=bool)
assert mask.shape == data.shape
ll = kappas * np.cos(data - mus) - np.log(2 * np.pi) - np.log(i0(kappas))
return np.sum(ll * mask, axis=-1)
def log_likelihoods(self, data, input, mask, tag):
from autograd.scipy.special import i0
# Compute the log likelihood of the data under each of the K classes
# Return a TxK array of probability of data[t] under class k
mus, kappas = self.mus, np.exp(self.log_kappas)
mask = np.ones_like(data, dtype=bool) if mask is None else mask
return np.sum(
(kappas * (np.cos(data[:, None, :] - mus)) - np.log(2 * np.pi) -
np.log(i0(kappas))) * mask[:, None, :],
axis=2)
def m_step(self, expectations, datas, inputs, masks, tags, **kwargs):
from autograd.scipy.special import i0, i1
x = np.concatenate(datas)
weights = np.concatenate([Ez for Ez, _, _ in expectations])
# convert angles to 2D representation and employ closed form solutions
x_k = np.stack((np.sin(x), np.cos(x)), axis=1)
r_k = np.tensordot(weights.T, x_k, (-1, 0))
r_norm = np.sqrt(np.sum(r_k**2, 1))
mus_k = r_k / r_norm[:, None]
r_bar = r_norm / weights.sum(0)[:, None]
# truncated newton approximation with 2 iterations
kappa_0 = r_bar * (2 - r_bar**2) / (1 - r_bar**2)
kappa_1 = kappa_0 - ((i1(kappa_0)/i0(kappa_0)) - r_bar) / \
(1 - (i1(kappa_0)/i0(kappa_0)) ** 2 - (i1(kappa_0)/i0(kappa_0)) / kappa_0)
kappa_2 = kappa_1 - ((i1(kappa_1)/i0(kappa_1)) - r_bar) / \
(1 - (i1(kappa_1)/i0(kappa_1)) ** 2 - (i1(kappa_1)/i0(kappa_1)) / kappa_1)
for k in range(self.K):
self.mus[k] = np.arctan2(*mus_k[k])
self.log_kappas[k] = np.log(kappa_2[k])
def vonmises_logpdf(data, mus, kappas, mask=None):
"""
Compute the log probability density of a von Mises distribution.
This will broadcast as long as data, mus, and kappas have the same
(or at least compatible) leading dimensions.
Parameters
----------
data : array_like (..., D)
The points at which to evaluate the log density
mus : array_like (..., D)
The means of the von Mises distribution(s)
kappas : array_like (..., D)
The concentration of the von Mises distribution(s)
mask : array_like (..., D) bool
Optional mask indicating which entries in the data are observed
Returns
-------
lps : array_like (...,)
Log probabilities under the von Mises distribution(s).
"""
try:
from autograd.scipy.special import i0
except:
raise Exception(
"von Mises relies on the function autograd.scipy.special.i0. "
"This is present in the latest Github code, but not on pypi. "
"Please use the Github version of autograd instead.")
D = data.shape[-1]
assert mus.shape[-1] == D
assert kappas.shape[-1] == D
# Check mask
mask = mask if mask is not None else np.ones_like(data, dtype=bool)
assert mask.shape == data.shape
ll = kappas * np.cos(data - mus) - np.log(2 * np.pi) - np.log(i0(kappas))
return np.sum(ll * mask, axis=-1)