HGangloff

'''
Equivalence between a gradient ascent over the Expectation Maximization
quantity Q and a gradient ascent over the model likelihood in the case of
training an Hidden Markov Chain with Gaussian Independent Noise
'''
import matplotlib.pyplot as plt
import numpy as np
import jax.numpy as jnp
import jax
from jax.scipy.stats import norm

HGangloff

'''
Gaussian Markov Random Field simulation using Fourier properties and base matrices for efficiency
References from Gaussian Markov Random Fields: Theory and Applications, Havard Rue and Leonhard Held

We treat the 2D case, with 0 mean, stationary variance and exponential correlation function
'''

import matplotlib.pyplot as plt
import numpy as np
from scipy.fftpack import fft2, ifft2

HGangloff

import numpy as np
from scipy.stats import norm
from ctypes import CDLL, c_int, c_double, c_char, c_void_p

def generate_observations(H, means, stds):
    X = ((H == 0) * (means[0] + np.random.randn(*H.shape) * stds[0]) +
        (H == 1) * (means[1] + np.random.randn(*H.shape) * stds[1]))

    return X

HGangloff

'''
Sequential Importance Resampling Particle Filter with Jax using jit and
lax.scan
The model equation for this simple application are:
X_{n} = 0.5 * X_{n-1} + 25 * X_{n-1} / (1 + X_{n-1}^2) + 8 * cos(1.2 * n) + U
Y_{n} = X_{n}^2 / 20 + V
where U~N(0, 10) and V~N(0, 1)
'''

import matplotlib.pyplot as plt

HGangloff

import numpy as np
import matplotlib.pyplot as plt

import jax.numpy as jnp
import jax
from jax import vmap, jit
from jax.scipy.signal import convolve2d

def color_image_graph(lx, ly, neigh_list):
    def first_available(color_list):

HGangloff

import numpy as np
import jax.numpy as jnp
import jax
from jax.scipy.stats import norm

def generate_observations(H, means, stds):
    X = ((H == 0) * (means[0] + np.random.randn(*H.shape) * stds[0]) +
        (H == 1) * (means[1] + np.random.randn(*H.shape) * stds[1]))

    return X