simplest y=mx + b kalman example

simple_kalman_example.py

import pykalman
import numpy as np

#model of the form -- I apologize for mathjax weirndess but too lazy to fix right now

#yt = β0,t +β1,txt +νt, νt ∼ N(0, σ2 ν)

#β0,t+1 = β0,t +ξt, ξt ∼ N(0, σ2 ξ)

#β1,t+1 = β1,t +ςt, ςt ∼ N(0, σ2 ς )

n_steps = len(dex_returns)

A = np.matrix([[1,0],[0,1]])
# Transition matrix is dependent on previous B0/B1 with no inherent cross dependency

C = map(lambda x: np.matrix(np.append(1,x[0])),np.reshape(dex_returns,(n_steps,1,1)))
# Observation matrix is tensor of [1,x_t] (of length equal to our observations)

# Initial hidden state means are 0 for intercept and 1 for slope

# Initial coverariance estimates aren't terribly critical, I've picked just two small numbers

# Transition covariance describes the B0 and B1 variances -- these can be estimated from 
# inverse of covariance matrix (preicison matrix)

kf = pykalman.KalmanFilter(transition_matrices=[A for i in xrange(n_steps)],
                  observation_matrices=C,
                  observation_covariance=np.eye(1)*1e-3,
                  transition_covariance = np.eye(2)*1e-5,
                  initial_state_mean=np.array([0,1]),
                  initial_state_covariance = np.eye(2)*1e-5)
(filtered_state_means, filtered_state_covariances)=kf.filter(np.reshape(single_name_returns,(n_steps,1)))