[Math] Hidden Markov Model

看上去不错的网站：http://iacs-courses.seas.harvard.edu/courses/am207/blog/lecture-18.html

SciPy Cookbook：http://scipy-cookbook.readthedocs.io/items/KalmanFiltering.html

良心视频：卡尔曼滤波器的原理以及在matlab中的实现

讲解思路貌似是在已知迭代结果的基础上做讲解，不是很透彻。

1. 用矩阵表示

2. 本质就是：二维高斯的协方差与sampling效果

3. 不确定性在状态之间的传递

4. 矩阵表示观察数据

5. Kalman系数

6. 噪声协方差矩阵的更新

7. Matlab实现

思考：

与数学领域 openBUGS 的估参的关系是什么？[Bayes] openBUGS: this is not the annoying bugs in programming

一个是对逐渐增多数据的实时预测；一个是对总体数据的回归拟合。

代码示例：纯python代码

# Kalman filter example demo in Python# A Python implementation of the example given in pages 11-15 of "An
# Introduction to the Kalman Filter" by Greg Welch and Gary Bishop,
# University of North Carolina at Chapel Hill, Department of Computer
# Science, TR 95-041,
# http://www.cs.unc.edu/~welch/kalman/kalmanIntro.html# by Andrew D. Strawimport numpy as np
import matplotlib.pyplot as pltplt.rcParams['figure.figsize'] = (10, 8)# intial parameters
n_iter = 50
sz = (n_iter,) # size of array
x  = -0.37727  # truth value (typo in example at top of p. 13 calls this z)
z  = np.random.normal(x,0.1,size=sz) # observations (normal about x, sigma=0.1)
# 已获得一组随机数Q = 1e-5 # process variance# allocate space for arrays
xhat =np.zeros(sz)         # a posteri estimate of x
P    =np.zeros(sz)         # a posteri error estimate
xhatminus =np.zeros(sz)    # a priori estimate of x
Pminus    =np.zeros(sz)    # a priori error estimate
K         =np.zeros(sz)    # gain or blending factorR = 0.1**2                 # estimate of measurement variance, change to see effect# intial guesses
xhat[0] = 0.0
P[0]    = 1.0

# 开始迭代
for k in range(1, n_iter):
    # time update
    xhatminus[k] = xhat[k-1]
    Pminus[k]    = P[k-1]+Q    # measurement update
    K[k]    = Pminus[k]/( Pminus[k]+R )
    xhat[k] = xhatminus[k]+K[k]*(z[k]-xhatminus[k])
    P[k]    = (1-K[k])*Pminus[k]plt.figure()
plt.plot(z,'k+',label='noisy measurements')
plt.plot(xhat,'b-',label='a posteri estimate')
plt.axhline(x,color='g',label='truth value')
plt.legend()
plt.title('Estimate vs. iteration step', fontweight='bold')
plt.xlabel('Iteration')
plt.ylabel('Voltage')plt.figure()
valid_iter = range(1,n_iter) # Pminus not valid at step 0
plt.plot(valid_iter,Pminus[valid_iter],label='a priori error estimate')
plt.title('Estimated $\it{\mathbf{a \ priori}}$ error vs. iteration step', fontweight='bold')
plt.xlabel('Iteration')
plt.ylabel('$(Voltage)^2$')
plt.setp(plt.gca(),'ylim',[0,.01])
plt.show()